# math.wiki (Full Content) > Complete open reference library for all of K-12 mathematics, from counting to calculus: 1467 lessons across 18 courses. Every topic is mapped to its prerequisites and what it leads to. Licence: CC BY-NC 4.0. - Concise index: https://math.wiki/llms.txt - Map and search: https://math.wiki/ - Sitemap: https://math.wiki/sitemap.xml ## Courses ### [Kindergarten](https://math.wiki/course/kindergarten/) Level: Elementary | 36 lessons in 10 chapters ### [Grade 1](https://math.wiki/course/grade-1/) Level: Elementary | 30 lessons in 11 chapters ### [Grade 2](https://math.wiki/course/grade-2/) Level: Elementary | 45 lessons in 11 chapters ### [Grade 3](https://math.wiki/course/grade-3/) Level: Elementary | 47 lessons in 12 chapters ### [Grade 4](https://math.wiki/course/grade-4/) Level: Elementary | 58 lessons in 12 chapters ### [Grade 5](https://math.wiki/course/grade-5/) Level: Elementary | 46 lessons in 12 chapters ### [Math 6](https://math.wiki/course/math-6/) Level: Middle school | 98 lessons in 13 chapters ### [Math 7](https://math.wiki/course/math-7/) Level: Middle school | 89 lessons in 13 chapters ### [Math 8](https://math.wiki/course/math-8/) Level: Middle school | 89 lessons in 13 chapters ### [Algebra 1](https://math.wiki/course/algebra-1/) Level: High school | 112 lessons in 13 chapters ### [Geometry](https://math.wiki/course/geometry/) Level: High school | 128 lessons in 12 chapters ### [Algebra 2](https://math.wiki/course/algebra-2/) Level: High school | 121 lessons in 13 chapters ### [Precalculus](https://math.wiki/course/precalculus/) Level: High school | 115 lessons in 12 chapters ### [Statistics](https://math.wiki/course/statistics/) Level: High school | 95 lessons in 11 chapters ### [Calculus 1](https://math.wiki/course/calculus-1/) Level: College | 71 lessons in 10 chapters ### [Calculus 2](https://math.wiki/course/calculus-2/) Level: College | 35 lessons in 7 chapters ### [Linear Algebra](https://math.wiki/course/linear-algebra/) Level: College | 39 lessons in 6 chapters ### [Discrete Mathematics and Proof](https://math.wiki/course/discrete-math/) Level: College | 76 lessons in 10 chapters ## Algebra ### [Absolute value equations](https://math.wiki/equations-with-absolute-value/) **Summary:** Distance from zero gives two cases; solve both and check. **Question:** A number is a certain distance from zero, but nobody says which side. How do you find it? ## When you reach for it The absolute value $|x|$ is how far $x$ sits from $0$ on the number line, with no regard for which side. So the equation $$|x| = 5$$ asks: which numbers are $5$ away from $0$? There are two, one on each side: $x = 5$ and $x = -5$. Both make the sentence true, since $|5| = 5$ and $|-5| = 5$. The pair is often written $x = \pm 5$. Reach for this move whenever the unknown sits inside absolute value bars: a score that is a certain number of points from a target, a loaf baked a certain number of minutes longer or shorter than the recipe says, a point a given distance from another point. Each of these states how far, and leaves out which way, so each has two candidate answers, one for each direction. ## The procedure ### Why there are two cases Absolute value has a two-part rule: $|u| = u$ when $u \ge 0$, and $|u| = -u$ when $u < 0$. (The absolute value graph draws that rule as a V.) Suppose $d$ is a positive number and $|u| = d$. Either $u \ge 0$, and then $u = d$; or $u < 0$, and then $-u = d$, which means $u = -d$. Going back the other way, both $u = d$ and $u = -d$ do have absolute value $d$. So, for any expression $u$ and any number $d > 0$, $$\begin{aligned} &|u| = d \\ &\text{exactly when } u = d \text{ or } u = -d \end{aligned}$$ The absolute value equation becomes two ordinary linear equations, one for each side of the number line. The size of $d$ decides how many solutions there are: ### [Absolute value inequalities](https://math.wiki/absolute-value-less-than/) **Summary:** Close to zero means trapped between two numbers: an and. **Question:** A number has to land within a set distance of a target, or stay more than that distance away. Which numbers do? ## When you reach for it "The jar must weigh within $8$ grams of $340$ grams" is a statement about distance, and absolute value measures distance: $|w - 340|$ is how far $w$ is from $340$ on the number line. So the condition is $|w - 340| \le 8$. Reach for this lesson whenever an absolute value sits on one side of $<$, $\le$, $>$ or $\ge$. You have already met the equation version in equations-with-absolute-value. There $|x| = 4$ has exactly two solutions, $x = 4$ and $x = -4$, the two numbers at distance $4$ from $0$; in general, for a positive number $a$, $|u| = a$ means $u = a$ or $u = -a$. An inequality asks instead for the numbers **nearer** than that, or **farther**. The answer is no longer two points but a whole stretch of the line, or two stretches. This lesson takes two days. **Part 1** is "less than": the numbers trapped between two ends, an "and". **Part 2** is "greater than", which gives an "or", then what happens when the right side is negative or zero, and how to write an absolute value inequality from a description or a graph. ## The procedure ### Part 1: less than, an "and" $|x| < 4$ asks for the numbers whose distance from $0$ is less than $4$. Walk away from $0$ in either direction: on the right you must stop before $4$, on the left before $-4$. So $$|x| < 4 \iff -4 < x < 4$$ The algebra agrees. If $x \ge 0$ then $|x| = x$, and the condition is $x < 4$. If $x < 0$ then $|x| = -x$, and the condition $-x < 4$ becomes $x > -4$ after multiplying by $-1$, which reverses the sign. Every number in $-4 < x < 4$ passes its own case, and every number outside fails it. The solution is one stretch, both ends at once: an "and", written as the chain from and-inequalities. ### [Adding and subtracting polynomials](https://math.wiki/add-polynomials/) **Summary:** Combine like terms across both. **Question:** Two polynomials each describe part of a quantity. How do you get one polynomial for the whole, or for the difference between them? ## When you reach for it A quantity is often built from parts, and each part is a polynomial. A school club prints T-shirts: what it takes in and what it spends both depend on the number of shirts, and its profit is one minus the other. A triangular frame has three sides, each a polynomial in a length $x$, and the timber it takes is their sum. Rather than work out each part every time you want a number, you can combine the polynomials once and get a single polynomial for the whole thing. Adding and subtracting polynomials is that combining. It is the same move as combining like terms, applied across two polynomials at once, with one extra piece of care for subtraction. The words *term*, *degree*, *standard form* and *leading coefficient* are used as in degree-of-a-polynomial. ## The procedure ### Like terms, and why only they combine Two terms are **like terms** when they have the same variables raised to the same powers; only the coefficients may differ. So $5x^2$ and $-3x^2$ are like terms, and so are $4a^2b$ and $a^2b$. But $x^2$ and $x$ are not, and neither are $a^2b$ and $ab^2$: the same letters, different powers on them. Like terms combine because of the distributive property, read backwards: $$\begin{aligned} 5x^2 + (-3x^2) &= \big(5 + (-3)\big)x^2 \\ &= 2x^2 \end{aligned}$$ Both terms count copies of the same thing, $x^2$, so you add the counts. Unlike terms have no common thing to count. Picture them as pieces: $x^2$ is a square of side $x$, $x$ is a strip $x$ long and $1$ wide, and $1$ is a unit square. ### [Adding and subtracting rational expressions](https://math.wiki/add-subtract-rational-expressions/) **Summary:** Rename over the common denominator, then combine the numerators. **Question:** Two fractions have polynomials in their denominators. What denominator can they share, so that they become one fraction? ## When you reach for it Rates, costs and formulas often arrive as fractions with a variable in the denominator. If Jacoba squeezes a jug of lemonade in $t$ minutes, she squeezes $\frac{1}{t}$ of a jug each minute, and the work of two people together is a sum of two such fractions. One fraction is easier to read, to evaluate, to graph and to set equal to something than a sum of several. The arithmetic is the arithmetic of number fractions from adding and subtracting fractions: rename both over a common denominator, then combine the numerators. Two things are new. The common denominator is now a polynomial, so it has to be built from factors. And a fraction can sit inside another fraction, which is called a complex fraction. The lesson runs over two days. Part 1 adds and subtracts rational expressions. Part 2 simplifies complex fractions. Both lean on simplifying a rational expression and its excluded values and on multiplying and dividing rational expressions. ## The procedure ### Part 1: sums and differences **With a common denominator.** For polynomials $P$, $Q$ and $R$, $$\frac{P}{R} + \frac{Q}{R} = \frac{P + Q}{R} \qquad \frac{P}{R} - \frac{Q}{R} = \frac{P - Q}{R}$$ for every value of the variable where $R \ne 0$. The reason is the distributive property: $\frac{P}{R}$ is $P \cdot \frac{1}{R}$, and $P \cdot \frac{1}{R} + Q \cdot \frac{1}{R} = (P + Q) \cdot \frac{1}{R}$. In a difference, $Q$ is one quantity and the minus sign acts on all of it. Put the second numerator in parentheses before combining, the way grouping-symbols mark a part to be treated as one number, and then remove the parentheses by changing the sign of every term inside. ### [Algebra](https://math.wiki/algebra/) **Summary:** Working with unknowns. Letters stand for numbers, and the rules of arithmetic still hold. **Question:** Letters stand in for numbers, and the rules of arithmetic still hold. What can you say, and solve, once you work with numbers you do not yet know? ## What this area is about Algebra is arithmetic with unknowns. A letter stands for a number, and every rule of arithmetic still applies to it, so expressions can be rewritten and equations solved without knowing the number first. The branch runs from single letters to quadratics and sequences. ## How these fit together **The language.** Expressions: variables, evaluating, and the rewriting moves, from combining like terms to factoring. **Solving.** Linear equations find the unknown when it appears only to the first power. Inequalities find whole stretches of numbers instead. Systems of equations satisfy two or more conditions at once. **Lines.** Linear relationships: slope and every form of a line's equation. **Beyond lines.** Polynomials add, multiply, factor and divide. Radicals and rational expressions handle roots and polynomial fractions. Quadratics solve and graph $ax^2 + bx + c$ with $a \ne 0$. **Patterns.** Sequences and series find the $n$th term of an ordered list and the sum of its terms. ## A path through it Expressions first. Then linear equations and polynomials. After linear equations: linear relationships and inequalities. Systems need both of those. Radicals and rational expressions follow linear relationships and polynomials, and sequences follow linear relationships. Quadratics come after radicals; quadratics, inequalities and systems refer to each other, so take them side by side. ### [Arithmetic sequences and the explicit formula](https://math.wiki/explicit-formula-arithmetic/) **Summary:** The nth term straight from n, which is a linear rule. **Question:** Each row of an amphitheater holds four more seats than the row in front. How many seats are in row 30, without counting through rows 2 to 29? ## When you reach for it An **arithmetic sequence** is a sequence in which every term is the term before it plus the same fixed number. That fixed number is the **common difference**, written $d$: $$d = a_{n+1} - a_n \quad \text{for every } n$$ The common difference may be positive (the terms climb), negative (they fall), zero (every term is the same) or a fraction. Terms, positions and the notation $a_n$ are set out in sequence-basics. Reach for the explicit formula when you want a term far down the list without building every term before it, when you want to know whether a given number appears in the sequence and in which position, or when you want to treat the sequence as a function of its position and read it the way you read any constant rate of change. ## The procedure **Finding the common difference.** Subtract each term from the one after it. The front row of an amphitheater has $18$ seats and each row behind has $4$ more: $$18,\ 22,\ 26,\ 30,\ 34,\ \dots$$ The differences are $22 - 18 = 4$, $26 - 22 = 4$, and so on, so $d = 4$. The order of the subtraction matters: it is always the later term minus the earlier one, which is why the falling sequence $7, 4, 1, -2$ has $d = -3$ and not $3$. If the differences are not all equal, as in $1, 2, 4, 8$ with differences $1, 2, 4$, the sequence is not arithmetic and nothing below applies. **Building the formula.** Write each term as the first term plus however many fours have been added to it. ### [Arithmetic series and partial sums](https://math.wiki/partial-sum/) **Summary:** The total of the first n terms. **Question:** The sequence 13, 19, 25, and onward climbs by 6 each time. What do its first 25 terms add up to, without doing 24 additions? ## Why this exists The sequence $13, 19, 25, 31, \ldots$ climbs by $6$ at every step. Its terms are easy to make, and its $25$th term is easy to find with the explicit formula: $13 + 24 \cdot 6 = 157$. But a sequence is often wanted for its **total**: every seat in a stand, every can in a display, all the money saved over half a year. Adding $25$ terms one at a time takes $24$ additions, each one a chance to slip, and $250$ terms would take $249$. Sigma notation writes such a total exactly, $$13 + 19 + 25 + \cdots + 157 = \sum_{k=1}^{25} (6k + 7)$$ but writing a sum down is not the same as finding it. This lesson finds it. For an arithmetic sequence the whole total comes from three numbers, the first term, the last term and how many terms there are, in one multiplication. ## Definition The **$n$th partial sum** of a sequence $a_1, a_2, a_3, \ldots$ is the total of its first $n$ terms. It is written $S_n$: $$S_n = a_1 + a_2 + \cdots + a_n = \sum_{k=1}^{n} a_k$$ "Partial" because it stops after $n$ terms; the sequence itself may go on forever. The partial sums form a sequence of their own, the running totals. For $13, 19, 25, \ldots$ they are $$\begin{aligned} S_1 &= 13 \\ S_2 &= 13 + 19 = 32 \\ S_3 &= 32 + 25 = 57 \end{aligned}$$ and so on, each running total being the one before plus the next term. An **arithmetic series** is the sum of terms of an arithmetic sequence, one in which each term is the one before plus the same common difference $d$. Its partial sums are what this lesson is about. "Series" is the word for a sum of the terms of a sequence; the sequence is the list, the series is the list added up. ### [Combining like terms](https://math.wiki/combine-like-terms-negatives/) **Summary:** Subtraction is adding the opposite, and the sign travels with the term. **Question:** An expression mentions the same kind of thing several times over. How do you count those up into fewer terms without changing what the expression is worth? ## When you reach for it Add up the three sides of a triangle whose sides are $3t + 2$, $5t - 4$ and $2t + 7$, and you get $$3t + 2 + 5t - 4 + 2t + 7$$ That is correct, but it is long: some $t$'s, some plain numbers, more $t$'s, more plain numbers, in no particular order. The same perimeter can be written as $10t + 5$, two terms instead of six. Rewriting an expression with as few terms as possible, by counting together the terms that are the same kind of thing, is **combining like terms**. You reach for it whenever an expression has been built by adding pieces together: a perimeter, a total of two kinds of item, the result of removing parentheses, either side of an equation. A shorter expression is quicker to evaluate, easier to compare with another one, and it is the form that solving an equation needs. The words *term*, *coefficient* and *constant* come from terms-coefficients-constants. This lesson runs over two days. Part 1 says which terms are alike, why they combine, and how to handle fraction and decimal coefficients. Part 2 brings in negative coefficients, several letters, and perimeters and totals. ## The procedure ### Part 1: like terms, and why they combine The **variable part** of a term is everything in it except the number in front: in $7x^2$ it is $x^2$, in $\frac{1}{2}mn$ it is $mn$. Two terms are **like terms** when their variable parts are the same, letter for letter, with each letter raised to the same power. Only the coefficients may differ. All constants are like terms with one another, since none has a variable part. ### [Completing the square](https://math.wiki/complete-the-square-leading-one/) **Summary:** Add and subtract the missing number, then rewrite. **Question:** A quadratic has the variable in two places. How do you get it down to one? ## When you reach for it You can solve $x^2 = 49$ in your head, because $x$ appears once: undo the square and $x = \pm 7$. You cannot do the same with $$x^2 + 6x = 16$$ because $x$ turns up twice, once squared and once not, and there is no single operation that undoes both. Completing the square is the rewrite that fixes exactly that. It turns the left side into one squared term, $(x + \text{something})^2$, so the variable appears **once**. After that the problem is one you could already do, by taking square roots. Reach for it when factoring fails, or does not obviously work, and you want exact answers rather than decimals read off a graph. It never fails: every quadratic equation can be pushed into the form $(x + k)^2 = d$, which is exactly why the quadratic formula exists, and it is derived this way. Reach for it also when you need more than the solutions: the same rewrite turns any quadratic expression into a square plus a number, which is how you find the vertex of a parabola and how you put the equation of a circle into center and radius form. ## The procedure The whole method rests on one pattern, the square of a sum: $$(x + k)^2 = x^2 + 2kx + k^2$$ Read it right to left. An expression $x^2 + 2kx + k^2$ is a perfect square, and the number inside the square, $k$, is **half** the coefficient of $x$. The constant at the end, $k^2$, is that half, squared. So given $x^2 + bx$, the piece that finishes the square is ### [Completing the square, leading coefficient not one](https://math.wiki/complete-the-square-leading-not-one/) **Summary:** Factor the coefficient out first. **Question:** The squared term has a number in front of it. How do you complete a square that does not start with a plain $x^2$? ## When you reach for it Completing the square works on $x^2 + bx$: take half of $b$, square it, add it, and the expression becomes $(x + \tfrac{b}{2})^2$. Now try the same move on $$2x^2 + 12x = 14$$ Half of $12$ is $6$, and $6^2 = 36$, but $2x^2 + 12x + 36$ is not a perfect square. The trouble is at the front. Every square of the form $(x + k)^2$ multiplies out to begin with $x^2$, exactly one of it. An expression that begins with $2x^2$ cannot match that pattern however you choose the constant. So one step comes first: **make the coefficient of $x^2$ equal to $1$**, and then complete the square exactly as before. That step is the whole of this lesson, together with the care it needs, because the way you remove the $2$ depends on whether you are working with an equation or with an expression. Almost every quadratic you meet in the wild has a leading coefficient other than one: the height of a thrown ball, $h = -5t^2 + 20t + 2$; the area of a fenced pen, $A = -2w^2 + 20w$; the general quadratic $ax^2 + bx + c$ itself. So this is the version of the method that gets used most. ## The procedure **In an equation, divide both sides by the leading coefficient.** Dividing both sides of an equation by the same number that is not zero gives an equation with exactly the same solutions, because the step can be undone by multiplying back. Every term on both sides gets divided, not only the $x^2$ term: ### [Compound inequalities: and, or](https://math.wiki/and-inequalities/) **Summary:** Both conditions hold; the solution is the overlap. **Question:** A number has to pass two tests at once, or pass at least one of two. Which numbers do? ## When you reach for it A single inequality fences the number line on one side: $x > 3$ is everything to the right of $3$. Many real conditions fence it on two sides, or allow two separate stretches. Cold-water fish are safe while their tank stays at least $18$ degrees **and** at most $25$ degrees. A fairground ride turns away anyone who is too short **or** too tall. Two inequalities joined by "and" or by "or" make a **compound inequality**: $$4x + 1 \ge -7 \quad \text{and} \quad 3x - 2 < 10$$ Each part is an ordinary linear inequality, solved with the moves of solving linear inequalities and two-step inequalities. What is new is how the two answers are put together, and that is decided entirely by the joining word. ## The procedure ### What "and" and "or" mean A number satisfies "$P$ and $Q$" exactly when it satisfies both parts. It satisfies "$P$ or $Q$" when it satisfies at least one of them, and a number that satisfies both counts too: in mathematics "or" always allows both. So the solution set of an "and" is the **overlap** of the two solution sets, and the solution set of an "or" is **everything either one covers**. In the language of sets, "and" is the intersection $\cap$ and "or" is the union $\cup$. ### Solve each part, then combine Solve each part on its own. Every step of solving an inequality (adding the same number to both sides, multiplying both sides by a positive number, multiplying by a negative number and reversing the sign) produces an inequality with exactly the same solutions. So replacing a part by its solved form does not change which numbers make the whole compound statement true. ### [Degree, terms and standard form](https://math.wiki/degree-of-a-polynomial/) **Summary:** The highest power, and the degree of each term. **Question:** Some expressions are built from a variable using only adding and multiplying. How do you describe one, and which single number tells you most about it? ## Why this exists Cannonballs and oranges are often stacked as square pyramids: one ball on top, a $2 \times 2$ layer under it, then $3 \times 3$, and so on. A stack with $n$ layers holds $1 + 4 + 9 + \dots + n^2$ balls, and that sum can be written in one line: $$C = \frac{2n^3 + 3n^2 + n}{6}$$ Counting the diagonals of a polygon with $n$ corners gives another such formula, $\frac{n(n - 3)}{2}$, met in angle sums of polygons. Both are built from a variable using nothing but multiplying and adding, so both are **polynomials**. Areas, volumes, counts and the paths of thrown objects keep producing expressions like these. To work with them you need words for their parts, and one number that tells you most about each: the **degree**, the highest power present. Because the cannonball formula has degree three, the stack grows roughly like a cube, close to $\frac{1}{3}n^3$ balls when $n$ is large, as Example 6 shows. ## Definition A **term** (also called a **monomial**) is a number, or a number multiplied by one or more variables, each raised to a whole-number power ($0, 1, 2, 3, \dots$). Examples: $7x^3$, $-y$, $4ab^2$, $\frac{x}{2}$ and $5$. The number in front is the term's **coefficient**. A **polynomial** is a term or a sum of terms. Subtracting a term counts as adding its opposite, so $5x^2 - 8x$ is $5x^2 + (-8x)$, a polynomial with two terms. Reading coefficients takes care in three places: ### [Deriving the quadratic formula](https://math.wiki/derive-the-quadratic-formula/) **Summary:** Complete the square on the general quadratic and the formula falls out. **Question:** Completing the square takes the same steps on every quadratic. Can you do it once, for all of them at the same time? ## Why this exists Completing the square solves every quadratic equation, but it does so the long way, and the same long way each time. Move the constant across. Divide by the leading coefficient. Halve the coefficient of $x$, square it, add it to both sides. Write the left side as a square, take square roots, solve. Only the numbers change from one problem to the next. When a calculation repeats with different numbers, the move in algebra is to do it once with letters standing for the numbers, the way you solve a formula for one of its variables. Run completing the square on $$ax^2 + bx + c = 0$$ with $a$, $b$ and $c$ left as letters, and what comes out is one formula that holds the answer to every quadratic equation there is. You substitute three numbers and the work is already done. Rules in words came long before the formula. Brahmagupta, in 628, described in words a rule that amounts to it for one arrangement of the equation, taking only the positive root. Al-Khwarizmi, around 820, solved quadratics by completing a square drawn as an actual picture, but he had no letters for the coefficients and did not use negative numbers, so he split quadratics into separate cases, each with its own rule. One line that covers every case needed algebra written in symbols, which grew up in the late sixteenth and seventeenth centuries. ## Definition For any numbers $a$, $b$ and $c$ with $a \neq 0$, the solutions of $ax^2 + bx + c = 0$ are ### [Direct variation](https://math.wiki/direct-variation/) **Summary:** A line through the origin: y is a constant times x. **Question:** A spring stretches further the more weight hangs on it, and a car goes further the more fuel it burns. When is one quantity a fixed number times the other, and what does that number look like on a graph? ## Why this exists Proportional relationships have already been met three ways: a table whose ratios share one constant of proportionality, an equation $y = kx$ with nothing added on the end, and a graph that is a straight line through the origin. Once slope is known, the three tests turn out to be one fact: **a proportional relationship is a line through the origin, and its constant is the line's slope.** This fact is called **direct variation**. It settles questions such as: one car's fuel use is given as a table and another's as a graph, so which goes further on a liter? Each becomes "which constant is bigger". The lesson runs over two days. Part 1 is the idea: $y = kx$ as a line through the origin with slope $k$, in a table, an equation and a graph. Part 2 uses it: comparing relationships given in different forms, reading $k$ as a unit rate, and finding a missing value. ## Definition **$y$ varies directly with $x$** when there is one fixed number $k$, not zero, such that every pair of values satisfies $$y = kx.$$ The number $k$ is called the **constant of variation**. It is the same number as the constant of proportionality: "$y$ varies directly with $x$", "$y$ is proportional to $x$" and $y \propto x$ all say the same thing. Two equivalent ways to say it, each useful in its own place: - **Ratio form.** For every pair with $x \ne 0$, the ratio $\frac{y}{x}$ is the same number $k$, and when $x = 0$, $y = 0$. - **Scaling form.** Multiplying $x$ by any number multiplies $y$ by the same number. Double $x$ and $y$ doubles; take a third of $x$ and $y$ becomes a third. ### [Distribute, then combine](https://math.wiki/distribute-and-combine/) **Summary:** Open the brackets first, then collect what matches. **Question:** An expression has parentheses with numbers in front, and terms of the same kind sit both inside and outside them. How do you bring it down to its shortest form without changing its value? ## When you reach for it Gaspard is ordering food for a dinner. Each guest gets a meal costing $15$ dollars and a drink costing $4$ dollars. There were going to be $g$ guests, but $3$ have canceled, so he pays for $g - 3$ meals and $g - 3$ drinks, and there is a delivery charge of $30$ dollars. The bill, in dollars, is $$15(g - 3) + 4(g - 3) + 30$$ That is correct, and it follows the story piece by piece, but it hides the two things Gaspard most wants to know: how much each guest adds to the bill, and what is left over. Rewritten, the same bill is $19g - 27$. Now it says that every guest adds $19$ dollars, a meal and a drink, and the fixed part is $-27$: the $30$ dollars of delivery minus the $3 \cdot 19 = 57$ dollars the canceled guests no longer cost. Getting from the first form to the second takes two moves you already have, done in a fixed order. First distribute: multiply the number in front of each set of parentheses into every term inside it. Then combine like terms. You reach for this whenever an expression has parentheses with numbers in front and you want its shortest equivalent form: before evaluating it, when comparing it with another expression, and above all when an equation with parentheses in it has to be solved. This lesson runs over two days. Part 1 says why the parentheses are opened first, works expressions with one and two sets of parentheses, and shows how to check an answer by substituting a number. Part 2 brings in a negative number in front of parentheses, subtracting a whole group in parentheses, fraction and decimal coefficients, and perimeters of shapes made from rectangles. ### [Equations from proportions and percents](https://math.wiki/proportion-equations/) **Summary:** Cross multiply, then solve the linear equation that results. **Question:** The unknown is buried inside a ratio, or hidden behind a percent change. How do you turn that into an equation you already know how to solve? ## When you reach for it Solving a proportion finds a plain unknown such as the $x$ in $\frac{x}{5} = \frac{3}{4}$: one multiplication and one division and it is done. Very often, though, the unknown is not alone. It sits inside an expression in a numerator, $$\frac{x + 2}{5} = \frac{3}{4}$$ or in a denominator, $$\frac{6}{x - 1} = \frac{2}{3}$$ and no single operation sets it free. The move in this lesson turns such a proportion into a two-step linear equation, which you can already solve. Percent problems lead to the same place. "A number increased by $20\%$ is $54$" describes an unknown that has been changed and asks you to undo the change. Writing the change as a multiplier turns the sentence into $1.2n = 54$, again a linear equation. So reach for this whenever two ratios are set equal and the unknown is tangled up in one of them, or whenever a percent has been added to or taken from an amount you do not know. The lesson runs over two parts: Part 1 is proportions with the unknown inside an expression, Part 2 is percent equations. ## The procedure ### Part 1: a proportion with the unknown inside an expression **Cross multiplying.** For any numbers with $b \ne 0$ and $d \ne 0$, $$\frac{a}{b} = \frac{c}{d} \quad\text{exactly when}\quad ad = bc$$ This is not a new rule. Multiplying both sides of $\frac{a}{b} = \frac{c}{d}$ by $bd$ is allowed by the multiplication property of equality, and it cancels each denominator: the left side becomes $ad$ and the right side becomes $bc$. Since $bd$ is not zero, the step can be undone by dividing by $bd$, so the two equations have exactly the same solutions. (What a proportion says has the same fact from the side of equal ratios.) ### [Equations of the form p(x + q) = r, from words](https://math.wiki/equations-of-the-form-p-x-plus-q/) **Summary:** Write equations like 3(x + 4) = 27 from a word problem and solve them by dividing first or by distributing first. **Question:** Several friends each buy the same bundle, and only the total is known. How do you work back to the price of one thing in the bundle? ## When you reach for it Dorothea and three friends go to a movie. Each of the four buys a ticket and a drink, and a drink costs $5$ dollars. Together they pay $48$ dollars. What does one ticket cost? Call the price of one ticket $x$ dollars. One person's bundle, a ticket and a drink, costs $x + 5$ dollars. There are four bundles, all the same, so the four together cost $4$ times that whole amount: $$4(x + 5) = 48$$ The parentheses carry the meaning: the drink is in every bundle, so the $4$ multiplies it as well as the ticket. Without parentheses, $4x + 5$ means four tickets and a single drink, because by the order of operations only the $x$ is multiplied. Reach for this whenever a situation is made of **equal groups**, each holding the unknown together with the same fixed amount, and the total is known: bundles bought by each person, teams that each gain the same number of players, the four sides of a square with a border, the sections of a fence. Every such equation has the shape $$p(x + q) = r$$ where $p$ is the number of groups, $q$ the fixed amount in each group, and $r$ the total. The numbers may be negative or fractions; only $p = 0$ is ruled out. An equation with no parentheses, such as $4x + 20 = 48$, is a two-step equation, and one of the two ways of solving below turns this equation into exactly that. ## The procedure There are two honest ways to solve $p(x + q) = r$: undo the multiplication by $p$ first, or open the parentheses first. Both are always allowed, both give the same answer, and the skill is in seeing which one the numbers favor. Over two days, Part 1 writes the equation from words and solves it both ways with positive numbers; Part 2 brings in negatives and fractions and asks which way to choose. ### [Equations with variables on both sides](https://math.wiki/collect-variables-one-side/) **Summary:** Subtract the smaller variable term from both sides, then solve. **Question:** The unknown turns up on both sides of the equals sign. How do you get it onto one side so the equation can be solved? ## When you reach for it A two-step equation such as $3x + 3 = 18$ has the unknown in one place, and you solve it by undoing what was done to $x$, last step first. A multi-step equation may have several $x$ terms, but they all sit on the same side, so combining them gets you back to two steps. Now look at $$5x + 3 = 2x + 18$$ The unknown is on **both** sides. Undoing does not work directly: subtracting $3$ and dividing by $5$ gives $x = \frac{2x + 15}{5}$, which still has $x$ on the right and so says nothing new. Something has to bring the two $x$ terms together first. Equations like this turn up whenever two quantities that depend on the same unknown are compared. Two phone plans each cost a fixed amount plus a charge per minute: for how many minutes do they cost the same? Each side of the equation is one of the two quantities, and the question "when are they equal?" is the equation itself. This lesson runs over two days. Day 1 is the move itself: collect the unknown on one side, finish as a two-step equation, and check by substitution. Day 2 adds parentheses and fractions, and builds equations from two quantities that become equal. ## The procedure ### Day 1: collecting the unknown on one side **Subtract a variable term from both sides.** In $5x + 3 = 2x + 18$, take $2x$ away from each side: $$5x + 3 - 2x = 2x + 18 - 2x$$ which simplifies to $$3x + 3 = 18$$ The unknown now appears once, and the rest is a two-step equation: subtract $3$ to get $3x = 15$, then divide by $3$ to get $x = 5$. ### [Equivalent expressions: the distributive property and like terms](https://math.wiki/equivalent-expressions-with-whole-numbers/) **Summary:** Two expressions are equivalent when they agree for every value; make them by distributing, factoring out a common factor and combining like terms. **Question:** Two people write different expressions for the same count. How can you tell that they will always give the same number, without trying every number there is? ## When you reach for it A garden center sets out plant pots in $3$ rows. Every row has some pots of basil, the same number in each row, and $4$ pots of mint. Call the number of basil pots in a row $x$. Kaia counts one row and multiplies by the number of rows: $$3(x + 4)$$ Rafa counts all the basil, then all the mint: $$3x + 12$$ They wrote different expressions, but they counted the same pots, so they must get the same total whatever $x$ is. Two expressions like these are called **equivalent**: they give the same value for every number you put in place of the letter. You reach for equivalent expressions whenever one form of an expression is more useful than another. A form with no parentheses is quicker to add to something else. A form with a factor at the front shows the rows, groups or packs hiding in a total. A form with fewer terms is easier to evaluate. And when two people (or two methods) produce different expressions, the question "are these equivalent?" is how you find out whether both are right. This lesson takes three days. **Part 1** makes equivalent expressions by distributing, with an area picture that shows why it works. **Part 2** runs the same move backwards and factors out a common factor. **Part 3** combines like terms, and then asks how to test whether two expressions are equivalent, and what a test can and cannot prove. ## The procedure ### Part 1: distributing Two expressions are **equivalent** when they have the same value for every value of the letter. Not for one value, not for most values: for every one. So $3(x + 4)$ and $3x + 12$ are equivalent only if no number at all makes them different. ### [Evaluating expressions and formulas](https://math.wiki/substitute-one-variable/) **Summary:** Put the number where the letter is, then compute. **Question:** An expression such as $6d + 15$ has a letter in it. Once you are told which number the letter stands for, how do you find the value of the expression? ## When you reach for it An expression with a letter in it, such as $6d + 15$, is a calculation waiting for one number. The letter marks the place where that number goes (what a letter stands for). Until the number is known, the expression has no single value; once it is known, it has exactly one. Putting the number in the letter's place is called **substituting**, and working out the result is called **evaluating** the expression. You reach for it whenever an expression or a formula has to give an actual number: - a rule for pay, cost or time, written in letters from words, and one particular case of it: how much for $8$ deliveries? - a formula for a shape, such as $A = lw$ for the area of a rectangle, and the measurements of one real rectangle; - a table of values, which is one expression evaluated again and again, once for each number in the top row. The lesson runs over two days. Part 1 puts one number into one letter, including a letter under a power. Part 2 uses formulas, expressions with two letters, and tables. ## The procedure ### Part 1: one letter, one number **Write the expression again with the number in place of the letter.** Take $6d + 15$ when $d = 7$. The letter $d$ stood for a number that was not yet known; now it is known, so the expression with $7$ in it describes the same quantity: $$6d + 15 \quad\text{becomes}\quad 6(7) + 15$$ Nothing else changes. Every operation stays where it was, and only the letter is swapped out. ### [Expression, equation and inequality](https://math.wiki/expression-vs-equation/) **Summary:** An expression is a number you have not worked out yet; an equation and an inequality are claims that can be true or false. **Question:** The cost of four cones is a number. "Four cones cost $12$ dollars" is a claim that could be wrong. How do you tell the two apart once they are written in symbols? ## Why this exists At the ice cream van a cone costs $3$ dollars. "The cost of four cones" names an amount: $4 \cdot 3$ dollars. "Four cones cost $12$ dollars" does something else. It says something **about** that amount, and you can check whether it is right. Change the $12$ to $13$ and you still have a sentence, only now a wrong one. A quantity is something you work out; a claim is something you check. In symbols the difference comes down to one mark: $4 \cdot 3$ is a quantity, and $4 \cdot 3 = 12$ is a claim. That mark sorts everything written in symbols into three kinds: **expressions**, **equations** and **inequalities**. ## Definition An **expression** is a way of writing a number. It is a single number, a single letter, or numbers and letters joined by operations: $+$, $-$, $\cdot$, $\div$, powers and grouping symbols. So $4 \cdot 3$, $k + 11$, $2^5$, $3(c + 2)$, $40$ and $w$ are all expressions. Its pieces have names, the terms, coefficients and constants, and a letter in it stands for a number in the ways variable-meaning describes. An expression has no verb. It says "this number" and nothing about that number. An **equation** is two expressions joined by an equals sign, $=$. It claims that the two sides are the same number. Read aloud, the sign becomes a verb ("is equal to", or just "is"), and the whole line becomes a sentence. An **inequality** is two expressions joined by one of the order signs. ### [Expressions](https://math.wiki/expressions/) **Summary:** Number sentences with letters in them, and the rules for rewriting them. **Question:** The same quantity can be written as $3(x + 4)$ or as $3x + 12$. What are the rules for changing one form into another without changing its value? ## What this area is about An expression is a calculation with letters in it, such as $15(g - 3) + 30$. It has no equals sign and claims nothing; it stands for a number once its letters do. This area is about reading expressions and rewriting them by the rules of arithmetic, so that the new form gives the same value for every number put in. ## How these fit together **Reading.** Variables says what a letter can stand for and names the parts of an expression. Evaluating expressions and formulas puts a number in and computes. Interpreting the parts of an expression reads terms and factors as quantities in a story. **Equivalence.** Equivalent expressions defines when two forms agree for every value. Combining like terms, the distributive property and distributing, then combining are the moves that keep that agreement. Rewriting to show meaning chooses the form that answers a question, as $1.09p$ does for $p + 0.09p$. **Products.** Monomials and the exponent rules and multiplying two binomials. **Factoring**, the distributive property run backward: the greatest common factor, then grouping for four terms with no factor common to all. ## A path through it Variables first. Then evaluating, equivalent expressions and monomials; combining like terms; the distributive property, then interpreting parts. After the distributive property: distributing and combining, binomials, rewriting for meaning, and the greatest common factor, with grouping last. ### [Factored form](https://math.wiki/factored-form-of-a-quadratic/) **Summary:** Written as a product, the roots are visible; three forms, three things made easy. **Question:** A quadratic can be written so that the places where its graph meets the x-axis are printed in the equation. What else does that form tell you, and what does it hide? ## Why this exists Here is one quadratic function written twice: $$\begin{aligned} f(x) &= x^2 - 4x - 5 \\ f(x) &= (x + 1)(x - 5) \end{aligned}$$ Multiply out the second and you get the first, so they are the same function with the same graph. But ask where the graph meets the $x$-axis and only one of them answers at once. From the first you have to factor, or reach for the quadratic formula. From the second you read it off: the output is $0$ when $x = -1$ and when $x = 5$, because a product is zero exactly when one of its factors is. A quadratic can be written in three standard ways, and each one puts a different feature of the parabola in plain sight. Vertex form shows the turning point. Standard form shows where the graph crosses the $y$-axis. **Factored form** shows the zeros, and, as it turns out, a great deal more besides. ## Definition A quadratic is in **factored form** when it is written $$f(x) = a(x - r)(x - s) \qquad (a \neq 0)$$ where $a$, $r$ and $s$ are numbers. It is a number times two linear factors. - $r$ and $s$ are the **zeros** of $f$: the inputs that give the output $0$. The graph meets the $x$-axis at $(r, 0)$ and $(s, 0)$. - $a$ is the same number as the leading coefficient in standard form, the coefficient of $x^2$. Its sign says which way the parabola opens: upward if $a > 0$, downward if $a < 0$. As with vertex form, the minus signs inside the parentheses are there on purpose, so that the numbers you read off are the zeros with their signs intact. A factor with a plus has to be read through that minus: ### [Factoring by grouping](https://math.wiki/factor-by-grouping/) **Summary:** Pair the terms, factor each pair, then factor the shared bracket. **Question:** Four terms share no factor, all four together. Can the sum still be written as a product? ## When you reach for it Taking out the greatest common factor works when one monomial divides every term. Now take $$xy + 2x + 5y + 10$$ The four terms share no factor except $1$: the first two contain $x$, the last two do not, and $10$ has nothing in common with $xy$. Yet this polynomial is a product. It equals $(x + 5)(y + 2)$, as multiplying the two binomials confirms. Factoring by grouping is the way back from the four terms to that product. Reach for it when a polynomial has four terms, no factor common to all of them, but pairs of terms that do share a factor. ## The procedure ### A binomial can be the common factor Look at $$x(y + 2) + 5(y + 2)$$ It has two terms, and both contain the factor $(y + 2)$. The distributive property says $aB + cB = (a + c)B$ for any numbers $a$, $c$ and $B$, and nothing stops $B$ from being the number $y + 2$. So $$\begin{aligned} &x(y + 2) + 5(y + 2) \\ &= (x + 5)(y + 2) \end{aligned}$$ This is exactly the move of taking out a monomial. The only difference is that the shared factor is a binomial, kept whole inside its parentheses, and what is left of each term, $x$ and $5$, forms the other factor. ### Making the common binomial appear Back to $xy + 2x + 5y + 10$. Addition can be grouped any way we like (the associative property), so split the four terms into two pairs and take out each pair's greatest common factor: $$\begin{aligned} &(xy + 2x) + (5y + 10) \\ &= x(y + 2) + 5(y + 2) \end{aligned}$$ ### [Factoring completely and choosing a strategy](https://math.wiki/factor-completely/) **Summary:** Take out the common factor first, then factor what is left, and check nothing more can go. **Question:** A polynomial has been factored once. How do you know when you are finished? ## When you reach for it You now have five ways to factor: taking out a common factor, a trinomial with leading coefficient one, a trinomial with any leading coefficient, grouping, and the special patterns. Each lesson handed you a polynomial already shaped for its method. Out in the open nobody says which method a polynomial needs, and one method is often not enough: $3x^2 - 75$ needs two, and $x^4 - 256$ needs the same pattern twice. This lesson is about the whole job. Reach for it whenever a polynomial must be factored and the result will be used: to cancel factors in a fraction, to read off where a product is zero, or to split a volume into its edges. Each of those needs every factor, so stopping early gives a wrong answer later. ## The procedure ### Day 1: what "completely" means, and the common factor first A polynomial with integer coefficients and degree at least $1$ is **prime over the integers** when its coefficients share no common factor other than $1$ and it cannot be written as a product of two polynomials with integer coefficients, each of degree at least $1$. So $x + 4$, $2x - 3$ and $x^2 + 9$ are prime; $2x + 8 = 2(x + 4)$ is not, and nor is $x^2 - 9 = (x + 3)(x - 3)$. A polynomial is **factored completely** when it is written as a monomial in front (a number times powers of the variables, possibly just $1$ or $-1$) times factors that are each prime over the integers. A prime factor may appear more than once, written as a power. The number in front is left as one number: $6(x + 2)$, not $2 \cdot 3(x + 2)$. ### [Factoring out the greatest common factor](https://math.wiki/factor-out-a-monomial/) **Summary:** Pull the shared factor to the front and write what is left in brackets. **Question:** Distributing turns a product into a sum. Given the sum, how do you get the product back? ## When you reach for it The distributive property multiplies an outside factor into every term of a sum in parentheses: $$4(2a + 5) = 8a + 20$$ That is the forward direction, from a product to a sum. This lesson runs the same equation from right to left. Given $8a + 20$, you find the number that both terms share, $4$, pull it out to the front, and write what is left of each term in parentheses. Rewriting a sum as a product this way is called **factoring out the greatest common factor**, or factoring out a monomial, because the shared factor is a single term such as $4$, $5m$ or $3y^2$. You reach for it whenever a product is more useful than a sum. A product can be read as equal groups: $6x + 9 = 3(2x + 3)$ says the amount is three copies of $2x + 3$. A product shows what a quantity is a multiple of. A product is what solving for one letter needs when that letter sits in two terms, and it is the first move before factoring by grouping or factoring a trinomial. The words *term*, *coefficient* and *constant* come from terms-coefficients-constants. This lesson runs over two days. Part 1 finds the greatest common factor of several terms, pulls it to the front, and checks the result by distributing back. Part 2 factors out a negative, factors expressions with fraction coefficients, and reads what the factored form says. ## The procedure ### Part 1: the greatest common factor of terms A **factor** of a term is anything that multiplies into it exactly. The term $12x^3y$ is $2 \cdot 2 \cdot 3 \cdot x \cdot x \cdot x \cdot y$, so $3$, $4x$, $x^2y$ and $6x^3$ are all factors of it. A **common factor** of several terms is a factor of every one of them, and the **greatest common factor** (GCF) is the largest one: the number part as large as possible and every letter to as high a power as possible. ### [Factoring special patterns](https://math.wiki/recognize-difference-of-squares/) **Summary:** a squared minus b squared factors into a sum times a difference. **Question:** Some polynomials are products in disguise. Which ones can you factor on sight, with no search at all? ## Why this exists Factoring a trinomial such as $x^2 + 11x + 30$ is a search: you hunt for two numbers whose product is $30$ and whose sum is $11$, as in factor-trinomials. Some polynomials need no search. Their shape alone names the factors, because they are the special products you already know how to multiply out, read in the other direction. The most useful one is the difference of two squares. The expression $16x^2 - 81$ has no middle term and a leading coefficient that is not $1$, so a search would be slow. Once you see that it is $(4x)^2 - 9^2$, the factors $(4x + 9)(4x - 9)$ can be written straight down. Two more shapes work the same way: the perfect square trinomial, and the sum or difference of two cubes. ## Definition A term is a **perfect square** when it is the square of a simpler term: $9x^2 = (3x)^2$, $x^4 = (x^2)^2$, $\frac{1}{4} = \left(\frac{1}{2}\right)^2$. A number coefficient must be a square number or the square of a fraction, and every variable must carry an even exponent. So $8x^2$ and $x^3$ are not perfect squares. A **difference of two squares** is an expression $a^2 - b^2$: two terms, each a perfect square, with subtraction between them. Here $a$ and $b$ stand for any terms, so $x^2 - 49$, $9x^2 - 25y^2$ and $x^4 - 1$ all qualify. A **perfect square trinomial** is the square of a binomial written out: $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$. The test has two parts. The first and last terms are perfect squares, $a^2$ and $b^2$, both added. The middle term is twice the product $ab$, with either sign. ### [Factoring trinomials](https://math.wiki/factor-trinomials/) **Summary:** Reverse the expansion: find two binomials that multiply to the trinomial. **Question:** Multiplying two binomials gives a trinomial. Handed only the trinomial, how do you find the two binomials, and how do you know you have finished? ## What this area is about Factoring a trinomial runs multiplication backwards: $x^2 + 13x + 22$ becomes $(x + 2)(x + 11)$, an equivalent expression written as a product instead of a sum. This area works with integer coefficients, where a trinomial may also turn out to be prime, with no such factoring at all. ## How these fit together All three children are procedures. Factoring when the leading coefficient is one handles $x^2 + bx + c$: find two integers that multiply to $c$ and add to $b$. Factoring when the leading coefficient is not one handles $ax^2 + bx + c$ with $a \ne 1$: split the middle term using two numbers with product $ac$ and sum $b$, then factor by grouping. Factoring completely and choosing a strategy puts every method together: take out the greatest common factor first, factor what is left, and stop only when each factor is prime over the integers. ## A path through it Leading coefficient one first, then any leading coefficient, then factoring completely, which draws on both. Factoring is what makes the zero product property solve quadratics, and it is the first step with rational expressions and limits found by factoring. ### [Factoring when the leading coefficient is not one](https://math.wiki/factor-trinomial-leading-not-one/) **Summary:** Split the middle term, then factor by grouping. **Question:** When x squared carries a coefficient, two numbers that multiply to the constant no longer give the middle term. What do you look for instead? ## When you reach for it You meet a trinomial $ax^2 + bx + c$ with integer coefficients and $a$ not $1$, and you want it as a product of two binomials. When $a = 1$, the shortcut is to find two numbers that multiply to $c$ and add to $b$. Try it on $2x^2 + 11x + 12$. The factor pairs of $12$ are $1, 12$ and $2, 6$ and $3, 4$, with sums $13$, $8$ and $7$. None is $11$, yet the trinomial does factor, as multiplying out shows: $$\begin{aligned} &(2x + 3)(x + 4) \\ &= 2x^2 + 8x + 3x + 12 \\ &= 2x^2 + 11x + 12 \end{aligned}$$ The middle term is $2 \cdot 4 + 3 \cdot 1$, not $3 + 4$. The $2$ in front of $x$ multiplies the $4$ before anything is added, so the leading coefficient is mixed into the middle term, and a search among the factors of $c$ alone cannot see it. Factored form is what later work needs: reading the zeros of a quadratic and solving equations with the zero product property both start from it. This lesson is about the factoring itself; solving comes later. ## The procedure This lesson runs over two days. Day 1 is the main method, splitting the middle term. Day 2 adds a second route and the other shapes a trinomial can take. ### Day 1: splitting the middle term Draw the product $(2x + 3)(x + 4)$ as the area of a rectangle. The four pieces are $2x^2$, $8x$, $3x$ and $12$. The middle term $11x$ is not one piece but two, $8x$ and $3x$, and there is a pattern in their coefficients: $8 \cdot 3 = 24$, and the corner pieces give $2 \cdot 12 = 24$ as well. So to rebuild the four pieces from $2x^2 + 11x + 12$, look for two numbers whose product is $ac = 2 \cdot 12 = 24$ and whose sum is $b = 11$. ### [Factoring when the leading coefficient is one](https://math.wiki/factor-trinomial-leading-one/) **Summary:** Find two numbers that multiply to the constant and add to the middle. **Question:** Multiplying two binomials gave $x^2 + bx + c$. Given only that answer, how do you find the two binomials? ## When you reach for it Multiplying two binomials turns a product into a sum: $$(x + 2)(x + 11) = x^2 + 13x + 22$$ Factoring runs the same road the other way. You are handed $x^2 + 13x + 22$ and asked for the two binomials that multiply to it. The answer is an **equivalent expression**: it has the same value as the trinomial for every $x$, written as a product instead of a sum. Reach for this move when a trinomial $x^2 + bx + c$, with leading coefficient $1$ and integer $b$ and $c$, would be more useful as a product: to see an area as a length times a width, to find where an expression equals zero, or to cancel a factor shared by the top and bottom of a fraction. Factoring out a common factor pulled one shared factor to the front. Here the terms share no factor, and the trinomial splits into two binomials instead. This lesson runs over two days. Day 1 builds the idea and handles a positive constant $c$. Day 2 handles a negative $c$, trinomials that do not factor, two variables, and terms written out of order. ## The procedure ### Where the two numbers come from Multiply two binomials that both begin with $x$, where $p$ and $q$ stand for any two numbers: $$\begin{aligned} (x + p)(x + q) &= x^2 + qx + px + pq \\ &= x^2 + (p + q)x + pq \end{aligned}$$ The first line is the distributive property, used twice. The second gathers the two $x$ terms, which is the distributive property read backwards: $qx + px = (p + q)x$. So the middle coefficient of the product is the **sum** of the two numbers, and the constant is their **product**. ### [Fitting a quadratic to data](https://math.wiki/quadratic-modeling/) **Summary:** Three points determine a parabola. **Question:** You have measured something three times, and it rises and then falls. Which parabola passes through all three measurements, and what does it predict in between and beyond? ## When you reach for it Earlier lessons on quadratics in context handed you the formula, a ball's height $h(t) = -16t^2 + 96t + 6$ say, and the work was to read its vertex or its roots. Real problems seldom arrive that way. What you have is a table: a height at three moments, the takings at three prices, a stopping distance at three speeds. The formula is what is missing. This lesson runs the other way, from points to equation. A quadratic $y = ax^2 + bx + c$ is fixed by three numbers, $a$, $b$ and $c$, and each point on its graph gives one equation those numbers must satisfy. Three unknowns need three equations, so three points are the right amount. Two are too few: through any two points runs a whole family of parabolas, some opening up, some down, some narrow, some wide. A third point, with an $x$ different from the other two and not on the line through them, keeps exactly one member of that family: Reach for this move when a situation is known to be quadratic, or behaves like one (it rises and then falls, or grows with the square of something), and you have measurements at three different inputs. ## The procedure Find the parabola through $(1, 2)$, $(2, 7)$ and $(4, 23)$. **Write the general quadratic, and treat $a$, $b$ and $c$ as the unknowns.** Every upright parabola is $y = ax^2 + bx + c$ with $a \ne 0$. Here the points supply $x$ and $y$, and the coefficients are what is unknown. ### [Geometric sequences and the explicit formula](https://math.wiki/explicit-formula-geometric/) **Summary:** The nth term straight from n, which is exponential. **Question:** A ball bounces lower each time, always by the same fraction. How high is the eighth bounce, without working out the seven before it? ## When you reach for it A phone tree passes news along fast. In round $1$ the organizer calls $3$ people. In round $2$ each of those $3$ calls $3$ more, so $9$ people are called. In round $3$ each of those $9$ calls $3$ more, so $27$ are called. How many are called in round $8$? You could keep multiplying by $3$ until you reach round $8$. The numbers $3, 9, 27, 81, \ldots$ form a **geometric sequence**, and its explicit formula gives any term straight from its position $n$, with no need to build the terms before it. Reach for it whenever a quantity is **multiplied** by the same number at every step: a ball that rebounds to the same fraction of its last height, a yeast culture whose mass doubles every hour, a message that each person passes to the same number of new people. It is the multiplying twin of the arithmetic sequence, where the same number is **added** at every step. ## The procedure ### The common ratio A **geometric sequence** is a sequence in which each term after the first is the term before it multiplied by one fixed number. That number is the **common ratio**, written $r$. Since $a_{n+1} = a_n \cdot r$, dividing both sides by $a_n$ gives $$r = \frac{a_{n+1}}{a_n}$$ so the ratio is any term divided by the one before it, and for a geometric sequence every such quotient is the same number. It can be a whole number, a fraction, a decimal or negative: | Sequence | First quotient | $r$ | |---|:-:|:-:| | $2, 14, 98, 686$ | $\frac{14}{2}$ | $7$ | | $80, 40, 20, 10$ | $\frac{40}{80}$ | $\frac{1}{2}$ | | $1000, 1100, 1210, 1331$ | $\frac{1100}{1000}$ | $1.1$ | | $2, -6, 18, -54$ | $\frac{-6}{2}$ | $-3$ | ### [Geometric series](https://math.wiki/geometric-series-applications/) **Summary:** Repayments, doubling, and the total of a shrinking sequence. **Question:** The terms 5, 15, 45, and onward triple each time. What do the first 8 of them add up to, and is there a way to know without adding them one by one? ## When you reach for it The sequence $5, 15, 45, 135, \ldots$ triples at every step. It is geometric, with first term $a_1 = 5$ and common ratio $r = 3$, so its $n$th term is $5 \cdot 3^{n-1}$. Its total is another matter. The first $8$ terms are $$5 + 15 + 45 + 135 + 405 + 1215 + 3645 + 10935$$ and adding them one at a time takes $7$ additions on numbers that grow fast. For $30$ terms the last one alone has fifteen digits. Reach for this skill whenever the thing wanted is a **total of terms that each multiply the one before by the same number**. Totals like that turn up wherever growth is by a percentage: a salary that rises by $2.5\%$ a year, savings deposits that each earn interest for a different number of years, the payments that clear a loan. They also turn up where something shrinks by a fixed fraction, like the heights of a bouncing ball. In sigma notation such a total is $\sum_{k=1}^{n} a_1 r^{k-1}$, and the move below turns it into one short formula. A sum of the terms of a geometric sequence is a **geometric series**. As with the arithmetic series, $S_n$ stands for the sum of the first $n$ terms. ## The procedure ### Multiply by the ratio and subtract Call the total of the first $8$ terms $S$. Multiplying every term by $3$ gives the very next term of the sequence, so $3S$ is the same list moved one place along: $$\begin{aligned} S &= 5 + 15 + 45 + \cdots + 10935 \\ 3S &= \phantom{5 + {}} 15 + 45 + \cdots + 10935 + 32805 \end{aligned}$$ ### [Graphing a line from a table of values](https://math.wiki/graph-from-a-table/) **Summary:** Pick inputs, compute outputs, plot, join. **Question:** A rule turns each input into an output. How do you see the whole rule at once, as a picture? ## When you reach for it An equation such as $y = 2x - 3$ is a rule: give it an input $x$ and it gives back an output $y$. Each input with its output is an ordered pair, and each pair is a point on the coordinate plane. The **graph** of the equation is the set of all the pairs $(x, y)$ that make it true. There are infinitely many, so you work out a few, set them out in a **table of values**, plot them, and look. Reach for it with an equation or a rule in words. The table also shows, before anything is drawn, whether the graph is a straight line, and it needs only evaluating an expression. ## The procedure The running example is $y = 2x - 3$. ### Choosing the inputs Every value of $x$ is allowed, so choose inputs that make the work easy. Small integers keep the arithmetic quick and put the points on grid corners. The input $0$ gives the point where the graph crosses the $y$-axis, and a negative input or two shows the left of that axis too. Here take $x = -1, 0, 1, 2, 3$. ### Computing the outputs Substitute each input and multiply before subtracting. Parentheses around a negative input keep its sign attached: $$x = -1: \quad y = 2(-1) - 3 = -2 - 3 = -5$$ The other inputs go the same way: | $x$ | $y = 2x - 3$ | |:-:|:-:| | $-1$ | $-5$ | | $0$ | $-3$ | | $1$ | $-1$ | | $2$ | $1$ | | $3$ | $3$ | Each row is one point: $(-1, -5)$, $(0, -3)$, $(1, -1)$, $(2, 1)$ and $(3, 3)$. ### Reading the steps in the table ### [Graphing a line from its equation](https://math.wiki/graph-from-slope-intercept/) **Summary:** Plot the intercept, then step by the slope. **Question:** You have the equation of a line, such as $y = 3x - 5$. How do you put its line on the grid without working out a whole table of values? ## When you reach for it You already know one way to draw a line from its equation: pick inputs, work out outputs, plot the points and join them, as in graphing a line from a table. That always works, but it finds points one at a time and uses none of what the equation already tells you. An equation in the form $y = mx + b$ shows its slope $m$ and its $y$-intercept $b$ at a glance, as slope-intercept form explains. Those two numbers are a complete set of drawing instructions: put your pencil on the $y$-intercept, then walk along the line by stepping out the slope. Reach for this move whenever you have the equation of a line and want its picture: to see where two lines meet, to see how fast a quantity grows or shrinks, or to check that an equation matches a situation. Day 1 covers equations already in the form $y = mx + b$. Day 2 covers the rest: equations to rewrite first, lines drawn from their two intercepts, the horizontal and vertical lines $y = c$ and $x = c$, and lines whose numbers are too big for an ordinary grid. ## The procedure ### Day 1: start at the intercept, step by the slope **Plot the $y$-intercept.** Putting $x = 0$ into $y = mx + b$ makes the $mx$ term vanish and leaves $y = b$, so $(0, b)$ is on the line, and it sits on the $y$-axis. **Step by the slope.** Write the slope as a fraction $m = \frac{p}{q}$; a whole-number slope such as $3$ is $\frac{3}{1}$. From $(0, b)$ count $q$ squares right and $p$ squares up, then do it again from the new point. ### [Graphing a linear inequality in two variables](https://math.wiki/graph-a-linear-inequality/) **Summary:** Draw the boundary line, dashed or solid, then shade the side that works. **Question:** An equation in x and y draws a line. What does an inequality in x and y draw? ## When you reach for it An equation such as $y = 2x - 1$ is true at some points of the plane and false at the rest, and the points where it is true form a line. Change the $=$ to $>$ and ask the same question: which points $(x, y)$ make $$y > 2x - 1$$ true? There are far too many to list, so they are drawn, and what gets drawn is a whole region: every point on one side of a line. Reach for this whenever a condition on two quantities is a limit rather than an exact target, such as an oven that can run for **at most** $12$ hours. On a number line, an inequality in one variable is drawn as a ray. In the coordinate plane, an inequality in two variables is drawn as a **half-plane**, the part of the plane on one side of a line. The edge comes from graphing a line, and the algebra uses the sign flip for dividing by a negative number. ## The procedure Part 1 graphs inequalities with $y$ alone on one side and explains why the answer is always one side of a line. Part 2 handles standard form, horizontal and vertical boundaries, reading an inequality off a graph, and situations where only whole-number points make sense. ### Part 1: the region on one side of a line **A solution is a point.** A solution of an inequality in $x$ and $y$ is an ordered pair that makes it true. Try three points with first coordinate $3$ in $y > 2x - 1$, where the right side is $2(3) - 1 = 5$. For $(3, 6)$, $6 > 5$ is true. For $(3, 2)$, $2 > 5$ is false. For $(3, 5)$, $5 > 5$ is false: the two sides are equal, and equal is not greater. ### [Graphing from standard form](https://math.wiki/graph-a-quadratic-from-standard-form/) **Summary:** Find the axis, the vertex and the intercepts, then sketch. **Question:** You are handed $y = ax^2 + bx + c$ and asked for its graph. Which few points pin the whole curve down, and how do you find them without filling in a table? ## When you reach for it You can graph any equation by making a table: pick some values of $x$, work out $y$, plot the points and join them, as in graphing from a table. For a parabola that works, but it is blind. You do not know which values of $x$ to pick, so the table can easily miss the turning point, or show one arm and not the other. An equation in **standard form**, $$y = ax^2 + bx + c \qquad (a \neq 0)$$ tells you more than a table does, if you know how to ask it. The sign of $a$ says which way the curve opens. The numbers $a$ and $b$ together say where its axis of symmetry is. The number $c$ is a point on the curve with no work at all. And since every parabola is symmetric, each point you find off the axis gives you a second one free. Reach for this skill when a quadratic arrives in standard form and you want its graph, or only its key features: where it turns, how high or low it gets, and where it crosses the axes. ## The procedure Work through $y = x^2 - 4x - 5$ as the running example. Here $a = 1$, $b = -4$ and $c = -5$, each coefficient taken with its sign. **Decide which way it opens.** When $x$ is far from $0$, the term $ax^2$ is much larger than $bx + c$, so it decides the sign of $y$. If $a > 0$, $y$ is large and positive far out on both sides: the parabola **opens upward** and the vertex is its lowest point. If $a < 0$, it **opens downward** and the vertex is its highest point. Here $a = 1 > 0$, so the curve opens upward. ### [Inequalities](https://math.wiki/inequalities/) **Summary:** Statements about greater and less, whose solutions are whole stretches of the line. **Question:** A rule says "at least" or "no more than" instead of "equals". How do you find, draw and write every number that passes? ## What this area is about A linear equation such as $2x + 1 = 7$ is usually true for one number. An inequality such as $x > 3$ is usually true for a whole stretch of the number line. This area is about finding that stretch, drawing it and naming it, and about the one move that differs from solving equations: multiplying or dividing both sides by a negative number reverses the sign. ## How these fit together **Pictures and words.** Inequalities on the number line shows a solution set with open or closed circles and shading. Writing an inequality for a condition turns "at most $23$ kg" into $w \le 23$. Interval and set-builder notation writes a stretch compactly, as $[3, \infty)$. **Solving.** Solving linear inequalities takes one, two and many steps, with the sign flip. **Two conditions.** Compound inequalities join two tests with "and" (the overlap) or "or" (either part). Absolute value inequalities read $|w - 340| \le 8$ as a distance and become compound inequalities. **Beyond lines.** Polynomial and rational inequalities use a sign chart built from zeros and undefined points. ## A path through it The number line first. Then writing a condition, interval notation, and solving linear inequalities; compound inequalities after solving, and absolute value inequalities after compound ones. Polynomial and rational inequalities wait for quadratic inequalities and graphs of rational functions. ### [Inequalities on the number line](https://math.wiki/graph-simple-inequality/) **Summary:** Open circle for strict, closed for inclusive, then shade the side that works. **Question:** A ride sign says riders must be at least $120$ cm tall. That allows not one height but endlessly many. How can one picture show every one of them? ## When you reach for it An equation such as $x = 3$ is true for exactly one number. An inequality is a claim that one amount is larger or smaller than another (expression-vs-equation), and it is usually true for a great many numbers. The claim $x > 3$ is true when $x$ is $4$, when $x$ is $20$, when $x$ is $3.5$ and when $x$ is $3.001$. It is false when $x$ is $3$ itself, and false for every number below $3$. A number that makes an inequality true when it is put in for the variable is a **solution** of the inequality. The solutions of $x > 3$ cannot be listed, not even with "and so on": between $3$ and $4$ alone sit $3.1$, $3.01$, $3.001$ and endlessly many more. The number line can hold them. Every number has a position on the line (number-line), so a whole stretch of the line can stand for a whole stretch of numbers, and one picture shows every solution at once. Reach for this graph whenever a condition allows a range of values rather than a single one: a minimum height for a ride, a weight limit for an elevator, a temperature below freezing. ## The procedure **The four symbols.** Two of them are the comparison signs already used to order numbers (comparison-symbols, order-rational-numbers); the other two add "or equal to". | Symbol | Read as | Example | Is $3$ a solution? | |:-:|---|:-:|:-:| | $<$ | is less than | $x < 3$ | no | | $>$ | is greater than | $x > 3$ | no | | $\le$ | is less than or equal to | $x \le 3$ | yes | | $\ge$ | is greater than or equal to | $x \ge 3$ | yes | ### [Infinite geometric series](https://math.wiki/infinite-geometric-sum/) **Summary:** First term over one minus the ratio. **Question:** The terms 12, 4, 4/3, 4/9 and onward never stop, but each is a third of the one before. Can adding infinitely many of them give an ordinary number, and which number? ## When you reach for it The sequence $12, 4, \frac{4}{3}, \frac{4}{9}, \ldots$ is geometric, with first term $a_1 = 12$ and common ratio $r = \frac{1}{3}$. It never ends, so the sum $$12 + 4 + \frac{4}{3} + \frac{4}{9} + \cdots$$ can never be finished by adding one term after another. Yet the running totals behave very calmly. They are $12$, then $16$, then $17\frac{1}{3}$, then $17\frac{7}{9}$, and each one is closer to $18$ than the last, without ever passing it. This lesson makes that precise: the infinite sum is $18$, exactly. Reach for this skill whenever the thing wanted is a **total of infinitely many terms, each the same multiple of the one before**. Such totals turn up in three common places. A repeating decimal such as $0.2\overline{4} = 0.2444\ldots$ is one, since its digits are an endless sum of shrinking pieces, and this skill turns it into the fraction $\frac{11}{45}$. Anything that repeats with a fixed fraction left over each time is another: a bungee jumper who springs back three quarters of the way after each drop, a medicine of which a quarter is still in the body when the next dose is taken. And in calculus functions are written as endless sums of powers of $x$, and the geometric series is the first and simplest of them. An **infinite geometric series** is the sum of all the terms of a geometric sequence, $$a_1 + a_1 r + a_1 r^2 + \cdots + a_1 r^{n-1} + \cdots$$ written in sigma notation as $\sum_{k=1}^{\infty} a_1 r^{k-1}$. The symbol $\infty$ on top says that the index never stops. ### [Interpreting the parts of an expression](https://math.wiki/interpret-parts-of-an-expression/) **Summary:** Reading terms, factors and coefficients as quantities, and seeing a complicated expression as built from simpler pieces. **Question:** A bank says your balance after $t$ years will be $2500(1.04)^t$ dollars. Before you work anything out, what is that expression already telling you? ## Why this exists A formula is usually met as a machine: put a number in, get a number out. But an expression written for a situation is also a description of it, and it can be read before any number goes in. Zinovia's savings account grows by $4\%$ a year, and the bank tells her that after $t$ years her balance will be $2500(1.04)^t$ dollars. Read piece by piece, it says what she started with, what happens to the balance each year, and how many years have passed. Nothing was calculated, yet doubling the deposit visibly doubles the balance, and a better rate changes only the $1.04$. Reading this way checks that a formula fits its story and says what a change will do before any sums. The names for the pieces come from terms, coefficients and constants; this lesson asks what they **mean**, and how small pieces join into bigger ones. ## Definition To **interpret** an expression is to say, for each of its parts, what quantity that part stands for in the situation, in which units, and how the parts combine into the whole. A **part** of an expression is any piece of it that gets worked out as a unit before being combined with the rest: - a **term**, one of the pieces added or subtracted at the top level, with its sign; - a **factor**, one of the pieces multiplied: in $3x(x - 4)$ the factors are $3$, $x$ and $x - 4$; - a **coefficient**, the number factor of a term, and a **constant**, a term with no letter; - a **group in parentheses**, which stands for one number however much is written inside it; - a **base** and an **exponent**: in $(1.04)^t$ the exponent $t$ says how many factors of the base $1.04$ are multiplied together. ### [Interval and set-builder notation](https://math.wiki/interval-notation/) **Summary:** Brackets and parentheses as a compact way to name a stretch of the line. **Question:** The condition $x \ge 3$ is met by endlessly many numbers, and there is no next number after $3$ to start a list with. How can a whole stretch of the number line be written down in a few symbols? ## Why this exists The condition $x \ge 3$ is met by $3$, by $4$, by $3.5$, by $1000$, and by numbers as close above $3$ as anyone cares to name. A roster cannot hold them: a list with an ellipsis needs a next entry, and there is no next number after $3$, since halfway between $3$ and any number above it is a smaller number still above $3$. A graph on the number line shows the set perfectly, with a filled dot at $3$ and shading to the right. But a drawing does not fit in a sentence. Mathematics uses two written forms instead. **Set-builder notation** writes the rule in symbols. **Interval notation** writes only the ends. A stretch of the line with no gaps is fixed by where it starts, where it stops, and whether it keeps each end, and interval notation records exactly those three things. ## Definition **Set-builder notation.** The set of all objects that meet a condition is written $$\{x \mid x \ge 3\}$$ and read "the set of all $x$ such that $x$ is greater than or equal to $3$". The braces are the same braces as a roster, but instead of a list they hold a letter standing for a typical member, a bar, and the condition. The elements are exactly the objects that meet the condition. Some books write a colon for the bar, $\{x : x \ge 3\}$, with the same meaning. The letter is only a placeholder: $\{t \mid t \ge 3\}$ is the same set as $\{x \mid x \ge 3\}$. Unless the condition says otherwise, the letter stands for a real number. The condition can narrow that down: $\{x \mid x \text{ is an integer and } x \ge 3\}$ is $\{3, 4, 5, \dots\}$, a different set from $\{x \mid x \ge 3\}$, which also holds $3.5$. ### [Inverse operations and the properties of equality](https://math.wiki/properties-of-equality/) **Summary:** Do the same thing to both sides and the equation still holds. This is the license for every step. **Question:** An equation says two amounts are equal, but one of them is hidden behind a letter. What can you do to the equation without breaking that equality? ## Why this exists A solution of an equation is a number that makes it true, and any guess can be tested by putting it back in: checking a solution. Testing says whether a guess is right. It does not say what to guess. For $x + 6 = 19$ a good guess comes quickly; for $27n = 1{,}053$ or $y + 3.85 = 11.2$ it does not. What is needed is a way to change an equation, one move at a time, into one so plain that it reads off its own answer, such as $x = 13$, without any move changing the answer. Two ideas do this. **Inverse operations** say which move frees the letter. **The properties of equality** say why the move is allowed: whatever is done to one side is done to the other. ## Definition **Inverse operations.** Two operations are inverse operations when each undoes the other: do one, then the other, and you are back where you started. - Adding a number and subtracting the same number are inverse operations: $(n + 6) - 6 = n$, and $(n - 6) + 6 = n$. - Multiplying by a number and dividing by the same number are inverse operations, as long as that number is not $0$: $(n \times 4) \div 4 = n$, and $(n \div 4) \times 4 = n$. This is how multiplication and division undo each other. **Properties of equality.** An equation says that two expressions name the same number. If $a = b$ is true, then each of these is true too, for any number $c$: | Property | If $a = b$, then | |---|---| | Addition property | $a + c = b + c$ | | Subtraction property | $a - c = b - c$ | | Multiplication property | $a \times c = b \times c$ | | Division property | $a \div c = b \div c$, when $c \neq 0$ | ### [Inverse, joint and combined variation](https://math.wiki/inverse-variation/) **Summary:** When one quantity goes up the other goes down by the same factor; the product stays fixed. **Question:** When one quantity shrinks as another grows, or depends on several others at once, how do you write that as a single equation with one constant in it? ## Why this exists Direct variation describes two quantities that grow together at a fixed rate: $y = kx$, so doubling $x$ doubles $y$. Many of the laws people actually meet run the other way, or depend on more than one thing at once. A fixed amount of work done faster takes less time. A gear with more teeth turns more slowly than the small gear driving it. Light from a lamp fades as you walk away from it, and it fades faster than the distance grows. The cost of painting a wall depends on its length and on its height, together. The pressure in a sealed tank rises when it is heated and falls when the tank is made bigger. Each of these is one quantity that is a constant times a product or quotient of others. Once the pattern has a name and an equation, one measured case is enough to find the constant, and the constant then answers every other case. That is the whole use of variation: a single number carries the law. ## Definition **Inverse variation.** $y$ *varies inversely with* $x$ when, for some fixed number $k \neq 0$, $$y = \frac{k}{x}, \qquad x \neq 0.$$ The number $k$ is the **constant of variation**. Multiplying both sides by $x$ gives the same statement as a product: $$xy = k.$$ So $y$ varies inversely with $x$ exactly when the product of every pair of matching values is the same number. The word *inverse* means the reciprocal: $y = k \cdot \frac{1}{x}$, so $y$ varies directly with $\frac{1}{x}$, which is why it is also written $y \propto \frac{1}{x}$. The same idea, met first through tables and stories, is inversely proportional relationships. ### [Linear equations](https://math.wiki/linear-equations/) **Summary:** Finding the number that makes a statement true, when the unknown is not raised to a power. **Question:** A number is hidden behind a letter, and the letter is never squared. How do you find the number, and how do you know you have all of them? ## What this area is about A linear equation in one unknown is one where, once each side is simplified, the unknown appears only to the first power: $ax + b = cx + d$. Solving it means finding every number that makes the sentence true, by moves that can never change the answer. ## How these fit together **The ground rules.** What a solution is says what you are looking for and how to check it. Inverse operations and the properties of equality say which moves are allowed (never dividing by $0$). **The core procedures**, each built on the one before: one-step equations, two-step equations $px + q = r$ with $p \ne 0$, multi-step equations that need tidying first, and equations with the unknown on both sides. **A result about outcomes.** No solution, one solution, or every number explains what it means when the unknown vanishes. **Applications of the core.** Equations of the form $p(x+q)=r$ from word problems, equations from proportions and percents, absolute value equations (two linear cases), and literal equations that rearrange formulas. ## A path through it What a solution is, then the properties of equality, one-step and two-step equations. After two-step equations come $p(x + q) = r$, proportions and multi-step equations. Multi-step equations lead to absolute value equations and to variables on both sides, which open the three outcomes and literal equations. Next: inequalities, linear relationships and rational equations. ### [Linear programming](https://math.wiki/linear-programming/) **Summary:** Maximize or minimize a quantity over a shaded region; the best value sits at a corner. **Question:** A shaded region holds every plan that keeps within the limits. Which of those endless plans makes the most money, or costs the least? ## When you reach for it model-constraints-with-inequalities turns the limits in a situation into inequalities, and overlap-of-inequality-regions draws them: the **feasible region** is the set of points that satisfy every constraint at once. That region usually holds infinitely many plans. This lesson answers the next question: which one is best? "Best" is measured by an amount built from the variables, such as a profit $P = 25x + 40y$ or a cost $C = 5x + 10y$. When that amount has the form $$z = ax + by$$ with $a$ and $b$ fixed numbers, it is called a **linear objective**, and the task of making it as large as possible (**maximizing**) or as small as possible (**minimizing**) over a region cut out by linear inequalities is **linear programming**. The word "programming" is old and means planning, not computer code. The move you reach for is the **corner principle**: if the objective has a best value on the region, that value is reached at a corner. So instead of searching infinitely many points, you find the few corners and compare. ## The procedure Part 1 (the first day) explains why corners decide everything and uses that on regions closed in on every side. Part 2 (the second day) handles the cases a closed region hides: a minimum on a region that runs off without end, two corners that tie, and an objective with no best value at all. ### Part 1: level lines and the corner principle **Name the variables, the constraints and the objective.** Say what $x$ and $y$ count, write each limit as an inequality, and write the objective as a formula in $x$ and $y$, with its units. ### [Linear relationships](https://math.wiki/linear-relationships/) **Summary:** Straight lines: constant rate, and every way of writing one. **Question:** One quantity changes at a steady rate as another grows. Why is the graph a straight line, and how many ways can you write its equation? ## What this area is about A linear relationship is one where the output changes by the same amount for every unit step in the input. Its graph is a straight line, and its rate of change is the line's slope. This area is about reading that line and its equation in each of their usual forms, and moving between them. ## How these fit together **The rate.** Slope is rise over run, the same between any two points of a non-vertical line. Direct variation is the case $y = kx$ with $k \ne 0$: a line through the origin. **Forms of the equation.** Slope-intercept form, $y = mx + b$, shows the slope and the $y$-intercept. Point-slope form starts from any point, or from two points. Standard form, $Ax + By = C$, fits a fixed total shared by two quantities. **Graphing.** From a table of values, point by point, and from the equation, intercept then slope. **Two lines.** Parallel lines that are not vertical share a slope; perpendicular lines have slopes whose product is $-1$, when neither line is vertical. Writing a parallel or perpendicular line puts both to use. **Using it.** Modeling with linear relationships. ## A path through it Start with slope and with graphing from a table. After slope: slope-intercept form and direct variation. Then graphing from the equation, and point-slope form. After point-slope form: standard form (which also needs the table) and parallel lines; then perpendicular lines, then writing either kind of line. Modeling comes after direct variation. ### [Literal equations: solving for one variable](https://math.wiki/solve-for-a-variable/) **Summary:** Treat every other letter as a number and isolate the one you want. **Question:** A formula gives one quantity once you know the others. What if you know that one, and need one of the others? ## When you reach for it The formula $d = rt$ says that distance is rate times time. Give it a rate and a time and it hands back a distance at once. Often the question runs the other way: you know how far a journey is and how fast you will go, and you want the time. You could put the two numbers in and solve the equation that results, but for twenty journeys that is twenty equations, each solved with exactly the same steps. The better move is to do those steps once, with the letters still in place: $$t = \frac{d}{r}$$ This is the same relationship, rearranged so that $t$ stands alone, and every later question about time is now a single division. An equation with more than one letter in it is called a **literal equation**, from the Latin *littera*, a letter. Formulas are the most familiar kind: $P = 2l + 2w$, $C = 2\pi r$, $A = P + Prt$. **Solving a literal equation for one letter** means rewriting it as an equivalent equation with that letter alone on one side and nowhere on the other. Reach for it when - one formula is to be used many times with a different quantity unknown, as in a table or a spreadsheet column; - you want to see how one quantity depends on the others: $w = \frac{P}{2} - l$ shows at a glance that, for the same perimeter, a longer rectangle is a narrower one; - an equation in $x$ and $y$, such as $3x + 2y = 12$, is wanted in the form "$y$ equals something in $x$", which is the form slope-intercept form and every graphing tool expect. ### [Maximum, minimum and projectile problems](https://math.wiki/maximize-or-minimize/) **Summary:** The vertex answers how much and when. **Question:** A situation has a best choice, the most area, the least cost, the highest point of a throw. How do you find it exactly, rather than by trying numbers, and how do you tell when the question wants the best value and when it wants something else? ## When you reach for it Some questions ask for the best of something. What price brings in the most money? How should $240$ meters of fence be laid out to hold the most ground? How high does a thrown ball go? Each has one best answer, and trying numbers only ever gets close to it. When the quantity you want to make large or small is a quadratic in one variable, its graph is a parabola, and a parabola turns exactly once. That turning point, the vertex, is the answer. Its first coordinate says **when** or **where** the best happens: the time, the width, the price. Its second coordinate says **how much**: the height, the area, the money. Reach for this move whenever a word problem says greatest, least, most, fewest, maximum, minimum, largest, smallest or highest, and the quantity involved is, or can be written as, a quadratic. The same models also carry questions that are not about a best value at all: when does the ball land, which price brings in exactly this much. Those are answered by a root, not the vertex, and the first part of the procedure is telling the two apart. The most common source of such a quadratic is a thrown object, whose height is a quadratic in time. Projectile motion has its own part of the procedure below. ## The procedure ### Quadratics in context: a root or the vertex? Some quantities rise and then fall, or fall and then rise: the height of a thrown ball, the profit as a price goes up, the area of a pen made from a fixed length of fence. Each is modeled by a quadratic. The work is knowing which piece of the parabola answers the question asked. ### [Modeling with linear relationships](https://math.wiki/linear-modeling/) **Summary:** Choose the variables, fit the line, read a prediction off it, and say where the model stops making sense. **Question:** Something in the world changes at a steady rate. How do you turn the story into an equation that predicts, and how do you know when to stop trusting it? ## When you reach for it An ice sculpture melts at a winter festival. A timber yard prices fence posts by length. A runner's split times are recorded on the way up a tower. Each time somebody wants a number nobody has measured: the sculpture's mass at midnight, the price of a post not on the list, the time to the top floor. A **model** is an equation that stands in for the situation, so a question about the world becomes a question about the equation. A **linear model** fits when the quantity changes at a constant rate: the same amount for every hour, every centimeter, every floor. Then the graph is a straight line, $y = mx + b$, with the rate as the slope $m$ and the starting value as $b$. Modeling also means checking that a line really fits, reading the slope and intercept back into the story, and saying where the equation stops describing anything real. Day 1 builds a model and predicts with it. Day 2 reads and sketches graphs made of straight pieces, and marks where a model stops. ## The procedure ### Day 1: building a model and using it **Name the two quantities, with units.** Write each as a full description, not a bare letter: "$t$, the number of hours since the sculpture was unveiled", "$m$, its mass in kilograms". A slope is a change in output divided by a change in input, so its unit is the output's unit **per** the input's unit, here kilograms per hour. The description also fixes where the input starts counting: from the unveiling, the starting mass sits at $t = 0$, exactly the intercept $b$. The input is usually the quantity you know or control, the output the one you want to predict. ### [Monomials and the exponent rules](https://math.wiki/multiply-monomials/) **Summary:** Multiply the numbers, add the exponents. **Question:** Single terms, each a number times some powers, are multiplied, divided or raised to a power. Why does the answer always come back as a number times powers, and what happens to the exponents? ## When you reach for it A **monomial** is a single term: a number, or a number multiplied by one or more variables, where each variable carries a positive whole-number exponent. The number in front is the monomial's **coefficient**. So $$7, \qquad x, \qquad -5a^4b^2, \qquad \frac{2}{3}y^3$$ are all monomials, with coefficients $7$, $1$, $-5$ and $\frac{2}{3}$. When no number is written, the coefficient is $1$, as in $x$, and a lone minus sign means $-1$, as in $-x^2$. A variable with no written exponent has exponent $1$: $x$ is $x^1$. By contrast $x + 2$ is not a monomial (it has two terms), and neither is $\frac{3}{x}$ (the variable sits in a denominator, which no whole-number exponent can express). Later in this lesson $\frac{3}{x}$ will be written $3x^{-1}$; that is a fine expression, but its exponent is negative, so it is still not a monomial. Monomials get multiplied all the time. A rectangle $4a$ long and $3a$ wide has area $(4a)(3a)$. A box with edges $2x$, $3x$ and $5x$ has volume $(2x)(3x)(5x)$. And every product of longer expressions, such as $2x^2(3x^3 - x)$ or $(x + 1)(x^2 - 4)$, is built out of products of single terms: multiplying a polynomial by a monomial and multiplying polynomials both come down to this move, done once for each pair of terms. They get divided just as often. If a rectangle has area $36a^5b^3$ and one side $4a^2b$, the other side is the quotient $\frac{36a^5b^3}{4a^2b}$. They get raised to powers: a square of side $3x$ has area $(3x)^2$, a cube of edge $4p^2$ has volume $(4p^2)^3$. And negative exponents turn up the moment a quotient leaves more factors below the fraction bar than above, or a very small quantity is written in scientific notation, such as $6 \times 10^{-4}$. ### [Multi-step equations](https://math.wiki/multi-step-combine-first/) **Summary:** Collect like terms on each side, then solve as a two-step equation. **Question:** An equation has parentheses, several terms with the unknown in them, or fractions and decimals everywhere. How do you turn it into one you already know how to solve? ## When you reach for it A two-step equation such as $3x + 7 = 25$ has the unknown in a single term, and two moves undo what was done to it. Many equations arrive less tidy than that. In $5x + 3 - 2x + 4 = 25$ the unknown appears in two terms. In $4(y - 3) + 2y = 30$ part of it is inside parentheses. In $\frac{x}{3} + \frac{x}{4} = 14$ it sits under fractions. Each of these still has the unknown on **one side only**, and each can be brought back to the two-step shape $$px + q = r$$ by tidying the side the unknown is on. An equation that needs this tidying first is called a **multi-step equation**. Once it has the shape $px + q = r$, the two-step method does the rest. The new work is the tidying, and every tidying step must be one that cannot change the answer. This lesson takes two days. **Part 1** multiplies out parentheses and combines like terms, then solves and checks. **Part 2** clears fractions and decimals before solving, and builds such equations from stories: a bill that adds several charges, a perimeter that adds four sides. ## The procedure ### Part 1: tidy each side, then solve Take $5x + 3 - 2x + 4 = 25$. Its left side is a sum of four **terms**. A term is one of the pieces of a sum, joined to the others by $+$ or $-$, and the sign in front of it belongs to it. A term may be a number such as $3$, a product such as $-2x$, or a product with parentheses such as $4(y - 3)$. **Like terms** have the same letter part: $5x$ and $-2x$ are like terms, and so are the constants $3$ and $4$. ### [Multi-step inequalities](https://math.wiki/multi-step-inequalities/) **Summary:** Distribute, collect, solve, flip if needed. **Question:** When the unknown sits on both sides of an inequality, tangled in parentheses, how do you find every number that makes it true? ## When you reach for it Two-step inequalities had the unknown in one place: $px + q > r$. Many limits are not that tidy. Two companies quote for a class museum tour, each with a booking charge and a price per student, and you want to know when the first is no more expensive than the second. Now the unknown appears on **both** sides: $$250 + 6s \le 80 + 10s$$ These are **multi-step inequalities**: the unknown may sit on both sides, inside parentheses, with fractions or decimals, and the sign is one of $<$, $>$, $\le$ or $\ge$. The equation versions, multi-step equations and equations with variables on both sides, used the same moves. The one change is the one one-step inequalities found: multiplying or dividing both sides by a negative number flips the inequality sign. The answer is a whole **solution set**. ## The procedure Every step below must leave the solution set exactly as it was, so each one comes with the reason it is allowed. ### Make each side as simple as it can be Clear parentheses with the distributive property and combine like terms on each side. In $5 - 2(x + 3) \ge 4x + 11$, the left side becomes $5 - 2x - 6$, which is $-2x - 1$: $$-2x - 1 \ge 4x + 11$$ For every $x$ the new left side is the same number as the old one, so the solution set is unchanged. Fractions are cleared by multiplying both sides by a common denominator, a positive number, so the direction stays. ### Gather the variable on one side ### [Multiplying and dividing rational expressions](https://math.wiki/multiply-divide-rational-expressions/) **Summary:** Factor, flip for division, cancel, multiply. **Question:** Two fractions with polynomials in them are multiplied or divided. How do you get one fraction in simplest form, and which values of the variable are still off limits? ## When you reach for it A rational expression is a fraction whose numerator and denominator are polynomials, such as $\frac{x^2 - 16}{2x + 6}$. You reach for this skill when two of them are multiplied or divided and you want the result as one fraction, as simple as it can be, together with the values of the variable where the original was never defined. It comes up whenever rates and ratios carry a variable: a speed times a time, a count per sheet times sheets per hour, the quotient of two functions. It is also the first half of fraction arithmetic with polynomials; adding and subtracting, which need a common denominator, come next. ## The procedure Everything here is the arithmetic of multiplying fractions and dividing fractions, applied to polynomials. That works because at any allowed value of the variable each polynomial is just a number, and number fractions obey these rules. ### Multiplying For polynomials $p, q, r, s$ with $q \ne 0$ and $s \ne 0$, $$\frac{p}{q} \cdot \frac{r}{s} = \frac{pr}{qs}$$ Numerators multiply, denominators multiply. Then the product is simplified by canceling common factors, using $\frac{ak}{bk} = \frac{a}{b}$ for $k \ne 0$, the same move that simplifying a rational expression rests on. **Factor before you multiply.** Only a factor of the whole numerator and the whole denominator can cancel. Multiplying out first buries those factors inside a large polynomial that then has to be factored again. So factor every numerator and denominator completely first, watching for the special patterns. ### [Multiplying polynomials](https://math.wiki/multiply-by-a-monomial/) **Summary:** Distribute the single term into every term. **Question:** One term multiplies a whole sum, or a sum multiplies a sum. How do you multiply out every piece and know that none was missed? ## When you reach for it A product written as factors, such as $3x^2(2x^3 - 5x + 4)$ or $(x + 1)(x + 2)(x + 3)$, hides what kind of polynomial it is. Multiplied out and collected, it shows its degree, its constant term and every coefficient, and it can be added to another polynomial term by term. Areas and volumes arrive as products, a length times a length, and multiplying out is how they become one polynomial. The only tool is the distributive property, used as often as needed, with the exponent rules doing each small product. The lesson runs over two days. Day 1 multiplies a polynomial by a single term, a monomial. Day 2 multiplies a polynomial by a polynomial, starting from the two-binomial case of multiply-two-binomials and carrying the same idea to longer factors and to three factors in a row. ## The procedure ### Day 1: a monomial times a polynomial The distributive property, $a(b + c) = ab + ac$, works for any number of terms in the parentheses. For three, group the first two (the associative property of addition) and use it twice: $$\begin{aligned} a(b + c + d) &= a\big((b + c) + d\big) \\ &= a(b + c) + ad \\ &= ab + ac + ad \end{aligned}$$ A fourth term is covered by the same step again. So a monomial times a polynomial is the monomial times **each** term, added up. Each small product is a product of two monomials. Reorder and regroup its factors (the commutative and associative properties of multiplication) so the numbers sit together and each letter sits with its own powers. Then multiply the numbers and add the exponents on each base, since $x^m \cdot x^n = x^{m+n}$. ### [Multiplying two binomials](https://math.wiki/multiply-two-binomials/) **Summary:** Every term in the first times every term in the second. **Question:** Two sums are multiplied together. Which products does that make, and how can you be sure none is missing? ## When you reach for it You already know how to multiply a number into a sum: $3(x + 5) = 3x + 15$, by the distributive property. Now both factors are sums. The product $(x + 3)(x + 5)$ is, for example, the area of a rectangle with sides $x + 3$ and $x + 5$, and you want that area as a single sum of terms with no parentheses. A sum of two terms is a **binomial**, and rewriting a product of binomials as a sum is called **expanding** it. Reach for it when parentheses stand in the way: comparing two areas, collecting terms in an equation, reading off the constant or the highest power, or checking a factorization. ## The procedure ### The distributive property, used twice For any value of $x$, the binomial $x + 5$ is just a number, and the distributive property holds for every number. So distribute the whole binomial $x + 5$ over the sum $x + 3$: $$(x + 3)(x + 5) = x(x + 5) + 3(x + 5)$$ This is $(a + b)c = ac + bc$, the distributive property with the common factor on the right (the commutative property of multiplication turns it into the familiar $c(a + b)$). Each piece is now a single term times a binomial, so distribute again: $$\begin{aligned} &x(x + 5) + 3(x + 5)\\ &\quad = x^2 + 5x + 3x + 15 \end{aligned}$$ where $x \cdot x = x^2$ by the exponent rules. Last, $5x$ and $3x$ are like terms, so they combine: $$(x + 3)(x + 5) = x^2 + 8x + 15$$ ### Why there are four products The two distributions made $x \cdot x$, $x \cdot 5$, $3 \cdot x$ and $3 \cdot 5$. Each of the $2$ terms of the first binomial met each of the $2$ terms of the second exactly once, so there are $2 \times 2 = 4$ products, the same count as choosing one of $2$ shirts and one of $2$ hats. That is the whole rule: **every term of the first times every term of the second**, then collect like terms. ### [No solution, one solution, or every number](https://math.wiki/no-solution-and-identity/) **Summary:** When the letters vanish, what is left tells you which of the three you have, and why. **Question:** Solving an equation usually ends with the unknown equal to one number. What has happened when the unknown disappears instead? ## Why this exists Most equations you solve end the same way: the unknown alone on one side, a number on the other, $x = 4$. Every step keeps the two sides equal, by the properties of equality, so the last line names the one number that makes the first line true. Sometimes that last line never arrives. You gather the variable terms on one side, as in solving with the unknown on both sides, and they all disappear, because $6x - 6x$ is $0$. What is left has no letter in it, such as $15 = 1$ or $-12 = -12$. That is not a slip in the working. The equation is telling you that either no number makes it true, or every number does. Knowing which of the three cases you are in, and why, finishes the job of solving linear equations. This lesson is taught over two days. Part 1 is what the vanished variable means. Part 2 is predicting the case before you solve, building equations with a chosen number of solutions, and seeing the three cases as two lines on a graph. ## Definition A **solution** of an equation is a number that makes it true when it replaces the unknown (what a solution is). A **linear equation** in one unknown is one where, after each side is simplified, the unknown appears only to the first power: each side has the form $ax + b$. Such an equation has exactly one of three kinds of answer. - **Exactly one solution.** $5x - 1 = 2x + 8$ is true for $x = 3$ and for no other number. - **No solution.** No number makes it true. $x + 5 = x + 2$ is false whatever $x$ is, because a number plus $5$ is never that number plus $2$. Such an equation is called a **contradiction**. - **Every number is a solution**, so there are infinitely many. $2(x + 1) = 2x + 2$ is true whatever $x$ is, because the two sides are one expression written two ways, by the distributive property. Such an equation is called an **identity**. ### [Number and shape patterns from a rule](https://math.wiki/patterns-from-a-rule/) **Summary:** A pattern is built from its rule, then studied for features the rule does not say, such as every term being odd. **Question:** Tejas starts at $7$ and keeps adding $3$. The rule says nothing about odd and even numbers, so why do his numbers take turns, odd, even, odd, even, forever? ## When you reach for it Tejas writes a list of numbers. He starts at $7$, and each new number is $3$ more than the one before: $$7,\ 10,\ 13,\ 16,\ 19,\ \ldots$$ A list like this is a **number pattern**. Each number in it is a **term**: $7$ is the first term, $10$ the second term, $13$ the third. The words "start at $7$, add $3$" are the pattern's **rule**. The rule tells you how to make the pattern, one term at a time, for as long as you like. Reach for a rule whenever something grows (or shrinks) by the same amount again and again: a savings jar that gets $3$ more dollars each week, a stack that loses $8$ books at each trip, a shape that gets $3$ more sticks of wood at each step. The rule builds the list. Then comes the more interesting part. Look hard at the list, and it shows things the rule never said. Tejas's terms take turns being odd and even. His rule says nothing about odd or even. A good question is always: *why* does that happen? ## The procedure ### Part 1 (day 1): building a pattern from its rule **A rule has two pieces: where to start and what to do each time.** "Start at $7$, add $3$" gives the first term, $7$, and the step that turns any term into the next one. Both pieces matter. "Add $3$" alone could start anywhere: from $7$ it gives $7, 10, 13, \ldots$ and from $20$ it gives $20, 23, 26, \ldots$, a different pattern with no terms in common with the first. **Each term comes from the one before it.** To get the $5$th term you need the $4$th, and to get that you need the $3$rd. So a pattern is built in order, and one slip spoils every term after it. A quick check at the end catches most slips (Example 1 shows one). ### [One-step equations](https://math.wiki/one-step-add-subtract/) **Summary:** Undo an addition by subtracting, and the other way around. **Question:** A number has had one thing done to it, and you know how it came out. How do you get back to the number? ## When you reach for it An equation such as $x + 18 = 52$ or $4x = 36$ hides a number behind a letter, and exactly one thing has been done to that number: one number added to it, taken from it, multiplied by it or divided into it. Such an equation is a **one-step equation**, because one move undoes what was done and leaves the letter alone. Reach for this move whenever you can say in one phrase what happened to the letter: "$18$ was added", "it was multiplied by $4$", "three quarters of it was taken". Many equations that come from a story have this shape. A price times a number of plates is a bill; a summer journey plus some extra minutes is a winter journey; part of a target is the distance swum so far. You already have both tools. The properties of equality say that doing the same thing to both sides keeps an equation true, as long as you never divide by $0$, and inverse operations say which thing to do. Checking a solution, by putting it back in, says whether the answer is right. This lesson puts them to work over three days: adding and subtracting (Part 1), multiplying and dividing, with fractions and decimals (Part 2), and writing the equation from a story (Part 3). ## The procedure Read what was done to the letter. Do the inverse of it to both sides. Check by putting the answer back into the equation you started with. Every case below is that one idea; what changes is which inverse to use and how to do the arithmetic. ### [One-step inequalities and the sign flip](https://math.wiki/one-step-inequalities/) **Summary:** Same moves as an equation, with one exception coming. **Question:** An equation is true for one number. When the condition says "at most" or "more than", which numbers pass, and which moves keep the condition true? ## When you reach for it An equation such as $x + 3 = 8$ is true for exactly one number, $x = 5$. Many conditions are not like that. A ride lets on anyone at least $120$ cm tall; a ferry ramp takes any vehicle up to its weight limit; a driver may go at any speed up to the limit. Each of these is an **inequality**, and an inequality is usually true for a whole stretch of numbers rather than one. Reach for this lesson when the unknown appears once, on one side, and a single operation stands between it and being alone: $x + 3 < 8$, $y - 7 \le -10$, $6x > 27$, $-4x \ge 28$. Solving means finding every number that makes the inequality true. The moves are the ones that solve one-step equations, doing the same thing to both sides to undo the operation, with one exception: multiplying or dividing both sides by a negative number reverses the inequality sign. Drawing the answer uses inequalities on the number line. This lesson is taught over two days. **Part 1** is what a solution set is, how to graph it, and the moves that never change the sign: adding or subtracting, and multiplying or dividing by a positive number. **Part 2** is multiplying or dividing by a negative number, and why that reverses the sign. ## The procedure ### Part 1: solution sets, and moves that keep the order **A solution is any number that makes the inequality true.** Take $x + 3 < 8$ and try some numbers. | $x$ | $x + 3$ | Is $x + 3 < 8$? | |:-:|:-:|:-:| | $-10$ | $-7$ | yes | | $4$ | $7$ | yes | | $4.9$ | $7.9$ | yes | | $5$ | $8$ | no, $8$ is not less than $8$ | | $6$ | $9$ | no | ### [Operations with radicals](https://math.wiki/multiply-and-divide-radicals/) **Summary:** Roots multiply and divide as one root. **Question:** A panel is $\sqrt{14}$ meters by $\sqrt{10}$ meters. How do you multiply, divide, add and subtract numbers like these and still get an exact answer? ## When you reach for it Square roots turn up in the middle of real calculations. A rectangle with sides $\sqrt{14}$ and $\sqrt{10}$ has an area, a right triangle whose legs are roots has a perimeter, and one length can be divided by another. Each of these is an expression with several roots in it, and the aim is to write it as simply as possible while keeping it exact, never a rounded decimal. Simplifying a square root already gives one root its simplest form: $\sqrt{90} = 3\sqrt{10}$. This lesson starts from there and combines roots with each other. There are three moves. Roots multiply under one root sign, they divide under one root sign, and they add only when they are multiples of the same root. After any of the three, the result is simplified again. ## The procedure ### Multiplying square roots For any numbers $a \ge 0$ and $b \ge 0$, $$\sqrt{a}\cdot\sqrt{b} = \sqrt{ab}$$ This is the product rule from simplify-square-roots, read from right to left. There it took a root apart; here it puts two roots together. **Why it holds.** The left side is a product of two non-negative numbers, so it is non-negative. Squaring it, and regrouping the factors (the commutative and associative properties of multiplication), $$\left(\sqrt{a}\cdot\sqrt{b}\right)^2 = \left(\sqrt{a}\right)^2\left(\sqrt{b}\right)^2 = ab$$ The right side, $\sqrt{ab}$, is by definition the non-negative number whose square is $ab$. Two non-negative numbers with the same square are the same number, so the two sides are equal. ### [Parallel and perpendicular lines](https://math.wiki/parallel-slopes/) **Summary:** Same slope, different intercept, and why they never meet. **Question:** Two lines on a grid look as if they will never meet. How can their equations settle it, without extending the drawing forever? ## Why this exists A drawing shows only a few squares of each line, and lines go on forever. The lines $y = 3x + 7$ and $y = 3.01x + 2$ look parallel on any ordinary sheet of graph paper, yet they cross at $(500, 1507)$, far off the page. Looking cannot decide whether two lines meet. The equations can. A line that is not vertical has a slope, the amount it rises for each step to the right, and a $y$-intercept, the height where it crosses the $y$-axis. Same slope and different intercepts: the lines never meet. Different slopes: they cross, exactly once. The same comparison tells how many solutions a system of two equations has. ## Definition Two different lines in a plane are **parallel** when they never meet, however far they are extended. In symbols, $\ell_1 \parallel \ell_2$ says that line $\ell_1$ is parallel to line $\ell_2$. This is the parallel lines of geometry, now read through coordinates. For two lines that are not vertical, written in slope-intercept form $$y = m_1 x + b_1 \qquad y = m_2 x + b_2$$ the lines are parallel exactly when $$m_1 = m_2 \quad \text{and} \quad b_1 \ne b_2$$ If the slopes match and the intercepts match too, the two equations describe one and the same line. Some books count a line as parallel to itself; this lesson does not, so that "parallel" matches "no solution" exactly. A vertical line has no slope, so vertical lines need their own statement: $x = h$ and $x = k$ are parallel exactly when $h \ne k$. ### [Point-slope form and the line through two points](https://math.wiki/write-from-point-and-slope/) **Summary:** Drop the point and slope into the form. **Question:** You know one point on a line and how steep it is, or only two points on it. How do you write its equation? ## When you reach for it Slope-intercept form, $y = mx + b$, writes a line from two facts: its slope $m$, and the number $b$ where it crosses the $y$-axis. Very often you are not told $b$. Point-slope form is the equation that starts from whichever point you happen to have. Reach for it when: - you know the slope of a line and one point on it, any point; - you know two points on a line, since the slope formula turns them into a slope and you are back in the first case; - you have a table of values that changes at a constant rate; - a situation gives two measurements of one quantity that changes steadily with another. This lesson runs over two days. Day 1 builds point-slope form and two routes from a point and a slope to $y = mx + b$. Day 2 starts from two points, a table or a situation, and uses a third point as a test. ## The procedure ### Day 1: point-slope form **Where the form comes from.** Suppose a line has slope $m$ and passes through a known point $(x_1, y_1)$. The small numbers are labels, not exponents: $x_1$ reads "$x$ one" and just means "the first $x$", a fixed number. Now let $(x, y)$ stand for any *other* point of the line. On a straight line the slope between any two of its points is the same, which is what similar slope triangles show. So the slope from $(x_1, y_1)$ to $(x, y)$ is $m$: $$\frac{y - y_1}{x - x_1} = m$$ Multiply both sides by $x - x_1$. This is allowed because $x - x_1$ is not zero: the point $(x, y)$ is a different point, and the line is not vertical, so its $x$ differs from $x_1$. The result is ### [Polynomial and rational inequalities with sign charts](https://math.wiki/polynomial-and-rational-inequalities/) **Summary:** Solving polynomial and rational inequalities by finding zeros and undefined points and testing the sign on each interval between them. **Question:** You can solve a quadratic inequality by finding its roots and testing the stretches between them. Does the same idea work when the expression is a cubic, a quartic, or a fraction with $x$ in the denominator? ## When you reach for it Quadratic inequalities rest on one fact: a quadratic can change sign only where it equals $0$. Between two neighboring roots the graph has no way to get from above the axis to below it, so one test number speaks for the whole stretch. That fact is not special to quadratics. The graph of any polynomial is one unbroken curve, so a polynomial of any degree changes sign only at its real zeros. A rational function, a quotient of two polynomials, has one more way to change sign: its graph can jump across a vertical asymptote, as you saw in graphing rational functions. Those two kinds of points, the zeros and the places the expression is undefined, cut the number line into stretches, and on each stretch the sign stays fixed. You reach for this move whenever an inequality has a polynomial of degree three or more, or a variable in a denominator: where is $(x + 3)(x - 2)^2(x - 6)$ negative, for which $x$ is $\frac{x + 2}{x - 5}$ at least $1$, which parts of a curved track sit above a given height. Day 1 handles polynomials. Day 2 adds denominators, and with them a step that cannot be skipped. ## The procedure **Compare with zero.** Move every term to one side, so the question is about the sign of a single expression $P(x)$: is it positive, negative, or zero? For a rational inequality, also combine everything into one fraction $\frac{N(x)}{D(x)}$ over a common denominator, as in adding and subtracting rational expressions. ### [Polynomial long division](https://math.wiki/polynomial-long-division/) **Summary:** Same procedure as with numbers, one power at a time. **Question:** You can divide 4729 by 13 one place at a time. Can a polynomial be divided the same way, one power at a time? ## When you reach for it Multiplying polynomials takes two factors and gives their product. Division runs the other way: you know a product and one factor, and you want the other. A box's volume and its floor area give its height. One known factor of a cubic leaves a quadratic, which you can factor further. Division also rewrites a fraction of polynomials whose top has the higher degree as a polynomial plus a fraction whose top has the lower degree, the way $\frac{17}{5}$ becomes $3\frac{2}{5}$. A single-term divisor goes term by term. A longer one needs long division, laid out as long division of whole numbers, one power at a time. ## The procedure ### Dividing by a monomial Dividing by $d$ is multiplying by $\frac{1}{d}$, so the distributive property splits a sum over a single term: $$\frac{a + b + c}{d} = \frac{a}{d} + \frac{b}{d} + \frac{c}{d}, \qquad d \neq 0$$ Each piece is then a quotient of monomials: divide the coefficients and subtract the exponents, since $\frac{x^m}{x^n} = x^{m - n}$ by the laws-of-exponents. ### Dividing by a polynomial Long division of $4729$ by $13$ starts at the biggest place. $13$ goes into $47$ three times, so it goes into $4700$ three hundred times, and what is left is passed on. Written as a chain of facts: $$\begin{aligned} 4729 &= 13 \cdot 300 + 829 \\ 829 &= 13 \cdot 60 + 49 \\ 49 &= 13 \cdot 3 + 10 \end{aligned}$$ Substituting each line into the one above gives $4729 = 13 \cdot 363 + 10$: quotient $363$, remainder $10$. The rounds stop when what is left is smaller than $13$. ### [Polynomials](https://math.wiki/polynomials/) **Summary:** Sums of powers of a variable, and how they add, multiply and factor. **Question:** Expressions built from a variable by adding and multiplying behave a lot like whole numbers. How do they add, multiply, factor and divide? ## What this area is about A polynomial is a sum of terms, each a number times a whole-number power of the variable, such as $2x^3 + 3x^2 + x$. Polynomials add, multiply and divide much as integers do, and factoring one into simpler polynomials is like factoring a number into primes. This area does that arithmetic and ends with what one evaluation, $p(c)$, says about factors and roots. ## How these fit together **The object.** Degree, terms and standard form names the parts; the degree is the highest power. **Operations.** Adding and subtracting polynomials combines like terms. Multiplying polynomials multiplies every term by every term. Special products are the squared binomial and sum times difference, recognized on sight. **Factoring.** Factoring trinomials reverses the expansion of two binomials. Factoring special patterns reads the special products backward, as $a^2 - b^2 = (a + b)(a - b)$, and adds the sum and difference of two cubes. **Division.** Polynomial long division works one power at a time; synthetic division is its shorthand when the divisor is $x - c$. **Results.** The remainder and factor theorems link $p(c)$, the remainder and the roots. Proving polynomial identities shows that two expressions agree for every value. ## A path through it Degree first; then adding and multiplying. After multiplying: factoring trinomials and long division. Special products come before both special patterns and identities. Trinomials and special patterns refer to each other, so take them together. Long division, then synthetic division, then the theorems. ### [Proving polynomial identities](https://math.wiki/prove-polynomial-identities/) **Summary:** Proving that two polynomial expressions are equal for every value, and using identities such as the one behind Pythagorean triples. **Question:** Two expressions agree at every number you try. How can you be sure they agree at every number there is? ## When you reach for it Work out $7^2 - 6^2 = 13$, $10^2 - 9^2 = 19$ and $31^2 - 30^2 = 61$. Each difference of two consecutive squares is odd, and each is the two numbers added together. A hundred more tries would agree, but a hundred tries are not every number. Algebra covers them all at once: for any number $n$, $$\begin{aligned} n^2 - (n - 1)^2 &= n^2 - (n^2 - 2n + 1) \\ &= 2n - 1 = n + (n - 1) \end{aligned}$$ An **identity** is a statement that two expressions are equal for **every** value of their variables. An **equation**, in the sense of something to solve, is true for **some** values, and solving finds which. $x^2 - 9 = (x - 3)(x + 3)$ is an identity; $x^2 - 9 = 0$ holds at $x = 3$ and $x = -3$ and nowhere else. Linear equations already showed the same split, when the letters vanished and left a statement true for every number. Reach for a proof whenever someone claims two polynomial expressions are always equal: a factoring pattern, a shortcut for arithmetic, a pattern spotted in a list of numbers. A proof turns "it worked every time I tried" into "it cannot fail". ## The procedure **Turn one side into the other.** Start from one side, usually the more complicated one, and rewrite it using only moves that hold for every number: the distributive property, the commutative and associative properties, collecting like terms, and identities already proved, such as the special products. Each move produces an expression equal to the one before for every value, so the first and last expressions in the chain are equal for every value. Ending at the other side proves the identity. ### [Quadratic expressions and parabolas](https://math.wiki/quadratic-expressions-and-parabolas/) **Summary:** What ax squared plus bx plus c looks like on a graph, and why it curves. **Question:** What does ax squared plus bx plus c look like, and why does it curve? ## What this area is about A linear expression graphs as a straight line. Add an $x^2$ and the line bends into a parabola: a U shape that turns around exactly once and is the same on both sides. This area is about what makes an expression quadratic and how to read the parabola it draws. ## How these fit together **The definition.** What makes an expression quadratic: degree two, so the variable is squared and no higher, and $a \neq 0$. **The shape.** The shape of a parabola explains why squaring gives a symmetric curve, and why $a > 0$ opens up while $a < 0$ opens down. **Its landmarks.** The vertex and the axis of symmetry are taken together. The axis is the line $x = -\frac{b}{2a}$ that the curve folds onto, and the vertex sits on that line as the highest or lowest point. **The procedure.** Graphing from standard form puts the landmarks together: axis, vertex, intercepts, then sketch. ## A path through it Definition, then shape, then the vertex and its axis, then graphing. Each one uses the one before. With the picture in hand, the real solutions of a quadratic equation are where the parabola meets the $x$-axis, which is what roots, zeros and $x$-intercepts makes precise. Factoring and square roots find those points, and this area is the ground for quadratic functions. ### [Quadratic inequalities](https://math.wiki/quadratic-inequalities/) **Summary:** Find the roots, then test which stretches of the line satisfy the inequality. **Question:** You can find where a quadratic equals zero. How do you find everywhere it is bigger than zero, or smaller? ## When you reach for it A linear inequality such as $2x - 6 > 0$ has one boundary, $x = 3$, and everything on one side of it works. A quadratic inequality, such as $$x^2 - 2x - 3 > 0$$ is different. Try a few numbers. $x = 5$ works, since $25 - 10 - 3 = 12$. $x = 0$ does not, since it gives $-3$. And $x = -2$ works again, since $4 + 4 - 3 = 5$. The numbers that work are not all on one side of anything. They come in stretches, and the job is to find where each stretch starts and stops. You reach for this move whenever the question is "for which $x$ is this quadratic positive, or negative?" rather than "for which $x$ is it zero?". That question turns up when a ball must be above a certain height, when an area must be at least some amount, when a square root must have something non-negative underneath it, and when an equation must have two real solutions. The answer is almost never a single number. It is a set of numbers, written in interval notation. ## The procedure The whole method rests on one fact about the graph of $y = ax^2 + bx + c$: it is a single unbroken curve. To get from above the $x$-axis to below it, the curve has to cross the axis, and it crosses only at its roots. So between two neighboring roots, and beyond the outermost ones, the quadratic keeps the same sign the whole way along. A quadratic has at most two real roots, so the number line breaks into at most three stretches, and each stretch is entirely in the answer or entirely out of it. ### [Quadratics](https://math.wiki/quadratics/) **Summary:** Expressions and equations where the variable is squared, and the parabolas they draw. **Question:** Square the variable and the straight line bends. What changes, and how do you still solve it? ## What this area is about A quadratic is an expression like $ax^2 + bx + c$, with the variable squared and no higher. Its graph is a parabola, a curve that turns once. Squaring makes it hard to solve: $x$ appears twice, and an equation can have two real answers, one, or none. ## How these fit together **The object.** Quadratic expressions and parabolas sets up the object, its graph and its vertex. Roots, zeros and $x$-intercepts shows that solutions, zeros and crossing points coincide. **Ways to solve** $ax^2 + bx + c = 0$. Solving quadratics with square roots handles a variable only inside a square, as in $x^2 = 25$. Solving quadratics by factoring is fastest when the expression factors. Completing the square always works: Completing the square for a plain $x^2$, then Completing the square, leading coefficient not one. The quadratic formula is that method done once, in general, and its discriminant, $b^2 - 4ac$, counts the real solutions in advance. **Forms of the graph.** Vertex form: reading, converting and graphing puts the turning point in the equation and covers moving and stretching the curve. Factored form puts the roots there instead. **Using it.** Quadratic inequalities, Maximum, minimum and projectile problems and Fitting a quadratic to data. ## A path through it Expressions and parabolas; square roots, factoring, roots and intercepts; completing the square, both cases, and the formula; vertex form, factored form, inequalities, best values; fitting last. ### [Radicals and rational expressions](https://math.wiki/radicals-and-rational-expressions/) **Summary:** Roots and fractions with variables in them. **Question:** Roots and fractions with polynomials in them both hide their simplest form. How do you rewrite them exactly, and solve equations that contain them without picking up false answers? ## What this area is about Two kinds of expression go past plain polynomials. A radical puts a quantity under a root, as in $\sqrt{245}$; a rational expression divides one polynomial by another, as in $\frac{x + 1}{x^2 - 4}$. Both are rewritten exactly, never rounded, and both carry restrictions: for a real value, a square root needs a quantity under it that is not negative, and a value that makes a denominator $0$ is excluded. Solving equations with either can produce an answer that fails the original equation, so every answer is checked. ## How these fit together **Radicals.** Simplifying radicals pulls perfect squares out from under the root, combines roots and rationalizes denominators. Solving radical equations isolates the root and raises both sides to the matching power; squaring can bring in false answers. **Rational expressions.** Simplifying rational expressions factors, cancels, and does the four operations while keeping excluded values excluded. Solving a rational equation multiplies through by the common denominator, then rejects any answer that makes a denominator $0$. **A model.** Inverse, joint and combined variation writes laws such as $y = \frac{k}{x}$ with one constant $k \ne 0$. ## A path through it The two strands are separate. Radicals: simplifying, then radical equations. Rational expressions: simplifying, then rational equations and inverse variation. After this: the quadratic formula, whose answers often leave a root, radical functions and rational functions. ### [Rate of change](https://math.wiki/rate-of-change/) **Summary:** How much the output moves for each step of the input. **Question:** When one quantity moves another, how much does the second change for each step of the first? ## Why this exists A taxi meter, a burning candle, a hiker's altimeter, a phone's battery icon: each shows one quantity changing as another moves on. The distance traveled moves the fare; the time passing moves the candle's height. Knowing only how much the output changed is not enough to compare two of these. A candle that burns down $6$ cm has burnt fast if that took $2$ hours and slowly if it took $12$. What lets you compare is the change in the output **for each one unit** of the input: centimeters per hour, dollars per kilometer, meters per minute. That number is the rate of change. You have met rates already in rates and unit rates, and in a proportional relationship the constant of proportionality was the whole story. Most relationships do not start from zero, though. A taxi fare has a charge before the car has moved, and a candle starts tall. The rate of change is the part of the relationship that says how fast, kept apart from where it started. ## Definition Take two quantities where the first, the **input**, decides the second, the **output**. Pick two values of the input. Between them, $$\text{rate of change} = \frac{\text{change in output}}{\text{change in input}}$$ Each change is the later value minus the earlier value, taken in the same order on top and underneath. The result says how much the output changes for each one-unit step of the input. Its units are the output's unit **per** the input's unit: a fare in dollars against a distance in kilometers has a rate in dollars per kilometer. A rate written without its units has lost half its meaning. ### [Rationalizing a denominator](https://math.wiki/rationalize-a-denominator/) **Summary:** Multiply top and bottom to clear the root below the line, using the conjugate when needed. **Question:** A fraction has a square root in its denominator. How can you make the bottom a whole number without changing the fraction's value? ## When you reach for it Dividing roots sometimes leaves a root below the line, as in $\frac{1}{\sqrt{2}}$. That is an exact, correct number. **Rationalizing the denominator** means rewriting such a fraction as an equal fraction whose denominator is a rational number, in practice an integer. The name says what happens: the denominator becomes rational. If the value does not change, why bother? Three reasons. **Estimating.** $\sqrt{2} \approx 1.41421$. To estimate $\frac{1}{\sqrt{2}}$ directly you would divide $1$ by $1.41421$, long division by a decimal with five places. Rewritten below, the same number is $\frac{\sqrt{2}}{2}$, and half of $1.41421$ is $0.70711$ at a glance. **Comparing and adding.** $\frac{1}{\sqrt{2}} + \sqrt{2}$ looks like two unlike things. Written as $\frac{\sqrt{2}}{2} + \frac{2\sqrt{2}}{2}$, it is two like radicals over a common denominator, and the sum is $\frac{3\sqrt{2}}{2}$. **One agreed form.** $\frac{1}{\sqrt{2}}$, $\frac{\sqrt{2}}{2}$, $\frac{2}{\sqrt{8}}$ and $\sqrt{\frac{1}{2}}$ are all the same number. If everyone writes it as $\frac{\sqrt{2}}{2}$, two answers can be compared by eye. So an expression with square roots counts as **simplest form** only when no radicand has a perfect-square factor other than $1$ (as in simplify-square-roots), no fraction sits under a root, and no root sits in a denominator. Reach for it whenever a calculation ends with a root below the line: a side of a square from its diagonal, a 45-45-90 triangle, or a quotient of two roots that did not divide evenly. ### [Recursive formulas](https://math.wiki/recursive-formula/) **Summary:** Define each term from the one before, plus a starting value. **Question:** If you know where a list of numbers starts and how each number turns into the next, is that enough to pin down the whole list? ## Why this exists Many quantities are easiest to describe by how they change from one step to the next. A library adds $40$ books every month. A waiting list loses a quarter of its names every week. A jar of sourdough starter doubles overnight, and then $100$ grams are taken out to bake with. It may be hard to say at once how much there will be in month $30$, but it is easy to say what next month looks like once you know this month. A recursive formula writes exactly that down. It names where the list starts and how each term turns into the next, and the whole sequence unrolls from there, one step at a time. You already know sequences and the notation $a_n$ from sequence-basics, and the explicit formulas for arithmetic and geometric sequences, which give $a_n$ straight from $n$. A recursive formula describes the same sequences the other way: step by step instead of all at once. ## Definition A **recursive formula** for a sequence has two parts: - a **starting value**, the first term given as a number, such as $a_1 = 7$; - a **recursive rule**, which says how to get each term from the term before it, such as $a_n = a_{n-1} + 4$ for $n \ge 2$. The symbol $a_{n-1}$ is read "$a$ sub $n$ minus one" and means the term just before $a_n$. Its position $n - 1$ is one less than $n$, so when $a_n$ is the fifth term, $a_{n-1}$ is the fourth. The rule $a_n = a_{n-1} + 4$ therefore says: each term is the term before it, plus $4$. The note "for $n \ge 2$" is there because the first term has no term before it; it comes from the starting value instead. ### [Remainder and factor theorems](https://math.wiki/remainder-and-factor-theorems/) **Summary:** Plugging in a number tells you the remainder, and a zero remainder means a factor. **Question:** Dividing a polynomial takes a page of work. What can one evaluation tell you instead, and how many roots must a polynomial have? ## What this area is about These are results about polynomials and their roots. They link three things: the value $p(c)$, the remainder when $p(x)$ is divided by $x - c$, and whether $x - c$ is a factor. Together they say where to look for roots, and how many there are. ## How these fit together All three children are theorems, and each is used as a tool. The remainder and factor theorems: dividing $p(x)$ by $x - c$ leaves the remainder $p(c)$, so $x - c$ is a factor exactly when $p(c) = 0$. The rational root theorem: for a polynomial with integer coefficients and $a_0 \ne 0$, any rational root in lowest terms has a numerator dividing the constant term $a_0$ and a denominator dividing the leading coefficient $a_n$. That gives a short list of candidates to test. The fundamental theorem of algebra: every polynomial of degree $n \ge 1$ with complex coefficients has a complex root, so it has exactly $n$ roots when repeats are counted. ## A path through it The remainder and factor theorems come first; the rational root theorem and the fundamental theorem both build on them. Next: whether a graph crosses or touches at a zero, and later irreducibility tests in abstract algebra. ### [Repeating patterns](https://math.wiki/repeating-patterns/) **Summary:** A repeating pattern has a core, like red-blue or circle-square-square, that repeats, so the next piece can be predicted. **Question:** How can a child look at a row of beads, blocks or claps and know what comes next? ## Why this exists A **repeating pattern** is a row of things built from one short block, the **core**, said over and over: red, blue, red, blue has the core red-blue. Once a child finds the core, the next piece is no longer a guess; it is whatever the core says comes next. Finding a core takes the same noticing a child uses to sort objects by one feature: these two are alike, that one is different. For the adult: a core has to show up at least twice before you can be sure of it. Red, blue, blue on its own could continue in many ways, so every pattern you build this week shows its core two or three full times. It is handy to name cores with letters (AB, AAB, ABB, ABC), where each letter stands for one kind of piece; the child does not need the letters and can say "two reds, then a blue". ## Day 1: The part that repeats **You need:** a dozen snap cubes or beads in two colors, say red and blue. Stand up, sit down, stand up, sit down, and invite the child to join in. Then ask, "What will I do next?" Now build a cube train: red, blue, red, blue, red, blue. Point to each cube as you say its color together, so the pattern is heard as well as seen. Say, "Red, blue is the part that repeats. We call it the core." Snap the train apart into its cores and line them up: every piece is the same red-blue pair. Build red, blue, red, blue, red and stop. Ask, "What comes next? How do you know?" A good answer names the core: "Blue, because it goes red, blue, and the red just came." ### [Rewriting an expression to show what it means](https://math.wiki/rewrite-an-expression-to-show-its-meaning/) **Summary:** Rewriting an expression in a different form shows something new about the situation, as a + 0.05a = 1.05a shows a 5 percent increase. **Question:** Two expressions give the same number every time. Why would you ever trade one for the other? ## When you reach for it Querida buys a lamp marked $p$ dollars, and a sales tax of $9\%$ is added at the till. The receipt lists the price and then the tax, so the natural way to write what she pays is $$p + 0.09p$$ The same amount can also be written $1.09p$. The two expressions are equivalent: whatever number $p$ is, they give the same result. What changes is what you can see. The first form says *the price, plus a tax worth $9\%$ of the price*. The second says *the total is $1.09$ times the price*, that is, $109\%$ of it, found with one multiplication. Reach for a rewrite when the form you have hides the thing you want to know: a percent change spread across a sum, a perimeter that is really two equal halves, a fixed fee tucked inside parentheses. The moves are ones you already have: distributing a number over a sum in parentheses, factoring out a common factor, and combining like terms, often one after the other. This lesson is about which form to aim for, and how to read it once you have it. ## The procedure ### A rewrite never changes the value Every rewrite uses properties that hold for all numbers: the distributive property, the commutative property (order does not matter in a sum or a product) and the associative property (grouping does not matter). So the new expression equals the old one for every value of the letter, and either one may be read as a description of the situation. ### [Roots, zeros and x-intercepts](https://math.wiki/roots-zeros-and-x-intercepts/) **Summary:** Three names for the same thing, seen in the equation, the function and the graph. **Question:** An equation has solutions, a function has zeros and a graph has x-intercepts. Why do all three turn out to be the same numbers? ## Why this exists The same numbers keep turning up under different names. Solve $x^2 - 2x - 3 = 0$ and you get $-1$ and $3$, the **roots** of the equation. Ask which inputs make $f(x) = x^2 - 2x - 3$ output zero and you get $-1$ and $3$ again, the **zeros** of the function. Draw the graph and it meets the $x$-axis at $-1$ and $3$, the **$x$-intercepts**. These are not three facts that happen to agree. They are one fact seen three ways: in the equation, in the function, and in the picture. Once you see why, you can move freely between the three. You can solve by factoring to find where a graph crosses the axis, read a rough answer off a graph before doing any algebra, and check a solution by looking at a picture. ## Definition Take a function $f$ and a real number $k$. - $k$ is a **root** of the equation $f(x) = 0$ when substituting $k$ for $x$ makes the equation true. "Root" and "solution" mean the same thing here; "root" is the older word, and it is the usual one for polynomial equations. - $k$ is a **zero** of the function $f$ when $f(k) = 0$: the input $k$ gives the output $0$. - The point $(k, 0)$ is an **$x$-intercept** of the graph $y = f(x)$ when the graph passes through it: the graph meets the $x$-axis there. Roots and zeros are numbers. An $x$-intercept is a point, and its $y$-coordinate is always $0$. Many books also call the number $k$ itself the $x$-intercept; the meaning is clear either way, but the point is the exact statement. ### [Sequences and series](https://math.wiki/sequences-and-series/) **Summary:** Lists of numbers with a rule, and what happens when you add them up. **Question:** A list of numbers follows a rule. How do you reach any term without counting through the others, and what do the terms add up to? ## What this area is about A sequence is an ordered list of numbers $a_1, a_2, a_3, \ldots$ made by a rule. This area asks two questions of it: what is the $n$th term, and what is the total of the first $n$ terms, or of all of them? ## How these fit together **Early patterns.** Repeating patterns repeat a core; patterns from a rule are built from a start and a step, then studied for features the rule does not state. **Language.** Sequences, terms and notation names the term $a_n$ and its index $n$. Sigma notation writes a sum such as $\sum_{k=1}^{n} a_k$ exactly. **Two families and two ways to define them.** Arithmetic sequences add a common difference $d$, a linear rule; geometric sequences multiply by a common ratio $r$, an exponential one. Recursive formulas give each term from the one before plus a starting value. **Sums.** Arithmetic series and partial sums total the first $n$ terms. Geometric series total the first $n$ terms of a geometric sequence, by a formula that needs $r \ne 1$. Infinite geometric series have a sum $\frac{a_1}{1 - r}$ only when $|r| < 1$. ## A path through it The two early pattern lessons stand on their own. Then sequence basics, arithmetic sequences, and geometric sequences, which lead to recursive formulas. Sigma notation follows arithmetic sequences; from it come arithmetic partial sums, and with geometric sequences, finite geometric series, then infinite ones. ### [Sequences, terms and notation](https://math.wiki/sequence-basics/) **Summary:** An ordered list with a rule; the nth term and its index. **Question:** A list of numbers runs on, or a pattern grows stage by stage. How do you name "the twentieth one" without writing out the first nineteen, and what kind of thing is such a list? ## Why this exists Walk along one side of a street where the houses carry the odd numbers: $1, 3, 5, 7, 9$ and on. Ask for "the fifth house" and the answer is house number $9$, not $5$. Two numbers are in play, the place in the row and the number on the door, and talking about the row without mixing them up needs a name for each. Often you want one far-off entry, the 40th house, without writing out everything before it. A sequence is the idea that makes that possible: an ordered list, treated as a function of position. ## Definition A **sequence** is an ordered list of numbers. Each number in the list is a **term**. Each term sits in a **position**, counted $1, 2, 3$ and so on from the start, and the position is called the **index** or **term number**, usually written $n$. The term in position $n$ is written $a_n$, read "a sub n". For the odd house numbers, $a_1 = 1$, $a_2 = 3$, $a_3 = 5$ and $a_5 = 9$. The small $n$ says *where*; the value $a_n$ says *what is there*. Any letter will do in place of $a$, such as $b_n$ or $t_n$, as long as each sequence keeps its own. **A sequence is a function.** Put in a position $n$ and out comes exactly one value, $a_n$. Its domain, the set of allowed inputs, is the counting numbers $\{1, 2, 3, \ldots\}$, because positions are counted. In function notation, $f(n)$ means the same as $a_n$: for the house numbers, $f(5) = 9$. - An **infinite sequence** has a term for every counting number. Three dots at the end, as in $1, 3, 5, 7, \ldots$, say the list never stops. - A **finite sequence** stops. Its domain is $\{1, 2, \ldots, N\}$ for some last position $N$, and its list has no dots at the end. ### [Sigma notation](https://math.wiki/read-sigma-notation/) **Summary:** The index, the limits, and the rule being summed. **Question:** A sum has fifty terms that all follow one rule. How do you write it down exactly, without the dots that leave the reader guessing? ## When you reach for it Adding up the terms of a sequence is one of the most common things done with it. Writing the sum out in full stops being practical once there are more than a handful of terms, and the usual shortcut, a few terms and then dots, $$2 + 5 + 8 + \cdots + 17$$ leaves the reader to guess the rule. Is the next term after $8$ really $11$? Where exactly does the sum stop? Dots show a pattern; they do not state one. **Sigma notation** states it. It names the rule that makes each term and the exact range of positions to use, so there is nothing left to guess. Reach for it whenever you need to write a sum of many terms that follow one rule: the total of a sequence's first $n$ terms, the area of a stack of thin rectangles, a total that builds up week by week. It is also the language of everything that comes after this lesson about series, so reading it fluently matters as much as writing it. This lesson uses the explicit formulas of arithmetic and geometric sequences, and the language of terms and positions from sequence-basics. ## The procedure ### Reading the notation Here is a sum in sigma notation: $$\sum_{k=1}^{6} (3k - 1)$$ It is read "the sum of $3k - 1$, for $k$ from $1$ to $6$". The large symbol $\Sigma$ is the capital Greek letter sigma, the Greek S, standing for "sum". The expression has four parts. - The **index** is the letter written under the sigma, here $k$. It counts the positions. - The **lower limit** is the number the index starts at, written after the equals sign under the sigma, here $1$. - The **upper limit** is the number the index stops at, written above the sigma, here $6$. - The **summand** is the expression to the right of the sigma, here $3k - 1$. It is the rule for the term in position $k$, exactly an explicit formula $a_k = 3k - 1$. ### [Simplifying a rational expression and where it is undefined](https://math.wiki/rational-expression-domain/) **Summary:** Any value that makes the denominator zero is excluded, and it stays excluded after canceling. **Question:** A fraction with $x$ in it can often be shortened by canceling. What does the short form keep, and what does it quietly lose? ## Why this exists Divide one polynomial by another and you get a fraction with a variable in it, such as $\frac{x^2 - 9}{x^2 + x - 6}$. Fractions like this turn up whenever a quantity is a share, a rate or an average: the cost per item, the time per job, the part of a paint mix that is one color. Two questions come with every one of them. Which values of $x$ are forbidden? A fraction with $0$ underneath has no value, and here the bottom changes as $x$ changes, so some values of $x$ break it. And can it be written shorter? Number fractions shorten by canceling a common factor, and so do these. This lesson answers both, and shows they are tied together: canceling can hide a forbidden value, so the forbidden values have to be found first and carried along. ## Definition A **rational expression** is a fraction whose numerator and denominator are both polynomials, with a denominator that is not the zero polynomial. $\frac{3}{x}$, $\frac{x + 1}{x^2 - 4}$ and $\frac{2x^2 - 8x}{6x}$ are rational expressions. So is any polynomial, such as $x^2 + 1$, since it is itself over $1$. An **excluded value** is a value of the variable that makes the denominator $0$. At an excluded value the expression has no value at all. The excluded values are exactly the solutions of the equation "denominator $= 0$". The numerator may be $0$: $\frac{0}{5} = 0$ is a perfectly good number. A rational expression is **simplified** when its numerator and denominator have no common factor other than $1$ and $-1$. A simplified form is written together with its **restrictions**: the excluded values of the expression you started from, stated as $x \neq \dots$. ### [Simplifying a square root](https://math.wiki/simplify-square-roots/) **Summary:** Pull out perfect square factors. **Question:** A square floor covers 245 square meters. Its side is the square root of 245, but can that number be written exactly and still show how big it is? ## When you reach for it Maximilian is marking out a square dance floor with an area of $245$ square meters. The side of a square is the number whose square is its area, so the side is $\sqrt{245}$ meters. That is exact, but it hides the size: is it near $15$, near $16$, a whole number? A calculator gives $15.6524\ldots$, which shows the size and is no longer exact. There is a third way to write it. Since $245 = 49 \cdot 5$, the floor can be cut into $49$ small squares of area $5$, seven along each side. Each small square has side $\sqrt{5}$, so the side of the floor is seven of those: $$\sqrt{245} = 7\sqrt{5}$$ $7\sqrt{5}$, read "seven root five", means $7 \cdot \sqrt{5}$. It is exactly the same number as $\sqrt{245}$, written so its size shows: $\sqrt{5} \approx 2.236$, so the side is about $7 \times 2.236 \approx 15.65$ meters. Reach for this rewrite whenever a square root is not a whole number and you want to keep it exact: the side of a square from its area, a length from Pythagoras' theorem, the answers that come out of the square root property and the quadratic formula. Simplified roots can be compared at a glance, estimated by hand, and recognized as multiples of the same root, which is what later lets them be added. ### The words | Word | What it names | In $\sqrt{245}$ | In $\sqrt[3]{250}$ | |---|---|:-:|:-:| | radical sign | the symbol itself | $\sqrt{\ }$ | $\sqrt[3]{\ }$ | | radicand | what sits under the sign | $245$ | $250$ | | index | the small number in the crook of the sign, saying which root | $2$, not written | $3$ | | radical | the whole expression | $\sqrt{245}$ | $\sqrt[3]{250}$ | ### [Simplifying radicals](https://math.wiki/simplify-radicals/) **Summary:** Pull perfect squares out from under the root. **Question:** A square root is exact but hides its size. How do you rewrite expressions with roots so they stay exact and become as simple as they can be? ## What this area is about An exact answer such as $\sqrt{245}$ can be written more usefully as $7\sqrt{5}$. This area is about rewriting square roots and expressions built from them in simplest form, never as rounded decimals. Everything rests on the product rule $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$, which holds for $a \ge 0$ and $b \ge 0$. ## How these fit together All three children are procedures, each using the one before. Simplifying a square root proves the product rule and pulls perfect square factors out from under the root. Operations with radicals multiplies and divides roots as one root, and adds or subtracts like radicals such as $3\sqrt{2} + 5\sqrt{2}$. Rationalizing a denominator multiplies top and bottom by the same nonzero number, often a root or a conjugate, so the denominator becomes rational: $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$. ## A path through it Simplifying a square root, then operations with radicals, then rationalizing a denominator. Next: radical equations, the quadratic formula, whose answers often carry a root, and special right triangles. ### [Simplifying rational expressions](https://math.wiki/rational-expressions/) **Summary:** Fractions with polynomials: factor, cancel, note what is not allowed. **Question:** A fraction has polynomials on top and bottom. How do you simplify it and combine it with others, and which values of the variable are never allowed? ## What this area is about A rational expression is a fraction whose numerator and denominator are polynomials, with a denominator that is not the zero polynomial, such as $\frac{x + 1}{x^2 - 4}$. Its arithmetic is the arithmetic of number fractions, with one extra duty: any value that makes a denominator $0$ is excluded, and it stays excluded after canceling. ## How these fit together **Definition and first procedure.** Simplifying a rational expression and where it is undefined defines excluded values, then factors and cancels common factors. The short form equals the original only where the original is defined: $\frac{x^2 - 9}{x - 3} = x + 3$ for $x \ne 3$. **Operations.** Multiplying and dividing rational expressions factor, flip the divisor, cancel and multiply, keeping every value excluded along the way. Adding and subtracting rational expressions rename each fraction over a common denominator, then combine the numerators, and simplify complex fractions. ## A path through it Start with simplifying and excluded values, which the multiplying lesson refers back to. Then multiply and divide, then add and subtract. Next: solving a rational equation and rational functions. ### [Slope](https://math.wiki/slope/) **Summary:** Rise over run: how steep a line is, and its sign. **Question:** Two lines both climb, but one creeps and one shoots up. What single number says how steep a line is, and why is it the same everywhere along the line? ## What this area is about Slope is rise over run: how many units a non-vertical line goes up for each unit it goes across, with a sign that says whether it climbs or falls. It is the rate of change of a linear relationship, and the $m$ in $y = mx + b$. ## How these fit together **The idea.** Rate of change is the change in output divided by the change in input, with units such as dollars per hour. Slope is that rate for a straight line. **Procedures.** Slope from a graph or a table counts the rise and the run between two points. The slope formula and the four kinds of slope computes $m = \frac{y_2 - y_1}{x_2 - x_1}$ from coordinates alone, valid when $x_2 \ne x_1$, and sorts lines into positive, negative, zero and undefined slope. **A result.** Slope triangles explains why any two points on a non-vertical line give the same slope: on a slanted line, the right triangles drawn along it are scaled copies of each other. ## A path through it Rate of change first, then slope from a graph or table, then the slope formula, and last slope triangles, which proves what the earlier lessons kept finding. Next: slope-intercept form, direct variation and, much later, the derivative at a point. ### [Slope from a graph or a table](https://math.wiki/slope-from-a-graph/) **Summary:** Count the rise, count the run, divide. **Question:** A straight line climbs across a grid, or a table lists pairs of numbers that change together. How do you put one number on how fast the second climbs as the first moves on? ## When you reach for it Two lines on the same grid can both go up from left to right and still look nothing alike: one creeps, the other shoots up. "Steep" and "gentle" are words. The **slope** is the number: how many units a straight line goes up for every one unit it goes across, with a sign that says whether it climbs or falls. Reach for it whenever a straight line or a list of pairs is in front of you and the question is "how fast?". On a graph you count squares; in a table you subtract. Either way the slope is the rate of change of the second quantity as the first moves on. The lesson takes two days. Part 1 reads slope off a graph. Part 2 reads it from a table, checks first that the table describes a straight line at all, and says what the number means once the quantities carry units. ## The procedure ### Part 1: slope from a graph **Rise and run.** Pick two points on a straight line on a coordinate grid. Walk from the left-hand point to the right-hand one in two straight moves: first across, then up or down. The move across is the **run**, the number of units traveled to the right. The move up or down is the **rise**, counted **positive going up and negative going down**. The slope of the line, written $m$, is $$m = \frac{\text{rise}}{\text{run}}$$ The two moves and the stretch of line between the points make a right triangle, with the run along the bottom and the rise up the side. Figure 1 draws two such triangles under one line. ### [Slope triangles: why a line has one slope everywhere](https://math.wiki/slope-triangles/) **Summary:** Any two right triangles drawn along the same line are scaled copies, so rise over run is the same wherever you measure it. **Question:** You measured how steep a line is between two of its points. Why would any other two points give the same number? ## Why this exists The slope of a line is its rise divided by its run between two of its points. But which two points? If two points on the left gave one number and two on the right gave another, there would be no such thing as "the slope of the line", only "the slope between these two points". In slope-from-a-graph and slope-from-two-points the same number came out whichever pair was chosen. The reason: every pair of points on a line makes a right triangle with the line, and all those triangles are scaled copies of one another, that is, similar. Similar triangles keep their sides in the same ratio, and rise over run is such a ratio. ## Definition Take a line that is neither vertical nor horizontal, and two points on it. Draw a horizontal segment from one point and a vertical segment from the other, so that they meet. The three segments make a right triangle whose longest side, the **hypotenuse**, lies along the line. This is a **slope triangle** of the line. The horizontal leg is the **run** and the vertical leg is the **rise**. In this lesson both are lengths, so both are positive numbers; the direction is read off separately. The slope of the line is $$m = \frac{\text{rise}}{\text{run}}$$ when the line goes up from left to right, and the same fraction with a minus sign in front when the line goes down from left to right. The same two points give two slope triangles, one on each side of the line, depending on which point the horizontal leg starts from. Both count, and the Results below cover both. ### [Slope-intercept form](https://math.wiki/read-slope-and-intercept/) **Summary:** In y = mx + b, m is the slope and b is where it starts. **Question:** A straight line crosses the y-axis at one height and climbs at one steady rate. Are those two numbers enough to say exactly which points are on it, and can you see them in its equation? ## When you reach for it A straight line is pinned down by two facts: where it crosses the $y$-axis, and how steeply it climbs. Written in the form $$y = mx + b$$ its equation shows both at a glance. The number $m$ is the slope, and $b$ is the **$y$-intercept**, the value of $y$ where the line crosses the $y$-axis. This way of writing a line is called **slope-intercept form**. Reach for it whenever you want to read a line rather than just own an equation for it: to say how fast one quantity changes with another and where it starts, to draw the line, to compare two lines, or to turn "an opening deposit of $50$ dollars, then $20$ dollars a week" into an equation. When the quantities carry units, $m$ is a rate of change, such as dollars per week, and $b$ is the **starting value**, the amount when the input is $0$. The lesson takes two days. Part 1 (Examples 1 to 4) shows why every line that is not vertical has an equation of this form, then reads and writes such equations. Part 2 (Examples 5 to 7) rewrites other equations into this form, reads $m$ and $b$ off a graph or a table, and gives both numbers their units. ## The procedure ### Part 1: why a line is $y = mx + b$ **The $y$-intercept.** A line that is not vertical is not parallel to the $y$-axis, so it crosses the $y$-axis at exactly one point. Every point of the $y$-axis has $x$-coordinate $0$, so the crossing point is $(0, b)$ for some number $b$, and that number is the $y$-intercept. Some books give the name to the point $(0, b)$ rather than to the number $b$; both carry the same information. ### [Slopes of perpendicular lines](https://math.wiki/perpendicular-slopes/) **Summary:** Slopes multiply to negative one: flip and change the sign. **Question:** Two lines on a grid cross. How can their equations alone tell you whether they cross at a right angle? ## Why this exists Parallel lines are easy to spot from their equations: they have the same slope. The other special way for two lines to sit is at a right angle, and that one is harder to see. A sketch of two lines crossing at $89^\circ$ looks exactly like a sketch of two lines crossing at $90^\circ$, and a protractor laid on a drawing measures the drawing, not the lines. Coordinates settle it exactly. Once each line has a slope, one multiplication decides whether the lines are perpendicular, with no measuring at all. That turns questions about square corners into arithmetic: whether a triangle given by its corners has a right angle, whether four points make a rectangle, which way the tangent to a circle runs. It is the test that lets algebra prove things about right angles. ## Definition Two lines are **perpendicular** when they cross at a right angle. With a symbol this is written $\ell_1 \perp \ell_2$. On the coordinate plane there is a test for this in terms of slope. Two lines that are not vertical, with slopes $m_1$ and $m_2$, are perpendicular exactly when their slopes multiply to $-1$: $$m_1 m_2 = -1$$ Dividing both sides by $m_1$ (which is not zero, since the product is not zero) gives the same fact the other way round. Each slope is the **negative reciprocal** of the other: $$m_2 = -\frac{1}{m_1}$$ To get the negative reciprocal of a number, take its reciprocal (flip the fraction) and change its sign. The negative reciprocal of $\frac{2}{3}$ is $-\frac{3}{2}$, the negative reciprocal of $-4$ is $\frac{1}{4}$, and the negative reciprocal of $1$ is $-1$. Both parts are needed: $\frac{2}{3}$ and $-\frac{2}{3}$ have opposite signs but multiply to $-\frac{4}{9}$, and $\frac{2}{3}$ and $\frac{3}{2}$ are reciprocals but multiply to $1$. ### [Solving a rational equation](https://math.wiki/solve-rational-equation/) **Summary:** Multiply through by the common denominator, then solve what is left. **Question:** An equation has the unknown in a denominator. How do you get it out, and how do you know the answer you get is real? ## When you reach for it A **rational equation** is an equation with a variable in a denominator, such as $\frac{6}{x} + 1 = \frac{5}{x - 2}$. They arise whenever an unknown is divided into something: the part of a road a snow plow clears in an hour is $\frac{1}{t}$ when the whole road takes $t$ hours, and the time to cover $30$ km at speed $v$ is $\frac{30}{v}$. Set such quantities equal to what is known and the equation is rational. Fractions make an equation hard to solve, so the first move is to get rid of them. This lesson runs over two days. Part 1 clears the denominators and checks what comes out. Part 2 uses the same move on problems about work, rates and speed. ## The procedure ### Part 1: clearing the denominators **List the excluded values first.** A value that makes any denominator zero gives no number on that side, so it can never be a solution. Factor each denominator and write down its zeros, as in excluded values of a rational expression. For $\frac{6}{x} + 1 = \frac{5}{x - 2}$ the list is $x \ne 0, 2$. **Multiply both sides by the least common denominator.** Find the LCD of every fraction in the equation, the way it is found in adding rational expressions, and multiply every term on both sides by it. Each denominator divides the LCD, so every fraction cancels and a polynomial equation is left. The move is allowed because multiplying both sides of a true equation by the same number keeps it true, one of the properties-of-equality. ### [Solving linear inequalities](https://math.wiki/solve-linear-inequalities/) **Summary:** Same moves as an equation, except the sign flips when you multiply or divide by a negative, and why. **Question:** A condition says "at most" or "more than" instead of "equals". How do you find every number that passes, and when does the sign turn around? ## What this area is about A linear inequality such as $3x + 5 \le 26$ is usually true for a whole stretch of numbers, not one. Solving it uses the same moves as solving an equation, with one exception: multiplying or dividing both sides by a negative number reverses the inequality sign. Every step must leave the solution set exactly as it was. ## How these fit together All three children are procedures, each building on the one before. One-step inequalities and the sign flip sets out what a solution set is, which moves keep it, and why the sign flips: $-2x < 6$ becomes $x > -3$. Two-step inequalities undo two operations in reverse order, as in $px + q > r$ with $p \ne 0$, watching for a negative $p$. Multi-step inequalities clear parentheses, collect the unknown from both sides, then solve, flipping the sign whenever both sides are multiplied or divided by a negative number. ## A path through it One-step inequalities, then two-step, then multi-step: each lesson leans on the one before. Next: compound inequalities, modeling constraints with inequalities and systems of inequalities. ### [Solving quadratics by factoring](https://math.wiki/zero-product-property/) **Summary:** If a product is zero, at least one factor is zero, and this is why factoring solves equations. **Question:** Two numbers multiply to zero. What does that force to be true about the numbers? ## Why this exists A linear equation such as $x - 4 = 0$ takes one step: add $4$ to both sides. A quadratic equation such as $$x^2 - 3x - 4 = 0$$ does not, because $x$ appears twice, once squared and once not, and no single operation undoes both at once. Now write the left side as a product. Factoring gives $$(x + 1)(x - 4) = 0$$ and the question changes. It is no longer "which $x$ makes this expression zero?" but "two numbers multiply to zero; what does that say about them?" That question has a sharp answer. A product can equal $12$ in endless ways: $1 \cdot 12$, $2 \cdot 6$, $3 \cdot 4$, $(-24)\left(-\frac{1}{2}\right)$, and so on. It can equal $0$ in only one way: one of the numbers has to be $0$. So the single hard equation splits into two easy ones, $x + 1 = 0$ and $x - 4 = 0$, each with $x$ appearing once. That is why factoring solves equations at all, and why, whenever the expression factors, it is the fastest method there is and the first one to try. This lesson follows the idea all the way through: the property itself, the last step that turns a factored equation into its answers, the whole method of solving a quadratic by factoring, and the area and dimension problems that the method is most often used for. ## Definition In plain words: **if the product of two numbers is zero, then at least one of the numbers is zero.** In symbols, for any real numbers $a$ and $b$: $$\text{if } a \cdot b = 0, \text{ then } a = 0 \text{ or } b = 0.$$ ### [Solving quadratics with square roots](https://math.wiki/square-root-property/) **Summary:** If something squared equals a number, that something is plus or minus the root. **Question:** You know that a number squared is 49. What could the number have been? ## Why this exists Most steps in solving an equation undo something. Adding $3$ is undone by taking $3$ away; multiplying by $5$ is undone by dividing by $5$. Each of those undoing steps gives back exactly one number, because nothing was lost on the way in. Squaring is different. $7^2 = 49$ and $(-7)^2 = 49$: two different numbers go in and the same number comes out. Squaring forgets the sign. So when an equation says $x^2 = 49$ and asks you to go backwards, there are two numbers that could have gone in, and an honest answer has to name both. Saying "$x$ is $7$" is only half the answer. The square root symbol alone does not do that. $\sqrt{49}$ means $7$ and only $7$: it is the principal square root, the one that is not negative. The square root property is the rule that recovers the other one. It is the first tool that solves a quadratic equation without factoring, and it works just as well when the answer is $\sqrt{7}$ as when it is a whole number, which factoring does not. The method covers every equation in which the unknown appears only once, inside a square. That square may be of a single letter, as in $x^2 = 25$, or of a whole expression in parentheses, as in $(x - 3)^2 = 16$. Either way the plan is the same: get the square alone on one side, take the root of both sides, and keep both signs. This lesson does the single-letter case first, then the squared binomial in *Solving when a binomial is squared*, at the end of the Intuition section below. ### [Solving radical equations](https://math.wiki/solve-one-radical/) **Summary:** Isolate the root, square both sides, solve. **Question:** The unknown is trapped under a root sign. How do you get it out, and why can the answer you get be false? ## When you reach for it A **radical equation** is an equation with the variable inside a root, such as $\sqrt{2x + 7} = x + 2$ or $\sqrt[3]{2x - 5} + 4 = 1$. They turn up whenever a quantity is a root of something unknown: the length of a slanted wire is a square root of a sum of squares, by Pythagoras, and a formula such as $T = 2\pi\sqrt{L/g}$ for a pendulum, read backwards, asks for the $L$ inside the root. None of the moves for multi-step equations can reach a variable under a root. One move can: a root is undone by the matching power, so raising both sides to that power frees the variable. The price is that squaring is not reversible, so the equation you end with can have solutions the original does not. Those are called **extraneous solutions**, and checking for them is part of the method, not an afterthought. This lesson runs over two days. Part 1 solves equations with one root: isolate it, raise both sides to a power, solve, check. Part 2 uses the same move on equations with two roots and on equations written with rational exponents. ## The procedure ### Part 1: one radical Take the equation $$\sqrt{2x + 7} = x + 2$$ **Isolate the radical.** Get the root alone on one side, using the ordinary moves of adding, subtracting, multiplying and dividing both sides. Here it is already alone. The reason for doing this first is that squaring has to remove the root completely. Squaring a sum such as $\sqrt{x + 5} + 1$ does not do that: $(a + b)^2 = a^2 + 2ab + b^2$, and the middle term $2ab$ still carries the root (Example 2). ### [Solving systems by elimination](https://math.wiki/elimination-by-adding/) **Summary:** When the coefficients are already opposites, add the equations. **Question:** Two equations, two unknowns, and neither letter stands alone. How can you make one of the letters disappear? ## When you reach for it Substitution is the natural move when one equation already gives a letter on its own, such as $y = 2x - 3$. Many systems do not arrive like that. Look at $$\begin{cases} 5x + 2y = 31 \\ 3x - 2y = 9 \end{cases}$$ No letter is alone, and freeing one brings fractions: the first equation gives $y = \frac{31 - 5x}{2}$. But read the system in columns instead of rows. The $y$ column holds $+2y$ and $-2y$, which cancel. If the two equations are added, side to side, the $y$ terms vanish and what is left, $8x = 40$, has only one letter in it. That is **elimination**: combine the two equations so that one letter drops out, solve the one-letter equation that remains, then go back for the other letter. It suits systems written in standard form, $Ax + By = C$, where the $x$ terms, the $y$ terms and the numbers already sit in columns. Whatever method is used, the answer is the same pair of numbers: the point where the two lines cross, as in solving by graphing. The lesson runs over three days: Day 1 adds or subtracts the equations, Day 2 multiplies an equation first, and Day 3 chooses between methods and deals with systems in which both letters vanish. ## The procedure ### Day 1: adding or subtracting the equations **Why adding two equations is allowed.** Suppose a pair $(x, y)$ makes both equations true. For that pair, $5x + 2y$ is not just an expression: it is the number $31$. In the same way $3x - 2y$ is the number $9$. Adding equal amounts to equal amounts gives equal results, so for that pair ### [Solving systems by graphing](https://math.wiki/what-a-system-solution-is/) **Summary:** A pair of numbers that makes both equations true at once. **Question:** Two equations each describe a whole line of points. Is there a point that makes both of them true at once, and how would you find it? ## Why this exists One equation in two unknowns does not pin anything down. The equation $x + y = 10$ is true for $(3, 7)$, for $(4, 6)$, for $(-2, 12)$ and for infinitely many other pairs, and its graph is a whole line of them (graphing a line). Real questions often bring two conditions at once. Two car-share schemes each set a price that depends on how long you drive; which trip length gives **both** schemes the same price? Each condition is a line, and a pair that fits both is a point on both lines. So the first way to answer is to draw the two lines and look where they cross. This lesson is about what that crossing means, why there is usually exactly one, and the two cases where there is not. ## Definition A **system of linear equations** is two or more linear equations in the same unknowns, to be satisfied together. A brace groups them: $$\begin{cases} x + y = 10 \\ x - y = 4 \end{cases}$$ A **solution of the system** is an ordered pair $(x, y)$ that makes **every** equation in it true when its numbers are substituted. A pair that makes only one equation true is not a solution of the system. To **solve** the system is to find all of its solutions. Here $(7, 3)$ is a solution: $7 + 3 = 10$ and $7 - 3 = 4$ are both true. The pair $(6, 4)$ is not: $6 + 4 = 10$ is true, but $6 - 4 = 2$, not $4$. A system of two linear equations has **one solution**, **no solution** or **infinitely many solutions**, and Part 2 shows there is no other possibility. A system with at least one solution is **consistent** and one with none **inconsistent**. ### [Solving systems by substitution](https://math.wiki/substitution-when-isolated/) **Summary:** Drop the expression for y into the other equation. **Question:** Two equations share two unknowns, and one of them already says what y equals. How do you turn that into numbers? ## When you reach for it A system of two equations in $x$ and $y$ asks for a pair of numbers that makes **both** equations true at the same time. Graphing shows that pair as the point where the two lines cross, but it reads the point off a picture, and a picture cannot tell $\frac{13}{5}$ from $2.6$ or from $2.55$. When you want the pair exactly, you need algebra. The trouble is that one equation in two unknowns has infinitely many solutions: $x + y = 7$ is true for $(0, 7)$, $(3, 4)$, $(10, -3)$ and every other pair on its line. What you already know how to solve is an equation in **one** unknown, such as $4x - 5 = 7$ or one with the variable on both sides. Substitution is the move that turns two equations in two unknowns into one equation in one unknown. Reach for it when one equation already gives one variable on its own: $$y = 3x - 5, \qquad x = 5 - 2y, \qquad l = w + 3$$ The last of these is what a word problem means when it says "the length is $3$ more than the width". Reach for it also when some variable has coefficient $1$ or $-1$, as $x$ does in $x - 3y = 4$, because one step of rearranging then leaves it alone. When every coefficient is something like $3$ or $5$, substitution still works, but elimination is usually less work. The lesson runs over two days. Part 1 is the move itself, when a variable is already alone. Part 2 is rearranging first, answers that are fractions, and what it means when the variable disappears. ### [Special products](https://math.wiki/polynomial-special-products/) **Summary:** The square of a binomial and the sum-times-difference, recognized on sight. **Question:** When the two binomials are the same, or differ only in one sign, what shape does their product always take? ## Why this exists Multiplying two binomials always works the same way: every term of the first times every term of the second, four products, then collect like terms. Two shapes of product turn up so often that the four products are worth studying once and for all. One is a binomial times itself, such as $(x + 5)^2$. The other is a sum times the difference of the same two terms, such as $(x + 5)(x - 5)$. In both, the two middle products are not just like terms but equal in size. In a square they are equal and add up; in a sum times a difference they are opposites and cancel. Because of that, the answer has a fixed shape that can be written down on sight, read backwards later to factor, and even used to do arithmetic such as $104^2$ in your head. ## Definition A **special product** is one of these three products of binomials, where $a$ and $b$ stand for any two terms (numbers, letters, or terms such as $3x$ or $2y$): - the **square of a sum**, $(a + b)^2$, which means $(a + b)(a + b)$; - the **square of a difference**, $(a - b)^2$, which means $(a - b)(a - b)$; - the **product of a sum and a difference** of the same two terms, $(a + b)(a - b)$. Two binomials like this are called **conjugates** of each other. A trinomial of the form $a^2 + 2ab + b^2$ or $a^2 - 2ab + b^2$ is a **perfect square trinomial**, because it is the square of a binomial. An expression $a^2 - b^2$ is a **difference of two squares**. ### [Standard form of a line](https://math.wiki/standard-form-meaning/) **Summary:** Ax + By = C, and when this form is the convenient one. **Question:** Two quantities share one fixed total, like money raised from selling two kinds of item. What is the natural equation for that, and how do you read its line straight from it? ## Why this exists Zainab's class is raising money for a trip. They sell boxes of shortbread at $5$ dollars each and T-shirts at $8$ dollars each, and they want to raise exactly $120$ dollars. Call the number of boxes sold $x$ and the number of T-shirts $y$. The boxes bring in $5x$ dollars, the shirts $8y$, and together they make $120$: $$5x + 8y = 120$$ The story hands you this equation directly: each quantity times its own price on the left, the total alone on the right. Two questions answer themselves. Only shortbread means $y = 0$, so $5x = 120$ and $x = 24$ boxes. Only T-shirts means $x = 0$, so $8y = 120$ and $y = 15$ shirts. Those are the two points where the line meets the axes. The same line in slope-intercept form is $y = -\frac{5}{8}x + 15$: correct, but it brings in a fraction and reads as though shirts depend on boxes, which the story never said. Slope-intercept form suits a quantity that starts somewhere and changes at a steady rate; when two quantities share a fixed total, the form above is the natural one, and lines move freely between it, slope-intercept form and point-slope form. ## Definition A linear equation in $x$ and $y$ is in **standard form** when it is written as $$Ax + By = C$$ where $A$, $B$ and $C$ are numbers and $A$ and $B$ are not both zero. Both variable terms are on the left, each times its own constant, and the constant stands alone on the right. On top of this there is a **convention**, an agreement about which of many correct equations to write, not a rule of mathematics: $A$, $B$ and $C$ are integers with no common factor bigger than $1$, and $A$ is not negative (when $A = 0$, $B$ is positive). The equations $10x + 16y = 240$, $5x + 8y = 120$ and $-5x - 8y = -120$ all have the form $Ax + By = C$ and all describe the fundraiser line; the convention picks $5x + 8y = 120$, so that two people writing the same line write the same equation. Some books use the name for any $Ax + By = C$. ### [Synthetic division](https://math.wiki/synthetic-division/) **Summary:** A shorthand for dividing by x minus a number. **Question:** Long division by x minus a number writes every power of x over and over. How little of it do you actually need? ## When you reach for it Long division of polynomials works for any divisor, but most of what it writes is repeated. When the divisor is linear, of the form $x - c$, nearly all of it can go: the powers of $x$, the copies of leading terms, and every subtraction. What is left is one row of coefficients, one number $c$, and a rhythm of multiply and add. That shorthand is **synthetic division**. Reach for it whenever you divide a polynomial by $x - c$, $x + c$ or $ax - b$, and above all to test many possible zeros quickly, which is how the rational root theorem is put to work. ## The procedure ### One division, done both ways Divide $x^3 + 2x^2 - 5x + 4$ by $x - 2$ by long division, but write every line in columns, one column per power, keeping only the coefficients. | | $x^3$ | $x^2$ | $x$ | $1$ | |---|:-:|:-:|:-:|:-:| | dividend | $1$ | $2$ | $-5$ | $4$ | | take away $x^2(x - 2)$ | $1$ | $-2$ | | | | what is left | | $4$ | $-5$ | | | take away $4x(x - 2)$ | | $4$ | $-8$ | | | what is left | | | $3$ | $4$ | | take away $3(x - 2)$ | | | $3$ | $-6$ | | remainder | | | | $10$ | The quotient is $x^2 + 4x + 3$ and the remainder is $10$. Now look at what the table repeats. The column headings already say which power each number belongs to, so the variables inside the division carry nothing. The first number of each "take away" row only cancels the leading number above it, which it always does, so it can go. The second number of each "take away" row is the quotient coefficient times $-2$, and it is taken away. Taking away $-2$ times a number is the same as adding $2$ times it, so write $2$, the number $c$ in $x - c$, and add instead. ### [Systems in three variables](https://math.wiki/three-variable-systems/) **Summary:** Eliminate one variable to get a two-variable system, then finish. **Question:** Three unknowns tied together by three equations. How do you pull the numbers apart when every equation holds all three letters at once? ## When you reach for it Some questions have three unknown numbers and three facts that tie them together. How many shots of each value did a player make? How was a sum of money split between three accounts? Which circle passes through three given points? Each fact becomes a linear equation, and every one of them may hold all three letters: $$\begin{cases} x + y + z = 3 \\ 2x - y + 3z = 17 \\ -x + 4y - 2z = -20 \end{cases}$$ A **solution** is an **ordered triple** $(x, y, z)$ that makes all three equations true at once. For the system above it is $(4, -3, 2)$: put those numbers in and each line comes out true, which you can check in your head. No single equation can give the answer, because each one has three unknowns. The move is the one from solving by elimination, used twice: remove one letter from two different pairs of equations. That leaves two equations in two letters, which you already know how to solve. Reach for it whenever the equations are linear (each letter to the first power, no letter multiplied by another) and there are as many equations as unknowns. ## The procedure ### Day 1: from three letters to two Number the equations so you can talk about them. $$\begin{cases} x + y + z = 3 & \text{(1)} \\ 2x - y + 3z = 17 & \text{(2)} \\ -x + 4y - 2z = -20 & \text{(3)} \end{cases}$$ **Choose one letter to remove, and remove it twice.** Any letter will do, but some are cheaper. Here $x$ has coefficient $1$ in (1) and $-1$ in (3), so adding those two removes it with no multiplying: ### [Systems in word problems](https://math.wiki/mixture-problems/) **Summary:** Two amounts and two totals: quantity and value. **Question:** You know how many there were in all and what they came to in all, but not how many of each kind. How do you find the two amounts? ## When you reach for it A museum knows how many people came in on Saturday and how much money it took, but not how many were adults. A shop knows how heavy its trail mix is and what it cost, but not how much of it is hazelnuts. Each time there are **two amounts you do not know** and **two separate facts** that tie them together. One fact alone allows many pairs: $230$ admissions could be $100$ adults and $130$ children, or $200$ and $30$. Two facts usually pin down one pair, and two equations that must hold at once are a system. Solving is the work of graphing, substitution and elimination. This lesson teaches the two ends: turning a story into two equations, and turning the solution back into an answer the story accepts. ## The procedure ### Day 1: a quantity equation and a value equation Sefton's jar holds only $10$-cent and $25$-cent coins. There are $32$ coins, worth $5.30$ dollars in all. How many of each are there? **Name the two unknowns, with their units.** Let $d$ be the number of $10$-cent coins and $q$ the number of $25$-cent coins. Saying what each letter counts is what lets you check that each equation adds like things to like things. **Write the quantity equation.** The story says how many coins there are: $d + q = 32$. Both terms count coins, so adding them is allowed. **Write the value equation.** Each kind of coin contributes (how many) $\times$ (value of one): | Coin | Number | Each (cents) | Total (cents) | |---|:-:|:-:|:-:| | $10$-cent | $d$ | $10$ | $10d$ | | $25$-cent | $q$ | $25$ | $25q$ | | Jar | $32$ | | $530$ | ### [Systems of equations](https://math.wiki/systems-of-equations/) **Summary:** Two conditions at once, and the point that satisfies both. **Question:** Two conditions hold at once, each on the same unknowns. Which values satisfy both, and how do you find them exactly? ## What this area is about One linear equation in $x$ and $y$, such as $x + y = 10$, is true for a whole line of pairs. A system adds a second condition, and its solution is every pair that makes both true at once. For two linear equations in two unknowns that is one pair, none (parallel lines) or infinitely many (the same line). This area finds those pairs by graph and by algebra, then widens to three unknowns, curves and inequalities. ## How these fit together **The idea.** Solving systems by graphing defines a solution as a crossing point and shows the three cases. **Procedures.** Substitution puts one equation's expression for a letter into the other. Elimination adds or subtracts equations, scaled if needed, so one letter cancels. **Extensions.** Systems in three variables eliminate one letter to get a two-variable system. Systems with a curve meet a line with a parabola, which can happen twice, once or not at all. Systems of linear inequalities answer with a shaded region instead of a point. **Using it.** Systems in word problems turn two totals, such as a count and a value, into two equations. ## A path through it Graphing first. Then substitution, and systems of inequalities. After substitution: elimination and systems with a curve. After elimination: word problems and three-variable systems. After this: matrices, which write a whole system as one equation, and fitting a quadratic to data, which solves a three-variable system. ### [Systems of linear inequalities](https://math.wiki/systems-of-inequalities/) **Summary:** Shade each region; the overlap is the answer. **Question:** Several limits must hold at once, each on two quantities. Which points survive all of them, and which of those is best? ## What this area is about A linear inequality in $x$ and $y$, such as $y > 2x - 1$, is true on one side of a line (and on the line itself for $\le$ or $\ge$). A system of linear inequalities asks for the points that make every inequality true at once. Its answer is a region of the plane, drawn rather than listed, and often a choice of the best point in it. ## How these fit together **Procedures for drawing.** Graphing a linear inequality in two variables draws the boundary line, solid for $\le$ or $\ge$ and dashed for $<$ or $>$, and shades the side that works. The overlap of two shaded regions puts several on one grid: the solution set is where all the shadings agree, and it may close into a polygon with corners found exactly. **Applications.** Writing and interpreting constraints turns the limits in a situation into equations and inequalities and decides which solutions make sense in context. Linear programming finds the largest or smallest value of an amount such as $P = 25x + 40y$ over that region; when a best value exists and the region has corners, it is reached at a corner. ## A path through it Graph one inequality, then the overlap of several. Both applications follow the overlap, and linear programming builds on writing constraints, so constraints come first. Finish here and you can describe every plan that keeps within a set of limits, and pick the best one: a first taste of optimization. ### [Systems with a curve](https://math.wiki/nonlinear-systems/) **Summary:** A line and a parabola can meet twice, once, or not at all. **Question:** A straight line and a parabola are drawn on the same axes. Where do they meet, and how many times can they? ## When you reach for it A system of two equations in $x$ and $y$ asks for every pair $(x, y)$ that makes both equations true at once. When both are lines there is at most one such pair, found by graphing, substitution or elimination. This lesson takes the next case: one equation is a line and the other is a parabola, such as $$y = x^2 - 2x - 3 \quad\text{and}\quad y = x + 1$$ A solution is still a pair that makes both equations true, so it is still a point on both graphs. What changes is how many there can be. A line and a parabola can cross twice, meet once, or miss each other, and the method has to find every meeting point, not stop at the first. Reach for it whenever two conditions hold at once and one of them is curved: where a straight beam strikes a curved surface, or simply where two graphs meet. Reading meeting points off a graph to solve $f(x) = g(x)$ is the same picture; this lesson adds the algebra that gives the points exactly. ## The procedure ### What a solution looks like A point lies on a graph exactly when its coordinates make the graph's equation true. So a pair satisfying both equations is a point lying on both graphs, and the solutions of the system are the meeting points of the line and the parabola. A line that is not vertical can cross the parabola at two points, touch it at one point and stay on one side of it, or miss it. A vertical line $x = h$ meets a parabola that opens up or down exactly once, because the parabola has exactly one height above each $x$. There is never a third meeting point; the substitution below shows why. ### [Terms, coefficients and constants](https://math.wiki/terms-coefficients-constants/) **Summary:** Naming the parts of an expression: the pieces added together, the number multiplying a letter, and the plain numbers. **Question:** In $5x^2 - 3x + 7$ the $5$, the $3$ and the $7$ do three different jobs. How do you point at one piece of an expression without rewriting the whole thing? ## Why this exists Even a short expression has parts doing different jobs. In $5x^2 - 3x + 7$ there are three numbers. The $5$ multiplies $x^2$. The $3$ multiplies $x$, and a minus sign comes with it. The $7$ is just $7$, whatever number $x$ stands for. To say anything about one of those pieces ("double the second one", "these two can be put together", "this part is the fixed charge") you need a name for it, or you end up rewriting the whole expression every time you want to point. The names also carry meaning. Suppose a phone plan costs $C = 8g + 20$ dollars a month when you use $g$ gigabytes. The $8$ is the price of each gigabyte, and the $20$ is what you pay even if you use nothing at all. Calling the $8$ a **coefficient** and the $20$ a **constant** says exactly that: one part grows with $g$, and the other part stays put. That split, a part that changes with the letter and a part that does not, sits inside almost every formula you will meet. ## Definition An expression is built by adding and subtracting pieces. Each piece is a **term**: whatever sits between the top-level plus and minus signs. Most terms are a single number, a single letter, or numbers and letters multiplied together, where a letter may carry a power as in $x^2$. In $5x^2 - 3x + 7$ the terms are $5x^2$, $-3x$ and $7$. **The sign in front belongs to the term.** Subtracting a number is the same as adding its opposite, so ### [The discriminant](https://math.wiki/discriminant-and-number-of-roots/) **Summary:** Positive gives two, zero gives one, negative gives none that are real. **Question:** Why does a quadratic equation sometimes have two solutions, sometimes one, and sometimes none at all, and how can you tell which before solving it? ## Why this exists Solve three quadratics that look almost the same and you can get three different kinds of answer. $x^2 - 4x + 3 = 0$ has two solutions, $1$ and $3$. $x^2 - 4x + 4 = 0$ has one, $2$. $x^2 - 4x + 5 = 0$ has none among the real numbers. Nothing in the look of the equations warns you which is which. Often the count is all you want. Does a thrown ball ever reach a given height? Are there two numbers with a given sum and product? Does a line meet a curve? Each asks whether a quadratic has a solution, not what it is, and solving in full is a lot of work to learn one word. The quadratic formula already holds the answer. Split its fraction in two: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}$$ The first piece, $-\frac{b}{2a}$, is the same for both roots: it is the middle, the point the two roots sit either side of. Everything that makes the two roots different from each other lives in the second piece, and inside it the only thing that can change the story is the number under the square root. Whether the roots are whole numbers, fractions, square roots or not real at all; how far apart they are; whether there are two of them or they merge into one; where the parabola meets the $x$-axis: all of it is settled by $b^2 - 4ac$. A number that settles that much deserves a name of its own. ## Definition ### What the discriminant is Take a quadratic equation in standard form, ### [The discriminant and the number of real solutions](https://math.wiki/discriminant-and-the-graph/) **Summary:** Two crossings, a touch, or a miss. **Question:** Before you solve a quadratic equation, can you tell how many real solutions it has, and whether its parabola meets the x-axis at all? ## Why this exists The quadratic formula solves every quadratic equation, but often a question asks for less than the full solution. Does this equation have a solution at all? Does a curve ever reach a given height? Will this trinomial factor? Each is really a question about how many solutions there are, and what kind, and one number answers it in a single line of arithmetic. That number is the part of the formula under the square root. This lesson gives it a name, shows why it counts the solutions, and shows the same count in the graph: how many times the parabola $y = ax^2 + bx + c$ meets the $x$-axis. ## Definition For a quadratic equation in **standard form** $$ax^2 + bx + c = 0 \qquad (a \neq 0)$$ the **discriminant** is the number $$D = b^2 - 4ac$$ It is exactly what sits under the root in the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Standard form comes first: every term on one side, $0$ on the other, and only then are $a$, $b$ and $c$ read off, each with its sign. The parabola $y = ax^2 + bx + c$ has the same discriminant, and its $x$-intercepts are the real solutions of the equation (roots-zeros-and-x-intercepts). When the two values given by $\pm$ are the same number, the equation has one **repeated root** (also called a double root). For example $(x - 5)^2 = 0$ has the single solution $5$, produced by both factors. ## Intuition Split the formula into a middle point and a step: ### [The distributive property with variables](https://math.wiki/distribute-over-a-sum/) **Summary:** Multiply the outside number into every term inside. **Question:** A number multiplies a sum whose parts cannot be added together, as in $4(x + 6)$. How do you carry out the multiplication anyway? ## When you reach for it The order of operations says to work inside parentheses first. In $4(10 + 6)$ that is easy: $10 + 6 = 16$, and $4 \cdot 16 = 64$. In $4(x + 6)$ it is impossible. The terms $x$ and $6$ are not like terms, so $x + 6$ is already as simple as it gets, and there is no single number to multiply by $4$. The distributive property gives another way in: multiply the $4$ into each term separately. $$4(x + 6) = 4x + 24$$ Reach for it whenever a factor multiplies a sum or difference in parentheses that cannot be simplified, and you want the expression without parentheses: to see what it contains term by term, to collect it with other terms, or to solve an equation such as $3(x + 4) = 27$. The form with parentheses is called the **factored form**, because it is written as a product. The result is the **expanded form**, because it is written as a sum of terms. The two give the same value for every value of the letter, so they are equivalent expressions. This lesson goes from factored to expanded and stops there. ## The procedure The property is the one met with plain numbers in the distributive property: for all real numbers $a$, $b$ and $c$, $$a(b + c) = ab + ac$$ A letter stands for a number, so the property applies to expressions with letters in them exactly as it does to numbers. This lesson takes two parts. Part 1 has a positive number outside a sum, whole, fraction or decimal. Part 2 has a negative outside and subtraction inside. A short look ahead at the end puts a letter outside the parentheses. ### [The fundamental theorem of algebra](https://math.wiki/fundamental-theorem-of-algebra/) **Summary:** A polynomial of degree n has exactly n roots, counting repeats and complex ones. **Question:** The real numbers ran out of answers at $x^2 + 1 = 0$, so $i$ was invented. Will some cubic or quintic need yet another new kind of number? ## Why this exists Every time a new kind of equation turned up with no answer, mathematicians built new numbers. $x + 5 = 2$ has no answer among the counting numbers, so negatives were admitted. $3x = 2$ needed fractions. $x^2 = 2$ needed irrational numbers, and $x^2 + 1 = 0$ needed $i$ and the complex numbers. So the worry is natural: maybe $x^5 + x + 3 = 0$, or some degree $50$ equation, needs a still larger number system. It does not. The complex numbers are enough, for every polynomial equation of every degree, and even when the coefficients themselves are complex. That is the fundamental theorem of algebra, and it turns "how many roots?" into a question with an exact answer: as many as the degree. ## Definition **The fundamental theorem of algebra.** Every polynomial of degree $n \ge 1$, with complex coefficients, has at least one complex root. Real numbers are complex numbers with imaginary part $0$, so this covers every polynomial with real or integer coefficients too, and the root it promises may happen to be real. A root $r$ has multiplicity $k$ when $(x - r)^k$ is a factor of the polynomial and $(x - r)^{k+1}$ is not. Counting $r$ that many times is called counting **with multiplicity**. On a graph, multiplicity decides whether the curve crosses or touches the axis. ## Intuition The theorem promises only one root. The factor-theorem turns one root into a factor, and the factor theorem works for a complex number $c$ exactly as for a real one: its proof divides by $x - c$ and never uses that $c$ is real, only the arithmetic of complex multiplication. The same goes for synthetic division by $x - c$. So a root $c_1$ of a degree $n$ polynomial $p$ gives ### [The overlap of two shaded regions](https://math.wiki/overlap-of-inequality-regions/) **Summary:** Where both shadings agree is the solution set. **Question:** One inequality keeps half of the plane. When two or more must hold at once, which points are left? ## When you reach for it One linear inequality in $x$ and $y$, such as $y < 2x + 1$, is true on one side of a line and false on the other; graph-a-linear-inequality draws it as a boundary line, solid or dashed, with the kept side shaded. A **system of linear inequalities** is two or more such inequalities, all required at once. A **solution** is an ordered pair that makes every one true, and the **solution set** is all of them. This is the "and" of and-inequalities moved from a line to a plane. There, $x > -2$ and $x \le 3$ each keep a ray and the answer is where the rays overlap. Here each inequality keeps a half-plane, and the solution set is where the half-planes overlap: where both shadings agree. Unlike a system of equations (what-a-system-solution-is), the answer is usually a whole region, and the crossing of two boundary lines is only a corner of it, which may or may not belong. Turning a situation into inequalities is the work of model-constraints-with-inequalities; here they are given. ## The procedure Part 1 (the first day) puts two inequalities on one grid. Part 2 (the second day) adds more, which can close the region into a polygon, and finds its corners exactly. ### Part 1: two inequalities on one grid **Graph both inequalities on the same axes.** Each boundary is solid for $\le$ or $\ge$ (equality is allowed, so the line is kept) and dashed for $<$ or $>$, and each kept side is shaded. A point solves the system exactly when it solves the first inequality and the second, so the solution set is exactly the part shaded twice. ### [The quadratic formula](https://math.wiki/quadratic-formula/) **Summary:** One formula that solves every quadratic, and where it comes from. **Question:** Is there one formula that solves every quadratic, and where does it come from? ## What this area is about Every equation $ax^2 + bx + c = 0$ with $a \neq 0$ is solved by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$ It is not a new idea. It is completing the square done once, on letters instead of numbers, so you never have to do it again. ## How these fit together **Where it comes from.** Deriving the quadratic formula completes the square on the general equation. Read it and the formula stops being something to memorize. **The procedure, and the judgment.** Using the quadratic formula is reading off $a$, $b$ and $c$, substituting with care over signs, and simplifying the radical. It also covers choosing how to solve: the formula always works, but factoring, square roots or completing the square is often faster. **The part under the root.** Take $a$, $b$ and $c$ real. The discriminant is $b^2 - 4ac$. The discriminant and the number of real solutions: when it is positive, zero or negative there are two, one or no real solutions, and the parabola crosses the $x$-axis twice, touches it or misses it. The discriminant goes further: a negative value gives two complex conjugate roots. ## A path through it Derivation, then use and choosing a method, then the discriminant with the graph, and last the full discriminant. The derivation assumes completing the square with any leading coefficient. The formula does the solving in word problems, and it turns up well outside quadratics: in the golden ratio, and in finding the eigenvalues of a $2 \times 2$ matrix. ### [The rational root theorem](https://math.wiki/rational-root-theorem/) **Summary:** The only possible fraction roots come from the first and last coefficients. **Question:** A polynomial with integer coefficients has a root somewhere. Before you start guessing, can you know which fractions are even possible? ## Why this exists The factor-theorem turns one root of a polynomial into a factor and lowers the degree by one. But it needs a first root, and a guess could be any number at all. For a polynomial with integer coefficients, the rational root theorem shrinks "any number" to a short, finite list read off the first and last coefficients. Test the list with synthetic-division, split off each root you find, and most cubics and quartics met by hand come apart completely. This lesson runs over two days. Day 1 is the theorem, why it is true, and listing and testing candidates (Examples 1 to 3). Day 2 is solving cubics and quartics completely, and where the theorem stops (Examples 4 to 7). ## Definition A rational number is a fraction $\frac{p}{q}$ of integers with $q \neq 0$. It is in **lowest terms** when $p$ and $q$ share no factor except $1$ and $-1$; we also take $q > 0$. A polynomial of degree $n$ is written $$f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$ with $a_n \neq 0$. Here $a_n$ is the leading coefficient and $a_0$ the **constant term**. A **rational root** of $f$ is a rational number $r$ with $f(r) = 0$. An integer $p$ **divides** an integer $a$, written $p \mid a$, when $a = pk$ for some integer $k$. The divisors of $6$ are $\pm 1$, $\pm 2$, $\pm 3$, $\pm 6$: the list from find-all-factors-of-a-number, with both signs. A **candidate** is a fraction $\frac{p}{q}$ with $p \mid a_0$ and $q \mid a_n$, $q > 0$. ### [The remainder and factor theorems](https://math.wiki/factor-theorem/) **Summary:** A zero remainder means you have found a factor, and a root. **Question:** Dividing a polynomial by x minus a number takes a page of work. Can you know the remainder without dividing, and tell at a glance when there is none? ## Why this exists polynomial-long-division and synthetic-division answer "what is $p(x)$ divided by $x - c$?" with a quotient and a remainder. Often only the remainder matters, and most of all whether it is $0$: a zero remainder means $x - c$ is a factor, and factors are what turn a cubic into pieces the zero-product-property can solve. Two theorems cut the work. The remainder is simply $p(c)$, one evaluation. And $x - c$ is a factor exactly when $p(c) = 0$. So one known zero of a polynomial splits off a factor and lowers the degree by one, which is how most cubics and quartics are factored by hand. ## Definition Dividing a polynomial $p(x)$ by $x - c$ produces a **quotient** $q(x)$ and a remainder $r$ with $$p(x) = (x - c)\,q(x) + r$$ for every value of $x$. The remainder always has lower degree than the divisor, and $x - c$ has degree $1$, so $r$ is a number. When $r = 0$, $p(x) = (x - c)\,q(x)$, and $x - c$ is a **factor** of $p(x)$; we also say $p(x)$ is **divisible** by $x - c$. A **zero** of $p$ is a number $c$ with $p(c) = 0$. The divisor fixes $c$ with its sign: $x + 2$ is $x - (-2)$, so $c = -2$. ## Intuition Take $p(x) = x^3 - x^2 - 4x + 4$ and divide by $x + 1$. Synthetic division by $-1$ on the coefficients $1, -1, -4, 4$ gives the bottom row $1, -2, -2, 6$, so $$p(x) = (x + 1)(x^2 - 2x - 2) + 6$$ Now simply evaluate: $p(-1) = -1 - 1 + 4 + 4 = 6$. The same number, and no accident. Put $x = -1$ into the identity and the factor $x + 1$ becomes $0$, wiping out the whole product and leaving only the remainder. ### [The shape of a parabola](https://math.wiki/parabola-shape/) **Summary:** Why squaring makes a symmetric curve, and which way it opens. **Question:** Square the input and plot what comes out. Why does the picture bend into the same U every time, and what decides which way the U points? ## Why this exists The graph of a linear equation is a straight line, and two points are enough to draw it. Try the same with $y = x^2$ and it fails at once. Plot $(0, 0)$ and $(2, 4)$, join them with a ruler, and the line says the point above $x = 1$ is at height $2$. It is at height $1$. So the graph of a quadratic is not a line. It is always the same shape, though: a U, or an upside-down U, the same on both sides and turning exactly once. This lesson explains why squaring produces that shape, and how to tell from the equation alone which way the U points and how narrow it is. ## Definition The graph of an equation $$y = ax^2 + bx + c \qquad (a \neq 0)$$ is called a **parabola**: the set of all points $(x, y)$ that make the equation true, just as the graph of $y = 2x + 1$ is. To see one, make a table and plot it. For $y = x^2$, take whole numbers from $-3$ to $3$: $$\begin{array}{c|ccccccc} x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline y = x^2 & 9 & 4 & 1 & 0 & 1 & 4 & 9 \end{array}$$ Join the seven points with a smooth curve, not straight pieces: the inputs in between have outputs too. The words that go with the picture: - The parabola **opens upward** when it is a U, and **opens downward** when it is an upside-down U. - The point where it turns around is the **vertex**. On an upward parabola it is the lowest point; on a downward one, the highest. - The vertical line the parabola folds onto, so that one half lands exactly on the other, is its **axis of symmetry**, or fold line. For $y = x^2$ it is the $y$-axis. ### [The slope formula and the four kinds of slope](https://math.wiki/slope-from-two-points/) **Summary:** Difference in y over difference in x, and why the order does not matter as long as it matches. **Question:** You know two points on a line by their coordinates, and there is no graph to count on. How steep is the line? ## When you reach for it Finding slope from a graph means counting squares up and squares across, and that needs a graph with the points on its grid. The points $(-140, 35)$ and $(260, -65)$ are far off any sheet of squared paper, and $\left(-\frac{1}{2}, \frac{1}{3}\right)$ does not sit on a grid corner at all. But the coordinates already contain the rise and the run. The rise is how far $y$ changes between the two points, the run is how far $x$ changes, and a change is found by subtracting. So the slope can be worked out from two points written as numbers, with no picture drawn. Reach for this whenever a line is known by two of its points: to say how steep it is, whether it rises or falls, whether three points lie on one line, or to find a missing coordinate. Day 1 is the formula: where it comes from, why the order of the points does not matter, and how it handles negative and fractional coordinates. Day 2 is what the slope tells you: the four kinds of slope, including the vertical line that has none, and two uses of the formula. ## The procedure ### Day 1: the slope formula **Naming the two points.** With two points in play, $x$ and $y$ alone cannot say which point is meant. So each point gets a small number written below and to the right, a **subscript**: $$(x_1, y_1) \qquad\text{and}\qquad (x_2, y_2)$$ read "x sub 1, y sub 1" and "x sub 2, y sub 2". The subscript is a name tag: $x_2$ means "the $x$-coordinate of the second point". It does not mean $x$ times $2$, and it is not a power. ### [The vertex and the axis of symmetry](https://math.wiki/vertex-of-a-parabola/) **Summary:** The turning point, the highest or lowest value the expression takes. **Question:** A parabola comes down, turns and goes back up, and its two sides look like mirror images. Where exactly does it turn, where exactly is the mirror, and can you find both from the equation without drawing anything? ## Why this exists Feed the numbers $0, 1, 2, 3, 4$ into $x^2 - 4x + 1$ and watch what comes out: $$\begin{array}{c|ccccc} x & 0 & 1 & 2 & 3 & 4 \\ \hline x^2 - 4x + 1 & 1 & -2 & -3 & -2 & 1 \end{array}$$ The outputs fall, stop, and climb again. Somewhere they reach a smallest value, and any question of the form "how high does the ball go", "what is the largest area", or "what is the least this can cost" is a question about that one turning point. Look again and a second pattern shows. The outputs repeat, in pairs: $1$ and $3$ both give $-2$, and $0$ and $4$ both give $1$, and every pair is centered on the same number, $2$. That is the shape of a parabola showing through: fold the graph along one vertical line and the two halves land on each other. If you know where that line is, you get half of every table for free, you know where the turning point must sit, and you can sketch the whole curve from one side of it. A table can only guess at either. The smallest output might sit between two of the inputs you happened to try, and the center of the pairs might be a fraction no table lands on. The vertex and the axis of symmetry are the exact answers: the point where the parabola turns and the line it is symmetric about, both found from the coefficients without drawing anything. ## Definition Every parabola $y = ax^2 + bx + c$, with $a \neq 0$, has two landmarks: a point where it turns, and a line it is symmetric about. The point lies on the line. ### [Translating between words and expressions](https://math.wiki/translate-words-to-expressions/) **Summary:** Turn five more than twice a number into an expression, and watch the order. **Question:** Five less than a number, is that n minus 5 or 5 minus n? And how do you say 3(n + 2) out loud so that the person listening writes down exactly that? ## When you reach for it Most mathematics reaches us in words before it reaches us in symbols. A price board says "a $12$-dollar entry fee, then $5$ dollars a route". A friend says "bring twice as many as last time". An **expression** says the same thing in a form you can calculate with, compare and, later, solve. Translating is the bridge, and it runs both ways: words into an expression when you meet a situation, and an expression back into words when you explain it or read it to someone who cannot see the page. Reach for it whenever a quantity depends on a number that is unknown or can change: the cost of hiring a paddleboard for $h$ hours, the routes climbed in a session, the area of a trampoline mat whose side has not been measured yet. The letter is the one from variable-meaning: it holds a number, and the expression says what is done to that number. Two things make this harder than swapping each word for a symbol. Some phrases name the numbers in the opposite order from the symbols: "$5$ less than $n$" is $n - 5$. And some phrases bundle several operations into one thing: "three times the sum of $n$ and $2$" is $3(n + 2)$. Both are about what the whole phrase means, which is why everything below asks what the words mean rather than which symbol a single word stands for. ## The procedure ### Part 1: words into expressions **Name the letter first.** An expression only means something once you know what its letter stands for. "Let $h$ be the number of hours" turns $15h$ into a cost; without that sentence, $15h$ is fifteen times a number nobody has named. In a bare phrase such as "the sum of $p$ and $11$" the letter is already given. ### [Two-step equations](https://math.wiki/two-step-with-negatives/) **Summary:** Signed coefficients and constants. **Question:** A number was multiplied, then something was added, and only the result is known. How do you get back to the number? ## When you reach for it A one-step equation such as $x + 5 = 26$ or $3x = 21$ has one thing done to the unknown, and one move undoes it. Many equations have two things done to it: $$3x + 5 = 26$$ Here $x$ was multiplied by $3$, and then $5$ was added. An equation of this shape is a **two-step equation**: $$px + q = r$$ where $p$, $q$ and $r$ are numbers, $p$ is not $0$, and the unknown appears exactly once. The number $p$ that multiplies $x$ is its **coefficient**, and $q$ is the **constant term**. The numbers may be negative, fractions or decimals. An equation such as $\frac{m}{5} + 2 = 9$ has the same shape, since dividing by $5$ is multiplying by $\frac{1}{5}$. You reach for this skill whenever an unknown has been scaled and then shifted: so much per visit plus a fixed charge, a steady rate of climb or descent from a starting height, points per level less a penalty. The question is always the same one, run backwards: what number went in, if this came out? This lesson takes three days. **Part 1** solves $px + q = r$ with positive numbers, with a bar picture to show why each step works. **Part 2** lets the coefficient and the constant be negative, fractions or decimals. **Part 3** writes the equation from a story, solves it, and checks the answer in the story. ## The procedure ### Part 1: undoing in reverse order Read $3x + 5 = 26$ as a record of what happened to a number. By the order of operations, the multiplication comes first and the addition second: start with $x$, multiply by $3$, add $5$, and the result is $26$. ### [Two-step inequalities](https://math.wiki/two-step-inequalities/) **Summary:** Undo two operations, watching for the flip. **Question:** A total is a fixed amount plus so much for each item, and it must stay on one side of a limit. Which numbers of items keep it there? ## When you reach for it In a two-step equation an unknown was multiplied, then shifted, and the result was known exactly: $3x + 5 = 26$, so $x = 7$. Very often the result is not pinned to one number but held on one side of a limit. A school trip must cost **at most** its budget. A fundraiser must raise **at least** its target. The clay a potter uses must stay **within** what is in the bin. Each time the total is a fixed amount plus a rate times the unknown, and the sentence is a **two-step inequality**: $$px + q > r$$ or the same with $<$, $\ge$ or $\le$ in place of $>$. Here $p$, $q$ and $r$ are numbers, $p$ is not $0$, and the unknown appears exactly once. The answer is not one number but a whole range of them, the **solution set**, which one-step inequalities introduced together with its graph on a number line. This lesson takes two days. **Part 1** solves $px + q > r$ and its relatives, watching for the one step that turns the inequality sign around, and graphs the answer. **Part 2** writes the inequality from a story, solves it, and reads the answer back in the story, including the case where only whole numbers make sense. ## The procedure ### Part 1: two steps, and the one place the sign can turn Solving an inequality uses the same two undoing steps as solving the matching equation, in the same reverse order: the last thing done to $x$ is the first thing undone. What is different is what each step must protect. A step is allowed only if it leaves the solution set exactly as it was, and one-step inequalities showed which steps do: ### [Using a letter for the unknown number](https://math.wiki/letter-for-an-unknown/) **Summary:** A letter stands for the number we do not know yet in an equation that models a story. **Question:** Eight crayon boxes hold sixty-four crayons in all. How do you write that story as one equation when one of its numbers is not known yet? ## When you reach for it Until now a missing number in an equation has been marked with a box. In unknown-number-in-an-equation the box sat in an adding or subtracting fact, such as $8 + \square = 15$, and was found by asking what number makes both sides the same. In equal-groups-problems the box moved into multiplication and division: $6 \times 4 = \square$, $8 \times \square = 56$, $\square \times 5 = 35$. This lesson changes only the mark. In place of the box, write a **letter**: $$8 \times c = 64$$ The letter does exactly the job the box did. It holds the place of one number that the story has not told us yet. Reach for a letter whenever a story has one number missing and you want to write the whole story down as a single equation before working anything out. Why change at all, when the box worked? Three reasons, each of which shows up later. - A letter can be chosen to remind the reader what it counts: $c$ for the number of crayons, $b$ for the number of baseball cards, $m$ for the number of dollars still missing. - A letter can be named in words. "Here $c$ stands for the number of crayons in one box" is a sentence a reader can check against the story. Two boxes in one equation cannot be told apart. - Letters are how all later mathematics writes an unknown. The thinking learned here carries straight into equations with more than one step. The lesson takes three days. Part 1 puts a letter into multiplication and division stories. Part 2 puts a letter into adding and subtracting stories with numbers up to $1{,}000$. Part 3 starts from a story alone and writes one equation that models it, with the letter wherever the unknown sits, in all four operations. ### [Using the quadratic formula](https://math.wiki/apply-the-quadratic-formula/) **Summary:** Identify a, b and c, substitute carefully, simplify the radical. **Question:** The quadratic formula solves every quadratic. How do you put numbers into it without losing a sign, and how do you tell when another method gets there faster? ## When you reach for it Completing the square solves every quadratic equation, but it makes you do the same algebra every time. Deriving the quadratic formula does that algebra once, with the letters $a$, $b$ and $c$ in place of numbers, and hands back the finished answer. The equation $$ax^2 + bx + c = 0, \qquad a \neq 0$$ has the solutions $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ So using the formula is not new algebra. It is substitution and arithmetic, and the whole skill is doing that arithmetic without dropping a sign or canceling something that cannot be canceled. Reach for it when a quadratic does not factor over the integers, when its coefficients are large or awkward, when you want a decimal answer, or when you are not sure whether it factors and do not want to hunt. It never fails: if the equation has solutions, the formula finds them, and if it has none that are real, the formula tells you so. When a quadratic factors quickly, factoring is faster, and when there is no $x$ term, taking square roots is. The part of the procedure headed "Choosing how to solve a quadratic" says how to tell, in a few seconds, which road is shortest. ## The procedure ### Substituting into the formula Take the equation $2x^2 = 3x + 5$. **Put the equation in standard form.** The formula was derived from $ax^2 + bx + c = 0$, with everything on one side and zero on the other. The letters $a$, $b$ and $c$ mean the coefficients of **that** equation and of no other arrangement. Here the $5$ is not yet $c$. Subtract $3x$ and $5$ from both sides, which keeps the equation true: ### [Variables](https://math.wiki/variables/) **Summary:** A letter that stands for a number we do not know yet, or for any number at all. **Question:** A letter sits where a number should be. What is it allowed to mean, and how do you read and write the expressions it lives in? ## What this area is about A variable is a letter that stands for a number: one number you have not found yet, an amount that changes, or a slot in a rule meant for many numbers. This area is about reading that letter correctly and naming the pieces of the expressions built around it, so that $3(n + 2)$ and $n + 5 = 12$ mean exactly one thing to everyone. ## How these fit together **A first step.** Using a letter for the unknown number replaces the box in an equation such as $8 \times \square = 64$ with a letter, to write a story as one equation. **Definitions.** What a letter stands for tells its three jobs apart: an unknown, as in $n + 5 = 12$, a quantity that changes, and a placeholder in a general rule. Terms, coefficients and constants names the parts of an expression such as $5x^2 - 3x + 7$. Expression, equation and inequality tells a quantity apart from a claim that can be true or false. **A procedure.** Translating between words and expressions turns "five less than a number" into $n - 5$, and back. ## A path through it Using a letter for the unknown stands on its own as the first step. Then what a letter stands for. After it, terms, coefficients and constants, then expressions versus equations; translating between words and expressions also follows what a letter stands for. Next: substituting a value, combining like terms and what a solution is. ### [Vertex form: reading, converting and graphing](https://math.wiki/vertex-form-meaning/) **Summary:** In a(x - h) squared plus k, the vertex is at h, k, and the sign tricks people. **Question:** A parabola's equation can be written so that its turning point sits in plain sight. How do you read it off, get any quadratic into that shape, and draw the whole curve from that one point? ## Why this exists The parabola $y = x^2 - 6x + 11$ turns somewhere, and nothing in the numbers $1$, $-6$ and $11$ tells you where. To find its vertex you need a formula, $-\frac{b}{2a}$, and then a substitution. Now look at the same parabola written another way: $$y = (x - 3)^2 + 2$$ Multiply it out and you get $x^2 - 6x + 9 + 2 = x^2 - 6x + 11$, so it is the same curve. But this time the vertex is printed in the equation: it is $(3, 2)$. No formula, no arithmetic. This way of writing a quadratic is **vertex form**, and it is the reason anyone bothers to complete the square. The catch is the sign. The $3$ appears as $-3$ in the parentheses, yet the vertex is at $x = 3$. Reading vertex form means knowing exactly why. Reading is the first of four things this lesson does with the form. The second is **converting**: rewriting any quadratic $ax^2 + bx + c$ in vertex form. The third is **graphing from vertex form**: drawing the whole parabola from its vertex and the number $a$, without a table of values. The fourth is seeing what the form says about every parabola at once: each one is the single curve $y = x^2$, moved, flipped and stretched. Those are the **transformations of a parabola**. ## Definition A quadratic is in **vertex form** when it is written $$y = a(x - h)^2 + k \qquad (a \neq 0)$$ where $a$, $h$ and $k$ are numbers. Each of the three does one job: - $(h, k)$ is the vertex, and the line $x = h$ is the axis of symmetry. - The sign of $a$ says which way the parabola opens: upward if $a > 0$, downward if $a < 0$. - The size of $a$ says how steep it is: the larger $|a|$, the narrower the curve. ### [What a letter stands for](https://math.wiki/variable-meaning/) **Summary:** A letter can hold one unknown number, or stand for any number at all. Which one it is depends on the sentence around it. **Question:** A letter sits where a number should be. Is it one number you have not found yet, or every number at once? ## Why this exists Arithmetic works with numbers you can see. In $7 + 5$ both numbers are on the page, and the only question is what they make. Many questions are not like that. You know that some number plus $5$ makes $12$, and you want the number. You know that your cousin is always $3$ years older than you, and you want to say so once, for every year, instead of listing $12$ and $15$, then $13$ and $16$, and so on without end. You know that the area of a rectangle is its length times its width, and you want to say it for every rectangle at once. Each of these needs a way to talk about a number without writing the number down. A letter does that. Writing $n$ says "a number goes here", and you can go on working with it before you know what it is, or even when it has no single value at all. The catch is that a letter can do three quite different jobs, and which one it is doing is decided by the sentence around the letter, never by the letter itself. ## Definition A **variable** is a letter that stands for a number. Which number, or which numbers, the letter may stand for is set by the problem it sits in. There are three ways that happens. **An unknown.** The letter stands for one particular number that the problem fixes but does not tell you directly. In $$n + 5 = 12$$ $n$ is not "any number". It is the number that makes the sentence true, and $7$ is the only one that does: $7 + 5 = 12$, a larger number gives more than $12$, and a smaller one gives less. The letter holds the place of a number while you work out what it is. ### [What a solution is, and checking one](https://math.wiki/what-a-solution-is/) **Summary:** A number that makes the sentence true. Check by putting it back in. **Question:** A sentence with a letter in it is neither true nor false until the letter has a value. Which values make it true, and how can you be sure a value you found really works? ## Why this exists Wilma keeps coins in a jar. She drops in $15$ more and counts $42$. How many were in the jar before? Calling the unknown number $c$, the story becomes one sentence: $$c + 15 = 42$$ As expression-vs-equation explains, an equation is a claim. But this claim has a letter in it, and a claim about "some number $c$" cannot be judged true or false until you say which number. Put in $c = 10$ and the claim is false, since $10 + 15 = 25$. Put in $c = 27$ and it is true. So the real question an equation asks is: **which numbers make me true?** Those numbers have a name, and there is a sure way to test any number you are handed or any number you work out: put it back in and see. ## Definition A **solution** of an equation or inequality in one letter is a value of the letter that makes the sentence true when it is put in place of the letter. To **check** a value, substitute it for the letter, work out each side as in substitute-one-variable, and compare. For an equation the two sides must come out equal. For an inequality the comparison it states ($>$ or $<$) must hold. While the comparison is still being worked out, the equals sign is written $\stackrel{?}{=}$, read "is it equal?": $$27 + 15 \stackrel{?}{=} 42 \qquad 42 = 42 \ \text{true}$$ To **solve** a sentence is to find all its solutions. Often a question hands you a short list of candidates written as a set, such as $\{3, 4, 5, 6\}$, and asks which of them are solutions; then the job is to check each one. ### [What makes an expression quadratic](https://math.wiki/what-makes-a-quadratic/) **Summary:** Degree two: the variable is squared and no higher. **Question:** What do the area of a square, the height of a thrown ball and the number of handshakes in a room have in common? ## Why this exists A linear expression such as $3x + 2$ changes by the same amount every time $x$ goes up by $1$, and its graph is a straight line. A great many quantities do not behave like that. The area of a square of side $x$ is $x^2$. The height of a ball thrown upward, $t$ seconds after it leaves the hand, is a number times $t^2$ plus a number times $t$ plus a starting height. The number of handshakes when $n$ people each shake hands once is $\frac{1}{2}n^2 - \frac{1}{2}n$. Each of these, written out, has its variable squared and no higher power. They form the next family after the linear ones, and the family is called **quadratic**, from the Latin *quadratus*, a square. Recognizing a member matters because the whole family behaves alike. Every quadratic graphs as a parabola, every quadratic equation has at most two solutions, and one short list of methods solves all of them. Before any of that, you have to be able to tell whether the expression in front of you is one, and it is often in disguise: written as a product, or with a square that cancels away. ## Definition A **quadratic expression** in $x$ is a polynomial in $x$ of degree exactly two. Once it has been multiplied out and like terms combined, the highest power of $x$ in it is $x^2$. Every quadratic expression can be written in standard form, highest power first: $$ax^2 + bx + c, \qquad a \neq 0$$ where $a$, $b$ and $c$ are numbers. $ax^2$ is the **quadratic term** and $a$ is the **leading coefficient**; $bx$ is the **linear term**; $c$ is the **constant term**. ### [Writing a parallel or perpendicular line through a point](https://math.wiki/write-parallel-or-perpendicular-line/) **Summary:** Take the slope you need, then use point-slope. **Question:** You have a line and a point. What is the equation of the line through that point that runs alongside the first one, or crosses it square? ## When you reach for it You are given a line and a point, and you want a second line through the point that either never meets the first (parallel) or crosses it at a right angle (perpendicular). The question comes up all over coordinate geometry. The altitude of a triangle is the line through a corner perpendicular to the opposite side. The tangent to a circle is the line through a point of the circle perpendicular to the radius. The shortest path from a point to a line runs along the perpendicular. The opposite sides of a rectangle are parallel and its neighboring sides are perpendicular. Two facts you already have do all the work. A line is fixed by one point on it and its slope, and point-slope form turns that point and slope into an equation. And the slope you need comes straight from the given line: parallel lines have **equal** slopes, and perpendicular lines have slopes that **multiply to $-1$**. So the whole move is: find the slope you need, then use the point. ## The procedure Take the line $\ell$ with equation $y = \frac{1}{2}x + 1$ and the point $P(2, 7)$, which is not on $\ell$ (since $\frac{1}{2} \cdot 2 + 1 = 2$, not $7$). We want the line through $P$ parallel to $\ell$ and the line through $P$ perpendicular to it. **Read the slope of the given line.** In slope-intercept form $y = mx + b$ the slope is the coefficient of $x$, so $\ell$ has slope $\frac{1}{2}$. If the line comes in standard form $Ax + By = C$, solve for $y$ first; if it comes as two points, use the slope formula. ### [Writing an inequality for a condition](https://math.wiki/write-an-inequality-for-a-condition/) **Summary:** Turn a condition such as "at least 12" or "less than 5 dollars" into an inequality and see that it has infinitely many solutions. **Question:** An airline says a checked bag may weigh at most $23$ kg. That one short rule allows endlessly many weights. How do you write it in symbols, and know whether a bag of exactly $23$ kg passes? ## When you reach for it Many rules do not name one number. An airline says a checked bag may weigh **at most** $23$ kg. A festival sells child tickets to anyone **under** $12$. A dog walker takes **no more than** $5$ dogs on a walk. Each rule draws a line at one number and lets through everything on one side of it. An equation cannot say that: $w = 23$ allows a single weight, while the rule allows $23$ kg, $22.5$ kg, $18$ kg and every weight below. An inequality can. The four symbols $<$, $>$, $\le$, $\ge$ and their graphs belong to graph-simple-inequality; this lesson is about getting from the words to the symbols. Reach for it whenever a rule sets a limit, a minimum, a maximum, a floor or a ceiling, and you want to say exactly which values pass. Once the rule is in symbols, testing a value means putting it in and seeing whether the claim is true (what-a-solution-is). ## The procedure Four decisions turn a condition in words into an inequality. Each is settled by asking what the situation allows, not by matching a key word. **Name the quantity.** Pick a letter and say in words what it stands for, units included: "let $w$ be the weight of the bag in kilograms". The rule is a claim about one quantity, so the letter must stand for exactly that quantity, and its units must match the number in the rule. This is the same habit as naming the unknown in translate-words-to-expressions. **Find the boundary.** The number the rule names, $23$ in the bag rule, is the boundary. On one side of it values pass; on the other they fail. ### [Writing and interpreting constraints](https://math.wiki/model-constraints-with-inequalities/) **Summary:** Turning the limits in a situation into equations and inequalities, and deciding which solutions are viable in context. **Question:** A budget, a clock and a bag of beads all say "this much and no more". How do you turn limits like these into mathematics, and then read off which plans are possible and which is best? ## When you reach for it Kiri makes bracelets and keyrings to sell at a craft stall. She has only so many beads, only so many hours, and she has promised a customer some keyrings. None of these limits tells her what to make. Each one rules some plans out, and the plans that survive all of them are her real choices. That is the situation this skill is for: two quantities to choose, and several limits on them. A limit written in mathematics is a **constraint**: an equation or inequality that the chosen quantities must satisfy. A plan that satisfies every constraint is **feasible**, and on a graph the feasible plans make up the **feasible region**, the overlap of the regions of all the constraints. A feasible plan is **viable** when it also makes sense in the situation: no one can make $22.5$ bracelets, even when every inequality allows it. Day 1 is writing the constraints; Day 2 is reading them on a graph. The graphing is graphing a linear inequality and finding where shaded regions overlap, used here rather than retaught. ## The procedure ### Day 1: turning limits into constraints **Name each unknown as a number.** Write "let $b$ be the number of bracelets Kiri makes this week", not "$b$ is bracelets": a variable stands for a number. **Write the amount each limit is about.** When each item uses a fixed amount, the total is a rate times a count: if a bracelet uses $20$ beads and a keyring $10$, then $b$ bracelets and $k$ keyrings use $20b + 10k$ beads. This is the rate-times-quantity reasoning of a linear model, and it is why every constraint here is linear. ## Calculus ### [Absolute and conditional convergence](https://math.wiki/absolute-and-conditional-convergence/) **Summary:** Whether a series still converges once every term is made positive. **Question:** A sailboat zigzags upwind and its sideways position settles down, yet the sideways distance it covers grows without end. When does a series converge only because its terms cancel, and what changes when they do? ## Why this exists Mihai sails a dinghy up a channel against the wind. A boat cannot sail straight into the wind, so he tacks: each leg carries him across the wind, alternately to the right and to the left, while he gains ground upwind. In a simplified model, chosen so the arithmetic comes out exactly, the sideways part of his $n$th leg is $\frac{2n+1}{n(n+1)}$ kilometers. Counting rightward as positive, his sideways offset from the start line after $N$ legs is the partial sum $S_N$ of $$\sum_{n=1}^{\infty} (-1)^{n+1}\frac{2n+1}{n(n+1)} = \frac{3}{2} - \frac{5}{6} + \frac{7}{12} - \cdots$$ while the sideways distance he has actually sailed is the partial sum $T_N$ of the same terms with every sign made positive. Dragos, watching from the shore, sees the boat's offset settle. Anca, who logs the distance sailed, sees her total keep growing. Both are right. The alternating series test needs signs that alternate strictly, and the comparison and integral tests need positive terms. A series such as $\sum \frac{\cos(3n)}{n^2+1}$ fits neither. Absolute convergence closes that gap: make every term positive, test that series, and convergence of the original follows. It also separates series that converge robustly from those that converge only by cancellation, and only the first kind may be added in any order. ## Definition Let $\sum a_n$ be a series whose terms may have either sign. - $\sum a_n$ **converges absolutely** if $\sum \lvert a_n \rvert$ converges. - $\sum a_n$ **converges conditionally** if $\sum a_n$ converges but $\sum \lvert a_n \rvert$ diverges. ### [Accumulation and net change](https://math.wiki/accumulation-functions/) **Summary:** The integral of a rate is the total amount accumulated. **Question:** An instrument reports only how fast an amount is changing. How do you get back the amount itself, its total change, and its largest value? ## What this area is about Many quantities are measured as rates: cars per hour, kilowatts, velocity. The integral of a rate over a time interval is the net change in the amount, and adding that change to a starting value gives the amount at any moment. This area puts the fundamental theorem of calculus to work on quantities in context. ## How these fit together All three apply the fundamental theorem of calculus: if $F'$ is continuous on $[a, b]$, then $\int_a^b F'(t)\,dt = F(b) - F(a)$, and for continuous $f$ the function $\int_a^x f(t)\,dt$ has derivative $f(x)$. **Amounts from rates.** The net change theorem in context totals water, cars or charge from a rate given by a formula, a table or a graph, with the units of the situation. **Motion.** Position, displacement and distance from velocity shows that integrating velocity $v$ gives displacement, while integrating speed $\lvert v \rvert$ gives total distance traveled. **The amount as a function.** Graphing an accumulation function studies $g(x) = \int_a^x f(t)\,dt$ as a function in its own right: it rises where $f > 0$, falls where $f < 0$, and peaks where $f$ crosses from positive to negative. ## A path through it No child requires another, so take them in any order. Net change in context is the gentlest start, then distance and displacement, then graphing an accumulation function, which also uses the first derivative test. ### [Alternating series and their error bound](https://math.wiki/alternating-series-error-bound/) **Summary:** The error after stopping is no bigger than the first term left out. **Question:** A balance pointer swings past its mark, back, and past again, each swing shorter than the last. Does it settle, and how close to the resting place is the pointer after a given number of swings? ## Why this exists Wirrin releases the pointer of a laboratory balance at the zero mark. It swings $4$ millimeters to the right, then $3$ back, then $2.4$ forward again. Wirrin models the $n$th swing as $\frac{12}{n+2}$ millimeters, each in the opposite direction from the one before. This is a simplified model, not the physics of a real balance, whose friction shrinks the swings much faster; it is chosen because swings that shrink this slowly are the hard case. Measuring positions to the right of the mark as positive, the pointer's turning points are the partial sums of $$\begin{aligned} &\sum_{n=1}^{\infty} (-1)^{n+1}\frac{12}{n+2} \\ &\quad = 4 - 3 + \frac{12}{5} - 2 + \frac{12}{7} - \cdots \end{aligned}$$ The swing lengths shrink like $\frac{1}{n}$, as slowly as the terms of the harmonic series, whose sum is infinite (partial-sums-of-a-series). Yet the turning points visibly close in on one position, because each swing takes back most of what the one before it gave. This lesson makes that cancellation exact: a test that proves such a series converges, and a bound on how far any turning point can be from the rest position. The nth-term-test can only prove divergence; this is a test that proves convergence for a series with negative terms. ## Definition An **alternating series** is a series whose terms alternate in sign: $$\sum_{n=1}^{\infty} (-1)^{n+1} b_n = b_1 - b_2 + b_3 - \cdots$$ ### [Antiderivatives and indefinite integrals](https://math.wiki/antiderivative-of-a-power/) **Summary:** Reversing the power rule by raising the exponent and dividing. **Question:** A rate is known, such as how many centimeters a kelp frond grows each day, and the amount is wanted, how long the frond is. How is a derivative run backward, and how many answers are there? ## Why this exists Every derivative rule so far answers one question: given an amount, find its rate. Science usually hands over the reverse. Gwion, a marine biologist, records how fast a kelp frond grows each day, not how long it is. Macsen reads a train's speed from its instruments and wants to know how far it travels. Each time the derivative is known and the function is wanted. Undoing a derivative is called **antidifferentiation**, and it raises two questions that differentiation never did. Is there more than one answer? Yes: adding a constant to a function changes nothing about its rate. Then how is the right answer picked out of all of them? By one known value, such as the frond's length on the first day. This lesson answers both questions for powers of $x$ and the functions built from them. ## Definition ### Day 1: antiderivatives and the indefinite integral A function $F$ is an **antiderivative** of $f$ on an interval $I$ if $F'(x) = f(x)$ for every $x$ in $I$. For example, $x^4$ is an antiderivative of $4x^3$ on $(-\infty, \infty)$, and so are $x^4 + 10$ and $x^4 - \pi$, because a constant has derivative $0$. The **indefinite integral** of $f$ is the whole collection of antiderivatives of $f$ on an interval. It is written $$\int f(x)\,dx = F(x) + C,$$ where $F$ is any one antiderivative and $C$ stands for an arbitrary constant, the **constant of integration**. The sign $\int$ is the integral sign, $f(x)$ is the **integrand**, and $dx$ names the variable. So $\int 4x^3\,dx = x^4 + C$. The same sign with limits attached, $\int_a^b f(x)\,dx$, is the definite integral of Riemann sums, a single number rather than a family of functions. Why the two share a symbol is the content of the fundamental theorem of calculus. ### [Antiderivatives of trig, exponential and log functions](https://math.wiki/antiderivatives-of-trig-functions/) **Summary:** Reading the trig derivative rules backwards. **Question:** The power rule run backward handles every power of x except one, and it says nothing about sine, the exponential or the arctangent. Which functions undo those derivatives, and what fills the gap left by one over x? ## Why this exists The power rule for antiderivatives finds $\int x^n\,dx$ for every real $n$ except $n = -1$, where it would divide by zero. Yet $\dfrac{1}{x}$ is a perfectly good function on each side of $0$, and its graph has a slope at every point, so something must have it as a derivative. The same lesson has nothing to say about $\sin x$, $e^x$ or $\dfrac{1}{1 + x^2}$, which turn up as rates all the time: a swinging weight's velocity is a sine, a population's growth rate is an exponential. Nothing new has to be invented. Every derivative rule already learned, read from right to left, is an antiderivative rule. The derivatives of the six trig functions, of exponentials, of the natural log and of the inverse trig functions together give a table of basic antiderivatives. With the sum and constant multiple rules, that table covers a large share of the integrals met before the special techniques of later lessons. ## Definition ### Day 1: reading the table backward As in the power lesson, $F$ is an antiderivative of $f$ on an interval $I$ when $F'(x) = f(x)$ for every $x$ in $I$, and $$\int f(x)\,dx = F(x) + C$$ names the whole family. A **basic antiderivative formula** is a derivative rule $F' = f$ written in this form. For example, $\dfrac{d}{dx}\big[\tan x\big] = \sec^2 x$ becomes $\int \sec^2 x\,dx = \tan x + C$. The **absolute value** inside $\ln\lvert x \rvert$ makes the logarithm defined for every $x \ne 0$: for $x > 0$ it is $\ln x$, and for $x < 0$ it is $\ln(-x)$. ### [Applications of derivatives](https://math.wiki/applications-of-derivatives/) **Summary:** What the derivative tells you about a function and the world. **Question:** A derivative is a rate at one instant. What can that one number tell you about the shape of a graph, the motion of an object, or the best possible choice? ## What this area is about The sign of $f'$ says where $f$ rises or falls, the sign of $f''$ says which way the graph bends, and the tangent line stands in for the curve near a point. So the derivative estimates, finds peaks and valleys, and describes motion. ## How these fit together **Lines from the derivative.** Tangent and normal lines turn a slope into a line. Linear approximation and differentials use that line to estimate values and errors, and Newton's method repeats it to close in on a root. **Shape.** Concavity and inflection points read the sign of $f''$. Maxima, minima and critical points finds where peaks and valleys can sit and tests each one. Curve sketching puts it all into one picture. **Theorems.** The mean value theorem says that if $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then $f'(c) = \frac{f(b) - f(a)}{b - a}$ for some $c$ in $(a, b)$. L'Hôpital's rule settles limits of the forms $\frac{0}{0}$ and $\frac{\infty}{\infty}$ through derivatives. **Applied problems.** Optimization and related rates build a function or an equation from a situation and read the answer from derivatives; position, velocity and acceleration reads motion along a line from the signs of $s'$ and $s''$. ## A path through it Tangent lines, linear approximation, concavity, then extrema. Newton's method needs linear approximation; optimization, motion, the mean value theorem and curve sketching need extrema. Related rates can follow implicit differentiation. L'Hôpital's rule is proved from linearization and Rolle's theorem, so it goes last. ### [Applications of integrals](https://math.wiki/applications-of-integrals/) **Summary:** Area, volume, distance, and total change. **Question:** An integral adds up infinitely many thin pieces. Which quantities, from an area or a volume to the work done lifting a load, can be built that way? ## What this area is about Every application here follows one pattern. Cut the quantity into thin pieces, approximate each piece by something simple (a rectangle, a disk, a shell, a straight segment), add the pieces, and let them shrink. The sum becomes an integral, usually evaluated by the fundamental theorem of calculus. ## How these fit together **Totals from rates.** Accumulation and net change, an area of its own, integrates a rate to get a total change or an amount. Work adds force times distance over small displacements. **Plane regions.** Area between curves integrates top minus bottom. Average value divides $\int_a^b f(x)\,dx$ by $b - a$, for $f$ continuous on $[a, b]$. **Volumes.** Known cross sections integrates the slice area $A(x)$. Disks and washers are the case where each slice of a solid of revolution is a circle or a ring; shells slice parallel to the axis instead. **Lengths and surfaces.** Arc length adds tiny straight segments along a curve, and surface area of revolution adds the thin bands that arc sweeps out. ## A path through it Net change first, then area between curves, which average value, arc length and disks all need. Then cross sections, disks, washers and shells, since each method refers to the ones before. Arc length, then surface area, which needs both arc length and disks. Work, which borrows the slicing habit of the volume methods, comes last. Next comes arc length and speed for parametric curves. ### [Approximating area with Riemann sums](https://math.wiki/area-under-a-curve-idea/) **Summary:** Approximating a curved region by stacking simple rectangles inside it. **Question:** A speedometer shows a speed at every instant, but distance is speed times time only while the speed holds still. How can rectangles, the one shape whose area is easy, measure the distance covered while the speed keeps changing? ## Why this exists A car holding $15$ meters per second for $6$ seconds covers $15 \cdot 6 = 90$ meters. On a graph of velocity against time that motion is a horizontal line at height $15$, and the distance is the area of the rectangle under it, width times height (area-of-rectangles). Once the speed changes, the graph curves, and no formula gives the area under a curve. Over a short stretch of time, though, the speed hardly changes. Treat it as constant there, take a rectangle for that stretch, and add the rectangles. Narrower stretches give a better estimate. The same holds for any rate: a flow in liters per minute, a power in kilowatts. The area under the graph of a rate is the total amount, so "how much area is under this curve?" and "how much accumulated?" are one question. ## Definition ### Day 1: the region and three sums Let $f$ be continuous with $f(x) \ge 0$ on $[a, b]$. The **region under the graph** is bounded above by $y = f(x)$, below by the $x$-axis, and on the sides by $x = a$ and $x = b$. Write $A$ for its area. Cut $[a, b]$ into $n$ pieces of equal width $$\Delta x = \frac{b - a}{n}, \qquad x_k = a + k\,\Delta x \quad (k = 0, 1, \dots, n).$$ Piece $k$ is $[x_{k-1}, x_k]$ and carries a rectangle of width $\Delta x$. Its height is a value of $f$ in that piece, chosen by one of three rules: the left end, the right end, or the midpoint $m_k = \frac{x_{k-1} + x_k}{2}$. $$\begin{aligned} L_n &= \sum_{k=1}^{n} f(x_{k-1})\,\Delta x, \\ R_n &= \sum_{k=1}^{n} f(x_k)\,\Delta x, \\ M_n &= \sum_{k=1}^{n} f(m_k)\,\Delta x. \end{aligned}$$ ### [Arc length](https://math.wiki/arc-length-formula-derivation/) **Summary:** Adding up tiny straight hypotenuses along the curve and taking a limit. **Question:** A curve is not made of straight pieces, so no ruler fits along it. How long is it? ## Why this exists A straight segment is measured by the Pythagorean theorem: a run of $\Delta x$ and a rise of $\Delta y$ give $\sqrt{(\Delta x)^2 + (\Delta y)^2}$. A circle's arcs are a fraction of the circumference, but only because a circle bends the same amount everywhere. A parabola or a hanging cable bends differently at every point, and no formula from geometry measures it. The way through is the usual one for integrals. Cut the curve into short pieces, replace each by a straight segment (the hypotenuse of a tiny right triangle), add, and let the pieces shrink. As with area, the sum is a Riemann sum and its limit is an integral. ## Definition Let $f$ have a derivative $f'$ that is continuous on $[a, b]$. Such a function is called **smooth** on $[a, b]$: its graph has no corners and no vertical tangents there. Split $[a, b]$ into $n$ pieces of width $\Delta x = \dfrac{b - a}{n}$, with endpoints $a = x_0 < x_1 < \dots < x_n = b$, and mark the points $P_i = (x_i, f(x_i))$ on the graph. Joining $P_0, P_1, \dots, P_n$ by straight segments gives a path **inscribed** in the curve (Figure 1). The **arc length** $L$ of the graph from $x = a$ to $x = b$ is the limit of the lengths of these paths: $$L = \lim_{n \to \infty} \sum_{i=1}^{n} \lvert P_{i-1} P_i \rvert$$ Here $\lvert P_{i-1} P_i \rvert$ is the length of the segment from $P_{i-1}$ to $P_i$. This is a definition: it says what "the length of a curve" means. The work below turns it into an integral. ### [Arc length and speed on parametric and polar curves](https://math.wiki/speed-of-a-parametric-curve/) **Summary:** The rate at which arc length is swept out as the parameter advances. **Question:** A pen plotter's pen is steered by two formulas of time, one for how far right it is and one for how far up. How fast is the pen moving along its line at a given moment, and how long is the line it draws, when that line loops and doubles back and is not the graph of any function? ## Why this exists Kawira programs a pen plotter: at each time $t$ (in seconds) the pen sits at $x(t)$ centimeters right of a corner and $y(t)$ centimeters up. Muthoni and Kamande draw with a spirograph, a toothed wheel rolling inside a ring with a pen through one of its holes; the pen's position is again a pair of formulas in one parameter. Two questions come up at once. How long is the line on the paper? How fast is the pen moving along it at a given instant? arc-length-formula-derivation answered the first question for a graph $y = f(x)$: add up tiny hypotenuses and take a limit. But a spirograph star loops back on itself and fails the vertical line test, so it is no graph, and even a curve that is a graph can be easier to handle in its parametric form. tangent-lines-to-parametric-curves took slopes straight from $\frac{dx}{dt}$ and $\frac{dy}{dt}$; this lesson takes length and speed from the same two rates. Day 1 does curves given by a parameter. Day 2 does polar curves $r = f(\theta)$ (polar-coordinates), which are parametric curves in disguise. ## Definition Let $x = x(t)$, $y = y(t)$ for $a \le t \le b$, with $x'$ and $y'$ continuous on $[a, b]$. As $t$ runs from $a$ to $b$, the point $\big(x(t), y(t)\big)$ moves along a path (parameter-and-orientation). The **arc length** traced from $t = a$ to $t = b$ is the limit of the lengths of inscribed polygons: choose $a = t_0 < t_1 < \dots < t_n = b$, join consecutive points by straight chords, add the chord lengths, and let the longest step $t_i - t_{i-1}$ shrink to $0$. This is the same definition as for a graph. ### [Area between curves](https://math.wiki/area-with-curves-that-cross/) **Summary:** Splitting the region at each intersection so the top curve stays on top. **Question:** A region is fenced in by two curves, and neither of them is the x-axis. How much area does it hold, and what changes when the curves cross? ## When you reach for it A region in the plane is fenced in by two or more curves, and you want its area: a flower bed between two parabolic edges, a leaf-shaped logo between two waves, the gap between a cubic and a line. None of these is a rectangle, triangle or circle, but each is filled by thin strips whose length can be read off the curves. signed-area showed that one integral measures area above the axis minus area below it. Here the axis is replaced by a second curve, and every piece of area is counted as positive. ## The procedure ### Day 1: vertical strips **Strips and a Riemann sum.** Let $f$ and $g$ be continuous on $[a, b]$ with $f(x) \ge g(x)$ there. Cut $[a, b]$ into $n$ pieces of width $\Delta x$ and pick a point $x_i^*$ in each. Over the $i$th piece stands a thin strip from the lower curve to the upper one, with height $f(x_i^*) - g(x_i^*)$ and width $\Delta x$ (Figure 1). The strips nearly fill the region, so $$A \approx \sum_{i=1}^{n} \big(f(x_i^*) - g(x_i^*)\big)\,\Delta x.$$ This is a Riemann sum for $f - g$ (definite-integral-as-a-limit). As the strips get thinner the sum tends to the definite integral, which exists because $f - g$ is continuous: $$A = \int_a^b \big(f(x) - g(x)\big)\,dx.$$ This holds when $f(x) \ge g(x)$ on all of $[a, b]$. Read it as top minus bottom, added across the region. With $g = 0$ it is the familiar area under a curve. Neither curve has to stay above the $x$-axis: moving both up by the same constant $c$ leaves every height $(f + c) - (g + c) = f - g$ unchanged, so only the gap between the curves matters. The integral is evaluated with an antiderivative (evaluating-a-definite-integral, part of the fundamental-theorem-of-calculus). ### [Area in polar coordinates](https://math.wiki/polar-area-between-curves/) **Summary:** Subtracting the inner sector area from the outer one over a shared range of angles. **Question:** A polar curve is drawn by a distance that changes as the direction turns. How much area does it sweep out, and how much lies between two such curves? ## When you reach for it A region whose edge is a polar curve, such as a cardioid, a petal of a rose or a loop of a limaçon (cardioid, rose-curves, polar-graph-of-a-circle), is awkward to cut into vertical strips: its rectangular equation is often of high degree and the top and bottom of a strip change formula partway across. The natural pieces for a polar region are thin wedges with their point at the pole. The area of a wedge of a circle is $\frac{1}{2}r^2\theta$ (sector-area-formula), and adding wedges under a curve leads to one integral in $\theta$. Reach for it whenever the edge of the region is given as $r = f(\theta)$, or when two polar curves bound a region between them, as in area-with-curves-that-cross for curves $y = f(x)$. ## The procedure ### Day 1: the area swept by one curve **The formula from thin sectors.** Let $f$ be continuous with $f(\theta) \ge 0$ on $[\alpha, \beta]$, where $0 < \beta - \alpha \le 2\pi$. The region $R$ is every point $(r, \theta)$ with $\alpha \le \theta \le \beta$ and $0 \le r \le f(\theta)$: the region swept by the segment from the pole to the curve as the direction turns from $\alpha$ to $\beta$. Cut $[\alpha, \beta]$ into $n$ pieces of width $\Delta\theta = \frac{\beta - \alpha}{n}$. Over one piece, the slice of $R$ lies between two circular sectors with the same angle $\Delta\theta$: one whose radius is the smallest value $m_i$ of $f$ on the piece, and one whose radius is the largest value $M_i$. So the slice has area between $\frac{1}{2}m_i^2\,\Delta\theta$ and $\frac{1}{2}M_i^2\,\Delta\theta$. Adding over all the pieces, ### [Average value of a function](https://math.wiki/average-value-geometric-meaning/) **Summary:** The height of the rectangle with the same area as the region under the curve. **Question:** A temperature changes at every instant of the day. What single number is its average, when there are infinitely many values to add up? ## Why this exists The mean of a list of numbers is their sum shared out evenly: add them, divide by how many there are. Hamdan reads a thermometer once an hour from 6 a.m. to 6 p.m. and averages the thirteen readings. Readings every minute would describe the day better, and readings every second better still, but the temperature has a value at every instant, and there is no way to add infinitely many numbers and divide by infinity. The average value of a function is the number these means settle toward as the readings get denser, and the definite integral is what computes it. ## Definition The **average value** of a function $f$ on an interval $[a, b]$, where $a < b$ and $f$ is integrable on $[a, b]$ (continuity is enough), is $$f_{\text{avg}} = \frac{1}{b - a}\int_a^b f(x)\,dx.$$ In words: the net signed area under the graph, shared out evenly over the length of the interval. It is a single number, not a function, and it depends on the interval as well as on $f$. ## Intuition **A limit of ordinary means.** Cut $[a, b]$ into $n$ pieces of width $\Delta x = \frac{b - a}{n}$ and sample $f$ at the right ends $x_i = a + i\,\Delta x$. The mean of the samples is $\frac{1}{n}\sum_{i=1}^{n} f(x_i)$. Since $\frac{1}{n} = \frac{\Delta x}{b - a}$, $$\frac{1}{n}\sum_{i=1}^{n} f(x_i) = \frac{1}{b - a}\sum_{i=1}^{n} f(x_i)\,\Delta x.$$ The sum on the right is a right Riemann sum (definite-integral-as-a-limit): each sample has become the height of a bar of width $\Delta x$ (Figure 1). As $n \to \infty$ the sum approaches $\int_a^b f(x)\,dx$, so the means of more and more equally spaced samples approach $f_{\text{avg}}$. For continuous $f$, sampling at left ends or midpoints gives the same limit. ### [Calculus](https://math.wiki/calculus/) **Summary:** Change and accumulation: derivatives and integrals. **Question:** How fast is something changing at this instant, and how much has piled up over time? Why do those two questions turn out to undo each other? ## What this area is about The derivative measures a rate at an instant; the integral adds up infinitely many thin pieces. Both are limits, and the fundamental theorem of calculus shows that each undoes the other. ## How these fit together **Foundation.** Limits say where a function is heading. **Change.** Derivatives define the rate and give the rules for finding it. Applications of derivatives use it for graphs, motion, estimates and optimization. **Accumulation.** Integrals build the definite integral and the techniques for evaluating it. Applications of integrals turn it into areas, volumes, lengths and work. **Further tools.** Sequences and series write functions as infinite polynomials. Differential equations find a function from a rule about its rate. Parametric, polar and vector valued functions reach curves that are not graphs. **College areas, not yet written.** Partial derivatives (functions of several variables and their extremes), multiple integrals (over regions in the plane and in space), vectors and space curves (lines, planes and curves in three dimensions) and vector calculus (fields, flow, and the theorems of Green, Stokes and Gauss). ## A path through it Limits first. Then derivatives and their applications together, since each borrows lessons from the other. Then integrals. The first-course lessons of differential equations come before applications of integrals, whose motion lesson solves initial value problems. Series, then parametric and polar calculus. ### [Choosing a convergence test](https://math.wiki/choosing-a-convergence-test/) **Summary:** Reading the shape of the terms to decide which test will settle the question. **Question:** Every convergence test answers the same question, but each one answers it only for series of a certain shape. Looking at a series, how do you tell which test will settle it, and what do you do when the first test you try says nothing? ## When you reach for it By the end of this unit there are nine ways to decide whether $\sum a_n$ converges: the nth-term-test, the geometric series test, the integral test and the p series test, the direct and limit comparison tests, the alternating series test, and the ratio and root tests. Each one was taught on series built for it. A series met in the wild comes with no label, and a test applied to the wrong shape usually does not give a wrong answer: it gives no answer, after a page of algebra. So the skill is reading. The form of $a_n$ (what grows, what shrinks, whether the sign changes, whether the whole term is raised to a power) points at the test whose hypotheses it meets and whose conclusion it can reach. This lesson is about that reading. Day 1 picks a test from the shape of the term and compares routes when more than one works. Day 2 handles series on which the most tempting test is silent, and sorts alternating series into absolutely and conditionally convergent ones. ## The procedure ### Day 1: reading the shape of the term **Look at the size of the terms first.** If $a_n$ does not tend to $0$, the series diverges, and nothing else is needed. This costs one limit, often one glance: a term that is at least $1$ for every $n$, or a ratio of polynomials of equal degree, settles the question at once. The test is one-sided. A limit of $0$ says nothing, and the reading goes on. ### [Choosing an integration technique](https://math.wiki/choosing-an-integration-technique/) **Summary:** Reading the shape of an integrand to decide which method to try. **Question:** Every method so far was taught with its own integrals in front of it. Faced with an integral that comes with no label, how do you tell which method it needs, and what happens when none of them can finish it? ## When you reach for it Each method lesson so far began by announcing its method: u-substitution, integration by parts, trigonometric-integrals, trigonometric-substitution, partial fractions, with division and completing the square as preparation. An integral met in the wild carries no such label. It may also yield to two methods at very different cost, need a rewrite before any method fits, or need one method to set up another. This lesson is about the reading, not the methods: looking at the shape of an integrand, choosing a route, and comparing routes. Day 2 adds the hard fact behind all of it. Some integrands have no antiderivative that can be written as a formula, and no amount of cleverness will produce one. Then the question changes from "what is the formula" to "what do we do instead". ## The procedure ### Day 1: reading the shape, and rewriting first **Rewrite before choosing.** Many integrals become basic forms after algebra: expanding a product, splitting a fraction over its denominator, dividing a top-heavy rational function, canceling a common factor, or applying an identity such as $\sin^2 x + \cos^2 x = 1$ or $1 + \cos 2x = 2\cos^2 x$. Each rewrite replaces the integrand by an equal function on the interval in question, so it changes nothing about the integral. It is always worth a look first, because when it works the integral is finished before a method has started. ### [Combining differentiation rules](https://math.wiki/choosing-differentiation-rules/) **Summary:** Deciding which rules apply and in what order for a layered expression. **Question:** A function such as x sin(x^2) divided by e^x is built with a product, a quotient and a function inside a function all at once. Which rule do you apply first, and how do you keep track of the rest? ## When you reach for it Each rule so far answers one shape: a power, an exponential, a sine, a product, a quotient, a composition. Real functions use several shapes at once. In $$y = \frac{x\sin(x^2)}{e^{x}}$$ there is a quotient, a product inside its numerator and a composition inside that product. No new rule is needed. What is needed is a way to see which rule applies first, to pass the smaller pieces on to the other rules in turn, and to pick the form of the function that makes the work shortest. Reach for this whenever a function is built from more than one operation, which from now on is almost always. The lesson runs over two days. Day 1 reads the structure of an expression, peels its layers, chooses between the quotient rule and a rewrite, simplifies, and checks an answer with numbers. Day 2 goes deeper: several rules nested inside each other, functions known only through a table or a graph, and the choice of least work. ## The procedure ### Day 1: reading the structure **The rules available.** With $f$ and $g$ differentiable: | Shape | Derivative | |---|---| | $x^n$ | $nx^{n-1}$ | | $e^x$ | $e^x$ | | $a^x$ | $a^x \ln a$ | | $\sin x$ | $\cos x$ | | $\cos x$ | $-\sin x$ | | $\tan x$ | $\sec^2 x$ | | $\sec x$ | $\sec x\tan x$ | | $\csc x$ | $-\csc x\cot x$ | | $\cot x$ | $-\csc^2 x$ | | $f(x)g(x)$ | $f'(x)g(x) + f(x)g'(x)$ | | $\dfrac{f(x)}{g(x)}$ | $\dfrac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$ | | $f\big(g(x)\big)$ | $f'\big(g(x)\big)\,g'(x)$ | ### [Comparing growth rates at infinity](https://math.wiki/growth-rates-at-infinity/) **Summary:** Exponentials outgrow powers and powers outgrow logarithms in the long run. **Question:** Two functions both grow without bound. How do you say, exactly, that one of them leaves the other behind, and which of the familiar families wins? ## Why this exists Pao, Txawj and Nou are comparing computer programs. For each one they measure the running time as a function of the input size $x$: how long the program takes on a list of $x$ items, say. A program whose time is $x^2$ and one whose time is $2^x$ both take longer and longer as $x$ grows, so "both go to infinity" says nothing useful. What they want to know is which one becomes hopeless first, and whether a head start on small inputs ever lasts. Plugging in a few values does not settle it. A function can trail for millions of inputs and then pull ahead for good. The question needs a limit: compare the two functions by their quotient, far out, where the head start no longer matters. That comparison sorts the functions met so far into a ladder, logarithms at the bottom, powers in the middle and exponentials at the top, and the ladder then hands over limits such as $\lim_{x \to \infty} x^2 e^{-x}$ with no computation at all. ## Definition Let $f$ and $g$ be positive for all large $x$. Compare them by the quotient $\dfrac{f(x)}{g(x)}$ as $x \to \infty$. - $f$ **grows faster than** $g$ (and $g$ grows slower than $f$) when $$\begin{aligned} &\lim_{x \to \infty} \frac{f(x)}{g(x)} = \infty, \\ &\text{equivalently} \quad \lim_{x \to \infty} \frac{g(x)}{f(x)} = 0. \end{aligned}$$ - $f$ and $g$ **grow at the same rate** when $\lim_{x \to \infty} \dfrac{f(x)}{g(x)} = L$ for a number $L$ with $0 < L < \infty$. ### [Completing the square inside an integral](https://math.wiki/completing-the-square-in-an-integral/) **Summary:** Reshaping a quadratic denominator so an inverse trig or log form appears. **Question:** The arctangent form fits one over one plus x squared exactly. What do you do with one over x squared plus six x plus ten, which matches nothing in the table? ## When you reach for it The arctangent and arcsine forms from antiderivatives-of-trig-functions fit $\dfrac{1}{1 + x^2}$ and $\dfrac{1}{\sqrt{1 - x^2}}$ exactly, and nothing else. An integrand such as $\dfrac{1}{x^2 + 6x + 10}$ matches no line of that table, and splitting the quotient does not help. But its denominator is a parabola that never reaches zero, and completing the square writes it as $(x + 3)^2 + 1$: the denominator $1 + x^2$ with $x$ replaced by $x + 3$. The graph is the arctangent integrand slid $3$ units left, and its antiderivative is $\arctan x$ slid the same way. Reach for this move when a quadratic with no real roots sits in a denominator, either alone or under a linear numerator, and when a quadratic that opens downward sits under a square root in a denominator. ## The procedure ### Reshaping the quadratic Completing the square rewrites any quadratic with leading coefficient $1$: $$x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 + \left(c - \frac{b^2}{4}\right).$$ Write $h = -\frac{b}{2}$. The constant $c - \frac{b^2}{4}$ equals $-\frac{1}{4}\left(b^2 - 4c\right)$, so it is positive exactly when the discriminant $b^2 - 4c$ is negative, that is, when the quadratic has no real roots. Then it is a square $k^2$ with $k > 0$, and $$x^2 + bx + c = (x - h)^2 + k^2.$$ A leading coefficient other than $1$ is factored out of the denominator first and comes outside the integral as a constant. Under a square root the quadratic usually opens downward: factor out $-1$, complete the square inside, and distribute the $-1$ back, which gives $k^2 - (x - h)^2$. ### [Concavity and inflection points](https://math.wiki/concavity-from-the-second-derivative/) **Summary:** A positive second derivative bends the graph up and a negative one bends it down. **Question:** A graph can rise while bending up or while bending down. How does the second derivative tell which way a graph bends, where the bend reverses, and whether a tangent line runs above or below the curve? ## Why this exists concavity-and-changing-rates described the bend of a graph with secant slopes and second differences, and error-estimation-with-differentials decided whether a tangent-line estimate ran high or low by looking at a picture. Both questions ask whether the slope $f'$ is increasing, and whether a function increases is read from its derivative. So the answer sits in the second-derivative. This lesson turns the bend of a graph into a sign of $f''$ computed from a formula, finds where the bend reverses, and settles which way a tangent-line estimate errs without a picture. ## Definition ### Day 1: concave up, concave down, inflection points Let $f$ be differentiable on an interval $I$. Then $f$ is **concave up** on $I$ if $f'$ is increasing on $I$, and **concave down** on $I$ if $f'$ is decreasing on $I$. This lesson takes the slope description as the definition. The other description, that the graph lies above each of its tangent lines on $I$ (concave up) or below them (concave down), follows from it and implies it back, as proved under "Why it is true". The secant form of concavity-and-changing-rates agrees too. Concavity is reported here on open intervals. A **point of inflection** is a point $(c, f(c))$ of the graph such that $f$ is continuous at $c$ and the concavity changes at $c$: $f$ is concave up on an open interval ending at $c$ and concave down on one starting at $c$, or the reverse. Nothing is asked of $f'$ or $f''$ at $c$ itself. ### [Continuity](https://math.wiki/continuity/) **Summary:** No holes, no jumps, no gaps: the limit equals the value. **Question:** A limit predicts where a function is heading, and the value says where it actually is. When do the two agree, and what can be proved about a function where they always do? ## What this area is about A graph drawn without lifting the pencil is the picture. The exact version uses limits: $f$ is continuous at $a$ when $\lim_{x \to a} f(x)$ exists and equals $f(a)$. Continuity lets a limit be found by plugging in, and it is the hypothesis behind the big existence theorems of calculus. ## How these fit together **Definitions.** Continuity at a point states the test above. Continuity on an interval asks for it at every interior input, adds one sided continuity at the ends of an interval such as $[a, b]$, and lists the familiar functions that are continuous wherever they are defined. **Classifying breaks.** Discontinuities and how to remove them sorts the failures into removable, jump, infinite and oscillating, using the two one sided limits, and repairs the removable ones. **Procedure.** Making a piecewise function continuous chooses constants so each rule of a piecewise function meets the next at its boundary. **Result.** The extreme value theorem says that a function continuous on a closed interval $[a, b]$ takes a largest and a smallest value there. ## A path through it Continuity at a point first, then on an interval. After that, discontinuities and then piecewise functions; the extreme value theorem can come any time after continuity on an interval. Next come the intermediate value theorem, extrema on a closed interval and differentiability. ### [Continuity at a point](https://math.wiki/continuity-at-a-point/) **Summary:** A function is continuous at a point when the limit there exists and equals the value. **Question:** A limit predicts the height a function heads toward near an input, and the value says where the function actually is at that input. When do the prediction and the value agree, and how can you tell without drawing anything? ## Why this exists A limit and a value answer two different questions. The limit $\lim_{x \to a} f(x)$ looks only at inputs near $a$, never at $a$ itself, and says what height the outputs are heading toward. The value $f(a)$ is what the function actually does at $a$. In sum-and-difference-limit-laws the two kept agreeing for polynomials and for rational functions with a nonzero bottom, which is why direct substitution worked; and in limits-by-factoring they kept disagreeing, because the function was not even defined at the point. A function whose value at $a$ is exactly the height its limit predicts is called **continuous at** $a$. At such a point the limit is found by plugging in, and continuity is the hypothesis behind the big theorems of calculus. The picture of a graph drawn without lifting the pencil is in continuous-and-broken-graphs; this lesson makes that picture exact with limits and works out its algebra. ## Definition In plain words: **a function is continuous at $a$ when its limit at $a$ exists and equals its value at $a$.** In symbols, $f$ is **continuous at** $a$ when $$\lim_{x \to a} f(x) = f(a)$$ This one equation carries three conditions, and all three must hold: - **Condition 1:** $f(a)$ is defined, so $a$ is in the domain of $f$. - **Condition 2:** $\lim_{x \to a} f(x)$ exists as a number. By two-sided-limit-exists-criterion, this means both one-sided limits exist and are equal. - **Condition 3:** the two numbers are equal, $\lim_{x \to a} f(x) = f(a)$. ### [Continuity on an interval](https://math.wiki/continuity-on-an-interval/) **Summary:** A function continuous at every point of an interval, with one sided continuity at the ends. **Question:** Continuity at a point checks one input at a time and needs inputs on both sides. What does it mean for a function to be continuous across a whole stretch of inputs, including the ends of that stretch, and which familiar functions pass that test everywhere they are defined? ## Why this exists continuity-at-a-point asks whether the limit at $a$ exists and equals $f(a)$, one input at a time. The intermediate value theorem and the extreme-value-theorem ask for more: continuity at every input of an interval, most often a closed interval $[a, b]$, ends included. The ends cause the trouble. The function $\sqrt{x}$ is defined only for $x \ge 0$, so at $0$ there are no inputs to the left, $\lim_{x \to 0} \sqrt{x}$ does not exist, and continuity at a point cannot be applied. Yet the graph starts at the origin and leaves it without a break. This lesson adds **one-sided continuity** to handle ends, defines continuity on every kind of interval, and lists the familiar functions that are continuous wherever they are defined, with the ways of combining them that keep continuity. ## Definition A function is **continuous from the right at $a$** when $$\lim_{x \to a^+} f(x) = f(a)$$ and **continuous from the left at $a$** when $$\lim_{x \to a^-} f(x) = f(a)$$ Each carries three conditions, as before: $f(a)$ is defined, the one-sided limit exists, and the two are equal. Let $a < b$. - $f$ is **continuous on the open interval $(a, b)$** when it is continuous at every number in $(a, b)$. The same wording covers $(a, \infty)$, $(-\infty, b)$ and $(-\infty, \infty)$. - $f$ is **continuous on the closed interval $[a, b]$** when it is continuous on $(a, b)$, continuous from the right at $a$, and continuous from the left at $b$. - $f$ is **continuous on $[a, b)$** when it is continuous on $(a, b)$ and from the right at $a$; **on $(a, b]$** when it is continuous on $(a, b)$ and from the left at $b$. Rays such as $[a, \infty)$ have one end to check. ### [Convergence of sequences](https://math.wiki/sequence-limit-laws/) **Summary:** Sums, products and quotients of convergent sequences behave as expected. **Question:** A list of numbers can creep toward a value, swing back and forth forever, or run off without bound. What does it mean, exactly, for the terms to settle on one number, and how can we prove that they settle when there is no formula to compute with? ## Why this exists A sequence (sequence-basics) is an endless list $a_1, a_2, a_3, \ldots$, given by a formula or by a rule that builds each term from the one before (sequence-from-a-rule). Calculus asks about the far end of the list: do the terms settle on one number? Levani takes $50$ milligrams of a medicine at equal intervals, and by the next dose his body has cleared $40\%$ of what was there. The amounts just after each dose are $50, 80, 98, 108.8, \ldots$, each found from the last by $a_{n+1} = 0.6a_n + 50$. Does the amount level off, and at what? There is no formula for $a_n$ to plug into. Day 1 makes "settles" exact and gives the laws that compute limits of sequences given by formulas, much as limits at infinity did for functions. Day 2 proves that a sequence converges without first knowing its limit: a sequence that only climbs, and never passes some ceiling, must settle. That fact rests on a property of the real numbers themselves. Both tools are needed next, where the sequence being tracked is the running total of an infinite sum. ## Definition In plain words: a sequence **converges** to $L$ if its terms eventually stay as close to $L$ as anyone demands. Not close once, but close from some point on, forever. In symbols: the sequence $\{a_n\}$ converges to the number $L$, written $$\lim_{n \to \infty} a_n = L \qquad \text{or} \qquad a_n \to L$$ if for every number $\varepsilon > 0$ there is a number $N$ such that ### [Convergence tests](https://math.wiki/convergence-tests/) **Summary:** Comparison, ratio, integral, alternating: tools for deciding whether a series settles. **Question:** Most series have no formula for their partial sums. How can we still decide whether adding infinitely many terms settles on a number? ## What this area is about A series $\sum a_n$ converges when its partial sums approach a number. Few series besides geometric ones have a formula for those sums. Convergence tests answer the question from the shape of the terms alone, each one only for series that meet its hypotheses. ## How these fit together **Tests for positive terms.** The integral test and p series compares a series with $\int_m^{\infty} f(x)\,dx$ when $f$ is continuous, positive and decreasing on $[m, \infty)$, and shows $\sum \frac{1}{n^p}$ converges exactly when $p > 1$. The direct and limit comparison tests judge a series of nonnegative terms by a known series it sits below or above, or stays in a fixed positive ratio with. **Signs that change.** Alternating series and their error bound shows that $\sum (-1)^{n+1} b_n$ converges when $b_n$ decreases to $0$, with error at most the first term left out. Absolute and conditional convergence asks whether $\sum \lvert a_n \rvert$ also converges. **Ratio and root.** The ratio and root tests settle series with factorials and powers when $L = \lim \left\lvert \frac{a_{n+1}}{a_n} \right\rvert$ is not $1$, and say nothing when $L = 1$. **Strategy.** Choosing a convergence test reads the shape of a term to pick the test that will work. ## A path through it Start with the integral test, since its $p$ series are the usual benchmarks for comparison. Comparison and alternating series come before absolute and conditional convergence, which comes before the ratio and root tests. Choosing a test comes last. ### [Critical points and local versus absolute extrema](https://math.wiki/critical-points/) **Summary:** Inputs where the derivative is zero or does not exist. **Question:** If a function has a peak or a valley somewhere inside its domain, what must its derivative be doing there, and how does that shrink the search for the largest and smallest values to a handful of points? ## Why this exists The extreme-value-theorem promises that a function continuous on a closed interval $[a, b]$ reaches a largest and a smallest value. It does not say where. There are infinitely many inputs in $[a, b]$, and we cannot test them one at a time. The derivative cuts the search down. At the top of a smooth hill the tangent line is horizontal, so the derivative is $0$. Turned into a theorem, this says that a peak or valley inside the domain can only sit where $f'(c) = 0$ or where $f'(c)$ does not exist. Those inputs, the critical points, are usually few, and they can be found by solving an equation. Algebra read peaks and valleys from graphs and tables (relative-maximum-and-minimum); here the formula locates them, with proof that none is missed. ## Definition Let $f$ have domain $D$. A number $c$ is an **interior point** of $D$ if some open interval $(c - \delta, c + \delta)$ with $\delta > 0$ lies inside $D$. The endpoints of $[a, b]$ are not interior points of $[a, b]$. **Absolute extrema.** $f$ has an **absolute maximum** at $c$ if $f(c) \ge f(x)$ for every $x$ in $D$, and an **absolute minimum** at $c$ if $f(c) \le f(x)$ for every $x$ in $D$. When we restrict $f$ to an interval $I$, "every $x$ in $D$" becomes "every $x$ in $I$". **Local extrema.** $f$ has a **local maximum** at $c$ if $c$ is an interior point of $D$ and $f(c) \ge f(x)$ for every $x$ in some open interval containing $c$. A **local minimum** is the same with $f(c) \le f(x)$. The words **relative maximum** and **relative minimum** mean the same thing. A local extremum only has to beat its neighbors; an absolute extremum has to beat everyone. ### [Curve sketching and connecting f, f' and f''](https://math.wiki/sketch-a-curve-from-sign-charts/) **Summary:** Assembling intercepts, asymptotes, rises, falls and bends into one picture. **Question:** Intercepts and asymptotes give the outline of a graph, but where exactly does it turn, and where does its bend change? And when all we have is the graph of a derivative, what does the function itself look like? ## When you reach for it Algebra already sketches graphs from zeros, asymptotes and end behavior (graph-polynomial-functions, graph-rational-functions); those sketches know where a graph meets the axis but can only guess where it turns and how it bends. Derivatives settle both: $f'$ places every peak and valley exactly, and $f''$ places every change of bend. Reach for this lesson when you want the true shape of a graph from its formula, or when all you have is the graph of $f'$ or of $f''$, or only a list of signs, and you want $f$. ## The procedure ### Day 1: from a formula to a sketch Every piece of the sketch is something already proved; the work is putting them in one picture. The **domain** comes first, because every later chart is split at its gaps. The **intercepts** come from $f(0)$ and the zeros of $f$, when they can be found. **Symmetry** halves the work: if $f(-x) = f(x)$ the graph is symmetric across the $y$-axis, and if $f(-x) = -f(x)$ it is symmetric through the origin. **Asymptotes** come from limits: a vertical asymptote at a point $a$ where a one-sided limit is infinite (vertical-asymptotes-from-limits), and a horizontal asymptote $y = L$ when $\lim_{x \to \infty} f(x) = L$ or $\lim_{x \to -\infty} f(x) = L$ (horizontal-asymptotes-from-limits). Then the two sign charts. The zeros of $f'$, the points where it does not exist, and the domain gaps split the line into pieces on which $f'$ keeps one sign (sign-charts-for-derivatives); $f$ is increasing where $f' > 0$ and decreasing where $f' < 0$, and the first-derivative-test reads each critical point as a peak, a valley or neither, which gives the local extrema. The same splitting for $f''$ gives concave up where $f'' > 0$, concave down where $f'' < 0$, and a point of inflection wherever the concavity changes at a point of the graph (concavity-from-the-second-derivative). Every point found is plotted at its exact height. ### [Definition of the derivative](https://math.wiki/derivative-definition/) **Summary:** The limit of the slope of a secant as the two points merge. **Question:** A rate of change needs two points, yet a speedometer shows a speed at one instant. How can a single point have a slope? ## What this area is about An average rate of change is the slope of a secant line through two points of a graph. Slide the second point toward the first and the secant slopes settle on one number, the slope of the curve at that point. That limit, $$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h},$$ when it exists, is the derivative. Every rule of differentiation is a shortcut for it. ## How these fit together **Definition.** The derivative at a point defines $f'(a)$ as the limit of the difference quotient and reads it as the slope of the tangent line and an instantaneous rate. **Procedure.** Finding a derivative from the definition evaluates the same limit by hand with $x$ left as a letter, giving a formula for $f'(x)$ at every point at once. **The derivative as a function.** The derivative as a function treats $f'$ as a function with its own domain and graph, sketched from a graph of $f$ or estimated from a table. ## A path through it The derivative at a point comes first. Finding a derivative from the definition comes next, and the derivative as a function last, since it builds on the formulas the other two produce. Finish here and the shortcuts begin: the power rule, the product rule and the second derivative all rest on this definition. ### [Derivatives](https://math.wiki/derivatives/) **Summary:** The rate of change at an instant, and the rules for finding it. **Question:** A speedometer reads a speed at a single instant. How is a rate of change at one point defined, and how can it be found for any formula without taking a limit every time? ## What this area is about The derivative $f'(a)$ is the limit of secant slopes, the slope of a curve at one point and the instantaneous rate of change there. Computing that limit by hand every time would be slow, so most of this area builds rules that give derivatives of whole families of functions directly. ## How these fit together **Definition and meaning.** Definition of the derivative, an area of its own, defines $f'$ as a limit. The derivative as a rate of change reads its units and sign. Differentiability shows where a continuous graph still has no derivative. **Rules.** The power, constant multiple and sum rules handle polynomials; the product rule, the quotient rule and the chain rule handle products, quotients and compositions. **Families.** Trig, exponential and log functions (an area), inverse functions and inverse trig functions. **Techniques.** Second and higher derivatives differentiate again. Implicit differentiation finds slopes on curves not solved for $y$. Combining rules decides the order for layered expressions. ## A path through it The definition first; the power rule, the rate interpretation and differentiability each build on it. Then the product, quotient and chain rules, in that order. Next the exponential and trig derivatives, then second derivatives, then implicit differentiation, which also uses tangent lines; the logarithm lesson comes after it. Inverse functions, then inverse trig. Combining rules draws on all of them, so it comes last. ### [Derivatives of exponential functions](https://math.wiki/derivative-of-exponential-base-a/) **Summary:** A base other than e brings a factor of the natural log of that base. **Question:** The rules so far differentiate powers of x, where the variable sits in the base. What happens when the variable moves up into the exponent, as in 2 to the x, and why does one base, e, come out as its own derivative? ## Why this exists The power rule differentiates $x^3$, where the base varies and the exponent is fixed. It says nothing about $3^x$, where the base is fixed and the exponent varies, and exponential functions are exactly what grows money, populations and bacteria, and what shrinks a radioactive sample. To know how fast such a quantity is changing at an instant, the derivative of $a^x$ has to be found from the definition of the derivative itself. That calculation turns up something no polynomial does: the derivative of an exponential is the same exponential multiplied by a constant. The constant depends on the base, and exactly one base makes it $1$. That base is the number $e$, and the constant for every other base turns out to be its natural logarithm, $\ln a$. ## Definition Let $a > 0$. Write the difference quotient of $f(x) = a^x$ and use the product of powers rule, $a^{x + h} = a^x a^h$: $$\begin{aligned} \frac{a^{x + h} - a^x}{h} &= \frac{a^x a^h - a^x}{h} \\ &= a^x \cdot \frac{a^h - 1}{h}. \end{aligned}$$ The factor $a^x$ does not involve $h$, so it passes outside the limit by the constant multiple law. What is left is a limit that depends on $a$ alone. Name it $$L(a) = \lim_{h \to 0} \frac{a^h - 1}{h}.$$ It is the derivative of $a^x$ at $x = 0$, since $a^0 = 1$: the slope of $y = a^x$ where it crosses the $y$-axis. **The number $e$** is the base for which this slope is exactly $1$: ### [Derivatives of inverse functions](https://math.wiki/inverse-function-derivative-formula/) **Summary:** The slope of the inverse is the reciprocal of the slope of the original at the partner point. **Question:** If we know how fast a function changes, how fast does its inverse change, even when the inverse has no formula we can write down? ## Why this exists Many useful functions arrive as the inverse of a function we already understand: the natural logarithm undoes $e^x$, arcsine undoes a slice of sine. Others have inverses that exist but have no formula at all. The function $f(x) = x^3 + x + 1$ takes every real value exactly once, so it has an inverse, but finding $f^{-1}(5)$ means solving the cubic $x^3 + x - 4 = 0$. We would still like to know how steep the graph of $f^{-1}$ is. This lesson shows that the slope of an inverse is always one number divided by a slope of $f$, read at the right point. ## Definition Recall from one-to-one-functions that a function has an inverse exactly when it is one-to-one: no output comes from two different inputs. A function that fails this, such as $x^2$, can be given an inverse by restricting its domain to a piece where it is one-to-one. For such an $f$, $$f^{-1}(b) = a \iff f(a) = b.$$ So the point $(a, b)$ lies on the graph of $f$ exactly when $(b, a)$ lies on the graph of $f^{-1}$, and the second graph is the first reflected in the line $y = x$ (graph-of-an-inverse). We call $(a, b)$ and $(b, a)$ **partner points**. The **derivative of the inverse at $b$**, written $(f^{-1})'(b)$, is the slope of the graph of $f^{-1}$ at the point $(b, a)$. Two other objects look similar and are different functions: $\dfrac{1}{f'(x)}$, the reciprocal of $f'$ at the same input, and the inverse of the function $f'$. Neither is $(f^{-1})'$ in general. ### [Derivatives of inverse trigonometric functions](https://math.wiki/derivative-of-inverse-trig-functions/) **Summary:** Arcsine, arccosine and arctangent have algebraic derivatives with no trig left in them. **Question:** Arcsine, arccosine and arctangent are built out of trigonometry. Why do their derivatives have no trigonometry left in them at all? ## Why this exists Odete stands in front of a tall billboard. Far away it looks small; right under the wall she sees it edge-on, and it looks small again. Somewhere between, the sign fills the widest angle of her view. That angle is a difference of two arctangents, and to say how fast it changes as she walks we need the derivative of arctangent. The inverse trigonometric functions turn a ratio back into an angle, so they appear wherever an angle depends on lengths: a camera tracking a moving object, a trapdoor opening, a sight line to a sign. Their derivatives also matter in the other direction. The functions $\dfrac{1}{1 + x^2}$ and $\dfrac{1}{\sqrt{1 - x^2}}$ contain no trigonometry, yet they turn out to be the derivatives of arctangent and arcsine. That fact is what later lets us find areas under these algebraic curves. ## Definition Recall the slices chosen in restricting-the-domain-for-an-inverse: | Function | Means | Domain | Range | |---|---|:-:|:-:| | $y = \arcsin x$ | $\sin y = x$ | $[-1, 1]$ | $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ | | $y = \arccos x$ | $\cos y = x$ | $[-1, 1]$ | $[0, \pi]$ | | $y = \arctan x$ | $\tan y = x$ | all reals | $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ | | $y = \operatorname{arcsec} x$ | $\sec y = x$ | $\lvert x \rvert \ge 1$ | $\left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right]$ | The arcsecant row is a convention, and books differ on it. This lesson uses $\operatorname{arcsec} x = \arccos\dfrac{1}{x}$, which gives the range shown; some books instead take $\left[0, \frac{\pi}{2}\right) \cup \left[\pi, \frac{3\pi}{2}\right)$, and their derivative formula looks different. ### [Derivatives of logarithms and logarithmic differentiation](https://math.wiki/derivative-of-natural-log-derivation/) **Summary:** Differentiating the exponential relationship implicitly gives the result. **Question:** The logarithm undoes the exponential, and the exponential is its own derivative. What is the slope of $y = \ln x$, and how does that one fact let us differentiate long products and powers such as $x^x$, where the variable sits in both the base and the exponent? ## Why this exists The derivative rules so far cover powers of $x$, the exponentials $e^x$ and $a^x$, and the trigonometric functions, with sums, products, quotients and compositions of them. The logarithm is missing, though it is just as basic: $y = \ln x$ means exactly $e^y = x$. The limit definition does not hand over its derivative easily, because $\ln(x + h)$ does not split into anything simpler. Implicit differentiation goes around the difficulty: the equation $e^y = x$ involves only the exponential, whose derivative is known. The answer is surprising. The derivative of a logarithm is not a logarithm or an exponential but the rational function $\frac{1}{x}$. That fact, joined to the laws of logarithms, then reaches functions no earlier rule touches. A product of six factors becomes a sum of six logarithms, and $x^x$, which is neither a power function (its exponent moves) nor an exponential (its base moves), becomes $x \ln x$, a plain product. ## Definition ### Day 1: the derivative of a logarithm For $x > 0$ the **natural logarithm** $\ln x$ is the number $y$ with $e^y = x$. So $\ln\left(e^x\right) = x$ for every real $x$ and $e^{\ln x} = x$ for every $x > 0$. The laws of logarithms (product-law-of-logs) hold for positive $A$ and $B$ and every real $r$: $$\begin{aligned} \ln(AB) &= \ln A + \ln B, \\ \ln\frac{A}{B} &= \ln A - \ln B, \\ \ln\left(A^r\right) &= r \ln A. \end{aligned}$$ ### [Derivatives of parametric curves](https://math.wiki/tangent-lines-to-parametric-curves/) **Summary:** Finding the slope at a parameter value and writing the line through that point. **Question:** A seat on a spinning ride is placed by two formulas of time, one for how far east it is and one for how far north. How steep is its path at a given moment, and which way does the path bend, when the path loops back and crosses itself and is not the graph of any function? ## When you reach for it Achola is studying the path of one seat on a teacup ride. A camera above the platform records where the seat is at each moment, and she fits two formulas to it: $x(t)$ for the distance east of a mark on the platform and $y(t)$ for the distance north, both in meters, with $t$ in seconds. That is a curve given by a parameter, as in parameter-and-orientation. The path loops, so it fails the vertical line test and is not the graph of any function $y = f(x)$. Even when the parameter can be removed (eliminate-the-parameter), the resulting equation is often not a function and is harder to differentiate than the two formulas themselves. Yet steepness is a local idea. At a single moment, a small piece of the path near the seat usually is the graph of a function, and that piece has a slope. This lesson finds that slope straight from $x(t)$ and $y(t)$, writes tangent lines, finds where the path is level or upright, and then finds which way the path bends. Reach for it whenever a curve is described by a parameter and you need its slope, a tangent line, or its concavity at a given parameter value. ## The procedure ### Day 1: the slope and the tangent line Suppose $x(t)$ and $y(t)$ have continuous derivatives, and at $t = t_0$ the horizontal rate $\frac{dx}{dt}$ is not $0$. Then $\frac{dx}{dt}$ keeps one sign for $t$ near $t_0$, so $x$ is strictly increasing or strictly decreasing there and each $x$ near $x(t_0)$ comes from exactly one $t$. On that small piece the curve is the graph of a function, $y = F(x)$, and $y(t) = F\big(x(t)\big)$. That $F$ is differentiable is the one-variable inverse function theorem, stated here without proof. The chain rule in the form of chain-rule-in-leibniz-notation now gives $\frac{dy}{dt} = \frac{dy}{dx}\cdot\frac{dx}{dt}$, and dividing by the nonzero $\frac{dx}{dt}$: ### [Derivatives of the trigonometric functions](https://math.wiki/derivative-of-the-other-trig-functions/) **Summary:** Each one follows from sine and cosine through the quotient rule. **Question:** Sine and cosine describe everything that goes around or back and forth. How fast are they changing at each moment, and what does that give for tangent, secant, cosecant and cotangent? ## Why this exists A point on a turning flywheel, the height of a tide, the swing of a weight on a spring: each repeats, and each is modeled by sine and cosine. Asking how fast such a thing is moving at one instant is asking for the derivative of $\sin x$ or $\cos x$. The power rule and the exponential rules say nothing about these functions, and computing a limit of difference quotients at every new point is no way to work. This lesson finds both derivatives once, from the definition of the derivative. The other four trigonometric functions are quotients of sine and cosine, so the quotient rule then gives theirs with no new limits at all. The lesson runs over two days: sine and cosine first, then the other four and their uses. ## Definition With $x$ measured in radians, the derivatives of the six trigonometric functions are $$\begin{aligned} \frac{d}{dx}\big[\sin x\big] &= \cos x \\ \frac{d}{dx}\big[\cos x\big] &= -\sin x \\ \frac{d}{dx}\big[\tan x\big] &= \sec^2 x \\ \frac{d}{dx}\big[\cot x\big] &= -\csc^2 x \\ \frac{d}{dx}\big[\sec x\big] &= \sec x \tan x \\ \frac{d}{dx}\big[\csc x\big] &= -\csc x \cot x \end{aligned}$$ Each holds at every $x$ where the function itself is defined. Every function starting with "co" has a minus sign in its derivative, and each pair of lines matches a function with its "co" partner: swap sine for cosine, tangent for cotangent, secant for cosecant, and put a minus sign in front. ### [Derivatives of trig, exponential and log functions](https://math.wiki/derivatives-of-trig-exp-log/) **Summary:** Sine to cosine, e to itself, log to one over x, and where each comes from. **Question:** The power rule handles $x^3$. What happens when the variable moves into an exponent, inside a sine, or inside a logarithm? ## What this area is about Growth, decay and anything that repeats are modeled by exponentials and trigonometric functions, and logarithms undo exponentials. None of these is a polynomial, so the power rule says nothing about them. This area finds their derivatives once, from the definition and the earlier rules, so they can be used like any other rule afterward. ## How these fit together All three are results, each a family of formulas with the argument behind it. **Exponentials.** Derivatives of exponential functions shows that $\frac{d}{dx}\big[a^x\big] = a^x \ln a$ for $a > 0$, so that $e^x$ is its own derivative. **Trigonometric functions.** Derivatives of the trigonometric functions finds $\frac{d}{dx}\big[\sin x\big] = \cos x$ and $\frac{d}{dx}\big[\cos x\big] = -\sin x$, with $x$ in radians, then gets the other four from the quotient rule. **Logarithms.** Derivatives of logarithms and logarithmic differentiation differentiates $e^y = x$ implicitly to get $\frac{d}{dx}\big[\ln x\big] = \frac{1}{x}$ for $x > 0$, then uses the laws of logarithms to differentiate long products and powers such as $x^x$. ## A path through it No child formally requires another, but the lessons are written in the order exponentials, trigonometric functions, logarithms, and each refers back to the ones before. The trigonometric derivatives also need the limit $\frac{\sin x}{x} \to 1$ and the quotient rule; logarithms lean on $e^x$ and implicit differentiation. ### [Differentiability: corners, cusps and piecewise functions](https://math.wiki/corners-cusps-and-vertical-tangents/) **Summary:** The three shapes where a continuous graph still has no single tangent slope. **Question:** A graph can be unbroken at a point and still have no slope there. Which shapes cause that, and how do you join pieces of a function so that it has a slope at every join? ## Why this exists derivative-at-a-point defined $f'(a)$ as the limit of the difference quotient and met one place where the limit fails: $|x|$ at $0$, a corner. Almost every result ahead begins "suppose $f$ is differentiable", so we need to know exactly where a derivative exists, what the graph looks like where it does not, and how that connects to continuity. Piecewise models of paths also need a slope at every join. Day 1 splits the derivative into a left half and a right half, sorts the ways a continuous graph can lack a derivative into three shapes, and proves that a derivative forces continuity. Day 2 turns this into a test for piecewise functions and uses it to choose parameters so that pieces join smoothly. ## Definition **One-sided derivatives.** The **left-hand derivative** and **right-hand derivative** of $f$ at $a$ are $$f'_-(a) = \lim_{h \to 0^-} \frac{f(a + h) - f(a)}{h}$$ $$f'_+(a) = \lim_{h \to 0^+} \frac{f(a + h) - f(a)}{h}$$ each provided the limit exists as a finite number. The left-hand derivative is the limit of the slopes of secant lines from $P(a, f(a))$ to points of the graph on its left; the right-hand derivative uses points on its right. **Differentiable.** A function is **differentiable at $a$** when $f'(a)$ exists (derivative-definition). It is differentiable on an open interval when it is differentiable at every point of it, and differentiable on a closed interval $[a, b]$ when it is differentiable on $(a, b)$ and $f'_+(a)$ and $f'_-(b)$ exist, the same pattern as continuity-on-an-interval. ### [Differential equations](https://math.wiki/differential-equations/) **Summary:** Equations about a rate, and finding the function that obeys them. **Question:** Many laws say only how fast something changes. How do you turn such a law into an equation, and what can you learn about the function that obeys it? ## What this area is about A differential equation relates an unknown function to its own derivatives, as $\frac{dy}{dt} = ky$ does. Its solutions are functions, not numbers. Some can be found by integrating; for the rest, pictures and step-by-step estimates still show how the solutions behave. ## How these fit together **Setting up.** Writing a differential equation turns a sentence about a rate into an equation, and differential equations and their solutions tests a proposed solution by substituting it. **Seeing and estimating.** Slope fields draw the slope the equation gives at each point. Euler's method follows those slopes in small steps to estimate a solution. **Solving.** Separable equations are undone by integrating both sides, then fixing the constant from a starting value. **Models.** Exponential growth and decay solves $y' = ky$ and Newton's law of cooling, $T' = k(T - A)$, and logistic growth adds a ceiling the population approaches. **Beyond a first course.** Three college areas, not yet written: first-order equations (slope fields, separable and linear equations, and when solutions exist), second-order linear equations (constant-coefficient equations, forcing, and oscillation) and systems and transforms (systems, phase portraits, and the Laplace and Fourier methods). ## A path through it Writing an equation, then checking a solution, then slope fields, which Euler's method and separable equations both need. Exponential models follow separable equations, and logistic growth follows exponential models, also using partial fractions and concavity. Euler's method can come last, since its examples reuse those models. ### [Differential equations and their solutions](https://math.wiki/verifying-a-solution-to-a-differential-equation/) **Summary:** Substituting a candidate function and its derivatives to check the equation holds. **Question:** An equation says how fast something changes, not what it is. What does it mean for a function to solve such an equation, and how can a proposed answer be tested without solving anything? ## When you reach for it A **differential equation** is an equation that involves an unknown function and one or more of its derivatives. Three examples: $$\begin{aligned} &xy' = 2y, \qquad y'' + 4y = 0, \\ &\frac{dT}{dt} = -0.1(T - 20). \end{aligned}$$ In each, the unknown is not a number but a function: $y$ as a function of $x$, $y$ as a function of $t$, the temperature $T$ as a function of time $t$. Equations like these are how science states its laws, because a law usually says how fast something changes (a rate, with units and a sign) rather than what it equals. Finding the unknown function can be hard, and later lessons build methods for it. Checking a proposed function is never hard: differentiate it, substitute, and see whether the equation becomes true. That check is the move of this lesson. Reach for it when a formula is offered and must be tested, when a family of solutions is known and the one member that starts at a given value is wanted, and when a form such as $e^{rx}$ is guessed and the constants that make it work are wanted. ## The procedure ### Day 1: equations, order and solutions The **order** of a differential equation is the order of the highest derivative that appears in it. | Equation | Highest derivative | Order | |---|:-:|:-:| | $xy' = 2y$ | $y'$ | $1$ | | $y'' + 4y = 0$ | $y''$ | $2$ | | $\dfrac{dT}{dt} = -0.1(T - 20)$ | $\dfrac{dT}{dt}$ | $1$ | | $y''' = x\,y'$ | $y'''$ | $3$ | ### [Discontinuities and how to remove them](https://math.wiki/jump-discontinuity/) **Summary:** A break where the left and right limits both exist but disagree. **Question:** When a function fails to be continuous at a point, is every failure the same kind of break, and which breaks can be repaired by choosing a better value at that one point? ## Why this exists continuity-at-a-point gives a yes or no answer: either $\lim_{x \to a} f(x)$ exists and equals $f(a)$, or the function is not continuous at $a$. A "no" hides very different pictures. A graph can be missing a single point, step from one height to another, run off to infinity, or swing faster and faster without settling. These breaks behave differently when the function is used. A missing point can be filled in, and the result is continuous; a step or an asymptote cannot be filled in by any choice of value. This lesson sorts the breaks into four kinds using only the two one-sided limits, so each kind follows from limits already computed in two-sided-limit-exists-criterion and vertical-asymptotes-from-limits. Day 1 classifies. Day 2 repairs the breaks that can be repaired, explains why the others cannot be, and follows breaks through compositions and through functions built from pieces. Reading breaks off a picture is done in continuous-and-broken-graphs and telling a hole from an asymptote in vertical-asymptotes; here the test is the limits themselves. ## Definition Let $f$ be defined at every number of some open interval around $a$, except possibly at $a$ itself. Then $f$ has a **discontinuity at $a$** when $f$ is not continuous at $a$. Write $L^- = \lim_{x \to a^-} f(x)$ and $L^+ = \lim_{x \to a^+} f(x)$ when these exist as numbers. The discontinuity is - **removable** when $L^-$ and $L^+$ exist and are equal, so $\lim_{x \to a} f(x)$ exists, but $f(a)$ is undefined or differs from it; - a **jump** when $L^-$ and $L^+$ both exist as numbers and $L^- \ne L^+$. The **size of the jump** is $\lvert L^+ - L^- \rvert$; - **infinite** when at least one one-sided limit is $\infty$ or $-\infty$; - **oscillating** in every remaining case: neither side runs off to infinity, and at least one one-sided limit does not exist because the values keep swinging. The swings may stay bounded, as for $\sin\frac{1}{x}$ at $0$, or grow without bound in both directions, as for $\frac{1}{x}\sin\frac{1}{x}$ at $0$, whose one-sided limits are neither numbers nor $\pm\infty$. ### [Dividing before integrating](https://math.wiki/long-division-before-integrating/) **Summary:** Splitting a top heavy rational integrand into a polynomial plus a proper remainder. **Question:** The table of antiderivatives has a line for one over x and a line for one over one plus x squared, but nothing for a fraction whose top is a bigger polynomial than its bottom. How can such a fraction be turned into pieces the table already knows? ## When you reach for it A **rational function** is a quotient $\dfrac{P(x)}{Q(x)}$ of two polynomials. Call it **top-heavy** when the degree of $P$ is at least the degree of $Q$, and **proper** when the degree of $P$ is less. An integral such as $$\int \frac{x^3 + x^2 + 4x + 9}{x^2 + 4}\,dx$$ matches no line of the table in antiderivatives-of-trig-functions, and splitting it term by term over the denominator, which worked for a one-term denominator such as $x^2$, leaves fractions like $\dfrac{x^3}{x^2 + 4}$ that are no better. What does work is the move from arithmetic: an improper fraction such as $\dfrac{17}{5}$ is $3 + \dfrac{2}{5}$, a whole number plus a proper fraction. Dividing the polynomials does the same thing, and leaves a polynomial, which the power rule integrates, plus a proper remainder fraction, which for the divisors in this lesson is a logarithm or an arctangent. Reach for division whenever the integrand is a top-heavy rational function. A proper integrand needs no division; it is already its own remainder. ## The procedure ### Dividing first polynomial-long-division gives, for polynomials $P$ and $Q$ with $Q$ not the zero polynomial, a quotient $S$ and a remainder $R$ with $$\begin{aligned} &P(x) = Q(x)\,S(x) + R(x), \\ &R = 0 \text{ or } \deg R < \deg Q. \end{aligned}$$ Dividing both sides by $Q(x)$, wherever $Q(x) \ne 0$, $$\frac{P(x)}{Q(x)} = S(x) + \frac{R(x)}{Q(x)}.$$ ### [Euler's method](https://math.wiki/eulers-method-error/) **Summary:** Smaller steps reduce the error, and concavity decides which way the estimate leans. **Question:** Most differential equations have no solution formula. How can the solution's value at a point still be found, how far off is the answer, and in which direction? ## Why this exists A slope field shows the shape of the solutions of $\dfrac{dy}{dx} = f(x, y)$, and separation of variables gives a formula for some of them. Most equations allow neither a formula nor an exact reading: the solutions of $\dfrac{dy}{dx} = x - y^2$, for example, cannot be written with the functions of this course. A model still has to answer "what is $y$ at $x = 2$?" with a number. Euler's method, described by Leonhard Euler in the eighteenth century, turns "follow the segments" into arithmetic. Because it only approximates, it comes with two questions, and they are the point of this lesson: how the error shrinks as the steps shrink, and whether the estimate lies above or below the truth. ## Definition Start at the known point and walk along the tangent line for a short run $h$. The equation gives the slope of that tangent, so the rise is slope times run. At the new point, compute the slope again and take another step. Each step is the tangent-line estimate of equation-of-a-tangent-line, restarted from wherever the last step landed. In symbols, for the initial value problem $\dfrac{dy}{dx} = f(x, y)$, $y(x_0) = y_0$, with step size $h$: $$x_{n+1} = x_n + h, \qquad y_{n+1} = y_n + f(x_n, y_n)\,h.$$ The number $y_n$ is the **Euler estimate** of $y(x_n)$, and $y(x_n) - y_n$ is its **error**. A negative $h$ steps to the left. The work is usually laid out in a table with columns $x_n$, $y_n$, the slope $f(x_n, y_n)$ and the change $f(x_n, y_n)\,h$. ### [Exponential growth and decay models](https://math.wiki/proportional-rate-of-change/) **Summary:** This single condition forces exponential behavior. **Question:** If a quantity changes at a rate proportional to its own size, what must the quantity be, and how do a growing population, a decaying sample, a bank balance and a cooling dish all follow from that one sentence? ## Why this exists Many quantities change at a rate set by how much of them there is. Each cell in a culture divides on its own schedule, so twice the cells make twice the new cells per hour. Each atom of carbon-14 has the same chance of decaying in the next year, so twice the atoms give twice the decays. A bank paying interest continuously adds money in proportion to the balance. modeling-with-differential-equations turned sentences like these into $\dfrac{dy}{dt} = ky$, and a cooling object into $\dfrac{dT}{dt} = k(T - A)$, without solving them. This lesson solves both once and for all. The solutions are exponential functions and nothing else, which is why one family of functions serves populations, radioactive dating, savings and cooling alike: the single assumption about the rate forces it. ## Definition ### Day 1: rate proportional to amount A quantity $y(t)$ **changes at a rate proportional to its size** if there is a constant $k$ with $$\frac{dy}{dt} = ky$$ for every $t$ in an interval. Where $y \neq 0$ the constant is $$k = \frac{1}{y}\,\frac{dy}{dt},$$ the rate of change per unit of amount, called the **relative growth rate** (or continuous growth rate). Its units are one over time: per hour, per year. The model is **exponential growth** when $k > 0$ and **exponential decay** when $k < 0$; when $k = 0$ the quantity is constant. A relative rate of $0.05$ per year is often stated as "$5\%$ a year, compounded continuously". ### [Extrema on a closed interval](https://math.wiki/closed-interval-method/) **Summary:** Comparing values at the critical points and at both endpoints to find the largest and smallest. **Question:** A function continuous on a closed interval is sure to have a largest and a smallest value there. With infinitely many inputs to search, how do we find them? ## When you reach for it The situation is always the same: a function $f$, a closed interval $[a, b]$, and the question "what is the largest value of $f$ on $[a, b]$, the smallest, and where are they reached?" These are the **absolute maximum** and **absolute minimum** of $f$ on $[a, b]$, defined in critical-points. If $f$ is continuous on $[a, b]$, the extreme-value-theorem promises that both exist, but not where. Critical points are the interior inputs where $f'(c) = 0$ or $f'(c)$ does not exist, and every local extremum sits at one. Together the two results turn an infinite search into a comparison of a few numbers. Unlike reading a graph (relative-maximum-and-minimum) or completing the square (maximize-or-minimize), this works for any function we can differentiate, with proof that nothing was missed. ## The procedure The method rests on two facts, both already proved. **The extremes exist.** Since $f$ is continuous on $[a, b]$, the extreme value theorem gives an input $c$ where $f$ takes its absolute maximum value $M$, and an input $d$ where it takes its absolute minimum value $m$. **They can only be in a few places.** Suppose the absolute maximum is at $c$ and $c$ is not an endpoint. Then $c$ lies in the open interval $(a, b)$, and $f(c) \ge f(x)$ for every $x$ in that open interval. That is exactly a local maximum at $c$, so $c$ is a critical point: if $f'(c)$ exists it equals $0$ by Fermat's theorem, and otherwise $f'(c)$ does not exist. The same holds for the minimum. So the absolute extremes are at the endpoints or at critical points in $(a, b)$, as critical-points showed. ### [Finding a derivative from the definition](https://math.wiki/derivative-from-first-principles/) **Summary:** Evaluating the limit of the difference quotient by hand for a given function. **Question:** The difference quotient gives the slope of every secant at once. If the width shrinks to nothing with x left as a letter, can one limit give the slope at every point at once? ## When you reach for it The derivative at a point is one limit for one number: pick $a$, take the limit of the secant slopes, get the slope of the curve at $a$. Ask about a second point and the whole limit is done again. Doing the limit once with $x$ left as a letter gives every slope at once. The result is a new function, the **derivative** $f'(x)$, whose value at each input is the slope of the graph of $f$ there. Reach for the definition when you have a rule and no differentiation rules yet, when you want to see where a rule comes from, or when a function is new enough that no rule covers it. The raw material is the difference quotient $\frac{f(x + h) - f(x)}{h}$, the slope of the secant from $x$ to $x + h$, which you already know how to simplify. The one new step is the limit. ## The procedure The derivative of $f$ is $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h},$$ at every $x$ where this limit exists. In Leibniz notation the same function is written $\dfrac{d}{dx}\big[f(x)\big]$, or $\dfrac{dy}{dx}$ when $y = f(x)$. Inside the limit $x$ is held fixed and only $h$ moves, so $x$ is treated like a constant throughout. Evaluating the limit by hand takes three moves. **Simplify the quotient for $h \ne 0$.** Substituting $h = 0$ straight away gives $\frac{f(x) - f(x)}{0} = \frac{0}{0}$, every time, because the two points have merged. As in limits by algebra, $\frac{0}{0}$ says a factor of $h$ is hiding in the numerator. The shape of $f$ says how to expose it: ### [Graphing an accumulation function](https://math.wiki/extrema-of-an-accumulation-function/) **Summary:** The accumulated amount peaks where the rate crosses from positive to negative. **Question:** You know how fast an amount is changing at every moment. When is the amount largest, how large is it, and what does its whole graph look like? ## When you reach for it Rauf, Togrul and Lale work the doors of an indoor arena. Turnstiles count people coming in, other counters count people going out, and the difference is a **net rate**: people per hour entering minus people per hour leaving. The rate is what they can measure; the number of people inside is what they care about. If $r(t)$ is the net rate and $P_0$ people were inside at time $0$, the number inside at time $x$ is $$P(x) = P_0 + \int_0^x r(t)\,dt.$$ You reach for this move whenever you hold the graph or formula of a rate and want the shape of the amount it builds: where it rises, falls and bends, where it peaks and how high. The graph you hold is the derivative of the function you want, so the reading is the one from sketch-a-curve-from-sign-charts, where $f$ was read from a graph of $f'$; what is new is that heights now come exactly, from area. The crowd is largest when the net rate crosses from positive to negative. ## The procedure ### Day 1: reading g from the graph of f Throughout, $f$ is continuous on $[a, b]$ and $$g(x) = \int_a^x f(t)\,dt, \qquad a \le x \le b.$$ **Heights come from area.** First, $g(a) = 0$. For any two inputs, additivity-over-intervals gives $$g(x_2) - g(x_1) = \int_{x_1}^{x_2} f(t)\,dt,$$ the signed area between the graph of $f$ and the $t$-axis from $x_1$ to $x_2$ (signed-area). Area above the axis raises $g$; area below lowers it. When $f$ is built from segments and circular arcs, every piece is a triangle, a trapezoid, a rectangle or part of a disk, so every value of $g$ is exact. ### [Implicit differentiation](https://math.wiki/implicit-differentiation-idea/) **Summary:** Treating y as an unnamed function of x so the chain rule applies to every y term. **Question:** An equation such as $x^3 + y^3 = 6xy$ draws a smooth curve, but it cannot be solved neatly for $y$. How can we find the slope of its tangent lines anyway? ## Why this exists Every derivative so far began with a formula $y = f(x)$. Many curves arrive instead as one equation in $x$ and $y$ together. The circle $x^2 + y^2 = 225$ fails the vertical-line-test, so no single formula $y = f(x)$ describes it. The curve $x^3 + y^3 = 6xy$ in Figure 2 is worse: solving it for $y$ means solving a cubic equation for each $x$. Yet both curves are smooth, and zooming in on any ordinary point shows something that looks like a line. Implicit differentiation finds the slope of that line without ever solving for $y$. Day 1 sets up the method and finds tangent lines. Day 2 adds horizontal and vertical tangents, second derivatives, the power rule for rational exponents, and the method's limits. ## Definition ### Day 1: y as an unnamed function of x An equation in $x$ and $y$ defines $y$ **implicitly** as a function of $x$ near a point $(a, b)$ of its graph when there is a differentiable function $f$ on an open interval around $a$ with $f(a) = b$, such that near $(a, b)$ the graph of the equation is exactly the graph of $y = f(x)$. Near $(9, 12)$ the circle above is the graph of $y = \sqrt{225 - x^2}$, and near $(9, -12)$ of $y = -\sqrt{225 - x^2}$. Near $(15, 0)$ it is the graph of no function of $x$ at all, since every vertical line just left of $x = 15$ meets it twice. **Implicit differentiation** finds $\dfrac{dy}{dx}$ without a formula for $f$: keep writing $y$ for the unknown $f(x)$, differentiate both sides of the equation with respect to $x$, and solve the result for $\dfrac{dy}{dx}$. Terms in $x$ alone are differentiated as usual. A term built from $y$ is a function of the function $f(x)$, so the chain rule attaches a factor $\dfrac{dy}{dx}$: ### [Improper integrals](https://math.wiki/improper-integral-infinite-limit/) **Summary:** Integrating out to infinity by taking a limit of ordinary integrals. **Question:** An oil well pumps a little less every year but never quite stops. Can the oil it will ever pump be a finite amount, and how do you add up a rate over a time that never ends? ## Why this exists A definite integral $\int_a^b f(x)\,dx$ is built on a closed, bounded interval: cut $[a, b]$ into finitely many pieces of finite width, add, take a limit. Many honest questions run past any bound. An oil well whose output declines but never stops: how much will it pump in all? A payment that arrives every year forever: what is it worth today? The area under $y = \frac{1}{x^2}$ to the right of $x = 1$: is it a number at all? None of these is a definite integral, because the interval $[a, \infty)$ cannot be cut into finitely many pieces of finite width. What *can* be done is to integrate out to a finite point $b$, which is an ordinary integral, and then ask what happens as $b$ moves off to infinity. That is a limit at infinity, and the whole idea of this lesson is to put the two together. ## Definition ### Day 1: integrals over an unbounded interval Let $f$ be continuous on $[a, \infty)$. The **improper integral** of $f$ from $a$ to infinity is $$\int_a^{\infty} f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx.$$ If the limit exists as a finite number, the integral **converges** and its value is that number. Otherwise it **diverges**: the limit may be $\infty$ or $-\infty$ (which the notation records but which is not a number), or there may be no limit at all, as with $\int_0^b \cos x\,dx = \sin b$, which keeps swinging between $-1$ and $1$. For $f$ continuous on $(-\infty, b]$, the left tail is defined the same way: ### [Improper integrals with an infinite integrand](https://math.wiki/improper-integral-unbounded-integrand/) **Summary:** Handling an integral where the function blows up at an endpoint. **Question:** A water clock drains through a hole in its floor, and the last of the water trickles out ever more slowly. The time to empty it is an integral whose integrand runs off to infinity at the bottom. Can such an integral still be a finite number? ## Why this exists The fundamental theorem computes $\int_a^b f(x)\,dx$ only when $f$ is continuous on the closed interval $[a, b]$, and in particular bounded there. Many natural integrands are not. A water clock is a vessel that drains through a small hole in its floor; the time it takes to empty turns out to be $\int_0^H \frac{c}{\sqrt{h}}\,dh$, and $\frac{c}{\sqrt{h}}$ runs off to infinity at the bottom, where $h = 0$. The integral $\int_0^1 \ln x\,dx$ has the same trouble at $0$. Each has a vertical asymptote at a finite point, and near it the Riemann sums have no limit, because one rectangle can be made as tall as you like. The cure is the one that improper-integral-infinite-limit used for an infinite interval: stay a little way back from the trouble, where the integral is ordinary, and then take a limit as you creep up to it. ## Definition Let $f$ be continuous on $(a, b]$ and unbounded near $a$. The **improper integral** of $f$ from $a$ to $b$ is $$\int_a^b f(x)\,dx = \lim_{t \to a^+} \int_t^b f(x)\,dx.$$ If $f$ is continuous on $[a, b)$ and unbounded near $b$, then $\int_a^b f(x)\,dx = \lim_{t \to b^-} \int_a^t f(x)\,dx$. As before, the integral **converges** if the limit exists as a finite number and **diverges** otherwise. If $f$ blows up at a point $c$ strictly inside $(a, b)$, split there: $$\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx.$$ The integral converges **only when both pieces converge**, each as its own one-sided limit, and then its value is the sum. If several bad points occur, or a bad point and an infinite limit of integration, split so that each piece has exactly one source of trouble at one end. ### [Increasing and decreasing from the derivative](https://math.wiki/sign-charts-for-derivatives/) **Summary:** Splitting the line at critical points and testing the sign of the derivative in each piece. **Question:** The derivative gives the slope at a single point. How can it tell us, over a whole stretch of inputs, that a function is going up or going down? ## When you reach for it A graph read by eye shows where a function rises and falls (increasing-and-decreasing-intervals did this from graphs and tables), but an eye cannot prove that a curve never turns back between two plotted points. The derivative can. If $f'(x) > 0$ at every point of an interval, $f$ is increasing on that whole interval, with no exceptions hiding between sample points. Reach for this whenever the question is "where does $f$ go up, and where does it go down?" for a function given by a formula or by the graph of its derivative. The answer is a list of intervals, and the tool for finding them is a **sign chart**: the line cut at a few special inputs, with the sign of $f'$ recorded on each piece. ## The procedure ### Day 1: the test and the sign chart **Words.** A function $f$ is **increasing** on an interval $I$ if $x_1 < x_2$ in $I$ always gives $f(x_1) < f(x_2)$, and **decreasing** on $I$ if $x_1 < x_2$ in $I$ always gives $f(x_1) > f(x_2)$. Both are statements about an interval, never about a single point. **The test.** Let $f$ be continuous on an interval $I$. - If $f'(x) > 0$ at every point inside $I$ (every point but the ends), then $f$ is increasing on $I$. - If $f'(x) < 0$ at every point inside $I$, then $f$ is decreasing on $I$. This was proved in rolles-theorem from the mean value theorem. In one line: for $x_1 < x_2$ in $I$ there is a $c$ between them with $f(x_2) - f(x_1) = f'(c)(x_2 - x_1)$, and the sign of the right side is the sign of $f'(c)$. The proof needs $f'(c)$ only at points strictly between $x_1$ and $x_2$, which is why nothing is asked of $f'$ at the ends of $I$. ### [Infinite limits and vertical asymptotes](https://math.wiki/vertical-asymptotes-from-limits/) **Summary:** A vertical asymptote is a place where a one sided limit runs off to infinity. **Question:** Near some inputs a function's outputs grow past every number anyone names. The limit is then not a number at all, so what exactly is being claimed when we write that it equals infinity, and how does that claim locate a vertical asymptote, even for functions that are not fractions of polynomials? ## Why this exists two-sided-limit-exists-criterion already met a side that runs away: the outputs of $\frac{1}{x + 1}$ passed $100$, then $1000$, as $x$ closed in on $-1$ from the right. "The limit does not exist" is true there but says too little: it does not separate outputs that grow past every bound from outputs that swing back and forth. The notation $\lim_{x \to a} f(x) = \infty$ records that extra information. This lesson makes it precise in the spirit of formal-limit-definition, with a bound $M$ in place of a tolerance $\varepsilon$, and then defines a vertical asymptote by one-sided limits, for any function. Reading asymptotes of rational functions off their factors is the Algebra 2 lesson vertical-asymptotes. ## Definition **Infinite limits.** In words: $\lim_{x \to a} f(x) = \infty$ means **the outputs can be made larger than any number anyone names, just by taking $x$ close enough to $a$**, on both sides, with $x \ne a$. In symbols: for every number $M > 0$ there is a $\delta > 0$ such that $$\begin{aligned} &f(x) > M \\ &\text{whenever } 0 < |x - a| < \delta. \end{aligned}$$ The other versions change only the inequality or the side: - $\lim_{x \to a} f(x) = -\infty$ means: for every $M > 0$ there is a $\delta > 0$ with $f(x) < -M$ whenever $0 < |x - a| < \delta$. - For $\lim_{x \to a^+}$, replace $0 < |x - a| < \delta$ by $a < x < a + \delta$; for $\lim_{x \to a^-}$, by $a - \delta < x < a$. ### [Infinite series and partial sums](https://math.wiki/partial-sums-of-a-series/) **Summary:** The running totals whose limit defines the sum of a series. **Question:** Adding infinitely many numbers can never be finished, so what could the sum of an infinite list possibly mean, and why do terms that shrink to zero sometimes add up to a number and sometimes grow past every bound? ## Why this exists Zviad stacks identical books at the edge of a table, one on top of another, and wants the top book to stick out as far as possible past the edge without the stack tipping. With one book the best he can do is half a book length. With more books he can do better, and the best stack (Figure 1) puts the overhangs, from the top down, at $\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \ldots$ of a book length. With $n$ books the top book reaches $$\frac{1}{2}\left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}\right)$$ book lengths past the edge. Can the overhang be made as large as he likes, given enough books? The question is about the sum $1 + \frac{1}{2} + \frac{1}{3} + \cdots$ continued forever, and an infinite sum cannot be computed by adding. It needs a definition. The definition turns an infinite sum into something already understood: the limit of a sequence (sequence-limit-laws). Add the first $n$ terms, as for the finite sums in partial-sum, and ask what those running totals do as $n$ grows. Day 1 sets up this definition, the laws it inherits from limits of sequences, and the telescoping series, whose running totals collapse to a short formula. Day 2 answers Zviad: the harmonic series grows past every bound, so the overhang can be as large as he likes, but it grows so slowly that a meter of overhang takes more than twelve thousand books. ## Definition In plain words: a series is an infinite sum of terms. Its partial sums are the running totals, and the series converges if those running totals settle on one number, which is then called its sum. ### [Integrals](https://math.wiki/integrals/) **Summary:** Adding up infinitely many infinitely thin pieces, and the surprise that undoes the derivative. **Question:** Adding up infinitely many thin pieces sounds impossible to finish. How does the integral do it exactly, and why does running a derivative backward turn out to be the key? ## What this area is about Two questions meet here. Which function has a given derivative? And what exact total do thinner and thinner rectangles approach? The fundamental theorem of calculus says they are one question: for $f$ continuous on $[a, b]$ with antiderivative $F$, $\int_a^b f(x)\,dx = F(b) - F(a)$. ## How these fit together **Antiderivatives.** Antiderivatives of powers and of trig, exponential and log functions run derivative rules backward. **The definite integral.** Riemann sums estimate with rectangles, the definite integral is their limit, and the fundamental theorem links it to antiderivatives (the last two are areas of their own). **Techniques.** u-substitution and integration by parts reverse the chain and product rules. Trig integrals, completing the square, trig substitution, dividing first and partial fractions reshape an integrand into known forms. Choosing a technique picks among them. **Estimates and limits.** Trapezoids and Simpson's rule join the heights at the cut points with segments or parabolas, and usually beat left and right sums. Improper integrals reach to infinity, and their unbounded cousins handle an integrand that blows up. ## A path through it Power antiderivatives, then trig, exponential and log ones. Separately, Riemann sums, then the definite integral; the fundamental theorem needs both strands. Then substitution, parts and trig integrals; completing the square before trig substitution; long division before partial fractions. The trapezoidal rule, then improper integrals over infinite intervals before unbounded integrands. Choosing a technique refers to all of them, so it is last. ### [Integration by partial fractions](https://math.wiki/partial-fractions-repeated-and-quadratic-factors/) **Summary:** Splitting a rational function into simple fractions, from distinct linear factors to repeated and irreducible quadratic ones, and integrating each piece. **Question:** No rule integrates five x plus one over x squared plus x minus two as it stands. How do you take a fraction like that apart into pieces you already know how to integrate? ## When you reach for it Adding fractions runs one way: $\dfrac{3}{x + 2} + \dfrac{2}{x - 1}$ combines over the common denominator into $\dfrac{5x + 1}{x^2 + x - 2}$. Integration needs the reverse. The combined fraction matches no rule, but each piece is a constant over a linear factor, whose antiderivative is a logarithm. Partial fractions is the method that recovers the pieces from the combined fraction. Reach for it when the integrand is a rational function (a polynomial over a polynomial) whose denominator factors, and no simpler move such as u-substitution fits. Day 1 covers denominators that split into distinct linear factors. Day 2 adds repeated linear factors and quadratic factors with no real roots. ## The procedure ### Day 1: distinct linear factors **Proper first.** A rational function $\dfrac{P(x)}{Q(x)}$ is *proper* when the degree of $P$ is less than the degree of $Q$. If it is not, divide first: the quotient is a polynomial, integrated term by term, and only the proper remainder is split. **The decomposition.** Suppose $Q(x) = c(x - r_1)(x - r_2)\cdots(x - r_n)$ with the $r_i$ distinct real numbers, and $P$ has degree less than $n$. Then there are unique constants $A_1, \dots, A_n$ with $$\begin{aligned} \frac{P(x)}{Q(x)} &= \frac{A_1}{x - r_1} + \frac{A_2}{x - r_2} \\ &\quad + \cdots + \frac{A_n}{x - r_n}. \end{aligned}$$ A factor $ax + b$ is $a\left(x + \frac{b}{a}\right)$, so the statement covers it too, with $\dfrac{A}{ax + b}$ as its piece. ### [Integration by parts](https://math.wiki/integration-by-parts-derivation/) **Summary:** Integrating both sides of the product rule and rearranging. **Question:** The product rule says how to differentiate $f(x)g(x)$. Read backward, what does it say about integrating a product such as $x\cos 5x$, or a lone $\ln x$, that no substitution can reach? ## Why this exists Substitution reverses the chain rule, so it handles a composite function multiplied by the derivative of its inside. Many integrands do not have that shape. In $\int x\cos 5x\,dx$ the factor $x$ is not a constant times the derivative of $5x$. In $\int \ln x\,dx$ there is only one factor, and no basic rule has $\ln x$ as a derivative. The rule left to reverse is the product rule. Reversing it does not finish an integral in one stroke: it trades $\int f g'\,dx$ for $\int f' g\,dx$, and the skill is to arrange the trade so the new integral is easier. That trade, made once, made repeatedly, or made twice in a loop that returns to the start, handles polynomials times exponentials or sines, logarithms, inverse trigonometric functions, and an exponential times a sine. ## Definition ### Day 1: the formula and the choice of parts Let $u = f(x)$ and $v = g(x)$ have continuous derivatives on an interval, and write $du = f'(x)\,dx$ and $dv = g'(x)\,dx$. **Integration by parts** is the identity $$\int f(x)g'(x)\,dx = f(x)g(x) - \int f'(x)g(x)\,dx,$$ written shortly as $\int u\,dv = uv - \int v\,du$. On $[a, b]$ it reads $\int_a^b u\,dv = \big[uv\big]_a^b - \int_a^b v\,du$. To use it, the integrand is split into two **parts**: $u$, which will be differentiated, and $dv$, which will be integrated (the $dx$ always goes with $dv$). The split is a choice, and a good one makes $v$ findable and $\int v\,du$ simpler than the integral we began with. So $u$ is a factor that differentiation simplifies: a polynomial drops a degree, and $\ln x$, $\arctan x$ and $\arcsin x$ become the algebraic $\frac{1}{x}$, $\frac{1}{1 + x^2}$ and $\frac{1}{\sqrt{1 - x^2}}$ (derivative-of-natural-log-derivation). And $dv$ is a factor whose antiderivative is known and no worse than itself: $e^{kx}$, $\sin kx$, $\cos kx$, a power of $x$, or plain $dx$. A logarithm or an inverse trigonometric function is nearly always $u$, since its antiderivative is the very thing being sought. ### [L'Hopital's rule](https://math.wiki/lhopital-zero-over-zero/) **Summary:** Replacing the ratio of two vanishing functions with the ratio of their derivatives. **Question:** Top and bottom of a quotient both shrink to zero, or both grow without bound, and no algebra cancels them. Can the rates at which they move decide the limit instead? ## When you reach for it Substitution into $\dfrac{\ln x}{x^3 - 1}$ at $x = 1$ gives $\dfrac{0}{0}$. That is not an answer: a quotient whose top and bottom both go to $0$ can approach any number, or none, depending on how fast each part shrinks. limits-by-factoring settles such a limit by canceling a common factor, and limit-of-sin-x-over-x settles $\dfrac{\sin x}{x}$ with a squeeze. Neither helps here: there is no factor to cancel between a logarithm and a polynomial. The limits $\dfrac{0}{0}$ and $\dfrac{\infty}{\infty}$, and the shapes $0 \cdot \infty$, $\infty - \infty$, $1^{\infty}$, $0^0$ and $\infty^0$ that can be rewritten as one of them, are called **indeterminate forms**: knowing the limits of the pieces does not determine the limit of the whole. L'Hopital's rule compares the rates at which top and bottom change, which is what the derivatives measure. ## The procedure ### Day 1: the rule for zero over zero **L'Hopital's rule.** Let $f$ and $g$ be differentiable on an open interval $I$ containing $a$, except possibly at $a$ itself, with $g'(x) \ne 0$ for every $x$ in $I$ other than $a$. Suppose $$\lim_{x \to a} f(x) = 0 \quad \text{and} \quad \lim_{x \to a} g(x) = 0.$$ If $\lim_{x \to a} \dfrac{f'(x)}{g'(x)}$ equals a number $L$, or $\infty$, or $-\infty$, then $\lim_{x \to a} \dfrac{f(x)}{g(x)}$ equals the same thing. The same holds for one-sided limits $x \to a^+$ or $x \to a^-$ (with $I$ an interval on that side), and for $x \to \infty$ or $x \to -\infty$. ### [Limits](https://math.wiki/limits/) **Summary:** What a function is heading toward, even where it never arrives. **Question:** A function can be undefined at a point and still head straight toward one height there. How do you name that height, compute it, and use it to say what a function does near a gap, at infinity, or across an interval? ## What this area is about The limit $\lim_{x \to a} f(x) = L$ says the outputs come as close to $L$ as you like for inputs close enough to $a$, with $x \ne a$. It ignores the value at $a$ itself, which is exactly what derivatives and integrals need: both are limits of quantities that cannot be computed at the point. ## How these fit together **Meaning.** What a limit is reads limits off graphs and tables. One sided limits split the approach into left and right. The epsilon delta definition makes "as close as you like" exact. **Computing.** The limit laws combine known limits. Factoring and rationalizing rewrite a quotient that gives $\frac{0}{0}$. The squeeze theorem traps a function between two others with the same limit, and proves $\lim_{x \to 0} \frac{\sin x}{x} = 1$ for $x$ in radians. **Infinity.** Infinite limits locate vertical asymptotes, limits at infinity horizontal ones, and growth rates rank logarithms, powers and exponentials. **Continuity.** Continuity asks that the limit equal the value. The intermediate value theorem says that if $f$ is continuous on $[a, b]$, it takes every value between $f(a)$ and $f(b)$. ## A path through it Graphs and tables, then one sided limits, then the limit laws. Factoring before the squeeze theorem, which refers back to it. Infinite limits, then limits at infinity. Continuity needs the laws, one sided limits and infinite limits; the intermediate value theorem needs continuity. Growth rates use limits at infinity and continuity. The epsilon delta definition needs only graphs, but it looks back at most of the area, so it fits best last. ### [Limits at infinity and horizontal asymptotes](https://math.wiki/horizontal-asymptotes-from-limits/) **Summary:** A horizontal asymptote is the value a function levels off toward far out on the axis. **Question:** A graph that levels off far to the right seems to settle at some height, but no table of large inputs can prove where. How do you compute that height exactly, and what do you do when the two ends of a graph settle in different places, or follow a slanted line instead of a level one? ## Why this exists limits-from-a-graph read a limit as $x \to \infty$ off a picture or a table: plug in $100$, $1000$, $10{,}000$ and watch the outputs. That is evidence, not proof, and it can mislead when the outputs drift slowly. asymptote-concept and horizontal-asymptotes name the level line a graph settles toward. This lesson makes the idea exact. It defines a limit at infinity with the same "every tolerance" precision as formal-limit-definition, proves the one basic fact everything rests on, $\frac{1}{x^p} \to 0$, and then computes limits at infinity with the limit laws and algebra. That algebra handles rational functions, square roots whose two ends behave differently, differences like $\sqrt{x^2 + 4x} - x$ where both pieces grow without bound, and slanted asymptotes found as limits. The lesson takes two days. Part 1 is the definition, the basic limit, and rational functions (Examples 1 to 4). Part 2 is square roots, differences that need a conjugate, and oblique asymptotes (Examples 5 to 7). ## Definition In plain words: **$f(x)$ has limit $L$ as $x \to \infty$ if the outputs can be kept as close to $L$ as anyone asks just by taking $x$ large enough.** In symbols, $\lim_{x \to \infty} f(x) = L$ means: for every tolerance $t > 0$ there is a number $N$ such that $$\begin{aligned} &|f(x) - L| < t \\ &\text{for every } x > N \text{ in the domain of } f. \end{aligned}$$ The limit as $x \to -\infty$ is the same with "$x > N$" replaced by "$x < N$". The domain of $f$ must contain inputs beyond every $N$ in the direction asked about. ### [Limits by algebra: factoring and rationalizing](https://math.wiki/limits-by-factoring/) **Summary:** Removing a common factor that makes both top and bottom zero so substitution works. **Question:** Substituting the target input into a quotient gives zero over zero. The limit is still a definite number. How do you find it, and why is changing the formula allowed? ## When you reach for it The limit laws turn most limits into arithmetic: for a polynomial $p$, $\lim_{x \to a} p(x) = p(a)$, and for a quotient of polynomials whose denominator is not zero at $a$, the limit is the value of the quotient at $a$. They say nothing when the denominator heads to $0$, and that case is not rare. Every derivative in calculus is a limit of a quotient whose top and bottom both shrink to $0$, so it is the case that matters most. Try the limit $$\lim_{x \to 4} \frac{x^2 - x - 12}{x - 4}.$$ Substituting $x = 4$ gives $\frac{16 - 4 - 12}{4 - 4} = \frac{0}{0}$. That is not an answer and not a statement that the limit fails. It is a report that the quotient laws do not apply, and the formula must be rewritten before they can. This lesson rewrites it in two ways: by **factoring** and canceling a common factor (part 1), and by **rationalizing** with a conjugate when a square root is involved (part 2). The lesson takes two days. Part 1 is the reading of a substitution, why canceling is allowed, and factoring; Examples 1 to 4 go with it. Part 2 is rationalizing; Examples 5 to 7 go with it. ## The procedure ### Substitute first, and read what comes out Substitute $x = a$ into the numerator and the denominator separately. Three things can happen. | Numerator at $a$ | Denominator at $a$ | What it means | |---|---|---| | any number | not $0$ | The limit is the quotient. Done. | | not $0$ | $0$ | No finite limit. Nothing cancels. | | $0$ | $0$ | The form $\frac{0}{0}$. Rewrite, then try again. | ### [Linear approximation and differentials](https://math.wiki/error-estimation-with-differentials/) **Summary:** Turning a small measurement error into an estimated error in a computed quantity. **Question:** A tangent line hugs its curve near the point of contact. How much can that line tell you about nearby values of the function, and how far does a small error in a measurement spread into a quantity computed from it? ## When you reach for it Reach for a linearization when you know a function's value and slope at one convenient input $a$ and want its value at a nearby input that is awkward to compute, such as $\sqrt[3]{66}$ or $e^{0.05}$. Reach for differentials when a quantity is computed from a measurement, the measurement carries an error, and you want to know how large the error in the computed quantity can be. Both rest on one fact from tangent lines: near the point of contact, the tangent line is the line that fits the curve best, closer than any other line through that point. Example 6 there, Hema's cube, already used the tangent line to estimate a volume. This lesson makes the idea a tool: when the estimate is high or low, how fast its error shrinks, and how an error in an input moves through a formula. Reading a derivative in context used "rate times a small change" for one step; here that step gets a name, $dy$, and a measure of how good it is. ## The procedure ### Day 1: the linearization **The line.** For a function $f$ that is differentiable at $a$, the **linearization** (or local linear approximation) of $f$ at $a$ is the tangent line written as a function, $$L(x) = f(a) + f'(a)(x - a).$$ It agrees with $f$ at $a$ and has the same slope there. For $x$ near $a$ we use it in place of $f$: $$f(x) \approx f(a) + f'(a)(x - a).$$ The input $a$ is a choice. Pick it close to the input you care about and with $f(a)$ and $f'(a)$ easy to find exactly: $a = 64$ for a cube root near $66$, $a = 0$ for $e^x$ with $x$ near $0$. The right-hand side then needs only arithmetic. The rules available so far are enough: the power rule for $x^n$ with any rational exponent, $\dfrac{d}{dx}\big[e^{kx}\big] = ke^{kx}$ and $\dfrac{d}{dx}\big[a^x\big] = a^x \ln a$ from derivatives of exponentials, and the product rule. ### [Logistic growth](https://math.wiki/logistic-carrying-capacity/) **Summary:** The ceiling a logistic population approaches but never passes. **Question:** A population grows at a rate proportional to its size while it is small, but the rate falls to zero as the population nears what its surroundings can support. What does that one equation force the population to do, and can it be solved exactly? ## Why this exists The model $\dfrac{dP}{dt} = kP$ of proportional-rate-of-change predicts that a population grows without bound, and no pond can hold infinitely many fish. Food, oxygen and space run short, and each fish then adds fewer young than before. logistic-growth-models read the resulting S-shaped curve from a given formula. This lesson starts one step earlier, from a statement about the rate alone: growth is proportional to size when the population is small and falls off as it nears a ceiling. From that equation, without being handed any formula, calculus finds the ceiling, shows that it attracts every positive solution, locates the moment of fastest growth, and then produces the formula itself. ## Definition ### Day 1: the logistic equation A population $P(t)$ follows the **logistic differential equation** with growth constant $k > 0$ and **carrying capacity** $K > 0$ if $$\frac{dP}{dt} = kP\left(1 - \frac{P}{K}\right).$$ Dividing by $P$ gives the **per-capita growth rate**, the rate of growth per individual: $$\frac{1}{P}\,\frac{dP}{dt} = k\left(1 - \frac{P}{K}\right).$$ It falls in a straight line from $k$ when $P$ is near $0$ to $0$ when $P = K$, and is negative above $K$. For a small population the equation is close to $\dfrac{dP}{dt} = kP$, exponential growth; near $K$ the growth stops. The equation is **autonomous**: the rate depends on $P$ only, not on $t$. As in sketching-a-slope-field, a value $c$ with zero rate gives an **equilibrium solution** $P = c$, and the equilibrium is **stable** if solutions starting near it on both sides approach it, **unstable** if they move away. ### [Making a piecewise function continuous](https://math.wiki/continuity-of-a-piecewise-function/) **Summary:** Choosing a parameter so the two pieces meet at the boundary. **Question:** A function is built from different rules on different intervals, and some of the rules hold an unknown constant. How do you choose the constants so each rule meets the next one where it hands over, and when is that impossible? ## When you reach for it A piecewise function uses one rule on one interval and another rule on the next: a tax bracket, a shipping charge, a water rate. Each rule is usually continuous on its own interval, so the function can break only at the **boundaries**, where one rule hands over to the next. When a rule holds an unknown constant, asking for continuity at the boundaries gives equations for it. The simplest case is in continuous-and-broken-graphs: joining $4 - x^2$ to $kx + 6$ at $x = 2$ gave one linear equation and $k = -3$. This lesson takes harder counts: two constants at two boundaries, a constant inside a square, an exponential or a sine, a piece that reads $\frac{0}{0}$ at its own boundary, and pieces no constant can join. Elnur, Gunay and Aygun work on a city's tiered water rate, where a new rule must join the old ones at the tier boundaries. ## The procedure Let $f$ use the rule $p$ just left of a boundary $c$ and the rule $q$ just right of it, with $f(c)$ given by whichever rule includes $c$, or by a line of its own. **Inside a piece there is nothing to solve.** On an open interval where one rule holds, $f$ agrees with that rule at every nearby input, and continuity at a point looks only at nearby inputs. So $f$ is continuous there exactly where the rule is (continuity-on-an-interval). The constants found must not break a piece from the inside. **At a boundary the three conditions become equations.** By continuity-at-a-point, $f$ is continuous at $c$ when $f(c)$ is defined, $\lim_{x \to c} f(x)$ exists, and the two are equal; by two-sided-limit-exists-criterion, the limit exists when the one-sided limits agree. Each one-sided limit sees only the rule on its side, so continuity at $c$ says ### [Maxima, minima and critical points](https://math.wiki/extrema/) **Summary:** Where the derivative is zero or undefined, and the tests that tell peak from valley. **Question:** A function has infinitely many inputs. How can its derivative point to the few places where a peak or a valley can sit, and tell which is which? ## What this area is about Algebra found highest and lowest points by reading a graph or completing the square. Calculus finds them from the derivative, with proof that none were missed. A smooth peak or valley inside the domain has a level tangent, so the search for extreme values shrinks from infinitely many inputs to a handful. ## How these fit together **Definitions and the key fact.** Critical points and local versus absolute extrema defines the inputs $c$ where $f'(c) = 0$ or $f'(c)$ does not exist, and proves that every local extremum sits at one. **Results.** Increasing and decreasing from the derivative turns the sign of $f'$ on an interval into rising or falling, using a sign chart cut at the critical points. The first and second derivative tests then sort each critical point into peak, valley or neither by whether $f'$ changes sign there; at a level tangent with $f''(c) \ne 0$, the sign of $f''(c)$ settles the same question. **Procedure.** Extrema on a closed interval finds the absolute maximum and minimum of a function continuous on $[a, b]$ by comparing $f$ at the critical points and at both endpoints. ## A path through it Critical points first. Then sign charts, then the derivative tests. The closed interval method needs only critical points, so it can come right after them or last. Finish here and you can do optimization, curve sketching from sign charts and Rolle's theorem. ### [Newton's method](https://math.wiki/newtons-method/) **Summary:** Using tangent lines over and over to close in on a root. **Question:** Most equations cannot be solved by algebra. If you can guess roughly where a root is, how can you sharpen the guess quickly? ## When you reach for it Reach for Newton's method when you need a root of $f(x) = 0$ to many decimal places, no formula gives it, $f$ is differentiable, and a graph, a table of values or a few steps of bisection give a reasonable first guess. Typical cases are a cubic such as $x^3 - 2x - 5 = 0$, an equation such as $e^x = 4x$, and a square root. When it works it is far faster than bisection. ## The procedure **One step, from the linearization.** Suppose $x_n$ is a guess near a root $r$ and $f'(x_n) \ne 0$. Near $x_n$ the graph is close to its tangent line, which written as a function is the linearization of $f$ at $x_n$: $$L(x) = f(x_n) + f'(x_n)(x - x_n).$$ We cannot solve $f(x) = 0$, but we can solve $L(x) = 0$, because $L$ is linear. Setting $L(x) = 0$ gives $f'(x_n)(x - x_n) = -f(x_n)$. Dividing by $f'(x_n)$, which is allowed because it is not zero, the solution is the next guess: $$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}.$$ So $x_{n+1}$ is where the tangent line at $(x_n, f(x_n))$ crosses the $x$-axis. From $x_{n+1}$ we do the same again. A first guess $x_0$ produces a sequence $x_0, x_1, x_2, \ldots$ in which every term comes from the one before by the same formula, an **iteration**. **How fast it closes in.** Write $e_n = x_n - r$ for the error. The following is stated here without proof. If $f'(r) \ne 0$, $f''$ is continuous near $r$, and $x_0$ is close enough to $r$, then the iterates converge to $r$, and there is a constant $K$ with ### [One sided limits and when a limit exists](https://math.wiki/two-sided-limit-exists-criterion/) **Summary:** A limit exists exactly when the left and right limits agree. **Question:** A function can head toward one height from the left and toward another from the right. When does it have a limit at all, and how do you find each side from a formula rather than a picture? ## Why this exists In limits-from-a-graph a limit was read off a picture, one side at a time, using the one-sided limits $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$; that lesson stated without proof that the limit exists exactly when they agree. Many functions are built from different rules on different sides of a point: a piecewise function, a quotient with an absolute value, a step function, a square root that stops at the edge of its domain. For these, a limit is found by splitting the question in two, settling each side from the formula that governs it, and comparing. This lesson proves the splitting is legitimate and carries it out with formulas. The lesson takes two days. Part 1 is the criterion itself, why it is true, and one-sided limits of piecewise functions found from their rules; Examples 1 to 4 go with it. Part 2 is absolute-value quotients, the greatest integer function, and limits at an endpoint of a domain; Examples 5 to 7 go with it. ## Definition The **left-hand limit** of $f$ at $a$ is $L$, written $\lim_{x \to a^-} f(x) = L$, when $f(x)$ is as close as you like to $L$ for every input $x$ that is less than $a$ and close enough to $a$. The **right-hand limit** $\lim_{x \to a^+} f(x) = M$ is the same with inputs greater than $a$. Each one asks about one side only, and neither ever looks at the input $a$ itself. The **two-sided limit** $\lim_{x \to a} f(x) = L$ asks that $f(x)$ be as close as you like to $L$ for every $x \ne a$ close enough to $a$, on either side. It needs $f$ to be defined at every input near $a$ on both sides, except perhaps at $a$. ### [Optimization](https://math.wiki/optimization-on-a-closed-interval/) **Summary:** Checking the ends of the sensible domain as well as the critical points. **Question:** A sheet of metal, a fixed volume, a speed limit. A situation asks for the most or the least of something but hands us no function. How do we build the function, decide which inputs make sense, and know for certain that the value we find is the best one? ## When you reach for it A problem asks for the largest or smallest value of some quantity (a volume, a cost, a time, a distance, a profit) and leaves a measurement up to us. closed-interval-method finds the absolute extremes of a continuous function on $[a, b]$, and first-derivative-test shows when a single critical point gives the absolute extreme on any interval. Optimization is the work that comes first: turning a description into one function of one variable, on a domain that makes physical sense, and then choosing the right tool. The general vocabulary of maxima and minima is in extrema. When the function turns out to be a quadratic, the vertex answers the question without calculus, as in maximize-or-minimize; a linear quantity over a region cut out by inequalities is linear-programming. Every problem below leads to a function that is not a quadratic, so the derivative is needed. ## The procedure ### Day 1: from words to a function on an interval Begin by naming each quantity that can change, with its units, and draw a picture when there is geometry. Then write the **objective**, the quantity to be made largest or smallest, as a formula. It usually involves two or more variables. The situation also ties them together by a **constraint**: a fixed amount of material, a set distance, a required volume. Solving the constraint for one variable and substituting leaves the objective as a function of a single variable. ### [Parametric, polar and vector valued functions](https://math.wiki/parametric-polar-and-vector-calculus/) **Summary:** Doing calculus on curves described by a parameter, by an angle, or by a position vector. **Question:** A path that loops and crosses itself is not the graph of any function. How do we find its slope, its length, the speed along it and the area it encloses? ## What this area is about Single variable calculus works on graphs $y = f(x)$. Many curves are not graphs: a seat on a spinning ride, a spirograph star, a spiral shell. They are described instead by a parameter, as $x(t)$ and $y(t)$, by an angle, as $r = f(\theta)$, or by a position vector $\mathbf{r}(t) = \langle x(t), y(t) \rangle$. This area carries derivatives and integrals over to all three. ## How these fit together **Parametric curves.** Derivatives of parametric curves gives the slope $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ wherever $\frac{dx}{dt} \ne 0$, with tangent lines and concavity. Arc length and speed takes the speed $\sqrt{x'(t)^2 + y'(t)^2}$ from the same two rates, for $x'$ and $y'$ continuous on $[a, b]$, and integrates it to get the distance traveled, which is the length of the curve when it is traced once. **Polar curves.** Slope of a polar curve treats $r = f(\theta)$ as the parametric curve $x = f(\theta)\cos\theta$, $y = f(\theta)\sin\theta$. Area in polar coordinates adds thin wedges, $\frac{1}{2}r^2\,d\theta$, and subtracts an inner curve from an outer one. **Vectors and motion.** Vector valued functions and motion in the plane makes velocity a vector, speed its length, and distance the integral of speed. ## A path through it Start with parametric derivatives. Slope of a polar curve and arc length both build on it, and motion in the plane comes after arc length. Polar area stands on its own and can come at any point. ### [Position, displacement and distance from velocity](https://math.wiki/total-distance-versus-displacement/) **Summary:** Integrating velocity gives displacement while integrating speed gives distance traveled. **Question:** A speedometer records how fast and in which direction something moves, moment by moment, but not where it is. From the velocity alone, how far did it end up from where it started, and how far did it actually travel? ## When you reach for it You reach for this whenever a motion along a line is known through its **velocity** $v(t)$ or its **acceleration** $a(t)$ rather than its position $s(t)$: a particle given by a formula, an elevator logged by a sensor, a dropped stone whose only known force is gravity. Three questions come up. Where is it at a later time? How far is it from where it started (the **displacement**)? How much ground did it cover (the **total distance**)? Integrals answer all three. In particle-at-rest-and-direction-change the position was known, and the distance came from adding $\lvert s(t_{i+1}) - s(t_i) \rvert$ between turning points. Here only the rate is known, so the same pieces come from integrals of $v$. ## The procedure ### Day 1: displacement, distance and position **Displacement.** Position is an antiderivative of velocity, since $s' = v$. If $v$ is continuous on $[a, b]$, the second fundamental theorem (evaluating-a-definite-integral) gives $$\int_a^b v(t)\,dt = s(b) - s(a).$$ This is the net change theorem (accumulation-in-context) read for motion. Like every integral it is a signed area: time spent moving in the negative direction counts against time spent moving in the positive one. **Position.** Rearranged, the same equation finds a later position from an earlier one: $$s(b) = s(a) + \int_a^b v(t)\,dt.$$ Velocity alone fixes position only up to a constant; one known position fixes the constant. ### [Position, velocity and acceleration](https://math.wiki/particle-at-rest-and-direction-change/) **Summary:** Reading zeros and sign changes of velocity as stops and reversals. **Question:** An object moves back and forth along a straight line. From a formula for where it is, how can we tell when it stops, which way it is going, when it turns around, whether it is speeding up, and how far it has really traveled? ## When you reach for it A paddleboard on a straight course, a bead on a wire, a piston in a cylinder: each moves along a single line, and its whole motion is recorded by one function, its position $s(t)$ at time $t$. Questions about the motion ("is it moving?", "which way?", "when does it turn back?", "is it gaining speed?", "how far did it go?") all become questions about the signs of $s'$ and $s''$. The derivative as a rate of change, with its units and sign, is interpreting-a-derivative-in-context; the second derivative as acceleration is in second-derivative, where Matus's quadcopter already computes height, velocity and acceleration. This lesson reads those derivatives as motion, and the tool it leans on is the sign chart of sign-charts-for-derivatives, now drawn for velocity. ## The procedure ### Day 1: rest, direction and turns **Position, velocity, acceleration.** Put a number line along the path, choose an origin and a positive direction (call it "right", or east). The **position** $s(t)$ is the signed number on that line where the object sits at time $t$. Its **velocity** is the instantaneous rate of change of position, and its **acceleration** is the rate of change of velocity: $$v(t) = s'(t), \qquad a(t) = v'(t) = s''(t).$$ Units follow the difference quotients. If $s$ is in meters and $t$ in minutes, then $\dfrac{s(t+h) - s(t)}{h}$ is in meters per minute, so $v$ is in meters per minute; and $\dfrac{v(t+h) - v(t)}{h}$ is in meters per minute per minute, written $\text{m/min}^2$. ### [Power series and the interval of convergence](https://math.wiki/endpoint-testing-for-power-series/) **Summary:** Checking each end separately, since the ratio test says nothing there. **Question:** A series whose terms carry a variable, such as $\sum c_n (x-a)^n$, is a different numerical series for every value of $x$. For which $x$ does it converge, and what shape can that set of $x$ have? ## When you reach for it A polynomial $c_0 + c_1 x + \cdots + c_N x^N$ has a value at every $x$. Letting the degree run on forever gives a **power series centered at $a$**, $$\sum_{n=0}^{\infty} c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots$$ with fixed numbers $c_n$, the **coefficients**, and the convention $(x-a)^0 = 1$ for every $x$, including $x = a$. For each fixed $x$ this is an ordinary numerical series, which may converge or diverge. Where it converges, its sum depends on $x$, so the series defines a function $f(x) = \sum c_n (x-a)^n$ whose domain is the set of $x$ where the series converges. A power series is the main example of a **series of functions** $\sum f_n(x)$, a series whose terms are functions of $x$; here $f_n(x) = c_n (x-a)^n$. The geometric series is the first example: $\sum_{n=0}^{\infty} x^n$ converges exactly when $\lvert x \rvert < 1$, and then its sum is $\frac{1}{1-x}$. The formula $\frac{1}{1-x}$ has a value at $x = 3$, namely $-\frac12$, but the series at $x = 3$ diverges, so the series and the formula are the same function only on $(-1, 1)$. You reach for this lesson whenever a series carries a variable. Before a power series can be differentiated, integrated, or used to compute values of a function, you must know for which $x$ it means anything. The answer is always an interval centered at $a$, found in two stages: the ratio-test gives the inside of the interval, and the two endpoints are tested one at a time. ### [Properties of definite integrals](https://math.wiki/additivity-over-intervals/) **Summary:** An integral over a long interval is the sum of the integrals over its parts. **Question:** Once a few integrals are known, how can others be found from them without going back to Riemann sums, and how much can be said about an integral that has no formula at all? ## Why this exists definite-integral-as-a-limit defines $\int_a^b f(x)\,dx$ as the limit of Riemann sums, and even $\int_1^3 x^2\,dx$ takes a page that way. Nobody computes every integral from scratch. Integrals are assembled from ones already known, using rules that every Riemann sum obeys and that therefore pass to the limit: the integral of a sum is the sum of the integrals, and an integral over a long interval is the sum of the integrals over its parts. Some integrals, such as $\int_0^1 \sqrt{1 + x^4}\,dx$, have no formula in the usual functions at all. Comparison rules still trap such a number between two values that can be computed, and say how large an integral can be from bounds on its integrand. ## Definition ### Day 1: the integral for every pair of limits For $a < b$ and $f$ integrable on $[a, b]$, the definite integral is the limit of Riemann sums (definite-integral-as-a-limit). It is extended to every pair of limits by two conventions, met first in signed-area: $$\int_b^a f(x)\,dx = -\int_a^b f(x)\,dx,$$ $$\int_a^a f(x)\,dx = 0.$$ The **properties of the definite integral** are the theorems below about this extended integral. Each one is a property of finite sums carried through the limit. ### Day 2: symmetry and bounds A function is **even** when $f(-x) = f(x)$ for every $x$ in its domain and **odd** when $f(-x) = -f(x)$ (test-symmetry-algebraically). Powers $x^2, x^4$ and $\cos x$ are even; $x, x^3$ and $\sin x$ are odd. A **bound** for $f$ on $[a, b]$ is a pair of numbers $m$ and $M$ with $m \le f(x) \le M$ for every $x$ in $[a, b]$. ### [Related rates](https://math.wiki/related-rates-with-geometry-formulas/) **Summary:** Linking changing lengths to changing areas or volumes. **Question:** A sphere of ice melts, losing volume at a known rate. How fast is its radius shrinking, when nobody has a formula for the radius as a function of time? ## When you reach for it Some quantities change together because an equation ties them at every instant: the radius and the area of a spreading circle, the depth and the volume of liquid in a glass, the two legs and the hypotenuse of a right triangle, an angle and the height it looks up to. One rate of change is known or easy to measure, and another is wanted. Usually neither quantity has a formula in terms of time, and none is needed. The move is to differentiate the equation with respect to time $t$. Every changing quantity is treated as an unnamed function of $t$, so each term picks up a factor such as $\dfrac{dr}{dt}$ by the chain rule, exactly as $y^2$ gave $2y\,\dfrac{dy}{dx}$ in implicit differentiation. Related rates is implicit differentiation with $t$ in the place of $x$. In the notation of chain-rule-in-leibniz-notation, the area of a circle changes at $$\frac{dA}{dt} = \frac{dA}{dr}\cdot\frac{dr}{dt}$$ so a rate in centimeters per second becomes a rate in square centimeters per second. The lesson runs over three days. Day 1 sets up the idea with circles, spheres, boxes and the Pythagorean theorem. Day 2 uses similar triangles to remove a variable whose rate is unknown. Day 3 brings in angles, two moving objects, and ways to check an answer. ## The procedure ### Day 1: an equation that holds at every instant Start by naming every quantity that changes as a function of time, and keep every quantity that stays fixed as a number. Then find an equation that ties the changing quantities together at every instant. For geometry this is usually a formula already known: $A = \pi r^2$ and $C = 2\pi r$ for a circle, $V = \frac{4}{3}\pi r^3$ for a sphere (volume-of-a-sphere) and $S = 4\pi r^2$ for its surface (surface-area-of-a-sphere), $V = Bh$ for a prism or cylinder with base area $B$ (volume-of-a-prism), and $x^2 + y^2 = z^2$ for a right triangle (pythagorean-theorem). ### [Representing functions as power series](https://math.wiki/integrating-a-series-term-by-term/) **Summary:** Term by term integration gives series for functions that resist antidifferentiation. **Question:** The geometric series gives $\frac{1}{1-x}$ as an infinite polynomial. Can the same be done for $\frac{x}{9+x^2}$, for $\ln(5 - x)$, for $\arctan\left(\frac{x}{3}\right)$, and for a function whose antiderivative cannot be written down at all? ## When you reach for it A power series $\sum c_n (x-a)^n$ defines a function on its interval of convergence, found as in endpoint-testing-for-power-series. This lesson runs the other way: start from a function and find a power series whose sum it is. Two reasons to want one. A series turns a function into arithmetic, so its values can be computed by adding terms. And a series can be differentiated and integrated term by term, which gives series for functions with no simple formula at all, such as an antiderivative that cannot be written with the familiar functions. Everything here grows from one identity, the geometric series read as a function of its ratio: $$\begin{aligned} \frac{1}{1-u} &= \sum_{n=0}^{\infty} u^n \\ &= 1 + u + u^2 + u^3 + \cdots \end{aligned}$$ for $\lvert u \rvert < 1$. At $u = \pm 1$ the terms do not tend to $0$, so the identity holds exactly on $(-1, 1)$ and nowhere else. Day 1 builds new series from it by algebra; Day 2 adds calculus. A different route, finding coefficients from derivatives as in taylor-coefficients-from-derivatives, belongs to taylor-series; the series found here are the same ones, because a function has only one power series at a given center (Exercise 24). ## The procedure ### Day 1: building series by algebra **Substitution.** The identity holds for every number $u$ with $\lvert u \rvert < 1$, so it holds when $u$ is replaced by any expression $g(x)$, for exactly those $x$ with $\lvert g(x) \rvert < 1$. With $u = -x^2$, for instance, ### [Second and higher derivatives](https://math.wiki/second-derivative/) **Summary:** The rate at which the first derivative itself is changing. **Question:** The derivative is itself a function, so it has a slope of its own. What does the rate of change of a rate of change measure, and what happens when you keep differentiating? ## Why this exists The derivative as a function turned the slopes of $f$ into a new function $f'$. Nothing stops us from doing it again. A speedometer reading is the derivative of position, but a driver pressing the gas pedal feels something else: how fast the speedometer reading itself is changing. That is the derivative of the derivative, and physics needs it, since a force produces acceleration, not velocity. Day 1 defines the second derivative, its notation and its meaning as acceleration, reads it from a graph, and computes it with the product and quotient rules. Day 2 keeps differentiating and finds the patterns: powers end in a factorial and then zero, sine and cosine cycle, exponentials repeat, and $\dfrac{1}{x}$ builds factorials with alternating signs. ## Definition ### Day 1: the derivative of the derivative If $f'$ is differentiable at $x$, the **second derivative** of $f$ at $x$ is the derivative of $f'$ there: $$f''(x) = \lim_{h \to 0} \frac{f'(x + h) - f'(x)}{h}$$ For this to make sense, $f'$ must exist on an open interval around $x$, not only at $x$. In Leibniz notation, applying $\dfrac{d}{dx}$ twice is written $$\frac{d}{dx}\left[\frac{dy}{dx}\right] = \frac{d^2y}{dx^2}$$ read "dee two $y$ dee $x$ squared". The $2$ sits on $d$ in the top and on $x$ in the bottom because the instruction $\dfrac{d}{dx}$ is applied twice; it is a symbol, not a square of anything. Continuing, the **third derivative** is $f''' = (f'')'$, and the **$n$th derivative** $f^{(n)}$ is the derivative of $f^{(n-1)}$, also written $\dfrac{d^ny}{dx^n}$. From the fourth on, primes give way to a number in parentheses: $f^{(4)}$, $f^{(5)}$. By convention $f^{(0)} = f$. The parentheses matter: $f^{(4)}$ is a derivative, $f^4$ is a power. ### [Separable differential equations](https://math.wiki/separable-with-an-initial-condition/) **Summary:** Solving for the constant after integrating, then choosing the branch through the starting point. **Question:** A differential equation gives the slope of its solutions but not the solutions. When can the equation be undone by integrating, and how is the one solution through a given point picked out? ## When you reach for it verifying-a-solution-to-a-differential-equation checked a proposed solution by substituting it, and sketching-a-slope-field drew the shape of solutions without any formula. Neither produces a formula. For one large and useful class of first-order equations, the ones whose right side splits into a factor in $x$ times a factor in $y$, a formula can be found by integrating, using nothing beyond the antiderivatives already known, u-substitution and the trigonometric antiderivatives. Many equations written in modeling-with-differential-equations are of this kind. Reach for separation when $\dfrac{dy}{dx}$ is given as a product $g(x)\,h(y)$, or can be rearranged into one, and you want every solution, or the one solution through a given starting point. ## The procedure ### Day 1: separating and integrating A first-order equation $\dfrac{dy}{dx} = f(x, y)$ is **separable** if its right side factors as $$\frac{dy}{dx} = g(x)\,h(y),$$ a function of $x$ alone times a function of $y$ alone. Either factor may be constant: $\dfrac{dy}{dx} = \cos x$ has $h(y) = 1$, and an autonomous equation $\dfrac{dy}{dx} = h(y)$ has $g(x) = 1$. The equation $\dfrac{dy}{dx} = x + y$ is not separable, since a sum does not split into such a product (Example 1 shows how to prove that). **Separating.** Where $h(y) \neq 0$, divide by $h(y)$ so that every $y$ sits on the left with the derivative: $$\frac{1}{h(y)}\,\frac{dy}{dx} = g(x).$$ ### [Sequences and series in calculus](https://math.wiki/series/) **Summary:** Infinite sums: when they settle, and how functions become polynomials. **Question:** Adding infinitely many numbers can never be finished. When does such a sum still have a value, and how can a function like sine be written as an infinitely long polynomial? ## What this area is about A series $\sum a_n$ has a sum when its partial sums $S_N = a_1 + \cdots + a_N$ approach a number. Deciding when that happens is the first half of this area. The second half lets the terms carry a variable, so that a series in powers of $x - a$ can stand for a function, and a calculator that only adds and multiplies can compute $e^x$ or $\sin x$. ## How these fit together **Foundations.** Convergence of sequences says when a list of numbers settles. Infinite series and partial sums defines a series' sum as the limit of its partial sums. **Deciding convergence.** The nth term test says a series whose terms do not approach $0$ diverges. The geometric series test says $\sum a r^n$ with $a \ne 0$ converges exactly when $\lvert r \rvert < 1$. Convergence tests handle the series with no formula for their sums. **Functions as series.** Power series converge on an interval, whose endpoints need separate checks. Representing functions as power series builds new series from the geometric one by substitution, differentiation and integration. Taylor series builds a series from a function's derivatives at one point. ## A path through it Sequences, then partial sums. The nth term test before the geometric series test, which uses it; then the convergence tests. Power series need the convergence tests, representing functions needs power series and geometric series, and Taylor series come last. ### [Signed area and integrals from geometry](https://math.wiki/signed-area/) **Summary:** Regions below the axis count as negative in a definite integral. **Question:** When a graph dips below the x-axis, how should the region trapped there count, and how can an integral be evaluated exactly from triangles, rectangles and circles alone? ## Why this exists Riemann sums approximate the area under a graph with rectangles, and the definite integral is the exact number those sums approach. Where the graph runs below the x-axis, a rectangle's height $f(x_i^*)$ is negative, so its term $f(x_i^*)\,\Delta x$ is negative too. The integral therefore counts area above the axis as positive and area below as negative. That reading gives the integral a meaning (net change, such as a displacement) and gives its exact value without any limit whenever the region is made of shapes with known areas. ## Definition Let $f$ be continuous on $[a, b]$ with $a < b$. The region between the graph of $y = f(x)$ and the x-axis, for $a \le x \le b$, splits into pieces lying above the axis and pieces lying below it. The definite integral $$\int_a^b f(x)\,dx$$ is the **signed area** (or **net area**) of that region: the total area of the pieces above the axis minus the total area of the pieces below it. The function $f$ is the integrand, and $a$ and $b$ are the lower and upper limits of integration. The definition of the integral as a limit of Riemann sums is in definite-integral-as-a-limit. This lesson uses the area reading, which that limit produces; it is taken here as the working meaning. The **total area** between the graph and the axis counts every piece as positive. Flipping the pieces below the axis up turns $f$ into $|f|$, so the total area is $\int_a^b |f(x)|\,dx$. ### [Slope fields](https://math.wiki/sketching-a-slope-field/) **Summary:** Drawing a short segment at each grid point with the slope the equation gives there. **Question:** An equation gives the slope of its solutions at every point, but no formula for them. How much of the solutions' shape can be seen from the slopes alone? ## When you reach for it A first-order differential equation solved for the derivative, $\dfrac{dy}{dx} = f(x, y)$, gives the slope of a solution curve at every point it passes through, before any formula for the solution is known. verifying-a-solution-to-a-differential-equation tested proposed solutions and met families of them, one curve for each value of a constant; modeling-with-differential-equations wrote such equations from descriptions of a rate. Many equations have no solution formula that can be found, and even when one exists, what usually matters is the shape: which way solutions head, where they level off, what they do in the long run. A slope field shows that shape. Reach for it to see a whole family of solutions at once, to sketch the solution through a given point, to match an equation with a picture, and to decide whether an equilibrium pulls nearby solutions in or pushes them away. ## The procedure ### Day 1: drawing a slope field At a point $(x_0, y_0)$ the equation says that a solution curve through it has slope $f(x_0, y_0)$ there. A **slope field** (or direction field) draws, at each point of a grid, a short segment centered at the point with that slope. Each segment is a small piece of the tangent line to the solution curve through its point, and computing it is substitution. By hand a small grid and a table of values are enough. Only direction is recorded, so all segments have the same length: slope $0$ is horizontal, slope $1$ rises at $45°$, a large slope is nearly vertical. ### [Slope of a polar curve](https://math.wiki/slope-of-a-polar-curve/) **Summary:** Differentiating the x and y forms with respect to the angle and dividing. **Question:** A curve is drawn by an angle and a distance, $r = f(\theta)$, rather than by $y$ as a function of $x$. How steep is it at a point, and how is that steepness different from how fast the distance $r$ is changing? ## When you reach for it Tsering has a nautilus shell sawn in half. The cut shows a spiral wall winding out from a center point, and its outline is close to $r = e^{\theta/3}$, with $r$ in centimeters measured from the center (the pole) and $\theta$ in radians. She wants the direction of the wall at a given point. The curve is given in polar form (polar-coordinates); it winds around the pole and is not the graph of any function $y = F(x)$. But every polar curve is a parametric curve with the angle as its parameter, and tangent-lines-to-parametric-curves already gives the slope of a parametric curve: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$, horizontal where only the top is $0$, vertical where only the bottom is $0$, and a limit to take where both are $0$. This lesson applies that to $r = f(\theta)$. Reach for it whenever a polar curve needs a slope, a tangent line, its level and upright points, its tangent lines at the pole, or the angle it makes with the radius. ## The procedure **A polar curve is a parametric curve.** The point with polar coordinates $(r, \theta)$ has rectangular coordinates $x = r\cos\theta$, $y = r\sin\theta$ (polar-to-rectangular-conversion). On the curve $r = f(\theta)$, substitute: $$x = f(\theta)\cos\theta, \qquad y = f(\theta)\sin\theta.$$ The angle $\theta$ plays the part of the parameter $t$. A negative value of $f$ needs no special care: that point is plotted on the opposite ray, and the formulas put it there, since both coordinates change sign. ### [Surface area of a solid of revolution](https://math.wiki/surface-area-of-revolution/) **Summary:** Adding the side areas of thin bands swept out by an arc. **Question:** Spin a curve about a line and it sweeps out a thin shell, like a bowl spun from sheet metal. How much area does the shell have? ## When you reach for it In a metal spinning workshop a flat disk of sheet metal is pressed against a form turning on a lathe until it takes the form's shape: a reflector, a bowl, a funnel. The finished piece is a **surface of revolution**, what a curve sweeps out when it makes one full turn about a line, the axis. The disk method measures the solid such a shape encloses. This lesson measures the shell itself: its area, which is what the metal, the polish or the plating must cover. Geometry gives the area of a cylinder, a cone and a sphere. For any other profile, reach for an integral built, like arc length, on the short piece of curve $ds$. ## The procedure **The band.** Replace a short piece of the profile by a straight segment. Turned about the axis, a segment sweeps a **band**: the side of a cone with its tip cut off, called a frustum. Its area comes from the cone's. A cone with base radius $r$ and slant height $\ell$ has side area $\pi r \ell$ (the lateral area of a cone). Let a band have end radii $r_1 < r_2$ and slant width $\ell$. Extend it to a whole cone. The tip that was cut off is a small cone of slant $\ell_1$, and the whole cone has slant $\ell_1 + \ell$. The two cones are similar, so $\dfrac{r_1}{\ell_1} = \dfrac{r_2}{\ell_1 + \ell}$. Cross-multiplying gives $r_2 \ell_1 = r_1 \ell_1 + r_1 \ell$, that is, $(r_2 - r_1)\ell_1 = r_1 \ell$. The band is the big cone minus the tip: ### [Tangent lines and normal lines](https://math.wiki/equation-of-a-tangent-line/) **Summary:** Using the derivative as the slope and the point of contact to build a line. **Question:** The derivative gives the slope of a curve at one point. How do you turn that single number into the line itself, and what can that line be used for? ## When you reach for it The derivative at a point defined the tangent line to $y = f(x)$ at $x = a$ as the limiting position of secant lines, with slope $f'(a)$, and the power rule now produces $f'(a)$ for any polynomial or power of $x$ in a line or two. Reach for the two together whenever a question asks for a line attached to a curve: the tangent at a point, the line perpendicular to it, a tangent with a required slope or through a point off the curve, a line tangent to two curves, or a quick estimate of a function value near a known point. ## The procedure **The tangent line.** A nonvertical line is fixed by one point and its slope. The tangent at $x = a$ passes through $(a, f(a))$ and has slope $f'(a)$, so point-slope form gives $$y - f(a) = f'(a)(x - a).$$ Both $f(a)$ and $f'(a)$ are numbers, found by putting the fixed input $a$ into $f$ and $f'$. Solving for $y$ gives the slope-intercept form $y = f'(a)\,x + \big(f(a) - a f'(a)\big)$; the intercept $f(a) - a f'(a)$ is what you compare when asking whether two tangent lines are the same line. **The normal line.** The **normal line** to the curve at $(a, f(a))$ is the line through that point perpendicular to the tangent. Recall from parallel-slopes that parallel nonvertical lines have equal slopes, and from perpendicular-slopes that two nonvertical lines are perpendicular exactly when their slopes multiply to $-1$. So when $f'(a) \ne 0$ the normal has slope $-\dfrac{1}{f'(a)}$: ### [Taylor and Maclaurin series](https://math.wiki/taylor-series/) **Summary:** Rebuild a function from its derivatives at one point. **Question:** A machine that can only add and multiply still reports $e^{0.7}$ and $\sin 0.3$. How can a function be rebuilt from its derivatives at a single point, and how far can we trust the result? ## What this area is about A tangent line matches a function's value and slope at one point. A polynomial can also match the bending, and every higher derivative, and the coefficients that do it are forced: $\frac{f^{(k)}(a)}{k!}$. Keep every term and the result is a power series. It equals the function at exactly those $x$ where the remainder $f(x) - P_n(x)$ tends to $0$ as $n$ grows. ## How these fit together **Definition.** Taylor polynomials builds the degree $n$ polynomial that matches $f$ and its first $n$ derivatives at the center $a$. **Results.** The common Maclaurin series writes down the series for $e^x$, $\sin x$, $\cos x$ and a few others, and builds new ones by substitution, multiplication and term by term calculus. The binomial series expands $(1 + x)^k$ for any real $k$, valid for $\lvert x \rvert < 1$. The Lagrange error bound says that if $\lvert f^{(n+1)} \rvert \le M$ between $a$ and $x$, then $\lvert f(x) - P_n(x) \rvert \le \frac{M\lvert x - a \rvert^{n+1}}{(n+1)!}$. **Procedure.** Using series to evaluate limits and integrals integrates a series term by term inside its interval of convergence to estimate integrals with no usable antiderivative, and settles hard limits. ## A path through it Taylor polynomials are the natural start, then the common Maclaurin series. Both the error bound, which proves those series equal their functions, and using series for limits and integrals come after them. The binomial series fits anywhere after Taylor polynomials. ### [Taylor polynomials](https://math.wiki/taylor-coefficients-from-derivatives/) **Summary:** Matching derivatives at the center forces each coefficient to be a derivative over a factorial. **Question:** A tangent line matches a function's value and slope at one point. If a polynomial also matched the bending, and the change in the bending, and so on, how close could it stay to the function, and what would its coefficients have to be? ## Why this exists The linearization $L(x) = f(a) + f'(a)(x - a)$ is the line that agrees with $f$ in value and slope at $a$. It estimates nearby values well, and its error shrinks like the square of the distance from $a$. But a line cannot bend. Wherever $f$ curves, the line peels away, and the only way to push the error down further is to stay ever closer to $a$. A polynomial can bend as much as you like. A quadratic can match value, slope and second derivative; a cubic can match the third derivative too. The question is which polynomial to take, and the answer turns out to be forced: once you decide how many derivatives to match at $a$, there is exactly one polynomial of that degree that does it, and its coefficients are derivatives divided by factorials. That polynomial is the Taylor polynomial. It turns values you cannot compute by hand, such as $\sqrt[4]{17}$ or $\arctan 1.1$, into a few multiplications, and it is how a calculator, an optics shop or a physics model replaces a hard function by an easy one near a point of interest. ## Definition Let $f$ have $n$ derivatives at the point $a$. The **Taylor polynomial of degree $n$ for $f$ about $a$** (also "centered at $a$") is $$\begin{aligned} P_n(x) &= f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 \\ &\quad + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n \\ &= \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x - a)^k. \end{aligned}$$ Here $f^{(0)} = f$, $0! = 1$ and $k! = 1 \cdot 2 \cdots k$. The number $c_k = \dfrac{f^{(k)}(a)}{k!}$ is the **$k$-th Taylor coefficient**. Plain words first: the coefficient of $(x - a)^k$ is the $k$-th derivative at the center, divided by $k!$. ### [The binomial series](https://math.wiki/binomial-series/) **Summary:** A power series for a binomial raised to any real exponent. **Question:** The binomial theorem expands $(1 + x)^n$ when $n$ is a counting number. What does $(1 + x)^{1/3}$ or $(1 + x)^{-1/2}$ expand to, and for which $x$ is the expansion true? ## Why this exists The binomial theorem expands $(1 + x)^m$ for a counting number $m$ into a polynomial whose coefficients are the counts $\binom{m}{n}$ from counting and Pascal's triangle. It says nothing about $(1 + x)^{1/3}$ or $(1 + x)^{-1/2}$: you cannot multiply $1 + x$ by itself a third of a time, and "choose $n$ from $-\frac12$" counts nothing. Yet these functions have Taylor polynomials like any other, and their coefficients turn out to follow the same pattern as the counting numbers, with $m$ replaced by any real $k$. The result, the **binomial series**, contains the geometric series as the case $k = -1$, turns roots and reciprocal powers into sums of powers, and simplifies any formula with a root or a $\frac32$ power in it when one quantity is small next to another. ## Definition For any real number $k$ and any integer $n \ge 0$, the **generalized binomial coefficient** is $$\binom{k}{n} = \frac{k(k-1)(k-2)\cdots(k-n+1)}{n!},$$ with $\binom{k}{0} = 1$. The top is a product of $n$ factors, starting at $k$ and stepping down by $1$. For example, $$\begin{aligned} \binom{-2}{3} &= \frac{(-2)(-3)(-4)}{3!} = -4, \\ \binom{1/3}{2} &= \frac{\frac13 \cdot \left(-\frac23\right)}{2} = -\frac19. \end{aligned}$$ When $k$ is a counting number these agree with the counts: $\binom{5}{2} = \frac{5 \cdot 4}{2} = 10$ (Exercise 18). The **binomial series** for exponent $k$ is the power series $$\begin{aligned} \sum_{n=0}^{\infty} \binom{k}{n} x^n = 1 &+ kx + \frac{k(k-1)}{2!}x^2 \\ &+ \frac{k(k-1)(k-2)}{3!}x^3 + \cdots \end{aligned}$$ ### [The chain rule](https://math.wiki/chain-rule-in-leibniz-notation/) **Summary:** Writing the rule as a product of rates makes the canceling pattern visible. **Question:** When one quantity depends on a second, and the second depends on a third, how fast does the first change when the third does? ## Why this exists The rules so far handle sums, constant multiples, powers, exponentials, products and quotients. They do not handle a function inside another, such as $(x^3 - 4x)^5$, $e^{x^2}$ or $\sqrt{x^2 + 16}$. Each is a composition: an inner rule is applied first and an outer rule is applied to its output. The obvious guess, differentiate the outer function and leave the inside alone, already fails on $(3x + 1)^2$. The guess gives $2(3x + 1)$. Expanding gives $9x^2 + 6x + 1$, whose derivative is $18x + 6 = 2(3x + 1) \cdot 3$. The missing factor $3$ is the rate at which the inside, $3x + 1$, changes. The chain rule says that this is always the repair: multiply by the rate of the inside. The same thing happens with units. Tumelo is descending on a scuba dive. The water pressure on him depends on his depth, and his depth depends on time. If the pressure rises $10$ kilopascals for every meter of depth and he sinks $4$ meters per minute, the pressure rises $10 \times 4 = 40$ kilopascals per minute. Rates in a chain multiply, the way unit conversions do. ## Definition **The chain rule.** If $g$ is differentiable at $x$ and $f$ is differentiable at $g(x)$, then the composition $f \circ g$ is differentiable at $x$, and $$(f(g(x)))' = f'(g(x))\,g'(x).$$ In words: the derivative of the outer function, evaluated at the inside, times the derivative of the inside. In Leibniz notation, write $u = g(x)$ and $y = f(u)$. Then ### [The common Maclaurin series](https://math.wiki/maclaurin-series-for-e-to-the-x/) **Summary:** Every derivative is the same at zero, so the coefficients are reciprocal factorials. **Question:** A Taylor polynomial matches a function's derivatives up to some degree and then stops. If the matching never stops, which series come out for $e^x$, $\sin x$ and $\cos x$, and how do a handful of such series turn into series for almost any function built from them? ## Why this exists taylor-coefficients-from-derivatives built Taylor polynomials: the degree $n$ polynomial whose value and first $n$ derivatives at the center $a$ match those of $f$, with coefficients $\frac{f^{(k)}(a)}{k!}$. A polynomial of fixed degree is good only near its center. Keeping every term gives a power series, and power series are functions in their own right, which can be added, multiplied, differentiated and integrated term by term (integrating-a-series-term-by-term). That lesson built series from the geometric series alone, which reaches rational functions, logarithms and the inverse tangent but not $e^x$, $\sin x$ or $\cos x$: no algebra on $\frac{1}{1-x}$ produces them. Their derivatives at $0$ repeat in a short cycle, though, so their coefficients can be written down at once. Day 1 does that. Day 2 collects six standard series and shows how substitution, multiplication, addition and calculus turn them into series for functions like $x^2\sin(3x)$ and $\cos\sqrt{x}$ without computing a single derivative. ## Definition ### Day 1: Taylor and Maclaurin series Let $f$ have derivatives of every order at $a$. The **Taylor series of $f$ centered at $a$** is $$\sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!}(x-a)^k$$ $$= f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots$$ Its partial sums are the Taylor polynomials $P_n(x)$ of taylor-coefficients-from-derivatives. A Taylor series centered at $0$ is a **Maclaurin series**. The definition makes no promise that the series converges at any $x \ne a$, or that where it converges its sum is $f(x)$; both have to be found. ### [The definite integral](https://math.wiki/definite-integral/) **Summary:** The limit of Riemann sums: exact area, with sign. **Question:** Every Riemann sum is only an estimate. Is there one exact number they all close in on, and what does it mean when part of the graph lies below the axis? ## What this area is about Rectangles under a curve give an estimate of an area or an accumulated amount, and thinner rectangles give a better one. When the Riemann sums approach a single number however the sample points are chosen, that number is the definite integral $\int_a^b f(x)\,dx$. It is exact, and it counts area below the $x$-axis as negative. ## How these fit together **Definition.** The definite integral as a limit of Riemann sums defines the integral as the limit of the sums as the number of pieces grows without bound, and computes a few exactly that way. **Meaning.** Signed area and integrals from geometry reads the integral of a function continuous on $[a, b]$ as area above the axis minus area below, and evaluates integrals exactly from triangles, rectangles and circles. **Results.** Properties of definite integrals gives the rules every Riemann sum obeys, such as $\int_a^b f\,dx + \int_b^c f\,dx = \int_a^c f\,dx$, linearity, and comparison bounds, so new integrals can be built from known ones. ## A path through it Start with the limit definition, which gives the other two their footing. Then signed area, and then the properties, which use the conventions for reversed limits that signed area sets up. Next comes the fundamental theorem of calculus, which evaluates these integrals without sums, and then net change in context and improper integrals. ### [The definite integral as a limit of Riemann sums](https://math.wiki/definite-integral-as-a-limit/) **Summary:** The exact accumulated amount defined as the limit of finer and finer approximations. **Question:** Every Riemann sum is only an estimate, and a finer one is better. Is there one exact number that all the estimates close in on, and can it be found without adding infinitely many rectangles by hand? ## Why this exists area-under-a-curve-idea trapped areas between left and right sums and watched the error fall as $n$ grew: for Leonor's velocity the sums crept toward $\frac{368}{3}$, a value taken on trust. A sum with a fixed $n$ is still an estimate. What is wanted is the exact amount, and a way to compute it. Limits supply both. A limit is the value an expression approaches as its input moves; here the input is the number of pieces $n$, and it grows without bound. If the Riemann sums approach one number however the rectangles are chosen, that number is the exact amount, and it gets a name and a symbol. ## Definition ### Day 1: the definite integral Let $f$ be defined on $[a, b]$ with $a < b$. Cut $[a, b]$ into $n$ pieces of equal width and pick a sample point $x_i^*$ in piece $i$: $$\Delta x = \frac{b - a}{n}, \qquad x_i = a + i\,\Delta x, \qquad x_{i-1} \le x_i^* \le x_i.$$ The **definite integral** of $f$ from $a$ to $b$ is $$\int_a^b f(x)\,dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*)\,\Delta x,$$ provided the limit exists and is the same number for every choice of sample points. When it is, $f$ is **integrable** on $[a, b]$. The general version allows unequal pieces. For a partition $P$: $a = x_0 < x_1 < \dots < x_n = b$ with widths $\Delta x_i = x_i - x_{i-1}$, the **mesh** $\|P\|$ is the largest width. Then $f$ is integrable with integral $I$ when every Riemann sum $\sum f(x_i^*)\,\Delta x_i$ is as close to $I$ as we like once $\|P\|$ is small enough, for every partition and every choice of sample points: ### [The derivative as a function](https://math.wiki/sketching-the-derivative-graph/) **Summary:** Turning the slopes of one graph into the heights of another. **Question:** The derivative at a point is one slope at one place. If you record the slope at every place, what kind of object do you get, and how do you read it off a graph or a table of measurements? ## When you reach for it The derivative at a point gives one number, $f'(a)$, the slope of the graph at one input. Finding a derivative from the definition does the same limit with $x$ left as a letter and gets a formula. This lesson treats that result as a function in its own right, with a domain and a graph of its own. Reach for it when the slope is needed everywhere at once, and especially when there is no formula. A graph of $f$ is enough to sketch the graph of $f'$; a graph of $f'$ is enough to sketch $f$, up to its height; a table of measurements is enough to estimate $f'$ at every row. Day 1 works with graphs and sets out the notation. Day 2 works with tables. ## The procedure ### Day 1: the derivative is a function The **derivative of $f$** is the function $f'$ whose value at each input $x$ is the slope of the graph of $f$ at $(x, f(x))$: $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$ Its domain is every $x$ where this limit exists, which can be smaller than the domain of $f$: $|x|$ is defined at $0$ but has no derivative there. **Slopes become heights.** To graph $f'$, read the slope of the graph of $f$ at an input $x$ and plot that number as a height above $x$ on a second set of axes. Each feature of $f'$ comes from a feature of $f$: | graph of $f$ at or near $x$ | graph of $f'$ | |:-:|:-:| | level tangent | on the $x$-axis: $f'(x) = 0$ | | rising as $x$ increases | on or above the $x$-axis | | falling as $x$ increases | on or below the $x$-axis | | steeper | farther from the $x$-axis | | straight piece of slope $m$ | horizontal piece at height $m$ | | corner | a gap: $f'(x)$ does not exist | ### [The derivative as a rate of change](https://math.wiki/interpreting-a-derivative-in-context/) **Summary:** Saying what a rate means for the situation, with its units and its sign. **Question:** A tank's volume is changing at -40 when t = 40. Minus forty what, and what does that number actually promise about the water? ## When you reach for it The derivative $f'(a)$ was built as the slope of a tangent line and as an instantaneous velocity, in derivative-at-a-point. Most functions met outside a calculus course are neither: they give dollars as a function of carpets, liters as a function of minutes, meters of depth as a function of days. Their derivatives are still rates, and the question is what a rate says about the situation. Reach for this lesson whenever a derivative comes with a story and someone has to say, in a sentence with units, what is happening and what is not. Part 1 (day 1) reads a derivative in words. Part 2 (day 2) covers marginal cost and revenue, rates in science, and rates given in a table or on a graph. ## The procedure ### Part 1: a derivative in words **Units.** The derivative is the limit of a difference quotient, $$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}.$$ The top is a change in $f$, measured in the units of $f$; the bottom is a change in $x$, measured in the units of $x$. Units divide like factors (the same bookkeeping as conversion factors), so every quotient carries the units of $f$ per unit of $x$, and so does their limit. If $V(t)$ is in liters and $t$ in minutes, $V'(t)$ is in liters per minute. When the input already has compound units, keep them whole: if $T(v)$ is a time in hours and $v$ a flow in liters per minute, $T'(v)$ is in hours per (liter per minute). Simplifying that unit would hide what it measures. ### [The derivative at a point](https://math.wiki/derivative-at-a-point/) **Summary:** A single number giving the slope of the curve at one place. **Question:** A rate of change needs two points, but a speedometer shows a speed at a single instant. How can one point have a slope, and how is that number found exactly? ## Why this exists A rate-of-change compares two moments: change in output over change in input. That is the slope of the line through two points of a graph, and it is an average. A funicular car climbing a hillside track speeds up as it leaves the bottom station, so "meters climbed per second" over a ten-second stretch says little about how fast the car moves at second $8$. Yet the speed at second $8$ is a real number; a speedometer would show it. estimate-a-rate-at-an-instant estimated such a number by computing averages over shorter and shorter intervals and watching where they settle. This lesson replaces the watching with a limit and gets the number exactly. The tool is the difference-quotient, $\dfrac{f(a + h) - f(a)}{h}$, the average rate of change of $f$ from $a$ to $a + h$. Let $h \to 0$ and the average over an interval becomes the rate at one point. That limit is the **derivative of $f$ at $a$**, the number at the center of calculus. Day 1 defines it, reads it as the slope of the tangent line, and uses it as an instantaneous velocity. Day 2 gives a second form of the definition, reads derivatives from graphs, shows a point where no derivative exists, and recognizes a limit that is secretly a derivative. ## Definition A **secant line** of a graph is a line through two points of it. Through $P(a, f(a))$ and $Q(a + h, f(a + h))$ the secant slope is the difference quotient $$m_{\text{sec}} = \frac{f(a + h) - f(a)}{h}$$ ### [The direct and limit comparison tests](https://math.wiki/limit-comparison-test/) **Summary:** Two positive series whose terms stay in a fixed ratio behave the same way. **Question:** A radio telescope adds up faint signals scan after scan, and no formula gives the running total. If each scan's share is smaller than the matching term of a series whose sum is known, must the total settle, and how close is it after a few scans? ## Why this exists Yousef, Samira and Leila run a radio telescope that points at one faint source again and again. Each scan adds a little more signal to a running total, and the share from scan $n$ shrinks as the receiver is retuned. Yousef's model gives scan $n$ a share of $\frac{1}{4^n + \sqrt{n}}$ units, so the total after $N$ scans is the partial sum $S_N$ of $$\sum_{n=1}^{\infty} \frac{1}{4^n + \sqrt{n}}.$$ No formula for $S_N$ exists, nothing telescopes, and the nth-term-test only reports that the terms tend to $0$, which decides nothing. But each share is smaller than $\frac{1}{4^n}$, and the geometric series $\sum \frac{1}{4^n}$ has sum $\frac{1}{3}$. A total built from smaller positive pieces never passes $\frac{1}{3}$, and a running total that only grows and never passes a ceiling has to settle. That is the direct comparison test. Its partner, the limit comparison test, settles series where the convenient inequality points the wrong way: two positive series whose terms stay in a settling ratio either both converge or both diverge. improper-integral-infinite-limit proved both tests for integrals. This lesson is about series: benchmarks that are geometric series and $p$-series (p-series-test), dropping or bounding part of a term, inequalities that hold only from some point on, and bounds on what the scans not yet made can add. ## Definition ### Day 1: direct comparison In plain words: a series of nonnegative terms that is smaller than a convergent series converges, and a series that is larger than a divergent series of nonnegative terms diverges. ### [The disk method](https://math.wiki/disk-method/) **Summary:** Slicing a solid of revolution into thin circles perpendicular to the axis. **Question:** Spin a flat region about a line and it sweeps out a solid whose every slice across the line is a circle. How do those circles add up to a volume? ## When you reach for it A solid of revolution is what a flat region sweeps out when it makes one full turn about a line, the **axis of revolution**. Turned wood is the everyday case: a spindle or a chair leg is cut by spinning a block about its center line while a chisel shapes its outline. Geometry gives the volume when the region is a rectangle, a right triangle or a half-disk. For any other outline, the curve itself has to be integrated. You reach for the disk method when the region lies against the axis all along its length, so that every slice perpendicular to the axis is a filled circle, a **disk**. It is the special case of volumes-with-known-cross-sections in which every cross section is a disk, with area $A = \pi R^2$. ## The procedure ### Day 1: disks along a coordinate axis **The slice is a disk.** Let $f$ be continuous on $[a, b]$, and let the region lie between the graph $y = f(x)$ and the $x$-axis for $a \le x \le b$. Turn it about the $x$-axis and cut the solid with the plane perpendicular to the axis at $x$. The region meets that plane in the segment from $(x, 0)$ to $(x, f(x))$, and a segment turned about one of its own endpoints sweeps a disk centered on the axis. Its radius is the distance from the axis to the curve, $$R(x) = |f(x)|,$$ and its area is $A(x) = \pi [R(x)]^2$ (Figure 1). Since $|f(x)|^2 = [f(x)]^2$, a region below the axis sweeps the same disks as its mirror image above it. ### [The epsilon delta definition of a limit](https://math.wiki/formal-limit-definition/) **Summary:** The precise statement that for every tolerance there is a window of inputs that meets it. **Question:** Saying that outputs "get close to" a number is a picture, not a test. What exact statement decides whether a limit is right, and how do you prove one? ## Why this exists In limits-from-a-graph a limit was read off a picture or a table: as $x$ approaches $a$ from both sides, the outputs $f(x)$ crowd around one height $L$, and we write $\lim_{x \to a} f(x) = L$. That works when the graph is honest, but "crowd around" is not something you can check. A table can mislead, and two people can look at the same picture and disagree about how close is close. Kham is a machinist. He cuts square metal plates, and a customer wants each plate to have an area of $9$ square centimeters. No cut is perfect, so the customer names a tolerance: the area may be off by at most $0.07$ square centimeters. Kham's question is the reverse one. How close to $3$ centimeters must he hold the side so that the area lands inside the tolerance? And if the next customer is stricter, can he always tighten his cut enough to satisfy them? That second question is the whole idea of a limit. Saying $\lim_{x \to 3} x^2 = 9$ means that whatever tolerance anyone names for the output, some tolerance on the input guarantees it. This lesson turns that sentence into a definition with no pictures in it, which is what lets limits be proved, and later lets the limit laws, continuity and the derivative rest on something solid. ## Definition Distance on the number line is measured with absolute value: $|f(x) - L|$ is how far the output is from $L$, and $|x - a|$ is how far the input is from $a$. By absolute-value-less-than, $|x - a| < \delta$ says the same thing as $a - \delta < x < a + \delta$. ### [The extreme value theorem](https://math.wiki/extreme-value-theorem/) **Summary:** A continuous function on a closed interval reaches a highest and a lowest value. **Question:** Before hunting for the largest and smallest value of a function on an interval, can we be sure there is a largest and a smallest value to find? ## Why this exists Many questions ask for the largest or smallest value of a quantity: the greatest area, the least cost, the highest point of a path. Each one assumes there is a largest or smallest value to find. That assumption can fail. On the interval $(0, 1]$ the function $f(x) = \dfrac{1}{x}$ has no largest value: $f(0.1) = 10$, $f(0.001) = 1000$, and every value is beaten by a value closer to $0$. So before any search for a highest or lowest value, there is an earlier question: does one exist? The extreme value theorem answers it for a large and common class of functions, the functions continuous on a closed interval. It is the second existence theorem about continuous functions, beside the intermediate value theorem of bisection-method. ## Definition Let $f$ be defined on an interval $I$. - $f$ has an **absolute maximum** on $I$ at $c$ if $c$ is in $I$ and $f(c) \ge f(x)$ for every $x$ in $I$. The number $f(c)$ is the **absolute maximum value**. - $f$ has an **absolute minimum** on $I$ at $d$ if $d$ is in $I$ and $f(d) \le f(x)$ for every $x$ in $I$. The number $f(d)$ is the **absolute minimum value**. Together these are the **absolute extreme values** of $f$ on $I$ (see extrema for the whole family). The value is unique, but it may be reached at more than one input: $\cos x$ on $[0, 4\pi]$ reaches its maximum $1$ at $0$, $2\pi$ and $4\pi$. Two words must be kept apart. $f$ is **bounded above** on $I$ if some number $B$ has $f(x) \le B$ for every $x$ in $I$. The smallest such $B$ is the **least upper bound**, written $\sup_{x \in I} f(x)$. An absolute maximum exists exactly when the least upper bound is itself a value $f(c)$. A function can be bounded without reaching its bound: $x^2$ on $[0, 1)$ stays below $1$ and comes as close to $1$ as we like, yet no input gives $1$. **Bounded below**, the greatest lower bound and the absolute minimum work the same way. ### [The first and second derivative tests](https://math.wiki/first-derivative-test/) **Summary:** A sign change of the derivative at a critical point tells whether it is a peak or a valley. **Question:** The critical points of a function are the only places a peak or a valley can sit, but not every critical point is one. How can we tell, without plotting, whether a critical point is a local maximum, a local minimum or neither? ## Why this exists critical-points proved that a local maximum or minimum can only sit at a critical point, but $x^3$ has a critical point at $0$ and no extremum there. sign-charts-for-derivatives found where a function rises and falls. Joining the two gives the first derivative test: a function that rises up to $c$ and falls after it has a peak at $c$. The second derivative test reaches the same verdict at a level tangent from one number, the sign of $f''(c)$, using concavity-from-the-second-derivative. Either test, at the only critical point of an interval, can give the largest or smallest value on the whole interval. ## Definition ### Day 1: a change of sign As in critical-points, a **local maximum** of $f$ at $c$ requires $c$ to be an interior point of the domain and $f(c) \ge f(x)$ for every $x$ in some open interval containing $c$; a local minimum reverses the inequality. An endpoint is never a local extremum, though it can be an absolute one. We say $f'$ **changes from positive to negative at $c$** if there is a $\delta > 0$ with $$\begin{aligned} f'(x) &> 0 \text{ on } (c - \delta, c), \\ f'(x) &< 0 \text{ on } (c, c + \delta). \end{aligned}$$ Nothing is asked of $f'(c)$: it may be $0$ or may not exist. "Negative to positive" is defined the same way, and $f'$ **does not change sign at $c$** if it is positive on both sides, or negative on both sides. ### Day 2: a level tangent The second derivative test applies to a critical point $c$ with $f'(c) = 0$ (a **stationary point**, where the tangent is horizontal) at which $f''(c)$ exists. Recall from second-derivative that ### [The first fundamental theorem of calculus](https://math.wiki/derivative-of-an-integral-with-variable-limits/) **Summary:** The chain rule attaches to the first fundamental theorem when the upper limit moves faster. **Question:** An integral with a moving end point is a function of that end point. How fast does it change as the end point moves, and can that be found without ever computing the integral? ## When you reach for it An integral $\int_a^b f(t)\,dt$ is a number. Let the upper end move and it becomes a function: the amount built up from $a$ to $x$. Pipaluk measures sea ice in a fjord. If $r(t)$ is the rate at which the ice thickens, in centimeters per day, then $\int_0^x r(t)\,dt$ is how much thicker the ice is on day $x$ than on day $0$. A natural question is how fast that amount changes as $x$ moves. The answer, the first fundamental theorem of calculus, is that the rate of change is the integrand at the moving end. You reach for it whenever an integral has a variable limit and you need its derivative, its tangent line, or its rising and falling, and especially when the integral itself cannot be written with the usual functions. It is the first half of the fundamental theorem of calculus, the link between the two halves of the subject. ## The procedure ### Day 1: the accumulation function Let $f$ be continuous on an interval $I$ and let $a$ be a fixed number in $I$. The **accumulation function** of $f$ from $a$ is $$A(x) = \int_a^x f(t)\,dt \qquad \text{for } x \text{ in } I.$$ The input of $A$ is the upper limit $x$. The letter $t$ is a **dummy variable** (definite-integral-as-a-limit): it runs from $a$ to $x$ inside the integral and is gone once the integral is a number, so $\int_a^x f(s)\,ds$ is the same function. A different letter is used for the dummy so that $x$ has one job, the end point. For $x < a$ the integral runs backward and $\int_a^x = -\int_x^a$; also $A(a) = 0$ (signed-area). ### [The fundamental theorem of calculus](https://math.wiki/fundamental-theorem-of-calculus/) **Summary:** Area under a rate is total change; integrals and derivatives undo each other. **Question:** Derivatives measure rates and integrals add up amounts. Why should these two separate ideas turn out to undo each other? ## What this area is about The definite integral is a limit of sums, which can take a page to compute. The fundamental theorem of calculus links it to the derivative, so that integrals can be found from antiderivatives in one line. It has two halves, and together they make differential and integral calculus one subject. ## How these fit together Both children are results, each a theorem with the procedures that follow from it. **The first theorem.** The first fundamental theorem of calculus says that if $f$ is continuous on an interval $I$ and $a$ is in $I$, then $A(x) = \int_a^x f(t)\,dt$ has $A'(x) = f(x)$ at every interior point $x$ of $I$. It also attaches the chain rule when the upper limit is itself a function of $x$. **The second theorem.** The second fundamental theorem of calculus says that if $f$ is continuous on $[a, b]$ and $F$ is an antiderivative of $f$ there, then $\int_a^b f(x)\,dx = F(b) - F(a)$. This is how nearly every exact integral is found. ## A path through it The first theorem comes first: it shows that every continuous function has an antiderivative, and the second theorem is proved from it. Next come accumulation and net change, substitution, areas between curves and improper integrals. ### [The geometric series test](https://math.wiki/geometric-series-test/) **Summary:** A geometric series converges exactly when the common ratio is smaller than one in size. **Question:** Geometric series are the one large family whose sums are known exactly, but they rarely arrive written as $\sum a r^n$. How do you see one through its disguise, read off its first term and ratio, and use it to judge the series around it? ## Why this exists Oyunaa strikes the A below middle C on a piano, a string that vibrates $220$ times a second. The hammer leaves $50$ millijoules of energy in the string, and in every cycle the string loses $0.8\%$ of the energy it holds at the start of that cycle, to the soundboard and the air. The energy given off in cycle $n$, in millijoules, is $$50(0.008)(0.992)^{n-1} = 0.4(0.992)^{n-1}$$ and the total given off is the series $\sum_{n=1}^{\infty} 0.4(0.992)^{n-1}$. Each term is the one before it times $0.992$: the series is geometric. Most series have no formula for their partial sums, and partial-sums-of-a-series could find one only by telescoping. Geometric series are the large family where the partial sums are known exactly, so they are the first thing to look for in a new series. They seldom arrive in the form $\sum a r^n$, though. They come as $\frac{5^{n+1}}{7^{n-1}}$, as $\frac{(-2)^n}{3^{2n+1}}$, starting at $n = 2$, or added to another geometric series. This lesson is about seeing them, reading off the two numbers that decide them, and using them as the yardstick that the later convergence tests measure other series against. ## Definition In plain words: a series is geometric when each term is the previous term times one fixed number. A series $\sum_{n=m}^{\infty} b_n$ with every $b_n \ne 0$ is **geometric with ratio $r$** if, for every $n \ge m$, $$\frac{b_{n+1}}{b_n} = r$$ ### [The integral test and p series](https://math.wiki/p-series-test/) **Summary:** The sum of one over n to a power converges exactly when the power is bigger than one. **Question:** The terms of a positive series often have no formula for their partial sums. When the terms are the values of a decreasing function, can the area under its graph decide whether the series has a sum, and say how close a partial sum comes to it? ## Why this exists Pekka sets up an idealized row of identical loudspeakers along a straight line, one every $3$ meters, the first $3$ meters from where Aatos stands, the row running on without end. A single speaker delivers sound intensity $\frac{k}{d^2}$ at distance $d$ (the inverse square law for a point source), so the total intensity reaching Aatos is $$\sum_{n=1}^{\infty} \frac{k}{(3n)^2} = \frac{k}{9}\sum_{n=1}^{\infty} \frac{1}{n^2}$$ Is it finite? The terms tend to $0$, so the nth-term-test is silent, and $\frac{1}{n^2}$ does not telescope the way the series in partial-sums-of-a-series do. Had the speakers been long line arrays, whose sound fades like $\frac{1}{d}$, the total would be a multiple of the harmonic series and infinite. The graph of $y = \frac{1}{x^2}$ passes through every term, and the area under it out to infinity is an improper integral already known to converge. The integral test turns such an area into a verdict on the series, and it settles every series $\sum \frac{1}{n^p}$ at once. ## Definition In plain words: when the terms of a series are the values at whole numbers of a positive, decreasing function, the series and the area under the graph are either both finite or both infinite. **The integral test.** Let $m$ be an integer and let $f$ be continuous, positive and decreasing on $[m, \infty)$, with $a_n = f(n)$ for every integer $n \ge m$. Then $\sum_{n=m}^{\infty} a_n$ and $\int_m^{\infty} f(x)\,dx$ both converge or both diverge. ### [The intermediate value theorem](https://math.wiki/bisection-method/) **Summary:** Halving an interval again and again to trap a root as closely as wanted. **Question:** A continuous function is negative at one end of an interval and positive at the other. Must it be zero somewhere in between, and if it must, how do you find where? ## When you reach for it Most equations cannot be solved by rearranging. No algebra isolates $x$ in $x\,2^x = 5$ or in $\ln x = 3 - x$. Two questions remain: is there a solution at all, and where is it? The first is answered by a theorem about continuous functions; the second by using that theorem again and again. **The intermediate value theorem.** Let $f$ be continuous on the closed interval $[a, b]$, and let $N$ be any number between $f(a)$ and $f(b)$. Then there is at least one number $c$ in $[a, b]$ with $f(c) = N$. In pictures: the graph of $f$ runs without a break from $(a, f(a))$ to $(b, f(b))$, and a horizontal line $y = N$ between those heights is below one end and above the other, so the graph must meet it. The theorem is stated here without proof. Its proof rests on the completeness of the real numbers, the fact that the number line has no gaps (every set of reals with an upper bound has a least upper bound), and belongs to a course in analysis. The most used case takes $N = 0$. If $f$ is continuous on $[a, b]$ and $f(a)$ and $f(b)$ have opposite signs, then $f$ has a zero strictly between $a$ and $b$: the theorem gives a zero in $[a, b]$, and neither end is a zero because $f$ is not zero there. This is how one shows that an equation has a solution, a root, without finding it. Notice what the theorem does not say. It promises at least one $c$, not exactly one, and it says nothing about where in $[a, b]$ the number $c$ lies. ### [The limit laws](https://math.wiki/sum-and-difference-limit-laws/) **Summary:** The limit of a sum is the sum of the limits when both limits exist. **Question:** Reading a limit off a graph or a table works, but it is slow and never quite certain. If you already know where two functions are heading, can you say exactly where their sum, product, quotient or composition is heading, without drawing anything? ## Why this exists In limits-from-a-graph a limit was read by sliding along a curve toward $x = a$ from both sides, or estimated from a table of inputs closing in on $a$. Both methods are evidence, not proof, and both are slow. Yet most functions in calculus are built from a few simple pieces by adding, multiplying, dividing and composing. If the limit of each piece is known, the **limit laws** give the limit of the whole by doing the same arithmetic on the pieces' limits. They turn a limit from something estimated into something computed exactly, and they show that for polynomials, and for many other functions, the limit at $a$ is simply the value at $a$. The lesson takes two days. Part 1 is the sum, difference, constant multiple and product laws, and what they say about polynomials; Examples 1 to 3 go with it. Part 2 is the quotient and root laws, direct substitution, and the limit of a composition; Examples 4 to 7 go with it. ## Definition In plain words: **when two functions each have a limit at $a$, the limit of their sum, difference, product or quotient is the sum, difference, product or quotient of their limits**, with one exception, a quotient whose bottom limit is $0$. In symbols, let $a$ be a number and suppose $$\begin{aligned} \lim_{x \to a} f(x) &= A \\ \lim_{x \to a} g(x) &= B \end{aligned}$$ where $A$ and $B$ are numbers, so both limits exist. Then $$\begin{aligned} \lim_{x \to a} \big[f(x) + g(x)\big] &= A + B \\ \lim_{x \to a} \big[f(x) - g(x)\big] &= A - B \\ \lim_{x \to a} \big[k\,f(x)\big] &= kA \\ \lim_{x \to a} \big[f(x)\,g(x)\big] &= AB \\ \lim_{x \to a} \frac{f(x)}{g(x)} &= \frac{A}{B} \end{aligned}$$ ### [The mean value theorem](https://math.wiki/rolles-theorem/) **Summary:** A smooth curve with equal endpoint heights must level off somewhere in between. **Question:** Two cameras time a car over a known stretch of highway and find an average speed above the limit. Without ever seeing the car speed, can they be sure its speedometer once showed that average? ## Why this exists A derivative is a local fact: $f'(c)$ describes $f$ in a tiny window around $c$. Most questions are about a whole interval. If the derivative is positive everywhere, does the function rise from one end to the other? If two functions have the same derivative, how different can they be? If a speed never passes $65$ miles per hour, how far can a car get in ten minutes? None of these follows from the definition of the derivative at a point alone. Something has to connect the slope at single points to the change over an interval. Biljana drives through a highway speed check that works this way. One camera records the moment her car passes, and a second camera a known distance down the road records it again. The authority divides distance by time and fines any average speed above the limit. Goce, a passenger, objects that the cameras never measured her speed at any instant. The mean value theorem answers him: a car whose average speed over the stretch is $75$ miles per hour was traveling at exactly $75$ miles per hour at some instant in between. ## Definition The lesson rests on two theorems already met. The extreme-value-theorem says a function continuous on $[a, b]$ reaches an absolute maximum and an absolute minimum there. **Fermat's theorem** (critical-points) says that if $f$ has a local maximum or minimum at a point $c$ inside an interval and $f'(c)$ exists, then $f'(c) = 0$. ### [The net change theorem in context](https://math.wiki/accumulation-in-context/) **Summary:** Totaling water, cars or charge from a rate given in the units of the situation. **Question:** A meter reads only how fast something is changing, such as cars per hour through a gate, kilowatts into a battery or barrels per hour through a pipe. How much did the amount itself change, and how much is there now? ## When you reach for it A rate is known and an amount is wanted. Wulan runs a parking garage and counts cars per hour at its gates. Ratri charges electric buses and reads kilowatts on the charger. Bagus collects rain from his roof and reads liters per hour off the downspout. The amount (cars, kilowatt-hours, liters) is what matters, yet the instruments report how fast it changes. interpreting-a-derivative-in-context read a rate as the derivative of an amount, in units of amount per time. This lesson runs the other way, from the rate back to the amount. The tool is the theorem of evaluating-a-definite-integral, read in the language of the situation. Day 1 takes rates given by formulas. Day 2 takes rates given by tables and graphs, and densities, where the variable is a position instead of a time. ## The procedure ### Day 1: the net change theorem **The theorem.** If $F'$ is continuous on $[a, b]$, then $$\int_a^b F'(t)\,dt = F(b) - F(a).$$ This is the fundamental theorem of evaluating-a-definite-integral, with $F$ as the antiderivative of $F'$, which it is by definition. Nothing new is proved here; what is new is the reading. If $F(t)$ is an amount at time $t$, then $F'(t)$ is the rate at which the amount changes, and the theorem says: **the integral of a rate over a time interval is the net change in the amount over that interval.** **Why the units multiply.** The integral is a limit of Riemann sums $\sum F'(t_k)\,\Delta t$ (definite-integral-as-a-limit). Each term is a rate times a short time, so it is about the change in the amount over that short time: kilowatts (kilowatt-hours per hour) times hours is kilowatt-hours. Adding the pieces adds the changes. So the units of $\int_a^b r(t)\,dt$ are the units of $r$ times the units of $t$, which are the units of the amount. ### [The nth term test for divergence](https://math.wiki/nth-term-test/) **Summary:** If the terms do not shrink to zero the series cannot converge. **Question:** A series whose terms stay near some fixed size, say three quarters, keeps piling up and can never settle on a total. What exactly must the terms of a series do for it to have a sum, and how much does that one fact decide? ## Why this exists Iroda models the growth of a stalagmite: in year $n$ it gains $\frac{3n}{4n+5}$ millimeters of height. Its height after $n$ years is the partial sum $S_n$ of the series $\sum_{n=1}^{\infty} \frac{3n}{4n+5}$, and no tidy formula for $S_n$ exists, so the telescoping of partial-sums-of-a-series has nothing to work on. Still, the yearly gains approach $\frac{3}{4}$ millimeter, and a total that keeps receiving about $\frac{3}{4}$ more each year cannot settle on any height. That observation, made exact, is the first convergence test. It asks only for the limit of the terms, a question about a sequence (sequence-limit-laws), and it works in one direction only: it can rule out a sum, never certify one. ## Definition In plain words: if the terms of a series do not shrink to $0$, the series has no sum. **The $n$th term test for divergence** (also called the divergence test). For a series $\sum_{n=1}^{\infty} a_n$: if $\lim_{n \to \infty} a_n$ does not exist, or exists and is not $0$, then the series diverges. "Does not tend to $0$" covers three cases: the terms tend to a nonzero number, the terms tend to $\infty$ or $-\infty$, or the terms have no limit at all. If $\lim_{n \to \infty} a_n = 0$, the test gives **no conclusion**. The series may converge or diverge, and another argument has to decide. ## Intuition Partial sums move in steps, and the steps are the terms: $S_n - S_{n-1} = a_n$. A sequence that settles on a number $S$ eventually stays inside any narrow band around $S$, and a sequence trapped in a band of width $2\varepsilon$ cannot take a step longer than $2\varepsilon$. So the steps must shrink to $0$. Turned around: if steps of length at least some fixed $c > 0$ keep coming, the partial sums never stay inside a band narrower than $c$, and they cannot settle. ### [The power rule and basic differentiation rules](https://math.wiki/derivative-of-a-polynomial/) **Summary:** Applying the power rule to each term of a polynomial. **Question:** Every derivative so far took a page of limit algebra. Is there a pattern in the answers that lets you read off the derivative of 4x^5 - 3x^2 + 7x - 9 without taking a limit at all? ## When you reach for it Finding a derivative from the definition works for any function, but each one costs a page of algebra: expand, cancel, let $h \to 0$. The answers it gave already rhyme. The derivative of $x^2$ is $2x$, of $x^3$ is $3x^2$, of $x^4$ is $4x^3$. The exponent comes down in front and drops by one. This lesson proves that pattern, the **power rule**, together with three rules for building functions out of pieces: the **constant rule**, the **constant multiple rule** and the **sum and difference rule**. Each is proved once from the limit definition $$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h},$$ and after that the limit never has to be taken again for these functions. Reach for these rules whenever a function is a sum of constant multiples of powers of $x$: every polynomial, and, after a rewrite, many expressions with roots and fractions. The lesson runs over three days. Part 1 proves the four rules. Part 2 differentiates polynomials and rewrites expressions so the rules apply. Part 3 extends the power rule to negative and fractional exponents and finds where it stops. ## The procedure ### Part 1: the rules, each proved from the definition **The constant rule.** If $f(x) = c$ for a constant $c$, then $f(x + h) - f(x) = c - c = 0$, so the difference quotient is $0$ for every $h \ne 0$ and its limit is $0$: $$\frac{d}{dx}\big[c\big] = 0.$$ The graph of a constant is a horizontal line, and a horizontal line has slope $0$ everywhere. ### [The product rule](https://math.wiki/product-rule-derivation/) **Summary:** Adding and subtracting a middle term splits the change of a product into two pieces. **Question:** The derivative of a sum is the sum of the derivatives. Is the derivative of a product the product of the derivatives, and if not, what is it? ## Why this exists The basic rules differentiate sums and constant multiples of powers. A product such as $x^3 e^x$ is neither, and it cannot be multiplied out. The first guess, that the derivative of a product is the product of the derivatives, fails on the simplest case there is. Take $f(x) = x$ and $g(x) = x$. Then $f(x)g(x) = x^2$, whose derivative is $2x$, while $f'(x)g'(x) = 1 \cdot 1 = 1$. These agree only at $x = \tfrac{1}{2}$. Units say the same thing. Chaltu runs a flower farm. Her revenue for a day is the price per stem times the stems sold that day. If time is in days, the price's rate is in dollars per stem per day and the sales rate is in stems per day per day. Their product is in dollars per day cubed, which is not a unit the rate of revenue (dollars per day, per day) can have. The rate of a product has to be built from different pieces, and this lesson finds them. ## Definition **The product rule.** If $f$ and $g$ are both differentiable at $x$, then the product $fg$ is differentiable at $x$ and $$(fg)'(x) = f'(x)\,g(x) + f(x)\,g'(x).$$ In words: the derivative of the first times the second, plus the first times the derivative of the second. The order of the two terms does not matter, since addition is commutative. In Leibniz notation, with $u$ and $v$ functions of $x$, $$\frac{d}{dx}\big[uv\big] = \frac{du}{dx}\,v + u\,\frac{dv}{dx}.$$ ## Intuition Picture $f(x)$ and $g(x)$ as the width and height of a rectangle, both positive, so that $f(x)g(x)$ is its area. Move from $x$ to $x + h$. The width grows by $\Delta f = f(x + h) - f(x)$ and the height by $\Delta g = g(x + h) - g(x)$. ### [The quotient rule](https://math.wiki/quotient-rule-derivation/) **Summary:** The quotient rule follows from the product rule applied to a reciprocal. **Question:** The product rule gives the derivative of f times g. What is the derivative of f divided by g, and where do its minus sign and its square come from? ## Why this exists The basic rules split a quotient only when the bottom is a single power of $x$. A function such as $\dfrac{2x + 3}{x^2 + 1}$ cannot be split that way, and the product rule, $(fg)' = f'g + fg'$, handles products, not quotients. The first guess, that the derivative of a quotient is the quotient of the derivatives, fails at once. Take $f(x) = x^2$ and $g(x) = x$. For $x \ne 0$ the quotient is $\frac{f(x)}{g(x)} = x$, whose derivative is $1$, while $\frac{f'(x)}{g'(x)} = \frac{2x}{1} = 2x$. The two agree only at $x = \tfrac{1}{2}$. This lesson finds the correct rule, and finds it from the product rule. ## Definition **The quotient rule.** If $f$ and $g$ are differentiable at $x$ and $g(x) \ne 0$, then $\frac{f}{g}$ is differentiable at $x$ and $$\left(\frac{f}{g}\right)'(x) = \frac{f'(x)\,g(x) - f(x)\,g'(x)}{g(x)^2}.$$ In words: the derivative of the top times the bottom, minus the top times the derivative of the bottom, all over the bottom squared. Unlike the product rule, the order matters: subtraction is not commutative, and swapping the two terms changes the sign. In Leibniz notation, with $u$ and $v$ functions of $x$, $$\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{v\,\dfrac{du}{dx} - u\,\dfrac{dv}{dx}}{v^2}.$$ The special case $f = 1$ is the **reciprocal rule**: $\left(\dfrac{1}{g}\right)' = -\dfrac{g'}{g^2}$ wherever $g$ is differentiable and not $0$. ## Intuition Start with the reciprocal. Where $g$ is positive and rising, $\frac{1}{g}$ is falling, so the derivative of $\frac{1}{g}$ must have the sign opposite to $g'$. That is the minus sign. The size of the change depends on how large $g$ already is: when $g$ is near $2$, a small change $\Delta g$ moves $\frac{1}{g}$ by about $\frac{\Delta g}{4}$, while near $10$ it moves it by only about $\frac{\Delta g}{100}$. That is the $g^2$ underneath. ### [The ratio and root tests](https://math.wiki/ratio-test/) **Summary:** Comparing each term with the one before it to judge convergence. **Question:** Factorials and powers make terms that are hard to compare with anything, yet the ratio of one term to the next is often simple. When does that ratio, or the nth root of a term, settle whether a series converges, and when does it say nothing? ## Why this exists Chipo runs a banquet hall with one long table that has a garden side and a wall side. At a banquet of $2n$ guests, each guest picks a side independently, the garden side with probability $p$. There are $\binom{2n}{n}$ ways to choose which $n$ guests sit by the garden (binomial-coefficients), so the table splits evenly, $n$ on each side, with probability (binomial-probability-of-exactly-k) $$\binom{2n}{n}p^n(1-p)^n = \frac{(2n)!}{(n!)^2}\bigl(p(1-p)\bigr)^n$$ Chipo hosts one banquet of each size $n = 1, 2, 3, \ldots$ and asks how many evenly split banquets to expect. Expected counts add, so the answer is the sum of these probabilities: an infinite series. For $p = \frac15$ the terms are $\frac{(2n)!}{(n!)^2}\left(\frac{4}{25}\right)^n$, a factorial count growing against a power shrinking. To use comparison one must guess a benchmark, and here the right one is not obvious. But the ratio of one term to the next is simple, and it settles toward $\frac{16}{25}$. A series whose terms shrink by a factor near $\frac{16}{25}$ at each step should behave like a geometric series with that ratio. The ratio test makes this precise, and the root test does the same with $\sqrt[n]{\lvert a_n \rvert}$, which suits terms raised to a power that grows with $n$. Both compare $\lvert a_n \rvert$ with a geometric series, so what they prove is absolute convergence. ## Definition ### Day 1: the ratio test ### [The second fundamental theorem of calculus](https://math.wiki/evaluating-a-definite-integral/) **Summary:** Finding an antiderivative and subtracting its values at the two ends. **Question:** A definite integral is defined as a limit of sums, which can take a page to compute. If an antiderivative of the integrand is known, can the integral be read off from it in one line? ## When you reach for it Finding $\int_0^8 \left(10 + \frac{t^2}{4}\right)dt$ from the definition (definite-integral-as-a-limit) took a formula for a sum of squares and a limit. The first fundamental theorem showed that the accumulation function $A(x) = \int_a^x f(t)\,dt$ of a continuous $f$ is an antiderivative of $f$. The second fundamental theorem turns this around: to find $\int_a^b f(x)\,dx$, take any antiderivative $F$ of $f$ and subtract, $F(b) - F(a)$. No sums and no limits. You reach for it whenever a definite integral has an integrand whose antiderivative you can write down: a polynomial or a power (antiderivative-of-a-power), a trig function, an exponential, $\frac{1}{x}$, or an inverse trig form (antiderivatives-of-trig-functions). It is the second half of the fundamental theorem of calculus, and it is how nearly every exact integral in the rest of calculus is found. ## The procedure ### Day 1: the theorem **Theorem (second fundamental theorem of calculus).** Let $f$ be continuous on $[a, b]$, and let $F$ be an antiderivative of $f$ on $[a, b]$: $F$ is continuous on $[a, b]$ and $F'(x) = f(x)$ for every $x$ in $(a, b)$. Then $$\int_a^b f(x)\,dx = F(b) - F(a).$$ The difference is written with an **evaluation bar**: $$F(x)\Big|_a^b = F(b) - F(a).$$ When $F$ has several terms they go in parentheses before the bar, so $\left(x^3 + x\right)\Big|_0^2 = 10 - 0 = 10$. - **Any antiderivative works.** Two antiderivatives of $f$ on an interval differ by a constant (a consequence of the mean value theorem), and the constant cancels: $\big(F(b) + C\big) - \big(F(a) + C\big) = F(b) - F(a)$. That is why the $+\,C$ is left out. - **Limits in either order.** With the conventions $\int_b^a = -\int_a^b$ and $\int_a^a = 0$ (signed-area), the formula also holds when the lower limit is the larger: both sides change sign when the limits swap, and both are $0$ when they are equal. - **The answer is a signed area**, because the integral is one. A negative answer means more area lies below the axis than above it. ### [The shell method](https://math.wiki/shell-method/) **Summary:** Building a solid from thin cylindrical shells parallel to the axis. **Question:** A region turned about a vertical line can be cut into thin vertical strips, and each strip sweeps out a thin tube. How do those tubes add up to a volume, and when is that easier than slicing across the axis? ## When you reach for it The disk-method and the washer-method cut a solid of revolution across the axis, so that every slice is a circle or a ring. That needs the radius of each slice as a function of the variable along the axis. Turn a region about the $y$-axis, and the slices are horizontal: their radii have to be read off as functions of $y$, which means solving the boundary $y = f(x)$ for $x$. For $y = x(x - 3)^2$ that is a cubic equation with two relevant roots at each height. The shell method cuts the other way: into thin strips **parallel** to the axis. A vertical strip turned about the $y$-axis sweeps a thin tube, a **cylindrical shell**, and its size is read straight from $f(x)$. You reach for shells when the boundary is given as $y = f(x)$ and the axis is vertical, or as $x = g(y)$ and the axis is horizontal, and solving for the other variable is hard or impossible. ## The procedure ### Day 1: shells about the $y$-axis **One shell.** A cylindrical shell is the solid between two cylinders with the same axis and the same height $h$, of radii $r_1 < r_2$. Its volume is the outer cylinder minus the inner one, as in volume-of-a-prism: $$\begin{aligned} \pi r_2^2 h - \pi r_1^2 h &= \pi (r_2 + r_1)(r_2 - r_1)\,h \\ &= 2\pi \left(\frac{r_1 + r_2}{2}\right) h\,(r_2 - r_1). \end{aligned}$$ With $r = \frac{r_1 + r_2}{2}$ the average radius and $\Delta r = r_2 - r_1$ the thickness, this is exactly ### [The squeeze theorem and the limit of sine x over x](https://math.wiki/limit-of-sin-x-over-x/) **Summary:** Near zero, sine of x and x itself are so close that their ratio approaches one. **Question:** Substituting $x = 0$ into $\dfrac{\sin x}{x}$ gives $\dfrac{0}{0}$, and no factoring can cancel a sine against an $x$. How can a limit be found when algebra has nothing to grab, and what does sine do near zero? ## Why this exists The limit laws find a limit from the limits of its pieces, and limits-by-factoring rescues a quotient that comes out as $\frac{0}{0}$ by canceling a common factor. Both fail on two kinds of function that calculus cannot do without. The first wobbles: $\cos \frac{2}{x}$ swings between $-1$ and $1$ infinitely often near $0$ and has no limit there, so the product law cannot be applied to $x^4 \cos \frac{2}{x}$. The second mixes a sine with a power of $x$: $\frac{\sin x}{x}$ is $\frac{0}{0}$ at $0$, and there is no common factor to cancel, because $\sin x$ is not a polynomial. The **squeeze theorem** handles both. Instead of computing a limit directly, it traps the function between two simpler functions heading to the same place. Day 1 of this lesson states and proves the theorem and uses it. Day 2 uses it, with a picture of the unit circle, to prove the central limit of trigonometric calculus: $$\lim_{x \to 0} \frac{\sin x}{x} = 1$$ That one fact, with $x$ in radians, is what makes the derivatives of sine and cosine possible. ## Definition **The squeeze theorem**, in plain words: if a function is trapped between two others near $a$, and the two outer functions approach the same number $L$, the trapped one has nowhere else to go, so it approaches $L$ too. In symbols: suppose that for every $x \ne a$ in some open interval around $a$, $$g(x) \le f(x) \le h(x)$$ and that $$\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$$ ### [The Taylor remainder and the Lagrange error bound](https://math.wiki/lagrange-error-bound/) **Summary:** A ceiling on the remainder built from the next derivative and the distance from the center. **Question:** A Taylor polynomial gives a number. Without knowing the true value, how far off can that number be, and does adding terms forever give the function back? ## Why this exists Lejla's thermistor meter turns a resistance into a temperature, and on the way its chip needs $e^{0.7}$. The chip can only add and multiply, so it evaluates a Taylor polynomial of $e^x$. In taylor-coefficients-from-derivatives the error of such a polynomial was watched by comparing with the true value, but the chip has no true value to compare with. It needs a guarantee computed from what is known in advance: "this polynomial is within $10^{-6}$." A second gap is left by maclaurin-series-for-e-to-the-x. The series for $\sin x$ converges for every $x$ (endpoint-testing-for-power-series gives the radius), but convergence says nothing about what it converges to. This lesson closes both gaps with one formula for the error. ## Definition ### Day 1: the remainder and its bound Let $P_n$ be the Taylor polynomial of degree $n$ of $f$ about $a$. The **remainder** is $$R_n(x) = f(x) - P_n(x)$$ so that $f(x) = P_n(x) + R_n(x)$ exactly. **Taylor's theorem with the Lagrange form of the remainder:** if $f$ has derivatives through order $n+1$ on an open interval containing $a$ and $x$, there is a number $c$ between $a$ and $x$ with $$R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x - a)^{n+1}$$ The remainder looks like the next term of the polynomial, except that the derivative is taken at an unknown point $c$ instead of at $a$. Since $c$ is unknown, what is used in practice is a bound. If $\lvert f^{(n+1)}(t) \rvert \le M$ for every $t$ between $a$ and $x$, then ### [The trapezoidal rule and Simpson's rule](https://math.wiki/trapezoidal-rule-accuracy/) **Summary:** Trapezoids sit above a concave up curve and below a concave down one. **Question:** The width of a field, or the depth of a river, is known only at evenly spaced marks. Joining the readings with straight lines, or with parabolas, estimates the area far better than rectangles do. How much better, and in which direction is the estimate wrong? ## Why this exists Curdin wants the area of a field whose far edge wanders; every $10$ meters along a straight fence he measures the width across. Ladina and Gian want the area of a river's cross-section; they wade across reading the depth with a pole. Either way the data is a table of heights at known positions, and the area wanted lies under the graph through them. Left, right and midpoint sums give each piece a rectangle with one height, yet each piece of a table has a measured height at both ends. Joining them with a straight segment makes a trapezoid, which follows a slanting edge exactly; joining three neighboring points with a parabola follows a bending edge too. These are the trapezoidal rule and Simpson's rule, and concavity tells which way the trapezoids err. ## Definition Let $f$ be continuous on $[a, b]$, cut into $n$ pieces of width $\Delta x = \frac{b-a}{n}$, with cut points $x_k = a + k\,\Delta x$ and heights $y_k = f(x_k)$. For $f \ge 0$ write $A$ for the area under the graph; its exact value is the integral $\int_a^b f(x)\,dx$ of definite-integral-as-a-limit. The trapezoid on piece $k$ has parallel sides $y_{k-1}$ and $y_k$ and width $\Delta x$, so its area is $\Delta x \cdot \frac{y_{k-1} + y_k}{2}$ (area-of-parallelograms-and-trapezoids). The **trapezoidal sum** adds them: $$T_n = \frac{\Delta x}{2}\big(y_0 + 2y_1 + 2y_2 + \dots + 2y_{n-1} + y_n\big).$$ Each inside height is a side of two trapezoids, so it counts twice. With pieces of different widths $\Delta x_k = x_k - x_{k-1}$, each trapezoid takes its own width: the estimate is $\sum_{k=1}^{n} \frac{y_{k-1} + y_k}{2}\,\Delta x_k$. ### [The washer method](https://math.wiki/washer-method/) **Summary:** Subtracting an inner circle from an outer one when the solid has a hole. **Question:** When a region is turned about a line it does not touch, the solid has a hole down its middle. How do you find the volume of a solid with a hole? ## When you reach for it The disk-method turns a region that lies against the axis of revolution, so every slice across the axis is a filled circle. Many solids are not like that. A pipe fitting, a bushing or a bead has a hole down its center, because the region that sweeps it out stays some distance away from the axis. Each slice across the axis is then a circle with a smaller circle cut out of its middle, shaped like the flat metal washer under a bolt. You reach for the washer method when the region lies between two boundaries, both on the same side of the axis, and at least part of it is separated from the axis by a gap. It is the disk method applied twice, once to the outer boundary and once to the inner one. ## The procedure ### Day 1: washers about a coordinate axis **The slice is a washer.** Let $f$ and $g$ be continuous on $[a, b]$ with $0 \le g(x) \le f(x)$, and turn the region between them about the $x$-axis. The plane perpendicular to the axis at $x$ meets the region in the segment from $(x, g(x))$ to $(x, f(x))$. Turning that segment about the axis sweeps every point whose distance from the axis lies between $g(x)$ and $f(x)$: a ring with $$\begin{aligned} R(x) &= f(x) \quad \text{(outer radius)}, \\ r(x) &= g(x) \quad \text{(inner radius)}. \end{aligned}$$ **The area of a ring.** The ring is the disk of radius $R$ with the disk of radius $r$ removed (Figure 1), so its area is ### [Trigonometric integrals](https://math.wiki/trigonometric-integrals/) **Summary:** Using identities to rewrite powers of sine, cosine, secant and tangent before integrating. **Question:** The basic table integrates $\sin x$ and $\sec^2 x$, but not $\sin^3 x\cos^4 x$ or $\tan^4 x\sec^4 x$. Which identity turns a product of trig powers into something substitution can finish? ## When you reach for it antiderivatives-of-trig-functions integrates a single trig function, and substitution integrates $\sin^4 x\cos x$ because the factor $\cos x$ is the derivative of the inside. Most products of trig powers have no such factor in plain view. In $\sin^3 x\cos^4 x$, $\sin^4 x\cos^2 x$ or $\tan^4 x\sec^4 x$, the integrand must first be rewritten, using an identity, until one factor is the derivative of a function and everything else is a function of that same function. You reach for these rewrites whenever the integrand is a product of powers of sine and cosine, a product of sines and cosines at different frequencies, or a product of powers of tangent and secant (cotangent and cosecant behave like tangent and secant). They also finish every integral produced by trigonometric-substitution, which turns a square root into exactly these products. ## The procedure ### Day 1: sine and cosine Three facts do all the work. The Pythagorean identity $\sin^2 x + \cos^2 x = 1$ (derive-the-pythagorean-identity) trades an even power of one function for a polynomial in the other. The derivatives $\frac{d}{dx}\sin x = \cos x$ and $\frac{d}{dx}\cos x = -\sin x$ supply the $du$. The power-reducing-formulas $\sin^2 x = \frac{1 - \cos 2x}{2}$ and $\cos^2 x = \frac{1 + \cos 2x}{2}$ halve a power by doubling the angle. **An odd power of sine.** In $\int \sin^m x\cos^n x\,dx$ with $m$ odd, write $m = 2k + 1$ and set one factor $\sin x$ aside: ### [Trigonometric substitution](https://math.wiki/trigonometric-substitution/) **Summary:** Replacing a variable with a trig function to clear a square root. **Question:** The root $\sqrt{49 - x^2}$ has no inside derivative sitting beside it, so u-substitution cannot touch it. Can the variable itself be renamed so that the root disappears? ## When you reach for it u-substitution works when the derivative of an inside expression sits in the integrand as a factor. In $\int \sqrt{49 - x^2}\,dx$ the inside is $49 - x^2$, its derivative $-2x$ is nowhere to be seen, and no choice of $u$ helps. Yet roots of this kind are everywhere: $\sqrt{49 - x^2}$ is a half circle, the area of an ellipse and the length of a parabola's arc both lead to such roots, and partial fractions leaves behind powers of $x^2 + a^2$ that call for the same move. Riemann sums can only approximate the area under a half circle; the exact value needs an antiderivative of the root. The idea is to rename $x$ itself as a trig function of a new angle $\theta$, chosen so that a Pythagorean identity turns the expression under the root into a perfect square. Reach for it when an integrand holds $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$ or $\sqrt{x^2 - a^2}$ with $a > 0$, a power of $a^2 \pm x^2$ or $x^2 - a^2$, or any quadratic that completing the square brings to one of these shapes, and a plain substitution does not already do the job. ## The procedure ### Day 1: substituting for x itself Ordinary substitution names a new variable as a function of $x$. Here $x$ is named as a function of the new variable, $x = g(\theta)$, and the question is why that is allowed. **Theorem (inverse substitution).** Let $g$ have a continuous derivative on an interval $J$ with $g'(\theta) \ne 0$ inside $J$, so that $g$ is one-to-one there and has an inverse $\theta = g^{-1}(x)$ on the interval $I$ of values $g$ takes. If $G'(\theta) = f\big(g(\theta)\big)\,g'(\theta)$ on $J$, then $G\big(g^{-1}(x)\big)$ is an antiderivative of $f$ on $I$. ### [u-substitution](https://math.wiki/u-substitution-indefinite/) **Summary:** Replacing the inner function and its differential, then changing back at the end. **Question:** The chain rule differentiates any composition. Can it be run backward, so that an integrand built from a function of a function and the inner derivative can be antidifferentiated in one step? ## When you reach for it The rules in antiderivative-of-a-power and antiderivatives-of-trig-functions antidifferentiate $x^n$, $\sin x$, $e^x$ and $\frac{1}{x}$ when the input is $x$ itself, or at most $kx$. They say nothing about $2x\sqrt{x^2 + 9}$ or $x^2\cos(x^3 + 1)$. Yet these integrands have a recognizable look: a function of an inside expression, multiplied by the derivative of that inside expression. That is exactly what the chain rule produces when it differentiates a composition. Substitution reads the chain rule backward. You reach for it when an integrand contains a composition whose inside, or something close to it, has its derivative sitting in the integrand as a factor, up to a constant. With the second fundamental theorem it also evaluates definite integrals, and it is the most used technique of integration after the basic table. ## The procedure ### Day 1: the chain rule read backward Suppose $F$ is an antiderivative of $f$, so $F' = f$. By the chain rule, $$\begin{aligned} \frac{d}{dx}\Big[F\big(g(x)\big)\Big] &= F'\big(g(x)\big)\,g'(x) \\ &= f\big(g(x)\big)\,g'(x). \end{aligned}$$ Read as a statement about antiderivatives, this is the whole method. **Theorem (substitution rule).** Let $g$ be differentiable on an interval $I$ with $g'$ continuous there, and let $f$ be continuous on an interval $J$ that contains every value $g(x)$ for $x$ in $I$. If $F$ is an antiderivative of $f$ on $J$, then on $I$ ### [Using series to evaluate limits and integrals](https://math.wiki/approximating-an-integral-with-a-series/) **Summary:** Integrating a series term by term to estimate a definite integral. **Question:** The integral $\int_0^{0.6} e^{-x^3}\,dx$ has no antiderivative to plug into, and the limit of $\frac{e^{x^2} - 1 - x^2}{x^4}$ at $0$ takes four rounds of L'Hopital's rule. Can a series settle both in a few lines, and say how accurate the answer is? ## When you reach for it Two kinds of problem call for this move. **A definite integral with no usable antiderivative.** The Fundamental Theorem of Calculus needs an antiderivative written with the familiar functions, and many integrands have none: $e^{-x^3}$, $\cos(x^2)$, $\frac{1 - e^{-x}}{x}$ (choosing-an-integration-technique states such results). If the integrand has a power series, integrating-a-series-term-by-term turns the integral into a numerical series. Its partial sums approximate the integral, and the error of a partial sum can be bounded, so the answer comes with a guaranteed accuracy. **A limit of an indeterminate form at the center of a series.** L'Hopital's rule differentiates top and bottom until one of them stops vanishing. Writing each function as its series does the same work at once: the lowest power that survives decides the limit, and the next power says how the expression approaches it. Both rest on the series of maclaurin-series-for-e-to-the-x: $e^x$, $\sin x$, $\cos x$ for every $x$, and $\ln(1+x)$, $\arctan x$ near $0$, with substitutions such as $x \to -x^3$. ## The procedure **Integrals.** Suppose $f(x) = \sum c_n x^n$ for $\lvert x \rvert < R$ and $[0, b]$ lies inside $(-R, R)$. Term by term integration gives $$\int_0^b f(x)\,dx = \sum_{n=0}^{\infty} \frac{c_n\,b^{n+1}}{n+1}$$ a series of numbers with sum exactly the integral. An integrand such as $\frac{1 - e^{-x}}{x}$, undefined at $0$, is given its limit there; its series then holds on the whole interval, and the integral is an ordinary one. ### [Vector valued functions and motion in the plane](https://math.wiki/speed-and-distance-in-planar-motion/) **Summary:** Speed is the length of the velocity vector and distance is the integral of that speed. **Question:** A sailplane circles in rising air while the wind carries it east. Its position is given by two formulas of time, one for how far east it is and one for how far north. How do we get its velocity, its speed and heading, tell whether it is gaining speed, and find how far it really flies? ## When you reach for it An object moving in a flat plane, a puck on an air hockey table or a sailplane seen from above, has two coordinates that change with time. Package them as one **position vector** $\mathbf{r}(t) = \langle x(t), y(t) \rangle$ and every question about the motion becomes a question about a derivative or an integral of that vector. Which way is it going, and how fast? Is it at rest, or moving straight up the page? Is it gaining speed? Given only its acceleration and a start, where is it later? How far has it flown, as against how far it ended up from where it began? particle-at-rest-and-direction-change did this on a line, where the sign of $v$ is the direction; speed-of-a-parametric-curve found the speed $\frac{ds}{dt} = \sqrt{x'(t)^2 + y'(t)^2}$ and the length of a parametric curve. Here velocity becomes a vector, speed is its length, and direction is the vector's direction. Vectors (component form, length, scalar multiples, the dot product) come from component-form-of-a-vector, magnitude-of-a-vector, scalar-multiple-of-a-vector, force-and-velocity-vectors, dot-product-as-magnitudes-and-cosine and orthogonal-vectors; adding vectors and scaling them are done component by component (vector-operations). ## The procedure ### Day 1: derivatives, velocity, speed and direction **Vector valued functions.** A **vector valued function** assigns to each $t$ in its domain a vector $\mathbf{r}(t) = \langle x(t), y(t) \rangle = x(t)\,\mathbf{i} + y(t)\,\mathbf{j}$. Drawn from the origin, the tip of $\mathbf{r}(t)$ traces the parametric curve $x = x(t)$, $y = y(t)$, with the orientation of parameter-and-orientation. A limit of $\mathbf{r}$ is taken component by component: $\lim_{t \to c} \mathbf{r}(t) = \langle \lim_{t \to c} x(t), \lim_{t \to c} y(t) \rangle$ when both limits exist. This is right because $\mathbf{r}(t)$ gets close to a vector $\mathbf{L}$ exactly when each component gets close to the matching component of $\mathbf{L}$: the distance between two vectors is at least each component difference and at most their sum. ### [Volumes with known cross sections](https://math.wiki/volumes-with-known-cross-sections/) **Summary:** Stacking squares, triangles or semicircles built on a base region. **Question:** A solid is not a prism, a pyramid or a ball, but every slice of it is a shape whose area you can work out. Is that enough to find its volume? ## When you reach for it Geometry gives the volume of a prism, a cylinder, a pyramid, a cone and a ball. Many solids are none of these: a block whose top follows a curve, a stone that tapers unevenly. What such a solid often does have is cross sections you can describe. A cross section is the flat shape a plane leaves when it cuts a solid. If every cut perpendicular to one line is a square, a semicircle or a triangle whose size is known at each position, the volume is an integral. You reach for this method when the area of each slice is known as a function of position, either directly, from a table of measurements, or because each slice is a known shape standing on a base region drawn in the plane. ## The procedure ### Day 1: volume as the integral of slice area **Slicing.** Place an $x$-axis so that the solid lies between the planes $x = a$ and $x = b$, and let $A(x)$ be the area of the cross section in the plane perpendicular to the axis at $x$. Cut the solid at $a = x_0 < x_1 < \dots < x_n = b$. The slab between $x_{i-1}$ and $x_i$ has thickness $\Delta x_i = x_i - x_{i-1}$. If it is thin, its cross sections hardly change across it, so it is nearly a prism with base area $A(x_i^*)$, for a sample point $x_i^*$ in the piece, and height $\Delta x_i$. A prism's volume is base area times height, so $$V \approx \sum_{i=1}^{n} A(x_i^*)\,\Delta x_i.$$ This is a Riemann sum for $A$ on $[a, b]$. When $A$ is continuous, these sums approach the integral of $A$ as the pieces get thinner (definite-integral-as-a-limit), and that limit is taken as the **definition** of the volume: ### [What a limit is: graphs and tables](https://math.wiki/limits-from-a-graph/) **Summary:** Finding what value a graph approaches near a point by looking at the curve from both sides. **Question:** A formula can refuse an input, or have its value there changed, and still have outputs near that input crowding around one height. How do you name that height, and how do you read it off a graph or a table? ## When you reach for it To know what a function does *at* an input, you evaluate it. Often the question is what it does *near* the input. The formula $\frac{x^2 - x - 6}{x - 3}$ refuses $x = 3$, yet its outputs near $3$ crowd around one height. In estimate-a-rate-at-an-instant a speed at an instant was estimated by average speeds over shrinking intervals, and the estimates closed in on one number. A graph settling onto a horizontal asymptote and the sum of an infinite geometric series are numbers that outputs approach without reaching them in one step. The **limit** is the single idea under all of these: the height a function heads toward as its input heads toward a target. The lesson takes two days. Part 1 is the idea of a limit, one-sided limits, and how a limit differs from the value of the function; Examples 1 to 4 go with it. Part 2 is limits that do not exist, limits as $x \to \infty$, and the infinite sum as a limit; Examples 5 to 7 go with it. ## The procedure ### Part 1: the idea of a limit **Definition.** Let $f$ be defined at every input near $a$ on both sides, except perhaps at $a$ itself. Suppose that for every input $x \ne a$ close enough to $a$, the output $f(x)$ is as close as you like to one number $L$. Then $L$ is the **limit of $f(x)$ as $x$ approaches $a$**, written $$\lim_{x \to a} f(x) = L$$ Three things in the definition matter. The input approaches $a$ from both sides. The input never equals $a$, so the value $f(a)$, if there is one, plays no part. And "as close as you like" means any closeness at all can be demanded, and it is met once $x$ is near enough to $a$. Making that sentence exact, with a named tolerance and a window of inputs, is the epsilon delta definition met in calculus. ### [Work as an integral](https://math.wiki/work-as-an-integral/) **Summary:** Adding force times distance over many tiny displacements. **Question:** Lifting a sack a fixed height takes force times distance. What does it take when the force changes along the way, or when different parts of the load travel different distances? ## When you reach for it In physics, a constant force $F$ pushing an object a distance $d$ along its line of action does **work** $W = Fd$. With $F$ in newtons and $d$ in meters the work is in newton-meters, called **joules**; with $F$ in pounds and $d$ in feet it is in **foot-pounds**. Lifting something at a steady speed takes a force equal to its weight, so raising a $20$ pound sack $4$ feet takes $80$ foot-pounds. In metric units weight is mass times $g = 9.8$ meters per second squared, so a $10$ kilogram mass weighs $98$ newtons. The formula $W = Fd$ stops working in two situations, and this lesson handles both with a definite integral: - the force changes as the object moves (a spring pulls harder the farther it is stretched, a leaking bucket gets lighter as it rises, a cable being wound up gets shorter); - different parts of the load travel different distances (water at the bottom of a tank has farther to go than water at the top). ## The procedure ### Day 1: a force that changes along the way **The integral.** An object moves along an axis from $x = a$ to $x = b$, pushed in the direction of motion by a force $F(x)$ that depends on position, with $F$ continuous on $[a, b]$. Cut $[a, b]$ into $n$ pieces of width $\Delta x$ and pick a point $x_i^*$ in the $i$-th piece. Over a short piece the force barely changes, so it is close to the constant $F(x_i^*)$, and the work over that piece is close to $F(x_i^*)\,\Delta x$. Adding the pieces gives ### [Writing a differential equation from a description](https://math.wiki/modeling-with-differential-equations/) **Summary:** Turning a sentence about how fast something changes into an equation. **Question:** A law of nature or a rule of thumb often says only how fast something changes, in words. How do you turn that sentence into an equation you can work with? ## When you reach for it Many statements about the world describe a rate, not an amount. "The herd grows faster the bigger it is." "A hot pan cools faster in a colder room." "Salt water runs in at the top and the mixture drains out the bottom." Each says how fast a quantity changes, and how fast depends on the quantity itself. Such a sentence becomes a **differential equation**: an equation for an unknown function that involves its derivative. You already read a derivative as a rate with units and a sign (interpreting-a-derivative-in-context); this lesson runs that skill backward, from the sentence to the equation. Solving the equations comes later. Here the job is to write them correctly and to read what they already say. ## The procedure ### Name the variables and their units Start by naming the unknown function, the independent variable and their units: "let $T$ be the pan's temperature in degrees Fahrenheit, $t$ minutes after it leaves the stove." Then $\frac{dT}{dt}$ is in degrees per minute, and every term on the right-hand side must carry those same units. That is the most useful single check on a model. ### Turn the phrases into symbols To say $z$ is **proportional to** $x$ means $z = kx$ for a constant $k$, the **constant of proportionality**; **inversely proportional to** $x$ means $z = \frac{k}{x}$; **jointly proportional to** $x$ and $y$ means $z = kxy$. When $z$ is a rate, these give the following forms. ## Fractions, ratio and percent ### [A fraction as division](https://math.wiki/fraction-as-division/) **Summary:** The bar in a fraction means divide, so three fourths is three things shared among four. **Question:** Three lasagnas, four hungry people, and nobody wants to be left out. How much does each person get, and what does a fraction have to do with sharing? ## Why this exists It is potluck night. Charu has brought $3$ lasagnas, and $4$ people will share them equally: Charu, Goro, Ijeoma and Goro's brother. How much lasagna does each person get? Whole-number division gives a strange answer. $3 \div 4$ is $0$ with $3$ left over: nobody gets a whole lasagna, and all three are left on the table. That is true, but nobody is going to leave the lasagnas uneaten. They will cut them. The question is how much each person ends up with, and the answer is less than one lasagna, so it cannot be a whole number. Fractions answer it. Each person gets $\frac{3}{4}$ of a lasagna. The same fraction that names "three of four equal pieces" also names "three wholes shared four ways". That is the whole idea of this lesson: the bar in a fraction means divide. Before this, you met fractions as parts of one whole (fraction-as-part-of-whole) and as parts of a group of objects (fraction-of-a-set: $\frac{1}{4}$ of a set of $12$ cookies is $3$ cookies). Sharing objects that can be counted out is the second picture. This lesson shares wholes that have to be cut. The lesson takes three days. Part 1 shares a few wholes among more people, so each share is less than one. Part 2 shares more wholes than people, and turns a division with a remainder into a mixed number. Part 3 reads word problems: which number is divided by which, and between which two whole numbers the answer lies. ### [A fraction as part of a whole](https://math.wiki/fraction-as-part-of-whole/) **Summary:** Split a whole into equal parts; the fraction says how many parts you have. **Question:** A sandwich is cut and shared. When does a piece really count as a half or a third, and how does one number say how much you have? ## What this area is about This is where fractions begin. A whole is cut into equal parts, and a fraction $\frac{a}{b}$ says that $a$ of those $b$ equal parts are taken. Two rules hold it up: the parts must be the same size, and the whole must be named, since half of a cupcake is not half of a cake. ## How these fit together **The ground rules.** Equal shares and naming the whole defines halves, thirds and fourths as pieces of the same size that fill the whole exactly. Equal shares that look different adds that same size does not mean same shape: two pieces are equal when they hold the same amount. **Three pictures of a fraction.** Area and length models cut a shape or a length into $b$ equal parts and shade $a$ of them, with $a$ at most $b$ here. A fraction of a set counts objects in a group instead, each object standing for one equal part. ## A path through it Equal shares first: everything else depends on it. The other three follow in any order. From here come unit fractions, the single pieces every fraction is built from, then fractions on the number line, where the length picture becomes a point, and later a fraction as division. ### [A fraction of a fraction of an amount](https://math.wiki/fraction-of-a-fraction-of-a-quantity/) **Summary:** Two cuts in a row, where the second one applies to what the first cut left. **Question:** Quang plants part of a berry patch with blueberries and puts netting over part of the blueberries. How much of the whole patch is under netting, and how many pounds is a part of a part of a harvest? ## When you reach for it Quang's family runs a berry farm. He plants $\frac{3}{4}$ of a berry patch with blueberries. Birds love blueberries, so he puts netting over $\frac{2}{3}$ of the blueberry part. How much of the whole patch is under netting? That is two cuts in a row. The first cut takes $\frac{3}{4}$ of the patch. The second cut takes $\frac{2}{3}$, but not of the patch: of what the first cut left, the blueberry part. Finding a part of a part is multiplying a fraction by a fraction, which a fraction times a fraction already explains with area models and a rule. This lesson uses that rule on real amounts: pounds of berries, acres of pasture, gallons of water. It also handles the story where the second cut is of what is **left over**, which needs one more step first. The lesson runs over three days. Day 1 finds a fraction of a fraction of one whole thing, with tape diagrams. Day 2 does the same with an amount, such as $60$ pounds, and shows two ways that always agree. Day 3 takes stories where the second cut is of "the rest". ## The procedure ### Day 1: two cuts in a row A **tape diagram** is a long bar that stands for the whole thing. Cutting it into equal parts shows a fraction of it. Draw the berry patch as a bar and cut it into $4$ equal parts. The blueberries are $3$ of those parts. Now the second cut works on those $3$ parts only. Two thirds of $3$ parts is $2$ parts, because $3$ parts split into $3$ equal groups gives $1$ part in each group, and $2$ groups are $2$ parts. ### [A fraction of a set](https://math.wiki/fraction-of-a-set/) **Summary:** A fraction that counts objects in a group instead of pieces of one object. **Question:** Three of the eight colored pencils in a box are red. Nothing has been cut into pieces, so how can a fraction say how much of the box is red? ## Why this exists In fraction-area-model the whole was one thing, a cake or a strip, cut into equal parts. Many wholes are not one thing. Vaiva's box holds $8$ colored pencils, and $3$ of them are red. Rokas's gumball machine holds $6$ gumballs, and $5$ are green. Milda has a bucket of $12$ seashells. Nobody cuts a pencil or a gumball. Still there is a question to answer: how much of the box is red? A fraction answers it. The whole is a **set**, a group of objects counted together, and the fraction names part of the set by counting objects. Each object does the job that one equal piece did in a cake. Three of the $8$ pencils are red, so three eighths of the pencils are red. This is the third picture of a fraction, beside the area picture and the length picture. It is also the picture behind sharing a set into equal groups, which ties fractions to division. ## Definition A **set** is a group of separate objects counted together as one whole: the pencils in a box, the gumballs in a machine, the seashells in a bucket. If a set has $b$ objects and $a$ of them are picked out, those objects are $$\frac{a}{b}$$ of the set. The bottom number, the denominator, counts every object in the set. The top number, the numerator, counts the objects picked out (see numerator-and-denominator). A set can also be arranged in **equal groups**, groups with the same number of objects in each. If a set is in $b$ equal groups, each group is $\frac{1}{b}$ of the set, and $a$ of the groups are $\frac{a}{b}$ of the set. ### [A fraction of a whole number](https://math.wiki/fraction-of-a-whole-number/) **Summary:** Splitting a number into equal groups and taking some of those groups. **Question:** A train trip is 120 miles long, and the train has gone two thirds of the way. How many miles is that, and what does it mean to take a fraction of a number? ## When you reach for it Zubin, Zlata and Anahera are taking a long train trip. All day, questions come up that ask for a part of an amount. Three fourths of the train's cars carry passengers: how many cars is that? The trip is $120$ miles, and the train has gone two thirds of the way: how far is that? Zlata reads for three fifths of an hour: how many minutes? Each question names a fraction and a whole number, joined by the word "of". The whole number is the amount you start with: a set of things you can count, or a measurement such as miles or minutes. The fraction says what part of it you want. In the last lesson, fraction-times-whole-number, a whole number counted copies of a fraction. Here the roles turn around. The fraction acts on the whole number: it cuts the amount into equal groups and takes some of them. By the end of Part 2 you will see that both readings give the same product. ## The procedure This lesson takes three days, one part a day. ### Part 1: a fraction of a set The train has $16$ cars. Three fourths of them are passenger cars, and the rest are baggage cars. How many passenger cars are there? Start with the denominator. Fourths means $4$ equal parts, so sort the $16$ cars into $4$ equal groups. Sorting a set into equal groups and asking how many land in each group is division (see division-as-sharing): $$ 16 \div 4 = 4 $$ Each group has $4$ cars, and each group is one fourth of the train. So one fourth of $16$ is $4$. ### [A fraction times a fraction](https://math.wiki/fraction-times-fraction/) **Summary:** Taking a part of a part, which multiplies the tops together and the bottoms together. **Question:** Junko paints part of a banner blue, and Eshan paints part of the blue part gold. What fraction of the whole banner is gold? ## When you reach for it The art club is painting paper banners for the school fair. Junko paints part of a banner blue. Then Eshan paints gold letters over part of the blue part. How much of the whole banner is gold? That is a part of a part, and finding it is multiplying a fraction by a fraction. You already know that a fraction times a whole number takes a fraction of that number: $\frac{1}{4} \times 8$ is one fourth of $8$ (fraction-times-whole-number). The same word "of" carries over. $\frac{1}{2} \times \frac{1}{3}$ means one half **of** one third. The answer is a fraction of the whole banner, and this lesson finds it with a picture first and then with a rule the picture explains: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$ Multiply the tops (the numerators) and multiply the bottoms (the denominators). The lesson takes four days. Day 1 is a part of a part when both parts are unit fractions. Day 2 is any two fractions and why the rule is true. Day 3 is the area of a rectangle whose sides are fractions. Day 4 is simplifying before you multiply, and fractions greater than $1$. ## The procedure ### Day 1: a part of a part Take $\frac{1}{2} \times \frac{1}{3}$. Draw the banner as a rectangle. Junko paints $\frac{1}{3}$ of it blue: cut the banner into $3$ equal columns and shade one. Now Eshan paints gold on $\frac{1}{2}$ of the blue part. Half of one column is found by cutting that column into $2$ equal pieces. Cut the other columns the same way, with one line straight across the whole banner. Now the banner is $3$ columns by $2$ rows, so it has $3 \times 2 = 6$ equal pieces. The gold part is $1$ of them. ### [A fraction times a whole number](https://math.wiki/fraction-times-whole-number/) **Summary:** Taking several copies of a fraction, which is the same as taking a fraction of that number. **Question:** Five cats at a cat rescue each get three fourths of a cup of food at feeding time. How much food is that in all, and why does it come to fifteen fourths of a cup? ## When you reach for it Volkan, Bohdan and Zanele help out at a cat rescue. At feeding time every cat gets the same amount of dry food, measured out with a scoop. Five cats each get $\frac{3}{4}$ of a cup. How much food has to come out of the bag? Equal groups call for multiplication. Recall that $5 \times 3$ means five groups of three (see multiplication-as-groups). Here each group is a fraction of a cup, so the question is $5 \times \frac{3}{4}$: five copies of $\frac{3}{4}$. Reach for this move whenever the same fraction is taken again and again: the same food for each cat, the same part of a can for each kitten, the same amount every day for a week. The whole number counts the copies. The fraction says how big each copy is. ## The procedure This lesson takes three days, one part a day. ### Part 1: a whole number times a unit fraction Bohdan's smallest scoop holds $\frac{1}{4}$ cup. A small kitten gets one scoop. Five kittens get five scoops, five copies of $\frac{1}{4}$. Five copies of a number is five times that number, so the food is $5 \times \frac{1}{4}$ cups. A whole number times a fraction means the same thing $5 \times 3$ means: that many copies of the number, added up. $$ \begin{aligned} 5 \times \frac{1}{4} &= \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} + \frac{1}{4} \\ &= \frac{5}{4} \end{aligned} $$ The sum is easy to read by counting pieces. Each copy is one fourth of a cup, and there are five copies, so there are five fourths. The count of pieces is the numerator, $5$. The size of a piece, fourths, is the denominator, $4$. ### [Adding and subtracting fractions](https://math.wiki/add-subtract-fractions-unlike/) **Summary:** Rename to same-size pieces first. **Question:** Two amounts are counted in pieces of different sizes. How can they be put together, or one taken from the other, as a single fraction? ## What this area is about A fraction counts pieces, and its denominator names their size. Counts can be added or subtracted only when they count the same piece, so this whole area rests on one move: rename the fractions in matching pieces, then count. The denominator is never added. ## How these fit together **The counting case.** Like denominators needs no renaming: for a whole number $n \neq 0$, $\frac{a}{n} + \frac{b}{n} = \frac{a+b}{n}$, because both count the same piece $\frac{1}{n}$. **Renaming first.** Tenths and hundredths is the first case with different pieces: each tenth becomes $10$ hundredths. Unlike denominators does the same for any two denominators, cutting both into a common piece before counting. **Judging and using it.** Estimating rounds each fraction to $0$, $\frac{1}{2}$ or $1$ to see roughly how big an answer must be. Word problems decide, from a story, whether pieces are being joined or taken away. ## A path through it Like denominators is the plain case, and tenths and hundredths the first renaming. The one fixed step is that unlike denominators comes before the word problems, which use it. Estimating can sit anywhere and is worth doing beside every exact sum. Next come adding and subtracting mixed numbers, and later operations with rational numbers, where the same renaming works with negative fractions too. ### [Adding and subtracting fractions with like denominators](https://math.wiki/add-fractions-like-denominators/) **Summary:** Counting unit fractions that are already the same size, so only the top numbers add. **Question:** At the class party, Basak eats some slices of a pizza and Taras eats some more slices of the same pizza. How much of the pizza did they eat together, written as one fraction? ## When you reach for it The class party has pizza. Every pizza is the same size and is cut into equal slices. Basak eats some slices from one pizza, and Taras eats some more from the same pizza. How much of the pizza is gone? Later, Vuk carries part of a pizza away to another table. How much is left on the tray? Questions like these join amounts or take them apart, and the amounts are fractions of one whole. When every fraction in the question has the same denominator, the pieces are all the same size. Then adding and subtracting is only counting pieces. You already know the other direction. In decompose-fraction-into-unit-fractions one fraction was broken into a sum of smaller parts. Adding reads that same kind of equation from the parts back to the total. ## The procedure This lesson takes three days, one part a day. ### Part 1: adding pieces of the same size A pizza at the party is cut into $8$ equal slices. Each slice is $\frac{1}{8}$ of the pizza, a unit fraction. Basak eats $3$ slices, which is $\frac{3}{8}$ of the pizza. Taras eats $4$ slices, which is $\frac{4}{8}$ of the pizza. Together they ate $$ \frac{3}{8} + \frac{4}{8} $$ Look at the picture and count. Basak's slices and Taras's slices are $7$ slices in all, and every one of them is an eighth of the pizza. So they ate $\frac{7}{8}$ of the pizza: $$ \frac{3}{8} + \frac{4}{8} = \frac{7}{8} $$ Here is the same thing written with unit fractions. Each fraction is a count of eighths, so write out the eighths and put them together: ### [Adding and subtracting fractions with unlike denominators](https://math.wiki/add-unlike-any-denominators/) **Summary:** Renaming both fractions in shared pieces before the counts can be added. **Question:** Aiyana practices at the barre for one half of an hour, then works on jumps for three tenths of an hour. Halves and tenths are different sizes, so how do you add them? ## When you reach for it Aiyana, Dohyun and Lanre take a ballet class. They keep a log of their practice time, in fractions of an hour, and they learn their routine a piece at a time, so they talk about fractions of a routine too. After class they make paper banners for the spring show, and each banner gets painted in parts. When two amounts are counted in the same pieces, adding them is easy. Two tenths and three tenths are five tenths, because tenths can be counted like apples (add-fractions-like-denominators). But Aiyana's log says $\tfrac{1}{2}$ hour at the barre and $\tfrac{3}{10}$ hour of jumps. Halves and tenths are different sizes. You reach for this lesson whenever two fractions with different denominators must be joined or one taken from the other. It has four parts, one for each day: - **Part 1:** why different pieces cannot be counted together, and two pictures that fix it: fraction strips and area models. - **Part 2:** adding when one denominator is a multiple of the other, and simplifying the answer. - **Part 3:** adding when neither denominator is a multiple of the other, using the product of the denominators or the least common denominator. - **Part 4:** subtracting, sums greater than $1$, and adding three fractions. ## The procedure ### Part 1: the pieces must match A fraction counts pieces. $\tfrac{1}{2}$ is $1$ piece of size one half. $\tfrac{3}{10}$ is $3$ pieces of size one tenth. Adding means putting the pieces together and saying how many there are. But $1$ big piece and $3$ small pieces are not "$4$" of anything. There is no single piece size to name. ### [Adding and subtracting mixed numbers with like denominators](https://math.wiki/add-and-subtract-mixed-numbers-like-denominators/) **Summary:** Mixed numbers with the same denominator are added and subtracted whole part and fraction part, trading a whole for fractions when needed. **Question:** At the school track meet, Ufuk runs $3\tfrac{2}{5}$ miles in the morning and $1\tfrac{4}{5}$ miles in the afternoon. How far is that in all, and how do you take one mixed number away from another when the fraction part you take is the bigger one? ## When you reach for it Ufuk, Agata and Habiba train for the school track meet. They count how far they run in miles, and the distances are rarely whole miles. Agata jogs $2\tfrac{1}{5}$ miles to warm up and then runs $1\tfrac{2}{5}$ miles more. Habiba still has part of a long run left to go. How far in all? How far is left? How much farther did one runner go than another? Each question joins two amounts or takes one away from another, and the amounts are mixed-numbers: a whole number and a fraction part together. A mixed number such as $2\tfrac{1}{5}$ means $2 + \tfrac{1}{5}$. When every fraction part has the same denominator, the pieces are all the same size, so the work comes down to two things you already know: adding and subtracting whole numbers, and adding and subtracting fractions with a common denominator (add-fractions-like-denominators). The one new move is trading. Sometimes the fraction parts add up to more than a whole, and sometimes a whole must be cut into pieces before you can take pieces away. Both trades rest on the two ways of naming an amount from mixed-to-improper. The lesson takes three days, one part a day: - **Part 1:** adding mixed numbers, wholes with wholes and fraction parts with fraction parts. - **Part 2:** adding when the fraction parts make more than a whole. - **Part 3:** subtracting mixed numbers, trading a whole when needed, and word problems. ## The procedure ### [Adding mixed numbers](https://math.wiki/add-mixed-numbers/) **Summary:** Adding the whole parts and the fraction parts, or rewriting both as improper fractions. **Question:** A trail is $3\tfrac{1}{6}$ miles long and the hill after it is $2\tfrac{1}{2}$ miles. How long are the two together, when sixths and halves are different pieces? ## When you reach for it Hania, Mingyu and Segun train on an obstacle course. It has trail sections measured in miles, and lanes such as a cargo net and a tunnel measured in yards. They also time themselves, in minutes, on each obstacle. Almost none of these amounts is a whole number. A trail is $3\tfrac{1}{6}$ miles; a rope climb takes $2\tfrac{1}{3}$ minutes. A **mixed number** such as $3\tfrac{1}{6}$ is a whole number and a fraction less than $1$ written side by side, and it means their sum: $3\tfrac{1}{6} = 3 + \tfrac{1}{6}$ (mixed-numbers). You reach for this lesson whenever two or more mixed numbers are put together: two trail sections laid end to end, two times run one after the other, three lanes in a row. When the fraction parts already share a denominator, the work is simpler and has its own lesson (add-and-subtract-mixed-numbers-like-denominators). Here the denominators differ, so the fraction parts must first be renamed in shared pieces, which is the work of add-unlike-any-denominators. This lesson does not reteach that renaming; it uses it. The lesson has three parts, one for each day: - **Part 1:** add the wholes, add the fractions, and put them back together. - **Part 2:** when the fraction parts make more than one whole, trade a whole's worth of pieces for a whole number. - **Part 3:** the other route, rewriting each mixed number as an improper fraction; when each route is easier; adding three mixed numbers; and word problems. ### [Adding tenths and hundredths as fractions](https://math.wiki/add-tenths-and-hundredths/) **Summary:** Tenths are rewritten as hundredths so a number of tenths and a number of hundredths can be added. **Question:** In March the rain gauge in the rainforest filled to 3 tenths of a meter, and in April to 45 hundredths of a meter. How do you add a number of tenths to a number of hundredths, when the pieces are not the same size? ## When you reach for it Leyla and Tendai help at a research station in a rainforest. Each month they read a rain gauge, a tall tube marked in meters, and write down how deep the water stood. Some months they write the depth in tenths of a meter, such as $\frac{3}{10}$ meter. Other months they read it more finely, in hundredths of a meter, such as $\frac{45}{100}$ meter. At the end of April they want the total for March and April together. The trouble is that the two amounts count pieces of different sizes. A tenth is a whole cut into $10$ parts; a hundredth is a whole cut into $100$ parts, so it is much smaller. Three of one and forty-five of the other are not $48$ of anything. You can only add counts when they count the same piece: that is adding fractions with like denominators, where $\frac{30}{100} + \frac{45}{100}$ is just $30 + 45$ hundredths. So you reach for this skill whenever an amount in tenths meets an amount in hundredths and the two must be put together: rainfall, lengths of rope or vine, and money, where a dime is a tenth of a dollar and a penny is a hundredth. It rests on tenths and hundredths, which showed that $\frac{7}{10} = \frac{70}{100}$, and on equivalent fractions in pictures: cutting every piece of a shaded shape into smaller equal pieces changes the numbers but not the amount. ## The procedure The lesson is taught in two parts, one a day. ### Part 1: write the tenths as hundredths, then add ### [Area and length models of a fraction](https://math.wiki/fraction-area-model/) **Summary:** A shape cut into equal pieces with some of them shaded, used to picture a fraction. **Question:** A sheet cake is cut into six equal pieces and four of them get frosting. How do you say, in one number, how much of the cake is frosted? ## Why this exists In equal-shares-in-a-fraction a whole was cut into equal shares and one share got a name: one half, one third, one fourth. But one share is rarely all that someone has. Lempi cuts a sheet cake into $6$ equal pieces and frosts $4$ of them. Pihla eats $3$ of the $8$ equal lengths of a fruit leather strip. "One sixth" and "one eighth" name a single piece. Nothing so far says "four of the sixths" in one number. A fraction says it. Behind every fraction is one picture: a whole cut into equal parts, with some of the parts shaded. The whole can be a flat shape, and then the parts are equal areas. Or it can be a length, and then the parts are equal lengths. Both pictures carry the rest of fractions, and the length picture is the one that later becomes the number line. ## Definition A whole is cut into $b$ equal parts, and $a$ of those parts are taken. Together they are the fraction $$\frac{a}{b}$$ of the whole. The bottom number counts the equal parts the whole is cut into. The top number counts the parts taken. In this lesson $a$ is never more than $b$: a whole cut into $6$ parts has only $6$ to take. A fraction is read by the name of its parts. The table gives the names used here. | Equal parts in the whole | One part | The parts are called | For example | |:-:|:-:|:-:|:-:| | $2$ | $\frac{1}{2}$, one half | halves | $\frac{1}{2}$, one half | | $3$ | $\frac{1}{3}$, one third | thirds | $\frac{2}{3}$, two thirds | | $4$ | $\frac{1}{4}$, one fourth | fourths, or quarters | $\frac{3}{4}$, three fourths | | $6$ | $\frac{1}{6}$, one sixth | sixths | $\frac{4}{6}$, four sixths | | $8$ | $\frac{1}{8}$, one eighth | eighths | $\frac{3}{8}$, three eighths | ### [Benchmark fractions](https://math.wiki/benchmark-fractions/) **Summary:** Familiar fractions such as one half and one fourth, used as reference points for judging others. **Question:** A measuring tape is covered in marks of different lengths. Which few fractions are worth knowing by heart, so that any other fraction can be placed next to one of them at a glance? ## Why this exists Kerem, Iryna and Dragana share a measuring tape. Between any two inch numbers it has a crowd of marks: one long mark in the middle, two shorter ones, and four very short ones. Nobody reads a tape by counting every tiny mark from the start. They look for the long marks first. "Just past the half." "Almost at the next inch." "Right on the quarter." Fractions need the same kind of long marks. A fraction such as $\tfrac{11}{18}$ is hard to picture on its own. Is it a lot? A little? Close to one whole? It becomes easy once it is set beside a fraction that everyone already knows well, such as $\tfrac{1}{2}$. Those well-known fractions are the **benchmarks**: fixed reference points, like the long marks on the tape, that every other fraction can be judged against. ## Definition A **benchmark fraction** is a familiar fraction used as a reference point for judging the size of other fractions. Between $0$ and $1$ the benchmarks are the five landmarks $$0, \quad \tfrac{1}{4}, \quad \tfrac{1}{2}, \quad \tfrac{3}{4}, \quad 1$$ Past $1$ the same landmarks repeat after every whole number: $1\tfrac{1}{4}$, $1\tfrac{1}{2}$, $1\tfrac{3}{4}$, $2$, and so on. A fraction is a point on the number line, as in fractions-on-number-line: the bottom number cuts each whole into that many equal pieces, and the top number counts the pieces. A benchmark is a point on the same line that we know without counting. ### [Benchmark percents and mental percent](https://math.wiki/benchmark-percents/) **Summary:** Ten, twenty five, fifty and one hundred percent, used as reference points. **Question:** Half of a crowd, or a tenth of it, takes no working at all. How far can a handful of easy percents like these take you without pencil and paper? ## Why this exists Ilya's magic show plays to an audience of $360$. How many people are half the audience? Almost anyone says $180$ at once. A tenth of the audience? $36$, just as fast. Neither needs a pencil, because halving and taking a tenth are things we already do in our heads. Most percents look less friendly. $37\%$ of $600$, or $15\%$ of $460$, seem to call for a written method. But $37\%$ is three lots of $10\%$ and seven lots of $1\%$, and $10\%$ and $1\%$ of a number are as easy to find as half of it. A few easy percents, the **benchmark percents**, are enough to build most others in the head, and to say roughly how big any percent of anything is. ## Definition The **benchmark percents** are $1\%$, $10\%$, $25\%$, $50\%$, $75\%$ and $100\%$. Four of them are the benchmark-fractions a quarter, a half, three quarters and one whole, the familiar fractions we use as reference points for judging others. The other two are a tenth and a hundredth. Each has a quick way to find it of a quantity. | Percent | Fraction | Of a quantity, found by | |:-:|:-:|---| | $100\%$ | $1$ | taking the whole | | $50\%$ | $\frac{1}{2}$ | halving | | $25\%$ | $\frac{1}{4}$ | halving, then halving again | | $75\%$ | $\frac{3}{4}$ | adding $50\%$ and $25\%$ | | $10\%$ | $\frac{1}{10}$ | dividing by $10$: each digit moves one place right | | $1\%$ | $\frac{1}{100}$ | dividing by $100$: each digit moves two places right | ### [Changing the units of a rate](https://math.wiki/convert-a-rate-to-other-units/) **Summary:** Rewriting feet per second as miles per hour by multiplying by carefully chosen ones. **Question:** A sloth's crawl is timed in feet per minute, but speeds are usually given in miles per hour. How do you rewrite a rate in other units without changing how fast it is? ## When you reach for it Tilda takes her pulse at rest and counts $72$ beats in a minute. How many times does her heart beat in an hour, or in a whole day? Stavros times a sloth crawling along a branch and finds it covers $11$ feet a minute, but animal speeds are usually given in miles per hour. Luzia's class reads that a glacier flows $0.48$ meters a day, and wants to know how far it creeps in one hour. Each time the rate is right, but its units do not fit the question. A rate compares two amounts of different kinds, as in unit-rate. Written as a fraction it has two units: the one on **top** names what is counted or measured (beats, feet, meters) and the one on the **bottom** names the "per" (per minute, per day). Either unit can be changed, or both. Reach for this move whenever a rate has to answer a question asked in other units, and whenever two rates must be compared but arrive in different units, such as two speeds, one in feet per minute and one in miles per hour. Like two unit prices, two rates can be compared only once their units match. The tool is the conversion factor: a fraction with the same amount on top and bottom, such as $\frac{60\text{ min}}{1\text{ h}}$, built from a customary or metric fact. Part 1 (day one) changes one unit at a time and explains why the bottom unit takes its factor the other way up. Part 2 (day two) changes both units, goes through middle units, compares rates, and checks each answer by its size. ### [Common denominators and the least common denominator](https://math.wiki/rename-two-fractions-to-a-common-denominator/) **Summary:** Rewriting both fractions so their pieces match, before comparing, adding or dividing. **Question:** One ribbon is measured in fifths of a yard and another in tenths. How can both lengths be written in the same size of piece without changing either one? ## When you reach for it Nil, Roksolana and Stojan share a sewing kit. It holds ribbons and fabric squares, and every length is written as a fraction of a yard. Nil's red ribbon is $\tfrac{2}{5}$ yard long. Roksolana's blue ribbon is $\tfrac{3}{10}$ yard long. Those two fractions count different pieces. The first counts fifths of a yard and the second counts tenths. A fifth is bigger than a tenth, so "$2$ pieces" and "$3$ pieces" do not say which ribbon is longer, and they cannot be put together into one count. It is like saying "$2$ feet and $3$ inches": the numbers are fine, but the units do not match. You reach for this move whenever two or more fractions have to be set side by side: to compare them, to add or subtract them, or just to see both on the same ruler. You rename every fraction so that all of them use the **same size of piece**. Nothing gets longer or shorter. Only the names change. You already have the tool. From multiply-to-make-equivalent: multiplying the top and the bottom of a fraction by the same counting number cuts every piece smaller and gives an equivalent fraction. This lesson asks which size of piece to cut to, so that two fractions land on the same one. It has three parts, one for each day: - **Part 1:** why the pieces must match, and what a common denominator is. - **Part 2:** when one denominator is a multiple of the other, and when neither is. - **Part 3:** the least common denominator, found by listing multiples, and renaming three fractions at once. ### [Comparing against a benchmark](https://math.wiki/compare-using-benchmarks/) **Summary:** Deciding which side of one half each fraction falls on and comparing from there. **Question:** Two smoothie cups of the same size are filled to three eighths and to four sevenths. Which cup holds more, and can you tell without cutting either cup into new pieces? ## When you reach for it Ayse, Solomiya and Tijana run a smoothie stand. Every cup at the stand is the same size, and a cup is often only partly filled: one customer wants $\tfrac{3}{8}$ of a cup to taste, another wants $\tfrac{4}{7}$ of a cup. Which order is bigger? One way is to cut both cups into matching pieces and count. That always works, and it is the subject of rename-two-fractions-to-a-common-denominator. But it takes work, and very often a quick look is enough. If one cup is less than half full and the other is more than half full, nothing else needs to be known. If both orders have three pieces, the order with the bigger pieces is bigger. If both cups are one piece short of full, the cup with the smaller gap is fuller. Reach for these quick comparisons whenever two fractions need to be compared and you do not want to rename them. Each one rests on a fraction you already know well, a **benchmark**, such as $\tfrac{1}{2}$ or $1$, met in benchmark-fractions. When none of them decides, you still have the slow way. ## The procedure A fraction counts pieces of one whole. The bottom number, the **denominator**, says how many equal pieces make the whole. The top number, the **numerator**, says how many of those pieces you have. Every comparison below reads those two numbers and decides which fraction is larger without renaming either one. ### Part 1: which side of one half? Before comparing anything, check that both fractions are of the same whole. ### [Comparing by finding a common denominator](https://math.wiki/compare-using-a-common-denominator/) **Summary:** Renaming both fractions in matching pieces so the counts can be compared directly. **Question:** One dog ate five sixths of its bowl and another ate seven ninths. The pieces are different sizes, so how can you tell which dog ate more? ## When you reach for it Chiyo, Obinna and Valeria run a pet sitting club. They feed dogs and cats, fill food bins, and walk dogs along the bike trail by the park. Their notes are full of fractions: a fraction of a scoop, of a bag, of a mile. Again and again they need to know which of two amounts is more. Some comparisons are quick. If the denominators match, the pieces are the same size, and the fraction with more pieces is larger (compare-fractions-same-denominator). If one fraction is less than $\tfrac{1}{2}$ and the other is more, a benchmark settles it (compare-using-benchmarks). But here is a harder one. Biscuit the dog ate $\tfrac{5}{6}$ of her bowl. Rocket ate $\tfrac{7}{9}$ of his, and the two bowls are the same size. Both fractions are a little less than $1$, so benchmarks do not help. The pieces are different sizes, so you cannot just count them. This lesson is about making the pieces match, so that counting works again. It has three parts, one for each day: - **Part 1:** rename both fractions to a common denominator and compare the counts. - **Part 2:** two other routes: a common numerator, or changing both fractions to decimals. - **Part 3:** putting a list of three or more fractions in order, and the cross product shortcut, which turns out to be the common denominator method in disguise. ## The procedure ### Part 1: make the pieces the same size A fraction counts pieces. $\tfrac{5}{6}$ means $5$ pieces, each one sixth of the whole. $\tfrac{7}{9}$ means $7$ pieces, each one ninth. A sixth is bigger than a ninth, so $5$ big pieces and $7$ small pieces cannot be compared just by looking at $5$ and $7$. ### [Comparing fractions](https://math.wiki/compare-fractions/) **Summary:** Same denominator, same numerator, or a benchmark like one half. **Question:** Two fractions of the same whole, and you want to know which is more. When is a glance enough, and when do you have to rename them? ## What this area is about Comparing fractions means saying which of two is greater, or that they are equal. It only makes sense for fractions of the same whole: half of a big cup is more than half of a small one. The three children are three ways to decide, from quickest to most general. ## How these fit together **Sharing a number.** Same denominator or same numerator: when the pieces are the same size, more pieces is more; when the counts are the same and not $0$, bigger pieces is more. So for whole numbers $a$, $b$ and $n \neq 0$, $\frac{a}{n} > \frac{b}{n}$ exactly when $a > b$. **Benchmarks.** Comparing against a benchmark places each fraction against $\frac{1}{2}$ or $1$. If one is below $\frac{1}{2}$ and the other above, nothing more is needed. **The general method.** A common denominator renames both fractions in matching pieces, which always works, then compares the counts. ## A path through it Same denominator or numerator first, then benchmarks, then a common denominator. Here the order is fixed: each method is built from the one before it. Afterwards you can compare and order rational numbers, negative ones included, and decide which of two ratios is larger. ### [Comparing fractions with the same denominator or numerator](https://math.wiki/compare-fractions-same-denominator/) **Summary:** When the pieces are the same size, the fraction holding more pieces is the larger one. **Question:** Two fractions share a bottom number, or share a top number. How can you tell which one is greater without measuring anything? ## When you reach for it Kittipong and Pranee are at a swim lesson. Kittipong swims $\frac{5}{8}$ of a length of the pool before he stops to rest, and Pranee swims $\frac{3}{8}$ of a length. Who swam farther? A cat eats $\frac{2}{3}$ of a can of food one day and $\frac{2}{6}$ of the same kind of can the next. On which day did it eat more? Each question asks you to compare two fractions: to say which one is greater, or that they are equal. This lesson handles the two cases where the fractions share a number. When the denominators match, the pieces are the same size, and the count of pieces decides. When the numerators match, the counts are the same, and the size of the pieces decides. Both cases can be seen two ways: in an area picture, a whole cut into equal parts as in fraction-area-model, and on a number line, where every fraction is a point, as in locate-a-fraction-on-the-number-line. A comparison is written with a symbol. The sentence $\frac{5}{8} > \frac{3}{8}$ reads "five eighths is greater than three eighths". The sentence $\frac{3}{8} < \frac{5}{8}$ reads "three eighths is less than five eighths". They say the same thing from opposite ends: the open side of the symbol faces the greater number and the point faces the smaller one. The sign $=$ says the two fractions are the same number. One condition sits under every comparison in this lesson: both fractions are fractions of the same whole. The same pool, the same size of can, the same hour. Part 4 shows what goes wrong without it. ### [Converting among fractions, decimals and percents](https://math.wiki/fraction-to-percent/) **Summary:** Finding how many hundredths a fraction comes to. **Question:** Three fifths of the seeds sprouted, $0.6$ of them sprouted, $60\\%$ of them sprouted: all three say the same thing. How do you turn any one of these forms into the others, and what happens when the division never comes out even? ## When you reach for it A seed packet from Bao's seed bank promises that $60\%$ of its seeds will sprout. The bank's test tray records that $0.6$ of the seeds sprouted, and a grower writes that three fifths came up. All three say the same thing in different spellings: a percent, a decimal and a fraction are three names for one number. You reach for a conversion whenever the name you have is not the one you need. Two amounts given in different forms cannot be set side by side until they share one, and putting them in order is order-rational-numbers. A percent has to become a decimal or a fraction before you can multiply with it, which is how a percent of a quantity is found. A fraction such as $\frac{5}{16}$ is exact but hard to picture, while $31.25\%$ says at once that it is a little under a third. And a ratio between two parts, such as "$3$ tire jobs for every $5$ gear jobs", often has to become a share of the whole. ## The procedure Every conversion runs through one fact from percent-as-rate-per-hundred: a percent is a number of hundredths. $60\%$ is sixty hundredths, $\frac{60}{100}$, and sixty hundredths written as a decimal is $0.60$, or $0.6$. So changing the name means writing the amount as hundredths and reading it off in the form you want. Figure 1 lines up the three names on one number line from $0$ to $1$, the whole, which is $100\%$. ### Part 1: three spellings of hundredths ### [Counting by unit fractions](https://math.wiki/count-by-unit-fractions/) **Summary:** Skip counting one third, two thirds, three thirds and on past the whole. **Question:** One fourth, two fourths, three fourths, four fourths. What comes next, and where is the count when it gets there? ## When you reach for it Counting is the first thing anyone does with whole numbers: one, two, three. Fractions can be counted the same way, once you know what is being counted. In numerator-and-denominator a fraction was copies of one piece, the unit fraction: $\frac{3}{8}$ is $3$ copies of $\frac{1}{8}$. Counting by a unit fraction takes those copies one at a time, out loud: one fourth, two fourths, three fourths, four fourths, five fourths, and on. You reach for this count whenever something grows or shrinks by equal pieces: a relay team covering a swimming lane a third at a time, a rider passing quarter-mile markers on a bike path, a child hopping along a hopscotch court of equal boxes. The count says where you are after any number of pieces. It also shows two things that are hard to see any other way. The count reaches one whole at a place you can predict: three thirds, four fourths, six sixths. And it does not stop there. Five fourths of a mile is a real place on a bike path, just past the first mile. It is the same move as skip-counting by twos or tens, with a fraction as the step. ## The procedure The lesson runs over three days. Part 1 counts up to one whole. Part 2 keeps counting past the whole and names where the count is. Part 3 counts backward. ### Part 1: counting up to one whole Start with the step. The unit fraction is the size of one step, and it comes from the whole: one length of a swimming lane split into $3$ equal parts gives steps of $\frac{1}{3}$ of a length. Every step in the count is that same size. ### [Deciding which of two ratios is larger](https://math.wiki/compare-two-ratios/) **Summary:** Putting two comparisons on the same footing before saying which one is stronger. **Question:** Two mixes are made with different amounts of everything. How can you tell which one is stronger? ## When you reach for it Bodhi makes tea with $5$ spoons of tea leaves to $6$ cups of water. Imani uses $4$ spoons to $5$ cups. Whose tea is stronger? Bodhi uses more leaves, but he also uses more water, so neither number settles it on its own. The two ratios have to be put on the same footing first: the same amount of water in both, or the same amount of leaves. Then a single comparison tells the whole story. Reach for this whenever two mixes, rates or shares are described with different amounts: two pots of tea, two compost heaps, two trays of seeds. The ratios may be part to part (leaves to water) or part to whole (seeds that sprouted to seeds planted), and both kinds are compared the same way. Part 1 (day one) matches one term of the two ratios; Part 2 (day two) compares shares of a whole as fractions and handles decimals. ## The procedure ### Part 1: why scaling is allowed Multiplying both numbers of a ratio by the same number gives the same mix (scaling a ratio). $10$ spoons to $12$ cups tastes exactly like $5$ spoons to $6$ cups: it is two pots of Bodhi's tea side by side. So each tea may be scaled to any size without changing how strong it is, and the whole skill is choosing sizes at which the two teas share one number. ### Part 1: matching the second term Make the water the same. Bodhi's water goes $6, 12, 18, 24, 30, \dots$ and Imani's goes $5, 10, 15, 20, 25, 30, \dots$. The first amount on both lists is $30$, a common multiple of $6$ and $5$. Scale Bodhi's ratio by $5$ and Imani's by $6$: ### [Decomposing a fraction into unit fractions](https://math.wiki/decompose-fraction-into-unit-fractions/) **Summary:** Writing a fraction as a sum of copies of its own unit fraction. **Question:** Five of the six portions of cornbread are left in the pan. How can that one amount, five sixths of the pan, be broken into smaller amounts and still be the same cornbread? ## When you reach for it Tolga bakes a pan of cornbread and cuts it into $6$ equal portions. After lunch, $5$ portions are left. That is $\frac{5}{6}$ of the pan. Now Kateryna wants some and Goran wants some. The $\frac{5}{6}$ has to be split into smaller amounts. How much can each person get? What equation tells the story? Questions like this need a fraction to come apart. To **decompose** a fraction is to write it as a sum of smaller fractions. In this lesson every part keeps the same denominator as the fraction you start with, so every part is made of the same size of piece. Reach for it when an amount has to be shared, split or grouped. Reach for it, too, before you add or subtract fractions: a sum of fractions is a decomposition read the other way. ## The procedure This lesson takes three days, one part a day. ### Part 1: a fraction is a count of unit fractions A **unit fraction** is one piece when a whole is cut into equal parts, a fraction with numerator $1$ (see unit-fractions). One portion of Tolga's pan is $\frac{1}{6}$ of the pan, because the pan is cut into $6$ equal portions. In $\frac{5}{6}$, the denominator $6$ names the size of the piece, sixths. The numerator $5$ counts how many of those pieces there are (see numerator-and-denominator). So $\frac{5}{6}$ means five pieces, each $\frac{1}{6}$ of the pan. Write those five pieces as a sum: $$ \frac{5}{6} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} $$ ### [Discounts, sales tax, tips and commission](https://math.wiki/discount-and-sale-price/) **Summary:** An amount taken off a price, and the price that is left behind. **Question:** A jacket marked 80 dollars is 25% off, and 8% sales tax is added at the till. What do you actually pay, and why is the answer not simply 17% off? ## When you reach for it Often the amount paid is not the number on the label. A pair of shoes is $30\%$ off. A bookshop adds sales tax at the till. A café bill has a space for a tip. A salesperson in a bike shop is paid a share of every bike sold. Each time, an amount is worked out as a **percent of some other amount**, and then either taken away or added on. That makes all four the same piece of mathematics, the question from the percent equation: a part is a rate times a whole. The work in each situation is deciding two things: - **which amount is the whole**, the base the percent is taken of, and - whether the part is **taken away** from it or **added** to it. | Situation | The whole | The part is | |---|---|---| | Discount | original price | taken off | | Sales tax | price paid | added | | Tip | bill | added | | Commission | sales | paid to the seller | The lesson takes two days. Part 1 is discounts, worked forwards and backwards. Part 2 is sales tax, tips, commission, and a discount followed by tax. ## The procedure ### Part 1: discount and sale price A shop marks a pair of shoes at $60$ dollars and puts up a sign: $30\%$ off. Four words name the amounts involved. - The **original price** is the price before the sale, here $60$ dollars. - The **discount rate** is the percent taken off, here $30\%$. - The **amount of discount** is how many dollars come off. - The **sale price** is what is left to pay. ### [Dividing by multiplying by the reciprocal](https://math.wiki/divide-fractions-invert-and-multiply/) **Summary:** Turning a division into a multiplication by flipping the second fraction over. **Question:** A jug holds six liters of sports drink and each bottle takes three quarters of a liter. How do you find how many bottles it fills, quickly, whatever the fractions are? ## When you reach for it At the drink station of a cycling race, riders grab small bottles of sports drink as they pass, and the volunteers fill those bottles from big jugs. Valentina has a jug holding $6$ liters, and each bottle holds $\frac{3}{4}$ of a liter. How many bottles can she fill? That is a division, $6 \div \frac{3}{4}$, in the "how many fit" sense of dividing fractions: how many fit. You could draw the jug as a bar, cut it into quarter liters and count groups of three: $24$ quarters make $8$ groups, so $8$ bottles. Drawing works for this one. It is slow for $\frac{8}{15} \div \frac{4}{9}$, and the riders will not wait. This lesson is the fast route. Dividing by a fraction gives the same answer as multiplying by that fraction turned upside down, so every fraction division becomes a fraction multiplication, which you already know how to do. Reach for it whenever a division has a fraction in it: a fraction divided by a fraction, a whole number divided by a fraction, or a fraction divided by a whole number. It also answers the sharing question of dividing fractions: sharing into equal parts, since both meanings are the same division. The lesson runs over two days. Day 1 is the move itself: the reciprocal, the rule, and whole numbers. Day 2 makes it quick and safe: simplifying before you multiply, judging the size of the answer before you start, and checking it by multiplying back. ### [Dividing fractions](https://math.wiki/divide-fractions/) **Summary:** How many of this piece fit in that one, and why invert and multiply works. **Question:** Dividing whole numbers means sharing out or counting how many fit. What do those mean when the numbers are fractions, and why does flipping the second one give the answer? ## What this area is about A division $a \div b$, with $b \neq 0$, asks for the number $q$ with $q \times b = a$. With fractions that question keeps its two everyday meanings, and one short rule answers both. ## How these fit together **Two meanings.** Sharing into equal parts asks how much one share is when an amount is split into $n$ equal shares. How many fit asks how many copies of one amount fit inside another, and the answer can be larger than the number you started with. **The rule and its reason.** Multiplying by the reciprocal turns every division into a multiplication: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$, where $b$, $c$ and $d$ are not $0$. Why flipping the second fraction works shows that dividing by a number and multiplying by its reciprocal do the same thing. **Using it.** Dividing mixed numbers renames each mixed number as an improper fraction first. Word problems decide which meaning a situation calls for and which amount goes first. ## A path through it Sharing first, then how many fit. After how many fit come the rule and why it works, in either order, and the word problems. Mixed numbers come after the rule. This area opens multiplying and dividing rational numbers, unit rates, and finding the whole from a fraction. ### [Dividing fractions: how many fit](https://math.wiki/fraction-division-measurement-meaning/) **Summary:** Asking how many copies of one amount fit inside another amount. **Question:** A toy store has 4 yards of ribbon and cuts it into ties one third of a yard long for its gift bags. How many ties does it get? ## Why this exists The toy store wraps every gift in a paper bag and closes the bag with a ribbon tie. Dyani has $4$ yards of ribbon on a spool, and each tie takes $\frac{1}{3}$ of a yard. How many ties can she cut? That is a division, but not the sharing kind. In the last lesson, fraction-division-partitive-meaning, an amount was split into a known number of equal shares, and the question was how big one share is. Here the size of each piece is known, and the question is **how many pieces fit**. You have met this reading with whole numbers: $12 \div 3$ also asks how many groups of $3$ fit in $12$ (division-as-sharing). What is new is that the piece is a fraction, and then something surprising happens: the answer is bigger than the number you started with. The lesson takes four days. Part 1 counts how many unit fractions fit in a whole number. Part 2 checks an answer by multiplying back and sets the two meanings of division side by side. Part 3 tells the idea as stories and explains why the answer comes out bigger. Part 4 divides one fraction by another by giving them the same bottom number and counting pieces, including a piece left over. ## Definition A **unit fraction** has $1$ on top, like $\frac{1}{3}$: one of $3$ equal parts of a whole (unit-fractions). Let $a$ and $d$ be two amounts measured in the same unit, with $d$ not $0$. Then $a \div d$ is the **number of copies of $d$ that fit in $a$**. If the copies do not come out even, the part left over is counted as a fraction of one copy of $d$. ### [Dividing fractions: sharing into equal parts](https://math.wiki/fraction-division-partitive-meaning/) **Summary:** Asking how much one whole group gets when an amount is shared out. **Question:** One third of a garden bed is still bare, and four children share it equally. How much of the whole bed does each child get? ## Why this exists The school garden has raised beds: long wooden boxes filled with soil, planted in sections. Ayasha's class has one bed, and $\frac{1}{3}$ of it is still bare. Four children want to plant the bare part, and the fair way is for each child to get the same amount. How much of the whole bed does each child get? That is a division. You already know division as sharing: $12 \div 4$ is how many each of $4$ people gets when $12$ things are shared out equally (division-as-sharing). You also know that sharing whole things among more people than there are things gives each person a fraction (fraction-as-division). What is new here is that the amount being shared is already a fraction. The bare third is cut into $4$ shares, and the answer has to say how big one share is compared with the **whole bed**, not just with the third. The lesson takes four days. Part 1 shares a unit fraction by drawing it. Part 2 finds the rule for any unit fraction and checks a share by multiplying back. Part 3 tells the same idea as a story, a tape diagram and a number line. Part 4 shares fractions that are not unit fractions, and shows that sharing among $n$ is the same as taking $\frac{1}{n}$ of the amount. ## Definition A **unit fraction** has $1$ on top, like $\frac{1}{3}$: one of $3$ equal parts of a whole (unit-fractions). Let $\frac{a}{b}$ be a fraction and $n$ a counting number ($1$, $2$, $3$ and so on). Then $\frac{a}{b} \div n$ is the size of one share when the amount $\frac{a}{b}$ is split into $n$ equal shares. The share is measured in the same whole as $\frac{a}{b}$. If $\frac{a}{b}$ is a fraction of a garden bed, the share is a fraction of that garden bed too. ### [Dividing mixed numbers](https://math.wiki/divide-mixed-numbers/) **Summary:** Rewriting each mixed number as an improper fraction before dividing. **Question:** A plank is five and a half meters long and each piece you need is three quarters of a meter. How many pieces can you cut, and how much is left over? ## When you reach for it You already know how to divide one fraction by another: multiply by the reciprocal of the second one, and why-inverting-works explains why that gives the right answer. Real measurements rarely arrive as plain fractions, though. A plank is $5\frac{1}{2}$ meters long, a shelf board $1\frac{1}{2}$ meters, a floor joist $2\frac{1}{4}$ meters. As soon as either number in a division is a **mixed number**, a whole number with a fraction beside it, this move is the one you need. Reach for it in two kinds of question. **How many fit?** How many $\frac{3}{4}$ m pieces can be cut from a $5\frac{1}{2}$ m plank (the "how many fit" meaning). **How long is each share?** A beam $8\frac{3}{4}$ m long cut into $5$ equal posts (the sharing meaning). Both are a division, and both are done the same way. Where they differ is in how the answer should be written, and that is part of this lesson too. ## The procedure Diego is building a ladder for a treehouse. He has a plank $5\frac{1}{2}$ m long and wants pieces $\frac{3}{4}$ m long. The question is $5\frac{1}{2} \div \frac{3}{4}$. **Estimate first.** Before any working, decide roughly what the answer must be. A piece is less than $1$ m, so more pieces fit than there are meters: more than $5\frac{1}{2}$. And $6 \div \frac{3}{4} = 8$, since every $3$ m holds $4$ pieces. The plank is a little shorter than $6$ m, so the answer is a little less than $8$: somewhere around $7$. The estimate costs a few seconds and catches the biggest slips, such as flipping the wrong fraction, which would give an answer smaller than $1$. ### [Equal shares and naming the whole](https://math.wiki/equal-shares-in-a-fraction/) **Summary:** The parts a fraction counts have to be the same size, or the name of the fraction means nothing. **Question:** Two children share one sandwich, and one cut makes two pieces. When is each piece really a half? ## Why this exists Two children share one sandwich. The grown-up makes one cut and hands each child a piece. If the cut is down the middle, nobody minds. If the cut is near one end, one child gets far more, and any child will say that is not fair. The words **half**, **third** and **fourth** carry exactly that idea of fair. In everyday talk "half" can mean any piece. In mathematics it cannot: a half is one of two pieces that are the same size. Without that rule, "I ate a half" would not tell you whether it was a crumb or nearly all. Even with that rule, "I ate a half" leaves one question open: a half of what? Half of a cupcake and half of a birthday cake are both halves, and nobody would call them the same amount. A fraction always belongs to a whole, and saying which whole is the second idea of this lesson, **naming the whole**. Every fraction a child meets later is counted in equal shares of a named whole, so these two ideas are the first ideas of the whole fractions story. ## Definition ### Equal shares A whole, such as a sandwich, a pizza or a strip of paper, is cut into **equal shares** when - every piece is the same size, and - the pieces make up the whole exactly: no gaps, no overlaps, nothing left over. When a whole is cut into equal shares: - with $2$ equal shares, each share is **one half** of the whole, and the shares are **halves**; - with $3$ equal shares, each share is **one third**, and the shares are **thirds**; - with $4$ equal shares, each share is **one fourth**, also called **one quarter**, and the shares are **fourths** or **quarters**. ### [Equal shares that look different](https://math.wiki/equal-shares-can-look-different/) **Summary:** Equal shares of the same whole can have different shapes and still be the same size. **Question:** Two people each get one half of the same cracker, but the pieces are different shapes. Did they get the same amount? ## Why this exists A child who has met equal shares knows that a half is one of two pieces of the same size, and that a square cut down the middle or from corner to corner gives two halves. That lesson showed one such pair. This one asks how far the idea goes. Hand a child a square cracker scored into small squares and ask for a fair share for two people. Most will snap it straight down the middle. But a staircase break, a hook, or a slanted cut can be just as fair, and the child who sees that has understood something deeper than "cut down the middle": **a share is fair because of how much it holds, not because of how it looks.** That raises two questions a child cannot answer by looking. How do you *show* that a staircase piece and a triangle piece hold the same amount? And if two halves of one cracker are always the same size, why is one half of a big cracker not the same as one half of a small one? This lesson answers both, over three days. ## Definition Recall from equal-shares-in-a-fraction and equal-shares-in-a-fraction: a whole is cut into **equal shares** when the pieces are all the same size and together make up the whole, with no gaps and no overlaps. Two equal shares are **halves**, three are **thirds**, four are **fourths**. Two pieces are **the same size** when they hold the same amount of the whole. They do **not** need to be the same shape. On a cracker or a paper square marked in small squares that are all alike, there are two ways to show two pieces are the same size: ### [Equivalent fractions](https://math.wiki/equivalent-fractions/) **Summary:** Different names for the same number: cut every piece the same way. **Question:** Two thirds of a pan, cut again, becomes four sixths. Is that a different amount, or only a different name for the same one? ## What this area is about Two fractions are equivalent when they name the same amount of the same whole, the same point on a number line. Cutting every piece the same way changes the name and not the amount, so one number has many fraction names. Choosing the right name is what makes fractions comparable and addable. ## How these fit together **The idea.** Seeing equivalent fractions shows it in pictures and on the number line: cut every piece in two and both numbers double, while the shaded amount stays put. **Two moves.** Multiplying renames a fraction in smaller pieces: $\frac{a}{b} = \frac{a \times k}{b \times k}$ for any whole number $k \neq 0$, with $b \neq 0$. Simplifying runs it backward, dividing the top and the bottom by a common factor; when the top and bottom share no factor but $1$, the fraction is in simplest form. **The working tool.** Common denominators renames two fractions in one size of piece, the smallest such being the least common denominator. ## A path through it Pictures first, then multiplying. After multiplying come simplifying and common denominators, in either order. With these you can compare and add fractions with unlike denominators, find a fraction between two fractions, and later scale ratios and convert fractions to percents. ### [Equivalent ratios and ratio tables](https://math.wiki/equivalent-ratios/) **Summary:** Scale both parts by the same factor and the ratio stays the same. **Question:** A recipe, a mix or a map scale works at one size. How do you get other amounts that keep exactly the same comparison? ## What this area is about Two ratios are equivalent when they describe the same comparison: $3 : 7$ and $9 : 21$ make the same cordial, one jug three times the size of the other. You get from one to the other by multiplying both amounts by the same positive number. This area is that one move, and three pictures that keep track of it. ## How these fit together **The core moves.** Scaling a ratio up or down multiplies both amounts by the same number, so the mix stays the same. Finding a missing value in a pair of equal ratios uses that move backwards: when three of four numbers are known, find the scaling number and apply it. **Ways to see it.** Ratio tables list equivalent ratios in rows, one column per quantity. The double number line draws two number lines one above the other, zero under zero, so matching points show equal ratios. Tape diagrams draw equal blocks for each part of a ratio, which makes a total easy to share out in that ratio. ## A path through it Scaling a ratio comes first, then finding a missing value, then ratio tables. The double number line can be taken at any point, and tape diagrams once part-to-part ratios are known. Finish here and you are ready for comparing two ratios, rates and unit rates, proportions and percent as a rate per hundred. ### [Estimating with fractions](https://math.wiki/estimate-fraction-sums/) **Summary:** Replacing each fraction with zero, one half or one whole to judge the answer quickly. **Question:** Before you find the exact value of $\tfrac{9}{10} + \tfrac{3}{7}$, can you say about how big it is, and use that to tell whether an answer makes sense? ## When you reach for it Bhavin, Emiko and Tunde work in a wood shop. Their boards are measured in inches and fractions of an inch: a strip $\tfrac{9}{10}$ inch thick, a cut $17\tfrac{5}{8}$ inches long. Many shop questions do not need an exact answer. About how thick is the glued stack? Is this board long enough for both pieces? Does the number on the cut list look right? Finding an exact sum of fractions with different denominators takes renaming and care (add-unlike-any-denominators). An **estimate** takes a few seconds. It answers "about how much" questions, and it catches an answer that cannot be right before any wood is cut. The tool is a short list of landmark fractions, the **benchmarks** $0$, $\tfrac{1}{2}$ and $1$ (benchmark-fractions are familiar fractions used as reference points). Replace each fraction with the benchmark it is nearest, then add or subtract the benchmarks. Those sums are easy: $1 + \tfrac{1}{2} = 1\tfrac{1}{2}$, $\tfrac{1}{2} + \tfrac{1}{2} = 1$. ## The procedure ### Part 1: rounding fractions to 0, one half or 1 A fraction between $0$ and $1$ has three benchmarks to choose from. To pick one, look at the top number (the numerator) and set it beside three numbers made from the bottom (the denominator): $0$, half the bottom, and the whole bottom. - If the top is closest to $0$, the fraction is near $0$. In $\tfrac{1}{9}$ the top is $1$, and half of $9$ is $4\tfrac{1}{2}$. The $1$ is much closer to $0$. - If the top is closest to half the bottom, the fraction is near $\tfrac{1}{2}$. In $\tfrac{3}{7}$, half of $7$ is $3\tfrac{1}{2}$, and $3$ is only $\tfrac{1}{2}$ away from it. - If the top is closest to the whole bottom, the fraction is near $1$. In $\tfrac{9}{10}$ the top is $9$, just $1$ short of $10$. ### [Finding a fraction between two fractions](https://math.wiki/fractions-between-fractions/) **Summary:** Between any two fractions there is always another one, so the line is never full. **Question:** Two fractions look like neighbors on the number line. Is there room for another number between them, and how do you name it? ## When you reach for it On a number line marked in fifths, $\frac{3}{5}$ and $\frac{4}{5}$ look like next-door neighbors: one mark, then the next, nothing in between. A measuring jug looks the same way, and so does a race clock. Each one is marked only so finely. This lesson is about what lives in the gap: whether there is a number between two fractions, and how to name one. You reach for it whenever an amount falls between two marks: water in a jug that sits above one line and below the next, a sprinter who finished after one runner and before another, a point on the number line between two labeled ones. It also leads to a surprise about the number line itself. Between any two different fractions there is always another fraction, so no two fractions are ever right next to each other. You already know how to place a fraction on the number line: cut each whole into equal pieces, as many as the denominator says, then count that many pieces from $0$ as the numerator says. Everything below uses that picture, together with equivalent-fractions and comparing fractions. ## The procedure ### Same denominator, with room between the numerators When two fractions have the same denominator, they count pieces of the same size, so the one with more pieces lies further right. Between $\frac{2}{9}$ and $\frac{7}{9}$ lie $\frac{3}{9}$, $\frac{4}{9}$, $\frac{5}{9}$ and $\frac{6}{9}$. Their numerators $3$, $4$, $5$ and $6$ are the whole numbers between $2$ and $7$, and each of them counts ninths. ### [Finding a missing value in a pair of equal ratios](https://math.wiki/missing-value-in-equivalent-ratios/) **Summary:** Working out the scaling number and applying it to the amount you already know. **Question:** Two ratios are meant to be equal, and you know three of their four numbers. What must the fourth one be? ## When you reach for it A pond keeper's rule says $5$ goldfish for every $800$ liters of water. Clio's pond holds $3200$ liters. How many goldfish can it take? The rule is a ratio, $5 : 800$, and Clio's pond has to keep the same comparison, so the two ratios are equal: $$5 : 800 = \,? : 3200$$ Three of the four numbers are known. This lesson finds the fourth. Reach for this move whenever a comparison is fixed (a recipe, a rule, a mixture, a ration of food) and the amounts change: a bigger picnic, a fuller hutch, a different pond. The fixed comparison is one ratio, the new situation is the other, and the question mark sits in the new one. How to read a ratio and keep its parts in order is in ratio language and notation. Why multiplying both parts by one number keeps a ratio the same is the work of scaling a ratio up or down. This lesson uses that fact the other way round: the second ratio is known to be a scaled copy of the first, and the job is to find out by how much. ## The procedure ### Across: from one ratio to the other Equal ratios are scaled copies of each other. One number, the **scaling number**, multiplies both parts of the first ratio to give the second. So two numbers in the same position (both goldfish, or both liters) are linked by the scaling number, and the position where both numbers are known tells you what it is. In Clio's pond the liters are known in both ratios: $800$ goes to $3200$. Ask "$800$ times what is $3200$?" Division answers that question: ### [Finding a percent of a quantity](https://math.wiki/percent-of-a-number-with-a-decimal/) **Summary:** Changing the percent into a decimal and multiplying by it. **Question:** A box of building bricks says that 35 percent of its bricks are red. How many red bricks is that, and is there one move that answers every question of this kind? ## When you reach for it Reach for this whenever a percent describes a share of a known amount and the question is how much that share comes to. A box holds $720$ bricks and $35\%$ of them are red. A triathlete covers $240$ km in a month and $6\%$ of it is swimming. A puppet theater sold $160$ tickets on Saturday and $115\%$ as many on Sunday. Each time the amount after the word "of" is known and the percent is known; what is missing is the amount the percent picks out. benchmark-percents answered questions like these for friendly percents, by halving or by building up from $10\%$ and $1\%$. That still works, and it stays the best way to estimate. This lesson turns the whole job into one multiplication, which works as well for $35\%$, $7.5\%$ or $115\%$ as it does for $50\%$. The lesson runs over two days: Part 1 multiplies by the percent written as a decimal, and Part 2 multiplies by it written as a fraction, when that is easier. ## The procedure ### Part 1: a percent is a multiplier A percent counts hundredths, so $35\% = \frac{35}{100} = 0.35$. (fraction-to-percent covers changing between the three forms.) The question "what is $35\%$ of $720$?" asks for thirty-five hundredths of $720$. One hundredth of $720$ is $720 \div 100 = 7.2$, and thirty-five of those hundredths make $35 \times 7.2 = 252$. Multiplying $720$ by $0.35$ counts exactly the same thirty-five hundredths in one go, so ### [Finding the whole from a part and a percent](https://math.wiki/find-the-whole-from-part-and-percent/) **Summary:** Working backwards when the amount you are given is only a slice of the total. **Question:** You can see how much you have, and you know what percent of the total it is. How big is the total? ## When you reach for it A percent situation has three numbers in it: a whole, a part of that whole, and the percent the part is of the whole. Know any two and the third follows, and the three percent questions are the three ways of choosing which one is missing. This lesson is the one that runs backwards. You know the **part** and its **percent**, and the **whole** is what you want. It turns up whenever a gauge or a progress bar shows a percent and you can see only how much has arrived so far. A download bar reads $20\%$ after $46$ megabytes. A charity's thermometer chart stands at $65\%$ with $1{,}560$ dollars raised. A phone battery reaches $15\%$ after $18$ minutes on the charger. Each time the question is the same: how big is the thing the percent is a percent **of**? How big is the file, what is the target, how long does a full charge take? The other two questions run forwards. Finding a percent of a quantity starts from the whole and multiplies. Finding what percent one number is of another starts from the part and the whole and divides. Here the whole is the unknown, so the move is to undo: work back from the slice you have to the $100\%$ you want. This lesson takes two days. **Part 1** reasons on a double number line, scaling the part up to $100\%$ the way you scale any ratio. **Part 2** writes the same fact as an equation, part equals percent times whole, and solves it, which works for any numbers at all. ### [Finding the whole when only a fraction of it is known](https://math.wiki/find-the-whole-from-a-fraction/) **Summary:** Working backwards from a part and the fraction of the whole it represents. **Question:** You know how much has been done, and what fraction of the job that is. How big is the whole job? ## When you reach for it A class is painting a mural on the school wall, one section at a time. Every question about it has three numbers in it: the **whole** job, a **part** of the job, and the **fraction** of the whole that the part is. They are tied together by one multiplication: $$\text{part} = \text{fraction} \times \text{whole}$$ When you know the whole and the fraction, you multiply, as in finding a fraction of a number: $\frac{3}{4}$ of a $16$ square meter mural is $12$ square meters. This lesson runs the other way. You can see the part, and you are told what fraction of the whole it is, but the whole is hidden. Beatriz has painted $12$ square meters, and that is $\frac{3}{4}$ of the mural. How big is the mural? Reach for this move whenever an amount comes with the words "which is ... of the whole", "that is ... of what we need", or "so far we have done ...". ## The procedure ### A tape diagram Draw the whole mural as a long bar, a **tape**. The fraction $\frac{3}{4}$ says the mural is cut into $4$ equal pieces and the part fills $3$ of them, so cut the tape into $4$ equal pieces and shade $3$. The $12$ square meters go over the shaded pieces. Now read the tape. Three equal pieces hold $12$ square meters, so one piece holds $12 \div 3 = 4$. The pieces are equal, which is what lets you share the $12$ evenly. The whole is $4$ of those pieces, so it is $4 \times 4 = 16$ square meters. ### [Finding what percent one number is of another](https://math.wiki/find-the-percent-one-number-is-of-another/) **Summary:** Dividing the part by the whole and reading the answer as hundredths. **Question:** Twenty-six of forty seeds came up in one bed, and thirty of fifty in another. How do you say how much of each bed came up so that the two can be set side by side? ## When you reach for it A count on its own says how many. It does not say how many **out of how many**, and that is usually the question that matters. If $26$ seeds sprouted in one bed and $30$ in another, the second bed had more seedlings, but the first bed may still have done better for its size: it all depends on how many seeds each bed was given. With $40$ seeds in the first bed and $50$ in the second, the first did better, $65\%$ against $60\%$. A percent answers "how many out of every hundred". Turning a part of a whole into a percent puts every whole on the same footing, a hundred, so parts of different wholes can be compared directly. That is why results of surveys, seed trials and sports seasons are so often reported as percents. You reach for this move whenever you know a **part** and the **whole** it comes from, and you want the **percent**. The question may say it outright ("what percent of the members chose Saturday?") or more loosely ("what share of its games did the team win?"). There are three numbers in any such situation, the part, the whole and the percent, and the three percent questions each find a different one. This lesson finds the percent. When the percent is given and the whole is missing, see find-the-whole-from-part-and-percent; when you want one equation that handles all three, see percent-as-an-equation. ## The procedure **Percent** means "per hundred", so $p\%$ is the fraction $\frac{p}{100}$. To say that a part is $p\%$ of a whole is to say that the part is the same fraction of the whole as $p$ is of $100$: ### [Fraction of a quantity](https://math.wiki/fraction-of-a-quantity/) **Summary:** Three fifths of 40: divide into parts, take some. **Question:** Two thirds of a 120 mile trip, or three fourths of a mural you can only see part of. How do you find a fraction of an amount, and how do you work back to the whole? ## What this area is about Every question here ties three numbers together: $\text{part} = \text{fraction} \times \text{whole}$. The word "of" means times. Given the fraction and the whole, you find the part; given the part and the fraction, you find the whole. A tape diagram, a bar cut into equal pieces, shows both directions. ## How these fit together **Finding the part.** A fraction of a whole number splits the amount into equal groups and takes some of them: $\frac{a}{b}$ of $N$ is $N \div b$, taken $a$ times, for whole numbers $a$, $N$ and $b \neq 0$. **Two cuts in a row.** A fraction of a fraction of an amount applies a second fraction to the part the first one took, which multiplies the two fractions, and also handles a second cut of the rest. **Working backward.** Finding the whole starts from a known part and its fraction: divide by the numerator to get one piece, then multiply by the denominator. This needs a fraction whose numerator is not $0$. ## A path through it A fraction of a whole number comes first. The other two build on it, in either order. This area feeds percents and ratios: finding a percent of a quantity and tape diagrams for sharing in a ratio both reuse the same part, fraction and whole. ### [Fractions](https://math.wiki/fractions/) **Summary:** A number that names part of a whole, and everything you can do with it. **Question:** A whole is cut into equal pieces. How does a count of those pieces become a number you can compare, add, multiply and divide? ## What this area is about A fraction $\frac{a}{b}$, for whole numbers $a$ and $b \neq 0$, names $a$ pieces of a whole cut into $b$ equal parts. From that picture, fractions become numbers: placed on a line, renamed, compared, and used in all four operations. ## How these fit together **What a fraction is.** Part of a whole demands equal parts and a named whole. Unit fractions are single pieces $\frac{1}{b}$; the numerator counts them and the denominator names their size. Fractions greater than one and whole numbers as fractions count past one. A fraction as division reads the bar as divide: $\frac{3}{4} = 3 \div 4$. **Fractions as numbers.** The number line gives each fraction a point, equivalent fractions give one point many names, and benchmarks like $\frac{1}{2}$ judge size. Comparing uses all three. **Operations.** Adding and subtracting counts matching pieces, renaming unlike ones first; mixed numbers write amounts like $2\frac{3}{4}$ and get added and subtracted too. Multiplying counts copies or takes a fraction of an amount, dividing shares or asks how many fit, and a fraction of a quantity finds a part or works back to the whole. ## A path through it Part of a whole, then unit fractions and the numerator and denominator (either order), fractions greater than one, and whole numbers as fractions. Next the number line with equivalent fractions, benchmarks, comparing; adding and subtracting, mixed numbers, multiplying, dividing, and a fraction of a quantity. A fraction as division fits anywhere after part of a whole. ### [Fractions greater than one (improper fractions)](https://math.wiki/fractions-greater-than-one/) **Summary:** A fraction whose top number is bigger than its bottom number, naming an amount past one whole. **Question:** A pie is cut into four equal pieces, and you have five of those pieces. How much pie is that, and how can a fraction name more than one whole? ## Why this exists In numerator-and-denominator a fraction counted pieces of one whole: the denominator names the size of the piece, and the numerator counts how many of those pieces there are. In count-by-unit-fractions the count went on past the whole: four fourths, then five fourths, six fourths. This lesson asks what an amount like five fourths is, and how to picture it. Think of an apple pie cut into $4$ equal pieces. Each piece is a fourth of the pie. You can have $1$, $2$, $3$ or all $4$ of them, and $\frac{4}{4}$ is the whole pie. But a bakery has more than one pie. Five pieces, each a fourth of a pie, is a real amount of pie: one whole pie and one piece of another. The fraction $\frac{5}{4}$ names it. A fraction can name an amount past one whole, as long as there is more than one whole to take pieces from. This comes up every time there is more than one of something: more than one bagel's worth of halves, more than one wheel's worth of cheese wedges. It also gives a quick test. Comparing the numerator with the denominator tells at a glance whether a fraction is less than one whole, exactly one, or more. The lesson runs over three days. Part 1 pictures an amount past one whole. Part 2 looks at fractions equal to one and sorts fractions into less than, equal to or more than one. Part 3 names an amount past one whole in words, as wholes and some more. ## Definition A fraction $\frac{a}{b}$ names $a$ copies of the unit fraction $\frac{1}{b}$, the piece you get by cutting one whole into $b$ equal parts. Nothing in that reading says $a$ must be smaller than $b$. ### [Fractions on the number line](https://math.wiki/fractions-on-number-line/) **Summary:** A fraction is a point on the line, a number like any other. **Question:** A fraction began as pieces of a pie. Where does it sit among the other numbers, and is there always room for one more? ## What this area is about Here a fraction stops being pieces of a thing and becomes a number with a place of its own. On a number line the stretch from $0$ to $1$ is one whole. Cut it into $b$ equal parts, for a whole number $b \neq 0$, and each part has length $\frac{1}{b}$; the point $\frac{a}{b}$ is $a$ of those jumps from $0$, past $1$ when $a > b$. ## How these fit together **Placing a fraction.** Locating a fraction is the procedure: mark equal parts between whole numbers, count unit fraction jumps from $0$, and read a point someone else has marked. **What lies between.** Finding a fraction between two fractions shows that between any two different fractions there is always another, found by cutting the pieces smaller. So no two fractions are neighbors, and the line never fills up. ## A path through it Locate a fraction first; finding one between two others uses those points. The number line is where the next ideas live: equivalent fractions are two names for one point, comparing fractions reads which point is further right, benchmark fractions are landmarks on it, and negative fractions extend it to the left of $0$. ### [Fractions, ratio and percent](https://math.wiki/fractions-ratio-percent/) **Summary:** Parts of a whole, comparisons by multiplying, and per-hundred. **Question:** Whole numbers count whole things. How do you say how much of a thing you have, how two amounts compare, or how big a share is out of every hundred? ## What this area is about Three ways of describing amounts that whole numbers alone cannot. A fraction names part of a whole. A ratio compares two amounts by how many times one fits in the other. A percent is a ratio measured against $100$. Each one builds on the one before. ## How these fit together Fractions is the foundation: a fraction $\frac{a}{b}$, for whole numbers $a$ and $b \neq 0$, is $a$ pieces of a whole cut into $b$ equal parts, and then a number on the line that can be renamed, compared, added, multiplied and divided. Ratio uses that multiplying. A ratio $a : b$ stays the same when both amounts are multiplied by one positive number, which gives equivalent ratios, unit rates, proportions, proportional relationships and scale drawings. Percent is a ratio with a fixed second amount: $p\%$ means $p$ for every $100$. It gives one scale for comparing parts of different wholes, and handles percent change, discounts, tax and interest. ## A path through it Fractions first, then ratio, then percent. Each depends on the one before: ratio needs multiplying fractions, and percent is defined as a ratio. Fractions lead on to rational numbers, ratio to similarity in geometry, and the whole area to probability, measuring how likely something is. ### [Inversely proportional relationships](https://math.wiki/inversely-proportional-relationships/) **Summary:** When one amount doubles the other halves, so the two always multiply to the same number. **Question:** Three workers take eight hours to lay a floor. Twice as many should take half as long, but what exactly stays the same, and how do you tell this pattern from an ordinary proportion? ## Why this exists Some pairs of amounts grow together: buy twice as many notebooks and you pay twice as much. That is a proportional relationship, which direct-variation writes as $y = kx$. Other pairs pull in opposite directions. More workers, less time. A faster drive, a shorter trip. More winners sharing a prize, a smaller share each. No fixed number times the number of workers gives the hours, so these are not proportional in the old sense. Something is still fixed, and it is a product, not a ratio. Three workers working $8$ hours do $3 \times 8 = 24$ worker-hours of work; six working $4$ hours also do $24$. "One goes up, the other goes down" is not enough on its own. A gift card loses $5$ dollars a week as it is spent, yet doubling the weeks does not halve what is left. This lesson is about the exact pattern, where doubling one amount halves the other. ## Definition Two quantities are **inversely proportional** when their product is always the same number. In letters, $y$ is inversely proportional to $x$ when $$xy = k$$ for every matching pair of values, where $k$ is a fixed number that is not $0$. Dividing both sides by $x$ gives the rule solved for $y$: $$y = \frac{k}{x}, \qquad x \neq 0$$ The number $k$ is still called the **constant of proportionality** (or constant of variation), but here it is a product rather than a ratio. "$y$ varies inversely with $x$" means the same thing, and the relationship runs both ways: $x = \frac{k}{y}$. ### [Locating a fraction on the number line](https://math.wiki/locate-a-fraction-on-the-number-line/) **Summary:** Counting unit fraction jumps from zero to find where a fraction lands. **Question:** Where on a line does the number three fourths sit, and how do you find that spot when nobody has marked it for you? ## When you reach for it Until now a fraction has been pieces of a whole: slices of a pie, wedges of a cheese wheel, as in fraction-area-model and fractions-greater-than-one. In count-by-unit-fractions the count ran along a bike path whose markers were already there. This lesson makes the markers yourself, and uses them to find the place where a fraction sits on the number line. On a number line every number is a point. $0$ is where the line starts, $1$ is one unit to the right, $2$ is one more unit on. A fraction is a number too, so it has a point of its own. The point $\frac{3}{4}$ sits between $0$ and $1$, three fourths of the way from $0$ to $1$. The point $\frac{5}{3}$ sits between $1$ and $2$. Seeing a fraction as one point on the same line as the whole numbers is the big idea here: it is not only a piece of something, it is a number with a place. You reach for this whenever a fraction measures a distance from a start along something straight: how far along a nature trail a bench stands, where a clip goes on a clothesline, where a float stopped on a parade route. It also pays off later. Once fractions are points on one line, the one farther to the right is the greater one, and the space between two points is a length. ## The procedure The lesson runs over four days. Part 1 cuts the stretch from $0$ to $1$ into equal parts. Part 2 finds a fraction less than one by jumping from $0$. Part 3 carries the jumps past $1$. Part 4 reads a point that someone else has marked. ### [Making an equivalent fraction by multiplying](https://math.wiki/multiply-to-make-equivalent/) **Summary:** Multiplying the top and the bottom by the same number renames a fraction in smaller pieces. **Question:** A quiche is cut into slices, and then every slice is cut again. The pieces change but the quiche does not, so what happens to the fraction? ## When you reach for it Ece, Halyna and Zoran are cooking brunch. There is a quiche, cut into slices, and there are waffles, which come cut into little squares. All morning the three of them talk in fractions: two thirds of a waffle, three quarters of the quiche. Then somebody cuts the food again. A slice of quiche is cut into thinner slices so more people can share. A strip of waffle is cut into squares. The food on the plate has not changed at all, but the pieces have, so the fraction gets a new name. You reach for this move whenever you want the same amount written with smaller pieces. You will need it to compare two fractions, to add them, and to write a fraction in hundredths. You already know the picture from equivalent-fractions-area-model: cutting every piece of a shape in two doubles both numbers of the fraction and changes nothing. This lesson cuts into any number of pieces and turns the picture into a rule. It has three parts, one for each day: - **Part 1:** cut every piece of a fraction model into the same number of smaller pieces. - **Part 2:** the rule, multiply the top and the bottom by the same number, and why it is true. - **Part 3:** find a missing top or bottom, and list a whole family of equal fractions. ## The procedure ### Part 1: cut every piece the same way Zoran has a waffle cut into $3$ equal strips. He spreads jam on $2$ of the strips, so jam covers $\tfrac{2}{3}$ of the waffle. Now he cuts every strip into $4$ squares. ### [Markup and margin](https://math.wiki/markup-percent/) **Summary:** The amount added to what something cost, given as a percent of that cost. **Question:** A seller buys something for one price and sells it for more. The extra is what percent? A percent of what it cost, or of what it sells for? ## When you reach for it Anyone who sells things has to get them first. A shop buys its stock from a supplier, a craft seller buys clay or beads or yarn, a café buys milk and coffee, a school fundraiser stall buys snacks in bulk. What the seller pays for one item is its **cost**. When the item is bought ready-made from a supplier, in bulk, the cost is also called the **wholesale price**. What a customer pays for it is the **selling price**. The gap between the two is the **profit** on the item: $$\text{profit} = \text{selling price} - \text{cost}$$ In this lesson profit means exactly that gap and nothing more: the price minus what the item itself cost. A shop's rent, a café's wages and a market stall's fee are left out of it, so it is not the same as the money the seller ends up keeping. A profit in dollars, like any change, says little on its own. Two dollars on a cup of coffee that cost $30$ cents to make is a lot; two dollars on a bicycle that cost $300$ dollars is almost nothing. So sellers state the profit as a percent. And here there is a real choice, because there are two natural wholes to measure it against. - The **markup** is the profit as a percent of the **cost**. - The **margin**, or **profit margin**, is the profit as a percent of the **selling price**. The two numbers are different, and they are different for the reason that runs through every percent question: a percent means nothing until you have named the whole. Both are the question from finding what percent one number is of another, with the profit as the part. Markup takes the cost as the whole; margin takes the price. ### [Mixed numbers and improper fractions](https://math.wiki/mixed-numbers/) **Summary:** A whole plus a part, and the same number written as one fraction. **Question:** Two whole loaves and three quarters of another, or eleven quarters. Which way should the amount be written, and how do you add and subtract amounts like it? ## What this area is about An amount bigger than one can be written two ways. A mixed number such as $2\frac{3}{4}$ is a whole number and a fraction less than $1$ side by side, and it means their sum, $2 + \frac{3}{4}$. An improper fraction such as $\frac{11}{4}$, whose numerator is at least its denominator, counts only pieces. Both name the same number, and each is easier for some jobs. ## How these fit together **Rewriting.** Mixed numbers and improper fractions turns the wholes into pieces and back again, and compares amounts written either way. **Adding and subtracting.** Because a mixed number is a sum, wholes can be combined with wholes and fraction parts with fraction parts. Like denominators is the case where the pieces already match, trading pieces for a whole when the fraction parts make more than one, and a whole for pieces when the fraction part taken is the bigger one. Adding mixed numbers and subtracting mixed numbers handle denominators that differ, renaming the fraction parts first. ## A path through it The one fixed step: rewriting comes before the like denominators case. Adding and subtracting with different denominators have no fixed place among the others. Next come multiplying and dividing mixed numbers, which lean on rewriting a mixed number as an improper fraction. ### [Mixed numbers and improper fractions: rewriting and comparing](https://math.wiki/mixed-to-improper/) **Summary:** Turning the whole part into pieces and adding them to the pieces already there. **Question:** A baker has two whole loaves and three quarters of another, all cut into quarters. How many quarter pieces is that, and how do you go back from a pile of pieces to whole loaves? ## When you reach for it Ozan, Levko and Jelena run a bakery stand. Some loaves sell whole. Others are cut into equal pieces first: quarters for the big loaves, fifths for the long thin ones. At the end of a morning the amounts left on the table are rarely whole numbers of loaves. There are two ways to write such an amount. One counts whole loaves and then a part of a loaf: $2\tfrac{3}{4}$ loaves, a **mixed number**, which means $2 + \tfrac{3}{4}$ (mixed-numbers). The other counts only pieces: $\tfrac{11}{4}$ loaves, eleven quarters, a fraction whose top is at least its bottom, called an **improper fraction** (fractions-greater-than-one). Both name the same amount of bread. You reach for this lesson whenever you need to switch between the two names. A count of pieces is easier to sell from: "I have $11$ quarter pieces." A count of wholes is easier to picture: "I have almost $3$ loaves." Later, adding, subtracting and multiplying mixed numbers all ask you to switch names in the middle of the work. And to say which of two amounts is bigger, it helps to write them the same way first. The lesson has three parts, one for each day: - **Part 1:** a mixed number to an improper fraction, by cutting every whole into pieces. - **Part 2:** an improper fraction to a mixed number, by packing pieces into wholes, using division with a remainder. - **Part 3:** mixed numbers on the number line, and comparing mixed numbers and improper fractions. ### [Multiplying as scaling](https://math.wiki/predict-size-of-a-fraction-product/) **Summary:** Judging a product against its starting number without doing the arithmetic. **Question:** A copy machine is set to make a picture three fourths as wide, or five thirds as wide. Before you press the button, can you tell whether the copy will come out bigger or smaller than the original? ## When you reach for it The school office has a copy machine that can make a picture bigger or smaller. You type in a size setting, written as a fraction, and every length in the copy is that fraction of the same length in the original. Yemi has a picture $12$ inches wide. With the setting $\frac{3}{4}$ the copy is $\frac{3}{4} \times 12$ inches wide. With the setting $\frac{5}{3}$ it is $\frac{5}{3} \times 12$ inches wide. Before she presses the button, Yemi wants to know one thing: will the copy be bigger or smaller than her picture? She can tell without multiplying at all. The answer sits in the setting, the number she multiplies by. This is **multiplication as scaling**, which means resizing. In a product such as $\frac{3}{4} \times 12$, think of $12$ as the thing being resized and $\frac{3}{4}$ as the **scale factor**, the amount it is resized by. Comparing the scale factor with $1$ tells you how the product compares with $12$. You already know how to work out a fraction of a number (fraction-times-whole-number) and a fraction of a fraction (fraction-times-fraction). This lesson is about judging the size of the answer before you find it. The lesson runs over three days. Day 1 reads a product as a resizing and compares it with the number being resized. Day 2 gives the rule for every case, a factor less than $1$, equal to $1$ or more than $1$, and shows why it is true. Day 3 shows that a fraction such as $\frac{4}{4}$ is a factor of exactly $1$, which explains why equivalent fractions have the same value, and uses the rule to compare and order products. ### [Multiplying fractions](https://math.wiki/multiply-fractions/) **Summary:** A fraction of a fraction, and why the numerators and denominators multiply. **Question:** Five copies of three fourths, or half of a third of a banner. What does it mean to multiply with fractions, and why do the tops and bottoms multiply? ## What this area is about Multiplying by a whole number counts copies, and multiplying by a fraction takes that fraction of an amount: the word "of" means times. Both readings lead to one rule, for whole numbers $a$, $c$ and nonzero $b$, $d$: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$ The children explain it with pictures before using it. ## How these fit together **Copies.** A fraction times a whole number takes several copies of one fraction, added up, and says which two whole numbers the product sits between. **A part of a part.** A fraction times a fraction cuts a rectangle both ways to find a fraction of a fraction, which is where the rule comes from. **Using the rule.** Multiplying mixed numbers splits a mixed number into its parts, or rewrites it as an improper fraction, before multiplying. Multiplying as scaling judges a product without computing it: for a positive number $N$, a factor less than $1$ gives less than $N$, a factor greater than $1$ gives more. ## A path through it A fraction times a whole number, then a fraction times a fraction. Mixed numbers and scaling both come after that, in either order. This area leads straight to a fraction of a whole number and dividing fractions, and later to volume with fraction edge lengths and probabilities on tree diagrams. ### [Multiplying mixed numbers](https://math.wiki/multiply-mixed-numbers/) **Summary:** Rewriting each mixed number as an improper fraction before multiplying. **Question:** Anvi pours $2\frac{1}{4}$ cups of seed into the backyard feeder every day. How much seed does she pour in $3$ days, and what if the number of days is a mixed number too? ## When you reach for it Anvi, Duc and Funmi keep bird feeders in their backyards. Anvi pours $2\frac{1}{4}$ cups of seed into her feeder every day. In $3$ days she pours $3$ groups of $2\frac{1}{4}$ cups, which is $3 \times 2\frac{1}{4}$ cups. Duc's seed bags weigh $12\frac{1}{2}$ pounds, and Funmi's ground tray is a rectangle whose sides are mixed numbers. Each of these asks for a product where at least one factor is a **mixed number**: a whole number and a fraction written side by side. A mixed number is a sum. $2\frac{1}{4}$ means $2 + \frac{1}{4}$, two wholes and one fourth more. The space between the $2$ and the $\frac{1}{4}$ is a hidden plus sign, never a times sign. It can also be written as one fraction, an **improper fraction**, with a top bigger than its bottom: $2\frac{1}{4} = \frac{9}{4}$, because $2$ wholes are $8$ fourths and one more fourth makes $9$ (mixed-to-improper). Those two ways of seeing a mixed number give two ways to multiply it. Seen as a sum, you multiply each part and add, which is the distributive property. Seen as one fraction, you use the rule you already have, tops times tops and bottoms times bottoms (fraction-times-fraction). This lesson takes three days. Day 1 multiplies a mixed number by a whole number or a fraction, by splitting it into parts. Day 2 multiplies two mixed numbers, by rewriting both as fractions, and checks the answer with a four-part area model. Day 3 estimates a product before working it out and uses the skill in problems about seed, recipes and areas. ### [Part-to-part and part-to-whole ratios](https://math.wiki/part-to-part-ratio/) **Summary:** A comparison of one group with another group, not with the total. **Question:** Two red flags for every five white ones. Does that mean two fifths of the flags are red? ## Why this exists Marisol is hanging bunting for a party: $2$ red flags, then $5$ white ones, then $2$ red and $5$ white again, all the way along. "Two to five" describes the pattern. But two to five of what? Ask what part of the bunting is red, and $\frac{2}{5}$ is the wrong answer. Each repeat of the pattern has $7$ flags, $2$ of them red, so the red part of the string is $\frac{2}{7}$. A ratio can compare a part with another part (red flags with white flags) or a part with the whole (red flags with all the flags). Both use a colon, and for the same string they give different numbers. This lesson names the two kinds, moves between them, extends the idea to three or more parts, and pins down when a ratio says the same thing as a fraction. ## Definition Take a collection split into parts, so that every item belongs to exactly one part. How a ratio is written and read is in ratio-language-and-notation; here the question is what is compared. - A **part-to-part ratio** compares one part with another part. Red flags to white flags is $2 : 5$. - A **part-to-whole ratio** compares one part with the whole collection. Red flags to all flags is $2 : 7$. - The whole is all the parts together, so its number is the sum of the parts' numbers: $2 + 5 = 7$. A **ratio of three quantities**, written $2 : 3 : 4$ and read "two to three to four", says that for every $2$ of the first there are $3$ of the second and $4$ of the third, all at once. Any two of its numbers, kept in order, compare those two parts, and the sum $2 + 3 + 4 = 9$ stands for the whole. Four or more parts work the same way. ### [Percent](https://math.wiki/percent/) **Summary:** Per hundred: a ratio with a fixed second part. **Question:** Eighteen of 25 seats are full on one ride and 34 of 50 on another. How do you compare parts of wholes that are not the same size? ## What this area is about A percent is a rate per hundred: $p\%$ means $p$ for every $100$, so $p\% = \frac{p}{100}$. Measuring every part against the same $100$ lets parts of different wholes sit side by side. Almost every question here ties three numbers together: part $=$ percent $\times$ whole. ## How these fit together **The idea.** What percent means defines it as a ratio whose second amount is always $100$. Converting among fractions, decimals and percents renames one number in all three forms, and percents above 100 and below 1 cover a comparison larger than the whole or a part too small for a whole percent. **Finding the numbers.** Benchmark percents such as $10\%$ and $50\%$ work in the head and give estimates. Finding a percent of a quantity multiplies by the percent as a decimal, and the three percent questions solve for whichever of part, percent or whole is missing. **Uses.** Percent change measures a rise or fall against a reference amount, usually the start. Percent in money situations covers discounts, tax, tips, markup and interest. ## A path through it What percent means first, then converting. A percent of a quantity follows converting; after it come percents above 100 and below 1, and the three percent questions. Percent change needs all three of those, and money situations come last. Benchmark percents fit any time after the first lesson and before money. Sampling and study design assumes percent. ### [Percent change](https://math.wiki/percent-change/) **Summary:** Increase or decrease, measured against where you started. **Question:** An amount went up or down. How do you say how big the change was, in a way that means the same thing whatever size the amount started at? ## What this area is about Subtracting says how much something changed, but not whether that was a lot. A rise of $120$ dollars is large on a rent of $400$ and small on a rent of $4{,}000$. Percent change measures the change against a reference amount, usually where it started, so changes to amounts of different sizes can be compared. ## How these fit together **The basic measure.** Percent increase and decrease divides the change by the original amount and writes the result as a percent. **The same move on a miss.** Percent error measures how far an estimate sits from the true value, as a percent of the true value. The distance is an absolute value, so the error is never negative. **The multiplier.** The multiplier for a percent change turns an increase of $r$ (as a decimal) into one multiplication by $1 + r$, and a decrease into one by $1 - r$. Dividing by it runs a change backwards. **Changes in a row.** Two percent changes one after the other multiply their multipliers. Adding the percents gives the wrong answer, because the second percent is taken of the amount after the first change. ## A path through it Start with percent increase and decrease. Percent error can come next or at any later point. The multiplier comes after percent increase, and successive changes after the multiplier. Finish here and discounts and sales tax, markup and margin, simple interest and growth and decay factors all build on it. ### [Percent error](https://math.wiki/percent-error/) **Summary:** How far a measured or predicted value sits from the true one, as a percent of the true one. **Question:** A guess, a prediction or an experiment gives one number, and the true value turns out to be another. How far off was it, in a way that means the same thing whether the true value is 40 or 4,000? ## When you reach for it Someone produced a number before the true one was known, and now the true one is known. Leocadia guessed how many marbles were in a jar, and then the jar was counted. An organizer predicted how many people would come to a concert, and then the tickets at the door were counted. A class worked out the time a pendulum takes to swing, and then compared it with the accepted value printed in a book. Subtracting tells you how far off the number was. As with any change, that amount alone cannot say whether the miss was large. Being $10$ marbles out is a poor guess for a jar of $40$ and an excellent one for a jar of $4{,}000$. To judge the miss, and to compare misses on things of different sizes, you measure it against the true value. That measurement, written as a percent, is the **percent error**. The arithmetic is the arithmetic of percent increase and decrease: a gap, divided by the amount it is measured against. What changes is the question. There is no "before" and "after" here, only a value that was supposed to be right and the value that actually is. ## The procedure Two numbers go in: the **estimate** (a guess, a prediction, an experimental result) and the **true value** (the actual count, what really happened, or the accepted value). Take a playing field that is truly $50$ meters long, which Noor paces out as $40$ meters. **Find the size of the error.** The error is how far the estimate sits from the true value: ### [Percent in money situations](https://math.wiki/percent-applications/) **Summary:** The everyday uses of percent in prices, wages, loans and taxes. **Question:** A price tag, a bill, a paycheck and a loan all carry a percent. Which amount is each percent taken of, and is the result added on or taken off? ## What this area is about Most percents met outside a classroom are about money: a sale, a tax, a tip, a markup, interest on savings or a loan. Each is a percent of a base amount, so the work repeats: decide which amount is the whole, find the part, then add it on or take it off. ## How these fit together All three are procedures built on the percent equation, part $=$ rate $\times$ whole. **At the till.** Discounts, sales tax, tips and commission take a percent of a price, a bill or a total of sales, then subtract it, add it, or pay it to the seller. **What a seller adds.** Markup and margin measure one profit two ways: markup as a percent of the cost, margin as a percent of the selling price. For a seller who makes a profit, the margin is the smaller of the two. **Money over time.** Simple interest is $I = Prt$: a percent $r$ of the principal $P$ for each of $t$ years, worked out on the principal only, so earlier interest never earns interest of its own. ## A path through it None of the three depends on another, so they can be taken in any order. Each assumes percent change, and discounts and simple interest also use two percent changes in a row. Finish here and you can check a receipt, tell a markup from a margin, and total a simple-interest loan or deposit. ### [Percent increase and decrease](https://math.wiki/percent-increase/) **Summary:** How much something grew, measured against what it was before. **Question:** A rent goes up by 120 dollars a month. Is that a big rise or a small one? How do you measure a change so that it means the same thing whatever size the amount started at? ## When you reach for it Something has changed, and you know what it was before and what it is now. An apartment's rent went from $800$ to $920$ dollars a month. A climbing gym had $240$ members and now has $317$. A city park grew from $12$ hectares to $15$. Subtracting tells you **how much** it changed, and sometimes that is all you need. But the amount alone cannot say whether the change was large. A rise of $120$ dollars is a lot on a rent of $400$ dollars and very little on a rent of $4{,}000$. To judge a change, and to compare changes to amounts of different sizes, you measure it against where it started. That measurement, written as a percent, is the **percent increase** (or **percent decrease**, if the amount went down). It is the question from finding what percent one number is of another, asked about a change: the change is the part, and the starting amount is the whole. This lesson is taught over two days. Part 1 is percent increase. Part 2 is percent decrease, what happens when a change is undone, and the difference between percent and percentage points. ## The procedure ### Part 1: percent increase Two numbers go in: the **original amount**, what the quantity was before, and the **new amount**, what it is after. Take the rent going from $800$ to $920$ dollars. **Find the amount of increase.** The increase is how far the new amount is above the original: $$920 - 800 = 120$$ So the rent went up by $120$ dollars. ### [Percents above 100 and below 1](https://math.wiki/percents-over-one-hundred/) **Summary:** A percent above one hundred names more than the whole, which happens whenever something grows. **Question:** A puppy can end its first year more than twenty times as heavy as it was at birth. If a percent counts hundredths of a whole, how can one amount be more than a hundred percent of another, and what does a percent smaller than one mean? ## Why this exists A percent first appears as a part of a whole: $18$ of $24$ tulip bulbs, $7$ of $20$ players. A part can never be bigger than its whole, so those percents run from $0\%$ to $100\%$. Many comparisons are not a part against its whole, though. A town is compared with the same town twenty years ago, a puppy with itself at birth, one dog with another, an office tower with the shed beside it. The second amount is not a piece of the first, and it can easily be the larger. Saying that a school now has $162.5\%$ of the students it opened with puts the two sizes side by side in one number, just as $75\%$ did for a part. At the other end, some parts are so small that a whole percent is too coarse to show them. In a town of $36{,}000$ people, $1\%$ is already $360$ people. A group of $27$ is far smaller than that, and rounding it to $0\%$ would say it is not there at all. Percents below one, such as $0.5\%$ or $0.075\%$, keep a group like that in view. Neither needs a new idea. A percent counts hundredths, and nothing about hundredths stops at one hundred of them or insists on at least one. ## Definition **Percent** means "per hundred", so a percent is a number of hundredths. For any number $p \ge 0$, whole or not, large or small, $$p\% = \frac{p}{100}$$ The percent sign divides whatever stands in front of it by $100$. When $p$ is above $100$ the result is above $1$, as in $250\% = 2.5$. When $p$ is below $1$ the result is below $\frac{1}{100}$, as in $0.5\% = 0.005$. ### [Proportion](https://math.wiki/proportion/) **Summary:** Two ratios set equal, and solving for the unknown part. **Question:** Two ratios are supposed to describe the same comparison, and one of their four numbers is missing. How do you write that down, and how do you find the missing number? ## What this area is about A proportion is a statement that two ratios are equivalent, written $a : b = c : d$ or $\frac{a}{b} = \frac{c}{d}$ with $b$ and $d$ not $0$. It is how a fixed comparison, such as stock to servings in a soup, gets carried from one size to another. This area says what the statement means, how to solve it, and how to build one from a word problem. ## How these fit together **The definition.** What a proportion says: two ratios of positive numbers are equivalent when the second is the first with both parts multiplied by the same positive number. **The procedure.** Solving a proportion finds the unknown that makes the two ratios equal: by scaling across from one ratio to the other, through a unit rate, by multiplying both sides, or by cross products, the last two working whatever the numbers are. **The setup.** Setting up proportions from word problems decides which number goes on top and which underneath, matching units on both sides, and first checks that the situation keeps a fixed ratio at all. ## A path through it What a proportion says comes first, then solving one, then setting one up from words. Finish here and proportional relationships, scale drawings, similar figures, percent problems and equations from proportions and percents all use it. ### [Proportional relationships](https://math.wiki/proportional-relationships/) **Summary:** When one quantity is always a fixed multiple of another, and how to spot it in a table, graph or equation. **Question:** When one quantity is always the same multiple of another, how do you spot it, in a table, a graph or an equation, and what is the one number that does the multiplying? ## What this area is about Cheese at $18$ dollars a kilogram: every weight times $18$ gives its price. Two quantities $x$ and $y$ are in a proportional relationship when one fixed number $k$ gives $y = kx$ for every pair. This area names that number and shows how to recognize the pattern in each form it comes in, plus the pattern where a product, not a ratio, stays fixed. ## How these fit together **The definition.** The constant of proportionality is $k$, which equals $\frac{y}{x}$ for every pair with $x \ne 0$: the unit rate. In a table, the test is whether that quotient is the same in every row with a non-zero input, and an input of $0$, if there is one, has output $0$. **Recognizing it.** In a graph, a proportional relationship is a straight line through the origin, and every such line other than the two axes is one. In an equation, it has the form $y = kx$, with $k$ fixed and not $0$, and nothing added on. **The other pattern.** Inversely proportional relationships keep the product fixed, $xy = k$ with $k \ne 0$: when one amount doubles, the other halves. ## A path through it The constant of proportionality comes first. Inverse proportion can follow it at any point. The graph test also follows it, and the equation test follows the graph test. Finish here and scale factor and direct variation build on it. ### [Rate and unit rate](https://math.wiki/rate-and-unit-rate/) **Summary:** A ratio of different kinds of things, scaled to per one. **Question:** Ninety boxes pass in six minutes. What one number says how fast the belt runs, and how do you use it to compare, predict and convert? ## What this area is about A rate is a ratio of two amounts measured in different units: boxes and minutes, dollars and liters, miles and hours. Its unit rate is how much of the first amount goes with exactly one of the second, found by division: $90$ boxes in $6$ minutes is $90 \div 6 = 15$ boxes per minute. One number like that answers questions about any amount. ## How these fit together **The definition.** Rates and unit rates says what a rate is and how to scale one to "per one". **Familiar rates.** Speed, distance and time is distance per unit of time; at a constant speed it gives how far something goes or how long it takes. Unit price and comparing rates is price per item or per unit of measure, used to decide which package size gives more for the money. **Procedures.** Unit rates when the amounts are fractions finds the amount per one when both amounts are fractions. Changing the units of a rate rewrites a rate, such as feet per minute as miles per hour, by multiplying by conversion factors equal to $1$, so the rate itself does not change. ## A path through it Rates and unit rates come first. The other four each build on it and can follow in any order. Unit price also uses comparing two ratios, and changing units uses converting customary units. Finish here and the constant of proportionality, rate of change, density and conversion factors build on it. ### [Rates and unit rates](https://math.wiki/unit-rate/) **Summary:** How much of one amount goes with exactly one of the other. **Question:** Three liters from a water cooler fill twelve cups. How many cups does one liter fill, and how much water goes into one cup? ## Why this exists Marek watches boxes roll past on a factory conveyor belt. In $6$ minutes he counts $90$ boxes. The ratio $90 : 6$ is true, but the question people ask about a belt is how fast it runs, and the clearest answer is how many boxes pass in exactly one minute: $15$. With that single number any stretch of time is easy. Ten minutes carry ten lots of $15$ boxes. The same need turns up wherever two different kinds of thing are tied together: cups poured for each liter in a water cooler, bottles packed into each crate at a bottle bank. A comparison of two amounts measured in different units is a **rate**, and bringing it down to exactly one of the second amount gives the **unit rate**, the most useful number in the comparison. ## Definition ### Day 1: rates and the unit rate A ratio compares two amounts. When the two amounts are measured in **different units**, such as boxes and minutes, cups and liters, or bottles and crates, the ratio is called a **rate**. The units travel with the numbers: "$90$ boxes in $6$ minutes", "$25$ cups for every $5$ liters". A rate can also be written as a fraction with its units, $$\frac{90 \text{ boxes}}{6 \text{ minutes}}$$ The word **per** means "for each" or "for every", and is often written as a slash, as in boxes/minute. The **unit rate** is the amount of the first quantity that goes with exactly $1$ of the second. For the belt it is $15$ boxes per minute, short for $15$ boxes for every $1$ minute. It is found by dividing the first amount by the second: ### [Ratio](https://math.wiki/ratio/) **Summary:** Comparing two amounts by how many times one fits in the other. **Question:** Three trumpets for every two drums. How do you describe how two amounts go together, so the description still holds when everything is scaled up or down? ## What this area is about A ratio compares two amounts by multiplying, not subtracting: $3 : 2$ says "for every $3$ of one, $2$ of the other". Multiply both amounts by the same positive number and the ratio stays the same. That one fact runs through rates, proportions and scale drawings. ## How these fit together **Definitions.** What a ratio is gives the language and notation, part-to-part and part-to-whole ratios, and how to tell which of two ratios is larger. Equivalent ratios and ratio tables scales both amounts by one positive number and keeps track of it in tables and diagrams. **Ratios of different units.** Rate and unit rate compares amounts in different units, such as boxes and minutes, and divides to get the amount for exactly one. **Results and procedures.** A proportion states that two ratios are equivalent, $\frac{a}{b} = \frac{c}{d}$ with $b$ and $d$ not $0$, and solves for a missing number. Proportional relationships are pairs of quantities with $y = kx$ for one fixed $k$, recognized in a table, graph or equation, set beside the inverse pattern $xy = k$. Scale drawings multiply every length by one positive scale factor and keep the angles, so a map gives real lengths. ## A path through it What a ratio is and equivalent ratios go together, in either order. Then rate and unit rate, then proportion, then proportional relationships, and scale drawings last. Ratio leads on to percent, a ratio out of $100$, and to similarity and partitioning a segment in a ratio in geometry. ### [Ratio tables](https://math.wiki/build-a-ratio-table/) **Summary:** Listing equivalent ratios in rows so the pattern between them is easy to see. **Question:** One mix is known. How do you lay out many more amounts of the same mix, and tell at a glance whether a list of amounts really keeps it? ## When you reach for it Beatrix is hanging wallpaper. The packet says to mix $3$ scoops of paste powder with $8$ liters of water. One wall needs a small batch and the whole room a large one, and every batch must be the same mix. Rather than work each batch out on its own, she lists them: | Scoops of powder | Liters of water | |:-:|:-:| | $3$ | $8$ | | $6$ | $16$ | | $9$ | $24$ | | $12$ | $32$ | | $15$ | $40$ | Every row equals $3 : 8$. A table like this is a **ratio table**: one column for each quantity, and every row the same ratio scaled by some number. Reach for one when a comparison stays fixed and you want many amounts at once, when an amount is awkward to reach in one step, or to test whether a list of pairs keeps one ratio. Why scaling both parts keeps a ratio the same is scaling a ratio up or down; one missing number in a pair of equal ratios is finding a missing value. A ratio table lines many such pairs up, so the rows can help each other. Part 1 (day one) builds tables and reads their patterns; Part 2 (day two) fills gaps and tests tables. ## The procedure ### Part 1: building rows by scaling Every row of a ratio table is the first row multiplied by one number, the same number in both columns. For $9$ scoops, from $3$ to $9$ is $\times 3$, so the water is $8 \times 3 = 24$ liters. The number may be less than $1$: half a batch is $\times \frac{1}{2}$, which gives $1.5$ scoops and $4$ liters. ### [Recognizing a proportional relationship in a graph](https://math.wiki/proportional-in-a-graph/) **Summary:** A straight line through the origin, and nothing else, is proportional. **Question:** A graph shows how far a cyclist has gone after each hour. How can you tell from the picture alone whether the distance is proportional to the time? ## When you reach for it A table of a journey makes you divide every pair of numbers to see whether one ratio holds. A graph lets you see the answer. Reach for this skill whenever two quantities are drawn against each other on the coordinate plane and the question is whether one is a fixed multiple of the other: whether distance is proportional to time, as it is for anything moving at a steady speed, or whether something (a head start, a stop, speeding up) has broken the pattern. The test is short. **The graph of a proportional relationship is a straight line through the origin, and every straight line through the origin, other than the two axes themselves, is the graph of a proportional relationship.** Both halves of the test matter: a straight line that misses the origin is not proportional, and neither is a curve that passes through it. A graph that passes also hands over the constant of proportionality: it is the height of the line one unit to the right of the origin. The lesson runs over two days. Part 1 is the test and why it works; Part 2 is reading a proportional graph. ## The procedure ### Part 1: the test, and why it works Throughout, $x$ is the quantity on the horizontal axis and $y$ the one on the vertical axis. For motion, $x$ is usually the time and $y$ the distance traveled. Recall that $y$ is proportional to $x$ when there is one number $k$ with $$y = kx$$ for every pair of values, so that every non-zero $x$ gives the same ratio $\frac{y}{x} = k$. The lesson on the constant of proportionality covers this, and how to spot it in a table. ### [Recognizing a proportional relationship in an equation](https://math.wiki/proportional-in-an-equation/) **Summary:** The form y equals k times x, with nothing added on the end. **Question:** An equation ties two quantities together. How can you tell, just by reading it, whether one is always the same multiple of the other? ## When you reach for it You already have two ways to test whether two quantities are proportional. In a table, every output divided by its input gives the same number, the constant of proportionality. On a graph, the points lie on a straight line through the origin (proportional-in-a-graph). Both tests need numbers worked out or points plotted first. An equation often comes before either. How far a car goes on its fuel, how many pages a printer has printed, what a phone plan charges for data: each can be written as a rule connecting two quantities, such as $$d = 28g$$ for the miles $d$ a car goes on $g$ gallons of fuel. This lesson is about reading such a rule and deciding, without a table or a graph, whether the relationship is proportional, and if it is, what the constant is. Reach for it in three situations. When an equation is given and the question is whether it describes a proportional relationship. When a situation is described in words and you want its equation, since every proportional situation has an equation of one simple shape. And when two quantities change together at a steady rate, so their graph is a straight line, and you need to know whether that line passes through the origin: a fixed fee, a starting amount or a warm-up time makes a relationship linear without making it proportional. The lesson takes two days. Part 1 reads and writes equations of the form $y = kx$. Part 2 tells a proportional equation apart from one that is only linear, or not linear at all. ### [Scale drawings](https://math.wiki/scale-drawings/) **Summary:** Maps and blueprints: the same shape at a different size, by a fixed factor. **Question:** A map, a floor plan and a model all show something at a different size. How can the same shape be bigger or smaller, and how do you read real lengths off the drawing? ## What this area is about A logo on a poster and on a badge must look like the same logo. Adding the same length to every side distorts the shape; multiplying every length by the same positive number keeps it, as long as the angles stay the same. That one number, and the scale printed on maps and plans, are what this area is about. ## How these fit together **The definition.** Scale factor: a figure $B$ is a scale copy of a figure $A$ when every length in $B$ is $k$ times the matching length in $A$, for one positive number $k$, and every angle matches. A $k$ greater than $1$ enlarges and a $k$ between $0$ and $1$ shrinks. **The procedure.** Using a scale on maps, plans and models reads a stated scale, such as $1$ inch to $25$ miles, as a rate, and uses it to go from a length on the drawing to the real length or back. ## A path through it Scale factor comes first, then reading maps and plans. Finish here and scale factor on lengths, perimeters and areas of scaled figures, slope triangles and dilations build on it. ### [Scale factor](https://math.wiki/scale-drawing-scale-factor/) **Summary:** The single number every length is multiplied by when one figure is enlarged or shrunk into a similar one. **Question:** A logo has to go on a large poster and on a small badge and still look exactly like itself. What has to happen to its lengths, and what single number describes the change? ## Why this exists A small company has a logo, drawn $5$ cm wide and $3$ cm tall. It wants the logo on a poster, much larger, and on a badge, much smaller, and in both places it must look like the same logo: nothing squashed, nothing pulled out of shape. Adding the same amount to every length does not do it. Adding $35$ cm to both sides turns the $5$ by $3$ logo into one $40$ cm wide and $38$ cm tall, nearly square. What keeps a shape is **multiplying**: if the width is multiplied by $8$, the height must be multiplied by $8$ too, and so must every other length in the drawing. That one multiplier is the subject of this lesson. The lesson takes two days. Part 1 is what a scale copy is and how to find and use its scale factor. Part 2 is testing whether a figure really is a scale copy, working back from a copy to the original, and writing the scale factor as a ratio. ## Definition A figure $B$ is a **scale copy** of a figure $A$ when there is one positive number $k$ such that - every length in $B$ is $k$ times the matching length in $A$, and - every angle in $B$ is equal to the matching angle in $A$. The number $k$ is the **scale factor** from $A$ to $B$. Matching lengths are lengths in the same place on the two figures: the bottom edge of one and the bottom edge of the other, the left side and the left side. Where the copy sits, and whether it has been turned or flipped over, does not matter. ### [Scaling a ratio up or down](https://math.wiki/scale-a-ratio-up/) **Summary:** Multiplying both amounts by the same number so the comparison stays the same. **Question:** A jug of cordial tastes just right. How do you make a bigger jug, or a smaller one, that tastes exactly the same? ## When you reach for it A ratio such as $3 : 7$ describes a mix: for every $3$ scoops of cordial there are $7$ scoops of water. (How a ratio is written and read aloud is in what a ratio is.) A recipe comes in one size, but jugs, bags of feed and wheelbarrows come in many. You reach for scaling whenever you need more or less of a mix that has to stay the same mix: a party bowl of cordial that tastes like the glass, a big batch of mortar as strong as the small one, a week of guinea pig food in the same balance as a day's. The same move answers two more questions. Are two mixes really the same, even though their numbers are different? And what are the smallest whole numbers that describe a mix, so it is easy to read and compare? The first question is about **equivalent ratios**, the second about **simplest form**. Every ratio in this lesson compares one part of a mix with another part, a part-to-part-ratio. ## The procedure ### Scaling up: copies of the group Read $3 : 7$ as a group: $3$ scoops of cordial that go with $7$ scoops of water. Two batches are two copies of that group, so $3 + 3 = 6$ scoops of cordial and $7 + 7 = 14$ scoops of water, the ratio $6 : 14$. Three batches are three copies, $9 : 21$. Every scoop of cordial in the big bowl still belongs to a group of $3$ that has its own $7$ scoops of water, so the bowl tastes the same as one glass. Making copies is multiplying, so ### [Seeing equivalent fractions in pictures and on the number line](https://math.wiki/equivalent-fractions-area-model/) **Summary:** Cutting every piece of a shaded shape in two doubles both numbers and changes nothing. **Question:** Two thirds of a pan of corn bread is left, and someone cuts every piece in two. Now it is four sixths of the pan. Did the amount of corn bread change, or only its name? ## Why this exists A fraction names an amount by counting equal pieces: in fraction-area-model, $\frac{2}{3}$ of a pan is $2$ of the $3$ equal pieces it was cut into. But the pieces are a choice. The same pan could have been cut into $6$ pieces, or $9$, and the same corn bread would be on it. So one amount can be counted in more than one way, and each way gives it a different name. Bolor has $\frac{2}{3}$ of a pan of corn bread left. Saran cuts every piece in two, and now the pan holds $\frac{4}{6}$ of its corn bread. Nobody ate any, and nobody baked more. Two different fractions are naming one amount. This lesson is about seeing when that happens, in a picture and on a number line, and making new names for a fraction by cutting every piece in two or by joining pieces in pairs. The same thing happens on a line. In locate-a-fraction-on-the-number-line each fraction got a point. A trail marked in fourths of a mile and also in eighths of a mile has a flag at $\frac{3}{4}$ of a mile and a marker at $\frac{6}{8}$ of a mile, and they stand at the same spot. Two names, one point. ## Definition Two fractions are **equivalent** when they name the same amount of the same whole. On a number line with one unit, that means they name the **same point**. We write an equals sign between equivalent fractions: $$\frac{2}{3} = \frac{4}{6}$$ This is read "two thirds equals four sixths". It does not say the pictures look alike: $3$ pieces and $6$ pieces are different cuts. It says the amount shaded, or the distance from $0$, is the same. ### [Setting up proportions from word problems](https://math.wiki/set-up-a-proportion-from-words/) **Summary:** Choosing what goes on top and what goes underneath so both sides line up. **Question:** A crew of 6 filled 15 bags of litter, and a crew of 14 is coming. Four numbers, one unknown. Which goes on top, which goes underneath, and does it matter? ## When you reach for it A litter-pick crew of $6$ volunteers fills $15$ bags in a morning. A bigger crew of $14$ is coming next week to work the same way, and the organizer wants to know how many bags they will fill. Four numbers are in play and three are known: $15$ bags, $6$ volunteers, $14$ volunteers, and the unknown number of bags, $b$. Solving a proportion finds a missing number once the proportion is written down. This lesson is about writing it down: which number goes on top, which goes underneath, and on which side. Getting that wrong produces no error message. The equation $\frac{15}{6} = \frac{14}{b}$ solves perfectly well, to $b = 5.6$, and claims that more than twice as many volunteers fill far fewer bags. The arithmetic is fine; the four numbers are in the wrong places. Reach for this lesson whenever a word problem describes two amounts that grow together at a fixed rate (bags per volunteer, pages per minute, goals per game), gives one complete pair, and asks about another. ## The procedure This lesson is taught over two days. Part 1 is placing the numbers: the rule about units, and why it works. Part 2 is the word problem around the proportion: whether a proportion fits at all, units that differ, a total in place of a part, and reading the answer back into the story. ### Part 1: putting each number in its place **Name the unknown, with its unit.** Here $b$ is a number of bags. The unit is what tells you where the letter belongs, so a letter without one is a setup waiting to go wrong. ### [Simple interest](https://math.wiki/simple-interest-formula/) **Summary:** Multiplying principal, rate and time to get the interest earned or owed. **Question:** Money left with a bank or lent to someone comes back with something extra. How much extra, and how do you work it out? ## When you reach for it Money can be rented. Put it in a savings account and the bank uses it, so the bank pays you for the use. Borrow it to buy a laptop or a bike and you pay the lender for the use. The payment for the use of money is called **interest**, and it is set as a percent of the money, for each year the money is out. Reach for the simple interest formula whenever a sum of money is lent or deposited at a stated rate, the interest is worked out on the original sum only, and you want to know one of four things: how much interest, how much in total, what rate, or how long. It is the most direct use of a percent of a quantity there is, repeated once for every year. The rates in this lesson are examples chosen to make the arithmetic clear. Real rates vary from lender to lender and from year to year. The lesson runs over two days. Part 1 names the five quantities, builds the formula $I = Prt$ and finds the total. Part 2 runs the formula backwards to find a missing rate, time or starting sum, handles rates given per month, and sets simple interest beside compound interest. ## The procedure ### Part 1: five words Every interest problem has the same cast. - The **principal**, $P$, is the sum of money deposited or borrowed at the start, in dollars. - The **rate**, $r$, is the percent of the principal paid as interest **for each year**. A rate of $3\%$ per year means $3\%$ of the principal for every year. In the formula it is written as a decimal or fraction: $3\% = 0.03$. - The **time**, $t$, is how long the money is out, **in years**. Eight months is $\frac{8}{12} = \frac{2}{3}$ of a year. - The **interest**, $I$, is the extra money, in dollars. A saver earns it; a borrower owes it. - The **total amount**, $A$, is the principal and the interest together: what the saver has at the end, or what the borrower must pay back. ### [Simplifying a fraction to simplest form](https://math.wiki/divide-to-make-equivalent/) **Summary:** Dividing the top and the bottom by a common factor renames a fraction in bigger pieces. **Question:** Twelve of the eighteen squares on a paint chip card are blue. Is there a shorter name for the same amount, using fewer and bigger pieces? ## When you reach for it Defne, Petro and Snezana are sorting paint chip cards. A paint chip card is a strip of stiff paper printed with small squares of color, every square the same size. The three of them describe the cards in fractions: twelve of the eighteen squares on this card are blue. That is true, but it is a mouthful. Twelve eighteenths is the same amount as two thirds, and two thirds is much easier to picture. In multiply-to-make-equivalent you cut every piece of a fraction into smaller pieces, which multiplies the top and the bottom by the same number. This lesson runs that move backward. Gluing small pieces together into bigger ones divides the top and the bottom by the same number, and the fraction gets a shorter name for the same amount. You reach for this move whenever a fraction is written with bigger numbers than it needs: to read an amount at a glance, to see that two fractions are equal, and to keep later work with fractions small. The lesson has three parts, one for each day: - **Part 1:** group the small pieces of a model into bigger pieces. - **Part 2:** the rule, and simplest form, reached in steps or in one step with the greatest common factor. - **Part 3:** check that a simplified fraction is equal, and see why the number you divide by must be a factor of both the top and the bottom. ## The procedure ### Part 1: group small pieces into bigger ones Snezana holds a paint chip card with $18$ small squares, in $3$ rows of $6$. The left $4$ columns are shades of blue, so $3 \times 4 = 12$ squares are blue, and the blue part of the card is $\tfrac{12}{18}$. ### [Solving a proportion](https://math.wiki/solve-a-proportion/) **Summary:** Finding the unknown amount that makes the two ratios equal. **Question:** A soup recipe serves 8 and you are cooking for 20. Three of the four amounts are known. How do you find the fourth? ## When you reach for it A soup recipe serves $8$ people and uses $6$ cups of stock. You are cooking the same soup for $20$. For it to taste the same, the stock has to compare with the servings exactly as it did before: the ratio of stock to servings must not change. Calling the unknown amount of stock $x$, that is the proportion $$\frac{6}{8} = \frac{x}{20}$$ with cups of stock on top and servings underneath on both sides. What a proportion says tested proportions in which all four numbers were known. Here one number is missing, and the question turns round: which number makes the proportion true? Finding it is **solving the proportion**. In this case there is exactly one answer. The cross products of a true proportion are equal, so $6 \cdot 20 = 8 \cdot x$, that is $8x = 120$, and only $x = 15$ does that. Reach for it whenever two quantities grow together at a fixed rate, and you know one matching pair and one more amount: a recipe scaled up for a crowd, how long a steady typist needs for a longer piece, a pulse counted for part of a minute and turned into beats per minute, a scale drawing turned into real lengths. If you have found a missing value in a pair of equal ratios with a ratio table, this is the same question, written as an equation so that it works for any numbers and not only friendly ones. ## The procedure Take the soup proportion $\frac{6}{8} = \frac{x}{20}$. There are four ways to finish it. The first two are quick when the numbers are friendly; the last two always work. ### [Solving percent problems with an equation or a proportion](https://math.wiki/percent-as-an-equation/) **Summary:** Turning part equals percent times whole into an equation that can be solved. **Question:** A label says a serving gives 15% of a day's iron, a candidate won 45% of the votes, a player made 34 of her 80 shots. Each sentence ties three numbers together. Is there one equation that finds whichever of the three is missing? ## When you reach for it Every percent statement ties together three numbers. "Halima received $45\%$ of the $460$ votes cast, which was $207$ votes" has a **whole**, $460$ votes, the amount the percent is taken of; a **percent**, $45\%$; and a **part**, $207$ votes, the piece that the percent picks out of the whole. In a real question one of the three is missing, and which one it is changes from problem to problem: - a label gives a nutrient's daily value and the percent in one serving: how much is in the serving? (the part is missing) - a candidate's votes and her share of the vote are known: how many votes were cast? (the whole) - a player's shots made and shots taken are known: what is her shooting percentage? (the percent) Each can be worked on its own terms: dividing the part by the whole finds the percent, and working backwards from a part finds the whole. This lesson replaces the three separate questions with one relationship, $$\text{part} = \text{percent} \times \text{whole}$$ written either as an equation or as a proportion. You reach for it when you would rather translate the sentence than first decide which kind of question it is: put the two numbers you know in their places, put a letter where the missing one goes, and solve. This lesson is taught over two days. Part 1 writes and solves the percent equation. Part 2 writes the same relationship as a proportion and solves it with the moves from solve-a-proportion. ### [Speed, distance and time](https://math.wiki/speed-as-a-rate/) **Summary:** Distance covered for each unit of time, the most familiar rate there is. **Question:** A snail, a balloon and an elevator all move, at very different paces. What one number says how fast each goes, and once you have it, how do you find how far it gets or how long it takes? ## Why this exists A garden snail crosses a paving slab. A hot air balloon drifts over fields. An elevator carries people up a tall building. Which one is fast? "It went $30$ cm" does not say, and neither does "it took $5$ minutes". A snail that crawls $30$ cm in an hour is slow; one that crawls $30$ cm in $5$ minutes is moving well, for a snail. How fast something goes depends on two amounts at once: the distance it covers and the time it takes. A unit-rate squeezes a pair like that into one number: how much of the first amount goes with exactly one of the second. Speed is that idea applied to moving things. It gives every snail, balloon and elevator one number, so they can be compared, and once that number is known, the distance after any time and the time for any distance follow from it. ## Definition ### Day 1: speed, distance and time The **speed** of a moving thing is the distance it covers for each one unit of time. A snail that crawls $30$ cm in $5$ minutes, at a steady pace, covers $6$ cm in each minute, so its speed is $6$ cm per minute. "Per" means "for each". A speed is always written as a distance unit per a time unit: centimeters per minute, meters per second, kilometers per hour. Something moves at a **constant speed** (a **steady** speed) when it covers the same distance in every stretch of time of the same length, however short the stretches are. The problems of the first day all assume a constant speed. ### [Subtracting mixed numbers](https://math.wiki/subtract-mixed-numbers/) **Summary:** Taking whole from whole and fraction from fraction. **Question:** Khanh's garden wagon holds five and one quarter cubic feet of mulch, and he shovels out two and two thirds cubic feet. How can he take two thirds away when there is only one quarter beside the wholes? ## When you reach for it Khanh, Nanette and Ojas keep a community garden. A big pile of mulch sits by the gate, and they haul it to the beds in a garden wagon. The wagon is measured in cubic feet, and the loads almost never come out even. A load might be $6\tfrac{7}{8}$ cubic feet, and a bed might take $3\tfrac{1}{4}$ of it. A mixed number such as $6\tfrac{7}{8}$ is a whole number and a fraction together: $6$ wholes and $\tfrac{7}{8}$ of another. You reach for this lesson whenever one mixed number is taken from another, or a mixed number is taken from a whole number. It has three parts, one for each day: - **Part 1:** wholes from wholes and fractions from fractions, when the fractions have different denominators. - **Part 2:** when the fraction being taken is bigger than the fraction you have, trading one whole for fraction pieces, and subtracting from a whole number. - **Part 3:** the improper-fraction route, counting up as a check, and stories that ask how much is left or how much more. ## The procedure ### Part 1: wholes from wholes, fractions from fractions Khanh's wagon holds $6\tfrac{7}{8}$ cubic feet of mulch. He spreads $3\tfrac{1}{4}$ cubic feet on the bean bed. How much is still in the wagon? A mixed number means a sum. $6\tfrac{7}{8}$ means $6 + \tfrac{7}{8}$, and $3\tfrac{1}{4}$ means $3 + \tfrac{1}{4}$. Taking away $3\tfrac{1}{4}$ means taking away $3$ wholes and also taking away $\tfrac{1}{4}$. Khanh can do those two jobs one after the other, in either order, and the wagon ends up the same. So subtract the wholes from the wholes and the fraction from the fraction. ### [Tape diagrams and sharing in a ratio](https://math.wiki/tape-diagram-for-ratios/) **Summary:** Equal blocks drawn for each part of a ratio, which makes a total easy to split. **Question:** Two gardeners agree to split every harvest 3 to 5. When 40 kg of potatoes come out of the ground, how much does each of them take home? ## Why this exists Fenna and Maeve rent an allotment together. Maeve does more of the digging, so they agree to split every harvest in the ratio $3 : 5$, Fenna to Maeve. One Saturday they lift $40$ kg of potatoes. The ratio says how the two shares compare, not how many kilograms each person takes, and $40$ is neither $3$ nor $5$. Splitting in half would be easy: two equal pieces. Splitting $3$ to $5$ becomes just as easy once the $3$ and the $5$ are turned into pieces of one size, because equal pieces can be counted out. A **tape diagram** is the picture that does this. ## Definition ### Day 1: the tape and its blocks A **tape diagram** for a part-to-part ratio draws one strip, or tape, for each quantity in the ratio, one under the other with their left ends lined up. Each strip is cut into **blocks**, as many as that quantity's number in the ratio, and every block in every strip is the same size. For Fenna and Maeve's $3 : 5$, Fenna's strip has $3$ blocks and Maeve's has $5$ (Figure 1). The order of the words in the ratio (ratio-language-and-notation) says which strip gets which number. Because the blocks are all the same size, each one stands for the same amount, whatever that amount turns out to be. Two more counts come straight from the picture: - the **whole tape** is all the blocks together, $3 + 5 = 8$ blocks, and stands for the total; - the **difference** is the blocks one strip has beyond the other, $5 - 3 = 2$ blocks. ### [The constant of proportionality and proportional tables](https://math.wiki/constant-of-proportionality/) **Summary:** The single number that every input is multiplied by to give its output. **Question:** Cheese costs the same for every kilogram, so one number turns any weight into its price. What is that number, and how can a table of prices show whether such a number exists at all? ## Why this exists At a market, a cheese stall sells one cheese by weight at $18$ dollars per kilogram. Customers ask for half a kilogram, $1.5$ kilograms, $0.3$ kilograms. The stallholder does not keep a separate price for every weight: one multiplication does them all, the weight times $18$. A garden center pays its staff the same way, the hours worked times the hourly wage. In each case one number turns every amount of the first quantity into the matching amount of the second, and once you have it every price and every pay packet is one step away. Not every price list works like this. A farm that charges an entry fee, or a shop that cuts the price for bulk buyers, has no single number that does the job. So there are two things to learn: what the number is and what it measures, and how to tell from a table or a description whether there is such a number at all. ## Definition Two quantities $x$ and $y$ are in a **proportional relationship** when there is one number $k$ such that every value of $y$ is $k$ times the matching value of $x$: $$y = kx \quad\text{for every pair}$$ We say that $y$ is **proportional to** $x$, and $k$ is the **constant of proportionality**. Dividing both sides by $x$, when $x \ne 0$, gives $$k = \frac{y}{x}$$ so the constant is the quotient of $y$ by $x$, and it is the same quotient for every pair. It is the unit rate of $y$ with respect to $x$: the amount of $y$ that goes with exactly one unit of $x$. Because of that it carries a unit, the unit of $y$ per unit of $x$. For the cheese, $x$ is the weight in kilograms, $y$ is the price in dollars, and $k = 18$ dollars per kilogram. ### [The double number line](https://math.wiki/double-number-line/) **Summary:** Two number lines drawn side by side whose matching points show equal ratios. **Question:** A map says that 2 centimeters stand for 7 kilometers. How can you see at once what 5 centimeters stands for, or which distance goes with any other? ## Why this exists Farah's hiking map has a note in the corner: $2$ cm on the map stand for $7$ km on the ground. A trail on the map measures $4$ cm, twice as long, so it is $14$ km. That much is scaling a ratio. But the next trail measures $5$ cm, and $5$ is not a whole number of $2$s. Map distances and ground distances are both lengths, and lengths live on a number line. So draw two number lines, one for each amount, and line them up so that every map distance sits straight above the ground distance it stands for. Then "what goes with $5$ cm?" becomes a question about a point you can see, sitting between two marks you already know. That picture is a **double number line**. ## Definition ### Day 1: two lines that share zero A **double number line** is two number lines drawn one above the other, each with its own label and unit, set up so that: - both lines start at $0$, and the two zeros sit one above the other; - the first pair of marks shows one ratio (see ratio-language-and-notation), its first amount on the top line straight above its second amount on the bottom line: in Figure 1, $2$ cm above $7$ km; - each line copies its own step again and again at equal spacing, so the next marks are two steps of each amount, then three steps, and so on: $4$ above $14$, then $6$ above $21$. Two numbers that sit one above the other **line up**, and every pair that lines up is a ratio equivalent to the first pair. ### [The multiplier for a percent change](https://math.wiki/percent-change-multiplier/) **Summary:** A rise of twenty percent is one single multiplication by one point two. **Question:** A herd grows by twenty percent. Is there one number you can multiply by to get its new size, and if you only know the new size, can you get back to the old one? ## Why this exists A herd of $50$ goats grows by $20\%$. To find its new size the long way, you find $20\%$ of $50$, which is $10$, and add it on: $50 + 10 = 60$ goats. That is two steps, and it works. Now turn the question round. A museum had $6{,}250$ visitors this month, which is $25\%$ more than last month. How many came last month? The long way breaks down here. The $25\%$ was $25\%$ of **last** month's number, which is the very number you do not know, so you cannot work out the increase and take it off. Both questions become one step once you notice that a percent change is a multiplication. Growing by $20\%$ leaves you with $120\%$ of what you had, and $120\%$ of an amount is $1.2$ times it. So the herd's new size is $50 \times 1.2 = 60$, and the museum's two months are tied by "last month $\times 1.25 =$ this month". Going forward you multiply by $1.25$; going back you divide by it. That single number is the **multiplier** for the change. This lesson builds on percent increase and decrease, which measures a change against the original amount, and on finding the whole from a part and a percent, which is the backward question in disguise. It is taught over two days. Part 1 is the multiplier itself and the new amount it gives. Part 2 runs it backwards to find the original amount. ## Definition Write the percent of the change as a decimal and call it $r$: a change of $20\%$ has $r = 0.2$, and a change of $4.5\%$ has $r = 0.045$. ### [The three percent questions](https://math.wiki/percent-find-the-whole/) **Summary:** If 12 is 40% of something, what is the something? **Question:** A part, a whole and a percent are tied together. Given any two of them, how do you find the third? ## What this area is about Every percent statement links three numbers: a whole, a part of it, and the percent the part is of the whole. "$207$ is $45\%$ of $460$" has all three. A real question hides one of them, and which one is hidden decides the move. Finding the part, a percent of a number, is the familiar case; this area handles the other two, and one equation that covers all three. ## How these fit together **The two missing-number moves.** Finding what percent one number is of another divides the part by the whole and reads the result as hundredths. Finding the whole from a part and a percent runs backwards: it scales the part up to the amount that goes with $100\%$. **One equation for all three.** Solving percent problems with an equation or a proportion writes part $= \frac{p}{100} \times$ whole, or the proportion $\frac{\text{part}}{\text{whole}} = \frac{p}{100}$, and solves for whichever number is missing. ## A path through it No child here requires another, so the three can be taken in any order. Finding the whole and the percent equation both assume solving a proportion; finding the percent assumes only changing a fraction to a percent. Finish here and you can run a percent change backwards and work out markup and margin. ### [Two percent changes one after the other](https://math.wiki/successive-percent-changes/) **Summary:** Multiplying the two multipliers together, which is not the same as adding the percents. **Question:** A drawing is enlarged by thirty percent and the copy is then reduced by thirty percent. Why is it not back to its old size? ## When you reach for it A quantity changes by a percent, and then the amount it has become changes by a percent again. A drawing goes through a photocopier twice. A collectible loses value one year and loses more the next. An insect population grows season after season. Each time the question is the same: what is the single change from the very start to the very end? The tempting answer is to add the percents, and it is wrong. percent-increase already showed the first sign of this: a rise and a fall of the same size do not land back where they began. The reason is that every percent is measured against the amount just before it, and after the first change that amount is no longer the original. This lesson stands on the multiplier for a percent change: an increase of $30\%$ is one multiplication by $1.3$, and a decrease of $30\%$ is one multiplication by $0.7$. So a chain of percent changes is a chain of multiplications, which collapses into one. The same move answers a question that looks different: what is $40\%$ of the $25\%$ of a group? That is a part of a part, the percent version of a fraction of a fraction of an amount. The lesson runs over two days. Part 1 combines two changes, and takes a percent of a percent. Part 2 repeats one change many times, undoes a change, and finds a change that is missing. ## The procedure ### Part 1: two changes, one multiplier A drawing is $20$ cm wide. A photocopier set to $130\%$ makes a copy $30\%$ wider. That copy goes back into the copier set to $70\%$, which makes a copy $30\%$ narrower. ### [Unit fractions](https://math.wiki/unit-fractions/) **Summary:** One piece when a whole is split into equal parts, the building block every other fraction is made of. **Question:** Every fraction is some number of equal pieces. What is the one piece, and how do the others get built from it? ## What this area is about A unit fraction is one piece when a whole is cut into equal parts: $\frac{1}{b}$, with numerator $1$ and a whole number $b \neq 0$ as denominator. Every other fraction is a count of these pieces: $\frac{a}{b}$ is $a$ copies of $\frac{1}{b}$. That single fact runs underneath counting, adding and placing fractions on a line. ## How these fit together Both children are procedures built on that one fact, used in two directions. **Building up.** Counting by unit fractions takes the copies one at a time, $\frac{1}{3}, \frac{2}{3}, \frac{3}{3}$, and keeps going past the whole, so $\frac{4}{3}$ is reached the same way as $\frac{2}{3}$. Only the numerator changes. It also counts backward. **Taking apart.** Decomposing a fraction writes one fraction as a sum of smaller fractions with the same denominator, such as $\frac{5}{6} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6}$ or $\frac{2}{6} + \frac{3}{6}$. ## A path through it The two children can be taken in either order. Next come adding fractions with like denominators, which reads a decomposition from the parts back to the total, fractions greater than one, a fraction times a whole number, and sharing a fraction into equal parts. ### [Unit price and comparing rates](https://math.wiki/unit-price/) **Summary:** The cost of one item or one unit of measure, used to compare package sizes. **Question:** A small bottle and a big bottle of the same detergent have different prices. Which one gives more for the money? ## When you reach for it Tova needs batteries. A pack of $8$ costs $6$ dollars and a pack of $12$ costs $8.40$ dollars. The bigger pack costs more, but it also holds more, so neither price settles the question on its own. What settles it is the price of **one** battery in each pack. The price of one item, or of one unit of measure such as one liter, is the **unit price**. It is the unit-rate of a price: dollars per battery, dollars per liter. Reach for it whenever the same thing is sold in different sizes and you want to know which size gives more for the money. The same move compares any two rates, not only prices: which of two pumps fills a tank faster, which of two printers is quicker. Bring each rate down to one unit, then compare the two single numbers. Part 1 (day one) finds and compares unit prices; Part 2 (day two) compares through the other unit rate, how much per dollar, and compares rates that are not prices at all. ## The procedure ### Part 1: the price of one A unit price is a price shared out equally over the items, so it is found by division, the way every unit rate is: $$\text{unit price} = \text{price} \div \text{number of units}$$ For Tova's pack of $8$, that is $6 \div 8 = 0.75$ dollars per battery, or $75$ cents. Sharing $6$ dollars equally over $8$ batteries is exactly what $6 \div 8$ means, which is why division is the right move. The units travel with the answer: **dollars per battery**. ### [Unit rates when the amounts are fractions](https://math.wiki/unit-rate-with-fractions/) **Summary:** Finding the amount per one when both sides of the rate are fractions. **Question:** You walked part of a mile in part of an hour. How far is that for one whole hour? ## When you reach for it A unit rate says how much of one quantity goes with exactly one of another: miles in one hour, cups of honey in one batch, how much of a lawn is mowed in one hour. When both amounts are whole numbers, finding it is one division: $6$ miles in $2$ hours is $6 \div 2 = 3$ miles per hour. Real measurements are rarely that tidy. A walk of $\frac{2}{3}$ of a mile takes a quarter of an hour. A painter covers $\frac{3}{8}$ of a fence in $\frac{3}{4}$ of an hour. A three-quarter batch of granola uses $\frac{1}{2}$ cup of honey. Neither amount is "one" of anything, and you cannot count up in steps of $1$ to reach a whole hour. Reach for this skill whenever both sides of a rate are fractions or mixed numbers and you want the amount that goes with one. The answer is still a division, and the rate can be written as a single fraction whose top and bottom are themselves fractions, such as $$\dfrac{\frac{2}{3}}{\frac{1}{4}}$$ This lesson shows why that division gives the amount per one, how to carry it out, and how to use the result. It is taught in two parts. Part 1 builds the idea with fractions of a mile and of an hour, and treats fractions built out of fractions as numbers in their own right. Part 2 brings in mixed numbers, jobs and areas, ingredients per batch, and comparing two rates. ## The procedure ### Part 1: from a fraction of an hour to a whole hour **A unit fraction of an hour.** Femi walks $\frac{2}{3}$ of a mile in $\frac{1}{4}$ of an hour, at a steady pace. Lay the hours and the miles side by side on a double number line, where points that sit one above the other go together. ### [Using a scale on maps, plans and models](https://math.wiki/read-a-map-or-plan-scale/) **Summary:** Turning a stated scale such as one inch to fifty miles into a rate you can use. **Question:** A map is a hand's width across, and the land it shows is many kilometers wide. How does a length on the paper tell you a length on the ground? ## When you reach for it Whenever a drawing stands for something too large to draw at full size (a road map, a city map, the plan of a house), or a model is built smaller than the real thing (a model train, a model ship). Each comes with a **scale**, a statement of how lengths on the drawing match lengths in reality. You reach for this skill when you know a length on one side and want the matching length on the other: "the towns are $3.5$ inches apart on the map, so how far apart are they?", or "the room is $15$ feet wide, so how long do I draw that wall?" One scale serves every length because the drawing is a scale copy: every length has been multiplied by the same number. scale-factor-and-length shows that this holds for every length you can measure, straight or curved. This lesson takes two days. **Part 1** reads a stated scale (in words, as a ratio, or as a scale bar) as a rate and uses it in both directions, including when the two sides of the scale are in different units. **Part 2** uses the same rate on floor plans and scale models, then redraws a plan at a new scale. ## The procedure ### Part 1: a scale is a rate A scale pairs a length on the drawing with the real length it stands for. "One inch represents $25$ miles" is the pair $1$ inch and $25$ miles, and like any ratio it gives a unit rate: $25$ miles of ground **per inch** of map. Scales are written in three ways. - **In words**, with a unit on each side: "$1$ inch represents $25$ miles", "$1$ cm to $3$ km", "$\tfrac{1}{4} \text{ inch} = 1 \text{ foot}$". The $=$ sign in the last is a habit of plans: the quarter inch stands for the foot, it does not equal it. - **As a ratio with no units**, such as $1 : 24{,}000$: one unit on the drawing for $24{,}000$ of **the same unit** in reality, whatever that unit is. - **As a scale bar**: a line drawn on the map, labeled with the real distance its length stands for. Measure the bar with a ruler and you have a scale in words. ### [What a proportion says](https://math.wiki/proportion-meaning/) **Summary:** A statement that two ratios name the very same comparison. **Question:** Two jugs of lemonade were mixed from different amounts. How can you tell, without tasting, whether they will taste the same? ## Why this exists Dalia mixes a jug of lemonade from $2$ cups of concentrate and $3$ cups of water. Tomas mixes his from $4$ cups of concentrate and $6$ cups of water. Tomas has more lemonade, but a glass from either jug tastes the same. What a glass tastes like does not depend on how much was made. It depends on how the concentrate compares with the water, and in both jugs that comparison is "$2$ cups of concentrate for every $3$ cups of water". A ratio records that comparison, and one comparison has many names: $2 : 3$, $4 : 6$, $10 : 15$, $1 : 1.5$. Equivalent ratios showed how to make new names by scaling both parts. A **proportion** is the sentence that says two names belong to the same comparison: $$2 : 3 = 4 : 6$$ Written as an equation, the claim can be tested, and it can be false. A painter matching a shade, a builder mixing concrete and a gardener diluting plant food all ask the same question: do these two sets of amounts make the same mixture? A proportion is how that question is written down, and when one of the four numbers is missing it becomes an equation to solve. ## Definition A **proportion** is a statement that two ratios are equivalent, that is, that they describe the same comparison. It is written either way: $$a : b = c : d \qquad\text{or}\qquad \frac{a}{b} = \frac{c}{d}$$ Both are read "$a$ is to $b$ as $c$ is to $d$". In the fraction form $b$ and $d$ must not be $0$. ### [What a ratio is](https://math.wiki/ratio-concept/) **Summary:** Two quantities in a fixed relationship, written 3 to 2 or 3:2. **Question:** Three trumpets for every two drums. How do you say how two amounts go together, in a way that still holds when everything gets bigger? ## What this area is about There are two ways to compare $3$ trumpets with $2$ drums. Subtracting says there is $1$ more trumpet. A ratio says how they go together: for every $3$ trumpets, $2$ drums, written $3 : 2$. Double the band and the difference changes, but the ratio does not. This area is what a ratio says, what it compares, and how to tell which of two ratios is larger. ## How these fit together **Definitions.** What a ratio is: language and notation gives the ways to say and write one: "$3$ to $2$", $3 : 2$, "for every $3$ trumpets, $2$ drums". Part-to-part and part-to-whole ratios separates comparing one part with another from comparing a part with the whole: $2$ red flags to $5$ white is $2 : 5$, but red flags are $\frac{2}{7}$ of all the flags, not $\frac{2}{5}$. **A procedure.** Deciding which of two ratios is larger scales both so they share one amount, then compares the other. ## A path through it Language and notation first, then part-to-part and part-to-whole ratios. Comparing two ratios needs equivalent ratios as well, so it comes after that area. Finish here and you are ready for scaling a ratio, tape diagrams, unit price and percent as a rate per hundred. ### [What a ratio is: language and notation](https://math.wiki/ratio-language-and-notation/) **Summary:** The ways a comparison of two amounts is written down and read out loud. **Question:** A band has 3 trumpets and 2 drums. What short phrase says how those two amounts go together, and still says it when the band gets bigger? ## Why this exists Priya's school band has $3$ trumpets and $2$ drums. There are two ways to compare those amounts. You can subtract: there is $1$ more trumpet than drum. Or you can say how they go together: for every $3$ trumpets there are $2$ drums. Now the school forms four sections, each set up like Priya's: $12$ trumpets and $8$ drums. The first comparison has changed, since there are now $4$ more trumpets than drums. The second has not: for every $3$ trumpets there are still $2$ drums, and the band sounds just as balanced. A recipe made bigger, a paint mixed in a bigger tin, a model built to scale: in each of them what stays the same is a "for every", not a difference. A **ratio** is the name and the notation for that kind of comparison. You already know how to compare amounts with $>$, $<$ and $=$, which say only which is bigger, and how to say one amount is so many times as many as another. A ratio packs a comparison into two numbers written in a fixed order. ## Definition ### Day 1: saying and writing a ratio A **ratio** compares two amounts by saying how many of one go with how many of the other. When there are $a$ of one thing and $b$ of another, the ratio of the first to the second can be said or written in three ways, and all three mean the same: - in words with "to": the ratio of trumpets to drums is $3$ to $2$; - with a colon: the ratio of trumpets to drums is $3:2$, read aloud "three to two"; - with "for every": for every $3$ trumpets there are $2$ drums. ### [What percent means: a rate per hundred](https://math.wiki/percent-as-rate-per-hundred/) **Summary:** A percent is a ratio whose second amount is always one hundred. **Question:** One carousel fills 18 of its 25 seats and a bigger one fills 34 of its 50. How can you tell which is fuller when the two wholes are not the same size? ## Why this exists Thiago runs carousels at a fair. On the noon ride his small carousel filled $18$ of its $25$ seats and his big one filled $34$ of its $50$. The big one carried more riders, but that does not say which was fuller: $18$ out of $25$ and $34$ out of $50$ are parts of wholes of different sizes. The way out is to measure both against the same whole. If each carousel had exactly $100$ seats and kept the same rate, how many would be filled? Multiplying $18$ out of $25$ by $4$ gives $72$ out of $100$, and multiplying $34$ out of $50$ by $2$ gives $68$ out of $100$. Now the two sit side by side, and the small carousel was fuller. Counting "for each hundred" is so useful that it has its own name and its own sign: a **percent**. ## Definition ### Day 1: for each hundred A **percent** is a rate per hundred: it says how many parts there are for every $100$ of a whole. The word comes from the Latin *per centum*, "for each hundred", the same *cent* as in $100$ years to a century or $100$ cents to a dollar. The sign $\%$ is read "percent". So $72\%$ means $72$ for every $100$. As a part-to-whole ratio it is $72 : 100$, and as a fraction it is $\frac{72}{100}$, seventy-two hundredths. The whole, whatever its size, counts as $100\%$. The picture of a percent is a **hundred grid**: a square cut into $10$ rows of $10$ small squares, $100$ in all, standing for the whole. A percent is the number of small squares shaded. Figure 1 shows $37\%$: three full rows of $10$ and $7$ squares more. ### [What the numerator and the denominator each tell you](https://math.wiki/numerator-and-denominator/) **Summary:** The bottom number names the size of the piece and the top number counts how many there are. **Question:** A fraction is written with two numbers, one above the bar and one below. What does each of them tell you, and why does it take both? ## Why this exists In fraction-area-model a fraction was a picture: a whole cut into equal parts, some of them shaded. The fraction $\frac{3}{8}$ was written with a $3$ on top and an $8$ below, and the two numbers were given names, numerator and denominator. This lesson is about what each of those numbers says, and why a fraction needs both. The two numbers do different jobs. One of them names a kind of piece, the way "inch" or "dollar" names a unit. The other counts how many of those pieces there are. "Three eighths" works like "three inches": the $3$ is a count, and "eighths" says what is being counted. Once that is clear, a lot follows. A fraction is so many copies of one small piece, the unit fraction. Cutting the same whole into more pieces makes each piece smaller. And the short reading "three out of eight" turns out to be true only some of the time. ## Definition In the fraction $$\frac{a}{b}$$ the number $b$ below the bar is the **denominator** and the number $a$ above it is the **numerator**. The line between them is the **fraction bar**. - **The denominator names the piece.** It says how many equal pieces make one whole. A whole made of $b$ equal pieces has pieces called $b$ths: halves for $2$, thirds for $3$, fourths for $4$, sixths for $6$, eighths for $8$. One such piece is the unit fraction $\frac{1}{b}$. - **The numerator counts the pieces.** It says how many of those pieces the fraction is made of. ### [Whole numbers written as fractions](https://math.wiki/whole-numbers-as-fractions/) **Summary:** Any whole number is a fraction with denominator one, and three thirds is exactly one. **Question:** Three pans of brownies, each cut into fourths, are all eaten. That is twelve fourths of a pan, and it is also exactly three pans. How can one amount be a fraction and a whole number at once? ## Why this exists So far whole numbers and fractions have looked like two kinds of number. Whole numbers count things: $3$ pans, $6$ trays. Fractions count pieces: $\frac{3}{4}$ of a pan, $\frac{5}{6}$ of a tray. In fractions-greater-than-one the pieces went past one whole, and $\frac{4}{4}$ turned out to be exactly $1$. That fact goes further. If three pans of brownies are each cut into fourths and every piece is eaten, then $12$ fourths of a pan are gone. That is a fraction, $\frac{12}{4}$. It is also exactly $3$ pans, with nothing left over. The same amount has two names. This lesson shows that every whole number can be written as a fraction, and how to tell when a fraction is really a whole number. Once whole numbers and fractions are seen as one family of numbers, they can sit on one number line, be compared with each other, and later be added and multiplied together. The lesson runs over two days. Part 1 writes whole numbers as fractions of one uncut whole, with denominator $1$, and looks again at the many names for $1$. Part 2 finds fractions such as $\frac{12}{4}$ and $\frac{6}{3}$ that equal whole numbers, in pictures and on the number line. ## Definition A fraction $\frac{a}{b}$ names $a$ copies of the piece you get by cutting one whole into $b$ equal parts (see numerator-and-denominator). **Denominator $1$.** Cutting a whole into $1$ part leaves it as it is: the one piece is the whole. So $\frac{1}{1}$ is one whole, and $\frac{n}{1}$ is $n$ whole things. For every whole number $n$, ### [Why flipping the second fraction works](https://math.wiki/why-inverting-works/) **Summary:** Dividing by a number and multiplying by its reciprocal undo exactly the same thing. **Question:** A scoop holds three quarters of a cup. How many scoops are in a bag of flour, and why does turning the fraction upside down and multiplying give the answer? ## Why this exists Dividing whole numbers has a plain meaning. $12 \div 3$ asks how many threes fit in twelve, and the answer is $4$. The same question still makes sense when the numbers are fractions: in a kitchen, a cook with $3$ cups of flour and a measuring cup that holds $\frac{3}{4}$ of a cup can ask how many scoops the flour makes. That meaning is the subject of dividing fractions: how many fit, and the sharing meaning is in dividing fractions: sharing into equal parts. Counting scoops on a picture works, but it is slow, and a picture cannot easily show $\frac{7}{8} \div \frac{2}{5}$. There is a short rule: turn the second fraction upside down and multiply. A rule handed over without a reason is easy to misremember (which fraction flips?) and impossible to trust. This lesson shows what dividing by a fraction means, counts the answer directly, and then proves three different ways that the rule must give the same answer. ## Definition **Division.** For any number $a$ and any number $b$ that is not $0$, $a \div b$ is the number that $b$ must be multiplied by to give $a$: $$a \div b = q \quad \text{means} \quad q \times b = a.$$ This is the same link between multiplication and division that holds for whole numbers (multiplication and division undo each other). In the kitchen it says: $a \div b$ is the number of $b$-cup scoops in $a$ cups, and a part of a scoop counts as that fraction of a scoop. ### [Word problems that add or subtract fractions](https://math.wiki/add-subtract-fraction-word-problems/) **Summary:** Reading a situation to decide whether pieces are being joined or taken away. **Question:** Takoda ran $\tfrac{4}{5}$ mile and Remi ran $\tfrac{1}{2}$ mile. Nothing was taken away from anyone, so why is "how much farther did Takoda run?" a subtraction? ## When you reach for it Phuong, Remi and Takoda swim on the same team, and on weekends they enter a road race. Their days are full of fractions: $\tfrac{3}{4}$ mile of warm-up, $\tfrac{2}{5}$ mile of kicking, a water bottle $\tfrac{7}{8}$ full, a race that is $1\tfrac{1}{2}$ miles long. You already know how to add and subtract fractions with different denominators: rename both in the same size of piece, then count (add-unlike-any-denominators). This lesson is about the step before that. A story does not come with a plus or minus sign. You have to read what is happening and decide which one the story needs. You reach for this lesson whenever a situation gives you fractions or mixed numbers and asks a question about them. The lesson has three parts, one for each day: - **Part 1:** joining and taking away, and the bar picture that shows which one a story is. - **Part 2:** comparing two amounts, and finding a missing part. Both can sound like adding, and both are solved by subtracting. - **Part 3:** stories with two steps, and an estimate to judge the answer. ## The procedure ### Part 1: joining and taking away Every story in this lesson is about parts and a whole. Some amount is the whole, and it is made of two parts. The question asks for one of the three numbers, and the other two are given. - If both **parts** are known and the **whole** is not, the parts are joined. You add. - If the **whole** and one part are known, the other part is what is left. You subtract. ### [Word problems that divide fractions](https://math.wiki/divide-fraction-word-problems/) **Summary:** Deciding whether a situation asks how many fit or how much each one gets. **Question:** A mill has flour, and the bakeries want it in bags, bins and batches. When does the order call for dividing, and which number goes first? ## When you reach for it At Catalina's flour mill, flour leaves in fractional kilograms: bags of $\frac{3}{4}$ kg, scoops of $\frac{3}{8}$ kg, a bin that is $\frac{2}{3}$ full. You already know how to compute a fraction division: multiply by the reciprocal, with mixed numbers renamed first, and why-inverting-works says why. What a bakery's order does not tell you is that it is a division. An order names the amounts, never the operation. This lesson is about that decision. Reach for it whenever a situation hands you two amounts and asks for a third: decide whether to multiply or divide, which amount goes first, and what the answer means once you have it. Day 1 is the decision itself, between the two meanings of how many fit and sharing into equal parts. Day 2 is problems with more than one step, answers that do not come out whole, and checking that an answer makes sense. ## The procedure ### Day 1: how many groups, or how much in one group **Every one of these problems is a multiplication with a gap in it.** Flour at the mill comes in equal groups (bags, bins, batches), and three numbers describe it: $$(\text{number of groups}) \times (\text{amount in one group}) = \text{total}$$ A word problem gives two of the three and asks for the third. Before computing anything, write this sentence with a $?$ in the gap. Which gap it is decides everything. - **The total is missing.** Multiply. Five bags of $\frac{3}{4}$ kg hold $5 \times \frac{3}{4}$ kg. - **The number of groups is missing.** This is the "how many fit" question. $? \times \frac{2}{3} = 2$ asks how many $\frac{2}{3}$ kg bags $2$ kg fills, and the division that undoes the multiplication is $2 \div \frac{2}{3}$. - **The amount in one group is missing.** This is the sharing question. $\frac{2}{3} \times ? = 2$ says that $2$ kg is $\frac{2}{3}$ of one group, so one whole group is $2 \div \frac{2}{3}$. ## Functions ### [A reasonable domain for a real situation](https://math.wiki/reasonable-domain-in-context/) **Summary:** In a real setting the domain shrinks to the inputs that make sense, such as whole numbers of people. **Question:** A formula will take any number you give it. When the formula describes something real, which of those numbers still make sense? ## Why this exists A formula is happy with any number. Feed $P(n) = 14n - 96$ the input $n = -2.5$ and it hands back $-131$ without complaint. But if $n$ is the number of books a shop has sold, $-2.5$ is nonsense: nobody sells minus two and a half books. So a function that describes a real situation carries, beside its rule, the set of inputs the situation allows. That set decides whether the graph is dots or an unbroken line, where it stops, which outputs can happen, and which solutions of an equation to keep. Earlier lessons met this idea in passing: reading a statement such as $C(12) = 40$ asked which inputs make sense, graphing from a table asked whether to join the dots, building a linear model asked where a model stops, and reading a set of pairs collected a domain that was handed over. This lesson asks where the domain comes from. ## Definition The **domain** of a function is the set of inputs it accepts, and the **range** is the set of outputs it gives (see domain-and-range). When a function describes a real situation, its **reasonable domain** is the set of inputs that make sense in that situation. The range then follows from it: it is the set of outputs the rule gives for inputs in the reasonable domain. The limits on a reasonable domain come from four places: - **What the input counts or measures.** A count of people, books or cars takes only the whole numbers $0, 1, 2, 3$ and so on. A time, a length or a mass can take any value in between. - **Where the story starts.** Time measured from the moment a hose is turned on cannot be negative. - **A limit in the world or in the rules.** A shop has only so many copies; a ride lasts a set number of minutes; a pool holds only so much. - **A limit on the output, carried back to the input.** A height cannot fall below $0$ and a pool cannot hold more than it holds, so the inputs that would give such outputs are left out. ### [Adding, subtracting and multiplying functions](https://math.wiki/sum-of-two-functions/) **Summary:** The sum function gives, at each input, the sum of the two outputs. **Question:** Two quantities each depend on the same input. How do you get one function that gives their total, their gap or their product at every input? ## When you reach for it A shop's revenue and its costs both depend on how many items it sells; the profit is one minus the other. A rectangle's width and length both depend on some measurement $x$; its area is one times the other. In each case two functions share an input, and the question is about a third quantity built from their two outputs at that same input. Rather than work out the two outputs and combine them afresh every time, you can build one new function whose output already is the total, the gap or the product. That new function has its own name, its own formula, its own table and its own graph, and it can be evaluated, graphed or solved like any other. This lesson builds sums, differences, products and constant multiples of functions given by formulas, by tables and by graphs. Dividing one function by another is left to quotient-of-two-functions, because there the bottom output can be zero. ## The procedure ### The new functions are defined output by output For two functions $f$ and $g$ whose outputs are numbers, and any constant $c$, the new functions $f + g$, $f - g$, $fg$ and $cf$ are defined by what they do to each input $x$: $$ \begin{aligned} (f + g)(x) &= f(x) + g(x)\\ (f - g)(x) &= f(x) - g(x)\\ (fg)(x) &= f(x)\,g(x)\\ (cf)(x) &= c\,f(x) \end{aligned} $$ Read the left side carefully in function-notation. The part in parentheses, $(f + g)$, is the name of the new function, in the way $f$ is a name, and the $(x)$ after it is the input being fed in. So $(f + g)(2)$ means: put $2$ into the function $f + g$. The right side says how that function works: put the same $2$ into $f$ and into $g$, then add the two outputs. The product $fg$ is sometimes written $f \cdot g$; both mean multiply the outputs. ### [Asymptotes](https://math.wiki/asymptotes/) **Summary:** Lines the graph approaches but never reaches: vertical, horizontal, slant. **Question:** Near an input it refuses, or far out to either side, a rational function's graph follows a line. Which line, and how can you tell from the formula? ## What this area is about An asymptote is a line a graph gets as close to as you like. For a rational function $f(x) = \frac{p(x)}{q(x)}$, with $p$ and $q$ polynomials, there are two kinds of place to look: near an input where the denominator is $0$, and far out to the left and right. The formula decides both, without plotting points. A graph may cross a horizontal or slant asymptote; it only has to settle toward the line far out. ## How these fit together **Near an input.** Vertical asymptotes sit at inputs where the denominator of the fraction in lowest terms is $0$. There the outputs grow without bound, and the lesson tells this apart from a single missing point. **Far out.** Horizontal asymptotes compare the degrees of $p$ and $q$: when the degree of $p$ is at most the degree of $q$, the graph levels off toward a horizontal line. Slant asymptotes handle a numerator one degree higher, where the graph follows a slanted line found by dividing. ## A path through it Vertical asymptotes can come at any point. Horizontal asymptotes come before slant ones. Next: graphing rational functions, then the tangent graph and logistic growth, which levels off toward a horizontal asymptote. ### [Average rate of change](https://math.wiki/average-rate-of-change/) **Summary:** How much the output changes per unit of input across an interval, the slope of the line joining two points. **Question:** A quantity speeds up, slows down, even turns around. What one number says how fast it changed between two moments? ## What this area is about When a graph is a straight line, one rate describes all of it. Most graphs are not straight, so the honest answer to "how fast?" depends on which stretch of inputs you mean. The average rate of change over an interval is the change in output divided by the change in input, $\frac{f(b) - f(a)}{b - a}$ with $a \ne b$: the slope of the line joining two points of the graph. ## How these fit together **Two procedures for one number.** Average rate of change from a table or graph reads the two outputs off rows or points, and average rate of change from a formula works them out from the rule. **One expression for every interval.** The difference quotient writes the rate over the interval from $x$ to $x + h$ as a single expression in $x$ and $h$. **A new idea.** Rates at an instant shrinks the interval around one point and watches the averages settle, which is how a rate can belong to a single moment. ## A path through it Tables and graphs or formulas first, in either order. Then the difference quotient, and last the rate at an instant, which needs it. This area feeds concavity, where the rate itself changes, and leads straight into calculus through the derivative at a point. ### [Average rate of change from a formula](https://math.wiki/average-rate-of-change-from-a-formula/) **Summary:** Evaluating the rule at both ends of an interval and dividing the change in output by the width. **Question:** You have the rule for a quantity. How fast did it change, on average, between two input values? ## When you reach for it A rule gives a quantity's value for any input: the height of a ball $t$ seconds after it is thrown, a savings balance after $t$ years, the users of an app after $m$ months. Often the question is not "how much?" but "how fast?": how many meters per second the ball climbed between two moments, how many dollars per year the balance gained. That number is the **average rate of change** over the stretch of inputs, the rate-of-change measured from one end of the stretch to the other. With the rule in hand you need no table and no graph. You evaluate the function at the two ends, then one subtraction and one division finish the job. When the values come from a table or a graph instead, the same quotient is read off two rows or two points, as in average-rate-of-change-from-a-table. For a straight-line rule the answer is the same on every interval. For anything else it changes with the interval, so the interval is part of the question: "the average rate of change of $h$ on $[0, 1]$", never just "the rate of change of $h$". ## The procedure The **average rate of change** of a function $f$ on the interval $[a, b]$, where $a \ne b$, is the change in output divided by the change in input: $$\frac{f(b) - f(a)}{b - a}$$ The numerator is how much the output changed between the two ends; the denominator is the width of the interval. Dividing turns a total change into a change **per unit of input**, so the units are output units per input unit: meters per second, dollars per year, users per month. ### [Average rate of change from a table or graph](https://math.wiki/average-rate-of-change-from-a-table/) **Summary:** Using two rows of a table to find the change in output divided by the change in input. **Question:** A quantity rose fast, then slowly, then fell a little. What one number says how fast it changed between two chosen moments? ## When you reach for it In rate of change one number described a whole relationship, because the rate was the same between any two inputs: a constant rate, and a straight graph. Most quantities are not like that. A sprinter speeds up and then holds her pace, a city grows for decades and shrinks for one, a snowman melts fast at first and slowly later. Ask "how fast did it change?" and the honest reply is "between which two moments?" The **average rate of change** answers once the two moments are chosen. You reach for it when you know a function's value at two inputs, as two rows of a table or two points on a graph, and want one number for how much the output changed per unit of input across that whole stretch. When the function is given by a rule, the same quotient comes from evaluating the rule: average rate of change from a formula. ## The procedure **The definition.** Take a function $f$ and two different inputs $a$ and $b$ where $f$ has values. The average rate of change of $f$ over the interval from $a$ to $b$ is the change in output divided by the change in input: $$\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$$ The Greek capital $\Delta$ (delta) is read "change in". In function notation $f(b)$ is the output at input $b$, so the top is the end value minus the start value. **Finding the two values.** In a table the two inputs pick out two rows; on a graph they pick out two points, $(a, f(a))$ and $(b, f(b))$, whose heights you read from the vertical axis. Nothing else enters the calculation: the two ends are compared and everything between them is ignored. ### [Building a model from a description](https://math.wiki/model-from-a-verbal-description/) **Summary:** Turning a written account of a situation into a rule, with each quantity named and its units stated. **Question:** A paragraph describes something changing. How do you turn its words into a rule you can calculate with? ## When you reach for it Often all you have is a description: "the elevator goes down at a steady $5$ meters per second", "the colony grows by $6\%$ a year", "the dose of a made-up medicine halves every $8$ hours". No table, no graph. To predict anything, the words have to become a rule: one quantity written as a function of another. This is also how you read a function out of a real situation at all. You decide which quantity depends on which (independent-and-dependent-variables) and check that each input gives exactly one output, as a function must. The words settle almost everything. They say what is measured and in what units, where the counting starts, and whether the change is a fixed amount or a fixed percentage, which decides between a line and an exponential. Day 1 turns a single description into a rule and tests it. Day 2 handles descriptions that need two facts combined, a period longer than one unit, numbers that do not belong, and a domain. ## The procedure ### Day 1: from words to a rule **Name every quantity in full, with its units.** Write "$t$ = seconds since the elevator left the platform", not "$t$ = time", and "$d$ = depth of the elevator below the surface, in meters", not "$d$ = distance". The full name does two jobs. It fixes where the input starts counting, which decides the starting value, since the start of a rule is its output when the input is $0$. And it fixes the units, so the rate has units too: meters per second, dollars per month. ### [Building a polynomial from its zeros](https://math.wiki/write-a-polynomial-from-its-zeros/) **Summary:** Turning each zero into a factor and choosing a leading coefficient to match one known point. **Question:** You know where a polynomial is zero. How do you get the polynomial back, and when is there only one? ## When you reach for it Graphing a polynomial starts from the rule and draws the curve. This move runs the other way. You know where a polynomial is zero, how many times, and perhaps one other point on its graph, and you want the rule. That happens when you read a formula off a drawn graph, when you design a curve that must meet the axis at chosen places, and whenever a problem says "a cubic with zeros $2$, $5$ and $7$" and the cubic has to be written down. For a quadratic this is factored form read backwards. Here it works at any degree, with repeated zeros, fractional zeros and non-real zeros. ## The procedure ### Each zero gives a factor By the factor theorem, $c$ is a zero of $p$ exactly when $(x - c)$ is a factor of $p$. A zero of multiplicity $m$ means $(x - c)^m$ is a factor (zeros and multiplicity). Suppose $p$ has degree $n$ and zeros $c_1, \dots, c_k$ with multiplicities $m_1, \dots, m_k$ that add up to $n$. Then $p$ contains the product $$(x - c_1)^{m_1}(x - c_2)^{m_2}\cdots(x - c_k)^{m_k}$$ and that product already has degree $n$. Dividing $p$ by it leaves a quotient of degree $0$, a constant. So, for some number $a \neq 0$, $$p(x) = a(x - c_1)^{m_1}\cdots(x - c_k)^{m_k}$$ Each factor carries the opposite sign of its zero: the zero $-2$ gives $(x - (-2)) = (x + 2)$. The fundamental theorem of algebra says a polynomial of degree $n$ has exactly $n$ zeros counted with multiplicity, complex ones included, so a list whose multiplicities add to $n$ is the whole list. ### [Building an exponential model from two points](https://math.wiki/exponential-model-from-two-points/) **Summary:** Using two known values to work out both the starting amount and the factor. **Question:** An algae bloom covers 36 square meters on day 1 and 972 on day 4. If it multiplies by the same factor every day, what was that factor, and where did it start? ## When you reach for it A quantity changes by the same factor every period: an algae bloom that grows by the same percent each day, a concentration that falls by the same percent each day. Its rule is $y = a \cdot b^x$, with a starting value $a$ and a factor $b$ per period, as in growth-factor-and-decay-factor. Often neither number is given. What you have instead is two measurements: the value at one time and the value at another. Two unknowns need two facts, and two readings supply exactly two. This move turns two points $(x_1, y_1)$ and $(x_2, y_2)$ into the whole model. It is the exponential partner of finding a line from two points with slope-from-two-points: a line is fixed by the difference of the outputs, an exponential by their ratio. ## The procedure **Two equations.** Each point lies on the graph of $y = a \cdot b^x$, so each makes a true equation: $$y_1 = a \cdot b^{x_1} \qquad y_2 = a \cdot b^{x_2}$$ **Divide to remove $a$.** Here $a \neq 0$, so the first equation can divide the second. The $a$ cancels, and powers of the same base divide by subtracting exponents: $$\frac{y_2}{y_1} = \frac{a \cdot b^{x_2}}{a \cdot b^{x_1}} = b^{x_2 - x_1}$$ One equation in one unknown is left. It says that over $x_2 - x_1$ periods the quantity is multiplied by $\frac{y_2}{y_1}$, which is the meaning of a factor. **Take a root to find $b$.** Write $n = x_2 - x_1$. The factor is positive, and a positive number has exactly one positive $n$th root, so ### [Change of base](https://math.wiki/change-of-base/) **Summary:** Compute any logarithm from the ones your calculator has. **Question:** A calculator has keys for the common logarithm and the natural logarithm, and no others. How do you find a logarithm in base 4, or base one half, or any other base? ## When you reach for it Some logarithms can be found by asking which power of the base gives the number, as in evaluating logarithms: $\log_4 64 = 3$ because $4^3 = 64$. Most cannot. Take $\log_4 50$. Since $4^2 = 16$ and $4^3 = 64$, and $50$ lies between them, the answer lies between $2$ and $3$, nearer to $3$. No whole number or simple fraction gives it, and a calculator has no key for base $4$. It has $\log$, the common logarithm with base $10$, and $\ln$, the natural logarithm with base $e$. The change of base formula closes that gap. It writes a logarithm in any base as a quotient of two logarithms in a base of your choosing, so one key on the calculator is enough for every base. Reach for it when: - you need the decimal value of a logarithm whose base is not $10$ or $e$; - two logarithms in different bases have to be compared, multiplied or added, and the laws of logarithms cannot help because each law works inside one base; - a base can be chosen that makes the answer exact, as with $\log_8 32$, where both numbers are powers of $2$; - a logarithm function such as $y = \log_{11} x$ has to be graphed or compared with $y = \ln x$. ## The procedure ### Where the formula comes from Let $b$ be an allowed base ($b > 0$, $b \neq 1$) and let $M$ be positive. Give the unknown logarithm a name: $$y = \log_b M$$ By the definition of a logarithm, this says the same thing as the exponential statement ### [Choosing a function family for data](https://math.wiki/choose-a-function-family/) **Summary:** Using differences, ratios and the shape of a scatter plot to decide which family fits the data. **Question:** You have a table of measurements and no rule. Is the rule behind it a line, a quadratic or an exponential, and can the numbers themselves tell you which? ## When you reach for it Before you can write a rule for data you have to know what kind of rule to write. Lines, quadratics and exponentials are three different families, and the work of fitting one (a line of best fit, a quadratic model, an exponential from a table) is wasted on the wrong one. Each family leaves a fingerprint in a table: something that stays the same from row to row. Reach for this move whenever you have a table or a scatter plot and no rule yet. ## The procedure ### What stays the same in each family Take a table whose inputs go up in equal steps of size $h$. For a linear function $f(x) = mx + b$, subtract each output from the next: $$f(x + h) - f(x) = m(x + h) + b - (mx + b) = mh$$ These **first differences** are all $mh$, the same number every time. That is the fact behind a constant rate of change. For a quadratic $f(x) = ax^2 + bx + c$ (with $a \ne 0$) the first differences are $$f(x + h) - f(x) = 2ahx + ah^2 + bh$$ which still contains $x$, so they change from row to row. But they change *linearly*, and by the first fact the differences of a linear expression are constant. So take differences of the differences, the **second differences**: every one equals $2ah \cdot h = 2ah^2$. For an exponential $f(x) = a \cdot b^x$, subtraction is the wrong tool. Divide each output by the one before: $$\frac{f(x + h)}{f(x)} = \frac{a \cdot b^{x + h}}{a \cdot b^x} = b^h$$ These **ratios** are all $b^h$, the growth or decay factor for one step. With $h = 1$ the ratio is $b$ itself. ### [Combining and composing functions](https://math.wiki/composition-of-functions/) **Summary:** Feed one function's output into another. **Question:** Two functions share an input, or one feeds the other. How do you build a single new function out of the two? ## What this area is about Functions can be combined into new ones. Two functions with the same input can be added, subtracted, multiplied or divided output by output, giving $f + g$ or $\frac{f}{g}$. Or one function's output can be fed into another, giving the composition $f(g(x))$. Each new function has its own rule and its own domain, and finding that domain is the part that needs care. ## How these fit together **Arithmetic of functions.** Adding, subtracting and multiplying functions combines outputs at the same input. Dividing functions and the domain of a combination does the same for division, removes the inputs where the bottom output is $0$, and finds which inputs any combination accepts. **Composition.** Evaluating a composition runs one number through the inner function and then the outer one, from formulas, tables or graphs. Composing functions algebraically substitutes the whole inner rule to get one formula. **The domain of a chain.** Domain of a composition says which starting inputs make it all the way through the chain. ## A path through it Sums and products, then quotients. Separately, evaluate a composition at a number, then compose algebraically. The domain of a composition comes last, after both quotients and algebraic composition. Composition is how inverses are checked, and it returns in calculus as the chain rule. ### [Combining transformations and the general form](https://math.wiki/combining-transformations/) **Summary:** Applying a shift, a flip and a stretch to one parent graph together. **Question:** A rule like y = -2(x - 5)^3 + 7 moves its parent graph several ways at once. Which moves, in what order, and where does each point land? ## When you reach for it Each lesson before this one taught a single move: shifts, reflections and stretches. Most rules do several at once. In $$y = -2(x - 5)^3 + 7$$ the parent $y = x^3$ is shifted, stretched, flipped and shifted again. Reach for this lesson when you need to read every move off such a rule, put the moves in the order they really happen, and carry single points of the parent across to the new graph. Those carried points are what graphing by transformations joins into a whole curve. ## The procedure ### The general form Every rule built from a parent $f$ by these moves can be written as $$g(x) = a\,f\big(b(x - h)\big) + k$$ with $a \ne 0$ and $b \ne 0$. Two numbers sit **outside** $f$ and act on the output, the height. Two sit **inside** and act on the input, the horizontal position. | Number | Where it sits | What it does to the graph | |---|---|---| | $a$ | outside, multiplies | vertical stretch by $\lvert a \rvert$ (a compression if $\lvert a \rvert < 1$); a reflection across the $x$-axis if $a < 0$ | | $k$ | outside, added | shift up $k$ (down if $k < 0$) | | $b$ | inside, multiplies | horizontal compression by $\frac{1}{\lvert b \rvert}$ (a stretch if $\lvert b \rvert < 1$); a reflection across the $y$-axis if $b < 0$ | | $h$ | inside, subtracted | shift right $h$ (left if $h < 0$) | Each row is one earlier lesson's result. What is new is how they fit together. ### Where one point goes ### [Comparing linear functions given different ways](https://math.wiki/compare-linear-functions-across-representations/) **Summary:** Deciding which of two linear functions grows faster when one is a graph, one a table and one a rule. **Question:** One linear function is a table and another is a graph or a sentence. How can you tell which grows faster, which starts higher, and whether one ever catches the other? ## When you reach for it Choices often come down to two linear functions set side by side, and they rarely arrive in the same form: one shop sends a table, another a graph; one plan is an equation, another a sentence. A table and a graph cannot be compared just by looking at them. Every linear function, whatever its form, is settled by two numbers: its **rate of change** $m$, the change in output for each one unit of input, and its **initial value** $b$, the output at input $0$. With them the rule is $f(x) = mx + b$. So the move is to turn each function into those same two numbers first, in the same units, and only then compare: which grows faster, which starts higher, and whether one ever overtakes the other. For two relationships that both start at $0$, one number settles the comparison, as in direct-variation. Here both functions may have a nonzero start, and the two numbers can pull in opposite directions. The lesson takes two days. Part 1 (Examples 1 to 3) gets the two numbers out of each form and compares them. Part 2 (Examples 4 to 6) uses them to decide who is ahead, and when, if ever, that changes. ## The procedure ### Part 1: the same two numbers from every form **From an equation.** When the equation already reads $y = mx + b$, the coefficient of the input is the rate and the constant is the initial value, as in read-slope-and-intercept. When it is written another way, such as $4x - 2y = 6$, solve for the output first: $y = 2x - 3$, so $m = 2$ and $b = -3$. The numbers $4$ and $-2$ in the original are neither, which is why standard form is rewritten before it is read. ### [Composing functions algebraically](https://math.wiki/compose-two-functions-algebraically/) **Summary:** Substituting the whole inner rule for the input of the outer rule and simplifying. **Question:** Two functions run one after the other. Is there a single rule that goes straight from the first input to the final output? ## When you reach for it In evaluating a composition at a number, a number went through the inner function and the answer went into the outer one. That works one input at a time. If you want the value at many inputs, or want to study the chained function as a function in its own right (its graph, its smallest value, where it equals something), you want one formula that does both steps at once. Composing **algebraically** finds that formula: the whole inner rule is substituted for the input of the outer rule, and the result is simplified. Reach for it when a question asks for $f(g(x))$ or $(f \circ g)(x)$ as an expression, when a situation happens in two stages and you want one rule from the start to the end (time gives a radius, the radius gives an area), and in reverse, when a complicated formula is easier to handle once you see it as a simple function with another simple function inside it. This lesson runs over two days. Day 1 is the move itself: substituting one rule into another, simplifying, checking with a number, and seeing that the two orders give different functions. Day 2 uses the move in situations with units, and runs it backwards, breaking a function into a composition. ## The procedure ### Day 1: substituting the whole inner rule Let $f(x) = x^2 - 6x$ and $g(x) = 2x + 3$. By definition, $$(f \circ g)(x) = f(g(x)) = f(2x + 3)$$ so the task is to evaluate $f$ at an expression. The rule for $f$ says: take whatever the input is, square it, and subtract six times it. The letter $x$ in $f(x) = x^2 - 6x$ only marks where the input goes. Here the input is the whole expression $2x + 3$, so every $x$ in the rule for $f$ is replaced by $(2x + 3)$, in parentheses, because the whole input is squared and the whole input is multiplied by $6$: ### [Compound interest and compounding periods](https://math.wiki/compounding-periods/) **Summary:** Splitting a yearly rate over more periods changes how much a balance grows in a year. **Question:** A bank quotes one yearly rate but adds the interest several times a year. How much does adding it more often really change what you end with? ## Why this exists With simple interest, interest is paid only on the money first put in. Most savings accounts and most loans do something else: each time interest is worked out, it is added to the balance, and from then on it earns interest of its own. That is **compound interest**, and over whole years it is the exponential growth you already know, with the factor $1 + r$ for a yearly rate $r$. But banks rarely add interest once a year. A savings account may add it every month, a credit card every month or every day, a bond twice a year. The advertised rate is still a yearly one, such as $5.4\%$ a year compounded monthly. So two questions arise that the once-a-year picture cannot answer. What does "$5.4\%$ a year, compounded monthly" actually do to a balance? And is $5.4\%$ compounded monthly worth more or less than $5.5\%$ compounded once a year? This lesson answers both, with one formula and one comparison number. ## Definition **Compound interest** is interest that is added to the balance at the end of each period, so that in later periods it earns interest too. Each such adding is a **compounding**, and the time between two of them is a **compounding period**. A rate quoted as $r$ a year **compounded $n$ times a year** means: the year is split into $n$ equal periods, and at the end of each one the balance grows by the fraction $\dfrac{r}{n}$ of itself. The yearly figure $r$ is called the **nominal rate** (from the Latin for "name": it is the rate in name). The common choices are ### [Concavity: when the rate of change itself changes](https://math.wiki/concavity-and-changing-rates/) **Summary:** A graph is concave up where its rate of change is increasing and concave down where it is decreasing, with points of inflection between. **Question:** Two graphs can both be rising, one faster and faster, the other more and more slowly. How do we say, and measure, which way a rate of change is itself changing? ## Why this exists Increasing and decreasing give the sign of a rate of change, up or down. They say nothing about how the rate itself behaves. A new game whose daily players climb by more each week is in a different situation from one whose players still climb, but by less each week; both graphs are increasing. Average rates and rates at an instant measure how fast an output changes. Concavity asks the next question: is that rate increasing or decreasing? On a graph the answer shows as the way the curve bends. ## Definition Write $m(a, b) = \frac{f(b) - f(a)}{b - a}$ for the slope of the secant from $x = a$ to $x = b$. A function $f$ is **concave up** on an interval if its secant slopes increase as the secant moves right: whenever $a < b < c$ lie in the interval, $$m(a, b) < m(b, c).$$ It is **concave down** on the interval if instead $m(a, b) > m(b, c)$ whenever $a < b < c$ lie in it. When $f$ has a rate of change at each point of the interval, concave up means exactly that this rate increases from left to right, and concave down that it decreases. (Calculus proves that the two forms agree.) A **point of inflection** is a point $(c, f(c))$ on the graph, with $f$ continuous at $c$, where the concavity changes: concave up on one side of $c$ and concave down on the other. ## Intuition A concave up graph bends like a cup opening upward; a concave down graph bends like a cap. Read left to right, a concave up graph keeps getting steeper in the upward direction: if it is rising, it rises faster; if it is falling, it falls less steeply. A concave down graph is the reverse. ### [Constant rate of change and linear tables](https://math.wiki/constant-rate-of-change/) **Summary:** A function is linear exactly when equal steps in the input always give equal steps in the output. **Question:** A table lists a few inputs of a function and their outputs. How can you tell whether one steady rule made every row, and how do you find that rule from the rows alone? ## Why this exists A press's sheet counter, the points on a loyalty card, a chair's height on a ski lift: each is a function, one output for each input. Often all anyone has is a table, a handful of rows written down at odd moments. Did one steady rule produce every row? And if it did, what is that rule, so that outputs nobody wrote down can be found? Rate of change gave the number that says how fast an output moves, and slope from a table gave the test for whether a table's points lie on one line. This lesson puts that test into the language of functions. A function whose rate never changes is a **linear function**; it always has a rule $y = mx + b$; and the rows of its table are enough to write that rule, even when the row for an input of $0$ is missing. ## Definition A function has a **constant rate of change** when the rate $$\frac{\text{change in output}}{\text{change in input}}$$ comes out the same number $m$ whichever two inputs you pick. A **linear function** is a function whose rule can be written $$y = mx + b$$ with $x$ the input, $y$ the output, and $m$ and $b$ fixed numbers. The number $m$ is the **rate of change**, and $b$ is the **initial value**: the output when the input is $0$. A table of values is **linear** when it could have come from a linear function, that is, when the rate of change between any two of its rows is the same number. ## Intuition Take this table. | $x$ | $1$ | $2$ | $4$ | $5$ | |---|:-:|:-:|:-:|:-:| | $y$ | $7$ | $10$ | $16$ | $19$ | ### [Continuous graphs, jumps and holes](https://math.wiki/continuous-and-broken-graphs/) **Summary:** A graph drawn without lifting the pen, compared with one that jumps or has a gap in it. **Question:** Some graphs can be traced from end to end without lifting the pencil, and some cannot. What exactly happens at the places where the pencil has to leave the paper? ## Why this exists Fill a bath and draw the depth of the water against time. The depth rises smoothly: it never skips from $10$ centimeters to $12$ without passing through every depth in between, and the drawn graph is one unbroken stroke. Now drop coins into a savings jar and draw the money in it against time. The amount sits still, then leaps by the value of a coin at the instant the coin lands. The graph is a row of flat pieces with gaps between them, and no pencil can draw it without lifting. Both are graphs of functions, yet they behave very differently. The unbroken one lets you say "at some moment the depth was exactly $11$ centimeters" without ever measuring it. The broken one forbids that: the jar may never hold certain amounts at all. To know which kind of statement a graph allows, you need to know where it breaks and how. Formulas break too, and the breaks are not always visible: the graph of $f(x) = \frac{x^2 - x - 20}{x - 5}$ looks exactly like a straight line, but one single point of that line is missing. This lesson names the ways a graph can break and shows how to find each one from a picture or a formula. ## Definition In plain words, a graph is **continuous** over an interval of inputs when you can trace the whole of it over that interval without lifting your pencil. Where the pencil has to leave the paper, the graph is **broken**, and the function is **discontinuous** at that input. ### [Cube root functions and roots as powers](https://math.wiki/cube-root-functions/) **Summary:** An odd root accepts every input, so its graph runs through the origin in both directions. **Question:** A square root turns away every negative input. What does a cube root do with them, and what could a power of one third mean? ## Why this exists A cube of side $s$ holds a volume of $s^3$. Often the question runs the other way: a box must hold $3375$ cubic centimeters, so how long is its edge? Undoing a cube is taking a cube root, and turning that undoing into a rule that accepts an input and returns an output gives the **cube root function**. It sits in the family of radical functions beside the square root, but it behaves differently. No real number squares to a negative, so $\sqrt{x}$ refuses every negative input and its graph is a half curve that starts at the origin. A negative number does have a cube root, since $(-2)^3 = -8$. So the cube root accepts every real input, and its graph runs across the whole page, through the origin, in both directions. There is a second way to write the same function. Exponents already have rules, such as $(a^m)^n = a^{mn}$. If a power of one third is to obey that rule, then $\left(x^{1/3}\right)^3 = x^1 = x$, which says $x^{1/3}$ must be a number whose cube is $x$. So a power of one third is a cube root, and a root function is a power function in disguise. ## Definition The **cube root function** is $$ f(x) = \sqrt[3]{x} $$ and its output is the one real number whose cube is the input: $$ y = \sqrt[3]{x} \quad\text{means}\quad y^3 = x $$ So $\sqrt[3]{8} = 2$ because $2^3 = 8$, and $\sqrt[3]{-8} = -2$ because $(-2)^3 = -8$. The small $3$ is the **index** of the root and the expression under the sign is the **radicand**. The square root $\sqrt{x}$ has index $2$, which is left unwritten; in general $\sqrt[n]{x}$, the $n$th root of $x$, is a number whose $n$th power is $x$. ### [Cubic, quartic and higher degree functions](https://math.wiki/cubic-functions/) **Summary:** Degree three functions, whose graphs run from one corner of the plane to the opposite corner. **Question:** The graph of $y = x^3$ is one curve. What shapes can the graph of any degree three function take, and what in its formula decides which one you get? ## Why this exists The cubing function $y = x^3$ levels off for an instant at the origin, and shifting or stretching it, as in graph-using-transformations, never adds a hill or a valley. Yet $y = x^3 - 3x$ has both. So the cubics are not all copies of one curve, but they are close: every cubic has one of exactly three shapes, one number made from its coefficients says which, and each is balanced about a single point where its bend changes direction. ## Definition A **cubic function** is a polynomial function of degree $3$: $$f(x) = ax^3 + bx^2 + cx + d, \qquad a \neq 0.$$ Degree $4$ gives a **quartic** and degree $5$ a **quintic**. As in degree-and-graph-shape, a turning point is a peak or a valley. A **chord** joins two points of a graph. A graph **bends upward** on an interval if the midpoint of every chord there lies above the graph, like a bowl, and **bends downward** if it lies below. A **point of inflection** is where the graph switches from one bend to the other. Every cubic has one of three **shapes**: - **no turn**: it only rises, or only falls, and never levels off; - **a flat point**: it only rises or only falls, but levels off for an instant at one point; - **two turns**: a peak and a valley. A cubic is in **factored form** when written $f(x) = a(x - r_1)(x - r_2)(x - r_3)$. Its zeros, the x-intercepts, are then $r_1$, $r_2$ and $r_3$. ## Intuition Look at $y = x^3 + px$. If $p > 0$, both $x^3$ and $px$ increase as $x$ does, so their sum does: no turn. If $p = 0$ it is $y = x^3$, with its flat point. If $p < 0$, the term $px$ pulls the graph down right of $0$ and up left of it. Near $0$, where $x^3$ is tiny, that pull wins, making a hill on the left and a valley on the right. Far out $x^3$ wins again. ### [Definition of a function](https://math.wiki/function-definition/) **Summary:** Each input gets one output. Tables, mappings, and why the rule matters. **Question:** Two quantities are paired up. When does knowing the first one settle the second, and what do we call that? ## What this area is about A function pairs each input with exactly one output. That single condition separates a function from any other pairing, and it is what lets you speak of *the* output for an input, build a table, or draw a graph and trust it. This area sets up the idea and its vocabulary. ## How these fit together **The picture.** Functions as input-output rules treats a function as a machine: one input in, exactly one output out. It also shows the four ways to present a function: words, a table, a graph or an equation. **The definition.** Relations and functions starts from any pairing at all, a relation, and says which relations are functions: those where no input has two outputs, whether or not a rule can be written down. **The vocabulary.** Independent and dependent variables names the two roles, the input you choose and the output that follows, and writes one in terms of the other. ## A path through it The input-output machine first, then independent and dependent variables. Relations and functions can come at any point after the machine. The rest of the branch builds on this: function notation such as $f(x)$, domain and range, and the vertical line test for reading the one-output rule off a graph. ### [Definition of a logarithm](https://math.wiki/exponential-and-log-forms/) **Summary:** Rewriting a logarithm statement as the power statement it stands for, and back again. **Question:** The unknown is the exponent, as in 2 to the x equals 12. What number is x, and how do you even write it down? ## When you reach for it Some equations put the unknown where none of the usual tools can reach it. In $$2^x = 12$$ the $x$ is an exponent. Adding, subtracting, multiplying, dividing, squaring and taking roots all act on the number $2^x$ as a whole; none of them brings the $x$ down out of the exponent. You can still say a good deal about the answer. Since $2^3 = 8$ and $2^4 = 16$, and powers of $2$ grow as the exponent grows, $x$ lies between $3$ and $4$. The graph shows there is exactly one such $x$. What is missing is a **name** for that number. It is not a fraction you can write down, and it is not a root, so the notation so far has nothing to call it. The **logarithm** is that name: $\log_2 12$ means "the exponent that turns $2$ into $12$". With it the equation is solved exactly, $$x = \log_2 12 \approx 3.58$$ So reach for this rewrite whenever the unknown sits in an exponent, as in an exponential function asked "when does it reach this value?". Reach for it in the other direction too: a statement about a logarithm is often easiest to understand, or to solve, once it is turned back into the power statement it stands for. ## The procedure ### The definition Let $b$ be a positive number other than $1$, and let $x$ be a positive number. The **logarithm of $x$ with base $b$**, written $\log_b x$, is the exponent that $b$ must be raised to in order to give $x$: $$\log_b x = y \quad\text{means}\quad b^y = x$$ ### [Degree, end behavior and the shape of a polynomial graph](https://math.wiki/degree-and-graph-shape/) **Summary:** The degree limits how often a polynomial graph can cross the axis and how many bends it can have. **Question:** Without plotting a single point, what can the formula of a polynomial tell you about the shape of its graph? ## Why this exists A table of values shows only the window you chose. Something has to say what happens outside it: whether the graph comes back down, how often it can cross the axis, how many hills it can have. For a polynomial, the degree answers most of this at once. The term with the highest power decides where both ends go, the degree caps the number of zeros and of turning points, and an odd or even degree decides whether the outputs can be every real number. ## Definition A **polynomial function** has the form $$p(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$$ where $n$ is a whole number, the coefficients are real numbers, and $a_n \neq 0$. As in degree-of-a-polynomial, $n$ is the **degree**, $a_n x^n$ the **leading term** and $a_n$ the **leading coefficient**. The leading term is the one with the highest power wherever it is written: in $8x - x^2 + 2x^3$ it is $2x^3$. - A **real zero** of $p$ is a real number $r$ with $p(r) = 0$, an $x$-intercept of the graph. - A **turning point** is a peak or a valley, where the graph switches between rising and falling, as in relative-maximum-and-minimum. - The **end behavior** is what the graph does far to the left and far to the right, in the arrows of end-behavior-from-a-graph: "as $x \to \infty$, $p(x) \to -\infty$" says that far to the right the graph falls without bound. ## Intuition Compare $p(x) = 3x^5 - x^3$ with its leading term alone. ### [Dividing functions and the domain of a combination](https://math.wiki/quotient-of-two-functions/) **Summary:** The quotient function divides one output by the other and is undefined where the bottom output is zero. **Question:** Two quantities depend on the same input, and you want one divided by the other at every input. Where does that ratio exist, and what happens where the bottom quantity is zero? ## When you reach for it A workshop's weekly cost and the number of items it makes both depend on how many items it makes; the cost per item is one divided by the other. A town's population and its area both change with time; the number of people per square kilometer is one divided by the other. Whenever two functions share an input and the question is about their ratio at that input, you want a quotient of functions. sum-of-two-functions built sums, differences and products output by output. Division works the same way, with one new difficulty: a number cannot be divided by zero. So a quotient of functions loses every input where the bottom output is zero, and finding exactly which inputs survive is half the work. The same question, which inputs does a combined function accept, is asked of sums and products too, and this lesson answers it for all of them. ## The procedure ### The quotient is defined output by output For two functions $f$ and $g$ whose outputs are numbers, the quotient function $\frac{f}{g}$ is defined by what it does to each input $x$: $$ \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} $$ This holds at every input where $f(x)$ and $g(x)$ both exist and $g(x) \ne 0$, and nowhere else. As in function-notation, the expression in parentheses $\left(\frac{f}{g}\right)$ is the name of the new function and the $(x)$ after it is the input. The condition $g(x) \ne 0$ is not a convention; it comes from what division means. To divide $f(x)$ by $g(x)$ is to find the one number that, multiplied by $g(x)$, gives $f(x)$. If $g(x) = 0$ and $f(x) \ne 0$, no number works, since every number times $0$ is $0$. If both are $0$, every number works, so there is no single answer to call the output. Either way the quotient has no value there. The top output being zero causes no trouble: $\frac{0}{g(x)} = 0$ whenever $g(x) \ne 0$. ### [Domain and range](https://math.wiki/domain-and-range/) **Summary:** Which inputs are allowed, and which outputs come out. **Question:** A function takes inputs and gives outputs. Which inputs does it accept, and which outputs does it ever produce? ## What this area is about Every function comes with two sets. The **domain** is the set of inputs it accepts; the **range** is the set of outputs it actually produces. Describing them is the first question to ask about any function, whether it arrives as a list of pairs, a graph, a formula such as $f(x) = \sqrt{10 - 2x}$, or a real situation. ## How these fit together **Procedures by form.** Domain and range from pairs and tables collects the inputs and outputs of a finite list. Domain and range from a graph reads how far a graph stretches left and right, down and up. **Conventions for formulas and situations.** The implied domain of a formula says that when no domain is stated, it is every real number for which the formula gives a real output. A reasonable domain for a real situation shrinks that set to the inputs that make sense, such as whole numbers of people. ## A path through it Pairs and tables first, then graphs. After graphs come the implied domain and the reasonable domain in context, in either order. With domain and range in hand you can describe one to one functions and inverses, find the domain of a rational function, and track how a transformation moves both sets. ### [Domain and range from a graph](https://math.wiki/domain-from-a-graph/) **Summary:** Reading how far a graph stretches left and right to describe every input it accepts. **Question:** A graph shows a function as a picture. Which inputs does it use, and which outputs does it ever reach? ## When you reach for it The **domain** of a function is the set of inputs it accepts, and the **range** is the set of outputs it actually produces. For a short list of pairs you collect them, as in domain and range from pairs and tables. A graph usually has infinitely many points, too many to list. What you can read instead is **how far the graph stretches**: left and right for the domain, down and up for the range. Reach for this whenever a function is given as a graph. It is the first look before reading a value from a graph: which inputs does the graph cover at all, and which heights does it ever reach? Answers are written as inequalities and in interval notation, which you already have. The lesson runs over two days. Day 1 reads a graph in one unbroken piece: its reach each way, its end dots and its arrows. Day 2 handles a graph in two pieces, a single missing point, a highest or lowest point in the middle, and a graph that levels off toward a height it never reaches. ## The procedure ### Day 1: the shadow on each axis Every point $(x, y)$ on a graph says one thing: the input $x$ gives the output $y$. So a number $a$ is in the domain exactly when the graph has a point directly above or below $a$ on the $x$-axis, and a number $b$ is in the range exactly when the graph has a point level with $b$ on the $y$-axis. Picture a light shining straight down onto the graph: its shadow on the $x$-axis is the domain. A light from the side throws a shadow on the $y$-axis, and that is the range. ### [Domain and range from pairs and tables](https://math.wiki/domain-from-a-set-of-pairs/) **Summary:** Collecting the inputs into the domain and the outputs into the range for a finite list of pairs. **Question:** A list of pairs can run to dozens of entries, with the same values turning up again and again. How do you say exactly which values go in and which come out? ## When you reach for it A relation pairs inputs with outputs. Very often the pairs are simply written down: a list of ordered pairs, a table of readings, or a mapping diagram with arrows. Before you can ask anything harder about such a relation (is it a function, what does its graph look like, can it be read backwards) you need two plain facts about it: which values go in, and which values come out. The **domain** of a relation is the set of all its inputs. The **range** is the set of all its outputs. Reach for this move whenever a relation is given as a finite collection of pairs, in any of its three usual dresses, and you need to say what it accepts and what it produces. ## The procedure ### Inputs first, outputs second In an ordered pair $(x, y)$ the first entry is the input and the second is the output. So for a relation written as a set of pairs, $$\begin{aligned} \text{domain} &= \text{the set of all first entries},\\ \text{range} &= \text{the set of all second entries}. \end{aligned}$$ For the relation $\{(1, 8), (3, 8), (5, 2)\}$ the first entries are $1$, $3$ and $5$, and the second entries are $8$, $8$ and $2$. The domain is $\{1, 3, 5\}$ and the range is $\{2, 8\}$. The order inside a pair matters, and that is the whole reason this works: $(1, 8)$ and $(8, 1)$ are different pairs, and in the first one $1$ goes in while in the second it comes out. ### Writing the answer as a set ### [Domain and zeros of a rational function](https://math.wiki/domain-of-a-rational-function/) **Summary:** Removing every input that would make the denominator zero. **Question:** A function is one polynomial divided by another. Which inputs can it take, and which of those give an output of zero? ## When you reach for it A **rational function** is a function whose formula is one polynomial divided by another, $$f(x) = \frac{p(x)}{q(x)}$$ where $p$ and $q$ are polynomials and $q$ is not the zero polynomial. Before you can graph one, solve with one, or use one as a model, you need two lists of numbers. The first is its **domain**: the inputs the formula can accept. A rational function has no square root and no logarithm in it, so the only way its formula can break is a division by zero. Its implied domain is therefore every real number except the zeros of the denominator. This is the same question the excluded values of a rational expression answers, now asked of a function. The second is its **zeros**: the inputs in the domain whose output is $0$. They are the places where the graph meets the horizontal axis. The two lists are tied together. A zero must be in the domain first, so the domain is always found first, and a number that the denominator excludes can never be a zero, even when the numerator is $0$ there too. ## The procedure Take the lead function of this lesson, $$f(x) = \frac{x + 6}{x^2 + 2x - 35}$$ ### The domain comes from the denominator The formula fails exactly when the denominator is $0$, because division by $0$ has no answer. Nothing else can go wrong: a polynomial accepts every real number, so the numerator, and the denominator itself, can always be worked out. To find the inputs that fail, solve ### [Domain of a composition](https://math.wiki/domain-of-a-composition/) **Summary:** The allowed inputs are those the inner function accepts whose outputs the outer function also accepts. **Question:** A composition runs a number through one function and then another. Which starting numbers make it all the way through? ## Why this exists A composition $f(g(x))$ is a two-step calculation: work out $g(x)$, then feed the answer to $f$. In composing functions algebraically the two steps were merged into one formula. Every function, though, comes with its inputs as well as its rule, and the merged formula does not say which inputs belong to the composition. Worse, it can say the wrong thing: $(\sqrt{x})^2$ simplifies to $x$, which accepts $x = -4$, although $\sqrt{-4}$ is not a real number and the chain never got started. The domain of a combination such as $f + g$ or $\frac{f}{g}$ was found by looking at both functions at the same input. A composition is different, because the second function never sees the input at all. It sees the output of the first one. So a second question appears: not "is this number allowed into $f$?" but "does $g$ turn this number into something allowed into $f$?" This lesson collects the inputs that pass both tests. ## Definition The **domain of the composition** $f \circ g$ is the set of inputs $x$ for which the whole chain works: $x$ is an input that $g$ accepts, and the output $g(x)$ is an input that $f$ accepts. In set notation, writing $\operatorname{dom} g$ for the domain of $g$: $$\operatorname{dom}(f \circ g) = \{\, x \in \operatorname{dom} g \;:\; g(x) \in \operatorname{dom} f \,\}$$ When a function is given only by a formula, its domain is understood to be every real number at which the formula gives a real number: no division by zero, and no even root of a negative number. When a function describes a situation, the situation can shrink its domain further (a time before the start, a negative length, half a person). ### [Doubling time and half life](https://math.wiki/doubling-time-and-half-life/) **Summary:** The fixed length of time a growing quantity takes to double or a decaying one takes to halve. **Question:** A dose of medicine in the blood falls to half every six hours, whether there are 400 milligrams or 4. How can one length of time describe the whole decline? ## Why this exists A growth or decay factor, from growth-factor-and-decay-factor, says what an amount is multiplied by in one period. People rarely talk that way, though. A doctor says a medicine has a half-life of six hours. A biologist says a yeast culture doubles every twenty-five minutes. A single length of time is easier to picture than a factor such as $0.8909$ per hour, and it answers the questions people actually ask: how long until the dose is mostly gone, how long until the culture is four times as big. That only works because of a special property of exponential change. The time to double, or to halve, is the same from every starting point. A dose of $400$ milligrams takes six hours to fall to $200$; the remaining $200$ takes six more hours to fall to $100$, and so on forever. One number describes the whole process. ## Definition Let a quantity change exponentially, $y = a \cdot b^t$ with $a > 0$. - If $b > 1$, its **doubling time** $D$ is the length of time it takes to double. Over any stretch of time of length $D$, the amount is multiplied by $2$. - If $0 < b < 1$, its **half-life** $H$ is the length of time it takes to fall to half. Over any stretch of time of length $H$, the amount is multiplied by $\tfrac{1}{2}$. Written with the doubling time or the half-life in place of the factor, the two models are $$y = a \cdot 2^{t/D} \qquad y = a\left(\tfrac{1}{2}\right)^{t/H}$$ where $a$ is the amount at $t = 0$, and $t$, $D$ and $H$ are measured in the same unit of time. The fraction $t/D$ counts how many doubling times have passed, and $t/H$ counts how many half-lives have passed. It need not be a whole number. ### [End behavior](https://math.wiki/end-behavior-from-a-graph/) **Summary:** Describing in words where the two ends of a drawn graph are going. **Question:** A graph runs off both edges of the page. Where does it go after that? ## When you reach for it Every drawn graph is a window. The page stops at its edges, but the inputs do not: there is always a larger $x$ to the right and a more negative $x$ to the left. Does a quantity keep growing out there? Does it shrink away to nothing? Does it settle at some steady value? The **end behavior** of a graph answers those questions. It describes where the graph goes at its two ends: far to the left, where $x$ is negative and very large in size, and far to the right, where $x$ is positive and very large. Reach for it whenever you describe the overall shape of a function. With the intercepts, where the graph rises and falls and its turning points, the ends are one of the key features of a graph. It is the same kind of description as in qualitative-graph-reading, aimed at the two ends. It also tells families apart quickly, because lines, parabolas and exponential curves each have their own kind of ends. ## The procedure ### Two ends, three things an end can do Describe each end on its own. For every graph in this lesson, each end does one of three things. - It **rises without bound**: going further out, the graph climbs and keeps climbing. Name any height, however large, and far enough out the graph passes it and stays above it. - It **falls without bound**: the same, downward, past any depth you name. - It **levels off toward a value**: going further out, the outputs get closer and closer to one number, and stay as close to it as you like. The graph flattens out along a horizontal line. ### [Evaluating a composition from numbers, tables and graphs](https://math.wiki/evaluate-a-composition-at-a-number/) **Summary:** Running a number through the inner function and feeding the answer into the outer one. **Question:** The output of one function is exactly what a second function needs as its input. How do you get from the first input to the final output? ## When you reach for it Many quantities depend on something that itself depends on something else. The height of a weather balloon depends on how long it has been rising, and the temperature of the air around it depends on its height. To know the temperature after $10$ minutes you have to go through the height: time gives height, and height gives temperature. Two functions are chained, and the output of the first is the input of the second. Running a number through one function and feeding the result straight into another is called **composing** the two functions, and the chained function is their **composition**. This lesson is about finding a composition's value at one particular number, when the two functions are given as formulas, as tables or as graphs. Each step is something you can already do: evaluating a function at a number from its formula, or reading a value off a graph. What is new is only the order of the steps and where the first answer goes. Reach for it whenever a function's parentheses hold another function, as in $f(g(3))$, whenever you see the small circle in $(f \circ g)(3)$, and whenever a problem happens in two stages, the result of the first being what the second works on. ## The procedure ### Inside first Let $f(x) = 3x - 7$ and $g(x) = 10 - x^2$. The expression $f(g(3))$ asks for $f$ evaluated at the number $g(3)$. The function $f$ cannot do anything until it has a number to act on, and that number is $g(3)$, so $g(3)$ is found first: ### [Evaluating a function at a number](https://math.wiki/evaluate-a-function-at-a-number/) **Summary:** Putting a number in for the input and working out the output the rule gives. **Question:** A rule turns every input into one output. How do you write "the output for this input" in a few symbols, and how do you run the rule forwards and backwards? ## When you reach for it A function is a rule that gives every allowed input exactly one output (input-output-machine). Once the rule is written as a formula, two questions come up again and again. **Forwards:** here is an input, what output does the rule give? **Backwards:** here is an output, which input or inputs give it? Both questions need a way to say "the output for this input" without a sentence. That is what **function notation** does: $f(5)$ is the output of the function $f$ when the input is $5$. Working it out is called **evaluating** the function at $5$, and it is substituting a number into a formula with a name attached. Running the rule backwards, solving $f(x) = 21$, turns into an equation of the kind you already solve (linear-equations). The lesson takes two days. **Part 1** (Examples 1 to 4) reads the notation and evaluates at whole numbers, negatives, fractions and simple expressions. **Part 2** (Examples 5 to 7) runs the rule backwards and reads the notation in real situations. ## The procedure ### Reading the notation (Part 1) Write $$f(x) = 4x - 7$$ and read it "$f$ of $x$ equals $4x$ minus $7$". It says three things at once: the function is called $f$; its input is called $x$; and the output for the input $x$ is $4x - 7$, so the rule is "multiply the input by $4$, then subtract $7$". The parentheses in $f(x)$ do **not** mean multiplication. $f$ is not a number; it is the name of a rule, and $f(x)$ is one symbol meaning "the output of $f$ at $x$". So $f(5)$, read "$f$ of $5$", is the output when the input is $5$. The parentheses hold the input, the way an address holds a house number. ### [Evaluating a function at an expression](https://math.wiki/evaluate-a-function-at-an-expression/) **Summary:** Putting a whole expression, such as x plus 2, in for the input and simplifying. **Question:** A function's input can be a whole expression, such as x plus 2 or 3a. What does the rule do with it, and why is f of (x plus 2) a different thing from f of x, plus 2? ## When you reach for it You can already evaluate a function at a number: with $f(x) = x^2 - 3x - 2$, the input $4$ goes in wherever $x$ stands, and $f(4) = 16 - 12 - 2 = 2$. That lesson put a short expression into a linear rule once. Here it is the main move, for any rule: the input is a whole **expression**, such as $x + 2$, $3a$ or $-x$, and the answer is a new expression rather than a number. You reach for it to say something about many inputs at once. How much does the output change when the input goes up by $1$? That compares $f(x + 1)$ with $f(x)$ for every $x$ together. What happens to a graph when every input is shifted? That is $f(x + 2)$. What such statements mean in a real situation, with units, is function notation in a real situation. ## The procedure ### The whole expression replaces every $x$ Take $f(x) = x^2 - 3x - 2$. The $x$ in the rule only marks where the input goes, so the rule could be written with an empty box in its place: $$f(\square) = (\square)^2 - 3(\square) - 2$$ Whatever goes in the box on the left goes into every box on the right. For $f(x + 1)$: $$f(x + 1) = (x + 1)^2 - 3(x + 1) - 2$$ The parentheses are needed because the rule squares the **whole input** and multiplies the **whole input** by $3$. Without them, $x + 1^2 - 3x + 1 - 2$ squares only the $1$ and multiplies only the $x$, a different expression. Nor is there a clash in using $x$ twice: the same function is $f(t) = t^2 - 3t - 2$, and $f(x + 1)$ means "put $t = x + 1$". ### [Evaluating a piecewise function](https://math.wiki/evaluate-a-piecewise-function/) **Summary:** Choosing the branch whose condition the input satisfies and using only that rule. **Question:** A rule uses one formula for some inputs and a different formula for others. Given an input, which formula do you use? ## When you reach for it Some rules change partway along. A kayak rental charges one price for each hour up to a limit and a different price after it; an electricity company charges more for each unit once a home has used a set amount. One formula cannot describe such a rule, but two formulas can, each with a note saying which inputs it is for. A **piecewise function** is a function written this way: several formulas, each paired with a condition on the input. Whenever a number goes into one, or a question asks which inputs give a stated output, the first job is the same: find which condition the input satisfies. After that it is ordinary evaluating a function at a number, with one formula. ## The procedure ### Reading the brace $$ f(x) = \begin{cases} x^2 & \text{if } x < 1 \\ 2x + 3 & \text{if } 1 \le x \le 4 \\ 20 - x & \text{if } x > 4 \end{cases} $$ The brace gathers three lines into one function named $f$. Each line is a **branch**: a formula, then a condition. The first line reads "$f(x)$ is $x^2$ when $x$ is less than $1$". The middle condition, $1 \le x \le 4$, is an "and" inequality: $x$ must be at least $1$ and also at most $4$. The brace stands for one function, not three. Each input gets one output, made by the one formula its condition picks out. Each condition is a stretch of the number line, as in the figure. ### Choosing the branch To find $f(-3)$, test $-3$ against each condition. The first, $-3 < 1$, is true. The second fails because $-3$ is less than $1$, and the third fails because $-3$ is not greater than $4$. So the first branch applies and ### [Evaluating logarithms, common and natural](https://math.wiki/evaluate-a-logarithm/) **Summary:** Asking what power of the base gives the number, for bases and numbers that work out whole. **Question:** Which power of 8 gives 16? Neither number is a power of the other, so how do you find the exponent, and what is a calculator doing when the number is not a neat power at all? ## When you reach for it The definition of a logarithm gives a name to an exponent: $\log_b x$ is the power that $b$ must be raised to in order to give $x$. A name is not yet a value. To **evaluate** a logarithm is to find that power as a number, either exactly, when the number turns out to be a neat power of the base, or as a decimal, when it does not. You reach for this whenever a logarithm appears and you need to know how big it is: in a formula, as the answer to an equation such as $10^x = 300$, as the output of a logarithm function, or on a scale that measures size by counting powers of ten. Two logarithms come up far more than any other, because they are the two on every scientific calculator. The **common logarithm** $\log x$ has base $10$, and the **natural logarithm** $\ln x$ has base $e \approx 2.718$, the base of the natural exponential function. ## The procedure ### Ask which power of the base gives the number Every evaluation starts from the definition, $$\log_b x = y \quad\text{means}\quad b^y = x$$ so evaluating $\log_b x$ means recognizing $x$ as a power of $b$. When it is a whole-number power you can read the answer off: $3^4 = 81$, so $\log_3 81 = 4$. The rules for exponents widen what "a power of $b$" can look like. - A **reciprocal** is a negative power, since $b^{-n} = \dfrac{1}{b^n}$. So $\log_3 \frac{1}{9} = -2$. - A **root** is a fractional power, by the meaning of unit fraction exponents: $\sqrt[n]{b} = b^{\frac{1}{n}}$. So $\log_3 \sqrt{3} = \frac{1}{2}$. - The number $1$ is $b^0$, by the zero exponent rule, and $b$ is $b^1$. So $\log_b 1 = 0$ and $\log_b b = 1$ for every allowed base. ### [Even and odd functions](https://math.wiki/test-symmetry-algebraically/) **Summary:** Replacing the input by its opposite, or both variables by their opposites, and seeing whether the equation survives. **Question:** Can you tell from a rule alone, without drawing anything, whether its graph is its own mirror image? ## When you reach for it Symmetry about the vertical axis and the origin describes two mirror images a graph can have: a fold along the $y$-axis, where each point $(a, b)$ has the partner $(-a, b)$, and a half turn about the origin, where the partner is $(-a, -b)$. A drawing can suggest either one, but it shows only a window, and only as exactly as a pencil line allows. The rule settles it exactly, for every input at once. Reach for this test when you have a formula and want to know its symmetry: before sketching, so that you plot only the half with $x \ge 0$ and mirror the rest; when checking a sketch; and later, when symmetry shortens a calculation. The same idea tests equations that are not functions, such as a circle. ## The procedure ### Even and odd functions The graph of $f$ is symmetric about the $y$-axis when every point $(a, f(a))$ has its partner $(-a, f(a))$ on the graph too. The point of the graph above $-a$ is $(-a, f(-a))$, so the partner is there exactly when $f(-a) = f(a)$. Asking this for every $a$ gives the definition. A function $f$ is **even** when $$f(-x) = f(x) \quad \text{for every } x \text{ in its domain.}$$ The graph is symmetric about the origin when every point $(a, f(a))$ has the partner $(-a, -f(a))$, which means $f(-a) = -f(a)$. A function $f$ is **odd** when $$f(-x) = -f(x) \quad \text{for every } x \text{ in its domain.}$$ Both definitions need a symmetric domain: whenever $x$ is in it, so is $-x$, or $f(-x)$ has no meaning. ### [Expanding and condensing logarithms](https://math.wiki/expand-a-log-expression/) **Summary:** Breaking one logarithm of a complicated expression into a sum and difference of simpler ones. **Question:** The laws of logarithms each break one product, quotient or power apart. How do they work together on an expression with all three at once, and how is a string of logarithms gathered back into one? ## When you reach for it The laws of logarithms each do one thing: a product inside a logarithm becomes a sum, a quotient becomes a difference, and a power becomes a multiple. Real expressions seldom need only one of them. The logarithm of $\frac{x^4\sqrt{y}}{z^2}$ has a quotient, a product, two powers and a root inside it, and getting it into its simplest pieces takes all three laws, used in the right order. Two jobs come up, and they are the same laws read in opposite directions. **Expanding** a logarithm means rewriting it as a sum and difference of multiples of logarithms, each of which has no product, quotient, power or root left in its argument. It is what you want when the separate pieces are easier to handle than the whole: when some of them are known numbers, when a function is easier to study term by term, or, later, in calculus, when a product of many factors is easier to differentiate as a sum. **Condensing** a logarithmic expression means gathering a sum and difference of multiples of logarithms, all with the same base, into a single logarithm with coefficient $1$. It is what you want when the pieces are awkward but the whole is simple, as in $2 \log 5 + \log 4$, where neither logarithm is a whole number but the combination is. It is also the first move in solving a logarithmic equation, where one logarithm on each side can be turned back into a power statement. ## The procedure ### [Exponential functions](https://math.wiki/exponential-functions/) **Summary:** The variable is in the exponent: growth by a constant factor. **Question:** Doubling, halving, interest earning interest. What kind of rule puts the variable up in the exponent, and what does its graph look like? ## What this area is about In a linear rule the input is multiplied by a fixed slope. In an exponential function $f(x) = a \cdot b^x$ (with $a \ne 0$, $b > 0$ and $b \ne 1$) the input is the exponent, so each step of $1$ in $x$ multiplies the output by the same factor $b$. This area builds those rules, draws their graphs, and meets the special base $e$. ## How these fit together **The idea.** Exponential growth and decay is its own area: growth and decay factors, doubling time and half life, and races against linear and polynomial growth. **Graphs.** Graphs of exponential functions draws $y = b^x$, which passes through $(0, 1)$ and runs flat along the $x$-axis on one side. Transforming an exponential graph shifts and stretches it, and moves its horizontal asymptote with it. **A procedure.** Writing an exponential function from a table or graph reads the starting value $a$ and the constant ratio $b$ back off the data. **Money and a new base.** Compound interest splits a yearly rate over several periods. Letting the number of periods grow without end leads to the natural exponential function $y = e^x$, built on the number $e$. ## A path through it Growth and decay first, then the graph. After the graph, write a rule from a table or graph and transform the graph, in either order. Compound interest also follows growth and decay, and $e$ comes after compound interest. Next, logarithms undo exponentials (once you also have inverse functions), exponential equations can be solved, and growth rates at infinity shows $b^x$ with $b > 1$ outgrowing every power of $x$. ### [Exponential growth and decay](https://math.wiki/exponential-growth-and-decay/) **Summary:** Doubling, halving, and everything in between. **Question:** Some quantities grow or shrink by the same percent every period. What single idea describes all of them, and how far can it be trusted? ## What this area is about A savings balance, a medicine dose in the blood, a phone's resale value: each changes by the same percent every period, so it is multiplied by the same number each time. That one multiplier, rather than a fixed amount added, is what makes change exponential. This area builds rules of the form $y = a \cdot b^x$ (with $b > 0$ and $b \ne 1$), reads them, and compares them with other kinds of growth. ## How these fit together **The core idea.** Growth and decay factors name the multiplier $b$: above $1$ for growth, between $0$ and $1$ for decay. Doubling time and half life turn that factor into one length of time. **A procedure.** Building a model from two points finds both $a$ and $b$ from two readings. **Comparisons.** Linear versus exponential growth races equal differences against equal factors, and linear, polynomial and exponential growth compared adds powers such as $x^2$ to the race. **A ceiling.** Logistic growth grows almost exponentially at first, then levels off at a carrying capacity. ## A path through it Start with growth and decay factors, then doubling time and half life, then the model from two points. Race a line against an exponential before the wider comparison with polynomials. Logistic growth comes last, after you can build an exponential model. Next come graphs of exponential functions, compound interest and solving exponential equations. ### [Features of a graph](https://math.wiki/features-of-a-graph/) **Summary:** The named landmarks any function graph can have, such as intercepts, turning points and asymptotes. **Question:** Two people look at the same graph. What words let them point at the same places on it, and agree on exactly what happens there? ## What this area is about Every function graph has landmarks: where it meets the axes, where it rises or falls, its peaks and valleys, where it breaks, and where its ends are heading. This area gives each landmark a name and a precise statement, as inputs, intervals or outputs, so a description can be checked, not guessed. ## How these fit together **Where it sits and moves.** Intercepts and zeros are where the graph meets the axes. Increasing, decreasing, positive and negative intervals name the stretches where it rises, falls, or sits above or below the axis. Turning points, maximums and minimums locate peaks and valleys: the largest or smallest outputs nearby or overall. **Shape.** Symmetry is a mirror fold along the $y$-axis or a half turn about the origin, and even and odd functions test for it from the rule alone. Continuous graphs, jumps and holes says where the pen must lift. **The far ends.** End behavior says where each end goes; an asymptote is a line the graph nears as closely as you like. **Rates.** Average rate of change is its own area: the slope $\frac{f(b) - f(a)}{b - a}$ for $a \ne b$. Concavity asks whether that rate is itself increasing or decreasing. ## A path through it Intercepts come before increasing and decreasing intervals, which come before turning points and average rates; concavity follows the average rate. Symmetry comes before even and odd functions. Continuity, end behavior and asymptotes fit anywhere. ### [Finding an inverse from a rule](https://math.wiki/find-an-inverse-algebraically/) **Summary:** Swapping the roles of input and output in the equation and solving for the new output. **Question:** A function is given by a formula. How do you find the formula that runs it backwards, from any output to the input that produced it? ## When you reach for it Reading a table backwards works when the function is a short list of pairs: find the output, read the input beside it. Most functions are formulas, and a formula has infinitely many pairs. Take $$f(x) = \frac{4x - 1}{x + 2}$$ Which input gives the output $3$? You could solve $\frac{4x - 1}{x + 2} = 3$ and get $x = 7$. Then someone asks about the output $5$, and you solve again, and again for every output after that. Solving once with the output left as a letter answers every one of those questions at a stroke, and the answer is a formula of its own: the **inverse function** $f^{-1}$. You reach for this whenever a quantity is known as a formula in one direction and the question starts from the other end: the fare is known and you want the distance, the volume is known and you want the edge, the euros received are known and you want the dollars handed over. The lesson runs over two days. Day 1 finds inverses of formulas and checks them by composition; Day 2 handles square roots, fractions and real quantities, where the domain has to be watched. ## The procedure ### Day 1: swap the roles and solve **What the inverse has to do.** As in the table lesson, the inverse is defined by one line read in two directions: $$f(a) = b \quad\Longleftrightarrow\quad f^{-1}(b) = a$$ The notation $f^{-1}$, read "$f$ inverse", names a function. The $-1$ is not an exponent: $f^{-1}(x)$ and $\frac{1}{f(x)}$ are different things (Example 1 compares them). ### [Finding the rule of a transformed graph or table](https://math.wiki/write-an-equation-from-a-transformed-graph/) **Summary:** Identifying the parent graph and every change applied to it, then assembling the rule. **Question:** You can see a curve, or a table of its values. It looks like a parabola, or a square-root curve, that has been moved. What rule draws exactly that curve? ## When you reach for it Combining transformations reads the moves off a rule, and graphing by transformations turns the rule into a curve. This lesson runs the other way: you have the curve, or a table of values, and you want the rule. Reach for it whenever a graph or a set of measurements has the shape of a parent you know, moved, stretched or flipped, and you need a formula that reproduces it. ## The procedure ### Name the parent from the shape Each parent has a shape you can recognize (squaring-parent-function, reciprocal-parent-function): | Shape | Parent | Its anchor point | |---|---|---| | a U with one turning point | $y = x^2$ | the vertex $(0, 0)$ | | an S that levels off for a moment | $y = x^3$ | the center $(0, 0)$ | | half a curve, leaving a start point | $y = \sqrt{x}$ | the start $(0, 0)$ | | two branches, each hugging two lines | $y = \frac{1}{x}$ | where the asymptotes $x = 0$ and $y = 0$ cross | ### The anchor gives $h$ and $k$ Write the rule as $y = a\,f(x - h) + k$. Putting $x = h$ gives $y = a\,f(0) + k$, and every parent above has $f(0) = 0$, so the anchor $(0, 0)$ lands at $(h, k)$. For $\frac{1}{x}$, which has no value at $0$, the asymptote $x = 0$ moves to $x = h$ and the asymptote $y = 0$ moves to $y = k$. So read the anchor's coordinates and they **are** $h$ and $k$. Mind the sign: an anchor at $x = -4$ means $h = -4$, and the rule shows $x + 4$. ### One more point gives $a$ ### [Function notation](https://math.wiki/function-notation/) **Summary:** f(x) means the output of f at x; reading and writing it. **Question:** How do you write "the output of this function for that input" in a few symbols, and what can those symbols say? ## What this area is about Function notation names a function and its output in one short phrase: $f(5)$ is the output of $f$ when the input is $5$. It is not multiplication. The notation lets you run a rule forwards or backwards, put whole expressions in for the input, and pack a sentence about a real situation, such as $C(12) = 40$, into a few symbols. ## How these fit together **The basic procedure.** Evaluating a function at a number puts a number in for the input and works out the output, and also asks the backwards question: which input gives this output? **Two extensions.** Evaluating a function at an expression puts in something like $x + 2$ and explains why $f(x + 2)$ is in general a different thing from $f(x) + 2$. Function notation in a real situation translates statements in notation into sentences with units, and back. ## A path through it Evaluate at a number first. The other two both build on it and can come in either order. Function notation is the language for the rest of the branch: reading values from a graph, average rate of change, combining and composing functions and modeling with functions all write in it. ### [Function notation in a real situation](https://math.wiki/function-notation-in-context/) **Summary:** Saying in words what a statement such as C of 12 equals 40 means for the quantities involved. **Question:** A statement such as C(12) = 40 packs a whole sentence into a few symbols. Which sentence is it, and how do you turn sentences about a real situation into notation and back again? ## When you reach for it Evaluating a function turns an input into an output. In a real situation the harder half is the meaning. A table or a chart hands you statements such as $P(10) = 5300$ or $U(3) = U(11)$, and the job is to say what they claim, in a sentence with units. Just as often a question asked in words has to be written in notation before anyone can answer it. This lesson is about that reading and writing, not about computing. ## The procedure ### Input, output and their units A function in a real situation pairs two quantities, and its description says what each measures: *$C(n)$ is the cost, in dollars, of $n$ skeins of wool.* That fixes the name, $C$; the input $n$, a number of skeins; and the output $C(n)$, an amount in dollars. The input is the quantity that is chosen and the output is the one that depends on it (independent-and-dependent-variables). Now read $C(12) = 40$. The number inside the parentheses is always the input, so it is $12$ skeins; the number on the other side is the output, $40$ dollars. The statement says *Twelve skeins of wool cost $40$ dollars.* The units are the check. The swapped reading, "$40$ skeins cost $12$ dollars", is a different claim, written $C(40) = 12$. When the input is a time, the description must also say where the clock starts. Let $T(t)$ be the temperature, in degrees Celsius, of a pie $t$ minutes after it comes out of the oven. Then $T(0) = 100$ says "the pie comes out of the oven at $100^\circ\text{C}$", and $T(20) = 60$ says "twenty minutes after it comes out, the pie is at $60^\circ\text{C}$". Without the starting point, $T(20)$ would not say twenty minutes after what. ### [Functions](https://math.wiki/functions/) **Summary:** Rules that turn one number into exactly one other, and the graphs they draw. **Question:** Cost depends on how much you buy, a population on the year, a balance on time. What single idea describes every one of these dependences, and how many different shapes can it take? ## What this area is about A function turns each allowed input into exactly one output, and its graph is the picture of that rule. This branch builds the idea, then studies the main families of functions, each with its own formula, graph shape and kind of growth, and the ways of making new functions from old ones. ## How these fit together **The idea.** What a function is gives the definition, notation, domain, range and graphs. Transformations shift, flip and stretch a graph by changing the rule. **The families.** Linear, quadratic, polynomial, rational, radical, piecewise and absolute value, exponential and logarithmic functions each grow or bend in a way their formula predicts. **New from old.** Combining and composing functions adds, divides or chains two functions. Inverse functions run one backward, when no two inputs share an output. **Using them.** Modeling with functions chooses a family and fits it to real data. ## A path through it What a function is and transformations come first, together. Linear functions come before piecewise functions and modeling. Polynomials come before rational functions, and both before exponentials. Composition and radical functions come before inverses, and inverses before logarithms; exponentials and logarithms support each other. Modeling waits for linear and exponential functions. Quadratics need only the function idea. Next come limits, calculus and trigonometry. ### [Functions as input-output rules](https://math.wiki/input-output-machine/) **Summary:** A function pictured as a machine that takes one number in and gives exactly one number back. **Question:** If you know what goes in, when can you be certain what comes out? ## Why this exists A great deal of mathematics is about one quantity settling another. The number of tickets you buy settles what you pay. The length of a square's side settles its area. The number of hours a candle has burned settles how tall it still is. "Settles" is doing real work in those sentences. It means that once you know the first number, there is nothing left to wonder about the second: no choice, no luck, no "it depends". Plenty of pairings do not work like that. Knowing how many hours someone studied does not settle their test score, and knowing a person's age does not settle their height. The idea of a **function** exists to say exactly which pairings settle the answer and which do not, so that when a pairing does, you can rely on it. The simplest picture of a function is a machine. Two more ideas make the picture usable. The instruction the machine follows is its **rule**, and an instruction has to meet a standard of its own before it can be the rule of a function. And one function can be written down in four different ways, as words, a table, a graph or an equation, each showing something the others hide. This lesson covers all three: the machine, the rule of a function, and the four ways to show a function. ## Definition Picture a machine with a slot on one side and a chute on the other. You drop an **input** into the slot, the machine does something to it, and an **output** comes out of the chute. What the machine does is its **rule**. You are told which inputs the machine accepts; call them its **allowed inputs**. ### [Graphing a function from a table](https://math.wiki/plot-a-function-from-a-table/) **Summary:** Making a table of inputs and outputs and plotting each pair as a point. **Question:** A rule gives one output for every input, but not every rule draws a straight line. How do you find out what its picture looks like? ## When you reach for it A function is a rule that gives exactly one output for each input it allows (functions as input-output rules). Each input with its output is an ordered pair, and each pair is a point on the coordinate plane. The **graph of a function** is the set of all its input-output pairs, plotted as points: the point $(x, y)$ is on the graph exactly when $y$ is the output the rule gives for the input $x$. That definition is also the method. There are usually infinitely many pairs, so you work out a few in a **table of values**, plot them, and look at the pattern. For a straight line, graphing a line from a table needs three points and a ruler. This lesson is for every other rule, where the table must also show which way the graph bends, whether to join the points at all, and what goes in the gaps. ## The procedure The running example is $y = x^2$: the output is the input multiplied by itself. ### Choosing the inputs Every number can be squared, so every $x$ is allowed. For a rule whose shape you do not yet know, take more inputs than a line would need, spread on both sides of $0$, and include $0$ itself. Here take the integers from $-3$ to $3$. A negative input goes in parentheses, so that its sign is squared with it: $(-3)^2 = (-3)(-3) = 9$. | $x$ | $y = x^2$ | |:-:|:-:| | $-3$ | $9$ | | $-2$ | $4$ | | $-1$ | $1$ | | $0$ | $0$ | | $1$ | $1$ | | $2$ | $4$ | | $3$ | $9$ | ### Plotting and reading the shape ### [Graphing a piecewise function](https://math.wiki/graph-a-piecewise-function/) **Summary:** Drawing each branch only over the inputs its condition allows. **Question:** A rule changes partway along the number line. How do you draw a graph that obeys each formula only where that formula is in charge? ## When you reach for it A piecewise function hands each input to one formula, chosen by a condition such as $x < 1$ or $x \ge 1$. Evaluating it deals with one input at a time. Graphing it shows every input at once: where the function rises and falls, where the formula changes, and whether the graph carries on unbroken at that change or leaps to a new height. You reach for it whenever a rule comes in pieces and you want to see the whole function, or read off its domain and range. ## The procedure ### Day 1: each piece over its own inputs Take $$f(x) = \begin{cases} x + 1 & \text{if } x < 1 \\ 4 - x & \text{if } x \ge 1 \end{cases}$$ The conditions split the $x$-axis at $x = 1$, the **boundary**. Every input to the left of $1$ goes to $x + 1$, and every input from $1$ onwards goes to $4 - x$. So the graph of $f$ is two familiar graphs, each cut down: the part of the line $y = x + 1$ above $x < 1$, and the part of the line $y = 4 - x$ above $x \ge 1$. Nothing else. Drawing either line in full would claim outputs the rule never gives. Each piece is a linear function, so two points fix it (graph-from-slope-intercept). The most useful point is at the boundary, so put $x = 1$ into each formula, even the one whose condition leaves $1$ out. The first gives $1 + 1 = 2$, so that piece heads toward $(1, 2)$. The second gives $4 - 1 = 3$, so that piece starts at $(1, 3)$. The inequality decides how each end is drawn. The condition $x \ge 1$ includes $1$, so $(1, 3)$ is on the graph: a **closed dot**, filled in. The condition $x < 1$ leaves $1$ out, so the first piece comes as close to $(1, 2)$ as you like without reaching it: an **open dot**, a hollow ring. This is the convention of inequalities on the number line, for the same reason: $\le$ and $\ge$ include the end, $<$ and $>$ do not. One more point on each piece fixes its direction, $(-1, 0)$ on the first and $(3, 1)$ on the second. Each condition has only one end, so each piece runs on forever, shown by an arrow. ### [Graphing by transforming a parent graph](https://math.wiki/graph-using-transformations/) **Summary:** Drawing a new graph by moving the parent graph instead of plotting many points. **Question:** Given a rule like y = 2(x + 3)^2 + 1, how do you draw its whole graph without plotting dozens of points, and which inputs and outputs does it have? ## When you reach for it You have a rule such as $y = 2(x + 3)^2 + 1$ and need its graph. Plotting it point by point takes many evaluations and still leaves you guessing at the shape between them. But the rule is a moved copy of a parent you already know, $y = x^2$, so its graph is a moved copy too. Move a handful of the parent's points and its key features, then draw the parent's shape through them. The same view answers a second question: which inputs does the rule accept, and which outputs does it give? The domain and range of a moved graph are the parent's, moved. The lesson is taught in two parts. Part 1 draws the graph; Part 2 finds the domain and range. ## The procedure ### Part 1: from the parent to the new graph Write the rule in the general form from combining-transformations, $$y = a\,f\big(b(x - h)\big) + k$$ factoring the inside first if it is not yet $b(x - h)$, and read off $a$, $b$, $h$ and $k$. Every point $(p, q)$ of the parent lands at $$(p,\ q) \;\mapsto\; \left(\frac{p}{b} + h,\ aq + k\right)$$ Every point of the parent moves by this one rule, so the graph moves as a single piece: the carried points fix where it is, and between them the curve keeps the parent's shape, stretched. That is why a few well-chosen points are enough. Which points? The parent's anchor and two or three more, chosen so the arithmetic is easy. The parents themselves are in squaring-parent-function and reciprocal-parent-function. ### [Graphing polynomial functions](https://math.wiki/graph-polynomial-functions/) **Summary:** Put zeros, multiplicity and end behavior together. **Question:** You know where a polynomial crosses the axis, where it only touches, and which way its two ends point. How do those pieces fit together into one picture of the whole graph, and how much of that picture can you trust? ## When you reach for it You reach for this when you need the shape of a polynomial graph without plotting dozens of points: to see where a quantity is positive, roughly where it peaks, or whether a picture on a screen is believable. Every piece comes from an earlier lesson. The behavior at each zero is zero-crosses-or-touches, the ends and the count of turns are degree-and-graph-shape, and the intercepts are x-intercepts-of-a-function. A sketch puts those pieces into one picture and adds a few computed points to set the heights. The lesson runs over two days. Part 1 sketches polynomials given in factored form. Part 2 starts from an expanded rule, factors it first, and handles factors that have no real zeros. ## The procedure ### Part 1: from factored form **The zeros.** A product is zero exactly when one factor is (zero-product-property), so each linear factor gives a zero. Its multiplicity says what the graph does there: an odd one crosses, an even one touches and turns back, and a multiplicity of $3$ or more lies flat as it meets the axis. **The ends.** Expanding the factors would multiply their leading terms, and nothing else reaches the highest power. So the leading term of $-2(x + 1)^3(x - 3)$ is $-2 \cdot x^3 \cdot x = -2x^4$ with no expanding at all, and the leading term decides the end behavior. **The $y$-intercept** is $f(0)$, the product of the factors' constant parts. **A sign chart.** A polynomial graph has no breaks (continuous-and-broken-graphs), so it cannot change sign without passing through a zero. The zeros cut the number line into intervals, and on each interval the sign is fixed. Find it from the sign of each factor, or from one value inside the interval. The sign flips at a zero of odd multiplicity and not at one of even multiplicity, which matches crossing and touching and makes a useful check. ### [Graphing rational functions](https://math.wiki/graph-rational-functions/) **Summary:** Find the holes, the asymptotes and the intercepts, then sketch. **Question:** You can find where a rational function breaks, where it levels off and where it meets the axes. How do those separate facts fit together into one picture of the whole graph? ## When you reach for it You reach for this whenever you want to see a rational function, a quotient $\frac{P(x)}{Q(x)}$ of two polynomials, as a whole picture rather than as a list of facts. Each piece has its own lesson: the refused inputs come from the domain, the breaks where the graph runs off the page from vertical-asymptotes, the level it settles to far out from horizontal-asymptotes, and the way it meets the axis at each zero from zero-crosses-or-touches. A graph is what you get when all of these are true at once. The pieces fit together because of one fact. Between two neighboring breaks the graph is a single unbroken curve (see continuous-and-broken-graphs), and on any stretch where it meets neither the axis nor a break it stays on one side of the axis. So once the breaks, the zeros and the sign on each stretch are known, the shape is forced up to details such as the exact height of a peak. Plotting many points by hand would still miss what happens near an asymptote; this method finds that first. This lesson takes two days. Day 1 (Examples 1 to 3) sketches functions whose factors each appear once, with no canceling: a horizontal asymptote at a ratio of leading coefficients, one at $y = 0$, and none at all. Day 2 (Examples 4 to 6) adds holes, factors that appear twice, and the reverse question: reading a formula off a graph. ## The procedure Each move below is allowed for a reason, and the reasons are what let you trust the sketch. ### [Graphing square root functions](https://math.wiki/graph-a-square-root-function/) **Summary:** Plotting the half curve that begins at its endpoint and rises more and more slowly. **Question:** A rule like f(x) = 2√(x − 3) + 1 refuses some inputs. Where does its graph begin, which way does it run, and which heights does it reach? ## When you reach for it Reach for this whenever a rule puts its input under a square root sign: $f(x) = 2\sqrt{x - 3} + 1$, a pendulum's period as a function of its length, the distance to the horizon as a function of height. Three questions come with every such rule, and this skill answers all three at once: which inputs it accepts, what its graph looks like, and which outputs it can produce. The graph is never a whole curve across the page. It is a **half curve**: it begins at a single point, its **endpoint**, and runs off in one direction only, steep at first and flattening as it goes. Drawing it well means finding that endpoint exactly, knowing which way the curve runs from it, and plotting a few points whose outputs come out as whole numbers. You already have the pieces. The square root function gave the parent graph of $y = \sqrt{x}$, through $(0, 0)$, $(1, 1)$, $(4, 2)$ and $(9, 3)$. The implied domain said a square root accepts only inputs that make the expression under it at least $0$. And graphing by transformations showed how to move a parent graph. This lesson puts them together for the square root family and adds what is special about it: the endpoint, the four directions, and the rule $\sqrt{c^2 x} = c\sqrt{x}$ that turns a squeeze into a stretch. ## The procedure The running example is the lead rule $$f(x) = 2\sqrt{x - 3} + 1$$ The expression under the square root sign, here $x - 3$, is the **radicand**. ### [Graphs of exponential functions](https://math.wiki/graph-exponential-functions/) **Summary:** The curve that starts flat and takes off, and its horizontal asymptote. **Question:** Doubling sends numbers soaring to the right. What does doubling look like to the left, where the inputs are negative? ## When you reach for it A rule such as $y = 2^x$ or $C = 5 \cdot 2^t$ describes a quantity multiplied by the same factor at every step, the growth and decay whose factor is studied in growth-factor-and-decay-factor. A table gives a handful of values. The graph shows all of them at once: where the quantity starts, how fast it climbs or falls, and the floor it never goes through. You reach for it to read off domain and range, to say what happens far out on each side, and to compare two rules at a glance. ## The procedure ### Day 1: the graph of $y = 2^x$ **A table that includes negative inputs.** As for any function, start with a table. Zero and negative inputs use zero and negative exponents: $2^0 = 1$ and $2^{-n} = \frac{1}{2^n}$, so for instance $2^{-3} = \frac{1}{8}$. | $x$ | $-3$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $3$ | |:-:|:-:|:-:|:-:|:-:|:-:|:-:|:-:| | $2^x$ | $\frac{1}{8}$ | $\frac{1}{4}$ | $\frac{1}{2}$ | $1$ | $2$ | $4$ | $8$ | Read left to right, each output is double the one before; read right to left, each is half. The negative inputs are no special case: they are what you get by halving backward from $2^0 = 1$. **Joining the points.** The input $x$ can be any real number, not only an integer. Take $x = \frac{1}{2}$: by product-of-powers, $2^{1/2} \cdot 2^{1/2} = 2^1 = 2$, so $2^{1/2}$ is the positive number whose square is $2$, that is $\sqrt{2} \approx 1.41$, a height between $2^0$ and $2^1$. Every input between two integers fills in a height between its neighbors, so the plotted points are joined by one smooth, unbroken curve, not by straight segments and not left as dots. ### [Graphs of functions](https://math.wiki/graphs-of-functions/) **Summary:** Plot input against output; the picture of the rule. **Question:** A rule turns each input into one output. How can a single picture show every one of those pairs at once, and what can you read from it? ## What this area is about The graph of a function $f$ is the set of all points $(x, f(x))$, one for each input $x$ in its domain. So a point $(a, b)$ lies on the graph exactly when $f(a) = b$. That one fact lets you go both ways: from a rule to a picture, and from a picture back to outputs, inputs and the story of what happened. ## How these fit together **Making the picture.** Graphing a function from a table works out some input and output pairs and plots each as a point. **Reading it.** Reading values off a graph finds the output for a given input, and every input that gives a chosen output. Graphs that tell a story reads the shape in words, with no exact numbers, and draws a shape from a story. **Naming its landmarks.** Features of a graph is an area of its own: intercepts, rising and falling intervals, turning points, symmetry, breaks, end behavior and rates of change. **Using two graphs.** Solving equations with graphs and tables finds the inputs where two graphs meet, the solutions of $f(x) = g(x)$, even when algebra cannot yet find them. ## A path through it Reading values comes before both the story graphs and the features of a graph. Plotting from a table and solving $f(x) = g(x)$ with graphs fit anywhere. Next, transformations move whole graphs by changing the rule, parent functions give the basic shapes to move, and piecewise functions draw different rules on different stretches. ### [Graphs of logarithmic functions](https://math.wiki/graph-log-functions/) **Summary:** The exponential's mirror image across y = x. **Question:** Every logarithm undoes an exponential. What does the graph of y = log_3(x - 2) + 1 look like, which inputs is it allowed, and which line does it run along? ## When you reach for it From exponential-and-log-forms you know that $\log_b x$ is the power $b$ must be raised to in order to give $x$. Read as a rule, $y = \log_b x$ is a function: it takes a positive number and returns an exponent. You also know the graph of $y = b^x$ from graph-exponential-functions. The two functions undo each other, so by graph-of-an-inverse the graph of the logarithm is the exponential's graph reflected in the line $y = x$. Everything about the logarithmic graph, its shape, the inputs it accepts and the line it runs along, comes from that one reflection. You reach for this whenever a rule has a logarithm in it and you want its picture: where the graph exists at all, where it shoots down (or up) along a wall, where it crosses the axes, and whether it rises or falls. A rule such as $y = \log_3(x - 2) + 1$ is the parent graph moved, in the sense of graph-using-transformations, so it is drawn by moving a few points and the asymptote, not by plotting point after point. ## The procedure **The logarithmic function.** For a base $b > 0$ with $b \ne 1$, the logarithmic function with base $b$ is $$f(x) = \log_b x, \qquad x > 0$$ where $y = \log_b x$ means exactly $b^y = x$. The common logarithm $\log x$ has base $10$, and the natural logarithm $\ln x$ has base $e \approx 2.718$, the base of the natural-exponential-function. Both are cases of the same graph. **Logarithms and exponentials undo each other.** Two facts follow straight from the definition: ### [Graphs that tell a story](https://math.wiki/qualitative-graph-reading/) **Summary:** Describing in words what a graph says happened, without reading exact numbers off it. **Question:** A graph has no numbers on it, only a shape. What can the shape alone tell you about what happened, and how do you draw that shape from a story? ## When you reach for it Many graphs come with no numbers worth reading: a sketch in a report, a curve on a weather app, a line drawn on a board to make a point. The shape still says a lot. When did the quantity grow, shrink or hold still? Quickly or slowly, steadily or not? When was it highest and lowest? This lesson reads those answers in words, then runs the other way and draws the shape from a story. Reading exact values off a graph with a scale is read-a-value-from-a-graph. The pictures here are the same graphs of functions, usually with time across. In linear-modeling graphs are read with numbers, rates and units; here only the shape is used, and curves join the straight pieces. ## The procedure ### Day 1: reading the shape **Read the axes first.** "Distance from home" and "speed" can draw the same shape and tell different stories, so nothing can be read until both axes are named. **Then read left to right**, because moving right means a later time. - A piece is **increasing** if it rises as you move right, **decreasing** if it falls and **constant** if it is flat. A flat piece says only that this quantity did not change. These words are made exact for stretches of input in increasing-and-decreasing-intervals. - A **steeper** piece means **faster** change: over the same width of time it climbs or drops through more height. This is slope without numbers, and it holds for falls too. - A **straight** piece means **steady** change, the same amount in every equal stretch of time, a constant-rate-of-change. A **curved** piece means the rate itself is changing: a rising curve that gets steeper grows faster and faster, one that flattens grows more and more slowly. See concavity-and-changing-rates and linear-and-nonlinear-functions. - The **highest point** and **lowest point** are where the quantity is greatest and least. Either may be reached more than once, or held along a flat piece. - A **corner**, where pieces of different steepness meet, is where something in the story changed. ### [Growth and decay factors](https://math.wiki/growth-factor-and-decay-factor/) **Summary:** The number the quantity is multiplied by each period, above one for growth and below one for decay. **Question:** A phone loses eighteen percent of its value every year. What single number describes how that value changes, year after year? ## Why this exists Many quantities change by the same percent every period. A phone's resale value falls $18\%$ a year. A cycling club's membership grows $4\%$ a year. After a pond is treated, the algae cover on it shrinks $20\%$ a day. In each case the amount gained or lost is different every period, because it is a percent of a different amount each time. What stays the same is the number the amount is multiplied by. percent-change-multiplier turns one percent change into one multiplication. When the same change repeats period after period, that multiplier describes the whole process, and it gets a name: the growth factor or the decay factor. With it, the value after three years, the change over a week, or the rate per month each takes one calculation. ## Definition A quantity has a constant **factor** $b$ per period when, at the end of every period, the amount is $b$ times what it was at the start of that period. - If $b > 1$, the quantity grows, and $b$ is a **growth factor**. - If $0 < b < 1$, the quantity shrinks, and $b$ is a **decay factor**. - If $b = 1$, the quantity stays the same. A **percent rate** $r$ per period, written as a decimal, gives the factor: $$b = 1 + r \ \text{(growth)} \qquad b = 1 - r \ \text{(decay)}$$ So $4\%$ growth a year is a factor of $1.04$ a year, and $18\%$ decay a year is a factor of $0.82$ a year. Read the other way, a factor of $0.88$ is a $12\%$ decay, because $0.88 = 1 - 0.12$, and a factor of $1.035$ is a $3.5\%$ growth, because $1.035 = 1 + 0.035$. ### [Horizontal asymptotes](https://math.wiki/horizontal-asymptotes/) **Summary:** A horizontal line the graph settles toward far out to the left and to the right. **Question:** Far out to the left and to the right, a fraction made of two polynomials keeps changing. Where is it heading, and can you tell without plotting a single point? ## Why this exists In asymptote-concept the graphs settled toward a line because a gap was multiplied by the same number between $0$ and $1$ at every step. A fraction of two polynomials, such as $$f(x) = \frac{6x^2 - x + 2}{3x^2 + 4}$$ has no such step. Both the top and the bottom grow without bound as $x$ moves away from $0$, so the fraction is a race between two huge numbers, and it is not obvious what their ratio does. A table of values can suggest an answer but cannot settle it, and plotting a few points far out is slow. This lesson shows that the answer is decided by one small piece of each polynomial: its leading term. Comparing the degrees of the top and the bottom says at once whether the graph levels off, and if it does, at what height. It also explains something a picture can hide: unlike the graphs in asymptote-concept, these graphs often cross the line they are settling toward. ## Definition A **rational function** is a function of the form $$f(x) = \frac{p(x)}{q(x)}$$ where $p$ and $q$ are polynomials and $q$ is not the zero polynomial (see rational-functions). Its domain is every input where $q(x) \neq 0$. The horizontal line $y = L$ is a **horizontal asymptote** of a function $f$ if the outputs $f(x)$ get as close to $L$ as you like when $x$ is taken far enough out on at least one side. In symbols, $$f(x) \to L \text{ as } x \to \infty \quad \text{or} \quad f(x) \to L \text{ as } x \to -\infty$$ ### [Increasing, decreasing, constant, positive and negative intervals](https://math.wiki/increasing-and-decreasing-intervals/) **Summary:** The stretches of input where a graph rises as you read left to right, and the stretches where it falls. **Question:** A graph climbs in some places and falls in others, and it sits above the axis in some places and below it in others. How do you say exactly where? ## Why this exists A graph tells a story (qualitative-graph-reading): the river rose, held steady, then fell. To use that story you need to say exactly when. "It rose for a while" cannot be checked; "it rose from day $0$ to day $3$" can. This lesson names four kinds of stretch, where the outputs go up, go down, stay put, and where they sit above or below zero, and states each one as an interval of inputs. ## Definition Let $f$ be a function and $I$ an interval of inputs inside its domain. - $f$ is **increasing** on $I$ if for any two inputs $a < b$ in $I$, $f(a) < f(b)$. Read from left to right, the graph rises. - $f$ is **decreasing** on $I$ if for any two inputs $a < b$ in $I$, $f(a) > f(b)$. The graph falls. - $f$ is **constant** on $I$ if every input in $I$ gives the same output. The graph is a flat, horizontal piece. - $f$ is **positive** on $I$ if $f(x) > 0$ for every $x$ in $I$: the graph is above the $x$-axis. It is **negative** on $I$ if $f(x) < 0$ for every $x$ in $I$: below the axis. Each interval is a set of **inputs**, $x$-values, written in interval-notation. The outputs say how high the graph is; the interval says where along the $x$-axis the behavior happens. **The endpoint convention.** This lesson writes rising, falling and level stretches with parentheses at every end: at a turning point, at a corner where a flat piece starts, and at an end of the domain. That is a choice, not something the definition forces: a parabola opening down is also increasing up to and including its top, since putting the top in breaks no comparison. Leaving the ends out keeps each stretch clear of the single points where the graph changes what it is doing. Positive and negative are different: a sign belongs to one input at a time, so the endpoint symbols follow the actual outputs. A zero is never included, since $0$ is neither positive nor negative, but an end of the domain with a nonzero output is included with a square bracket. ### [Independent and dependent variables](https://math.wiki/independent-and-dependent-variables/) **Summary:** The input variable is chosen freely and the output variable depends on what was chosen. **Question:** When two amounts change together, which one do you get to choose, and how do you write the other in terms of it? ## Why this exists Abeni runs a dog walking business: the more dogs she walks, the more she earns. Valen drops an ice cube into a glass: the longer it sits, the less ice is left. Each has two amounts that change together, but they do not play the same part. Abeni decides how many dogs to take, and her earnings follow. Nobody decides how much ice is left; it follows from the minutes that have passed. One amount is set first and the other is worked out from it. Saying which is which lets you write the relationship down: one short equation that turns any number of dogs into earnings, a table built from it, and a picture on a grid. The machine picture is the idea underneath: one amount goes in, the other comes out. ## Definition When two quantities change together and one is worked out from the other: - the **independent variable** is the quantity whose value is chosen, or which runs on by itself, like time. Nothing in the situation works it out; - the **dependent variable** is the quantity whose value is worked out from the independent variable. Its value **depends on** the value chosen. A quick test is the sentence "the ___ depends on the ___", with the dependent variable in the first blank. "The earnings depend on the number of dogs" makes sense; "the number of dogs depends on the earnings" describes nothing Abeni does. Each variable gets a letter, as in what a letter stands for, and a unit. For Abeni, at $6$ dollars a dog, let $n$ be the number of dogs and $e$ her earnings in dollars: ### [Intercepts and zeros of a function](https://math.wiki/x-intercepts-of-a-function/) **Summary:** The inputs where a graph meets the horizontal axis, the places where the output is zero. **Question:** Where does a graph meet the two axes, and what do those meeting points say about the function? ## Why this exists roots-zeros-and-x-intercepts gave three names to one set of numbers (root, zero, x-intercept) and found them for quadratics in factored form. Nothing about those names belongs to quadratics. Every function has a graph, and the two places a graph can meet the axes carry meaning in almost any setting. Where the graph meets the horizontal axis the output is $0$: a lift is level with the ground floor, a phone has used up its data. Where it meets the vertical axis the input is $0$: the clock has just started, so the output there is the starting amount. This lesson finds both kinds of point for any function, from a graph, from a table and from a rule. ## Definition Let $f$ be a function. - An **x-intercept** of the graph of $f$ is a point where the graph meets the horizontal axis. Every point on that axis has second coordinate $0$, so an x-intercept is a point $(k, 0)$ with $f(k) = 0$. - A **zero** of $f$ is an input $k$ with $f(k) = 0$. The zero is the number $k$; the x-intercept is the point $(k, 0)$. - The **y-intercept** of the graph of $f$ is the point where the graph meets the vertical axis. Every point on that axis has first coordinate $0$, so the y-intercept is $(0, f(0))$. It exists only when $0$ is in the domain of $f$. Finding $f(0)$ is evaluating the function at one number. Finding the zeros is solving the equation $f(x) = 0$, which may have no solution, one, or many. When the variables have other names, such as $t$ and $h$, the same points are called the **horizontal intercept** and the **vertical intercept**. ### [Interpreting the numbers in a model](https://math.wiki/interpret-model-parameters/) **Summary:** Saying what the starting value, the rate or the factor means in the units of the situation. **Question:** A model says a rowing club has $40(1.15)^t$ members $t$ years after it was founded. What do the $40$ and the $1.15$ tell you about the club? ## When you reach for it A model is a rule with numbers in it, and each number stands for something in the situation. A canoe hire shop charges $C = 8h + 12$ dollars for $h$ hours. A rowing club has $P = 40(1.15)^t$ members $t$ years after it was founded. Before a model can be used, each number in it has to be read back as a full sentence with units: "each hour of hire adds $8$ dollars", "the club was founded with $40$ members". For a line, linear-function-in-context reads the slope and intercept of one model, and interpret-the-slope-of-a-regression-line does the same for a line fitted to data. This lesson is mostly about exponential models, where the meaning is easier to miss: a factor of $0.97$ hides a loss of $3\%$, and a $4$ inside an exponent can be a length of time. It also sets the two kinds side by side. ## The procedure **Name the quantities first.** Every reading needs the input and the output with their units: $h$ in hours and $C$ in dollars, $t$ in years and $P$ in members. Without them a sentence can only say "it goes up by $8$", which is not yet a meaning. **The starting value comes from an input of $0$.** In $y = mx + b$, putting $x = 0$ leaves $y = b$. In $y = a \cdot b^t$, putting $t = 0$ leaves $a \cdot b^0 = a \cdot 1 = a$. So the number standing alone is the output when the input is $0$, in the output's units. Then ask whether an input of $0$ belongs to the situation at all. If it does not, the number still fixes the model, but it may describe nothing real (reasonable-domain-in-context). ### [Inverse from a table or list of pairs](https://math.wiki/inverse-from-a-table/) **Summary:** Swapping the input and output columns to get the inverse pairing. **Question:** A price list tells you what a parcel of a given weight costs. How do you read it backwards, from a price to the weight that was paid for? ## When you reach for it A courier's price list has no formula behind it, only five rows. Call the price of a parcel that weighs $w$ kilograms $P(w)$ dollars: | weight $w$ (kg) | $1$ | $2$ | $5$ | $10$ | $20$ | |:-:|:-:|:-:|:-:|:-:|:-:| | price $P(w)$ (dollars) | $6$ | $9$ | $14$ | $22$ | $35$ | Read from top to bottom, the table answers "what does this weight cost?": $P(5) = 14$. Very often the question comes the other way round. A receipt says $22$ dollars: how heavy was the parcel? With no formula there is nothing to solve. You find $22$ in the price row and read the weight that sits with it, $10$ kilograms. Doing that for every price builds a second function, one that takes a price and returns a weight. It is called the **inverse** of $P$ and written $P^{-1}$, read "$P$ inverse", so that $P^{-1}(22) = 10$. You reach for this move whenever a function is given as a table or a list of pairs and the question starts from an output. ## The procedure **What the inverse has to do.** The inverse sends each output back to the input it came from. For a function $f$ and its inverse $f^{-1}$, that is one statement read in two directions: $$f(a) = b \quad\Longleftrightarrow\quad f^{-1}(b) = a$$ Everything below follows from this line. **Turn every pair round.** A function given as a table is nothing more than its pairs: the parcel list is the five pairs $(1, 6)$, $(2, 9)$, $(5, 14)$, $(10, 22)$ and $(20, 35)$. The line above says that $(a, b)$ is a pair of $f$ exactly when $(b, a)$ is a pair of $f^{-1}$. So the inverse is found by turning every pair round, and in a table that means swapping the two rows (or the two columns): ### [Inverse functions](https://math.wiki/inverse-functions/) **Summary:** Swap input and output; the function that undoes another. **Question:** A function turns inputs into outputs. When can it be run backwards, and how do you find the function that does it? ## What this area is about An inverse function undoes another: if $f$ sends $a$ to $b$, its inverse $f^{-1}$ sends $b$ back to $a$. Not every function has one. It works only when no two inputs share an output, and this area covers that test, how to find an inverse from a table, a formula or a graph, and what to do when a function fails the test. ## How these fit together **When an inverse exists.** One to one functions and the horizontal line test is the condition: each output comes from a single input. **Procedures by form.** Inverse from a table swaps the input and output columns. Finding an inverse from a rule swaps the roles of $x$ and $y$ and solves. **A result.** The graph of an inverse is the original graph reflected across the line $y = x$. **A repair.** Restricting a domain keeps a one to one piece of a function, such as half a parabola, so that piece can be inverted. ## A path through it One to one functions first, then inverses from a table, then from a rule. After the rule come the graph of an inverse and restricting a domain, in either order. Next, the inverse of the exponential is the logarithm, drawn in graphs of logarithmic functions, and restricting the domain of a repeating function makes the inverse trigonometric functions possible. ### [Judging and comparing models](https://math.wiki/judge-a-models-fit/) **Summary:** Comparing predicted outputs with the real ones to see where the model starts to drift. **Question:** Two models both run close to the data. How do you tell which one to trust, and how far beyond the data you can trust it? ## When you reach for it You have data and more than one candidate rule for it: a line and an exponential, an exponential and a logistic curve. Differences and ratios (choose-a-function-family) may already have pointed at a family, and technology (fit-a-function-to-data-with-technology) will hand you a best fit inside any family you name. What neither does is tell you how well a finished model matches what was measured, where it starts to miss, or whether to believe it outside the data. That is this move. Put the models side by side on the same data, compare each prediction with the reading it was meant to match, and read the misses: their size, their signs, and how they change as the input grows. ## The procedure ### Predicted against actual For each data point $(x, y)$, a model predicts $\hat{y}$, its output at that $x$. The **residual** is what was measured minus what was predicted: $$r = y - \hat{y}$$ A positive residual means the model came in too low, a negative one too high (residuals, interpret-a-residual). Laying the readings, each model's predictions and the residuals in one table puts every model on the same footing: same inputs, same readings, same units. ### Reading the pattern Real readings carry noise: a gauge rounds, a ruler slips. A model that has the right shape leaves residuals that look like that noise: small, with signs mixed and no long runs, and no steady growth in size. A residual plot (residual against input) shows this at a glance. Two patterns mean something is wrong. ### [Laws of logarithms](https://math.wiki/product-law-of-logs/) **Summary:** The logarithm of a product is the sum of the logarithms, because powers with the same base add. **Question:** Multiplying powers of the same base means adding their exponents, and a logarithm is an exponent. So what is the logarithm of a product, a quotient or a power? ## Why this exists Multiplying is harder than adding. Adding two long numbers is a column of single-digit sums; multiplying them is a grid of products that then has to be added up as well. Raising a number to a high power is harder again. Before electronic calculators, people did long multiplications by looking up the logarithms of the two numbers in printed tables, **adding** them, and looking the sum up backwards. The slide rule does the same thing with lengths. Both rest on one fact: the logarithm of a product is the sum of the logarithms. That fact still matters now that calculators do the arithmetic. It is what lets a logarithm be broken into simpler pieces, what lets an unknown exponent be brought down to where it can be solved for, and what turns curving exponential data into a straight line. All of it comes from three laws, and each of the three is a law of exponents read through the definition of a logarithm. The lesson is taught over two days. The first takes the product and quotient laws, the second the power law and what it makes possible. ## Definition Let $b$ be an allowed base ($b > 0$ and $b \neq 1$), let $M$ and $N$ be positive numbers, and let $p$ be any real number. The **laws of logarithms** are these three. The **product law**: the logarithm of a product is the sum of the logarithms of the factors. $$\log_b(MN) = \log_b M + \log_b N$$ The **quotient law**: the logarithm of a quotient is the logarithm of the top minus the logarithm of the bottom. ### [Linear and nonlinear functions](https://math.wiki/linear-and-nonlinear-functions/) **Summary:** A function is linear when equal steps in the input give equal steps in the output; tell linear from nonlinear in tables, graphs and equations. **Question:** Some rules draw a straight line and some draw a curve. Can you tell which from the equation, the graph or the story alone, before plotting a single point? ## Why this exists Constant rate of change gave a test for a table: a function is linear exactly when equal steps in the input always give equal steps in the output, and then its rule is $y = mx + b$. Graphing a function from a table showed that many rules, such as $y = x^2$, draw curves instead. But a function rarely arrives as a table. It arrives as an equation, such as $5x + 2y = 12$, or as a picture, or as a sentence about a garden or a jar of coins. Making a table and plotting it every time would work, slowly. This lesson is about telling linear from nonlinear straight from the form the function comes in, and knowing why the quick look gives the right answer. The lesson has two parts. **Part 1** reads equations. **Part 2** reads graphs and situations, and builds a nonlinear function from a story. ## Definition A function is **linear** when its rule can be written $$y = mx + b$$ for two fixed numbers $m$ and $b$. The number $m$ may be negative, a fraction or $0$, and $b$ may be $0$. A function whose rule cannot be written this way, for any choice of $m$ and $b$, is **nonlinear**. Three descriptions of a linear function say the same thing, and each is the one to use for a different form of the function: - **the rule** can be written $y = mx + b$; - **the steps**: equal steps in the input always give equal steps in the output, the constant rate of change $m$; - **the graph** is a straight line that is not vertical. ### [Linear functions](https://math.wiki/linear-functions/) **Summary:** Constant rate of change, straight-line graph, and modeling with them. **Question:** Some quantities change by the same amount at every equal step. What does that one steady rate let you say about a function, however it reaches you? ## What this area is about A linear function changes its output by the same amount for every equal step in its input. That steady rate means two numbers settle the whole function: the rate of change $m$ and the initial value $b = f(0)$, so that $f(x) = mx + b$. Its graph is a straight line. This area is about finding those two numbers from a table, a graph or a situation, and saying what they mean. ## How these fit together **The defining test.** Constant rate of change and linear tables says when a table comes from a linear function: equal input steps always give equal output steps. Linear and nonlinear functions applies that test to equations, graphs and descriptions, before plotting anything. **Reading the two numbers.** Reading a linear function from its graph takes the initial value and the rate off a line and writes $f(x) = mx + b$. Linear functions in context builds the function from a description and names what $m$ and $b$ mean, with units. **Comparing.** Comparing linear functions given different ways reduces a table, a graph and an equation to the same two numbers so they can be set side by side. ## A path through it Constant rate of change first, then reading a function from its graph. After that, linear versus nonlinear, functions in context and comparing across representations, with comparing last. Finish here and you are ready for piecewise functions built from linear pieces, and for choosing a function family for data, where a line is the first family to try. ### [Linear functions in context](https://math.wiki/linear-function-in-context/) **Summary:** Naming what the starting value and the rate of change mean for the quantities in a situation. **Question:** A situation changes by the same amount at every step. How do you turn its description into a function, and what do the two numbers in that function say about the situation? ## When you reach for it A situation rarely arrives as a rule. It arrives as a price list, a pair of measurements, a table, a graph or a sentence. When one quantity changes by the same amount for every equal step in another, it is described by a linear function, $$f(x) = mx + b$$ and the whole description boils down to two numbers: the **starting value** $b = f(0)$, the output when the input is $0$, and the **rate of change** $m$, the change in output for each one unit of input. Building the function means finding those two numbers; using it means saying, in the words and units of the situation, what each number and each output stands for. Reading the two numbers off a picture is taught in linear-function-from-a-graph, and turning a story into a model and judging its fit in linear-modeling. This lesson works from any mix of descriptions, then asks what happens when a number changes. It takes two days. Day 1 (Examples 1 to 4) builds the function from words, from two input-output pairs, from a table and from a graph. Day 2 (Examples 5 and 6) changes the numbers, compares two functions, and reads values and limits. ## The procedure ### Day 1: building the function and naming its numbers **Which quantity is the input.** The output is the quantity that depends on the other. A cost depends on how many days a car is hired, so days are the input and cost the output. "$A$ as a function of $B$" makes $B$ the input, and a rate "per seat" or "per hour" names the input in its second half. A question that runs backwards, such as "how long until the tank is empty?", does not swap them: it becomes an equation to solve, $f(x) = 0$, as in function-notation-in-context. ### [Linear versus exponential growth](https://math.wiki/compare-rates-of-change-of-families/) **Summary:** Linear functions grow by equal differences over equal intervals, exponentials by equal factors, and an increasing exponential eventually passes any linear function. **Question:** One pay plan adds the same raise every day, another raises the pay by the same percent every day. Which one ends up paying more, and when? ## Why this exists A summer job offers two pay plans. Plan A pays $80$ dollars on day $0$, the first day, and adds a raise of $4$ dollars every day after. Plan B pays only $40$ dollars on day $0$, then raises the pay by $10\%$ every day. For the first week Plan A is far ahead. Is it the better plan? The question is not really about these numbers. It is about two kinds of change: adding the same amount each step, and multiplying by the same factor each step. linear-and-nonlinear-functions tells a line from a curve. This star puts the two most common kinds of change side by side, shows how to tell them apart from a table, and asks which one wins in the long run. ## Definition **Equal intervals** of input are steps of the same length $h$: from $x$ to $x + h$, from $x + h$ to $x + 2h$, and so on. A function **grows linearly** when, over equal intervals, it changes by **equal differences**: $f(x + h) - f(x)$ is the same number wherever the interval starts. Linear functions $f(x) = mx + c$ behave this way. A function **grows exponentially** when, over equal intervals, it changes by **equal factors**: $\dfrac{g(x + h)}{g(x)}$ is the same number wherever the interval starts. Exponential functions $g(x) = a \cdot b^x$, with $a \neq 0$, $b > 0$ and $b \neq 1$, behave this way, and $b$ is the factor for one unit (growth-factor-and-decay-factor). In a table with equally spaced inputs, subtracting each output from the next tests for equal differences, and dividing each output by the one before tests for equal factors. Differences of outputs divided by differences of inputs are the average rates of change already met; for a linear function they all equal the slope. ### [Linear, polynomial and exponential growth compared](https://math.wiki/exponential-versus-linear-growth/) **Summary:** Linear growth adds the same amount each step while exponential growth multiplies by the same factor. **Question:** A quantity that adds the same amount each step, one that grows like a square or a cube, and one that multiplies by the same factor each step. Wait long enough, and which one is biggest? ## Why this exists compare-rates-of-change-of-families raced a line against an exponential and found that an increasing exponential always gets past an increasing line in the end. A line is the slowest kind of growth there is, though. A quadratic such as $x^2$ also grows faster and faster: its steps get longer every time, just as an exponential's do. A cube grows faster still, and $x^{10}$ faster again. Is there a polynomial that keeps up with $2^x$, or even with $1.01^x$? The question matters whenever a model is chosen. Three different rules can fit the same few early measurements and then disagree wildly later on. Knowing which kind of growth wins in the long run, and why, says how far each model can be trusted. This star sets linear, polynomial and exponential growth side by side and proves the answer: an increasing exponential outgrows every polynomial, whatever its degree and however large its coefficients. ## Definition The three kinds of growth compared here are: - **linear growth**, $f(x) = mx + c$ with slope $m > 0$; - **polynomial growth**, a polynomial $p(x)$ of degree $k \geq 1$ with a positive leading coefficient, so $p(x)$ grows without bound as $x$ grows (degree-and-graph-shape); linear growth is the case $k = 1$; - **exponential growth**, $g(x) = a \cdot b^x$ with $a > 0$ and $b > 1$. A quantity **grows without bound** when it eventually passes every number you name: for every number $M$, the quantity is bigger than $M$ for every large enough $x$. ### [Logarithmic scales](https://math.wiki/logarithmic-scales/) **Summary:** Scales such as pH, decibels and earthquake magnitude, where each step up multiplies the quantity. **Question:** The quietest sound a person can hear and a sound loud enough to hurt differ in strength by a factor of about a million million. How can one scale with everyday-sized numbers hold both? ## Why this exists Some quantities in nature range over many powers of ten. The intensity of a sound you can just hear is about $10^{-12}$ watts per square meter, and a sound strong enough to hurt the ear is about $1$ watt per square meter, a trillion times stronger. The concentration of hydrogen ions in a liquid, which decides how acidic it is, runs from about $1$ mole per liter down to about $10^{-14}$. The shaking recorded from a small earthquake and from a huge one differ by a factor of millions. Measured on an ordinary scale, numbers like these are awkward in two ways. They need scientific notation just to be written down, and on an ordinary axis everything except the very largest values is squashed into a speck next to zero. What people usually want to know about such quantities is also not "how much more" but "how many times more". A scale that counts **factors** rather than amounts answers that question directly, and the tool that turns factors into counts is the common logarithm: it counts how many factors of ten a number holds. ## Definition A **logarithmic scale** measures a positive quantity by a constant multiple of the logarithm of how many times bigger it is than a fixed reference value. If the quantity is $x$, the reference value is $x_0 > 0$ and $c$ is a nonzero constant, the reading on the scale is $$S = c \log\left(\frac{x}{x_0}\right)$$ where $\log$ is the common logarithm, base $10$. The reference value fixes where the scale reads $0$ (when $x = x_0$, $S = c \log 1 = 0$), and the constant $c$ fixes the size and direction of one step. ### [Logarithms](https://math.wiki/logarithms/) **Summary:** The exponent you need: the inverse of the exponential. **Question:** When the unknown is an exponent, what number is it, and how do you calculate with numbers like that? ## What this area is about For a base $b > 0$ with $b \ne 1$ and a number $x > 0$, $\log_b x$ is the exponent $b$ must be raised to in order to give $x$. So $\log_b x = y$ says exactly what $b^y = x$ says, and $y = \log_b x$ undoes $y = b^x$. Logarithms turn multiplying into adding, which is why they solve equations with the unknown in an exponent and measure quantities that span many powers of ten. ## How these fit together **Definition and values.** Definition of a logarithm turns a logarithm into a power statement and back. Evaluating logarithms finds the value, exactly or with the $\log$ and $\ln$ keys, and change of base reaches any other base from those two. **The laws.** The laws of logarithms turn the logarithm of a product, quotient or power into a sum, difference or multiple. **Procedures.** Graphing logarithmic functions reflects the exponential's graph in the line $y = x$ and moves it. Expanding and condensing uses the laws together. Matching bases and taking a logarithm of both sides solve exponential equations. Solving logarithmic equations drops any answer that puts a number that is not positive inside a logarithm. **Applications.** Logarithmic scales such as pH and decibels, and semi-log plots, which straighten exponential data. ## A path through it The definition first, then evaluating and the laws. The laws open change of base, expanding and both ways of solving exponential equations; graphs come before solving logarithmic equations. Logarithmic scales lead into semi-log plots. ### [Logistic growth: when growth levels off](https://math.wiki/logistic-growth-models/) **Summary:** Growth that starts nearly exponential and slows toward a carrying capacity, its S-shaped graph and its horizontal asymptotes. **Question:** Two hundred elk are released into a valley that can feed about five thousand. The herd grows fast at first, but it cannot keep multiplying forever. What function grows like an exponential early on and then levels off? ## Why this exists An exponential model, such as one built with exponential-model-from-two-points, multiplies by the same factor every period, forever. That is a fine description of a small elk herd in an empty valley with plenty of grass. It cannot stay true. The valley holds only so much food, so as the herd grows, each year's growth must slow, and the count must settle near some ceiling. A model for this needs three features at once: nearly exponential growth while the count is small, slower growth as the count nears a ceiling, and a graph that flattens out at that ceiling instead of crossing it. The logistic model has all three. The same shape describes many other things that spread until they run out of room: a piece of gossip moving through a school, a new product reaching the people who might buy it. ## Definition A **logistic model** is a function of the form $$P(t) = \frac{C}{1 + Ae^{-kt}}$$ with constants $C > 0$, $A > 0$ and $k > 0$. Here $e$ is the base of the natural-exponential-function. - $C$ is the **carrying capacity**, the level the model levels off at. - $A$ is fixed by the starting value: $P(0) = \frac{C}{1 + A}$. - $k$ sets how fast the change happens. A larger $k$ makes a steeper S. The **halfway time** $t^{*}$ is the time when $P(t^{*}) = \frac{C}{2}$. The same family is often written with a base other than $e$, as $\frac{C}{1 + Ab^{t}}$ with $0 < b < 1$. Since $b^{t} = e^{t \ln b}$, this is the form above with $k = -\ln b$. ### [Maximum, minimum and range of a quadratic function](https://math.wiki/maximum-or-minimum-of-a-quadratic/) **Summary:** The vertex gives the largest output for a downward parabola and the smallest for an upward one. **Question:** A parabola turns around exactly once. What does that one turning point tell you about every output the function can give? ## When you reach for it A quadratic function $f(x) = ax^2 + bx + c$, with $a \ne 0$, has a parabola for its graph, and a parabola turns around exactly once, at its vertex. Reach for this lesson when a question about outputs is really about that point: the smallest or largest output and where it occurs, the range, where the graph falls and rises, the outputs from a limited set of inputs, and how fast the function changes when, unlike a line, its rate keeps changing. ## The procedure ### The vertex gives the least or greatest output Every quadratic can be written in vertex form $$f(x) = a(x - h)^2 + k$$ where $(h, k)$ is the vertex. Finding the vertex is its own lesson; the question here is what it says about outputs. The square $(x - h)^2$ is never negative, and it is $0$ only when $x = h$. Suppose $a > 0$. Then $a(x - h)^2 \ge 0$ for every $x$, so $$f(x) = a(x - h)^2 + k \ge k$$ with equality only at $x = h$. So $k$ is the **minimum value** of $f$: no input gives less, and only $x = h$ gives exactly $k$. If $a < 0$, a non-negative square times a negative number is never positive, so $a(x - h)^2 \le 0$, $f(x) \le k$, and $k$ is the **maximum value**. Keep two things apart. The **value** is an output: $k$, the vertex's second coordinate. **Where** it occurs is an input: $h$, the first coordinate. "The minimum is $-5$" and "the minimum occurs at $x = 2$" answer different questions. This minimum or maximum is **absolute**: least or greatest over every real input, not only nearby ones. A relative maximum or minimum only has to beat its neighbors, and a general function may have several. A quadratic has one turning point, so its relative extreme is also its absolute one. The other kind does not exist at all: an upward parabola has no maximum, because $a(x - h)^2$ grows past any number as $x$ moves far enough from $h$. ### [Modeling with functions](https://math.wiki/function-modeling/) **Summary:** Choosing a function family and fitting its numbers so that one rule describes real data. **Question:** Real data comes with no rule attached. How do you choose one, fit it, and know when to believe it? ## What this area is about A model is a function chosen to describe something real: a line, a quadratic, an exponential such as $40(1.15)^t$. Modeling is the whole cycle around that choice. You decide which family fits, write the rule from data or words, say what each number in it means, and judge where its predictions can be trusted. ## How these fit together **Building a model.** Choosing a function family for data uses differences, ratios and the shape of a scatter plot to pick a family. Building a model from a description turns a written account into a rule with named quantities and units. **Reading a model.** Interpreting the numbers in a model says what the starting value, the rate or the factor means in the situation. **Trusting a model.** Where a model holds separates safe predictions inside the data from risky ones outside it. Judging and comparing models compares predictions with real values to choose between candidates. ## A path through it Choose a function family first. Then build a model from a description and interpret its numbers. After interpreting comes where a model holds, and judging a model's fit last. From here, fitting with technology hands the arithmetic of a best fit to a calculator or spreadsheet. ### [Modeling with polynomial functions](https://math.wiki/polynomial-modeling/) **Summary:** Fitting a polynomial to data that rises and falls more than once, such as the volume of a folded box. **Question:** Your measurements change direction more than once, or turn once but lopsidedly, so no line or parabola passes through them. Which polynomial does, how do you find it, and for which inputs can you believe it? ## When you reach for it A line only rises or only falls. A parabola turns once, symmetrically. The depth of a river, measured from bank to bank, starts at $0$, reaches a deepest point that need not be in the middle, and returns to $0$. A channel with a sandbar gets deep, shallow, then deep again. When data turns more often than a parabola can, or turns once but off-center, try a polynomial of higher degree. Three questions come with every such model: which degree, which polynomial passes through the data, and for which inputs it means anything. Choosing a function family tested lines, quadratics and exponentials, and quadratic modeling is the same work one degree down. ## The procedure ### Which degree? **Count the turns.** A polynomial of degree $n$ has at most $n - 1$ turning points (degree-and-graph-shape), so data that turns twice needs degree at least $3$, and three turns need degree at least $4$. **Take differences on equally spaced inputs.** Subtract each output from the next, then do the same to those differences, and so on. Constant first differences mean a line, constant second differences a quadratic, constant third differences a cubic, and constant $n$th differences degree $n$. The reason is that one round of differences lowers the degree by one. For a term $ax^n$ and a step $h$, $$a(x + h)^n - ax^n = anh\,x^{n-1} + (\text{lower powers of } x)$$ because the $x^n$ terms cancel and the next term of $(x + h)^n$ is $nhx^{n-1}$. Lower terms of the polynomial only drop lower. So each round takes degree $n$ to $n - 1$ and multiplies the leading coefficient by $nh$, and after $n$ rounds a constant is left: ### [One to one functions and the horizontal line test](https://math.wiki/one-to-one-functions/) **Summary:** A function is one to one when no two different inputs share the same output. **Question:** A function turns each input into one output. When can you run it backwards, from an output to the single input that made it? ## Why this exists A function runs one way. You give it an input and it hands back exactly one output. Very often, though, the question arrives the other way round. A thermometer reads $77$ degrees Fahrenheit: what is that in Celsius? A box office took $84$ dollars for tickets at $12$ dollars each: how many tickets were sold? Each time you know the output and want the input. For the temperature this works. The rule $F = \frac{9}{5}C + 32$ turns $25$ degrees Celsius into $77$ degrees Fahrenheit, and no other Celsius reading gives $77$, so the answer is $25$ and only $25$. Squaring shows what can go wrong. If a squaring machine prints $9$, the input was $3$ or $-3$, and nothing in the $9$ says which. The machine has thrown information away, and no amount of cleverness gets it back. So before trying to run a function backwards, you need to know whether it ever sends two different inputs to the same output. The functions that never do are called one to one, and they are exactly the functions that can be undone by an inverse function. ## Definition A function $f$ is **one to one** when different inputs always give different outputs. In symbols, for any two inputs $a$ and $b$ in the domain of $f$: $$a \neq b \;\Rightarrow\; f(a) \neq f(b)$$ Turned round, the same statement says that equal outputs can only come from equal inputs. This is the form used in proofs: $$f(a) = f(b) \;\Rightarrow\; a = b$$ ### [Parent functions](https://math.wiki/parent-functions/) **Summary:** The simplest member of each family, the plain graph every other member is a moved or stretched copy of. **Question:** Most graphs you meet are a few basic shapes moved, stretched or flipped. Which basic shapes are they, and what does each look like? ## What this area is about Each family of functions has a plainest member, its parent: $y = x$ for lines, $y = x^2$ for parabolas, $y = \sqrt{x}$ for square root functions, and so on. Shifting, stretching or reflecting a parent's graph gives other members of its family, so knowing a handful of parent graphs by heart saves plotting each new function from scratch. For each parent, the things to know are its shape, its domain and range, its intercepts and its symmetry. ## How these fit together All three children are definitions, each pairing two parents. The identity and constant functions are $y = x$, which gives back its input, and $y = c$, which gives the same output for every input. The squaring and cubing functions are $y = x^2$, which loses the sign of its input, and $y = x^3$, which keeps it. The square root and reciprocal functions are $y = \sqrt{x}$, which accepts no negative input, and $y = \frac{1}{x}$, which does not accept $0$; both graphs show the gap. The absolute value parent, $y = |x|$, is the absolute value graph. ## A path through it The identity and constant functions first, then squaring and cubing, then the square root and reciprocal functions. Next: graphing by transforming a parent graph, and the families built on these parents, such as higher degree polynomials, square root functions and transformed reciprocal graphs. ### [Piecewise and absolute value functions](https://math.wiki/piecewise-and-absolute-value-functions/) **Summary:** Different rules on different stretches of the input. **Question:** A phone plan charges one rate up to a limit and another rate past it. How can a single function follow different rules on different stretches of its input? ## What this area is about Not every function is one formula. A piecewise function pairs several formulas with conditions on the input, such as $x < 0$ and $x \ge 0$. As long as the conditions do not overlap, each input in the domain meets exactly one formula and gets exactly one output. Two famous functions are built this way: the absolute value $|x|$, and the staircase that rounds every input down to an integer. ## How these fit together **The general tool.** Piecewise functions is an area of its own: evaluating, graphing, writing the formulas from a graph, and building one from a threshold in words. **Two special cases.** The absolute value graph writes $|x|$ as $x$ for $x \ge 0$ and $-x$ for $x < 0$, which draws a V with its corner at the origin. Step functions and the greatest integer function give a staircase that stays flat and jumps by $1$ at every integer, the shape of a charge for every started unit. **A procedure.** Transforming the absolute value graph shifts, flips and stretches the V into $y = a|x - h| + k$ (with $a \ne 0$), with its corner at $(h, k)$. ## A path through it Piecewise functions come first. Then the absolute value graph and step functions, in either order, and transforming the V after the absolute value graph (it also needs the general moves of graphing with transformations). Next, making a piecewise function continuous chooses a parameter so two pieces meet, once you have limits and continuity. ### [Piecewise functions](https://math.wiki/piecewise-functions/) **Summary:** Read which rule applies, then evaluate or graph it. **Question:** A function can use one formula for some inputs and another formula for the rest. How do you read one, draw one and build one? ## What this area is about A piecewise function is written as several formulas, each paired with a condition on the input, such as $x < 1$ or $x \ge 1$. The conditions do not overlap, so each input in the domain is handed to exactly one formula and gets exactly one output. Every skill in this area starts by finding the formula in charge. ## How these fit together **From formulas to values and pictures.** Evaluating a piecewise function picks the piece whose condition the input meets and uses only that formula. Graphing a piecewise function draws each piece only over its own inputs, and shows whether the graph joins up or jumps where the formula changes. **From a picture back to formulas.** Writing the formulas from a graph finds the equation of each piece and the interval of inputs it covers. **Applications.** Piecewise functions in context builds one from a threshold in words, such as overtime pay or shipping by weight band, and says what happens exactly at the threshold. ## A path through it The order is fixed: evaluate, then graph, then write the formulas from a graph, then context. Next: the absolute value graph and step functions, both piecewise functions in disguise, and later making a piecewise function continuous. ### [Piecewise functions in context](https://math.wiki/piecewise-functions-in-context/) **Summary:** Rules that change at a threshold, such as tax bands or shipping charges by weight. **Question:** A price, a wage or a tax follows one rule up to some amount and a different rule after it. How do you write that as one function, and what happens right at the change? ## When you reach for it Many everyday rules change at a set amount. A worker's hourly pay goes up after $40$ hours in a week. A courier charges by weight band. An ice rink stops charging once a month's sessions pass a cap. Each rule is written in words, and one formula cannot hold it, because the formula itself changes at the threshold. The tool is a piecewise function: one formula for each stretch of inputs, each paired with a condition. This lesson is about building one from a description, reading the threshold correctly, and saying what the function's behavior at the threshold means for the people paying or being paid. Naming the quantities and units follows function-notation-in-context. ## The procedure ### Finding the pieces Read the description for its **thresholds**, the amounts where the rule changes, and for what the rule says on each side. Two shapes come up again and again. **The rate changes for the part beyond the threshold.** Overtime pay and income tax work this way: the first $40$ hours are paid at one rate, and only the hours beyond $40$ at the higher rate. The second piece is then $$ \begin{aligned} &(\text{output at the threshold}) \\ &\quad + (\text{new rate}) \times (\text{input} - \text{threshold}). \end{aligned} $$ The first term is everything earned up to the threshold; $\text{input} - \text{threshold}$ counts only the amount past it, which is all the new rate applies to. ### [Polynomial functions](https://math.wiki/polynomial-functions/) **Summary:** Graphs of polynomials: their shape, their ends, and where they cross zero. **Question:** Without plotting a single point, how much of a polynomial's graph can you know from its formula? ## What this area is about A polynomial function, such as $f(x) = x^3 - 3x$, adds up constant multiples of whole-number powers of $x$. Its graph is one smooth, unbroken curve, and a few facts about the formula fix much of its shape. The leading term decides where the two ends go. A polynomial of degree $n \ge 1$ has at most $n$ real zeros and at most $n - 1$ turning points. The factors give the zeros and what the graph does at each one. ## How these fit together **Results about shape.** Degree, end behavior and shape reads the ends and the limits on zeros and turns from the degree. Zeros and multiplicity says that a zero of odd multiplicity crosses the axis and one of even multiplicity touches it and turns back. **A family up close.** Cubic, quartic and higher degree functions sorts the shapes those graphs can take. **Procedures.** Graphing polynomial functions puts zeros, multiplicity and end behavior into one sketch. Building a polynomial from its zeros runs the other way, from the zeros and one more point to the formula. **Modeling.** Modeling with polynomial functions fits data that turns more often than a parabola can, or turns once off-center, and asks for which inputs the model means anything. ## A path through it Degree and end behavior, and zeros and multiplicity, first, in either order. Then higher degree shapes and graphing; building from zeros needs only multiplicity. Modeling comes after graphing. ### [Quadratic functions](https://math.wiki/quadratic-functions/) **Summary:** The family whose graphs are parabolas, where the output depends on the square of the input. **Question:** A parabola turns around exactly once. What do that turning point and the intercepts say about a function's outputs, and about the situation it describes? ## What this area is about A quadratic function has the form $f(x) = ax^2 + bx + c$ with $a \ne 0$. Its graph is a parabola that opens up when $a > 0$ and down when $a < 0$. The algebra of parabolas, such as the vertex and axis of symmetry and factored form, comes earlier. This area reads those features as facts about a function: its outputs, its range, and what it says about a real situation. ## How these fit together **Choosing a form.** Reading a quadratic in its three forms shows which of standard, vertex and factored form answers a given question, and compares two quadratics given in different ways. **A result about outputs.** Maximum, minimum and range uses the vertex: it gives the smallest output when $a > 0$ and the largest when $a < 0$, and that fixes the range. **Applications.** Quadratic functions in context says what the intercepts, the vertex and the axis of symmetry mean, with units, for areas, profits and thrown objects. ## A path through it The three forms first, then maximum, minimum and range, then context, which needs the vertex as a highest or lowest point. A quadratic is also the degree $2$ case of the polynomial functions, whose shapes are treated in general in their own area. ### [Quadratic functions in context](https://math.wiki/quadratic-function-in-context/) **Summary:** Areas, profits and thrown objects, whose graphs rise to a peak and then fall. **Question:** Someone hands you a quadratic that describes something real. What does each part of its graph say about that thing? ## When you reach for it Someone gives you a quadratic that describes something real: a leaping dolphin, a firework, the area inside a loop of rope. It may arrive as a rule, a table or a graph. You already know how to find its features: the intercepts, the vertex and the axis of symmetry, and the vertex as a highest or lowest point. In a situation each feature has a meaning, with units. Reach for this skill when the question is "what does this number say about the situation?" rather than "what is this number?". Nothing here is solved by algebra. Every intercept is read from a graph, a table, or a rule given in factored or vertex form. Exact solving comes later, with factoring and the quadratic formula. ## The procedure One made-up model runs through this section. Let $t$ be the time in seconds since a dolphin began its run upward, and $h$ its height in meters **relative to the water surface**: positive above, negative below. The model is $$\begin{aligned} h(t) &= -5t^2 + 8t - 1.4\\ &= -5(t - 0.2)(t - 1.4) \end{aligned}$$ The two forms agree, since $(t - 0.2)(t - 1.4) = t^2 - 1.6t + 0.28$, and $-5$ times that is $-5t^2 + 8t - 1.4$. ### The sign of a: the shape of the story Here $a = -5 < 0$, so the parabola opens downward. In words: the height **rises, reaches one top, then falls**. With $a > 0$ the quantity would fall to one low point, then rise. The size of $a$ says how sharply the arch bends, not where its top is. ### [Radical functions](https://math.wiki/radical-functions/) **Summary:** The family whose rule takes a root of the input, with graphs that start at an edge and curve away. **Question:** A function that takes a root of its input may turn some inputs away. Where does its graph begin, how does it bend, and what does it describe? ## What this area is about A radical function puts its input under a root, as in $f(x) = \sqrt{x}$ or $g(x) = \sqrt[3]{x}$. With a square root, only inputs that make the expression under the root at least $0$ are allowed. For a rule such as $f(x) = a\sqrt{x - h} + k$ the graph is a half curve: it begins at an endpoint and runs off in one direction, flattening as it goes. A cube root accepts every real number, so its graph runs in both directions. ## How these fit together **Graphs.** Graphing square root functions finds the domain, the endpoint and the range of $f(x) = a\sqrt{x - h} + k$ with $a \ne 0$, and draws the half curve. Cube root functions and roots as powers graphs the cube root, which accepts every input, and reads $x^{1/3}$ as $\sqrt[3]{x}$. **Applications.** Radical functions in context reads and builds models such as wave speed $v(d) = \sqrt{9.8d}$, where four times the depth gives twice the speed, along with pendulums and braking distances. ## A path through it Square root graphs first, then context. Cube roots can come at any point, and read most easily after square roots. Next: restricting a domain to get an inverse, where $\sqrt{x}$ undoes $x^2$ only on the inputs $x \ge 0$. ### [Radical functions in context](https://math.wiki/radical-functions-in-context/) **Summary:** Pendulum periods, braking distances and other quantities that grow like a root. **Question:** Water four times as deep carries a wave only twice as fast. How do you read, use and build a rule whose output grows like the square root of its input? ## When you reach for it A wave rolling toward a beach slows down as the water gets shallower. For water that is shallow compared with the length of the wave, its speed in meters per second is $$v(d) = \sqrt{9.8d}$$ where $d$ is the depth of the water in meters. Rules like this one, with the input under a square root sign, describe many quantities: the time a pendulum takes to swing, the distance to the horizon from a height, the speed a car was going from the length of its braking marks, the side of a square from its area. You reach for this skill whenever such a rule describes a real situation and you need to say what it means, use it in both directions, or write it down in the first place. The reading comes from function-notation-in-context: $v(20) = 14$ is a sentence about water and waves, not only about numbers. The shape comes from graph-a-square-root-function: a half curve from an endpoint, steep at first and flattening. This lesson joins the two and adds what a situation brings to a square root rule: units, a domain set by the world rather than by the formula, the habit of running the rule backwards, and the one fact that makes square root quantities recognizable, that multiplying the input by $4$ only doubles the output. ## The procedure The running example is the wave rule $v(d) = \sqrt{9.8d}$, depth $d$ in meters, speed $v$ in meters per second. ### Read the rule: input, output and units ### [Rates at an instant from shrinking intervals](https://math.wiki/estimate-a-rate-at-an-instant/) **Summary:** Estimating the rate of change at a single point by computing average rates over smaller and smaller intervals around it. **Question:** A speedometer shows a speed at one instant, but speed is distance over time and an instant has no time in it. How can a rate belong to a single moment? ## Why this exists An average rate of change needs two inputs: change in output over change in input. A speedometer says how fast the car is going now, at one input, where there is no change at all and the fraction would be $\frac{0}{0}$. The way out is a whole family of intervals around the instant, narrower and narrower. Each has an ordinary average rate. If those averages settle on one number as the width shrinks, from both sides, that number is the rate at the instant. The difference quotient already showed averages crowding toward such a number; this lesson says what it is, how to estimate it well, how to read it on the graph, and when there is none. ## Definition Fix an input $a$. For a small width $h \ne 0$, the average rate of $f$ between $a$ and $a + h$ is the difference quotient $$\frac{f(a + h) - f(a)}{h}.$$ With $h > 0$ the interval $[a, a + h]$ lies to the right of $a$; with $h < 0$ the interval $[a + h, a]$ lies to the left. Call these the **right estimates** and the **left estimates**. The **rate of change of $f$ at $a$**, also called the **instantaneous rate**, is the number that the right estimates and the left estimates both approach as $h$ shrinks toward $0$. If the two sides approach different numbers, or either side approaches no number, $f$ has no rate of change at $a$. A third kind of interval sits evenly around $a$, from $a - h$ to $a + h$, with width $2h$. Its average rate is the **centered average**: ### [Rational functions](https://math.wiki/rational-functions/) **Summary:** One polynomial over another, and the gaps and asymptotes that creates. **Question:** Divide one polynomial by another and the graph breaks apart and runs off along lines. Where are the breaks, which lines does it follow, and what does it mean when it models something real? ## What this area is about A rational function is a quotient $f(x) = \frac{p(x)}{q(x)}$ of two polynomials, with $q$ not the zero polynomial. Its formula breaks only where $q(x) = 0$, and near those inputs, or far out to the left and right, its graph often draws closer and closer to a line, an asymptote. This area finds those places from the formula and puts them together into a graph. ## How these fit together **Where it lives.** Domain and zeros of a rational function removes every input that makes the denominator $0$ and finds the inputs whose output is $0$. **The lines it follows.** Asymptotes is an area of its own: vertical asymptotes at refused inputs that do not cancel away, and horizontal or slant asymptotes far out, all read from the formula. **Procedures.** Graphing rational functions combines holes, asymptotes and intercepts into one sketch. Transforming the reciprocal graph handles any rule of the form $y = \frac{a}{x - h} + k$ with $a \ne 0$, moving both asymptotes of $y = \frac{1}{x}$. **Applications.** Rational functions in context models average cost, concentration and shared work, where one quantity is divided by another that changes. ## A path through it Asymptotes come before graphing a rational function, and the domain comes before working in context. Transforming the reciprocal graph can be taken at any point once you have the parent graph and graphing with transformations. ### [Rational functions in context](https://math.wiki/rational-function-in-context/) **Summary:** Average cost, concentration and shared work, where one quantity is divided by another that changes. **Question:** One quantity is divided by another that keeps changing. How do you write that as a function, and what does the formula tell you about the situation? ## When you reach for it Many quantities in the world are one amount **per** another amount. The average cost of an item is the total cost divided by the number of items. The concentration of a mixture is the amount of what is dissolved divided by the volume of liquid. The time a shared job takes comes from adding rates, and a rate is a job divided by a time. When the amount on the bottom changes with the input, the quantity is a rational function of that input: one polynomial divided by another, and in these models usually one straight-line expression divided by another. Reach for this when a question asks what such a ratio is at some input, which input gives a chosen ratio, or what happens to the ratio as the input grows. Answering it uses three things you already have: function notation read in words, the domain of a rational function, and solving a rational equation. ## The procedure Take the lead model of this lesson. Theodric runs a pottery workshop. Setting up a batch of mugs (the kiln, the molds, the glaze) costs $2400$ dollars however many mugs are made, and each mug then costs $9$ dollars more in clay and glaze. ### Build the ratio from the two amounts Write the top and the bottom separately, each as a function of the input, before dividing. Let $x$ be the number of mugs. The total cost is the fixed $2400$ plus $9$ for each mug, $2400 + 9x$ dollars, a linear function of $x$ because the cost per mug is constant. The number of mugs is $x$. The average cost per mug is the total divided by the number: ### [Reading a linear function from its graph](https://math.wiki/linear-function-from-a-graph/) **Summary:** Taking the starting value and the steady change off the picture and writing the rule. **Question:** A graph shows a straight line. How do you turn the picture into a rule that gives the output for any input, including inputs the picture never shows? ## When you reach for it A function is often handed to you as a picture: a line on a grid, with inputs along the bottom and outputs up the side. The picture shows the shape at a glance, but it cannot tell you the output for an input beyond its edge, and it gives exact values only where the line crosses a grid corner. A rule does both. When the graph is a straight line, the function has a constant rate of change, and two numbers settle everything: the **initial value**, $f(0)$, the output when the input is $0$; and the **rate of change**, the amount the output changes for each one unit of input. With those two numbers the rule is $$f(x) = mx + b$$ where $m$ is the rate of change and $b = f(0)$ is the initial value, written in function notation. Measuring a slope on a grid is in slope-from-a-graph, and reading $m$ and $b$ in $y = mx + b$ is in read-slope-and-intercept. This lesson reads the line as a function: what its two numbers are worth, in which units, and what to do when the picture does not show input $0$ at all. It takes two days: Part 1 (Examples 1 to 3) reads the two numbers, on any scale and in any window; Part 2 (Examples 4 and 5) handles graphs that start later than input $0$ and graphs that fall. ## The procedure ### Part 1: the two numbers and the rule **The initial value.** Input $0$ sits on the vertical axis, because every point of that axis has first coordinate $0$. So when the horizontal axis starts at $0$, the initial value is the height at which the line meets the vertical axis: the point $(0, f(0))$. ### [Reading a quadratic in its three forms](https://math.wiki/quadratic-function-representations/) **Summary:** Standard, vertex and factored form each show a different feature of the very same parabola. **Question:** The same parabola can be written three ways. Which way answers the question you were actually asked? ## When you reach for it Most questions about a quadratic function ask about one feature of its graph. How high or low does it go? Where does it cross the $x$-axis? Where does it start, at $x = 0$? A quadratic can be written in three forms, and each one puts a different feature in plain sight while hiding the others. Reach for this skill when you have a quadratic in one form and a question that another form answers, and when you must compare two quadratic functions that arrive in different dress: one as an equation, the other as a graph, a table or a sentence. The conversions themselves are taught elsewhere: converting to vertex form and factored form. This lesson is about choosing which one to do. ## The procedure ### What each form shows | Form | Written | Shows at once | |---|---|---| | standard | $ax^2 + bx + c$ | $y$-intercept $(0, c)$ | | vertex | $a(x - h)^2 + k$ | vertex $(h, k)$, axis $x = h$, greatest or least value $k$ | | factored | $a(x - p)(x - q)$ | zeros $p$ and $q$, axis $x = \frac{p + q}{2}$ | Each feature sits where a piece vanishes. At $x = 0$ every term with $x$ in it is $0$. The square in vertex form is never negative and is $0$ only at $x = h$. A product is $0$ exactly when one of its factors is. In every form, $a$ is the same number, because each form multiplies out to a single $x^2$ term, $ax^2$. Its sign gives the direction: $a > 0$ opens up, so the vertex is a least value, and $a < 0$ opens down, so the vertex is a greatest value. ### [Reading values off a graph](https://math.wiki/read-a-value-from-a-graph/) **Summary:** Finding the output for a given input, and the inputs for a given output, from the picture alone. **Question:** You have the graph of a function but not its rule. How do you get the output for a given input, or every input that gives a certain output, from the picture alone? ## When you reach for it Often a graph is all you have: a printout from a heart monitor, a chart of prices, a curve in a report. There is no rule to put numbers into, yet the graph holds every pair of input and output the function has. This lesson reads those pairs off the picture. Two questions come up, and they go in opposite directions. - **Given an input, what is the output?** In function notation: find $f(3)$. - **Given an output, which inputs give it?** In function notation: find every $x$ with $f(x) = 2$. The first always has at most one answer. The second can have none, one, two or many, and finding all of them is most of the work. When there is a rule, putting the number into the rule gives an exact output. Reading a graph is for when the picture is what you have, or when you want to see every answer at once. Saying in words what a graph's shape tells you happened is qualitative-graph-reading; here we read numbers. ## The procedure ### What a point on the graph says The graph of a function $f$ is the set of all points $(x, y)$ with $y = f(x)$: each input paired with its own output. So the two statements $$f(a) = b \qquad \text{and} \qquad (a, b) \text{ is on the graph of } f$$ say exactly the same thing, one in function-notation and one as a picture. Every reading in this lesson translates one into the other. The point $(4, 3)$ on the graph of $f$ is the statement $f(4) = 3$; the statement $f(-2) = 0$ is the point $(-2, 0)$. ### [Reflecting a graph across the axes](https://math.wiki/reflection-across-the-x-axis/) **Summary:** Taking the opposite of the whole output turns the graph upside down. **Question:** How do you turn a graph upside down, or make it face the other way, by changing only its rule? ## When you reach for it You already have two ways to move a graph without changing its shape: add a number to the output and the graph slides up or down, replace $x$ by $x - h$ and it slides sideways (shifting a graph). Both keep the graph the right way up. Sometimes the graph you need has the right shape but is upside down, or faces the other way: a parabola that opens downward, a square root curve that runs to the left. That is a reflection, and a single minus sign in the rule makes it. Where the minus sign goes decides which way the graph flips. You also reach for it in reverse, when a graph in front of you looks like a parent graph turned over and you want to name what was done to it. ## The procedure ### Flipping the outputs: $-f(x)$ Start with a function $f$ and build $g(x) = -f(x)$. For each input, $g$ does whatever $f$ does and then takes the opposite of the answer. The inputs are untouched and every output changes sign, so if $(a, b)$ is on the graph of $f$, then $(a, -b)$ is on the graph of $g$: $$(a, b) \;\longmapsto\; (a, -b)$$ A point above the horizontal axis lands the same distance below it, at the same $x$, and a point below lands above. That is the reflection of a point across the $x$-axis, done to every point of the graph at once. So the graph of $y = -f(x)$ is the graph of $f$ **reflected across the $x$-axis**, with the axis as the mirror. Two things never change. A point on the axis has $b = 0$, and $-0 = 0$, so the zeros of $f$ stay where they are. And the inputs are the same, so the domain is the same. The range is what flips: every output is replaced by its opposite. ### [Relations and functions](https://math.wiki/relation-concept/) **Summary:** A relation is any pairing of inputs with outputs, with no rule about how many outputs one input may have. **Question:** Pairs link one thing to another, a library card to its holder, a house to its ZIP code. When does knowing the first thing settle the second? ## Why this exists A great deal of information comes in pairs. A library lists each card number beside the person who holds the card. A postal service gives every house a ZIP code. A school records the class each student is in. Hand a librarian a card number and there is one name on file: the card settles who holds it. Tell a mail carrier a ZIP code and there may be many houses with it: the ZIP code does not settle the house. Turning a pairing around can change the answer too: a person who holds two cards has no single card number. Mathematics needs a word for any pairing at all, and a sharper word for the pairings in which the first thing settles the second. These are **relation** and **function**. The input-output machine pictured a function as a box with a rule inside. This lesson looks at a function from the outside, as nothing more than its pairs, which works even when no rule can be written down. ## Definition An **ordered pair** $(x, y)$ is two things written in a fixed order. In a relation the first entry is the **input** and the second the **output**. On the coordinate plane the same kind of pair names a point, $x$ across and $y$ up. A **relation** is any set of ordered pairs. There is no rule about which pairs may appear: an input may have one output or several, and several inputs may share an output. A relation can be written as a list in braces, in roster notation: $$\{(1, 4),\ (2, 5),\ (2, 7)\}$$ ### [Restricting a domain to get an inverse](https://math.wiki/restrict-a-domain-to-get-an-inverse/) **Summary:** Cutting a function down to a one to one piece so that an inverse can exist. **Question:** A function sends two different inputs to the same output, so it cannot be run backwards. Can you keep part of it and run that part backwards? ## When you reach for it Take the function $$f(x) = (x - 5)^2 - 3$$ It sends $3$ to $(-2)^2 - 3 = 1$ and it sends $7$ to $2^2 - 3 = 1$. Asked "which input gives the output $1$?", there are two honest answers, so $f$ is not one to one and has no inverse function. The algebra of finding an inverse says the same thing in its own way: solving $y = (x - 5)^2 - 3$ for $x$ produces a $\pm$, two inputs for one output. Very often only one of the two answers is wanted. A ball passes the same height on the way up and on the way down, and the question is about the way down. A price above the best price and a price below it give the same profit, and the shop only considers raising its price. In each case you throw away the part of the domain you do not need, keep a piece on which the function is one to one, and invert that piece. The piece is a new function with the same rule and a smaller domain, and it has an inverse even though the original does not. You reach for this move whenever a graph fails the horizontal line test, whenever solving for the input throws up a $\pm$, and whenever a question only makes sense on part of the domain. It is also how the square root came to exist: $\sqrt{x}$ is the inverse of squaring, once squaring is cut down to the numbers that are zero or more. ## The procedure ### Find where the function repeats itself Before cutting, find out how the outputs repeat. For $f(x) = (x - 5)^2 - 3$, step a distance $d$ to either side of $5$: ### [Semi-log plots: straightening exponential data with logarithms](https://math.wiki/semi-log-plots/) **Summary:** Taking the logarithm of the output turns exponential data into a line, and logs of both variables turn a power law into one. **Question:** Exponential data curves upward more and more steeply. Is there a way to look at it so that it lies along a straight line, with its factor showing up as a slope? ## When you reach for it A table of values climbs faster and faster. Is it exponential, and if so, what are its starting value and factor? exponential-model-from-two-points builds a model from two readings, but a real table has many readings, and two of them cannot tell you whether the rest agree. A graph can, but the eye is poor at telling one upward curve from another. It is very good at telling a straight line from a bend. Logarithms turn the question into one about a line. They change multiplying into adding, so a quantity multiplied by the same factor at every step has a logarithm that goes up by the same amount at every step. Plot the logarithm of the output and exponential data lies on a line; take logarithms of both variables and a power law $y = ax^p$ lies on one too. In this lesson $\log$ is the common logarithm, base $10$. ## The procedure **Why the logarithm straightens an exponential.** Suppose $y = a \cdot b^x$ with $a > 0$ and $b > 0$. The logarithm of a product is the sum of the logarithms, and the logarithm of a power brings the exponent down (product-law-of-logs): $$\log y = \log a + x \log b$$ Call $Y = \log y$. Then $Y = (\log b)\,x + \log a$, a linear function of $x$ with slope $\log b$ and intercept $\log a$. In words: each step of $1$ in $x$ multiplies $y$ by $b$, so it adds $\log b$ to $\log y$. Equal ratios in $y$ become equal differences in $\log y$. **Plot the logarithm.** Make a second table of the points $(x, \log y)$ and plot it on ordinary axes. Or plot $y$ itself on **semi-log paper**, where the horizontal axis is ordinary and the vertical axis is a logarithmic scale: the value $y$ is placed at height $\log y$, so the two pictures are the same. ### [Shifting a graph up, down, left and right](https://math.wiki/vertical-shift/) **Summary:** Adding a number to the whole output slides the graph up or down by that amount. **Question:** If you already know the graph of a function, can you get the graph of the same rule with a number added, outside or inside, without plotting it all again? ## When you reach for it You know the parent graphs: $y = x^2$ and $y = x^3$, $y = \sqrt{x}$ and $y = \frac{1}{x}$. A great many functions are one of these with a number added somewhere: $x^2 + 4$, $\sqrt{x - 3}$, $(x + 1)^3$. Each of their graphs is the parent graph slid, with its shape unchanged, to a new place. A slide like this is called a **translation**. Reach for this skill when a rule is a parent with a constant added to its output or to its input and you want the graph without plotting from scratch, or when a graph has a parent's exact shape in a new position and you want its rule. You have met the idea in geometry as translating a figure and for parabolas in vertex form. Here it works for every function at once. ## The procedure ### Adding to the output: a vertical shift Take a function $f$ and a constant $k$, and define $$g(x) = f(x) + k$$ For each input, $g$ does exactly what $f$ does and then adds $k$. So at every $x$ the graph of $g$ is $k$ higher than the graph of $f$: a point $(a, b)$ on the graph of $f$ becomes the point $(a, b + k)$ on the graph of $g$. A positive $k$ moves the graph up, a negative $k$ moves it down, and since the input is untouched, nothing moves sideways. ### Changing the input: a horizontal shift Now take a constant $h$ and define $$g(x) = f(x - h)$$ This is $f$ evaluated at the expression $x - h$: $g$ first subtracts $h$ from its input and only then hands the result to $f$. Suppose $(a, b)$ is on the graph of $f$, so $f(a) = b$. Which input makes $g$ give the same output $b$? $g$ must hand $f$ the number $a$, so its own input has to be $a + h$: ### [Slant asymptotes](https://math.wiki/slant-asymptotes/) **Summary:** When the numerator is one degree higher than the denominator, the graph follows a slanted line far out. **Question:** When the top of a fraction outgrows the bottom by one power of x, the graph never levels off. Does it still settle down to something simple far out? ## Why this exists Horizontal asymptotes settle the far-out behavior of a rational function when the numerator's degree is at most the denominator's. When the numerator has the higher degree, the outputs grow without bound and there is no horizontal line to settle toward. Yet something simple remains. Take $$f(x) = \frac{3x^2 - 2x + 4}{x - 1}.$$ At $x = 101$ it gives $f(101) = 304.05$, and at $x = 1001$ it gives $3004.005$. Those are almost exactly $3x + 1$, which gives $304$ and $3004$. The graph keeps climbing, but it climbs along a straight line, and the line can be found exactly. A slanted line is the next simplest thing after a horizontal one, and naming it finishes the picture of the asymptotes a rational function can have. ## Definition A line $y = mx + b$ with $m \neq 0$ is a **slant asymptote** (also called an oblique asymptote) of the graph of $f$ when the vertical gap between the graph and the line shrinks toward $0$ far out: $$\begin{aligned} &f(x) - (mx + b) \to 0 \\ &\text{as } x \to \infty \text{ or as } x \to -\infty. \end{aligned}$$ For a rational function $f(x) = \frac{P(x)}{Q(x)}$ this happens exactly when the degree of $P$ is one more than the degree of $Q$ and $Q$ does not divide $P$ exactly. Long division then writes $$f(x) = \underbrace{mx + b}_{\text{quotient}} + \frac{R(x)}{Q(x)},$$ where the remainder $R \neq 0$ has lower degree than $Q$, and the quotient line $y = mx + b$ is the slant asymptote. ### [Solving by taking a logarithm of both sides](https://math.wiki/take-a-log-of-both-sides/) **Summary:** Applying a logarithm to both sides brings an unknown exponent down where it can be solved for. **Question:** The unknown sits up in the exponent, and the two sides are not powers of one base. How do you bring it down to where it can be solved for? ## When you reach for it Some equations with the unknown in the exponent give way to a trick: write both sides as powers of one base and set the exponents equal, which is solving by matching bases. That trick needs luck. Look at $$7^x = 40$$ There is no whole-number power of $7$ that makes $40$, since $7^1 = 7$ and $7^2 = 49$, and $40$ is not a power of $7$ in any tidy way. The answer is some number between $1$ and $2$, nearer $2$, and trying decimals by hand would only ever close in on it. The way out is the power law of logarithms, $\log_b\left(M^p\right) = p\log_b M$. It takes an exponent and turns it into a factor. So if you take the logarithm of both sides of an equation, an unknown that was up in the exponent comes down to the main line, where ordinary algebra can reach it. Reach for it whenever the unknown is in an exponent and the bases do not match: to find when a balance reaches a target, when a population passes a size, when a warm loaf cools to a temperature, or how long a radioactive sample takes to lose half its mass. In the lessons on compounding-periods and doubling-time-and-half-life questions like these were answered by trying whole numbers of periods one after another. With logarithms they are answered exactly. ## The procedure ### Why you may take a logarithm of both sides If two numbers are equal, then doing the same thing to both gives equal results. That is all an equation step ever is. A logarithm is a function: each positive input has exactly one output. So if $M = N$ and both are positive, then ### [Solving exponential equations by matching bases](https://math.wiki/solve-when-bases-match/) **Summary:** If both sides are powers of the same base, then the two exponents must be equal. **Question:** The unknown is up in the exponent, where no adding or dividing can reach it. When can you bring it down without a calculator? ## When you reach for it An **exponential equation** has the unknown in an exponent, as in $$4^{x + 1} = 8^{x - 2}$$ Nothing you know from linear equations reaches up there. Subtracting $1$ or dividing by $4$ does not touch the $x$, because the $x$ is not a term or a factor; it is a count of how many times $4$ is multiplied. There is one situation where the exponent can be brought down at once, with no calculator: when both sides are powers of the **same base**. If $2^{\text{something}} = 2^{\text{something else}}$, the two somethings must be equal, and that is an ordinary equation. The skill is mostly the rewriting that gets you there: seeing that $4$ is $2^2$ and $8$ is $2^3$, that $\frac{1}{27}$ is $3^{-3}$, that $\sqrt{5}$ is $5^{1/2}$. Reach for it when the numbers in the equation are all powers of one small number: $2$, $4$, $8$, $16$, $32$, $\frac{1}{2}$; or $3$, $9$, $27$, $81$, $\frac{1}{3}$; or $5$, $25$, $125$. When they are not (for example $2^x = 5$, since $5$ is no whole or fractional power of $2$ you can write down), the exponent is a logarithm, and the move is taking a logarithm of both sides. ## The procedure ### The one fact the method rests on For a base $b$ with $b > 0$ and $b \neq 1$, $$b^S = b^T \quad\text{exactly when}\quad S = T$$ One direction is plain: if $S = T$ then $b^S$ and $b^T$ are the same number. The useful direction is the other one, that two equal powers of the same base must have equal exponents. There are two ways to see it. ### [Solving f(x) = g(x) with graphs and tables](https://math.wiki/solve-f-equals-g-graphically/) **Summary:** The x-coordinates where two graphs meet are the solutions of f(x) = g(x), found from a graph, a table or successive approximation. **Question:** Two rules each turn an input into an output. At which inputs do they give the same output, even when no algebra you know can find them? ## When you reach for it An equation $f(x) = g(x)$ asks for every input at which two functions give the same output. Often algebra cannot answer that yet: in $2^x = x + 3$ the variable sits in an exponent on one side and in a plain sum on the other, and no rearranging gets $x$ on its own. Graphs and tables answer it anyway, exactly when the solutions are tidy numbers and to any accuracy you like when they are not. Reach for it when the two sides are different kinds of expression, when you want to count the solutions before hunting for them, or when one quantity catches up with another. It extends solving a system by graphing from lines to curves, and asks only for the inputs. ## The procedure ### Why the meeting points hold the answer The graph of $f$ is the set of points $(a, f(a))$ and the graph of $g$ is the set of points $(a, g(a))$. If $a$ solves $f(x) = g(x)$, those two points are the same point, so the graphs meet above $x = a$. Conversely, if the graphs meet at $(a, b)$, the point is on the graph of $f$, so $f(a) = b$, and on the graph of $g$, so $g(a) = b$; then $f(a) = g(a)$ and $a$ is a solution. **The solutions of $f(x) = g(x)$ are exactly the $x$-coordinates of the points where the graphs of $y = f(x)$ and $y = g(x)$ meet.** The $y$-coordinate of a meeting point is the output the two functions share there. It is not a solution: the equation asks for inputs. For $x^2 = x + 2$, the graphs of $y = x^2$ and $y = x + 2$ meet at $(-1, 1)$ and $(2, 4)$, so the solutions are $x = -1$ and $x = 2$. Substituting confirms them: $(-1)^2 = 1 = -1 + 2$ and $2^2 = 4 = 2 + 2$. ### [Solving logarithmic equations](https://math.wiki/extraneous-log-solutions/) **Summary:** An answer must be dropped if it makes any logarithm in the original equation take a value that is not positive. **Question:** The unknown sits inside a logarithm. How do you get it out, and why can the algebra hand back an answer that does not work? ## When you reach for it A **logarithmic equation** is an equation with the unknown inside a logarithm, such as $$\log_2 x + \log_2(x - 6) = 4$$ You reach for the moves in this lesson whenever the unknown is in the argument of a logarithm, the number the logarithm is taken of. (When the unknown is in an exponent instead, as in $5^x = 30$, the tools are matching the bases or taking a logarithm of both sides.) Two facts you already have do the solving. A logarithm statement is a power statement in disguise, so $\log_b M = c$ can be rewritten as $b^c = M$. And the laws of logarithms turn a sum or difference of logarithms into a single one. What is new is a third fact, and it is the reason this lesson exists: a logarithm only accepts positive arguments, and the laws quietly forget that. Rewriting an equation with the laws can make room for values the original equation never allowed. The algebra then finds them along with the real solutions, and each one has to be caught and dropped. Such a value is called an **extraneous solution**: it solves a rewritten equation but not the one you were given. ## The procedure ### Where the equation lives $\log_b M$ is defined only when $M > 0$, for a base $b > 0$ with $b \ne 1$. So before solving, read off the **domain of the equation**: the values of $x$ that make every argument in it positive. For the equation above, $$x > 0 \quad\text{and}\quad x - 6 > 0$$ ### [Step functions and the greatest integer function](https://math.wiki/greatest-integer-function/) **Summary:** The function that rounds every input down to the nearest whole number. **Question:** How do you write and draw a rule that stays flat for a while and then jumps, such as a charge for every started half hour? ## Why this exists Some quantities do not change smoothly. Your age in whole years stays the same for a full year and then jumps by $1$ on your birthday. An e-scooter scheme that charges for every started half hour charges the same for a $31$-minute ride as for a $59$-minute one, and more for a $61$-minute one. The graph of either stays flat, leaps up, and stays flat again. No straight line does that. A piecewise function can, with one flat piece per stretch, but a rule that steps at every whole number would need endlessly many pieces. This lesson gives that whole staircase one name, the **greatest integer function**, and uses it to write any rule that charges by the started unit. The staircases in plot-a-function-from-a-table and read-a-value-from-a-graph now get a formula. ## Definition For any real number $x$, the **greatest integer function** gives the greatest integer that is less than or equal to $x$. It is written $\lfloor x \rfloor$ and also called the **floor** of $x$. In plain words, it rounds $x$ down to the integer at or below it. "Down" means to the left on the number line, not toward zero, which matters for negative numbers: $$\lfloor 3.7 \rfloor = 3 \qquad \lfloor 5 \rfloor = 5 \qquad \lfloor -2.3 \rfloor = -3$$ The integers at or below $-2.3$ are $-3, -4, -5, \ldots$, and the greatest of them is $-3$. An integer is its own floor. Some books write $[x]$ for $\lfloor x \rfloor$. ### [Stretches and compressions](https://math.wiki/vertical-stretch-and-compression/) **Summary:** Multiplying the whole output by a number pulls the graph away from the horizontal axis or squashes it toward it. **Question:** A parent graph comes in one size. What does multiplying its output, or its input, by a number do to it? ## When you reach for it The parent graphs of the squaring and cubing functions and the square root and reciprocal functions come in one size. Real rules rarely do: $3x^2$, $\frac{1}{2}x^3$, $\sqrt{2x}$. Each is a parent with a number multiplying either the whole output or the input, and its graph is the parent pulled or squashed in one direction. Reach for this move to draw such a graph without plotting it point by point, to say what a number in a rule does, and to name the move when a graph or table has a parent's shape but the wrong size. It also describes scaling in the world: turning up the volume multiplies how far a speaker cone moves, and playing a recording faster squeezes the same motion into less time. ## The procedure ### Multiplying the output: $a\,f(x)$ Let $g(x) = a\,f(x)$ with $a > 0$. At every input, $g$ works out what $f$ gives and then multiplies it by $a$. So if $(p, q)$ is on the graph of $f$, then $g(p) = a\,f(p) = aq$, and the point moves like this: $$(p,\ q) \;\longmapsto\; (p,\ aq)$$ Every point moves straight up or down, and its distance from the $x$-axis is multiplied by $a$. Points on the $x$-axis have $q = 0$ and $a \cdot 0 = 0$, so they stay where they are. - If $a > 1$, the graph is **stretched vertically by a factor of $a$**: pulled away from the $x$-axis. - If $0 < a < 1$, it is **compressed vertically by a factor of $a$**: squashed toward the $x$-axis. ### [Symmetry about the vertical axis and the origin](https://math.wiki/symmetry-about-the-y-axis/) **Summary:** A graph with this symmetry matches itself when the page is folded along the vertical axis. **Question:** Fold the page along the vertical axis, or turn it upside down. When does a graph land exactly on itself, and what does that tell you about it? ## Why this exists Pania weaves borders on a loom, and she draws each design as a curve on squared paper first. She never charts the whole curve. Some borders are mirror images left to right, so she charts the right half and the left half follows. Others look the same when the cloth is turned upside down, and again half the chart is enough. Many graphs are made of two matching halves in just these ways. Seeing it halves the work, and it is a check: a graph that should match itself and does not has a mistake in it. ## Definition A graph is **symmetric about the vertical axis** when folding the page along the $y$-axis lays the graph exactly onto itself. Point by point: whenever $(a, b)$ is on the graph, so is $(-a, b)$, at the same height on the other side of the axis. A graph is **symmetric about the origin** when turning the page half a turn ($180^\circ$) about the origin lays the graph exactly onto itself. Point by point: whenever $(a, b)$ is on the graph, so is $(-a, -b)$, as far away on the far side of the origin. $$\begin{aligned} \text{vertical axis:}&\quad (a, b) \mapsto (-a, b) \\ \text{origin:}&\quad (a, b) \mapsto (-a, -b) \end{aligned}$$ The first image is the reflection of a point over the y-axis, the second the rotation of a point by 180 degrees about the origin. Symmetry is a claim about **every** point: each one needs its partner. ## Intuition In Figure 1, fold along the $y$-axis and the right half of the W lands on the left half. Each dashed guide joins a point to its partner: it is level, and the axis cuts it in half. ### [Testing graphs and equations for functions](https://math.wiki/vertical-line-test/) **Summary:** If a vertical line hits the graph twice, one input has two outputs, so it is not a function. **Question:** A graph has endless points and an equation has endless solutions. How can you tell whether any input has been given two outputs without listing them all? ## When you reach for it A function gives each input exactly one output. Relations and functions sets that definition out, and when the pairs come as a list you test it by looking for an input that appears twice with different outputs. A graph is different: a curve holds endless points, and an equation such as $x = y^2$ has endless solutions. You cannot list them and scan. This lesson gives two ways to answer the question anyway. **On a graph**, slide a vertical line across it and count how many times it meets the graph. **From an equation**, fix an input, solve for $y$, and count how many values come out. Reach for it whenever a graph or an equation is offered as a function: before naming it $f$, before reading a value off the graph, and before asking for its domain. Throughout, $x$ is the input, measured across, and $y$ is the output, measured up, as in graphs of functions. ## The procedure ### Why a vertical line Every point on the vertical line $x = a$ has first coordinate $a$. So the points where that line meets a graph are exactly the points of the graph whose input is $a$. Counting crossings is counting outputs. - **No crossing** means the input $a$ has no output. That is allowed: $a$ is simply not in the domain. - **One crossing** means $a$ has exactly one output. - **Two or more crossings**, say at $(a, b)$ and $(a, c)$ with $b \ne c$, means the single input $a$ is paired with two different outputs. The graph is not a function. ### [The absolute value graph](https://math.wiki/graph-the-absolute-value-parent/) **Summary:** Plotting the V shaped graph whose point sits at the origin. **Question:** What does a graph look like when every output is a distance, never a direction? ## When you reach for it A reminder first. The absolute value $|x|$ is how far $x$ sits from $0$ on the number line, with no regard for which side: $|5| = 5$ and $|-5| = 5$. Asking for $|-2|$ gives one distance. The graph of $y = |x|$ gives every distance at once: each input $x$ is drawn at the height of its distance from $0$. You reach for it whenever a quantity measures how far something is from a fixed point and ignores the direction, such as how far a room's temperature is from the temperature a thermostat is set to. It is also the plainest member of a whole family, one of the parent-functions, so knowing its shape lets you sketch $|x| + 2$ or $|x - 4|$ without a long table. ## The procedure **Write the rule in two pieces.** A number at or to the right of $0$ is its own distance from $0$, so $|x| = x$ when $x \ge 0$. A negative number is the same distance from $0$ as its opposite, so $|x| = -x$ when $x < 0$; since $x$ is negative here, $-x$ is positive, as a distance must be. That makes $|x|$ a piecewise function: $$ |x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases} $$ The condition tells you which line of the rule to use, exactly as in evaluating a piecewise function: for $x = -3$ the second line applies, and $-(-3) = 3$. **Make a table and plot it.** A new function is first met by plotting points, and since the rule changes at $0$, the table needs inputs on both sides of $0$ as well as $0$ itself. ### [The difference quotient](https://math.wiki/difference-quotient/) **Summary:** Writing the average rate of change over an interval of width h as a single expression. **Question:** The average rate of change takes one interval at a time. Can a single expression give it for every interval at once? ## When you reach for it The average rate of change from a formula answers one question at a time: name an interval, evaluate the rule at both ends, divide the change in output by the width. Ask about a second interval and the work starts over. The difference quotient does that work once, for every interval. Call the left end $x$ and the width $h$, so the interval runs from $x$ to $x + h$. The average rate on it is then a single expression in $x$ and $h$, and any particular interval is just a pair of numbers put into it. Reach for it when you want to see how the average rate depends on where the interval sits and how wide it is, and above all when you want to watch it as the width shrinks. That last use, narrower and narrower intervals around one input, is how a rate at a single instant is found. ## The procedure On the interval from $x$ to $x + h$ the outputs are $f(x)$ and $f(x + h)$, and the width is $(x + h) - x = h$. The average rate of change, change in output over change in input, is therefore $$\frac{f(x + h) - f(x)}{h}, \qquad h \ne 0.$$ This is the **difference quotient** of $f$. Geometrically it is the slope of the secant line through $(x, f(x))$ and $(x + h, f(x + h))$, exactly as in the slope from two points. Finding it for a given rule is three moves, each with its own reason. **Evaluate at $x + h$.** The whole expression $x + h$ goes in wherever $x$ stands in the rule, as in evaluating a function at an expression. For $f(x) = x^2$ that gives $f(x + h) = x^2 + 2xh + h^2$. ### [The graph of an inverse](https://math.wiki/graph-of-an-inverse/) **Summary:** The inverse graph is the original graph reflected across the line where output equals input. **Question:** You have the graph of a function and want the graph of its inverse. Do you have to find a formula first, or does the picture already contain the answer? ## Why this exists Finding an inverse from a rule turns a formula into the formula that runs it backwards. Many functions never arrive as a formula. A graph read off an experiment, a curve sketched from a description, a broken line drawn from a few measured points: each is a function you can see but cannot solve. And even when there is a formula, solving it gives no picture of what the inverse looks like. The graph turns out to need no algebra at all. The inverse undoes $f$, so every pair it holds is a pair of $f$ turned round. Turning every pair round has one geometric meaning: reflect the whole graph in the line $y = x$. With that one fact you can draw an inverse you have no formula for, see at a glance whether an inverse exists, read off its domain and range, and predict where the two graphs meet. ## Definition The **graph of the inverse** $f^{-1}$ is the set of points $(b, a)$ for which $(a, b)$ is a point on the graph of $f$. In words: take every point of the graph of $f$ and swap its two coordinates. This comes straight from the definition of the inverse, which reads one fact in two directions: $$f(a) = b \quad\Longleftrightarrow\quad f^{-1}(b) = a$$ The left side says $(a, b)$ is on the graph of $f$. The right side says $(b, a)$ is on the graph of $f^{-1}$. The two graphs are drawn on the same axes, with $x$ as the input of both. The **line $y = x$** is the graph of the identity function: every point on it has output equal to input. It is the mirror for inverses. ### [The identity and constant functions](https://math.wiki/identity-function/) **Summary:** The function that gives back its input unchanged, drawn as the line through the origin at slope one. **Question:** Most rules do something to their input. What do the two laziest rules look like, the one that hands back exactly what it was given and the one that hands back the same number whatever it was given? ## Why this exists A function is a rule that takes an input and hands back exactly one output, as relations and functions sets out. Most rules do some work: double the input, add $3$, square it. Two rules sit right at the bottom, doing the least a rule can do. One hands back exactly what it was given. The other ignores what it was given and always hands back the same number. The first is the "do nothing" of functions, playing the part that $0$ plays for adding and $1$ for multiplying, the identity elements. Its graph, the line $y = x$, is the parent that every other line is a tilted and lifted copy of, and the mirror that swaps inputs with outputs. The second is the flat line. It settles a question that tables keep raising: may different inputs share an output? They may, and a constant function is the extreme case, where every input shares one. ## Definition The **identity function** hands back its input unchanged. For every real number $x$, $$f(x) = x$$ A **constant function** hands back one fixed number $c$ for every input. For every real number $x$, $$g(x) = c$$ Both accept every real number as an input, so each has all real numbers as its domain, unless a situation limits the inputs. Their ranges, the sets of outputs they actually produce, are very different. The identity function produces every real number, since any number $r$ comes out when $r$ goes in. A constant function produces only $c$, so its range is the set with one member, written $\{c\}$. ### [The implied domain of a formula](https://math.wiki/implied-domain/) **Summary:** When no domain is stated, the domain is every input the formula can accept without breaking. **Question:** A formula arrives with no list of allowed inputs. Which numbers can you actually put into it? ## Why this exists Most functions in algebra arrive as a single line, such as $f(x) = \sqrt{10 - 2x}$, with nothing said about which inputs are allowed. Yet some are not. Put in $x = 7$ and the formula asks for $\sqrt{-4}$, which is not a real number, so the function has no output there. Listing the allowed inputs every time would be tedious, so mathematics uses a convention instead: **when no domain is stated, the domain is every real number the formula can accept.** That set is the implied domain. Finding it means hunting down every input that breaks the formula, and in this course a formula can break in only two ways. ## Definition The **implied domain** of a function given by a formula (also called its natural domain) is the set of all real numbers $x$ for which the formula gives a real number as output. In this course, two things stop a formula from giving a real output: - **a denominator equal to $0$**, because division by zero has no answer; - **an even root of a negative number**, such as $\sqrt{-4}$ or $\sqrt[4]{-1}$, because no real number has a negative square or fourth power. Everything else is safe for every real input: adding, subtracting, multiplying, whole-number powers, and odd roots such as $\sqrt[3]{x}$. So a polynomial has the implied domain of all real numbers, $(-\infty, \infty)$. A domain can also be set in two other ways, and both narrow the implied one. - A **stated domain** is chosen by whoever defines the function and written beside it. The function $f(x) = x^2$ for $0 \le x \le 3$ is a different function from $f(x) = x^2$, because it accepts fewer inputs. - A **context domain** comes from what the variable means. If $x$ counts people, $x = 2.5$ makes the formula work but makes no sense; that is a reasonable domain for a real situation. ### [The number e and the natural exponential function](https://math.wiki/natural-exponential-function/) **Summary:** The exponential function with base e, the one whose growth rate always equals its own height. **Question:** If a bank added interest not every month or every day but at every instant, how much would a balance grow, and what number sits behind the answer? ## Why this exists Compound interest left a question hanging. A yearly rate $r$ compounded $n$ times a year multiplies a balance by $\left(1 + \dfrac{r}{n}\right)^n$ each year, and that factor grows as $n$ grows, but by less and less each time. Quarterly beats once a year, monthly beats quarterly, daily beats monthly. So what happens if the year is split into more and more periods without end, so that interest is added at every instant? Does the balance grow without limit, or does it settle on some ceiling? And if there is a ceiling, how is it computed? The answer turns on one number, found by asking the simplest version of the question: $1$ dollar at $100\%$ a year, compounded more and more often. That number is called $e$. It then turns out to do something no other base does. The curve $y = e^{x}$ climbs at every point at a slope equal to its own height, which is why $e$, and not $10$ or $2$, is the base that calculus, physics, biology and finance reach for when something grows or shrinks continuously. ## Definition **The number $e$** is the number that $$\left(1 + \frac{1}{n}\right)^{n}$$ settles toward as $n$ grows without bound (written $n \to \infty$). To nine decimal places, $$e \approx 2.718281828$$ It is irrational: its decimals go on forever without repeating, so every decimal written for it is an approximation. The letter $e$ was chosen by Leonhard Euler, who worked out many of its properties. ### [The square root and reciprocal functions](https://math.wiki/reciprocal-parent-function/) **Summary:** The function one over x, whose graph is two curves hugging the two axes. **Question:** What does a graph look like when its rule refuses some inputs, such as a negative under a square root or a zero in a denominator? ## Why this exists The parents met so far, the line $y = x$, the parabola $y = x^2$ and the cubic $y = x^3$ and the V of $y = |x|$, accept every real number. Their graphs run unbroken from far left to far right. Two of the most common operations do not behave that way. A square root undoes squaring, and no real number squares to a negative, so $\sqrt{x}$ refuses every negative input. Division by zero has no answer, so $\frac{1}{x}$ refuses the input $0$. Each refusal shows on the graph. The graph of $y = \sqrt{x}$ begins at a point and goes only one way. The graph of $y = \frac{1}{x}$ splits into two pieces that hug the axes without touching them. With these two, the list of parent-functions is long enough to set side by side, and that comparison is what lets you name a family from a graph, a table or a rule at a glance. A sound engineer meets both: a tone's period is the reciprocal of its frequency, and an amplitude factor is the square root of a power factor. ## Definition The **square root function** is $f(x) = \sqrt{x}$. For $x \ge 0$, $\sqrt{x}$ is the one number that is at least $0$ and squares to $x$, as in square-and-cube-roots. Its domain is $x \ge 0$. The **reciprocal function** is $g(x) = \frac{1}{x}$. For $x \ne 0$, $\frac{1}{x}$ is the one number that multiplied by $x$ gives $1$. Its domain is every real number except $0$. An **asymptote** of a graph is a line the graph comes as close to as you like, without end, as it runs off along it (what an asymptote is). The graph of $\frac{1}{x}$ has two: the $y$-axis, which is the line $x = 0$, and the $x$-axis, which is the line $y = 0$. ### [The squaring and cubing functions](https://math.wiki/squaring-parent-function/) **Summary:** The function that squares its input, whose graph is the plain upward parabola. **Question:** Squaring a number forgets its sign and cubing keeps it. What does that one difference do to the two graphs? ## Why this exists The identity function draws a straight line, and any two of its points fix it. Many rules are not like that. A square deck panel of side $s$ has area $s^2$, and a cube of concrete of side $s$ has volume $s^3$; neither grows by the same amount for each step in $s$, so neither graph is a line. Rather than plot every such rule from scratch, it pays to know two graphs by heart: the graph of squaring and the graph of cubing. Each is one of the parent-functions, the plainest member of a family. Every quadratic function is a moved and stretched copy of the first, and the second is where cubic functions start. ## Definition The **squaring function** and the **cubing function** are $$ f(x) = x^2 \qquad\text{and}\qquad g(x) = x^3 $$ The first multiplies its input by itself once, the second twice: $f(-2) = (-2)(-2) = 4$ and $g(-2) = (-2)(-2)(-2) = -8$. Their graphs are all the points $(x, x^2)$ and all the points $(x, x^3)$. The graph of $y = x^2$ is a parabola. The **key points** are the ones at $x = -1$, $0$ and $1$, where the outputs are small integers, with $x = \pm 2$ to show how fast each graph climbs: | $x$ | $-2$ | $-1$ | $0$ | $1$ | $2$ | |:-:|:-:|:-:|:-:|:-:|:-:| | $x^2$ | $4$ | $1$ | $0$ | $1$ | $4$ | | $x^3$ | $-8$ | $-1$ | $0$ | $1$ | $8$ | The points are joined by smooth curves, since every input in between has an output too. The **domain** of a function is the set of inputs it accepts, and the **range** is the set of outputs it gives, both read off a graph as in domain and range from a graph. Any real number can be squared or cubed, so both domains are all real numbers. The ranges differ, and that difference is the main fact of this lesson. ### [Transformations of functions](https://math.wiki/transformations-of-functions/) **Summary:** Shift, flip and stretch a graph by changing the formula. **Question:** Once you know the graph of one parabola, do you have to plot every moved copy of it point by point, or can you just move the graph you already have? ## What this area is about Most graphs you meet are a few basic shapes slid, flipped or stretched. Each change to a rule matches a change to its graph. For $k > 0$, adding $k$ to the output slides the graph up $k$ units; for $h > 0$, replacing $x$ by $x - h$ slides it right $h$ units. Negating the whole output flips the graph upside down, and multiplying the output by $a > 1$ stretches it away from the horizontal axis. This area reads such moves off a rule, draws the result, and runs the process backward. ## How these fit together **The shapes.** Parent functions is an area of its own: the plainest graph in each family, such as $y = x^2$, $y = \sqrt{x}$ and $y = \frac{1}{x}$. **Single moves.** Shifting a graph slides it up, down, left or right. Reflecting across the axes turns it upside down or makes it face the other way. Stretches and compressions multiply the output or the input by a number. **Procedures.** Combining transformations reads several moves off one rule, in the order they happen. Graphing by transforming a parent graph draws the curve and finds its domain and range. Finding the rule of a transformed graph goes from a curve or table back to the rule. ## A path through it Shifts, then reflections, then stretches, then combining them. Graphing by transformations needs those and the parent functions; finding the rule from a graph comes last. Next, the same moves draw exponential, absolute value and square root graphs, and the trigonometric graphs. ### [Transforming an exponential graph](https://math.wiki/transform-exponential-graphs/) **Summary:** Shifting and stretching the curve, which also moves its horizontal asymptote. **Question:** The graph of y = 2^x hugs the x-axis. Where does the graph of y = 2^(x+3) - 5 level off, and which way does it run, without plotting it point by point? ## When you reach for it You know the graph of $y = b^x$ from graph-exponential-functions: it passes through $(0, 1)$, runs flat along the $x$-axis on one side and grows without bound on the other. A rule such as $y = 2^{x+3} - 5$ is that graph moved, in exactly the sense of graph-using-transformations. So instead of plotting it point by point, you move a few points of the parent and, above all, its asymptote. You reach for this whenever an exponential rule carries extra numbers: a number added to the input, a number multiplying the power, a number added at the end. It also explains models that level off at something other than $0$. A bowl of soup does not cool toward $0$ degrees but toward the temperature of the room, so its graph is an exponential graph whose asymptote has been lifted. ## The procedure **The general form.** Every moved exponential in this lesson is written as in combining-transformations: $$y = a \cdot b^{\,x - h} + k$$ with $b > 0$, $b \ne 1$ and $a \ne 0$. The parent is $y = b^x$. The number $h$ shifts the graph right ($h > 0$) or left ($h < 0$), the number $a$ stretches it vertically and flips it over if $a < 0$, and $k$ shifts it up ($k > 0$) or down ($k < 0$). In $y = 2^{x+3} - 5$, the $x + 3$ is $x - (-3)$, so $h = -3$, $a = 1$ and $k = -5$. **Carry a few points.** The easiest points of the parent are $\left(-1, \frac{1}{b}\right)$, $(0, 1)$ and $(1, b)$. Each point $(p, q)$ of the parent lands at ### [Transforming the absolute value graph](https://math.wiki/transform-absolute-value-graphs/) **Summary:** Shifting, flipping and stretching the V to match a given rule. **Question:** The V-shaped graph of the absolute value can be slid, flipped and stretched. Where does its corner land, how steep are its arms, and which heights does it reach? ## When you reach for it The absolute value graph is a V with its corner at the origin and arms of slope $-1$ and $1$. The unit's general moves, shifting, flipping and stretching, put into one rule by combining-transformations, work on it exactly as on any parent. Applied to $|x|$ they give the family $$ y = a|x - h| + k, \qquad a \ne 0 $$ Reach for this lesson when such a rule has to be drawn, when a V on a grid needs its rule, or when an absolute value equation or inequality is easier to see than to solve. A V is also the simplest model for a quantity that rises at a steady rate up to one point and falls at the same rate after it, such as a made-up signal meter that reads higher as you drive toward a transmitter and lower after you pass it. Real radio signals do not fade in straight lines; the V is a simplified model. ## The procedure ### The vertex is $(h, k)$ The bars $|x - h|$ are $0$ at $x = h$ and positive at every other input. So at $x = h$ the output is $a \cdot 0 + k = k$, and the corner, the **vertex**, sits at $(h, k)$. The sign inside the bars works as for every parent in graph-using-transformations: $|x + 2|$ is $|x - (-2)|$, so $h = -2$. ### The arms have slopes $a$ and $-a$ To the right of the vertex $x - h \ge 0$, so $|x - h| = x - h$ and the rule is the line $y = a(x - h) + k$, with slope $a$. To the left $x - h < 0$, so $|x - h| = -(x - h)$ and the rule is $y = -a(x - h) + k$, with slope $-a$. Each arm is a ray. One step to either side of the vertex the height changes by $a$, so $(h - 1, k + a)$ and $(h + 1, k + a)$ are on the graph, and those three points are enough to draw it. ### [Transforming the reciprocal graph](https://math.wiki/transform-the-reciprocal-function/) **Summary:** Shifting and stretching one over x, which moves both of its asymptotes. **Question:** Stretch or slide the graph of one over x, and both of the lines it hugs move with it. Where do they go, and how do you draw the curve around them without plotting a long table? ## When you reach for it You already know the graph of $y = \frac{1}{x}$ from the reciprocal function: two curves, one in the first quadrant and one in the third, each hugging both axes without touching them. A great many rules are that same graph moved: $$\begin{aligned} g(x) &= \frac{3}{x + 2} - 1, \\ C(n) &= \frac{240}{n} + 5, \\ f(x) &= \frac{4x - 1}{x - 3}. \end{aligned}$$ Reach for this move whenever a rule is, or can be rewritten as, a number over $x - h$ plus a number. From the rule alone you get both asymptotes, the domain, the range, the shape and a handful of exact points, without a table of dozens of values. It is the parent graph method applied to the one parent whose key features are two lines rather than a single point. The same move gives inverse variation its graph as a function. The rule $y = \frac{a}{x}$ is the reciprocal stretched by $a$, and when a fixed amount is added (a set-up time, a cost everyone pays) the graph is that curve slid up or across. ## The procedure ### One form for every move From graphing by transformations: a vertical stretch by $|a|$ (with a reflection in the $x$-axis when $a < 0$), then a shift right by $h$ and up by $k$, turns the graph of $y = f(x)$ into the graph of $y = a f(x - h) + k$. With $f(x) = \frac{1}{x}$, and since $a \cdot \frac{1}{x - h} = \frac{a}{x - h}$, every such move of the reciprocal has the form $$y = \frac{a}{x - h} + k, \qquad a \neq 0.$$ ### [Turning points, maximums and minimums](https://math.wiki/relative-maximum-and-minimum/) **Summary:** An output that is the largest or smallest in its own neighborhood, even if not for the whole function. **Question:** A graph has several hills. Which one is "the maximum", and how do you say where it is? ## Why this exists The elevation profile of a mountain stage in a cycling race has several summits. Ask "where is the top?" and there are two fair answers: the top of each climb, and the top of the whole stage. A rider going over a lower summit has still reached a top, even though a higher one comes later. Mathematics keeps both answers by giving them different names, **relative** and **absolute**. It also insists on two numbers each time: how high (an output) and where (an input). "The highest point of the route was $1{,}840$ m, at $62$ km" says both; "the top is $62$" leaves the reader guessing which. ## Definition Let $f$ be a function and $c$ an input in its domain. - $f(c)$ is a **relative maximum** (also called a **local maximum**) if $f(c)$ is at least as large as every output $f(x)$ for inputs $x$ near $c$, on both sides of $c$. - $f(c)$ is a **relative minimum** (or **local minimum**) if $f(c)$ is no larger than every output for inputs near $c$, on both sides. - $f(c)$ is the **absolute maximum** if $f(c) \ge f(x)$ for every $x$ in the domain, and the **absolute minimum** if $f(c) \le f(x)$ for every $x$ in the domain. - A **turning point** is a point where the graph switches from rising to falling (a peak) or from falling to rising (a valley), reading left to right. "Near $c$" means within some interval around $c$, however small, that reaches to both sides of it. The plurals are maxima and minima. ### [Vertical asymptotes](https://math.wiki/vertical-asymptotes/) **Summary:** A vertical line the graph shoots up or down along, sitting at an input the function cannot accept. **Question:** Some functions refuse one particular input, and close to it their outputs explode. Where does that happen, which way does the graph go on each side, and how is it different from a single missing point? ## Why this exists A fraction cannot have zero as its denominator. So a formula such as $f(x) = \frac{x + 1}{x^2 - 9x + 14}$ refuses certain inputs: here $x = 2$ and $x = 7$, where $x^2 - 9x + 14 = 0$. Knowing that an input is refused (see implied-domain) tells you there is no point on the graph above it. It does not tell you what the graph does *near* that input, and that is where the interesting behavior is. Put in $x = 1.999$, just short of $2$. The numerator is about $3$ and the denominator is a tiny number, about $0.005$, so the output is about $600$. Move closer and the output grows past a thousand, past a million, without any ceiling. On the other side of $2$ the outputs grow without bound in the negative direction. The graph does not merely have a gap at $x = 2$: it runs off the top and bottom of every page, hugging the vertical line $x = 2$ as it goes. In continuous-and-broken-graphs this was the third way a graph can break. An asymptote in general is a line the graph gets ever closer to as it heads off without end. A vertical asymptote is the case where the heading-off happens in the outputs, upward or downward, while the inputs close in on one number. This star pins down when that happens, which way the graph goes on each side, and how to tell it from the other kind of refused input, a single missing point called a hole. ## Definition **In words.** A vertical line $x = a$ is a **vertical asymptote** of the graph of $f$ when, as the input closes in on $a$ from at least one side, the outputs grow without bound, either upward or downward. ### [What a function is](https://math.wiki/function-concept/) **Summary:** A rule that gives exactly one output for each input. **Question:** Price depends on weight, distance depends on time, height depends on age. What exactly does it mean for one quantity to depend on another, and how do we write and picture that? ## What this area is about A function is a rule that gives each allowed input exactly one output. That one condition is what lets you speak of *the* output for an input, write it as $f(x)$, and draw a graph you can trust. This area sets up the idea, the notation, the sets of inputs and outputs, and the picture. ## How these fit together **The definition.** Definition of a function is an area of its own: functions as input and output rules, relations and which of them are functions, and independent and dependent variables. **The notation.** Function notation writes the output of $f$ at the input $x$ as $f(x)$, evaluates it at numbers and at expressions, and reads it in real situations. **Inputs and outputs.** Domain and range are the inputs a function accepts and the outputs it produces, found from pairs, graphs, formulas and situations. **The picture.** Graphs of functions plot every pair $(x, f(x))$ and read values, stories and landmarks off the result. Testing graphs and equations for functions checks, without listing every point, whether some input has two outputs. ## A path through it The definition first, then notation. After that, domain and range and graphs of functions can come in either order, and the vertical line test follows graphs. Next comes every family of functions: linear, quadratic and exponential functions, and new functions built from old ones, such as inverses. ### [What an asymptote is](https://math.wiki/asymptote-concept/) **Summary:** A line the graph gets ever closer to as it heads off without end, showing where the graph is heading. **Question:** A bottle of juice from the fridge keeps warming, but it never gets warmer than the room. What is its graph heading toward? ## Why this exists A bottle of juice comes out of the fridge at $6^\circ\text{C}$ into a room at $22^\circ\text{C}$. It warms quickly at first, then more and more slowly. It never gets warmer than the room, yet if you wait long enough it gets as close to room temperature as you like. Its graph does not stop at a last point, and it does not run off the top or bottom of the page. It flattens out along the line at height $22$. A phone charging toward full behaves the same way, and so does the distance left in a walk that always covers half of what remains. Each graph is heading toward a line it gets ever closer to. Such a line is called an **asymptote**, from a Greek word meaning "not falling together". Naming it is the quickest way to say where a graph is heading far out, and with end behavior it is one of the features of a graph worth reading off. ## Definition A **horizontal asymptote** of a graph is a horizontal line $y = k$ that the graph gets as close to as you like as $x$ grows without bound, or as $x$ is negative and grows in size without bound. "As close as you like" carries the meaning. Name any distance, however small: $0.1$, $0.001$, a millionth. Far enough out, every point of the graph is within that distance of the line, and it stays within it from then on. A graph that passes near $k$ once and moves away again, or that hovers a fixed distance from $k$, does not have $y = k$ as an asymptote. ### [Where a model holds: domain, interpolation and extrapolation](https://math.wiki/interpolation-and-extrapolation/) **Summary:** Reading a value inside the range of the data is far safer than predicting one outside it. **Question:** A model that fits the data will give an answer for any input you hand it. For which inputs can that answer be believed? ## Why this exists A model is fitted to a handful of measurements and then asked about cases nobody measured: a pumpkin's mass in a week it was not weighed, the drinks a kiosk will sell on a hotter day than any on record, next year's winning time. The formula answers every such question in the same tone. It cannot tell you which of its answers rest on data and which are guesses about a world it never saw. That has to come from outside the formula: from where the data lie, and from what the situation allows. line-of-best-fit, interpret-the-slope-of-a-regression-line, linear-modeling, quadratic-modeling and reasonable-domain-in-context each met this in passing. This lesson gathers those warnings into one idea. ## Definition Measured data cover a stretch of inputs, from the smallest input measured to the largest. Using a model at an input inside that stretch, the two ends included, is **interpolation**. Using it at an input outside the stretch is **extrapolation**. (Latin *inter*, between, and *extra*, outside.) When a question gives an output and asks for the input ("when will it reach $30$ kg?"), the input the model lands on decides which it is. The **domain of a model** is the set of inputs on which the model is trusted to describe the situation. It is stated with the model, as part of it: $$D = 6t - 45, \quad 18 \le t \le 30$$ The formula alone accepts every number, so a model stated without its domain is incomplete. The domain is never wider than the reasonable domain the situation allows, and when the model was fitted to data it is usually the data's stretch, or a little beyond it if there is a reason to trust a short step further. ### [Writing a piecewise rule from a graph](https://math.wiki/write-a-piecewise-rule-from-a-graph/) **Summary:** Finding the equation of each segment and the interval of inputs it covers. **Question:** You have the graph of a function whose rule changes partway along. How do you get the formulas back, and say exactly which inputs each one is in charge of? ## When you reach for it Graphing a piecewise function goes from the rule to the picture. This lesson goes the other way. You have a graph made of pieces, perhaps a record of a machine's speed or a sketch someone else drew, and you want its rule: a formula for each piece and the inputs that piece covers. With the rule in hand you can evaluate the function at inputs between grid lines, where reading the picture is only a guess. ## The procedure Take the graph of $f$ below. **Find the boundaries.** Scan from left to right for every input where the graph changes: a dot, a break or a corner. Here there is one, $x = 1$. Each stretch between boundaries is one piece, and each piece gets one line of the rule. **A formula for each piece.** Each piece here is straight, so it is part of a line, and a line is fixed by its slope and one point. Pick two points where the piece crosses grid corners, so the coordinates are exact, and find the slope as rise over run (slope-from-two-points). The left piece passes through $(0, 1)$ and $(1, 3)$, so its slope is $\frac{3 - 1}{1 - 0} = 2$. It crosses the $y$-axis at $1$, so its formula is $y = 2x + 1$ (read-slope-and-intercept). The right piece passes through $(1, 0)$ and $(3, -2)$, slope $\frac{-2 - 0}{3 - 1} = -1$. It never reaches the $y$-axis, so use point and slope instead: $y - 0 = -1(x - 1)$, so $y = 1 - x$. An open dot may serve as one of the two points. It is not on the graph, but it is on the line the piece lies on, and the formula describes that line. ### [Writing an exponential function from a table or graph](https://math.wiki/exponential-function-from-a-table-or-graph/) **Summary:** Reading the starting value and the constant ratio from a table or graph and writing the exponential rule. **Question:** A table of values gets multiplied by the same number at every step. How do you turn the table back into the rule that made it? ## When you reach for it You have some outputs of an exponential function, in a table or as points on a graph, and you want the rule $$y = a \cdot b^x$$ that produced them. With the rule you can find any output, not only the listed ones, and its two numbers say what is going on: $a$ is the starting value and $b$ is the growth or decay factor, the number the output is multiplied by for each step of $1$ in $x$. Reach for it also before you know the data are exponential. The arithmetic that finds $b$, dividing each output by the one before, is the same arithmetic that tells you whether an exponential rule fits at all. ## The procedure Everything rests on two facts about $y = a \cdot b^x$, where $a \neq 0$, $b > 0$ and $b \neq 1$. **At $x = 0$ the output is $a$.** Any nonzero number to the power $0$ is $1$, so $y = a \cdot b^0 = a \cdot 1 = a$. On a graph, that is the height where the curve crosses the $y$-axis. **A step of $1$ in $x$ multiplies the output by $b$.** Raising the exponent by $1$ adds one more factor of $b$: $$a \cdot b^{x+1} = (a \cdot b^x) \cdot b$$ So dividing any output by the one before it gives $b$, and gives the same $b$ every time. ### A table with steps of 1, starting at 0 Read $a$ from the row $x = 0$. Divide each output by the one before. If every quotient is the same number, that number is $b$. Checking all the quotients is part of the work, not a formality: any two numbers have a quotient, so one quotient proves nothing, and only agreement across the whole table shows the factor is constant. ### [Zeros and multiplicity](https://math.wiki/zero-crosses-or-touches/) **Summary:** A zero of odd multiplicity crosses the axis and one of even multiplicity touches it and turns back. **Question:** The polynomials $(x - 1)(x + 3)$ and $(x - 1)^2(x + 3)$ have exactly the same zeros. Why does one graph cross the axis at $1$ while the other only touches it there? ## Why this exists The zero-product-property turns a factored polynomial into its zeros. Both $(x - 1)(x + 3)$ and $(x - 1)^2(x + 3)$ are zero at $x = 1$ and $x = -3$ and nowhere else, so their lists of x-intercepts are the same. Their graphs are not. The first passes through the axis at $1$; the second comes down, touches the axis at $1$, and goes back up. The list has lost how many times each factor appears. Keeping that count, the multiplicity, tells you what the graph does at each zero without plotting a point. ## Definition Let $p$ be a polynomial and $c$ a real zero of $p$, so that $x - c$ is a factor of $p$ by the factor-theorem. The **multiplicity** of $c$ is the number of times the factor $x - c$ appears: the largest positive integer $k$ for which $(x - c)^k$ is a factor of $p$. Equivalently, $$p(x) = (x - c)^k \, q(x), \quad q \text{ a polynomial with } q(c) \neq 0.$$ The condition $q(c) \neq 0$ says that no copy of $x - c$ is left inside $q$. A zero of multiplicity $1$ is a **simple zero**, of multiplicity $2$ a **double zero**, of multiplicity $3$ a **triple zero**. In $(x - 1)^2(x + 3)$, the zero $1$ has multiplicity $2$ and $-3$ has multiplicity $1$. A quadratic whose discriminant is zero has a single double zero: $x^2 + 10x + 25 = (x + 5)^2$ has one at $-5$, and its parabola touches the axis there, at its vertex. ## Intuition Near a zero $c$, the factor $(x - c)^k$ is tiny and changes fast, while $q(x)$ hardly moves from $q(c)$. So near $c$ the graph looks like $y = (x - c)^k$ stretched by $q(c)$: everything comes down to $y = x^k$ near the origin. ## Geometry ### [AA similarity](https://math.wiki/aa-similarity/) **Summary:** Two pairs of congruent angles are enough to make two triangles similar. **Question:** Two triangles have two angles that match. Without measuring a single side, can you be sure they have the same shape? ## Why this exists To show two triangles are similar straight from the definition you need six facts: three pairs of congruent angles and three side ratios that agree. In most real problems some of those are out of reach. The height of a tree, the width of a river, a side hidden inside a diagram: these are the lengths you want to find, so they cannot be part of the evidence. Angles are often much easier to come by. A corner two triangles share, two lines that cross, a line drawn parallel to another, the sun's rays falling at the same slant on everything at once: each of these hands you a pair of congruent angles without a ruler. AA similarity says that two such pairs are enough. Once two angles match, the shape is settled, and every side ratio follows. This one test is behind measuring heights with shadows, behind the triangle proportionality theorem, and behind the fact that the sine of an angle is a single number. ## Definition Two triangles are similar when one is a scaled copy of the other: matching angles congruent, and matching sides all in one ratio $k$, the **scale factor**. We write $\triangle ABC \sim \triangle DEF$, and the order of the letters is the matching: $A$ with $D$, $B$ with $E$, and $C$ with $F$. **AA similarity** (angle, angle). If two angles of one triangle are congruent to two angles of another triangle, $$\angle A \cong \angle D \quad\text{and}\quad \angle B \cong \angle E$$ ### [Acute, right, obtuse and straight angles](https://math.wiki/acute-right-obtuse-straight/) **Summary:** Angles sorted by size, from smaller than a square corner up to a full half turn. **Question:** Angles come in every size from a sliver up to a half turn. Which sizes are worth a name, so that you can tell two angles apart without measuring either? ## Why this exists Once you can measure an angle you can give it a number. Most of the time you do not need the number. A carpenter checking a corner or a reader glancing at a diagram wants to know something simpler first: is this angle smaller than a square corner, exactly a square corner, or bigger? Two sizes act as landmarks for that question. One is the square corner, the right angle. The other is the angle made by a straight line, the straight angle, which is two right angles side by side. Naming the landmarks, and the two ranges on either side of the first one, gives four words that sort every angle between two rays. They are the words the rest of geometry is written in: complementary angles are defined by the right angle, supplementary angles by the straight angle, and triangles are sorted by whether they contain a right or an obtuse angle. The right angle came first, long before degrees. Euclid defines it without any number at all: when one line stands on another and makes the two angles beside it equal, each of those equal angles is right. Degrees came later and put the landmark at $90^\circ$. ## Definition An angle is sorted by its measure, in degrees. For an angle $\angle A$: - $\angle A$ is **acute** when $0^\circ < m\angle A < 90^\circ$. It is narrower than a square corner. - $\angle A$ is **right** when $m\angle A = 90^\circ$. It is exactly a square corner, and a drawing marks it with a small square at the vertex instead of an arc. - $\angle A$ is **obtuse** when $90^\circ < m\angle A < 180^\circ$. It is wider than a square corner but its arms do not yet form a line. - $\angle A$ is **straight** when $m\angle A = 180^\circ$. Its two arms point in opposite directions and together make one straight line through the vertex. ### [Adjacent angles and linear pairs](https://math.wiki/linear-pairs/) **Summary:** Two adjacent angles whose outer sides form a straight line, so their measures add to 180 degrees. **Question:** Two angles sit side by side at one corner. When is the angle they make together simply the two of them added, and what happens when their outer sides make a straight line? ## Why this exists Measuring an angle means lining up a protractor, and a figure hands you most of its angles without a protractor anywhere near it. What makes that possible is a very ordinary situation: two angles sitting next to each other at one corner, like two slices cut side by side from the same pizza. When two angles sit like that, the bigger angle they make together is just the two of them added. Know two of the three and you get the third by adding or subtracting. But it only works when the angles really are side by side, so the arrangement needs a name and a careful definition. The name is **adjacent**. One arrangement of adjacent angles comes up more than any other. A straight line is the one angle you never have to measure: its two halves point in opposite directions, so the turn from one to the other is half of a full turn, $180^\circ$, every time. Let a single ray leave a point on that line. The straight angle is cut into two adjacent pieces, and you still know their total. So you only ever need to measure one of them; the other is whatever is left of $180^\circ$. That pair, a **linear pair**, turns up wherever a line meets anything at all: under every crossing of two lines, beside every corner of a triangle when a side is extended, and at each point where a transversal cuts a line. Nearly every missing-angle problem, from a single crossing to a whole figure, is these two facts used again and again. ### [Alternate and same side angles](https://math.wiki/alternate-interior-angles/) **Summary:** Angles on opposite sides of the transversal and between the two lines. **Question:** A line cuts across two parallel lines. Which angles at one crossing are forced to match an angle at the other, and which are forced to make up $180^\circ$ with it? ## Why this exists When a single line crosses two others, it makes eight angles, four at each crossing. From vertical angles and linear pairs you already know that the four angles at one crossing come in only two sizes. But nothing so far connects the first crossing to the second. Each crossing is its own little island of information. Parallel lines build the bridge. If the two lines are parallel, certain angles at the top crossing are forced to equal certain angles at the bottom one, and certain others are forced to add up to a straight angle with them. A single measurement then fixes all eight angles, and the same facts, read backwards, prove that two lines are parallel without ever following them to see whether they meet. This lesson names those bridges and puts them to work. Alternate interior angles are the most used, because they sit right next to each other in the space between the lines, which is where the triangles and parallelograms of later geometry live. Alternate exterior angles are the same bridge on the outside of the figure. Same side interior angles are the odd one out: they are usually not equal at all, one sharp and one wide, yet they are tied just as tightly, because they always add to $180^\circ$. That last pair is the oldest of them. When Euclid wrote down the assumption that flat geometry rests on, the one now called the parallel postulate, he phrased it with this pair: if the two interior angles on one side add to less than two right angles, the lines meet on that side. So this pair is the one that decides whether two lines are parallel in the first place. ### [Angle bisectors of a triangle](https://math.wiki/angle-bisector-distance-theorem/) **Summary:** A point is the same distance from both sides of an angle exactly when it lies on the bisector. **Question:** A point sits inside an angle. When is it exactly as far from one side as from the other, and what does that tell you about the three angles of a triangle? ## Why this exists Two straight fences meet at the corner of a field and make an angle. A sprinkler is to stand inside the corner, exactly as far from one fence as from the other. Where can it go? You could try a spot, measure to each fence, and move it a little. That finds one good spot at a time and never tells you whether you have found them all. The answer in this lesson finds every one at once: the good spots are exactly the points of the angle bisector, the ray that cuts the corner into two equal angles. Every point on it works, and no other point inside the corner does. Now make the field a triangle, with three fences. Is there a spot the same distance from all three? Each pair of fences has its own bisector, and the surprise is that the three bisectors of a triangle always cross at a single point. That point is the same distance from every side, so a circle drawn around it can touch all three sides at once. The rest of that story belongs to the incenter; this lesson proves the facts it stands on. ## Definition The **distance from a point to a line** is the length of the perpendicular segment from the point to the line, as in distance-from-a-point-to-a-line. The end of that segment on the line is the **foot** of the perpendicular. The **distance from a point to a side** of an angle is its distance to the line containing that side. A point $P$ is **equidistant from the sides** of $\angle ABC$ when its perpendicular segments to line $BA$ and to line $BC$ have the same length. ### [Angle sums of polygons](https://math.wiki/polygon-interior-angle-sum/) **Summary:** Splitting a polygon into triangles shows its interior angles add to 180 degrees times two fewer than its sides. **Question:** Every triangle's angles add to the same number. Does every pentagon, every hexagon, every shape with straight sides have a number of its own, and if so, what is it? ## Why this exists The angles of a triangle always add to $180^\circ$, whatever its shape. That is the triangle angle sum, and it is what lets you find a third angle from the other two. Most shapes with straight sides are not triangles. A floor tile, a field boundary, the outline of a nut on a bolt: each has four, five, six or more corners, and the same question comes up. If you know all the angles but one, is the last one forced? It is, and the reason is that every polygon can be cut into triangles. Count the triangles and you have the angle sum. The same counting gives two more facts this lesson also covers: how many diagonals a polygon has, and what its exterior angles add to. The second has a surprise in it. The interior angles add to more and more as the sides increase, but the exterior angles always add to the same thing. ## Definition A **polygon** is a closed figure made of straight segments, its **sides**, joined end to end. Each side meets the next at a **vertex**, sides meet only at the vertices they share, and no two neighboring sides lie along one straight line. A polygon with $n$ sides has $n$ vertices and is called an $n$-gon; the everyday names, pentagon, hexagon and the rest, are in naming polygons. The **interior angle** at a vertex is the angle between the two sides that meet there, measured on the inside of the polygon. The **angle sum** of a polygon is the total of its $n$ interior angles. ### [Angle types and pairs](https://math.wiki/angle-types-and-pairs/) **Summary:** Acute, right, obtuse; complementary, supplementary, vertical. **Question:** Angles rarely appear alone. What does an arrangement of two of them force to be true? ## What this area is about A single angle is just a number of degrees. Almost everything useful about angles comes from how two of them sit together: sharing an edge, sharing a vertex, or lying along one straight line. Certain arrangements force certain measures, and that is what lets you find an angle you were never told. ## How these fit together **Names.** Acute, right, obtuse and straight angles sort angles by size. Congruent angles, angle addition and bisectors says when two angles match and how measures add when an angle is split. Perpendicular lines cross at a right angle; parallel lines lie in one plane and never meet. **Measure pairs.** Complementary and supplementary angles add to $90^\circ$ or $180^\circ$, wherever the two angles are drawn. **Arrangements.** Adjacent angles and linear pairs and vertical angles each describe how two angles can sit, with a fact about their measures. **Using them.** Finding unknown angle measures chains these facts, and a full turn of $360^\circ$, to reach an angle nobody measured, solving for $x$ when measures are expressions: the star to aim at. ## A path through it Sizes, congruent angles, perpendicular and parallel lines first. Then complementary and supplementary angles, linear pairs, vertical angles, and last the angle chase. Finish here and you can work out unmarked angles from the ones you are given. Triangles, parallel lines and transversals and every proof about angle pairs assume it. ### [Angles and how to name them](https://math.wiki/naming-angles/) **Summary:** Reading and writing an angle by its vertex or by three points, with the vertex in the middle. **Question:** Several angles can share one corner of a figure. How do you say which one you mean, so that nobody has to guess? ## When you reach for it An angle is two rays that start from the same point. The point is the **vertex**, and the two rays are the **sides**. What makes an angle worth studying is the opening between the sides: how far one ray has to turn, the shorter way round, to lie along the other. An angle as an amount of turn and measuring angles give that opening a number of degrees. Before a turn can be measured, or two of them compared, you have to say which angle you mean. One angle drawn on its own needs no name: you can point at it. Pointing stops working once a figure has more than one angle. A triangle has three corners; two crossing lines make four angles at one point. To write "this angle is $40^\circ$" or "these two angles are equal", you need a name that picks out exactly one angle and that any reader can trace back to the figure. ## The procedure There are three ways to name an angle. All three are in Figure 1. **By its vertex alone: $\angle B$.** The symbol $\angle$ is read "angle", and the letter is the vertex. This is allowed only because exactly one angle in Figure 1 has its vertex at $B$, so the letter already picks it out. **By three points: $\angle ABC$.** Take a point on one side, then the vertex, then a point on the other side. The **middle letter is always the vertex**; the outer two tell you which rays are the sides, $\overrightarrow{BA}$ and $\overrightarrow{BC}$. This name works in every figure, however crowded, and it is the one to use when in doubt. ### [Angles made by chords, secants and tangents](https://math.wiki/angles-outside-a-circle/) **Summary:** Two secants, a secant and a tangent, or two tangents give an angle equal to half the difference of the arcs. **Question:** An angle at the center of a circle equals its arc, and an angle on the circle is half its arc. What happens when the vertex moves inside the circle, or outside it, or when one side is a tangent? ## Why this exists A central angle has its vertex at the center of a circle, and it measures the same as its arc. An inscribed angle has its vertex on the circle, and it measures half its arc. Those are only two places for a vertex. Two chords can cross anywhere inside a circle, two secants can meet anywhere outside it, and a side of an angle can be a tangent that touches the circle instead of crossing it. Every one of these angles cuts off arcs, and the arcs decide the angle. This lesson finds the rule for each position. A vertex inside the circle gives half the **sum** of two arcs, a vertex outside gives half their **difference**, and a tangent behaves like a chord whose two ends have run together. Together with central and inscribed angles, that covers every angle whose two sides cross or touch a circle. ## Definition Recall from chords, secants and tangents that a **secant** is a line that meets a circle at two points, and a **tangent** is a line that meets it at exactly one point, its point of contact. The angles in this lesson have their sides along chords, secants or tangents, and they are sorted by where the vertex sits. A **tangent-chord angle** has its vertex at the point of contact $A$ of a tangent, one side along the tangent and the other along a chord $\overline{AB}$. Its **intercepted arc** is the arc from $A$ to $B$ that lies inside the angle. In Figure 1, $\angle QAB$ intercepts the heavy minor arc $\overset{\frown}{AB}$, and the angle on the other side of the chord, $\angle PAB$, intercepts the major arc. ### [Arc length](https://math.wiki/arc-length-formula/) **Summary:** Taking the fraction of the full turn an arc covers and applying it to the circumference. **Question:** You can find the distance all the way around a circle. How long is just a piece of the rim, cut off by an angle at the center? ## When you reach for it The circumference formula, $C = 2\pi r$, gives the distance all the way around a circle. Often you need only part of the way around: the curved crust of a slice of pizza, the path of a clock hand's tip over twenty minutes, the bend of a running track, the distance between two cities on the same line of longitude. Each of these is an **arc**, a piece of the circle between two of its points, and what you want is its **arc length**: the distance measured along the curve. Reach for this move when you know how far around the circle the arc goes, as an angle at the center, and how big the circle is, as a radius or a diameter. It also runs backwards. Knowing an arc's length and either one of the other two, you can find the third. From arcs and central angles you already have one number attached to an arc: its measure, in degrees. This lesson adds a second, its length. The two describe different things, and the first job is to keep them apart. ## The procedure ### Arc measure and arc length are different quantities The **measure** of an arc, written $m\overset{\frown}{AB}$, is in degrees. For a minor arc it equals the central angle $\angle AOB$ that cuts it off, where $O$ is the center, and for a major arc it is $360^\circ$ minus that angle. It says how far around the circle the arc goes, as a share of a full turn. It says nothing at all about how big the circle is. The **length** of an arc, written $s$ here, is a distance, in centimeters or meters or miles. It is how far you would walk going along the rim from one end of the arc to the other. ### [Arcs and central angles](https://math.wiki/minor-and-major-arcs/) **Summary:** A chord splits a circle into a shorter arc and a longer one, named with two or three letters. **Question:** Two points on a circle cut it into two curved pieces. How do you say which piece you mean, and how much of the circle it is? ## Why this exists A polygon has corners, so every piece of it has a name: side $\overline{AB}$, angle $\angle B$. A circle has no corners. Put two points $A$ and $B$ on it and the circle falls into two curved pieces, and both of them run from $A$ to $B$. "The curve from $A$ to $B$" no longer says which one you mean. There is a second problem. A curved piece cannot be laid against a ruler, and how long it is depends on how big the circle is. What stays the same when a circle grows is how much of the full turn the piece takes up: a piece that is a quarter of one circle is a quarter of any circle. This lesson settles both. It names the pieces, called **arcs**, so that each name picks out exactly one, and it measures them in degrees, by the angle they make at the center. That single choice ties angles and circles together. Almost every later fact about circles, from arc length to the inscribed angle theorem, is stated in arc measures. ## Definition Take a circle with center $O$, written $\odot O$, and two different points $A$ and $B$ on it. An **arc** is an unbroken piece of the circle. The points $A$ and $B$ split the circle into two arcs, and each of them has **endpoints** $A$ and $B$. An arc is a curve, made of points on the circle and including its endpoints. It is not the region inside. A **central angle** is an angle whose vertex is the center $O$ and whose sides pass through two points of the circle. For $A$ and $B$ it is $\angle AOB$, and its sides run along the radii $\overline{OA}$ and $\overline{OB}$. ### [Area of a circle](https://math.wiki/area-of-a-circle/) **Summary:** Cut into wedges, lay them into a near-rectangle, read off pi r squared. **Question:** A circle has no straight sides to multiply. How much flat surface is inside one? ## Why this exists The area of a rectangle comes from counting rows of unit squares, and every polygon after that is found by cutting it into rectangles and triangles. A circle defeats both moves. It has no straight side to call a base, and however you cut it, curved edges are left over. Unit squares laid on a circle never fit at its edge: some poke out, some leave gaps. Yet round things need areas all the time: how much mat a trampoline has, how much paint a round sign takes, how much lawn a sprinkler soaks. The answer is one short formula, and it needs only the radius. ## Definition The **area of a circle** is the amount of flat surface inside it, measured in square units, the way any area is: square centimeters, square meters, square feet. Strictly, the circle is the curve itself and the region inside it is a disc, but everyday speech says "the area of a circle" for the area of that inside region, and so does this lesson. For a circle of radius $r$, the area $A$ is $$A = \pi r^2$$ where $\pi$ is the same number that links a circle's distance around to its distance across, in the circumference formula $C = \pi d = 2\pi r$. Here $r^2$ means $r \times r$: square the radius first, then multiply by $\pi$. When the diameter $d$ is given instead, the radius is half of it, $r = \frac{d}{2}$, so $$A = \pi\left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}$$ $\pi$ is about $3.14159$, and its decimals never end or repeat, so no decimal is exactly $\pi$. An answer such as $49\pi\text{ ft}^2$ is **exact**. A decimal such as $153.9\text{ ft}^2$ is **rounded**, and is written with $\approx$. The exact form keeps every digit; the rounded form tells you how big the area actually is. ### [Area of a polygon from coordinates](https://math.wiki/area-of-a-polygon-on-the-coordinate-plane/) **Summary:** Boxing a plotted polygon inside a rectangle and subtracting the corner pieces. **Question:** The corners of a shape are given as coordinates, and some of its sides run slantwise across the grid. How much space does it cover? ## When you reach for it A zoo map, a garden plan or a video game level drawn on a coordinate grid often gives a region only as a list of its corners. To find its area you need lengths, and coordinates hand you lengths in just two directions: straight across, from the first numbers, and straight up, from the second numbers. A slanted length needs the distance formula, which comes much later. So the skill is to find an area using **only lengths that run along grid lines**. That is always possible, whatever the polygon and in whichever of the four quadrants its corners sit. This lesson runs over two days. Day 1 handles a polygon with a side along a grid line, where a formula from area-of-triangles or area-of-parallelograms-and-trapezoids applies once its base and height are read off the coordinates. Day 2 handles a polygon whose sides slant, by boxing it inside a rectangle and taking away the corner pieces. ## The procedure ### Day 1: lengths along grid lines **Find the sides that run along grid lines.** Two corners (vertices) with the same second number are joined by a horizontal side; two with the same first number, by a vertical side. Its length is how far apart the other two numbers are, found as in horizontal-and-vertical-distance: add the two absolute values when the ends lie on opposite sides of an axis, subtract the smaller absolute value from the larger when they lie on the same side. ### [Auxiliary lines](https://math.wiki/auxiliary-lines/) **Summary:** A segment added to a figure to create the triangles or parallel lines an argument needs. **Question:** A proof is stuck because the figure is missing the triangle or the parallel line it needs. When may you draw one in, and what may you say about it once it is there? ## Why this exists Some proofs cannot start from the figure they are given. The triangle angle sum is about three angles in three corners, and nothing in a bare triangle puts them side by side. The Base Angles Theorem needs two triangles to compare, and an isosceles triangle is only one. In both, the proof begins by drawing something new: a parallel through the top vertex, a segment that splits the triangle in two. A line, segment or point added to a figure this way is an **auxiliary line**, from the Latin for "helping". It is the most inventive step in geometry, and also the easiest place to cheat, because a line drawn with a pencil can quietly be given properties nothing guarantees. So each added line comes with a reason it exists, and the proof may use only what that reason provides. ## Definition An **auxiliary line** is a line, ray, segment or point added to a figure during a proof. It is not part of the Given, so it is drawn **dashed**, and it must be **justified**: the proof names the fact that guarantees such an object exists. In Figure 1 the left triangle has a line through $A$ parallel to $\overline{BC}$. It exists without any postulate, since copying an alternate interior angle makes one, and it is unique by the parallel postulate. The right triangle has $\overline{AD}$ bisecting $\angle A$, which exists because every angle has exactly one bisector. The matching arcs at $A$ are allowed: the bisector was made that way. ### [Bisecting a segment and an angle](https://math.wiki/construct-a-perpendicular-bisector/) **Summary:** Two equal arcs from the endpoints cross on the perpendicular bisector, which also locates the midpoint. **Question:** With a compass and a straightedge that has no markings, how can you find the exact middle of a segment, or cut an angle exactly in half? ## When you reach for it You have a segment $\overline{AB}$ and you want its midpoint, or the line that crosses it at a right angle through that midpoint, its perpendicular bisector. Or you have an angle and you want the ray that splits it into two equal angles. With a ruler and a protractor you would measure and halve. A construction does it with two tools and no measuring at all: - a **straightedge**, which draws the line through two points and has no marks, so it cannot measure; - a **compass**, which draws a circle or an arc about a chosen center and holds its width, so it can carry one length to another place, as in copying a segment. A measurement is only as good as the scale it is read from. A segment $7.3$ cm long has its middle at $3.65$ cm, and a ruler marked in millimeters cannot show that. A construction finds the point by a reason instead, and the reason is exact: the drawing is as good as the pencil, but the point it aims at is exactly the middle. These two moves are building blocks for most of what follows in constructions: the center of the circle through three points, the center of the circle inside a triangle, the perpendicular through a point, and angles such as $45^\circ$ and $22.5^\circ$. ## The procedure ### Bisecting a segment The construction rests on one fact: **A point that is the same distance from $A$ and from $B$ lies on the perpendicular bisector of $\overline{AB}$.** ### [Building shapes from sticks and clay](https://math.wiki/build-shapes-from-sticks-and-clay/) **Summary:** Shapes are built from sticks for sides and clay balls for corners, so the parts that make a shape can be counted. **Question:** How many sticks and clay balls does it take to build a triangle, a square, a rectangle or a cube, and how does a built shape turn into a drawing? ## When you reach for it A flat shape is made of straight sides that meet at corners. Build one with craft sticks for the sides and small balls of clay for the corners, and both parts can be held, counted and moved: a triangle takes $3$ sticks and $3$ balls, a square or rectangle $4$ and $4$. Reach for this once a child can count the sides and corners of a shape; building is the same counting done with the hands, and it ends with drawing the shape on paper. **You need, all week:** craft sticks or drinking straws (some cut in half, so there are long and short ones), and play dough or clay rolled into balls about the size of a marble. Each day takes ten to fifteen minutes. ## Day 1: Building a triangle Show how a ball holds two stick ends together: push one stick into each side of the ball. Say, "The ball is a corner. The sticks are sides." Then build a triangle together, one stick and one ball at a time, and close it by pushing the last stick into the first ball. Ask, "How many sticks did we use? How many balls?" Count each with a finger. Then let each child build a triangle from any three sticks: a long one, a short one, whatever is at hand. Line the triangles up and ask what is the same about all of them. Every one has $3$ sticks and $3$ balls, however tall, wide or turned it is. **Practice.** 1. Triangle A. How many sticks and balls? **Answer:** $3$ sticks and $3$ balls. 2. Triangle B. How many sticks and balls? **Answer:** $3$ sticks and $3$ balls, even though it is tall and thin. 3. Triangle C. How many sticks and balls? **Answer:** $3$ sticks and $3$ balls. 4. You have $3$ sticks but only $2$ balls. Can you build a triangle? **Answer:** Not yet. One corner has nothing to hold it, so you need $1$ more ball. 5. Build a triangle from two long sticks and one short stick. **Answer:** It works: a tall, thin triangle, still with $3$ sticks and $3$ balls. ### [Cavalieri's principle](https://math.wiki/cavalieris-principle/) **Summary:** Two solids whose cross sections match in area at every height have the same volume, even if one leans. **Question:** A stack of coins pushed into a lean is still the same coins. How do you find the volume of a solid that leans, when no box of unit cubes fits it? ## Why this exists The volume of an upright prism comes from stacking. The base is one layer, the prism is that layer repeated all the way up, and so $V = Bh$ (volume-of-a-prism). The argument leans on the solid standing straight: each layer sits exactly on the one below, and a box of unit cubes fits. Now push the stack sideways. A pile of coins, a deck of cards, a ream of paper: lean it over and it is still the same coins, the same cards, the same paper. Nothing was added and nothing was taken away, so the volume has not changed. But the leaning shape is no longer a box, no grid of unit cubes fits it neatly, and $V = Bh$ was worked out for the upright stack only. Cavalieri's principle turns the coin stack into a tool. It compares two solids slice by slice, at every height, and says when their volumes must agree. With it, leaning prisms and cylinders get their volumes, pyramids and cones can be compared with one another, and the volume of a sphere comes within reach without calculus. It is named after Bonaventura Cavalieri, an Italian mathematician who worked with it in the 1630s. ## Definition A **cross section** of a solid is the flat shape where a plane cuts through it (cross-sections-of-prisms-and-pyramids). Choose a direction to call "up" and cut with planes at right angles to it. The cross section at height $y$ has an area, written $A(y)$. **Cavalieri's principle.** Suppose two solids lie between the same two parallel planes. If every plane parallel to those two cuts the solids in cross sections of equal area, then the two solids have equal volume. ### [Centroid](https://math.wiki/centroid/) **Summary:** Where the three medians meet, two thirds of the way along each from its vertex, and the balance point of the triangle. **Question:** A triangle cut from stiff card balances on a pencil point at exactly one spot. Where is that spot, and can it be found with a ruler or with arithmetic instead of by trial? ## Why this exists Cut a triangle out of stiff card and try to balance it flat on the point of a pencil. Almost everywhere it tips over. There is exactly one spot where it stays level, and finding it by trial takes patience and gives only a rough answer. Geometry finds it exactly, in two quite different ways. With a ruler: draw the medians, the segments from each corner to the midpoint of the opposite side, and the spot is where they cross. With arithmetic: if the corners have coordinates, average them, and the average is the spot. Neither of those ought to work as well as it does. Two lines that are not parallel cross at one point, but a third line drawn through the picture almost always misses that point. That the third median **always** passes through the crossing of the other two is a theorem, not a lucky drawing. That an average of three corners lands on all three medians at once is a second surprise. This lesson proves both, and shows that they are one fact seen two ways. The point is the **centroid**. It is one of the classic centers of a triangle, beside the circumcenter and the incenter, and it is the one that says where the triangle's weight sits. ## Definition A **median** of a triangle joins a vertex to the midpoint of the opposite side, and every triangle has three. The **centroid** of a triangle is the point where its three medians meet. It is usually named $G$. In Figure 1, $D$, $E$ and $F$ are the midpoints of $\overline{BC}$, $\overline{CA}$ and $\overline{AB}$ (matching tick marks show the equal halves), so $\overline{AD}$, $\overline{BE}$ and $\overline{CF}$ are the medians, and they meet at $G$. ### [Chord theorems](https://math.wiki/perpendicular-from-center-bisects-chord/) **Summary:** A radius drawn perpendicular to a chord cuts that chord and its arc in half. **Question:** The center of a circle cannot be seen on the circle itself, but its chords can. What does a chord give away about where the center is, and how long the chord must be? ## Why this exists A circle is drawn around a center, but once it is drawn the center leaves no mark on it. A broken plate, the rim of a round table, the curve of a stone arch: each is a piece of a circle whose center is nowhere to be seen. What can be reached are chords, the straight segments between two points of the curve, because those can be measured with a ruler. The theorems here tie each chord to the hidden center. The center always sits on the line that cuts a chord in half at right angles, so two chords are enough to find it. And how long a chord is depends on nothing but its distance from the center, so equal chords sit equally far out, equal chords cut equal arcs, and a chord, its arc and its distance from the center can each be worked out from the others. ## Definition Fix a circle with center $O$ and radius $r$, and a chord $\overline{AB}$. - The **central angle** of the chord is $\angle AOB$, the angle at the center between the radii to its ends. When $\overline{AB}$ is a diameter this is a straight angle, $180^\circ$. - The **arcs** of the chord are the two arcs it cuts the circle into: the minor arc $\overset{\frown}{AB}$, measuring the same as $\angle AOB$, and the major arc, measuring $360^\circ$ minus that. When the chord is a diameter both are semicircles. - The **distance from the center to the chord** is the length of the perpendicular segment from $O$ to the line through $A$ and $B$, the distance from a point to a line. Its foot is called $M$ below; Result 1 shows that $M$ is the midpoint of the chord. A diameter is at distance $0$. - Two circles are **congruent** when their radii are equal. ### [Chords, secants and tangents](https://math.wiki/chord/) **Summary:** A segment whose two endpoints both sit on the circle. **Question:** A straight line and a circle lie on the same page. In how many points can they meet, and what decides which? ## Why this exists A circle already has one named segment that runs across it: a chord, with both ends on the curve. But a segment stops at its ends, and a straight line does not. Lay a ruler across a coin and three different things can happen. The ruler can cut straight across, going in at one point and out at another. It can rest against the edge and touch it at a single point. Or it can lie beside the coin and miss it altogether. Those three cases have names, secant, tangent and a line that misses, and every later fact about circles is phrased in them: the angles that chords and tangents make, the lengths of their pieces, the right angle a tangent makes with a radius. This lesson sets out the words and answers the question the coin raises: before you draw anything, how can you tell which case you are in? One number decides it, the distance from the center to the line, compared with the radius. ## Definition Take a circle with center $O$ and radius $r$, and a straight line in the same plane. - A **chord** is a segment whose two endpoints both lie on the circle. A chord that passes through the center is a diameter. - A **secant** is a line that meets the circle at exactly two points. The part of a secant between those two points is a chord, and every chord lies on exactly one secant, the line through its two ends. The difference is that a chord is a segment and has a length, while a secant is a line and runs on past the circle in both directions. - A **tangent** is a line that meets the circle at exactly one point. That point is the **point of tangency**, and the line is said to be tangent to the circle **at** that point. A segment or ray is also called tangent when it lies along a tangent line and contains the point of tangency. - A line can also meet the circle at **no** points. It has no special name; it misses. - **Concentric circles** are circles in the same plane that share a center and have different radii. ### [Circles](https://math.wiki/circles/) **Summary:** Circumference, area, and the angle and chord theorems inside a circle. **Question:** A circle has no corners and no straight sides. What can you still measure on it, and what does its roundness force to be true? ## What this area is about A circle is every point in a plane at one distance $r$ from a center. That single condition fixes its size, the lengths and areas of its pieces, and a surprising number of angle and length facts about lines that cut or touch it. ## How these fit together **Parts and names.** Radius and diameter, chords, secants and tangents and arcs and central angles name what can be drawn on a circle. **Size.** Circumference defines $\pi$ and gives $C = 2\pi r$; area gives $A = \pi r^2$. Arc length takes a fraction of the circumference, and sectors and segments a fraction of the area (a segment is a sector minus a triangle). **Angle results.** The inscribed angle theorem and angles made by chords, secants and tangents tie angles to arcs. Tangent properties start from the right angle a tangent makes with the radius at its point of contact. A quadrilateral is cyclic, its corners on one circle, exactly when its opposite angles add to $180^\circ$. **Length results.** Chord theorems and segment lengths in circles. **Coordinates.** The equation of a circle and finding the center and radius. ## A path through it Radius and diameter first, then chords and circumference. Circumference comes before area and arcs; arcs before arc length and the inscribed angle theorem; area, arcs and chords before sectors. Tangent properties follow chords. After the inscribed angle theorem and tangent properties come angles made by chords, secants and tangents, and the chord theorems, then segment lengths. The equation comes before finding the center and radius. Cyclic quadrilaterals have no fixed place. ### [Circumcenter](https://math.wiki/circumcenter/) **Summary:** Where the perpendicular bisectors meet, the one point the same distance from all three vertices. **Question:** Three points not on one line. Is there always a circle through all three, and where is its center? ## Why this exists Pick three points that are not on one line. Is there a circle that passes through all three? Someone holding three pieces of a broken plate wants to know where the plate's center was. Someone choosing a site for a radio mast that must be the same distance from three towns is asking the same thing. Both want one point that is equally far from three given points. For two points the answer is already known. By the perpendicular bisector theorem, the points equally far from $A$ and $B$ are exactly the points of the perpendicular bisector of $\overline{AB}$, a locus that is a straight line. Three points give three such lines, one for each pair. Two lines that are not parallel always meet, so two of the three conditions can always be met at once. The real question is whether the third line passes through that same point. It always does, as the perpendicular bisector lesson proves. That point is the circumcenter, and it settles the plate, the mast and every other "equally far from three points" question in one move. This lesson is about what that point does: the circle it carries, where it sits for each shape of triangle, and how to find it. ## Definition Three or more lines are **concurrent** when they all pass through one point. That point is their **point of concurrency**. Two lines that are not parallel always meet, so the word only has bite for three or more lines: concurrency says that a third line goes through the point the first two already share. ### [Circumference](https://math.wiki/circumference/) **Summary:** Pi as the ratio of around to across, and the formula that follows. **Question:** Measuring across a circle is easy and measuring around it is awkward. Is the distance around always the same multiple of the distance across, and if so, what is that number? ## Why this exists The distance across a round thing is easy to measure: lay a ruler over the middle. The distance around it is not. A ruler will not bend, a tape slips off a curved edge, and nobody can wrap a tape around a pond or a Ferris wheel. So it would be very useful if the distance around were always the same multiple of the distance across: one easy measurement times one fixed number would give the hard one. That is exactly how it turns out, for every circle there is, and the fixed number is called pi. This lesson is about that number and what it lets you work out, from the length of a fence to the distance a bicycle travels. The lesson is taught over two days. **Part 1** is the number itself and the formula it gives. **Part 2** runs the formula backwards and uses it on pieces of circles and on turning wheels. ## Definition The **circumference** of a circle is the distance once around it: the length of the circle itself. It is a length, measured in centimeters, meters or any other unit of length. It is written $C$. The diameter $d$ is the distance across the circle through its center, and the radius $r$ is the distance from the center to the circle, so $d = 2r$. **Pi**, written $\pi$, is the circumference of a circle divided by its diameter: $$\pi = \frac{C}{d}$$ This defines one number, not one number per circle, because the division gives the same answer for every circle (see Why it is true). Its value begins ### [Classifying figures from coordinates](https://math.wiki/classify-a-quadrilateral-on-the-plane/) **Summary:** Using slopes and distances to decide which quadrilateral four plotted points form. **Question:** Four points are plotted on a grid. How can you be sure which shape they make, without trusting your eyes? ## When you reach for it A drawing can only suggest a shape. Two sides that look parallel may differ in slope by a hair, and a corner that looks square may be $87^\circ$. Once the corners are given as coordinates, though, there is no need to guess: every property in the quadrilateral family tree can be tested with arithmetic. - Parallel sides are a question about **slopes**. - Right angles are a question about **slopes** too: perpendicular lines have slopes that multiply to $-1$. - Equal sides and equal diagonals are a question about **lengths**. - Diagonals that cut each other in half are a question about **midpoints**. Reach for this whenever points are plotted and a question asks what figure they make: is this triangle right, is it isosceles, is this quadrilateral a parallelogram, a rectangle, a rhombus, a square, a trapezoid or a kite? The answer is then not "it looks like one" but a short argument that it meets the definition, or that it fails one property every such figure must have. ## The procedure ### Three measurements Everything rests on three formulas you already have. The slope of the segment from $(x_1, y_1)$ to $(x_2, y_2)$ is $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ and it is undefined when $x_1 = x_2$, for a vertical segment. Two segments are **parallel**, $\parallel$, when their slopes are equal and they do not lie on one line, or when both are vertical and not on one line. Two segments are **perpendicular**, $\perp$, when their slopes multiply to $-1$, or when one is horizontal and the other vertical (perpendicular-slopes). That last case has to be checked separately, since a vertical slope cannot be multiplied. ### [Classifying shapes in a hierarchy](https://math.wiki/two-dimensional-shape-hierarchy/) **Summary:** Flat shapes, triangles and four-sided shapes included, are sorted into a family tree where each group sits inside the one above it. **Question:** A square is also a rectangle, a rhombus and a parallelogram. How can one shape belong to so many groups, and how do the groups fit inside one another? ## Why this exists The museum of shapes has a sorting game. Each visitor gets a stack of cards, and each card shows a shape along with the facts about it: how many sides, which sides are equal, which sides are parallel, and how big the angles are. Around the room stand bins with labels such as "rectangle", "rhombus" and "triangle". Fatoumata runs the game, and Kahiau and Murat play it. Kahiau picks up a card showing a square and stops. It fits the "rectangle" bin, because it has four right angles. It fits the "rhombus" bin, because its four sides are equal. It fits "parallelogram" and "quadrilateral" too. A shape can belong to more than one group at once, as in shapes-in-more-than-one-category. So which bin is right? All of them are. The groups are not separate piles. They fit inside one another, like boxes inside boxes. A map of which group sits inside which is called a **hierarchy**. With that map, one look tells you every group a shape belongs to, and so every fact it is promised. ## Definition A **property** of a shape is a fact about its sides or angles, such as "has four right angles" or "has two pairs of parallel sides". A **category** is the group of all shapes that have a certain list of properties. A category $B$ is a **subcategory** of a category $A$ when every shape in $B$ is also in $A$. This happens when $B$ asks for every property that $A$ asks for, and more. The rectangles are a subcategory of the parallelograms, because a rectangle is a parallelogram that also has four right angles. ### [Classifying triangles](https://math.wiki/classify-triangles-by-angles/) **Summary:** Sorting triangles into acute, right and obtuse. **Question:** Every triangle has three sides and three angles. Which two comparisons sort all of them into families, and what does each name tell you? ## When you reach for it Triangles come in endless shapes, but two plain questions sort every one of them into families. **How do its three sides compare with each other?** That gives scalene, isosceles or equilateral. **How does its largest angle compare with a square corner?** That gives acute, right or obtuse. Every triangle gets one name from each list, as in "an isosceles right triangle" or "a scalene obtuse triangle". You reach for this sort whenever a triangle is in front of you, drawn or described, and you need to say what kind it is, because the kind tells you what you are allowed to use. An isosceles triangle hands you two equal sides, and an equilateral one hands you all three. A right triangle opens the Pythagorean theorem and right triangle trigonometry. A triangle with no right angle, acute or obtuse, is called oblique and needs other tools. Some theorems apply to one family only, such as the base angles theorem for isosceles triangles. In a proof, "let $\triangle ABC$ be acute" is a real assumption, and it is worth knowing what it rules out. You also use the names backwards. Told that a triangle is isosceles with a perimeter of $84$, you can write one equation with one unknown. Told that a triangle has an angle of $110^\circ$, you know at once what its other two angles must be like. ## The procedure The two sorts are separate questions, each with its own test. They are taken one at a time below, and then put together. ### [Compass and straightedge constructions](https://math.wiki/constructions/) **Summary:** Copy a segment, bisect an angle, drop a perpendicular, with no ruler markings. **Question:** With only a compass and a straightedge that has no markings, what can you draw exactly, and how do you know it is exact? ## What this area is about A construction is a drawing made with two tools: a straightedge, which draws the line through two points and cannot measure, and a compass, which draws circles and holds one width. Nothing is read off a scale, so every step can be justified by a definition or a theorem, and the result is exact for every starting figure, not just the one on the page. ## How these fit together **The basic moves.** Copying a segment and an angle carries a length or an angle to a new place. Bisecting a segment and an angle finds a midpoint and splits an angle in half. **Building on them.** Perpendicular and parallel lines through a point, constructing a triangle from given parts, the equilateral triangle, square and hexagon in a circle, constructions with circles and dividing a segment into $n$ equal parts are procedures that reuse the basic moves. **Why they work.** Why a construction works is the concept behind all of them: each step creates equal distances or one fixed line, and a proof, often through congruent triangles, turns those into the claimed property. ## A path through it Copying comes first, before bisecting and before constructing a triangle from given parts. Bisecting comes before perpendicular and parallel lines, constructions with circles, and why a construction works. Perpendicular and parallel lines come before the equilateral triangle, square and hexagon, and before dividing a segment into equal parts. ### [Complementary and supplementary angles](https://math.wiki/complementary-angles/) **Summary:** Two angles whose measures add to a right angle. **Question:** A square corner or a straight line is cut in two. If you know one piece, what is the other? ## Why this exists Two shapes turn up in almost every figure in geometry. One is the square corner: the corner of a page, a wall meeting a floor, the two lines of a plus sign. Each is a right angle, exactly $90^\circ$. The other is the straight line, which as an angle measures $180^\circ$. Put a ray through either one and you cut it into two pieces. You do not need to measure both. The pieces have to fill the whole between them, so once you know one, the other is whatever is left of $90^\circ$, or of $180^\circ$. That trade happens so often, in rectangles, in right triangles, on every pair of perpendicular lines, every time a ray stands on a line or a side of a triangle is extended, that each kind of pair gets a name of its own. The names are worth having even when the two angles are nowhere near each other. Two angles can be the two pieces of one corner or one line in spirit without ever touching on the page, and the names let you say so in one word. ## Definition ### Complementary angles Two angles are **complementary** when their measures add to $90^\circ$. Each angle is then the **complement** of the other. In symbols, $\angle A$ and $\angle B$ are complementary when $$m\angle A + m\angle B = 90^\circ$$ The two angles do not have to touch. In Figure 1 the $35^\circ$ and $55^\circ$ angles on the left share a vertex and a side, so they are adjacent and together they fill a right angle. The pair on the right sit apart and point different ways, and they are just as complementary. The word is about the two numbers, not about where the angles are drawn. ### [Composing and decomposing shapes](https://math.wiki/compose-and-decompose-shapes/) **Summary:** Building a new flat shape out of smaller ones and cutting a shape into pieces. **Question:** How can a child put small flat shapes together to make a bigger one, and take a shape apart into smaller ones? ## When you reach for it Shapes can be put together to make bigger shapes (composing) and taken apart into smaller ones (decomposing). Two triangles can make a square, two squares make a rectangle, and a square folded and cut makes triangles or smaller squares. Reach for this once a child can name circles, triangles, squares and rectangles; it shows that a shape is made of parts, which is where halves, fourths and area will later come from. Each day takes ten to fifteen minutes. Pattern blocks are ideal if you have them; paper shapes cut from card work just as well. Let the child move the pieces with their own hands every time. ## Day 1: Two triangles make a new shape **You need:** several pairs of paper triangles, each pair cut from one paper square by cutting it corner to corner (cut these ahead of time). Give the child one pair. Say, "Put these two triangles together so their sides touch all the way along. What shape did you make?" Then, "Can you make a different shape with the same two triangles?" Matching a different pair of sides, or turning one triangle around, gives a new shape. Ask each time, "How many sides? How many corners? What is its name?" There is no single right answer here. The point is that the same two pieces can make several shapes, and that a shape's name comes from its sides and corners, not from the pieces inside it. **Practice.** For the first three, ask what shape the two triangles make. ### [Composing transformations](https://math.wiki/composition-of-transformations/) **Summary:** Applying one transformation and then another, where the order can change the result. **Question:** A figure is slid and then flipped, or turned about one point and then about another. Where does it end up, and is there one motion that does the whole job? ## When you reach for it A single translation, reflection or rotation is easy to describe. Many moves are not one of those on its own. A drawing program rotates a shape and then drags it somewhere else. A row of footprints in sand goes left foot, right foot, left foot, and each print is the one before it flipped and moved forward. A triangle on a grid sits where it does because it was turned and then slid, or flipped and then turned. **Composing** two motions means doing one and then doing the other to the result. You reach for it in three situations. - A sequence of motions is given, and you need to know where the figure ends up. - A sequence is given, and you want **one** motion with the same effect: to name it, to undo it, or to repeat it. - Two figures are given, and you need a sequence that carries one onto the other. That is exactly what it takes to show the figures are congruent. The lesson runs over two days. Part 1 is doing motions in a row: following points, writing the combined coordinate rule, seeing when the order matters, and undoing a sequence. Part 2 is naming the single motion a sequence amounts to, which brings in the glide reflection, the one rigid motion that is not a slide, a turn or a flip. ## The procedure ### Part 1: one motion after another **Composition.** If $f$ and $g$ are motions of the plane, their composition $g \circ f$ sends each point $P$ first through $f$, to $P' = f(P)$, and then through $g$, to $P'' = g(P')$: ### [Composite solids and density](https://math.wiki/volume-of-composite-solids/) **Summary:** Splitting a shape such as a silo into familiar solids and then adding or subtracting the pieces. **Question:** A grain silo is a can with a dome on top, and a pipe is a can with a hole down its middle. No single formula fits either. How do you find how much they hold, and how heavy they are? ## When you reach for it Most solid objects are not one textbook shape. A grain silo is a cylinder with a dome for a roof; a candle may be a cylinder that ends in a cone; a pipe is a cylinder with a narrower cylinder missing from its middle. No single formula gives their volumes, but each is built from solids that do have formulas, and volume behaves exactly as area does in composite figures: cut a solid into pieces that do not overlap, and the volumes of the pieces add up to the volume of the whole. Reach for it whenever a solid is "this shape on top of that shape" or "this shape with that shape taken out". The formulas for the pieces come from earlier lessons: prisms in volume-of-a-prism, cylinders in volume-of-a-cylinder, pyramids and cones in volume-of-a-pyramid, and balls and half balls in volume-of-a-sphere. A volume says how much space an object fills. Often the real question is how heavy it is, and that needs one more fact about what it is made of: its **density**, the mass packed into each cubic centimeter. With a density, a volume turns into a mass, a mass turns back into a volume, and objects made of different materials can be compared. This lesson takes two days. **Part 1** splits solids into pieces and adds or subtracts their volumes. **Part 2** brings in density: finding a mass from a volume, a volume from a mass, comparing objects, and weighing hollow ones. Examples 1 to 4 and Exercises 1 to 13 belong to Part 1; Examples 5 to 7 and Exercises 14 to 22 belong to Part 2. ### [Congruence statements and correspondence](https://math.wiki/congruence-statements-and-correspondence/) **Summary:** Writing a congruence statement so that the order of the letters names which parts match. **Question:** Two triangles are exactly the same size and shape. When you write that down, how does the reader know which corner of one goes with which corner of the other? ## When you reach for it Look at the two triangles in Figure 1. The one on the right is the one on the left, flipped over and turned. They are the same size and the same shape, so they are **congruent**. But saying "the triangles are congruent" leaves out the part you actually need: which side of one matches which side of the other, and which angle matches which angle. A triangle has six parts, three sides and three angles. To say two triangles are congruent is to say that their parts can be paired off, vertex to vertex, so that **all six pairs match**. **Definition.** Two triangles are **congruent** when their vertices can be matched one to one so that each pair of matched sides is congruent and each pair of matched angles is congruent. The matching of vertices is called a **correspondence**. In Figure 1 it is $$A \leftrightarrow D \qquad B \leftrightarrow E \qquad C \leftrightarrow F$$ and the marks show the six pairs it produces: matching ticks for sides, as for congruent segments, and matching arcs for angles, as for congruent angles. Writing out six pairs every time would be slow, so geometry packs the whole correspondence into the order of the letters: $$\triangle ABC \cong \triangle DEF$$ This is a **congruence statement**. The first letter on the left goes with the first letter on the right, the second with the second, the third with the third. One line holds all six facts, and a reader can list them without seeing the figure. ### [Congruent angles, angle addition and bisectors](https://math.wiki/congruent-angles/) **Summary:** Two angles are congruent when they have equal measure. **Question:** Two angles sit in different places, a ray splits an angle into pieces, a fold halves a corner. When are two angles the same size, how do the pieces add up to the whole, and what does a ray that cuts an angle exactly in half tell you? ## Why this exists Look at the two angles in Figure 1. One has short arms and opens up and to the right. The other has arms nearly twice as long and opens to the left. Are they the same? They are not the *same angle*: they sit in different places, and one is made of different rays from the other. But they open by exactly the same amount, and in almost every problem that is the only thing that matters. Geometry needs a word for "same size and shape, wherever it is drawn", so that it can say this without pretending the two angles are one object. That word is **congruent**, and angles are the first place most people meet it. The same word does the same job for segments (same length), and later for whole triangles, where it carries most of the weight of a proof. Two more facts about angle sizes are needed before any real angle problem can be written down as a proof. You already know that angle measures add: put two angles side by side and the angle around both is the total. But "it looks like they add" is not a reason anyone can write down, and every step of a proof has to point at something already agreed. The **angle addition postulate** is that agreed thing. It says exactly when two measures add to a third, and it is the one fact you cite every time you split an angle, glue two together, or work out a piece from the whole. The third fact is about splitting an angle *equally*. Fold a paper corner so that one edge lies along the other, and the crease splits the corner into two equal parts. A carpenter joining two boards at a corner cuts each board along that same crease, so the two cut ends meet flush. The crease is the **angle bisector**, and it turns up far more often than the paper corner suggests, because it is the only line that treats the two sides of an angle exactly alike. ### [Congruent triangles](https://math.wiki/congruent-triangles/) **Summary:** Same shape, same size: every side and angle matches. **Question:** Two triangles are copies of each other. How do you say which parts match, and how few matching parts does it take to be sure? ## What this area is about Two triangles are congruent when their vertices can be matched so that all six pairs of parts, three sides and three angles, are congruent. Six checks is a lot. This area is about writing the matching down, and which shorter lists of matching parts do, or do not, force one. ## How these fit together **Notation.** Congruence statements and correspondence: letter order packs the matching, so $\triangle ABC \cong \triangle DEF$ pairs $A$ with $D$, $B$ with $E$ and $C$ with $F$. **Short lists.** The criteria sit next door: SSS, SAS and ASA are postulates, and AAS is a theorem from ASA and the Third Angle Theorem. Why AAA and SSA fail covers the other two: three angles fix the shape but not the size, and two sides with an angle not between them can build two different triangles. Hypotenuse leg congruence is the right-angle case of SSA, which is safe: a theorem for right triangles, proved from the Pythagorean theorem and SSS. **Using a congruence.** Once two triangles are congruent, every matched part is, including pairs nobody checked. That fact lives with the proofs, in proving parts congruent after the triangles. ## A path through it Congruence statements first, then the criteria next door. Then why AAA and SSA fail, and last hypotenuse leg congruence, best read once you have seen SSA fail. Finish here and you can prove triangles congruent, and then use it to prove sides and angles equal. The AAA lesson points on to similarity, where same shape is enough. ### [Conic sections](https://math.wiki/conic-sections/) **Summary:** Slice a cone and you get a circle, an ellipse, a parabola, or a hyperbola. **Question:** A circle, an oval, a dish-shaped curve and a curve in two pieces look unrelated. What do they have in common? ## What this area is about Cut a double cone with a flat plane and, when the plane misses the vertex, the cut is a circle, an ellipse, a parabola or a hyperbola, depending only on its tilt. These four curves are the conic sections. Each can also be described by distances in the plane, and that description turns it into an equation in $x$ and $y$. ## How these fit together **The family.** Conic sections from a double cone shows all four as slices of one solid. Focus and directrix gives one planar rule for the three non-circles: the points with $PF = e \cdot PL$, where $e < 1$ gives an ellipse, $e = 1$ a parabola and $e > 1$ a hyperbola. **One curve at a time.** The ellipse keeps the sum of distances to two foci fixed, the hyperbola keeps their difference fixed, and the parabola as a locus keeps a point equally far from a focus and a directrix. Each gets its standard equation. **The procedure.** Identifying and translating conics reads an expanded equation $Ax^2 + Cy^2 + Dx + Ey + F = 0$, with $A$ and $C$ not both $0$, and says which curve it is and where it sits. ## A path through it Two strands meet at the end. One runs from the double cone to focus and directrix, then to the parabola as a locus. The other runs from the ellipse to the hyperbola. Identifying conics comes last, after both the parabola and the hyperbola. The circle itself is studied in the circles area, and the identifying lesson uses its equation as well. ### [Conic sections from a double cone](https://math.wiki/the-double-cone/) **Summary:** Every conic comes from cutting a double cone at a different angle. **Question:** A flashlight beam on a wall can make a round spot, a stretched oval, or a patch that never closes. What decides which one you get? ## Why this exists A circle closes up and is round. An ellipse closes up and is stretched. A parabola is one open curve, and a hyperbola is two open pieces. They look like four unrelated curves, yet Greek geometers, Apollonius of Perga above all, studied all four as cuts through one solid, long before coordinates existed. In cross sections of solids, a cone cut straight across gave a circle and a cut on a slant gave an ellipse. This lesson finishes that story. Cut a cone that runs on without end in both directions, and only the **tilt** of the plane changes from one curve to the next. One comparison of two angles decides the curve, which is why the four are one family, the **conic sections**. ## Definition Take a line, the **axis**, and a second line crossing it at a point $V$ at an angle $\alpha$, with $0^\circ < \alpha < 90^\circ$. Spin the second line all the way around the axis, keeping the angle fixed. The surface it sweeps out is a **double cone** (a right circular double cone). - $V$ is the **vertex**. - Each position of the spinning line is a **side line**. Every side line passes through $V$ and makes the angle $\alpha$ with the axis, the cone's **half-angle**. - The two halves that meet at $V$ are the **nappes**. Both run on without end. A **conic section** is the set of points where a plane meets the double cone. The plane's tilt is measured by $\beta$, the **angle between the plane and the axis**: the smallest angle the axis makes with any line in the plane, so $0^\circ \le \beta \le 90^\circ$. A plane perpendicular to the axis has $\beta = 90^\circ$; a plane parallel to the axis, or containing it, has $\beta = 0^\circ$. ### [Constructing a triangle from given parts](https://math.wiki/construct-a-triangle-from-three-sides/) **Summary:** Arcs of the two other lengths swung from the ends of the first meet at the missing vertex. **Question:** You are handed three lengths, or two lengths and an angle, or two angles and a length. Is there a triangle with exactly those parts, how do you draw it with only a compass and a straightedge, and is the one you draw the only one? ## When you reach for it A triangle has six parts: three sides and three angles. The congruence-criteria say that some sets of three parts already decide all six: three sides (SSS), two sides and the angle between them (SAS), two angles and the side between them (ASA). This lesson is the other half of that statement. A criterion says two triangles with those parts must match. A construction shows that a triangle with those parts exists, and draws it. Reach for it whenever a figure has to be built from given parts: laying out a triangle whose sides are given lengths, deciding whether three lengths can be the sides of a triangle at all, or starting a larger construction whose first move is a triangle with chosen sides. The tools are those of every construction: an unmarked straightedge and a compass that holds its opening. The given parts arrive as drawn segments and drawn angles. The two moves used throughout, setting the compass to a segment and swinging it from a new point, and copying an angle onto a ray, are taught in copying a segment and an angle. Numbers such as $6$ cm appear in the examples only so the results can be checked; no step reads a scale. With a ruler and protractor instead, the same questions are the subject of draw-triangles-from-conditions. In $\triangle ABC$ the side opposite each vertex is named by the matching small letter, so $a = BC$, $b = CA$ and $c = AB$. ## The procedure ### [Constructions with circles](https://math.wiki/construct-the-circumscribed-circle/) **Summary:** Two perpendicular bisectors locate the circumcenter, which is the center of the circle. **Question:** With only a compass and a straightedge, how do you draw the circle through the three corners of a triangle, the circle inside it that touches all three sides, and the lines from an outside point that just touch a circle? ## When you reach for it Three situations call for a circle whose center nobody has marked: - You have three points, the corners of a triangle, and you want the one circle that passes through all three. It is the **circumscribed circle** of the triangle, and its center is the circumcenter. - You have a triangle and you want the largest circle that fits inside it, touching each side once. It is the **inscribed circle**, and its center is the incenter. - You have a circle and a point outside it, and you want the lines through the point that touch the circle at exactly one point each. They are the **tangents** from the point. A compass draws a circle about a center you already have; it cannot find one. So in each case the work is to locate a point exactly, using only the moves of bisecting a segment and an angle, and then the compass does the rest. The same construction finds the center of anything round from a piece of its rim, such as a broken plate (Example 6). All three rest on one idea: a point fixed by equal distances. A point the same distance from two points lies on a perpendicular bisector; a point the same distance from two lines lies on an angle bisector; and a point where a line meets a radius at a right angle is a point of tangency. ## The procedure ### The circle through three corners The construction rests on one fact, the perpendicular bisector theorem: **A point is the same distance from $A$ and from $B$ exactly when it lies on the perpendicular bisector of $\overline{AB}$.** ### [Converse of the Pythagorean theorem](https://math.wiki/pythagorean-converse/) **Summary:** If the squares add up, the angle is right. **Question:** A triangle's three sides satisfy the Pythagorean equation. Does that alone force a square corner? ## Why this exists The Pythagorean theorem starts from a right angle and ends with an equation: if the corner is square, then the squares on the two short sides add up to the square on the long one. But people often meet the problem the other way round. A corner has been marked out on a floor, or drawn in a diagram, and nobody knows whether it is square. What they can do is measure lengths, and lengths are far easier to measure than angles. So the question is whether the equation can be used backwards. Measure two sides and the gap between their ends. If the squares add up, is the corner square? That is not a formality. A true statement turned round is often false: every square is a rectangle, but not every rectangle is a square. Turning a theorem round gives its **converse**, and each converse has to be proved on its own (a theorem and its converse). Here the turned-round statement turns out to be true, and it is the reason a string knotted into $12$ equal spaces, pulled tight into a triangle with sides of $3$, $4$ and $5$ spaces, makes a square corner every time. ## Definition The **converse** of a statement "if $P$, then $Q$" is the statement "if $Q$, then $P$" (conditionals and their converses). The Pythagorean theorem says: if a triangle has a right angle, then the squares on the two sides that meet at it add up to the square on the side opposite it. Its converse starts from the lengths and ends at the angle. ### [Converses and biconditional definitions](https://math.wiki/theorem-and-its-converse/) **Summary:** A true theorem can have a false converse, so each direction needs its own proof. **Question:** A theorem says that if one thing is true of a figure, then another is. Once it is proved, what do you know about the statement that runs the other way, and why may a definition always be run backwards when a theorem may not? ## Why this exists Every line of a proof needs a reason, and the reasons come in kinds that behave differently when you turn them around. The most common reason in a geometry proof is a definition. Told that $M$ is the midpoint of $\overline{AB}$, you write $AM = MB$, reason "definition of midpoint". Shown that a point $M$ on $\overline{AB}$ has $AM = MB$, you write "$M$ is the midpoint of $\overline{AB}$", and the reason is again "definition of midpoint". The same reason runs in both directions. Theorems do not behave like that. Two theorems from the same chapter of geometry look alike: - If two angles are vertical angles, then they are congruent. - If two lines cut by a transversal are parallel, then their corresponding angles are congruent. Turn each one around. "If two angles are congruent, then they are vertical angles" is false: two $30^\circ$ angles drawn in different corners of a page are congruent and nowhere near each other. "If corresponding angles are congruent, then the lines are parallel" is true, and it is the standard way to show that two lines are parallel. Same shape of sentence, opposite fates. So a theorem and its converse are two claims, each needing its own proof, and the converses that come back true become geometry's tests. A definition runs both ways for a different reason: it is not a fact someone found out but an agreement about what a word will mean, and the agreement covers both directions at once. ### [Coordinate geometry](https://math.wiki/coordinate-geometry/) **Summary:** Geometry done with numbers on a grid. **Question:** Once every point of the plane has a pair of numbers, can lengths, areas and even proofs be done by arithmetic instead of by eye? ## What this area is about Put a figure on a grid and each corner becomes a pair of numbers $(x, y)$. Lengths, areas, midpoints and right angles can then be computed, with no ruler and no guessing from the drawing. ## How these fit together **The setting.** The coordinate plane gives every point its pair $(x, y)$. **Lengths.** Horizontal and vertical distance measures a segment that runs along a grid line by subtracting coordinates. The distance formula measures a slanted segment as the hypotenuse of a right triangle. **Points on a segment.** The midpoint formula finds the point halfway between two points, and runs backwards to a missing endpoint. Partitioning a segment finds the point that cuts it in a ratio $a : b$. **Measuring figures.** Area of a plotted polygon uses only grid-line lengths: a base and height read off the grid, or a box with its corner pieces taken away. Perimeter and area adds the distance formula, so slanted sides can be measured too. **Proving.** Classifying a plotted quadrilateral tests slopes, lengths and midpoints to name the shape for certain. Placing a figure uses letters for coordinates, so one computation proves a theorem for every such figure. ## A path through it The coordinate plane, then horizontal and vertical distance and the area of a plotted polygon. Next the distance formula, then perimeter and area with slanted sides. Then the midpoint formula, partitioning, classifying quadrilaterals, and last placing a figure for a proof. ### [Coordinate proofs with variables](https://math.wiki/placing-a-figure-on-the-coordinate-plane/) **Summary:** Choosing axes and simple coordinates so the figure stays general and the algebra stays short. **Question:** Checking a theorem on one triangle with numbers proves nothing about the next triangle. How do you put every triangle on the grid at once, and where should it go so the algebra stays short? ## When you reach for it The midsegment theorem says that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. You can test it on the triangle with vertices $(2, 8)$, $(-4, 0)$ and $(10, 2)$: find the midpoints, compare slopes, compare lengths, and it works. But that is one triangle. There are infinitely many others, and no number of numerical checks covers them all. A **coordinate proof** fixes this by putting a triangle on the grid with **letters** for its coordinates instead of numbers. The letters can stand for any values, so a computation done with them is done for every triangle at once. The distance formula, the midpoint formula and the slope formula then turn statements about the figure into algebra: - two segments are **congruent** when the distance formula gives the same length; - two segments **bisect each other** when they have the same midpoint; - two lines are **parallel** when their slopes are equal, and **perpendicular** when the product of their slopes is $-1$ (parallel lines, perpendicular lines). Reach for a coordinate proof when the theorem is about lengths, midpoints, parallel lines or right angles. The skill this lesson is about is the step before the algebra: **where to put the figure**. A good placement leaves the figure completely general and still makes most coordinates zero. A bad one either drowns the proof in letters or, worse, quietly proves the theorem for only some figures. ### [Coordinates and distance in three dimensions](https://math.wiki/three-dimensional-coordinates/) **Summary:** Locating a point in space with three numbers instead of two. **Question:** Two numbers pin down any spot on a flat page. How many does it take to pin down a spot in a room, and once you have them, how far apart are two such spots? ## Why this exists The coordinate-plane gives every point of a flat page a name: two numbers, one across and one up. That is enough for a map, a graph or a floor plan. It is not enough for a light hanging from a ceiling, a drone over a field or a corner of a box. Two lamps can sit over exactly the same spot on the floor and still be in different places, because one hangs lower than the other. A third number is needed, for height. The distance-formula has the same limit. It measures a straight line drawn on the page. The straight line from a socket near the floor to a lamp near the ceiling leaves the floor, and the plane formula has nowhere to put the climb. Three coordinates give space a name for every point, and with them comes a distance formula for space, built from the Pythagorean theorem used twice. ## Definition Pick a point $O$, the **origin**, and three number lines through it, each one perpendicular to the other two, all with the same unit of length. These are the **coordinate axes**: the $x$-axis, the $y$-axis and the $z$-axis. The usual way to set them up is the **right-hand system**: point the fingers of your right hand along the positive $x$-axis and curl them toward the positive $y$-axis; your thumb then points along the positive $z$-axis. In the pictures here the $y$-axis runs to the right, the $z$-axis runs up and the $x$-axis comes out of the page toward you, drawn slanting down and to the left. ### [Copying a segment and an angle](https://math.wiki/construct-a-congruent-segment/) **Summary:** Setting the compass to a given length and marking that same length on a new ray. **Question:** You have a segment, or an angle, drawn on paper, and no ruler marks and no protractor. How do you make an exact copy of it somewhere else? ## When you reach for it A **construction** is a drawing made with exactly two tools. The **straightedge** draws the line through two points you already have. It carries no marks, so it cannot measure anything. The **compass** draws a circle, or a piece of one called an **arc**, about a center you choose, and it holds its opening when you lift it. Nothing else is allowed: no scale on a ruler, no protractor, no judging by eye. The restriction is the point. A measured drawing is only as good as the ruler and the eye that read it. A construction is exact, because every step is justified by a definition or a theorem, and the justification does not depend on how steady your hand is. Copying is the first move of all constructions. Reach for it whenever a later construction needs a length or an angle you already have, placed somewhere new: laying lengths end to end to add them, building a triangle with given parts, or making an angle equal to one somewhere else in the figure, which is how a parallel line through a point is drawn. A copy is a congruent figure placed where you want it. From congruent-segments, two segments are congruent when they have the same length; from congruent-angles, two angles are congruent when they have the same measure. So copying a segment means producing a segment of the same length, and copying an angle means producing an angle of the same measure, without ever finding out what that length or measure is. ### [Corresponding angles](https://math.wiki/corresponding-angles/) **Summary:** Angles in matching positions at the two crossings, congruent when the lines are parallel. **Question:** One line cuts across two parallel lines. How much does the angle at the first crossing tell you about the angle at the second? ## Why this exists Two lines that never meet are hard to work with directly. You cannot measure an angle between them, because there is no angle between them. Nothing connects one to the other. A third line that crosses both, a **transversal**, is the connection. It makes a crossing on each line, four angles at each, eight in all. The question is how the two groups of four are related. When the two lines are parallel, the answer is as strong as it could be: the two crossings are copies of each other. Every angle at the first crossing reappears, at the same size and in the same position, at the second. That single fact is how an angle gets carried from one place in a figure to another, and most of what is known about the angles of a triangle and of a parallelogram is built on it. ## Definition A **transversal** is a line that crosses two other lines at two different points. In Figure 1 the transversal $t$ crosses line $l$ and line $m$, making four angles at each crossing. They are numbered the usual way: upper left, upper right, lower left, lower right, so $\angle 1$ to $\angle 4$ at $l$ and $\angle 5$ to $\angle 8$ at $m$. Two of the eight angles are **corresponding angles** when they sit in the **same position** at the two crossings: on the same side of the transversal, and on the same side of their own line. $\angle 2$ is above $l$ and to the right of $t$; $\angle 6$ is above $m$ and to the right of $t$. They correspond. The four corresponding pairs are ### [Cross sections of solids](https://math.wiki/cross-sections-of-prisms-and-pyramids/) **Summary:** Naming the flat shape left behind when a plane slices through a polyhedron. **Question:** Cut straight through a solid, a block of tofu, a cucumber, a grapefruit. What flat shape does the knife leave behind, and what decides it? ## When you reach for it Cut a cucumber straight across and the knife leaves a round disc; cut it on a long slant and the disc stretches into an oval. The solid did not change. The direction of the cut did. That flat face is a **cross section**, also called a **plane section**: the flat shape where a plane meets a solid. The plane is an ideal knife, perfectly flat and perfectly thin, and the cross section is everything it passes through, edge and inside together. Reach for it when a question is about the inside of a solid rather than its outside (surface area): what shape a cut face will be, how a solid is built from thin slices, or which solid could have produced a given set of slices. The solids are the ones named in three-dimensional-shapes: prisms and pyramids, cylinders and cones, and spheres. Stand the solid on its base. A cut **parallel to the base** is level, like a slice across a loaf. A cut **perpendicular to the base** goes straight down. A cut **on a slant** is neither. ## The procedure Naming a cross section comes down to seeing which faces of the solid the cut crosses, and what each crossing looks like. Three facts do nearly all the work. ### Three facts behind every cut **A flat cut through a flat face leaves a straight line.** Two planes that meet, meet in a straight line (intersections-of-lines-and-planes). So where the cut crosses one flat face, it leaves one straight segment, one side of the cross section. For a polyhedron, a solid whose faces are all flat polygons, this settles two things. The cross section is a polygon, so never a circle. And it has **one side for each face the cut crosses**, so no more sides than the solid has faces (naming-polygons-by-sides). ### [Cylinders and cones](https://math.wiki/cylinders/) **Summary:** Two congruent parallel circular bases joined by a curved surface. **Question:** A soup can and a traffic cone both stand on a circle. What are their parts called, and why are they not counted with boxes and pyramids? ## Why this exists Boxes and pyramids are easy to describe: count the flat faces, measure the straight edges. A soup can, a hockey puck and a traffic cone have no straight edges round their sides. They are round, and each is built on a circle. To describe these shapes, compare them, and later work out how much they hold, we need names for their parts and a clear sense of what each measurement means. Two solids cover many of them: the **cylinder** and the **cone**. ## Definition A **cylinder** is a solid with two **bases** that are circles of the same size, lying in parallel planes, joined by a **curved surface**. The curved surface is made of straight segments, all parallel to each other, each joining a point on the edge of one base to a point on the edge of the other. - The **radius** $r$ of the cylinder is the radius of either base. Its **diameter** is $d = 2r$. - The **axis** is the segment joining the centers of the two bases. - The **height** $h$ is the distance between the two bases, measured at right angles to both. A **cone** is a solid with one circular **base** and one point, the **apex**, that is not in the plane of the base. Its **curved surface** is made of all the straight segments from the apex to the edge of the base. - The **height** $h$ is the distance from the apex straight down to the plane of the base. - The **slant height** $\ell$ is the distance from the apex to the edge of the base, measured along the curved surface. ### [Deductive reasoning](https://math.wiki/deductive-reasoning/) **Summary:** Drawing a conclusion that must follow from statements already accepted as true. **Question:** A protractor says two angles in a drawing are equal. What would it take to know they are equal in every drawing of that kind, including the ones nobody has drawn? ## Why this exists There are several ways to come to believe something in geometry. Someone you trust can tell you. You can try it on a few drawings and see that it works. You can look at a well-made diagram and be convinced by it. Or you can argue from statements already accepted, one justified step at a time, until the claim has no way out. The first three are how most geometry is first met, and they are useful. But none of them settles a claim about **every** triangle, or every pair of parallel lines, because there are endlessly many of those and a protractor reads to about a degree. The fourth way does settle it. That way is **deductive reasoning**, and it is what turns a geometric guess into a theorem. ## Definition **Deductive reasoning** reaches a conclusion that must be true whenever the statements it starts from are true. Each step is justified by a statement already accepted, and in geometry those come in four kinds: - **Givens**: what the problem says about this figure, such as $l \parallel m$. - **Definitions**: what a word means, such as "a midpoint cuts a segment into two equal parts". - **Postulates**: basic facts accepted without proof, such as the segment addition postulate. - **Theorems**: facts already proved by deduction, such as the Vertical Angles Theorem. The sorting of statements into these kinds is its own lesson, definitions, postulates and theorems. The logic underneath is already in place. A step that applies a theorem to the case at hand is modus ponens, which geometry books call the law of detachment: the reason is a conditional, and an earlier line supplies its if-part. Joining two conditionals end to end is hypothetical syllogism, which geometry books call the law of syllogism. A deduction in geometry is these valid steps, run on facts about angles and segments, ending in $\therefore$ ("therefore") and the conclusion. ### [Defining and non-defining attributes of shapes](https://math.wiki/defining-attributes-of-shapes/) **Summary:** Three straight sides make a triangle, but its color, size and which way it turns do not. **Question:** A huge red triangle and a tiny blue one turned the other way are both triangles. What makes a shape a triangle, and what about it does not matter? ## Why this exists Ask a young child to point at a triangle and they will often pick the one that sits flat on its bottom with its point up, like a roof. Turn that same cutout upside down and some children will say it is not a triangle any more. Make it very large, or very thin, and the doubt grows. The child has learned the name from a picture, and a picture carries a lot of things that have nothing to do with the name: its color, its size, which way it happens to be turned. A name in mathematics comes with a rule. "Triangle" means a closed flat shape with $3$ straight sides, and nothing else. Anything that obeys the rule is a triangle, however big, however colored, however turned. Anything that breaks the rule is not, however much it looks like one. The things a shape must have to earn its name are its **defining attributes**. Everything else about it is a **non-defining attribute**: true of this one shape, but free to change. Once a child sorts what matters from what does not, they can name a shape they have never seen before, in any position, and they can make one from a description. That is the whole point of a shape name, and it is how every later lesson on shapes talks. The lesson runs over four parts, one each class day. | Part | What it teaches | Examples | Exercises | |:-:|---|:-:|:-:| | $1$ | What a defining attribute is, with the triangle | $1$, $2$ | $1$ to $7$ | | $2$ | Flat shapes, and building and drawing them from their attributes | $3$, $4$ | $8$ to $12$ | | $3$ | The same shape in other positions and sizes | $5$, $6$ | $13$ to $16$ | | $4$ | Solid shapes: flat faces and curved surfaces | $7$, $8$ | $17$ to $22$ | ### [Dilations](https://math.wiki/center-of-dilation/) **Summary:** The fixed point that every other point moves directly toward or directly away from. **Question:** How do you make a copy of a figure that is exactly the same shape but a different size, and how do you say where the copy goes? ## Why this exists A photocopier set to $150\%$, a projector throwing a slide onto a wall, a plan redrawn at a larger scale: each makes a copy with the same shape and a different size. A scale factor says how much bigger the copy is, since every length is multiplied by one number. It does not say where the copy goes, or how to draw it. "Twice as big" fits a copy on the left of the page and a copy on the right equally well. A dilation settles both questions with one rule. It names a single fixed point, the **center**, and moves every other point straight away from it or straight toward it, each one by the same factor. That decides where every point lands, and the copy that comes out has every length multiplied by the scale factor and every angle unchanged. A dilation is the move that makes "the same shape" something you can construct, and together with the rigid motions it is how similar figures are defined. This lesson takes two days. Part 1 is the dilation itself: the center and the scale factor, what a dilation keeps and what it changes, and how to find the center of a dilation that has already been drawn. Part 2 puts dilations on the coordinate plane, first with the center at the origin and then with the center anywhere. ## Definition Choose a point $O$, the **center**, and a positive number $k$, the **scale factor**. The **dilation** with center $O$ and scale factor $k$ moves the points of the plane like this. ### [Distance from a point to a line](https://math.wiki/distance-from-a-point-to-a-line/) **Summary:** The shortest distance is measured along the perpendicular segment from the point to the line. **Question:** A point sits off to one side of a straight line. Of all the ways to walk from the point to the line, which one is shortest, and how long is it? And when two lines never meet at all, how far apart are they? ## Why this exists Between two points there is no argument about distance: you measure the straight segment joining them. Between a point and a line there is. A house stands back from a straight road. Walk to the road at a slant and you cover one length; walk at a steeper slant and you cover another. A line has infinitely many points, so there are infinitely many segments from the house to the road. For "the distance from the house to the road" to mean anything, one of those segments has to be singled out, the same one for everybody. The shortest is the natural choice, and the surprise is how easy it is to find: it is the one that meets the road at a right angle. The same question comes back for two lines that never meet. Railway rails are a fixed gauge apart, and a shelf is a fixed height above the one below it, yet two parallel lines have no crossing point to measure from. You could drop the perpendicular from a point of one line to the other, but two worries stand in the way. That segment is square to the line it lands on; is it square to the line it started from, or has it been measured along a slant? And would a different starting point give a different number? The **perpendicular transversal theorem** settles the first worry and a short congruence argument settles the second, and together they make "the distance between two parallel lines" a single number. ## Definition Let $P$ be a point and $\ell$ a line. Drop the perpendicular from $P$ to $\ell$: the line through $P$ that meets $\ell$ at a right angle. It crosses $\ell$ at a point $Q$, called the **foot of the perpendicular**. ### [Distance on a number line](https://math.wiki/distance-on-a-number-line/) **Summary:** Finding how far apart two marked points are by taking the absolute value of their difference. **Question:** Two points on a line are named by numbers, and one of the numbers is negative. How far apart are the points? ## When you reach for it A ruler already measures by subtracting. A pencil lying from the $3$ cm mark to the $11$ cm mark is $11 - 3 = 8$ cm long, and you never needed its end to sit at zero. A number line is a ruler that carries on past zero to the left, so every point on it has a number, its **coordinate**, and the length of a segment should come from the coordinates of its two ends in the same way. It does, but negative coordinates raise two questions the ruler never did. Which of $-8$ and $-3$ is the larger one to subtract from? And what does subtracting a negative number measure? This skill answers both with one rule that works for any two points, in either order, on either side of zero. Reach for it whenever two positions are measured from the same zero: temperatures above and below freezing, heights above and below sea level, a bank balance before and after it went overdrawn, and the coordinates of points along a line in geometry. Every distance in coordinate geometry starts here. ## The procedure Name two points $A$ and $B$ on a number line, with coordinates $a$ and $b$. The **distance** between them, written $AB$, is the length of the segment joining them: the number of units you cover walking from one to the other. A distance is a length, so it is never negative. **When both points are to the right of zero, subtract.** Put $A$ at $2$ and $B$ at $9$. $B$ is $9$ units from zero and $A$ is $2$ units from zero, in the same direction. The first $2$ of those $9$ units are shared, so what separates the points is the rest: $9 - 2 = 7$. Counting the spaces between the marks gives the same $7$. ### [Dividing a segment into equal parts](https://math.wiki/divide-a-segment-into-equal-parts/) **Summary:** Marking equal steps along a slanted ray and joining back with parallel lines. **Question:** Halving a segment takes two pairs of arcs. How do you cut a segment into three, five or seven exactly equal pieces when nothing can be measured? ## When you reach for it You have a segment $\overline{AB}$ and you want the points that cut it into $n$ equal pieces: three, five, seven, any whole number of them. Or you want the one point that splits it in a given ratio, such as $2:3$. A ruler cannot do this exactly. A segment $10$ cm long has its thirds at $3.333\ldots$ cm, which no scale shows. And bisecting, however often it is repeated, only reaches halves, quarters, eighths and so on, never a third. The way round it is that equal pieces are easy to make somewhere else. A compass held at one width lays off equal steps along any line, as in copying a segment. The trouble is that those steps do not fit $\overline{AB}$: their total is whatever it happens to be. Parallel lines fix that. They carry the equal steps from a slanted ray across onto $\overline{AB}$, shrinking or stretching every one of them by the same factor, which is exactly what the triangle proportionality theorem says parallel lines do. The tools are the usual two: a straightedge with no marks, and a compass. ## The procedure ### Dividing into $n$ equal parts The figure uses $n = 5$; nothing in the reasoning depends on it. **From $A$, draw a ray that leaves $\overline{AB}$ at an angle.** Any angle will do, as long as the ray does not lie along the line $AB$. Its job is to hold the equal steps, and once the last step is joined back to $B$ it must make a triangle with $\overline{AB}$. ### [Drawing triangles from given sides and angles: one, many or none](https://math.wiki/draw-triangles-from-conditions/) **Summary:** Draw triangles from given side lengths and angle measures with ruler and protractor, and decide whether the conditions give one triangle, many or none. **Question:** You are told some of a triangle's sides and angles. Can you draw it, and is it the only triangle that fits? ## When you reach for it A plan for a triangle never lists everything about it. A garden plan gives the three edges of a bed; a tent plan gives two edges of a panel and the angle between them; a roof truss is given by its span and the slope of the roof at each end. Before anything is cut, two questions need answers: can a triangle with these measurements exist, and is it the only one? If two different triangles fit, the plan has not said enough. The way to find out is to draw it with a ruler, a protractor and a compass, one piece at a time, watching at each step whether you had a choice. Every set of conditions ends in one of three ways. - **Exactly one triangle.** Every triangle that fits can be cut out and laid exactly on any other, perhaps after turning it over. Where it sits on the page and which way it faces do not count. Triangles that match like this are congruent. - **Many triangles.** Triangles that really differ, in shape or in size, all fit. - **No triangle.** The pieces cannot be made to close. Part 1, the first day, takes the conditions that always settle the question: three sides, two sides with the angle between them, and two angles with the side between them. Part 2, the second day, takes the rest: three angles, two angles with a side not between them, and two sides with an angle not between them. ## The procedure **The three tools.** A ruler draws and measures a length, a protractor an angle. A compass set to $5$ cm with its point on $A$ draws points that are all exactly $5$ cm from $A$. All such points lie on one circle, and a short piece of it is an **arc**. So "the corner is $5$ cm from $A$" means "the corner is on this circle", and the arc shows every place the corner could be. ### [Equation of a circle](https://math.wiki/circle-equation-from-center-and-radius/) **Summary:** Turning a stated center and radius into the standard equation. **Question:** A circle is every point at one fixed distance from a center. What single equation picks out exactly those points? ## When you reach for it With a compass, a circle is easy: fix the point, fix the opening, turn. On the coordinate plane that is often not enough. You may need to say for certain whether the point $(7, 3)$ is on a circle or only close to it, to find exactly where a straight road crosses the edge of a signal's range, or to hand the circle to algebra so that it can be combined with lines and other curves. For all of these the circle has to become an **equation**: a condition on $x$ and $y$ that is true at every point of the circle and false at every other point of the plane. The definition says what that condition must be. A circle is the set of all points at one fixed distance, the **radius**, from one fixed point, the **center**. The distance-formula turns "at distance $r$ from the center" into algebra, and what comes out is a short equation that names the center and the radius in plain sight. Reach for it when you know the center and the radius, or can find them: the radius from the center and any point on the circle, the center and radius from the two ends of a diameter. Reach for it, run backwards, when an equation is already in this form and you want the center and radius. And reach for it when you need to decide whether a point is inside, on or outside a circle, or where a line meets one. ## The procedure ### From the definition to the equation Let the center be the point $(h, k)$ and the radius be a positive number $r$. A point $(x, y)$ is on the circle exactly when its distance from $(h, k)$ is $r$. The distance formula writes that distance out, so the point is on the circle exactly when ### [Equilateral triangle, square and hexagon in a circle](https://math.wiki/construct-an-equilateral-triangle/) **Summary:** Two circles of the same radius drawn through each other's centers meet at the third vertex. **Question:** With only a compass and a straightedge that has no markings, how can you draw a triangle with three exactly equal sides, or a square or a regular hexagon with every corner on a given circle? ## When you reach for it You need a $60^\circ$ angle and have no protractor. Or you need a triangle whose three sides are exactly equal, or a regular polygon drawn exactly inside a circle: a hexagon for a tiling pattern, a square, a triangle. The tools are the two used in copying a segment and bisecting: a straightedge, which draws the line through two points and has no marks, and a compass, which holds one width. All three constructions come from one fact about the compass: **every point it draws is the same distance from its center.** A triangle with three equal sides is a pair of circles, and a regular polygon in a circle is a set of equal angles at the center. For a regular polygon with $n$ sides in a circle, the angle at the center between two neighboring corners is $$\frac{360^\circ}{n}$$ which is $120^\circ$, $90^\circ$ and $60^\circ$ for $n = 3$, $4$ and $6$. Building one of these polygons means building that angle at the center, and the compass makes $60^\circ$ and the perpendicular makes $90^\circ$. ## The procedure ### An equilateral triangle on a given side Start with a segment $\overline{AB}$. The triangle will have $\overline{AB}$ as one side. **Open the compass to $AB$: the point on $A$, the pencil on $B$. Draw the circle about $A$.** Every point of it is the distance $AB$ from $A$. It passes through $B$. **Without changing the width, draw the circle about $B$.** Every point of it is the distance $AB$ from $B$, and it passes through $A$. ### [Faces, edges and vertices](https://math.wiki/faces-edges-and-vertices/) **Summary:** The flat surfaces of a solid, the lines where two faces meet, and the corners where edges meet. **Question:** A box and a pyramid feel different in your hands. What exactly is different about them, and how can you say it so someone who cannot see them knows which is which? ## Why this exists A child who can name solid shapes can point at a box and say "box", and at a can and say "can". That is a good start, but a name says nothing about why two solids are different. Ask a child to describe a pyramid to a friend on the phone and the name runs out fast. Flat shapes already have words for their parts: sides and corners (defining-attributes-of-shapes). A triangle is the shape with $3$ straight sides. Solids need the same kind of words, and there are three of them. A **face** is a flat surface, an **edge** is a line where two faces meet, and a **vertex** is a corner where edges meet. With those words, and a count of each, a child can say exactly what makes a cube a cube and a pyramid a pyramid, can tell solids apart in a bag by touch alone, and can check that a solid built from paper or playdough is the one that was meant. Counting the parts is also harder than it looks. Some faces are underneath and some edges are around the back, so a child who looks once from the front counts too few, and a child who turns the solid around without a plan counts some parts twice. The habit this lesson builds is to count with a plan: top, bottom and around, marking each part once it is counted. ## Definition A **solid** takes up room: it can be held, and it has an inside you cannot see. Its outside is made of surfaces. - A **face** is a flat surface of a solid. It lies flat on a table when the solid sits on it, and pressed into soft playdough it leaves a flat print with the shape of the face. - An **edge** is a straight line where two faces meet. Run a finger across a box from one face to the next and the finger goes over an edge. - A **vertex** is a point where edges meet: a corner of the solid. The word for more than one is **vertices**. "Corner" is the everyday word and is fine to use alongside it. ### [Finding the center and radius](https://math.wiki/find-center-and-radius-by-completing-the-square/) **Summary:** Rewriting an expanded circle equation into standard form so the center and radius can be read off. **Question:** A circle's equation has been multiplied out, and its center and radius have vanished into the numbers. How do you get them back? ## When you reach for it The standard form of a circle, $(x - h)^2 + (y - k)^2 = r^2$, shows the center $(h, k)$ and the radius $r$ at a glance. Multiply it out and they disappear. Take the circle with center $(-2, 5)$ and radius $3$: $$(x + 2)^2 + (y - 5)^2 = 9$$ Expanding both squares gives $x^2 + 4x + 4 + y^2 - 10y + 25 = 9$, and collecting everything on the left, $$x^2 + y^2 + 4x - 10y + 20 = 0$$ This is the same circle, every point of it and no other, but nothing in the equation says $(-2, 5)$ or $3$. The $4$ and the $25$ that came out of the squares have been combined with the $-9$, and all three are now hidden inside the $20$. An equation in this shape, $x^2 + y^2 + Dx + Ey + F = 0$ for some numbers $D$, $E$ and $F$, is called the **general form** of a circle. It is how circles usually arrive when they come out of a calculation rather than from a stated center: fitting a circle through three points, describing a set of points by a condition on their distances, or any work that multiplied out parentheses. Reach for this move whenever an equation has an $x^2$ and a $y^2$ with the same coefficient and no $xy$ term, and you need the center and the radius: to draw the circle, to compare it with another circle, to test points against it, or to find out whether the equation describes a circle at all. The move is completing the square, done twice: once for the $x$ terms and once for the $y$ terms. ### [Finding unknown angle measures](https://math.wiki/find-unknown-angle-measures/) **Summary:** Using named angle pairs in a figure to work out a measure that was not given. **Question:** A figure gives you some angles and asks for one that nobody measured. How do you get from the angles you know to the one you want? ## When you reach for it A figure hands you one or two angle measures and asks for an angle that was never measured. A protractor on the page would only measure the drawing, and a drawing is a sketch. The question is what the figure **forces** to be true, not what it happens to look like. What it forces comes from a short list of facts about angles that share a vertex. Each fact has its own lesson; here they are used together. - **Parts add to the whole.** When a ray lies inside an angle, the two smaller angles add to the big one. This is angle addition. - **A right angle is $90^\circ$.** Two angles that together fill a right angle are complementary: their measures add to $90^\circ$. - **A straight line is $180^\circ$.** Two neighbors whose outer sides make a straight line are a linear pair, and their measures add to $180^\circ$. More generally, angles that sit side by side and fill one side of a straight line add to $180^\circ$. - **A full turn is $360^\circ$.** The angles all the way round a point, with no gaps and no overlaps, add to $360^\circ$, the number of degrees in a full turn. - **Vertical angles are equal.** Where two lines cross, the angles opposite each other have equal measures. That is the vertical angles theorem. Each fact turns part of a picture into an equation. Reach for this skill when the angle you want is joined to the angles you know by a chain of these facts, or when the measures are written as expressions in $x$, so that finding the angle means solving an equation first. One fact with plain numbers is finding a missing angle; this lesson is what that becomes when the figure has more in it. ### [Flowchart proofs and choosing a format](https://math.wiki/choosing-a-proof-format/) **Summary:** Two column, paragraph, flowchart and coordinate forms all carry the same argument in different clothes. **Question:** The same theorem can be proved in a table, in sentences, in boxes joined by arrows, or with coordinates. How do the arrows show which step leans on which, which form should you write, and does the choice change what has been proved? ## Why this exists A proof is an argument: a chain of statements, each with a reason pointing back to things already settled. Geometry writes it down in four common ways: the two-column proof, the paragraph proof, the flowchart proof and the coordinate proof. Each form shows some things and hides others. A two-column proof writes its statements in one order, top to bottom, but most steps use only one or two of the lines above them, and two steps that have nothing to do with each other sit one above the other as if the second came out of the first. A paragraph can bury a reason inside a sentence. And one route can need facts the reader has not met when another needs none. Knowing all four, and how to carry a proof between them, lets you pick the form that shows the argument best. ## Definition - A **two-column proof** numbers the statements down the left and writes a reason beside each, naming the earlier lines it uses. Giving a reason for every step is the skill under it. - A **paragraph proof** writes the same statements and reasons as sentences. The reason rides on a word such as "by", "because" or "so" (writing a proof as prose). - A **flowchart proof**, or flow proof, puts each statement in a box with its reason written under it, and draws an arrow into a box from every statement its reason uses. The Given is at the top, the Prove at the bottom, and following the arrows from one to the other is following the argument. - A **coordinate proof** places the figure on the coordinate plane, gives letters to the coordinates that are free to vary, and proves the theorem by computing (coordinate proofs). ### [Focus and directrix](https://math.wiki/focus-directrix-definition/) **Summary:** Each conic is the set of points whose distances to a fixed point and a fixed line keep a fixed ratio. **Question:** A circle needs only a center and a radius. Is there one rule, with a single number to turn, that draws every other conic? ## Why this exists Cutting a double cone showed that the ellipse, parabola and hyperbola are one family: tip the plane a little and the curve changes kind. But that picture needs a cone in space, and each curve's own lesson describes it a different way. The ellipse is a constant **sum** of distances to two foci, the hyperbola a constant **difference**, and the parabola is equal distances to a point and a line. This lesson gives one rule for all three, in the plane: a point, a line, and one number. Turning that number moves through ellipse, parabola and hyperbola in order, the way tilting the plane did, and gives every conic a short polar equation with a focus at the pole. ## Definition Fix a point $F$, the **focus**, a line $L$ that does not pass through $F$, the **directrix**, and a number $e > 0$, the **eccentricity**. The **conic** with this focus, directrix and eccentricity is the set of all points $P$ in the plane with $$\frac{PF}{PL} = e, \qquad \text{that is,} \qquad PF = e \cdot PL$$ where $PL$ is the distance from $P$ to the line $L$, measured along the perpendicular. It is a locus: every point that meets one condition. - If $e < 1$, the conic is an **ellipse**. - If $e = 1$, the conic is a **parabola**. - If $e > 1$, the conic is a **hyperbola**. Write $d$ for the distance from $F$ to $L$. The letter $e$ here is a ratio, not the number $2.718\ldots$ of exponential growth. ## Intuition ### [Geometric means in a right triangle](https://math.wiki/geometric-mean-altitude-relation/) **Summary:** The altitude is the geometric mean of the two pieces it cuts the hypotenuse into. **Question:** Drop a segment from the square corner of a right triangle straight to the long side. How is its length tied to the two pieces it cuts that side into? ## Why this exists A right triangle with one extra segment drawn in, the altitude from the right angle to the hypotenuse, has six lengths in it: the two legs, the hypotenuse, the altitude, and the two pieces the altitude cuts the hypotenuse into. The Pythagorean theorem ties three of them together, the legs and the hypotenuse, and says nothing about the other three. The altitude needs a tie of its own. It splits the triangle into two smaller triangles with the same angles as the whole, so the figure holds three similar triangles, and similar triangles give proportions. The proportions that come out have a particular shape: one length appears twice, once at the bottom of a ratio and once at the top of the ratio it equals, as in $\frac{p}{h} = \frac{h}{q}$. A number that sits in the middle of a proportion like that is called a **geometric mean**, and this figure is full of them. The payoff is practical. Any two of the six lengths, with a few limits, give the other four by multiplying and taking square roots, with no angle measured. The altitude fact is old: it appears in Euclid's *Elements* as a corollary of Proposition 8 of Book VI, and Proposition 13 of the same book uses it to construct a length that is the "mean proportional" between two given lengths. ## Definition **The altitude to the hypotenuse.** Let $\triangle ABC$ have its right angle at $C$, so that $\overline{AB}$ is the hypotenuse. The altitude from $C$ is the segment from $C$ perpendicular to $\overline{AB}$. It meets $\overline{AB}$ at a point $D$, the **foot** of the altitude, and in a right triangle the foot always lands strictly between $A$ and $B$ (the reason is set out here). So the altitude cuts the hypotenuse into two pieces, sometimes called the segments of the hypotenuse. The usual names for the six lengths are ### [Geometric proof](https://math.wiki/geometric-proof/) **Summary:** Arguing from definitions and accepted facts to a certain conclusion. **Question:** A drawing shows a fact for one figure. What would it take to know it for every figure of that kind? ## What this area is about A geometric proof argues from the Given, through definitions and results already accepted, to the Prove, with a reason on every line. Once proved, a theorem holds for every figure that meets its hypothesis, and can itself be cited. ## How these fit together **Ways of knowing.** Inductive reasoning proposes a pattern from examples; deductive reasoning makes the conclusion follow. **The machinery.** Given, prove and the diagram sets a proof up. A theorem and its converse and properties of congruence supply kinds of reasons, and justifying each step puts one on every line. Auxiliary lines may be added, each with a reason. **First theorems.** Angle pairs, proving lines parallel, alternate interior angles and the triangle angle sum. **Forms.** Indirect proof shows something never happens; choosing a format compares two-column, paragraph, flowchart and coordinate proofs. **Proof families.** Congruent triangles, parallelograms, rigid motions and dilations, similarity and circles. ## A path through it The two kinds of reasoning, then Given and Prove, converses, properties of congruence, justifying each step and auxiliary lines. Next angle pairs, parallel lines, alternate interior angles and the angle sum, then indirect proof and formats. Last congruent triangles, parallelograms, transformational proofs, similarity and circles. This area leads to justifying a construction, which proves that a compass and straightedge figure is exactly what it claims to be. ### [Geometry](https://math.wiki/geometry/) **Summary:** Shapes, space, and the reasoning that proves facts about them. **Question:** Shapes can be measured with a ruler. How do you know a fact about a shape holds for every one of its kind, even the ones nobody has drawn? ## What this area is about Geometry studies shapes and space: what figures are, how they move, how big they are, and the reasoning that proves facts about them. It runs from naming a triangle in kindergarten to proofs, equations of curves and solids in high school. ## How these fit together **Starting objects.** Points, lines and planes starts from those three undefined terms and builds segments, angles and parallel lines from them. Two-dimensional shapes names and sorts flat figures. **Figures.** Triangles carry most of plane geometry's theorems. Circles add arcs, tangents and their angle facts. Solids take shapes into three dimensions, with surface area and volume. **Motions and matching.** Transformations slide, flip and turn figures without changing lengths or angles. Similarity adds scaling by a factor $k > 0$. **Methods.** Coordinate geometry computes with $(x, y)$. Geometric proof argues from definitions to conclusions that hold for every figure. Constructions draw exactly with compass and straightedge. **Beyond.** Conic sections are the curves a plane cuts from a double cone when it misses the vertex: circle, ellipse, parabola or hyperbola. ## A path through it These areas interleave rather than stack. Points, lines and planes and the first flat shapes come first. Then coordinate geometry and triangles, which draw on each other, and transformations; the regular-polygon lessons in two-dimensional shapes wait for transformations. Then circles, solids and similarity, in that order. Last geometric proof, then constructions; conic sections also come late, after circles and solids. ### [Given, prove and the diagram](https://math.wiki/given-and-prove/) **Summary:** Every proof starts by sorting what is assumed from what must be shown, beside a labeled figure. **Question:** A theorem is a sentence about every triangle, or every pair of lines. How do you turn it into something a proof can start from, and which of the things you can see in its figure are facts? ## Why this exists Theorems are usually stated in words: "the base angles of an isosceles triangle are congruent", "every point on the perpendicular bisector of a segment is equally far from its ends". Words like these speak about every triangle of a kind at once, and that is their point. But a proof cannot work on the base angles of no triangle in particular. Each step has to point: this angle, that segment, the place where these two lines meet. So before its first step, every proof sorts the sentence into what may be assumed and what must be shown, names the figure with letters, and draws it. Euclid did this in every proposition. In Book I, Proposition 15 he states the result in words, then writes "Let the straight lines $AB$ and $CD$ cut one another at the point $E$", then "I say that the angle $CEA$ equals the angle $DEB$", and only then argues. Proclus, writing on Euclid in the fifth century, named these two steps; in English they are the *setting-out* and the *specification*. School geometry calls them the **Given** and the **Prove**. The drawing brings a problem of its own. A proof is a chain of reasons, each one a definition, postulate or theorem or something you were given, and a figure shows far more than the words do. A segment in it has a length you could measure, two lines in it meet at an angle you could read with a protractor, and two lines that do not meet look as if they never will. If every one of those appearances counted as a fact, a proof could stop at "look". If none counted, you could not even say which point lies on which line without writing it out. Geometry settles this with a rule about what a figure is allowed to say, and every proof you write leans on it. ### [Hinge theorem](https://math.wiki/hinge-theorem/) **Summary:** Two triangles with two pairs of congruent sides have the longer third side where the included angle is larger. **Question:** Two triangles have two sides the same length, but the angle between those sides is wider in one of them. Which triangle has the longer third side, and can a proof be sure? ## Why this exists The congruence criteria answer one question: when do two triangles have to be copies of each other? SAS says that two sides and the angle between them settle the whole triangle. Change that angle and the triangle changes, so SAS has nothing more to say. But something is still true. Open a pair of compasses wider and the pencil lands further from the point. Open a door further and its edge moves further from the frame. Two sides of fixed length, joined at a hinge, have a gap between their free ends, and that gap grows as the hinge opens. The hinge theorem makes that a proved fact about triangles, and its converse runs the other way: from two gaps, which opening was wider. They are the first tools that compare two triangles which are **not** congruent. ## Definition In a triangle, the **included angle** of two sides is the angle of the triangle whose vertex is the point the two sides share. In $\triangle ABC$ the included angle of $\overline{AB}$ and $\overline{AC}$ is $\angle BAC$, written $\angle A$ when there is no doubt which angle at $A$ is meant. The third side, $\overline{BC}$, is the side **opposite** that angle. The hinge theorem is about two triangles $\triangle ABC$ and $\triangle DEF$ in which two pairs of sides are congruent, $$AB = DE \qquad\text{and}\qquad AC = DF$$ so that $\angle A$ and $\angle D$ are the included angles of matching pairs and $\overline{BC}$ and $\overline{EF}$ are the two third sides. ### [Horizontal and vertical distance](https://math.wiki/horizontal-and-vertical-distance/) **Summary:** Finding the length of a segment that runs straight across or straight up by subtracting coordinates. **Question:** Two places on a grid map sit on the same grid line, one on each side of an axis. How far apart are they, without a ruler? ## When you reach for it On a grid map, many lengths run along grid lines: a trail drawn straight down the page, a runway laid straight across it, the side of a fenced rectangle. In the first quadrant such a length is the larger of two numbers minus the smaller. Once points sit in all four quadrants, one end may lie left of an axis and the other right of it, and that subtraction would subtract a negative. This lesson finds such lengths by counting squares and, with no grid to count, from absolute values. Part 1 measures one segment. Part 2 uses four of them as the sides of a rectangle: its perimeter, its area and a missing corner. ## The procedure ### Part 1: the length of a segment straight across or straight up **Spot the shared number.** Two points with the same second number are at the same height, so the segment joining them is **horizontal**: it runs straight across, along one grid line. Two points with the same first number sit one above the other, so the segment is **vertical**. The shared number only says which grid line the segment lies on. The length is how far apart the other two numbers are. For $(-4, 3)$ and $(5, 3)$, the $3$ puts the segment $3$ units above the horizontal axis, and the length is the gap between $-4$ and $5$. **Count, when there is a grid.** Each square is one unit, so the length is the number of squares between the ends. For points far apart, or with no grid drawn, counting is slow. ### [Hypotenuse leg congruence](https://math.wiki/hl-congruence/) **Summary:** Right triangles are congruent when their hypotenuses match and one pair of legs matches. **Question:** Side, side and an angle not between them usually fail to fix a triangle. When the angle is a right angle, do they? ## Why this exists SSS, SAS, ASA and AAS each name three matching parts that force two triangles to be congruent. SSA is not on the list: two sides and an angle that is not between them can fit two different triangles (why side side angle fails). Right triangles are the exception that matters most. The right angle often comes for free, and the lengths easiest to measure are the longest side and one other: a wire and the height of the pole it holds. That is side, side and an angle not between them, and for right triangles it is enough, because the Pythagorean theorem makes two sides of a right triangle decide the third. This lesson proves it as the **HL theorem**. ## Definition In a right triangle the two sides that meet at the right angle are the **legs**, and the side opposite the right angle is the **hypotenuse**. **Hypotenuse leg congruence**, or **HL**, says: two right triangles are congruent when the hypotenuse of one is congruent to the hypotenuse of the other and a leg of one is congruent to a leg of the other. All three facts, the right angles included, must be known, not just seen in a drawing. Read around the triangle, those are an angle (the right angle), a side touching it (the leg) and a side facing it (the hypotenuse). The angle is not between the two sides, so HL is the right-angle case of SSA, and that case is safe (it is not the only one: see Boundaries). SSS, SAS and ASA are postulates on this map; HL is a **theorem**, proved below from the Pythagorean theorem and the SSS postulate. ### [Identifying and translating conics](https://math.wiki/identify-a-conic-from-its-equation/) **Summary:** Reading the squared terms and their signs to tell circle, ellipse, parabola and hyperbola apart. **Question:** An equation in $x$ and $y$ with squared terms draws some curve in the plane. Which curve is it, and where does it sit, before anything is graphed? ## When you reach for it Each of the four conic-sections has its own standard form, taught in its own lesson: the circle, the ellipse, the hyperbola and the parabola. A standard form tells you at a glance what the curve is and where it sits. But an equation rarely arrives that way. Expand any standard form, move everything to one side, and you get an equation of the shape $$Ax^2 + Cy^2 + Dx + Ey + F = 0,$$ with $A$ and $C$ not both zero. This is the **general second degree equation** (with no $xy$ term; see Boundaries). It hides the type and the position of the curve. This lesson undoes the hiding in two moves. Day 1: read the type from $A$ and $C$ alone. Day 2: complete the square to put the equation back in standard form and read its center or vertex. ## The procedure ### Day 1: the squared terms name the curve Tidy the equation first: expand every product, move every term to the left side, and combine like terms. Only then read the coefficients $A$ of $x^2$ and $C$ of $y^2$. To see why these two numbers decide everything, expand each standard form and look at its squared terms. Expanding $(x - h)^2$ gives $x^2 - 2hx + h^2$: the square produces one $x^2$, and the shift $h$ only feeds the $x$ term and the constant. So the shifts never touch $A$ or $C$. - A circle $(x - h)^2 + (y - k)^2 = r^2$ expands with $x^2 + y^2$: $A = C$. - An ellipse $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$, multiplied by $a^2b^2$, starts $b^2x^2 + a^2y^2$: $A$ and $C$ have the same sign, so $AC > 0$. - A hyperbola $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ starts $b^2x^2 - a^2y^2$, and the form that opens up and down starts $-b^2x^2 + a^2y^2$: opposite signs, so $AC < 0$. - A parabola $(x - h)^2 = 4p(y - k)$ or $(y - k)^2 = 4p(x - h)$ squares only one variable: one of $A$, $C$ is zero, so $AC = 0$. ### [Incenter](https://math.wiki/incenter/) **Summary:** Where the angle bisectors meet, the one point the same distance from all three sides. **Question:** A triangle has three sides. Is there a point the same distance from all three, and a circle inside the triangle that touches every side? ## Why this exists A carpenter has a triangular board and wants to cut the largest possible round tabletop from it. A town has a triangular park bounded by three straight roads and wants to put a fountain as far from the traffic as it can. Both are asking for a point deep inside a triangle that keeps its distance from all three sides, and the fair choice is a point the same distance from every side. For two sides the answer is already known. By the angle bisector theorem and its converse, the points inside an angle that are equally far from its two sides are exactly the points of its bisector. A triangle has three angles and so three bisectors, and that lesson proved that they always cross at a single point. This lesson is about that point: the circle it carries, where that circle touches the sides, and how its radius is tied to the triangle's area and perimeter. It is one of the classic triangle centers. The circumcenter is equally far from the three corners. The incenter is equally far from the three sides. ## Definition The **incenter** of a triangle is the point where its three angle bisectors meet. It is usually called $I$. (British texts often spell it incenter.) The **distance from a point to a side** is the length of the perpendicular segment from the point to the line containing that side (distance-from-a-point-to-a-line). A line is **tangent** to a circle when it meets the circle at exactly one point, called the **point of tangency**. ### [Indirect proof](https://math.wiki/indirect-proof-in-geometry/) **Summary:** Assuming the opposite of what is to be shown and following it until something impossible appears. **Question:** Some facts in geometry say that something never happens. Two lines never cross twice, a triangle never has two right angles, no side of a triangle is as long as the other two together. How can a proof show that something never happens? ## When you reach for it Most proofs run forward, from the Given to the Prove. That stalls when the Prove says something does **not** happen. "$\ell \parallel m$" means the lines never meet, however far they run. "A triangle has at most one right angle" means a second one is never there. There is nothing to work on: no meeting point to name, no second right angle to measure. An **indirect proof** supplies the missing object. It assumes, for the length of the argument, that the Prove is false: the lines do meet, at some point $X$. Then it reasons from the Given and the assumption, using only results already established, until it reaches something that cannot be true. Every step was valid and the Given is true, so the assumption failed, and the Prove holds. Reach for it when the Prove says "not", "never", "at most one" or "parallel", and its negation hands you a point, a line or an angle to name. The logic is already done: what the proof reaches is a contradiction, and proof by contradiction is the method in general. This lesson uses it in geometry. The best-known fact of this shape is about lengths. Lay sticks of length $3$, $4$ and $8$ on a table. The two short ones end to end reach only $7$, so they cannot meet the far end of the long one, and no triangle closes. Every triangle behaves this way. The **triangle inequality** says that in a triangle $ABC$ $$AB < AC + CB$$ and the same for each of the other two sides: one side is always shorter than the other two together. The road from $A$ to $B$ by way of $C$ is never shorter than the straight one. ### [Inductive reasoning](https://math.wiki/inductive-reasoning/) **Summary:** Noticing a pattern in examples and proposing that it always holds. **Question:** A pattern has held in every case you have tried. Does that make it true, how do you hunt for the case that breaks it, and what is the pattern good for? ## Why this exists Nobody finds a theorem by starting at the proof. Somebody first notices something: the angles of every triangle they have measured add to about $180^\circ$, or the odd numbers keep adding up to square numbers. The noticing comes first, and it has a name. **Inductive reasoning** is the step from "it happened in these cases" to "maybe it always happens". It is how most mathematics is discovered. It is also, taken alone, unreliable, and the whole skill is knowing exactly how far it can be trusted. In geometry the cases are usually drawings: you draw a few figures, notice something, and state it for every figure of that kind. But the figures you draw are the easy ones: symmetric, upright, acute. A claim can hold in all of them and still be false. So a geometric conjecture is tested by hunting for the least tidy figure the hypothesis allows. If one breaks the claim, the next question is how to repair it. If none does, and you can see why, that reason is the start of a proof. Inductive reasoning makes the guess, and this lesson follows the guess until it is broken, repaired or proved. ## Definition Inductive reasoning draws a **general** statement from **particular** cases. You look at examples, notice a pattern, and state that the pattern holds in every case, including the ones you have not looked at. The statement it produces is a conjecture: a claim that looks true and has not been proved. Inductive reasoning cannot do more than that. However many cases agree, the conclusion is **probable, never certain**, because the next case has not been checked. ### [Inscribed polygons and cyclic quadrilaterals](https://math.wiki/cyclic-quadrilateral/) **Summary:** A quadrilateral with all four vertices on one circle has opposite angles that add to 180 degrees. **Question:** Any three points not on one line sit on exactly one circle. Which four-sided shapes can have all four corners on one circle, and how can you tell from the shape alone? ## Why this exists Pick any three points that are not on one straight line. There is always a circle through all three, and only one: its center is the circumcenter, the point equally far from all three. So a triangle can always be fitted into a circle with every corner touching it. Add a fourth point and that luck runs out. The circle is already fixed by the first three points, and the fourth either lands on it or does not. Most of the time it does not. So a question appears that did not exist for triangles: which quadrilaterals, and which polygons in general, have all their corners on one circle? Finding the center and measuring every distance answers it for one shape at a time. This lesson finds something better. A quadrilateral sits on a circle exactly when its **opposite angles add to $180^\circ$**, so the angles alone decide, with no center and no measuring. The same idea explains why every rectangle and every regular polygon fits in a circle, and why a parallelogram that is not a rectangle never does. ## Definition A polygon is **inscribed in a circle** when every one of its vertices lies on the circle. The circle is then **circumscribed about** the polygon, and is called its **circumscribed circle** or **circumcircle**. A polygon that can be inscribed in some circle is called **cyclic**. A **cyclic quadrilateral** is a quadrilateral whose four vertices lie on one circle. Being on the circle is a statement about distances. If the circle is $\odot O$ with radius $r$, then a polygon $A_1 A_2 \ldots A_n$ is inscribed in it exactly when ### [Isosceles and equilateral triangles](https://math.wiki/base-angles-theorem/) **Summary:** The two angles opposite the congruent sides are congruent. **Question:** Two sides of a triangle are equal. What does that force about its angles, does it work the other way round, and what if all three sides are equal? ## Why this exists Three side lengths fix a triangle, but most of the time you are handed less, and the question is how much of the rest you can still work out without a protractor. Here is a case that comes up constantly: two sides of a triangle have the same length. That is a fact about **sides**, yet it forces two of the three angles to be equal, and then one angle is enough to find the other two. Triangles with two equal sides are everywhere in geometry, often without being drawn as such: two radii of a circle, two sides of a regular polygon, the two rafters of a roof. Very often the information arrives the other way round. Parallel lines, an angle bisector or a quick angle chase hand you two equal angles, and what you want is a length. The converse runs in that direction: it is how you prove a triangle is isosceles without measuring a single side, and how a surveyor finds the width of a river without crossing it. Push both directions to a triangle whose three sides are all equal and every angle is known at once: mark three sides equal and you have been handed $60^\circ$, $60^\circ$ and $60^\circ$ for nothing. This lesson proves all of it from the congruence criteria and the triangle angle sum. ## Definition A triangle is **isosceles** when at least two of its sides are congruent. The names of its parts come from those two sides. - The two congruent sides are the **legs**. - The third side is the **base**. - The angle between the two legs, opposite the base, is the **vertex angle**. - The two angles at the ends of the base are the **base angles**. ### [Line symmetry](https://math.wiki/line-symmetry/) **Summary:** A figure has line symmetry when a reflection maps it exactly onto itself. **Question:** When can you fold a picture so that one half lands exactly on the other half? ## Why this exists In Fiadh's art class everyone is making butterfly pictures. Fiadh paints one wing in wet paint, folds the paper along a pencil line, presses, and opens it. A second wing has appeared, an exact copy, flipped over. Her picture now has a special property: fold it on that crease and every part of one wing lands on the matching part of the other. Many things share that property: letters, leaves, flags, a square. It lets you describe a shape with half the work, because half of it and the fold line tell you the rest. It also lets you check a drawing: if a design is meant to match itself, every point on one side must have a partner on the other, and partners can be checked by counting squares. In general, a **symmetry** of a figure is a move that leaves it looking exactly as it did. This lesson is about the folding kind. The turning kind is rotational symmetry. ## Definition A **line of symmetry** of a figure is a line you can fold the figure along so that the two sides match exactly. Every point of the figure on one side lands on a point of the figure on the other side, and nothing is left over on either side. A figure with at least one line of symmetry has **line symmetry**. A line of symmetry is sometimes called a **mirror line**: stand a mirror on it and the mirror shows the missing half. When the figure is folded on a line of symmetry, each point $P$ lands on a point $P'$ (read "$P$ prime"). The two are **matching points**. A point on the line stays where it is, so it is its own match. ### [Lines and planes in space](https://math.wiki/intersections-of-lines-and-planes/) **Summary:** Two lines meet in at most one point and two planes meet in a line. **Question:** Could two straight lines cross twice? When two flat surfaces cut through each other, what shape is the place where they meet? And out in space, must two lines that never meet be parallel? ## Why this exists A triangle, a circle and two crossing lines all fit on one flat sheet. A box, a ball and the room you are sitting in do not, and no amount of flat geometry describes them. Geometry needs a setting that holds many planes at once, stacked, tilted and crossing, with points above and below each: **space**. It also needs words that keep the skin of an object apart from the object, since the paint on a box and the sand inside it are measured in different ways. Every figure in space is built from lines and planes, and the first thing to know about any two of them is where they touch. If two lines could cross in two places, or two walls could meet in a scattered handful of points, you could never say how many corners a shape has, or whether an edge you see is one straight edge. The answer is short, and it is what lets you name the corners and edges of a box, find where two graphs cross, and later describe every solid there is. On a flat sheet of paper, two lines either cross or run side by side forever, and the second kind are parallel. On paper, "never meet" and "parallel" mean the same thing. In space they do not. A road running east over a bridge and a road running north underneath it never meet, since one is always above the other, but nobody would call two roads at right angles parallel. Space has a third way for two lines to sit, and it needs its own name. ## Definition ### [Locus of points](https://math.wiki/locus-of-points/) **Summary:** A locus is the set of every point that meets a stated condition, such as staying a fixed distance from a point. **Question:** You know the rule a point has to obey. Where are all the points that obey it? ## Why this exists A pirate's note says the gold is buried "ten paces from the old oak". That does not tell you where to dig. It gives a whole ring of places. A second line, "and as far from the east post as from the west post", gives a second set of places. The gold must be in both sets at once, so you dig only where they cross. Geometry is full of questions shaped like this. You are not handed a point; you are handed a **rule** a point must obey, and asked where such points can be. The answer is rarely one point. It is a whole figure, and naming that figure exactly is the first step to using it. Even the circle is best understood this way, as every point one fixed distance from a center (what a circle is), and so are the parabola, the ellipse and the hyperbola. ## Definition A **locus** (plural **loci**) is the set of **all** the points that meet a stated condition, and **only** those points. Both halves matter. To say that a figure is the locus you must know two things: - every point on the figure meets the condition, and - every point that meets the condition is on the figure. A figure that fails the first has extra points in it. A figure that fails the second has missed some. A locus is also stated for a setting, usually a plane, sometimes all of space, and the same rule can give a different figure in each. ## Intuition Picture every point of the page testing itself against the rule and lighting up if it passes. The locus is the shape the lit points make. ### [Medians and altitudes](https://math.wiki/medians-of-a-triangle/) **Summary:** A median joins a vertex to the midpoint of the opposite side. **Question:** From one corner of a triangle, which line to the opposite side cuts the triangle fairly in two, and which one measures how tall the triangle stands? ## Why this exists Pick a corner of a triangle and draw a segment from it to the side across the way. There are endlessly many such segments, one for every point of that side, and most of them are of no special interest. Two of them are. One lands exactly halfway along the opposite side. It cuts the triangle into two pieces that usually look nothing alike, and yet always have the same area. That is the **median**. The other lands so that it meets the opposite side squarely, at a right angle. It is the shortest way from the corner to that side, and so it is the honest answer to "how tall is this triangle?" That is the **altitude**, and it is the height in the area formula $A = \tfrac12 bh$. Without it, "height" has no meaning for a triangle that is leaning over. Every triangle has three of each, one from every corner, and this lesson is about what they are, where they fall, and what they let you find. ## Definition In $\triangle ABC$, the side across from vertex $A$ is $\overline{BC}$, and it is called the **opposite side** of $A$. A **median** of a triangle is a segment from a vertex to the midpoint of the opposite side. In Figure 1, $M$ is the midpoint of $\overline{BC}$, so $BM = MC$ (the tick marks say so), and $\overline{AM}$ is the median from $A$. An **altitude** of a triangle is a segment from a vertex, perpendicular to the line containing the opposite side, that ends on that line. The point where it ends is its **foot**. In Figure 1, $\overline{AD}$ is the altitude from $A$: $\overline{AD} \perp \overline{BC}$ (the square mark says so), and $D$ is its foot. ### [Midpoints and segment bisectors](https://math.wiki/midpoint-of-a-segment/) **Summary:** The point that cuts a segment into two congruent pieces. **Question:** A segment has two ends. Where exactly is the point that is just as far from one end as from the other, and which lines cut the segment there? ## Why this exists Cut a plank into two equal pieces, hang a picture in the center of a wall, find the center of a circle from the two ends of a diameter, meet a friend halfway between your two houses. Each time the question is the same: along the straight path from one place to another, where is the point that splits it fairly? That point is the **midpoint**. It is the simplest special point a segment has, and it keeps coming back. The center of a circle is the midpoint of every diameter. A median of a triangle runs from a corner to the midpoint of the opposite side. A great many constructions begin by finding one. In a figure, though, you rarely meet a midpoint on its own. You meet it because something passes through it: a road that crosses a bridge at its center, a diagonal that crosses another diagonal, a line drawn from the corner of a triangle to the middle of the far side. So the useful question is not only "where is the middle?" but "does this line go through the middle?" A line that does is a **segment bisector**, and the moment you know a line is one, you know two lengths are equal. That single fact turns up everywhere from triangles to parallelograms to circles, which is why it gets its own name. ## Definition ### The midpoint The **midpoint** of a segment $\overline{AB}$ is the point $M$ that lies on $\overline{AB}$ and is the same distance from $A$ as from $B$: $$AM = MB$$ ### [Midsegments of triangles and trapezoids](https://math.wiki/triangle-midsegment/) **Summary:** The segment joining two side midpoints is parallel to the third side and half as long. **Question:** Join the midpoints of two sides of a triangle. What does that new segment know about the third side? ## Why this exists Every side of a triangle has a midpoint, and finding it is easy: measure and halve, fold the paper, or average the coordinates. A median joins one of those midpoints to the opposite vertex. Join two of the midpoints to each other instead and something unexpected happens. The new segment knows the direction and the length of the third side, a side it never touches: it runs exactly parallel to it and is exactly half as long, whatever the shape of the triangle. That makes two midpoints a cheap source of facts. A parallel line hands you equal angles, and a length you can measure hands you one you cannot reach. The same idea, used twice, settles the line across the middle of a trapezoid: it is parallel to both bases, and its length is their average. ## Definition A **midsegment of a triangle** is a segment whose two endpoints are the midpoints of two sides of the triangle. A triangle has three sides, so there are three ways to choose two of them, and every triangle has exactly three midsegments. In the figure, $D$ is the midpoint of $\overline{AB}$ and $E$ is the midpoint of $\overline{AC}$, so $\overline{DE}$ is a midsegment. The side it does not touch, $\overline{BC}$, is called its **third side**. Matching tick marks show equal lengths: $AD = DB$ and $AE = EC$. A midsegment is not a median. A median has one end at a vertex and the other at a midpoint; a midsegment has both ends at midpoints and no end at a vertex. ### [Modeling with solids: density and design](https://math.wiki/modelling-with-solids/) **Summary:** Describing real objects as solids and using volume, surface area and density to answer design questions. **Question:** A log, a pipe, a roof and a can are not textbook shapes. How do you use volume and area to say how heavy, how much material, or which design is best? ## When you reach for it A tree trunk is not a cylinder, a roof is not quite a pyramid, and a city is not a circle. But each is close enough to one that the formulas you know for cylinders, spheres, composite solids and the surface of pyramids and cones give answers that are good enough to act on. Reach for this move whenever a real object raises a question about size: - **How heavy is it?** That needs a volume and the density of what it is made of. - **How much material wraps it?** Paint, roofing, sheet metal, canvas and labels are surface area. - **How crowded is it?** People per square kilometer, trees per hectare: a count divided by an area. - **Which shape is best?** A fixed amount must be held, and one design uses less material than another. The mathematics inside each answer is a formula you already have. What is new is the step before the formula: deciding which solid stands in for the object, which of its measures the question is really about, and what the units say. ## The procedure ### Choosing the solid A **model** is a solid that is simple enough to have a formula and close enough to the object that the formula's answer is useful. A log is a cylinder; a silo is a cylinder with a half sphere on top; a pipe is a cylinder with a cylinder taken out; a tent with one center pole is a pyramid. Every model throws something away: the bark's roughness, the slight taper of the trunk, the rounded rim of a can. That is allowed exactly when what is thrown away is small next to what is kept, and it is worth saying in one line what the model ignores, because that line is what tells you later whether the answer is likely too big or too small. ### [Naming and classifying polygons](https://math.wiki/naming-polygons-by-sides/) **Summary:** Pentagon, hexagon, octagon and the rest, each named by how many sides it has. **Question:** A stop sign has eight straight sides and a yield sign has three. What makes a shape a polygon, and how does counting its sides give it a name? ## When you reach for it Grainne, Ronan and Bronagh work in a shop that cuts road signs out of flat metal sheets. A sign's shape carries its message before anyone reads a word. A stop sign has eight sides. A yield sign is a triangle standing on one point. A school zone sign has five sides and is shaped like a house. A warning sign is a square turned onto one corner, like a diamond. The round sign before a railroad crossing has no straight sides at all. To talk about these shapes, the shop needs names, and the names come from one plain count: **how many straight sides does the shape have?** That count gives the name: triangle, quadrilateral, pentagon, hexagon, and on up. A second question then sorts shapes that share a name: **does the shape have a dent?** A shape with no dent is convex, and a shape with a dent is concave. You reach for these names whenever a flat shape with straight sides is in front of you and you need to say what it is, sort it, or know which facts about it apply. The names also open the families inside them. Triangles are sorted further in classify-triangles-by-angles, quadrilaterals in properties-of-special-quadrilaterals, and the hierarchy of shapes fits all of those families together. ## The procedure The lesson runs over two days. Part 1 says what a polygon is and names it by its sides. Part 2 sorts polygons into convex and concave, names the regular ones, and puts the names together. ### [Naming and describing solid shapes](https://math.wiki/name-solid-shapes/) **Summary:** Telling solid shapes apart by name and spotting them in everyday objects. **Question:** How does a child tell a ball from a can from a block, and call each one by its name? ## When you reach for it A **flat shape** lies on paper, like a drawing or a paper cutout. A **solid shape** is thick: a child can pick it up, turn it over and look at it from every side. Four solids come first, each with a name: the **cube**, the **sphere**, the **cone** and the **cylinder**. A child describes them by what they do (roll, stack, slide) and by their flat parts, called **faces**. The flat shapes a child already knows by name turn up again as those faces. Each day takes about ten to fifteen minutes. Use real things the child can hold; the pictures are for the practice at the end of each day. ## Day 1: Flat or solid? **You need:** a ball, a box (a tissue box or small block), and a circle and a square cut from paper. Put the paper circle and the ball side by side. Ask, "How are these the same? How are they different?" Both look round from above. Then let the child pick each one up. The paper circle is so thin it lies flat on the table; the ball fills the hand and can be turned all the way around. Say: "The circle is a flat shape. The ball is a solid shape." Do the same with the paper square and the box. Then play "Flat or solid?": point to things around the room, and the child says which. A picture in a book is flat even when it shows a ball, because the page is flat; the ball itself is solid. **Practice.** Point to each picture. 1. Picture A: flat or solid? **Answer:** flat. It is a circle, shaded like a paper cutout. 2. Picture B: flat or solid? **Answer:** solid. The curved line around its middle shows a ball, round all the way around. 3. Picture C: flat or solid? **Answer:** flat. It is a square. 4. Picture D: flat or solid? **Answer:** solid. You can see its front, its top and its side. 5. A sticker of a star: flat or solid? **Answer:** flat. It lies flat on the page. ### [Naming circles, triangles, squares and rectangles](https://math.wiki/name-basic-shapes/) **Summary:** Telling everyday flat shapes apart and calling each one by name. **Question:** How does a child learn to call a circle, a triangle, a square and a rectangle by name, whatever its size and whichever way it is turned? ## When you reach for it Young children often learn shape names from one picture each: a triangle that sits flat with its point up, a square that sits flat on a side. Then a triangle pointing down, or a square standing on its corner, gets a new name, or none. The aim of these four days is that a name belongs to the shape itself, so a triangle stays a triangle when it is huge, tiny, thin, or turned any way at all. Each day takes ten to fifteen minutes. Talk about shapes the way the child sees them ("it's pointy", "it's round all the way") and add the name each time; careful counting of sides and corners comes in the next lesson. ## Day 1: Circles and triangles **You need:** a cup or jar lid, a sheet of paper, crayons, scissors. Trace around the bottom of a cup and say, "This is a circle. It is round all the way around, with no straight parts and no points." Trace a smaller lid and a bigger plate: all circles. Then fold a square of paper corner to corner and cut along the fold. Hold up one piece: "This is a triangle. It has three straight sides and three points." Turn it around in your hands and ask, "What is it now?" It is still a triangle. Show the picture below and ask about each shape, "Is it a circle, a triangle, or neither? How do you know?" Two of the shapes are traps. A pizza slice has a curved crust, so it is not a triangle; a triangle needs every side straight. A shape with a dent cut into it has an extra point, four instead of three, so it is not a triangle either. ### [Nets of solids](https://math.wiki/net-of-a-prism/) **Summary:** Unfolding a prism into its two bases and the rectangle that wraps around them. **Question:** How do you make a box, a tent or a tin from one flat sheet, and how can you tell before cutting whether the pattern will close up? ## When you reach for it A cereal box, a tent and a tin all start as flat sheets. Before cutting, you want to know what shape to cut so that, folded up, it closes into the solid with every face covered once. That flat pattern is a **net**. You reach for one when you design a package or a model, when you want to see every face of a solid at once, and before adding up the areas of the faces in surface-area-from-a-net. This lesson takes two days. Part 1 (Examples 1 to 4) is nets of prisms: what a net is, which patterns fold, and which lengths must match. Part 2 (Examples 5 to 7) is nets of pyramids and cylinders. The solids and their faces, edges and vertices are named in prisms, polyhedra and cylinders. ## The procedure ### Part 1: nets of prisms **What a net is.** A net of a solid is a flat shape in one piece, made of the solid's faces joined edge to edge, that folds along those joins into the surface of the solid with every face covered exactly once, no gap and no overlap. The joins are the **folds**, dashed in these figures. Folding bends the sheet at a fold and nowhere else, so a net has exactly one piece for each face, each the same shape and size as its face. You can also get a net by cutting a cardboard box along some of its edges until it lies flat in one piece; the uncut edges become the folds. Different cuts give different nets. A cube has eleven, counting a net and its mirror image or a turned copy as one. ### [Ordered pairs and the first quadrant](https://math.wiki/ordered-pairs/) **Summary:** Two numbers in a fixed order that name one point, the first across and the second up. **Question:** One number says where something is on a line. A map or a floor spreads out in two directions. How do two numbers pin down one spot, and why does it matter which number comes first? ## Why this exists A number-line gives every number a place, and one number is enough to say where something is on a line: the mark at $7$. A flat place is different. A museum floor, a school campus or a sheet of paper spreads out in two directions, across and up. Saying $7$ no longer tells you where to go: $7$ steps which way? The fix is to use two number lines at once, one running across and one running up, and to name a spot with two numbers: how far across, and how far up. Ruolan's museum map works this way. So does the campus map at Yamini and Chukwuma's school. Once every spot has a pair of numbers, you can find a place from its numbers, write down the numbers of a place you see, and work out how far apart two places are. The two numbers must always be read in the same order, or one pair could send two people to two different places. That is why they are called an **ordered** pair. This lesson takes three days. Part 1 builds the grid and the pair. Part 2 plots and names points, including points on the axes and grids that count by $2$s, $5$s or $10$s. Part 3 uses grid maps to describe places and find distances, with coordinates that can be halves. ## Definition The **coordinate plane** here is a flat grid made from two number lines that are perpendicular: they meet at a right angle, at the $0$ of each. - The **$x$-axis** is the number line running across, from left to right. - The **$y$-axis** is the number line running up. - The **origin** is the point where they meet. It is $0$ on both. ### [Orthocenter and the Euler line](https://math.wiki/orthocenter/) **Summary:** Where the three altitudes meet, a point that lands outside the triangle when the triangle is obtuse. **Question:** The three altitudes of a triangle have no obvious reason to agree. Why do they always pass through one point, and where does that point sit? ## Why this exists A triangle has several families of three special lines, and each family that meets in one point does so for a reason you can see. A point on the perpendicular bisector of a side is equally far from the two ends of that side, so where two perpendicular bisectors cross, the point is equally far from all three vertices, and it must lie on the third bisector as well. That is the circumcenter. The angle bisectors meet for the same kind of reason, with distances to the sides in place of distances to the vertices, at the incenter. The medians meet at the centroid, two thirds of the way down each one. The altitudes are another family of three: from each vertex, the line that meets the opposite side at a right angle. Here no such reason is in sight. An altitude is not the set of points equally far from anything, and in an obtuse triangle two of the altitudes do not even touch the triangle except at their vertex. Yet draw the three altitude lines of any triangle carefully and they pass through one point, every time. This lesson shows why, works out where that point sits, and finds something less expected still: the point where the altitudes meet, the centroid and the circumcenter always lie on one straight line, with the centroid exactly a third of the way from one end. ## Definition The **altitude** from a vertex, as defined in medians-of-a-triangle, is the segment from that vertex perpendicular to the line containing the opposite side; it ends at its **foot** on that line. In an obtuse triangle the altitudes from the two acute vertices fall outside the triangle and do not cross each other at all, so this lesson works with the whole lines rather than the segments. ### [Parallel lines](https://math.wiki/parallel-lines/) **Summary:** Two lines in a flat plane that never meet, however far they go, are parallel. **Question:** Two straight avenues run side by side across a town map. How can you be sure they will never meet, however far the map goes? ## Why this exists Peadar is drawing a map of his town. The avenues run side by side from one end of town to the other. Clodagh writes on lined paper, where the printed lines run across the page side by side. Cian copies pictures onto grid paper. All three keep meeting the same kind of pair: two straight lines that run side by side and never cross. The rails of a straight railroad track are like that, and so are the top and bottom edges of a door. The last two lessons were about lines that cross: kinds of angles and perpendicular lines, which cross at a square corner. This lesson is about lines that never cross at all. The lesson takes two days. **Part 1** is the meaning, the symbol and the marks, on a grid and in everyday things. **Part 2** is telling parallel from not parallel: by the gap, by grid steps, and in rectangles, segments and rays. ## Definition **Part 1: parallel lines.** Think of a flat surface, like a sheet of paper, that goes on forever in every direction. Two lines on the same flat surface are **parallel** when they never meet, however far they go. In symbols, $\ell \parallel m$, read "$\ell$ is parallel to $m$". The symbol is two upright strokes side by side, a picture of the idea. With a slash, $\ell \nparallel m$ means "$\ell$ is not parallel to $m$". On a drawing, parallel lines carry small matching **arrow marks** partway along, as in Figure 1. Lines with one arrow mark are parallel to each other. A second set of parallel lines gets two arrow marks each, so the sets cannot be mixed up. The arrowheads at the very ends still mean what they meant in points, lines, rays and segments: the line keeps going. ### [Parallel lines and transversals](https://math.wiki/parallel-lines-and-transversals/) **Summary:** A line crossing two parallels makes equal and supplementary angle pairs. **Question:** One line crosses two parallel lines. How many of the eight angles do you actually need to measure? ## What this area is about A transversal is a line that crosses two other lines, making eight angles at two crossings. When the two lines are parallel, any two of those eight angles are either equal or add to $180^\circ$. Measure one angle and you know all eight. ## How these fit together **The assumption.** The parallel postulate says there is exactly one parallel through a point off a line. Every result below that starts from parallel lines leans on it. **Named pairs.** Corresponding angles are equal when the lines are parallel. Alternate and same side angles gathers the rest: alternate interior and exterior angles are equal too, and same side interior angles are the odd one out, adding to $180^\circ$. The same star chains these facts to fill in a whole figure. **Right angles and distance.** Distance from a point to a line is measured along the perpendicular from the point. That star also holds the perpendicular transversal theorem: a transversal perpendicular to one of two parallel lines is perpendicular to the other, so one perpendicular drops square to both. Parallel lines stay the same distance apart. ## A path through it Take the postulate, then corresponding angles, then alternate and same side angles with their problems; each pair follows from corresponding angles with vertical angles or linear pairs. Distance from a point to a line comes last. This area lets you prove lines are parallel, show the angles of a triangle add to $180^\circ$, and work out the properties of a parallelogram. ### [Parallelograms, rectangles, rhombuses and squares: what each one guarantees](https://math.wiki/properties-of-special-quadrilaterals/) **Summary:** Each special four-sided shape promises certain things, and anything true of every parallelogram is also true of every rectangle, rhombus and square. **Question:** A parallelogram, a rectangle, a rhombus and a square all have four sides. What does each name promise, and why does a square keep every promise the others make? ## Why this exists At the kite festival, every toy that flies is called a kite. In this lesson the toys are always called **toy kites**, and the word **kite** on its own means a shape, which comes in Part 4. A toy kite's sail can be almost any shape: a parallelogram, a rectangle, a rhombus, a square, a trapezoid, or a kite shape. Golnaz sews edge tape around the sails, and Dumisani judges the sail contest. Neither of them wants to measure every side and every corner of every sail. A shape's name can save that work, because each name is a **promise**. Once you know a sail is a parallelogram, you know things about it that nobody measured. The second saving is bigger. These names fit inside one another, as in shapes-in-more-than-one-category: every square is also a rectangle, and every rectangle is also a parallelogram. So a promise made to every parallelogram is kept by every rectangle and every square too. You learn a fact once, at the top, and it holds all the way down. ## Definition Two words from earlier lessons do most of the work. Two sides are **parallel** when the lines they lie on never meet, however far they go, like the two rails of a track (parallel-lines). Two sides are **perpendicular** when they meet in a right angle, a square corner of $90^\circ$ (perpendicular-lines-geometry). We write $AB \parallel CD$ for "side $AB$ is parallel to side $CD$". A **quadrilateral** is a flat shape with four straight sides. Six special quadrilaterals have names: ### [Partitioning a segment in a given ratio](https://math.wiki/partition-a-segment-in-a-ratio/) **Summary:** Finding the point that divides a segment so its two pieces have a chosen ratio. **Question:** The midpoint cuts a segment into two equal pieces. How do you find the point that cuts it into pieces in some other ratio, such as one part to three? ## When you reach for it The midpoint of a segment cuts it into two equal pieces, and the midpoint formula finds it from the coordinates of the ends. Often the cut you need is not in the middle. A valve goes a third of the way along a straight pipe. A walker at steady speed has used $4$ of her $10$ minutes. The point where the medians of a triangle meet sits two thirds of the way along each one. Each time the question is the same: where is the point that cuts a segment into two pieces whose lengths are in a given ratio? Stated exactly: given points $A$ and $B$ and positive numbers $a$ and $b$, find the point $P$ on $\overline{AB}$ with $$AP : PB = a : b$$ This is called **partitioning the directed segment from $A$ to $B$ in the ratio $a : b$**. The word *directed* matters. The first number belongs to the piece next to $A$, so the cut is counted from $A$; counted from $B$, the same ratio gives a different point unless $a = b$. A directed segment is exactly a segment with that choice of start built in. ## The procedure ### What the ratio says $AP : PB = a : b$ means the whole segment is made of $a + b$ equal parts, of which $AP$ takes $a$ and $PB$ takes $b$. So $P$ is the fraction $$k = \frac{a}{a + b}$$ of the way from $A$ to $B$. This one number does all the work. For the ratio $1 : 3$ it is $\frac{1}{4}$, not $\frac{1}{3}$: there are four parts, and the first piece is one of them. For $2 : 3$ it is $\frac{2}{5}$. Equivalent ratios give the same fraction, since $4 : 6$ gives $\frac{4}{10} = \frac{2}{5}$, so they give the same point, as they should. ### [Parts of a triangle](https://math.wiki/parts-of-a-triangle/) **Summary:** Vertices, sides and interior angles, and which side is included between two named angles. **Question:** A triangle has three corners, three sides and three angles. How do you point at exactly the part you mean, however the triangle is turned on the page? ## Why this exists Turn a triangle over and "the top corner" becomes the bottom one. Stretch it and "the long side" becomes the short one. Any description that leans on how a triangle happens to be drawn breaks the moment the triangle is drawn again. Geometry's fix is to put a letter on each corner and to describe every other part using those letters. A part of the triangle then stops being "this one, here" and becomes a fact about letters: the side across from $A$, or the angle between $\overline{AB}$ and $\overline{AC}$. Those descriptions survive turning, stretching and being copied into a proof. Much of what comes later is written in exactly this language. The rules that say when two triangles must match are sentences such as "two sides and the angle between them" and "two angles and the side between them" (see the congruence criteria), and a sentence like that is only exact if you know which side or angle "between" points to. This lesson makes the vocabulary exact. ## Definition Take three points that do not lie on one line. Join each pair of them with a segment. The figure you have drawn is a **triangle**. - The three points are its **vertices** (one of them is a **vertex**, singular). - The three segments are its **sides**. - At each vertex, the two sides that meet there make an angle that opens toward the inside of the triangle. These are the **interior angles**, and "the angles of a triangle" always means these three. ### [Perimeter and area from coordinates](https://math.wiki/perimeter-and-area-on-the-coordinate-plane/) **Summary:** Computing the perimeter and area of polygons drawn on the coordinate plane with the distance formula and decomposition. **Question:** A polygon is known only by the coordinates of its corners. How long is its boundary, and how much space does it enclose? ## When you reach for it A polygon drawn on a grid with its corners at known coordinates is a polygon you can measure without a ruler. Two questions come up again and again: how far it is around, its perimeter, and how much of the plane it covers, its area. When every side runs along the grid, both are counting. Once a side is slanted, counting squares stops working: the side cuts through squares, and its length is not a whole number of anything. This lesson combines two tools you already have, the distance formula for lengths and boxing and cutting for areas, into one method that measures any polygon from its corner coordinates alone. Reach for it whenever a figure is given by points rather than by measurements: a plot of land on a site plan, a shape on a design grid, or a triangle in a geometry problem where you need its area to find something else. ## The procedure Throughout, the vertices of a polygon are listed **in order around it**, so that each vertex is joined to the next and the last is joined back to the first. That order is what says which pairs of points are sides. ### Perimeter: one distance per side The perimeter is the sum of the side lengths. For a side from $(x_1, y_1)$ to $(x_2, y_2)$: **If the side is horizontal or vertical,** its length is a difference of coordinates, $|x_2 - x_1|$ or $|y_2 - y_1|$. The absolute value is there because a length is never negative, whichever end you start from. ### [Perpendicular and parallel lines through a point](https://math.wiki/construct-a-perpendicular-at-a-point-on-a-line/) **Summary:** Marking equal distances either side of the point and then bisecting the segment between them. **Question:** With only a compass and a straightedge that has no markings, how do you draw the line through a given point that meets a given line at a right angle, or the one that never meets it at all? ## When you reach for it You have a line $\ell$ and a point $P$, and you want one of three lines through $P$: - the line perpendicular to $\ell$, when $P$ is on $\ell$, as at the corner of a square or a right triangle; - the line perpendicular to $\ell$, when $P$ is off $\ell$. This is called **dropping a perpendicular**, and the point where it lands, the **foot**, is the point of $\ell$ closest to $P$. The distance from a point to a line is measured along it, and the altitude of a triangle is one; - the line parallel to $\ell$, when $P$ is off $\ell$. The tools are the same as in every construction: a straightedge, which draws the line through two points and cannot measure, and a compass, which draws arcs and holds a width. All three come down to a move already in hand, bisecting a segment. The trick each time is to make a segment on a line whose perpendicular bisector is the line you want. ## The procedure Every step below leans on one fact from bisecting a segment: **A point that is the same distance from $A$ and from $B$ lies on the perpendicular bisector of $\overline{AB}$.** So two different points, each equidistant from $A$ and $B$, fix the perpendicular bisector. ### The perpendicular at a point on the line **With the point on $P$, swing one arc that crosses $\ell$ on both sides of $P$.** Call the crossings $A$ and $B$. One width made both, so $PA = PB$, and $A$, $P$ and $B$ lie on one line with $P$ between them. So $P$ is the midpoint of $\overline{AB}$. Any width will do. ### [Perpendicular bisector of a segment](https://math.wiki/perpendicular-bisector-of-a-segment/) **Summary:** The line that crosses a segment at its midpoint at a right angle. **Question:** Fold a paper segment so that its two ends land exactly on each other. Where does the crease fall, and why there? ## Why this exists A segment has exactly one middle point, its midpoint. Plenty of lines pass through that point, and each cuts the segment into two equal halves. Plenty of other lines meet the segment squarely, at a right angle. Neither family on its own picks out a single line. Ask for both at once, through the middle **and** square, and the choice collapses to one line: the crease you get by folding one end of a segment onto the other. It later turns out to hold exactly the points equally far from both ends, which makes it the tool for finding the center of a circle through three points. ## Definition The **perpendicular bisector** of a segment $\overline{AB}$ is the line that passes through the midpoint $M$ of $\overline{AB}$ and is perpendicular to $\overline{AB}$. The name carries both conditions. **Bisector** says it passes through the midpoint, so it cuts $\overline{AB}$ into two equal pieces, $AM = MB$; any line, ray or segment doing that is a segment bisector. **Perpendicular** says it meets $\overline{AB}$ at a right angle. On a figure, matching tick marks on $\overline{AM}$ and $\overline{MB}$ say the halves are equal, and a small square at $M$ marks the right angle. In Figure 1, line $\ell$ carries both marks, so every point of $\ell$, such as $P$, lies on the perpendicular bisector of $\overline{AB}$. A ray or segment lying along this line and crossing $\overline{AB}$ at $M$ is also called a perpendicular bisector of $\overline{AB}$. ### [Perpendicular bisectors of a triangle](https://math.wiki/perpendicular-bisector-theorem/) **Summary:** A point is the same distance from two endpoints exactly when it lies on the perpendicular bisector between them. **Question:** Which points are exactly as far from one point as from another, and is there a point that is equally far from three? ## Why this exists Two friends live at points $A$ and $B$ and want a meeting place that is fair: exactly as far from one home as from the other. The midpoint of $\overline{AB}$ works. So does a point far off to the side, if it happens to be placed just right. There are infinitely many fair points, and testing them one at a time with a ruler would never end. The perpendicular bisector was defined by how it is built: the line through the midpoint, at a right angle. That definition says nothing about distance to $A$ or to $B$. This lesson shows that the two ideas pick out the very same line. Every point of the perpendicular bisector is equally far from $A$ and $B$, and no other point is. A question about distances, with infinitely many candidates, becomes a line you can draw. Then add a third friend. A triangle has three sides and so three perpendicular bisectors, and the same fact shows they all pass through one point: the one place equally far from all three corners. ## Definition A point $P$ is **equidistant** from $A$ and $B$ when $$PA = PB$$ Here $PA$ is the length of $\overline{PA}$, the distance from $P$ to $A$. The **perpendicular bisectors of a triangle** are the perpendicular bisectors of its three sides. Each one is a whole line through the midpoint of its side, at a right angle to that side. It belongs to the side, not to the opposite corner, and in general it does not pass through any vertex. ### [Perpendicular lines](https://math.wiki/perpendicular-lines-geometry/) **Summary:** Two lines that meet at a right angle. **Question:** Two lines cross. When do they cross squarely, and how much does one square corner force on the other three? ## Why this exists A carpenter checking a door frame, a surveyor laying out a plot, and anyone drawing a graph all need the same thing: two lines that meet **squarely**, neither leaning toward the other. Square corners are what let a room be a box, a page be a rectangle, and a grid give every point one address. Most crossings are not square. When two lines cross they make four angles in two equal pairs, the vertical angles, and usually one pair is sharp and the other is blunt. There is exactly one way for the two pairs to be the same size, and that special crossing gets its own name. It is also the crossing that measures things honestly: the height of a triangle, the distance from a point to a line, and the axes of the coordinate plane are all built on it. ## Definition Two lines are **perpendicular** when they cross to form a right angle, an angle of exactly $90^\circ$. In symbols, $\ell \perp m$, read "$\ell$ is perpendicular to $m$". On a figure, a small square drawn in the corner marks the right angle. The word describes the **angle** between the lines, not which way they point. Neither line in Figure 1 is level, and they are perpendicular all the same. Segments and rays are called perpendicular when the lines they lie on are. So $\overline{AB} \perp \overline{CD}$ means the line through $A$ and $B$ meets the line through $C$ and $D$ at a right angle. ## Intuition Take a sheet of paper, fold it once to make a straight crease, then fold again so that the first crease lands exactly on top of itself. Open the paper. The second crease crosses the first, and the two angles on either side of it along the first crease are the same size, because the fold laid one exactly on top of the other. Those two angles sit side by side along a straight line, so together they make $180^\circ$. Two equal angles making $180^\circ$ are $90^\circ$ each. ### [Planning a congruence proof](https://math.wiki/plan-a-congruence-proof/) **Summary:** Marking the figure, listing what is given, and choosing which criterion those given parts can reach. **Question:** The Given hands you one or two facts, and every congruence criterion needs three matching pairs. Where do the others come from? ## When you reach for it You are asked to prove two triangles congruent. The four criteria from the congruence criteria are ready to cite: the SSS postulate, the SAS postulate, the ASA postulate and the AAS theorem. Each one needs three pairs of corresponding parts. The Given almost never lists three pairs. It says that a point is a midpoint, that a ray is a bisector, that two segments are parallel, and leaves you to turn those into pairs of parts. Planning is that turning. Before the first line of the two-column proof is written, you find all three pairs, know the reason for each, and know which criterion they fit. The proof after that is short, usually five to eight lines. Without a plan, it is easy to write six true lines that never reach a criterion. Often the two triangles do not sit apart. They share a side or an angle, or their sides run along one line so that a piece belongs to both, and the eye sees small triangles before it sees the two that the Prove names. Planning works the same way for them, with a few moves of its own. The procedure below first plans for two triangles that sit side by side or meet at a point, then, under **Proofs with overlapping triangles**, adds what overlap needs. Either way the lesson stops at the line that says the triangles are congruent; claiming a further part from that line is the next lesson, though Example 11 gives a first taste of it. ## The procedure ### [Plotting in all four quadrants](https://math.wiki/plot-points-with-negative-coordinates/) **Summary:** Extending plotting to all four quadrants. **Question:** A map has places west and south of its starting point as well as east and north. How can a pair of numbers name a point on every side? ## When you reach for it An ordered pair names a point with two numbers: the first says how far across from the origin, the second how far up. With only positive numbers and zero, that reaches just one corner of the plane, above and to the right of the origin. Plenty of pictures need more. A treasure map measured from a well in the middle of an island has places on every side of the well. A robot vacuum that starts from a dock in the middle of a hall wanders behind the dock as well as in front of it. A town measured from its town hall has streets to the west and to the south. Negative numbers on a number line already point the other way from zero. This lesson puts them on both axes, so every point of the plane gets a pair of numbers, names the four regions the axes make, and chooses how much each grid square stands for. ## The procedure The skill is taught over two days, one part each. ### Part 1: plotting and reading points anywhere **The full grid.** Take a horizontal number line, with negative numbers to the left of zero, and a vertical number line, with negative numbers below zero, and cross them at right angles at their zeros. These are the **axes**: the **horizontal axis** (often called the $x$-axis) and the **vertical axis** (the $y$-axis). They cross at the **origin**, $(0, 0)$. Each axis is a whole number line, so it keeps equal steps for equal amounts on both sides of zero. **The two numbers mean what they meant before.** In a pair such as $(-5, 3)$ the first number says how far to move from the origin across, along the horizontal axis, and the second says how far to move straight up or down. What is new is that each number now carries a direction in its sign: ### [Points that differ only in sign: reflections across the axes](https://math.wiki/points-reflected-across-the-axes/) **Summary:** Changing the sign of one coordinate reflects a point across an axis, and changing both reflects it through the origin. **Question:** Two mirrors standing along the axes turn one point into four. Knowing the first point's pair of numbers, what are the pairs of its mirror images? ## Why this exists A kaleidoscope works with mirrors. Stand two flat mirrors at right angles, drop a bead between them, and you see four beads: the bead, its image in each mirror, and in the corner an image of an image. A pattern designer working on a grid wants the same trick on paper: draw one shape, and have its mirror copies appear in the other three corners. Plotting in all four quadrants gives every point of the plane a pair of numbers. This lesson shows that copies made by mirrors along the axes need no measuring at all. A point and its mirror image have the same two numbers, except that one or both have changed sign. Changing the sign of the second number flips a point across the horizontal axis, changing the first flips it across the vertical axis, and changing both turns it half a turn round the origin. ## Definition To **change the sign** of a number is to replace it with its opposite, the number the same distance from $0$ on the other side of $0$. Changing the sign of $3$ gives $-3$, changing the sign of $-3$ gives $3$, and changing the sign of $0$ leaves $0$, because $0$ is its own opposite. The **mirror image** of a point across a line is the point straight across the line from it, on the other side and the same distance away. If the paper were folded along the line, the point and its mirror image would land on top of each other. A point on the line is its own mirror image. Moving a point to its mirror image is **reflecting** it across the line, which is called the **line of reflection** or mirror line. The new point is the **image**, often named with a prime mark: the image of $P$ is $P'$, read "$P$ prime". ### [Points, lines and planes](https://math.wiki/points-lines-planes/) **Summary:** The undefined starting objects, and what is built from them. **Question:** Geometry has to start somewhere. What do you take for granted, and what can you build from it? ## What this area is about Every definition uses other words, so a chain of definitions has to stop somewhere. School geometry stops at three: point, line and plane. They are not defined, only described, and every other figure is built from them. ## How these fit together **The starting objects.** Points, lines, rays and segments names the first four pieces: a line goes on forever both ways, a ray one way from an endpoint, and a segment stops at two endpoints. Points, lines and planes explains why three terms are left undefined, and names collinear and coplanar points. **How they sit together.** Lines and planes in space: two different lines meet in at most one point, and two different planes that meet do so in a line. It also holds skew lines, which exist only in three dimensions. **Built pieces.** Segments, rays and length gives length, and how lengths add along a line. Angles and how to name them: two rays from one point. Locus of points: a figure described by a condition. **The first results.** Angle types and pairs and parallel lines and transversals are where these objects start forcing facts about each other. ## A path through it Points, lines, rays and segments, then the undefined terms and intersections in space. Then angles, and angle types and pairs before segments and length, whose perpendicular bisector needs perpendicular lines. Then parallel lines and a locus. This is the ground floor: triangles and everything after are built from it. ### [Points, lines and planes](https://math.wiki/undefined-terms-in-geometry/) **Summary:** Point, line and plane are accepted without definition so that everything else can be defined from them. **Question:** Every definition is made of other words. Where does geometry's chain of definitions stop? ## Why this exists Look up "point" in a dictionary and you might read "an exact position". Look up "position": "the place where something is". Look up "place": "a particular point or area". Three steps, and you are back where you started. Every dictionary does this. A definition explains a word with other words, and with a finite stock of words the chain can only loop back or stop at words that are never defined. A loop is fine for everyday speech, since you learned "place" by being in places, but useless for mathematics, where every claim has to be traced back to something. So geometry stops the chain on purpose: it picks a few words, declines to define them, and builds everything else on them. Those words are the **undefined terms**, and in school geometry there are three: **point**, **line** and **plane**. The first words defined from them say where points sit. Mark three dots on a page and try to rule one straight line through all of them. Two dots always allow it, but the third has to sit in exactly the right place, and a dot placed at random almost never does. Points that do line up are called **collinear**. The same question comes up one size larger: a flat plane can always be laid through three points, but a fourth point usually sticks out above or below it. The lucky case there is **coplanar**. Whether points line up decides what they can build. Three noncollinear points are the corners of a triangle; four noncoplanar points are the corners of a solid. Many later definitions, from the plane $ABC$ to skew lines, have one of these words inside them. ### [Points, lines, rays and segments](https://math.wiki/points-lines-rays-and-segments/) **Summary:** A line goes on forever both ways, a ray goes on one way from an endpoint, and a segment stops at two endpoints. **Question:** A star map joins stars with straight marks, and a telescope's line of sight goes on into space. How do you say exactly where a straight path starts, where it stops, and whether it ever stops at all? ## Why this exists Nevena, Bindi and Jedda run an astronomy club. On their star map each star is a tiny dot, and the shapes of the constellations are drawn by joining some of the dots with straight marks. When Bindi aims the club's telescope, she looks from where she stands, past one star, and on out into space. To talk about these drawings, the club needs exact words. "The straight bit" is not enough. One straight mark stops at two stars. Bindi's line of sight starts at her eye and never stops. Another straight path might go on forever in both directions. These are three different things, and they get three different names. The place where a star sits gets a fourth name. Those four words are **point**, **line**, **ray** and **segment**. Every shape in geometry is built from them. A triangle's sides are segments. An angle is made of two rays. A square has corners that are points. The lesson takes two days. **Part 1** is the four objects: what each one is, how it is drawn, and how it is named with letters and symbols. **Part 2** is using the names: counting the segments and rays in a figure, and finding them in drawings. ## Definition **Part 1: the four objects.** A **point** is an exact location. It has no size at all; the dot you draw only shows where it is. A point is named with one capital letter, such as point $A$. A **line** is a straight path that goes on forever in **both** directions. It has no ends. It is drawn with an arrowhead at each end to show that it keeps going. A line is named by any two points on it, with a two-headed arrow over the letters: $\overleftrightarrow{BC}$, read "line $BC$". A line can also be named by one lowercase letter, such as line $m$. ### [Polygons](https://math.wiki/polygons/) **Summary:** Closed shapes with straight sides, named by how many. **Question:** Shapes with straight sides come with three, four, five or more corners. What do they share, and what changes as the number of sides grows? ## What this area is about A polygon is a flat, closed figure made of straight sides that meet only at their ends. Counting the sides names it, and the same count fixes how much its interior angles add up to. That angle total in turn helps decide which polygons can cover a floor with no gaps. ## How these fit together **Names.** Naming and classifying polygons says what a polygon is, names it by its number of sides (pentagon, hexagon, octagon), and sorts polygons into convex and concave, regular and not. **A result.** Angle sums of polygons cuts a polygon into triangles to show that the interior angles of a polygon with $n$ sides add to $(n - 2) \cdot 180^\circ$, and also counts diagonals and exterior angles. **A use.** Tessellations asks which shapes tile the plane: where tiles meet corner to corner at a point, their angles must add to exactly $360^\circ$. ## A path through it Angle sums come before tessellations, since the tiling question is settled by the angles at each corner. Naming polygons has no fixed place in that order. Finish here and you are ready for regular polygons and their angles, cyclic quadrilaterals and polyhedra. ### [Polyhedra](https://math.wiki/polyhedra/) **Summary:** Solids whose faces are all flat polygons. **Question:** Some solids are built only from flat pieces and some curve. What makes the flat-sided ones a family of their own, and is there a rule hiding in how many faces, edges and corners they have? ## Why this exists Put a sugar cube, a soup can and a marble on a table. The sugar cube is made only of flat pieces with straight creases between them. The can has two flat ends, but its side curves. The marble curves everywhere. A solid of the first kind can be described completely by flat polygons: you can count its pieces, name their shapes, cut them from card and fold them up. A solid with a curve cannot be taken apart into flat pieces that way. Flat-sided solids turn up wherever nature or people build with straight cuts. A grain of table salt grows as a tiny cube. A quartz crystal grows as a six-sided column with a pointed tip. A jeweler cuts a gemstone into many flat, shiny faces, and a geodesic dome is a round-looking roof made of flat triangles. This lesson names that family and counts its parts. The counting turns up a surprise: for every one of these solids without a hole through it, the numbers of faces, edges and corners obey one simple rule. ## Definition A **polyhedron** is a closed solid whose whole surface is made of flat polygons. The plural is **polyhedra**, and the name is Greek for roughly "many faces". Its parts have names (the earlier lessons faces-edges-and-vertices and name-solid-shapes meet them on everyday solids): - A **face** is one of the flat polygons that make up the surface. - An **edge** is a straight segment where two faces meet. - A **vertex** (plural **vertices**) is a corner point where edges meet. ### [Position words: above, below, beside, in front of, behind](https://math.wiki/position-words/) **Summary:** Words like above, below, beside, in front of and behind say where one object is compared with another. **Question:** How do we say where something is, compared with something else? ## Why this exists Position words say where one thing is compared with another: the lamp is above the table, the cat is behind the couch. Every one of them needs two things, the thing being placed and the thing it is placed against, so "the ball is behind" says nothing until you add "behind the box". Over three days the child meets the words in pairs (above and below, then beside and next to, then in front of and behind), first with their own body, then with objects, then in pictures. No counting is needed for any of it. Ask the child to say the whole sentence, "the book is below the table", not just "below", because the second thing is part of the meaning. ## Day 1: Above and below **You need:** a table, a small ball, a book. Start with the body. Say, "Put your hands above your head. Now put them below your chin." Then "above your knees", "below your knees". Next, hold the ball up over the table without letting it touch: "The ball is above the table." Put the book on the floor under the table: "The book is below the table." Point out that above and below do not need touching: a light hanging from the ceiling is above the table even with a big gap between them. "On" is the word for when they do touch, and "under" works for below with or without touching; both are fine to use too. Then turn the sentence around. Ask, "The ball is above the table. So where is the table, compared with the ball?" The answer is "below the ball". Every above has a below that goes with it. ### [Prisms and pyramids](https://math.wiki/prisms/) **Summary:** Two congruent parallel bases joined by parallelogram faces. **Question:** A tissue box, a glass prism from a science lab and a pyramid tea bag are all built from flat faces. What sorts them into prisms and pyramids, and how does one polygon, the base, decide how many faces, edges and vertices the whole solid has? ## Why this exists Polyhedra, solids made only of flat polygons, come in endless shapes, and counting the parts of each new one from scratch is slow. But two families turn up again and again, in boxes, buildings, packaging and molds. In one, the same polygon sits at both ends with flat walls joining them. In the other, one polygon sits at the bottom and triangles rise from its sides to a point. Naming these families, prisms and pyramids, lets a few words describe a whole solid, and it turns counting into a rule: know the base, and you know how many faces, edges and vertices there are. ## Definition A **prism** is a polyhedron with two faces, its **bases**, that are exact copies of each other (the same shape and size, matching corner to corner) and lie in parallel planes: flat, facing each other, the same distance apart everywhere, like the floor and ceiling of a room. Every other face joins a side of one base to the matching side of the other and is a parallelogram. These are the **lateral faces** ("lateral" means "at the side"). The edges joining matching corners of the two bases are the **lateral edges**. A **pyramid** is a polyhedron with one polygon face, its **base**, and one point off the base, its **apex**. Every other face is a triangle made by one side of the base and the apex. These triangles are its lateral faces, and the edges from the apex to the corners of the base are its lateral edges. ### [Proofs about circles](https://math.wiki/proofs-about-circles/) **Summary:** Arguments that lean on radii being congruent and on the isosceles triangles they create. **Question:** A circle gives a proof almost one fact to start from, that every radius has the same length. How far does that one fact go? ## What this area is about Every point of a circle is the same distance $r$ from its center. Proofs about circles start from little else, and it goes a long way: two radii make an isosceles triangle. This area proves circle facts you may only have measured, so each can be cited in a two-column proof. ## How these fit together **The base.** Proving the inscribed angle theorem shows that an inscribed angle measures half the arc it intercepts, $m\angle APB = \tfrac{1}{2}\, m\overset{\frown}{AB}$. Its first case is the isosceles triangle made by two radii. The same lesson cites it twice to prove that a cyclic quadrilateral's opposite angles add to $180^\circ$, and proves the converse. **Built on it.** Proving the power of a point relationships uses the theorem for equal angles; similar triangles give equal products such as $PA \cdot PB = PC \cdot PD$. **Apart.** Proving a tangent is perpendicular to the radius needs no arcs: an indirect proof shows the radius to the point of contact is perpendicular to the tangent. ## A path through it The inscribed angle theorem first, the tangent theorem any time before power of a point. Power of a point goes last. It also needs similar triangle proofs, the tangent theorem for $PT^2 = OP^2 - r^2$, where $\overline{PT}$ is a tangent from an outside point $P$ and $O$ the center, and the cyclic quadrilateral converse for its own converse. No later star builds on these proofs yet. They back facts that cyclic quadrilaterals and tangent and radius state and use. ### [Proofs about parallelograms](https://math.wiki/proofs-about-parallelograms/) **Summary:** Congruent triangles made by a diagonal supply almost every parallelogram property. **Question:** A parallelogram is defined only by its parallel sides. How do its equal sides and halved diagonals follow, and how do you prove a shape is one? ## What this area is about The definition of a parallelogram says only that both pairs of opposite sides are parallel. Every other fact about it has to be proved, and nearly every proof uses one move: draw a diagonal, find two congruent triangles, and read off their matching parts. ## How these fit together **Properties of every parallelogram.** Proving properties of a parallelogram: opposite sides are congruent, by ASA across one diagonal, and the diagonals bisect each other, using both diagonals and the first theorem. **Running them backwards.** Proving a quadrilateral is a parallelogram turns those properties into tests: both pairs of opposite sides congruent, both pairs of opposite angles congruent, diagonals bisecting each other, or one pair of opposite sides both parallel and congruent, $\overline{AB} \parallel \overline{DC}$ and $\overline{AB} \cong \overline{DC}$. **Special parallelograms.** Proving a parallelogram is a rectangle or a rhombus adds one fact: a right angle or congruent diagonals for a rectangle; two congruent neighboring sides, perpendicular diagonals or a diagonal that bisects an angle for a rhombus; one of each for a square. **A near relative.** Proving properties of an isosceles trapezoid: only one pair of sides is parallel; the same method plus two altitudes shows that congruent legs force congruent base angles and diagonals. ## A path through it Properties, tests, then the rectangle and the rhombus. The trapezoid goes last: it uses the parallelogram tests and, for its converses, the rectangle. ### [Proofs about similar figures](https://math.wiki/proofs-about-similarity/) **Summary:** Arguments that establish proportional sides once two pairs of angles have been matched. **Question:** Congruence proofs end with equal sides. How do you prove that sides are in proportion instead? ## What this area is about A congruence proof ends with equal sides. Many results in geometry are about sides that are not equal but in proportion, and those are proved by showing two triangles similar and then reading a proportion off the similarity. ## How these fit together **The method.** Proving two triangles similar names a correspondence, finds the pairs, and cites AA, SAS or SSS similarity. **Theorems it proves.** The triangle proportionality theorem shows that in $\triangle ABC$ a line parallel to $\overline{BC}$, meeting $\overline{AB}$ at $D$ and $\overline{AC}$ at $E$ strictly inside those sides, cuts them in the same ratio, $\frac{AD}{DB} = \frac{AE}{EC}$, and that a cut in the same ratio at such points is parallel. The Pythagorean theorem with similar triangles drops the altitude to the hypotenuse, which cuts the right triangle into two smaller similar copies, turns two proportions from them into products, and adds those to get $a^2 + b^2 = c^2$. **A proof by area.** The rearrangement proof reaches the same $a^2 + b^2 = c^2$ with no similarity at all, by filling one square two ways with four copies of a right triangle. ## A path through it Start with proving triangles similar: the other two similarity proofs are built on it. Then take the triangle proportionality theorem and the Pythagorean proof in either order. The rearrangement proof needs nothing from the other three, so it can come at any point. ### [Properties of a parallelogram](https://math.wiki/parallelogram-properties/) **Summary:** Opposite sides are parallel and congruent, opposite angles are congruent, and neighboring angles are supplementary. **Question:** Two pairs of parallel lines cross and box in a four-sided shape. What does the parallelism alone force to be true about its sides, its angles and its diagonals? ## Why this exists Draw two parallel lines, then two more parallel lines across them. Where the four lines cross they box in a four-sided shape, and you chose only two things: two directions. You never asked for any lengths to be equal or any angles to match. Yet the shape that comes out has equal opposite sides, equal opposite angles, neighboring angles that add to $180^\circ$, and diagonals that cut each other exactly in half. That is a great deal of information for free, and all of it comes from the angle facts about a transversal crossing two parallel lines. It is also why the shape turns up wherever something must move while staying parallel to itself. A folding arm made of four bars, with opposite bars of equal length, is a parallelogram at every position as long as its bars never cross, so as it folds the top bar keeps the direction of the bottom one. This lesson is about what a parallelogram guarantees, so that when a figure says "parallelogram" you know at once which lengths and angles you get for nothing. ## Definition A **parallelogram** is a quadrilateral whose two pairs of opposite sides are parallel. The vertices are named in order around the shape. In parallelogram $ABCD$ the sides are $\overline{AB}$, $\overline{BC}$, $\overline{CD}$ and $\overline{DA}$, and the definition says $$\overline{AB} \parallel \overline{DC} \qquad\text{and}\qquad \overline{AD} \parallel \overline{BC}$$ ### [Properties of equality and congruence](https://math.wiki/properties-of-congruence/) **Summary:** Congruence of segments and angles behaves like equality, which is why a shared side can be cited. **Question:** Everyone can see that $x = 5$ follows from $2x = 10$, and that a side two triangles share matches itself. So why does a proof have to say so, and what gives it the right to? ## Why this exists Solving $2x = 10$ is something you can do in your head. A proof will not let you. It wants every line to come with a reason, as deductive reasoning requires, because the reader has to be able to check it without trusting you. For the algebra in a proof, the reasons are a short, fixed list. They say what you are allowed to do to an equation and still have a true equation: add the same thing to both sides, replace a thing by something equal to it, and a few more. The oldest of them open Euclid's *Elements* as "common notions": things equal to the same thing are equal to each other, and equals added to or taken from equals give equals. They are things so plain that nobody needs to be argued into them. Once they have names, a proof can say which one it is using. That is all a reason is. "Addition property of equality" is not a spell; it is a label for a move that the reader already believes. Geometry adds a second kind of line, about figures rather than numbers. In Figure 1, $\overline{CA} \cong \overline{CB}$ and $D$ is the midpoint of $\overline{AB}$. The segment $\overline{CD}$ splits the figure into two triangles, $\triangle ACD$ and $\triangle BCD$, and it is a side of both. To show the two triangles congruent by matching three pairs of sides, a proof needs three statements. Two come from the Given. The third is that the side of $\triangle ACD$ called $\overline{CD}$ is congruent to the side of $\triangle BCD$ called $\overline{CD}$. Pull the triangles apart, as in the lower half of Figure 1, and it becomes clear that this is a real pair: one side in each triangle. It is also clear that the pair matches. But a proof accepts no line on the grounds that it is clear, and the line $\overline{CD} \cong \overline{CD}$ needs a reason too. ### [Proportions in similar triangles](https://math.wiki/similar-triangle-proportions/) **Summary:** Set up the ratio of matching sides and solve for the missing one. **Question:** Two triangles are known to have the same shape. How do the sides you can measure give you the one you cannot? ## When you reach for it Two triangles are known to be similar, most often because two pairs of their angles are congruent. Some of their sides have been measured and one has not. Every side of one triangle is the same multiple of its partner in the other, so the measured sides give the missing one through a proportion. The move turns up in two kinds of problem, one per day. On the first day the triangles are drawn on the page: side by side, turned or flipped, or one inside the other. On the second they are in the world, and the missing side is a length nobody can put a tape measure on, such as the height of a lamp or the width of a river. Finding a length from other lengths, without measuring it, is called **indirect measurement**. Every method of it finds a small triangle that can be measured completely and a large triangle with the same angles that contains the length wanted. ## The procedure ### The matching comes from the angles A similarity statement such as $\triangle PQR \sim \triangle STU$ is a list of matched vertices: $P$ with $S$, $Q$ with $T$ and $R$ with $U$. A side is named by its two ends, so the matched vertices fix the matched sides: $PQ$ with $ST$, $QR$ with $TU$, and $RP$ with $US$. Every pair is in one ratio, the **scale factor** $k$: $$\frac{ST}{PQ} = \frac{TU}{QR} = \frac{US}{RP} = k$$ When the triangles are drawn with arcs, the vertices carrying the same arcs are matched, wherever they sit on the page. Said another way, matching sides sit opposite matching angles. ### [Proving a parallelogram is a rectangle or a rhombus](https://math.wiki/prove-a-parallelogram-is-a-rectangle/) **Summary:** Congruent diagonals or one right angle forces all four angles to be right. **Question:** A parallelogram is already halfway to a rectangle and halfway to a rhombus. What is the least extra fact that forces four right angles, and the least that forces four equal sides? ## When you reach for it A proof or a problem already has a parallelogram, or can get one from the tests for a parallelogram, and now needs it to be something more special: a rectangle, a rhombus, or both at once, a square. The targets are four right angles or four congruent sides, and neither is easy to reach directly. A proof rarely hands over four angles or four lengths. What it usually supplies is one right angle, a pair of equal diagonals, one pair of equal sides, a right angle where the diagonals cross, or a diagonal that cuts a corner in half. A parallelogram already carries a lot of structure, and this lesson finds how little has to be added to it for each target. **The words, exactly.** As in rectangles, rhombuses and squares: **Rectangle.** A **rectangle** is a parallelogram with four right angles. **Rhombus.** A **rhombus** is a parallelogram whose four sides are all congruent. **Square.** A **square** is a parallelogram that is both a rectangle and a rhombus: four right angles and four congruent sides. These definitions are **inclusive**. The rectangle definition asks about angles and says nothing about sides, so a square, with four right angles, is a rectangle. The rhombus definition asks about sides and says nothing about angles, so a square is a rhombus too, and every rhombus test in this lesson is passed by squares as well as every rectangle test. The lesson axioms, definitions and theorems states both definitions without the word "parallelogram": a rectangle as a quadrilateral with four right angles, and a rhombus as a quadrilateral whose four sides all have the same length. Each pair of wordings picks out exactly the same shapes, so either can be used. One way around is immediate in both cases: a parallelogram with four right angles is a quadrilateral with four right angles, and the same for four congruent sides. The other way around is a theorem in each case, and each is one of the tests below. A quadrilateral with four right angles is a parallelogram by the argument of the Three Right Angles Test. A quadrilateral $ABCD$ with four congruent sides has $\overline{AB} \cong \overline{CD}$ and $\overline{BC} \cong \overline{DA}$, both pairs of opposite sides congruent, so it is a parallelogram by the Opposite Sides Converse in proving a quadrilateral is a parallelogram; that is the Four Sides Test. Here the parallelogram wordings are the ones in force, because they say what is already known and what is still missing. ### [Proving a quadrilateral is a parallelogram](https://math.wiki/prove-a-quadrilateral-is-a-parallelogram/) **Summary:** Any one of several conditions on the sides, the angles or the diagonals is enough. **Question:** A parallelogram is defined by two pairs of parallel sides, and parallel is hard to see. Which easier facts about the sides, the angles or the diagonals are enough to guarantee it? ## When you reach for it A proof or a problem needs a quadrilateral to be a parallelogram, and nobody has said that it is. The definition asks for both pairs of opposite sides to be parallel, and parallel is a hard thing to show head on. Lengths, angles and midpoints are much easier to get. The lessons before this one went the other way. Starting from a parallelogram, they proved that opposite sides are congruent and that the diagonals bisect each other. This lesson turns those facts round: which of them, on its own, is enough to **force** a parallelogram? Each answer is the converse of a theorem already proved, and a converse is a new statement that needs its own proof, since a true theorem can have a false converse (a theorem and its converse). Four tests work, and the definition itself makes a fifth: | Test | What must be shown about $ABCD$ | |---|---| | definition | $\overline{AB} \parallel \overline{DC}$ **and** $\overline{AD} \parallel \overline{BC}$ | | Opposite Sides Converse | $\overline{AB} \cong \overline{CD}$ **and** $\overline{BC} \cong \overline{DA}$ | | Opposite Angles Converse | $\angle A \cong \angle C$ **and** $\angle B \cong \angle D$ | | Diagonals Converse | $\overline{AC}$ and $\overline{BD}$ bisect each other | | One Pair Test | $\overline{AB} \parallel \overline{DC}$ **and** $\overline{AB} \cong \overline{DC}$, the same pair | Half of any row is not enough, and the Boundaries show a shape for each half that fails. ### [Proving a tangent is perpendicular to the radius](https://math.wiki/prove-the-tangent-radius-theorem/) **Summary:** Every other point of the tangent line lies outside the circle, so the radius has to be the shortest path. **Question:** A line touches a circle at exactly one point. Why does that force the radius to meet the line at a right angle? ## When you reach for it A line is **tangent** to a circle at a point $P$ when $P$ is the only point the line and the circle share. Nothing in that sentence mentions a right angle. Yet in almost every problem about tangents the first thing written down is that the radius to the point of contact is perpendicular to the line. This lesson proves that right angle from the one-point definition, using nothing but a perpendicular from the center, two congruent triangles and the Pythagorean theorem. It also proves the converse, so the statement can be used in both directions: **Theorem.** Let $\ell$ be a line, and let $P$ be a point of $\ell$ that lies on a circle with center $O$. Then $\ell$ is tangent to the circle at $P$ **if and only if** $OP \perp \ell$. Reach for the first half, tangent gives perpendicular, when a figure shows a tangent and a proof or calculation needs a right angle or a right triangle. Reach for the second half, perpendicular gives tangent, when you have to show that a given line really is a tangent. The proof of the first half is also a clear example of indirect proof: assume the opposite, and follow it until it contradicts the definition. The lesson tangent-radius-perpendicular uses this statement freely. Here is why it is true. ## The procedure An "if and only if" is two claims, and each needs its own proof. The claims are the theorem and its converse: tangent gives perpendicular, and perpendicular gives tangent. ### [Proving alternate interior angles congruent](https://math.wiki/prove-alternate-interior-angles/) **Summary:** Combining the corresponding angles postulate with a pair of vertical angles. **Question:** Corresponding angles along parallel lines are known to be equal. How do you turn that into a proof that alternate interior angles are equal, one that later proofs can quote? ## When you reach for it Twice, in two different ways. The first time is once only: to earn the theorem. Alternate interior angles states it, shows with a half turn why it holds, and sketches the argument in prose; deductive reasoning (Example 4) deduces it as a chain of reasons, and a flowchart proof (Example 2) draws the same chain as boxes and arrows. Here it is written as a two-column proof, every line resting on a reason already accepted, so that afterwards it can be a reason itself. On this map the order is fixed: the parallel postulate first, then the Corresponding Angles Theorem proved from it, then this theorem proved from that. The second time is every time after. Whenever a proof has to carry an angle across the gap between two parallel lines, one of its lines will read "Alternate Interior Angles Theorem", and most of this lesson is about writing that line so it is actually justified. ## The procedure The set-up follows Given, prove and the diagram, with the angles numbered as in Figure 1. **Given.** $l \parallel m$; the line $t$ crosses $l$ and $m$. **Prove.** $\angle 3 \cong \angle 6$ and $\angle 4 \cong \angle 5$. The arrowheads on $l$ and $m$ are the Given. No arcs mark $\angle 3$ and $\angle 6$, because their congruence is what has to be earned. **Plan.** $\angle 3$ and $\angle 6$ sit at different crossings, and the only reason on the list that reaches across the gap is the Corresponding Angles Theorem. So go through an angle that is linked to both: $\angle 2$ is opposite $\angle 3$ at the same crossing, and it sits in the same position as $\angle 6$, above its line and to the right of $t$. For the second pair, $\angle 8$ does the same job. ### [Proving angle pair theorems](https://math.wiki/prove-vertical-angles-congruent/) **Summary:** Using two linear pairs that share an angle to show the opposite angles must match. **Question:** Every right angle matches every other, angles with matching partners match, and opposite angles at a crossing match. Each looks too obvious to prove. What does each proof need, and which has to come first, so that later proofs may lean on them? ## When you reach for it A proof needs two angles to be congruent, and the figure gives no measure for either. Three situations come up again and again, and each has a theorem that settles it in one line: - both angles are right angles, wherever they sit; - each angle has a partner it is supplementary or complementary to, and the two partners are congruent or are one and the same angle; - the two angles sit opposite each other where two lines cross. These are the **proofs about angle pairs**. Each fact is already known as a fact: vertical angles showed why opposite angles match, and congruent angles did the algebra of equal measures. Here each is proved with a reason on every line, as in giving a reason for every step. Once proved, its name may stand as the reason for a single line of any later proof, so each short proof here saves many lines elsewhere. **Right Angle Congruence Theorem.** All right angles are congruent. **Congruent Supplements Theorem.** If two angles are supplementary to the same angle, or to two congruent angles, then they are congruent. **Congruent Complements Theorem.** If two angles are complementary to the same angle, or to two congruent angles, then they are congruent. **Vertical Angles Theorem.** If two angles are vertical angles, then they are congruent. **The order they are proved in.** Right angles first, straight from the definitions, as the warm-up. Then congruent supplements and complements, from the definitions and the properties of equality, with no vertical angles anywhere. Then vertical angles, whose shortest proof quotes the Congruent Supplements Theorem. The order matters: a proof may lean only on theorems proved before it, so the vertical angle proof is sound only because the supplements theorem was proved without vertical angles. ### [Proving lines parallel](https://math.wiki/proving-lines-parallel/) **Summary:** Use the transversal angle pairs in reverse. **Question:** Two lines run across a figure and look as if they never meet. Which angles are enough to prove that they never will? ## When you reach for it Whenever the Prove says two lines are parallel. The definition is no help there: "they never meet" is a claim about points as far off as you like, and no figure can show them all. What a figure does offer is angles, eight of them where a transversal crosses the two lines. The facts in parallel-lines-and-transversals run from parallel lines to angles. This lesson runs them backwards, as converses: from the right pair of angles to parallel lines. It is the last step of every proof that a quadrilateral is a trapezoid or a parallelogram, and the reason the compass construction of a parallel line works. ## The procedure Number the angles as in Figure 1: at each crossing upper left, upper right, lower left, lower right, so $\angle 1$ to $\angle 4$ at $l$ and $\angle 5$ to $\angle 8$ at $m$. **The five tests.** The last line of the proof is $l \parallel m$, and it stands on one of these. | The pair | proves $l \parallel m$ when it is | |---|---| | corresponding, such as $\angle 2$ and $\angle 6$ | congruent | | alternate interior, such as $\angle 3$ and $\angle 6$ | congruent | | alternate exterior, such as $\angle 1$ and $\angle 8$ | congruent | | same side interior, such as $\angle 4$ and $\angle 6$ | supplementary | | the angles $t$ makes with $l$ and with $m$ | both right | The first four are cited as "converse of the Corresponding Angles Theorem" and so on, one for each pair. The fifth is cited as "two lines perpendicular to one line are parallel". ### [Proving parts congruent after the triangles](https://math.wiki/proof-with-cpctc/) **Summary:** Proving two triangles congruent first so that a matching side or angle can then be claimed. **Question:** The Prove names a side or an angle that nothing in the Given mentions. How can a pair of triangles hand it to you? ## When you reach for it Look at Figure 1. $E$ is the midpoint of $\overline{AD}$ and of $\overline{BC}$, and you are asked to prove $\overline{AB} \parallel \overline{DC}$. Nothing in the Given mentions either segment, and no definition turns "midpoint" into a fact about angles at $A$ or $D$. But the angles at $A$ and $D$ that would make the segments parallel are angles of $\triangle AEB$ and $\triangle DEC$, and those two triangles can be proved congruent from what you do know. Once they are, every pair of their corresponding parts is congruent, including the pair you wanted. That is the move: **prove the triangles congruent first, then claim the part**, and then, often, one more step. Reach for it whenever the Prove names a side or an angle, or something built from one (a midpoint, a bisector, parallel lines, perpendicular lines), and the Given says nothing about it directly. Almost nobody wants congruent triangles for their own sake. A navigator wants the distance to a ship; a proof wants two segments equal, or two angles. The triangles are a tool for reaching a part that could not be measured or was not given. Writing the first half is proving triangles congruent, and choosing which triangles to use is planning a congruence proof. This lesson is about the second half and how far it can carry you: the result that lets a part out of a congruence, the procedure that uses it, and the **two stage proof**, where the part won from one pair of triangles is what proves a second pair congruent. ### [Proving properties of a parallelogram](https://math.wiki/prove-opposite-sides-congruent/) **Summary:** One diagonal splits the parallelogram into two congruent triangles. **Question:** A parallelogram is defined by its sides being parallel, and nothing in that says how long they are or where its diagonals cross. What makes opposite sides equal in length, and why do the diagonals cut each other exactly in half? ## When you reach for it A proof has a parallelogram in it and needs one of two facts the definition does not give. The definition says only that the opposite sides lie on parallel lines. Two segments on two parallel lines can be any lengths at all, and two diagonals drawn across the shape could, for all the definition says, cross anywhere inside it. Both facts have to be **earned**. - **Opposite sides are congruent.** Reach for this when a proof needs one side equal in length to the side opposite, or when a problem gives sides in terms of a letter. - **The diagonals bisect each other.** Reach for this when a proof needs the crossing of the diagonals to be a midpoint: to find a diagonal from part of it, to find a missing corner in coordinates, or to show that a segment through the crossing is cut in half there. These are the first two of the properties of a parallelogram to be proved, and both proofs follow the pattern of proofs about parallelograms: draw a diagonal, let it cut the figure into triangles, show two of the triangles congruent, and read the wanted parts off them with corresponding parts. The first theorem is proved with one diagonal. The second is proved with both, and uses the first as one of its steps. Once proved, the words **Opposite Sides Theorem**, and "diagonals of a parallelogram bisect each other", may stand as the reason for one line in any later proof, which is how they are used in the examples below. ### [Proving properties of an isosceles trapezoid](https://math.wiki/prove-isosceles-trapezoid-properties/) **Summary:** Base angles and diagonals shown congruent by dropping altitudes or by using congruent triangles. **Question:** An isosceles trapezoid is defined by its two legs being equal in length. Why should that force its base angles, and its two diagonals, to be equal as well? ## When you reach for it A proof or a problem has a trapezoid in it whose two legs are the same length, and it needs one of the facts that follow: the two angles at one base are equal, or the two diagonals are equal. Or it runs the other way, and the job is to show that a trapezoid **is** isosceles because its base angles or its diagonals match. The definition gives none of this. It speaks only of the legs, and the equal angles and equal diagonals have to be earned, the way the equal sides of a parallelogram were earned in proving opposite sides congruent. **The words, exactly.** As in trapezoids and what a definition does, a **trapezoid** here is a quadrilateral with *at least one* pair of parallel sides. The pair of parallel sides we use are its **bases**, and the other two sides are its **legs**. The two angles at the ends of one base are a pair of **base angles**. So a trapezoid $ABCD$ with $\overline{AB} \parallel \overline{DC}$ has two pairs: $\angle D$ and $\angle C$ at the base $\overline{DC}$, and $\angle A$ and $\angle B$ at the base $\overline{AB}$. An isosceles trapezoid is a trapezoid whose legs are congruent. **Why one more hypothesis is needed.** With "at least one pair", that definition lets in more than it seems to. Every parallelogram is a trapezoid, and if one pair of its opposite sides is called the bases, its legs are congruent by the Opposite Sides Theorem. But a parallelogram's base angles are supplementary, not equal, unless it is a rectangle; Figure 7 in the Boundaries has one with base angles $60^\circ$ and $120^\circ$. So the theorems below cannot rest on congruent legs alone. Every one of them assumes all three of these about $ABCD$: ### [Proving the base angles theorem](https://math.wiki/prove-base-angles-congruent/) **Summary:** Adding a bisector or a median to split an isosceles triangle into two congruent halves. **Question:** The Base Angles Theorem needs two triangles to compare, and an isosceles triangle is only one. Which line cuts it into two halves that can be proved congruent, and why may that line be drawn? ## When you reach for it The Base Angles Theorem says that if two sides of a triangle are congruent, the angles opposite them are congruent. That lesson proves it by Pappus's method, matching the triangle with itself, and draws no line. This lesson gives the two proofs most textbooks use. Each adds one auxiliary line from the vertex to the base, cutting the triangle into two halves, and proves the halves congruent. Reach for them when a proof needs more than the base angles: the congruent halves also show that the added line halves the base, halves the vertex angle, and meets the base at right angles. It is proving parts congruent after the triangles, with the triangles made first. ## The procedure **Given.** $\overline{AB} \cong \overline{AC}$ in $\triangle ABC$. **Prove.** $\angle B \cong \angle C$. **Choosing the line.** Three segments from $A$ to $\overline{BC}$ come to mind: the bisector of $\angle A$, the segment to the midpoint of $\overline{BC}$, and the perpendicular from $A$. Each exists and is unique (auxiliary-lines lists why). Any segment $\overline{AD}$ from $A$ to the base makes $\triangle ABD$ and $\triangle ACD$ with two pairs of matching sides already: $\overline{AB} \cong \overline{AC}$ from the Given, and $\overline{AD} \cong \overline{AD}$ by the reflexive property. The line's definition has to supply a third pair. Once the halves are congruent, the base angles follow, because corresponding parts of congruent triangles are congruent. ### [Proving the inscribed angle theorem](https://math.wiki/prove-the-inscribed-angle-theorem/) **Summary:** Handling the case through the center first, then reaching the others by adding or subtracting. **Question:** An angle with its vertex on a circle seems to measure half the arc it opens onto, wherever the vertex sits. How do you prove that for every such angle, when the center of the circle can land on a side of the angle, inside it, or outside it, and what does it then force on four points of one circle? ## When you reach for it Inscribed angles states the fact: an angle whose vertex is on a circle and whose sides are chords measures half the arc it opens onto. Measure a few and it is easy to believe. Proving it is harder than it looks, because the one argument that works cleanly needs the center of the circle to sit on a side of the angle, and in most drawings it does not. So the proof is a proof by cases, and this lesson is about writing it. The plan fits in one sentence: **handle the angle with the center on a side first, then reach every other angle by adding two of those or subtracting one from another.** Once it is written, "Inscribed Angle Theorem" can stand in the Reasons column of a two-column proof, and the rest of proofs about circles leans on it: right angles in semicircles, equal angles on one arc, cyclic quadrilaterals, and the angles made by chords and secants. You reach for the same pattern again whenever a fact about circles is easy in one special position. Prove it there, then draw an auxiliary diameter to cut the general figure into special pieces. The first big use of the theorem is also proved here in full. A cyclic quadrilateral is one whose four vertices lie on one circle. Draw a few and measure: the angles at opposite corners always add to $180^\circ$. You reach for that fact when four points of a figure are known to lie on a circle and you need an angle. Each angle of the quadrilateral is an inscribed angle, so its proof is one citation of the Inscribed Angle Theorem, used twice. ### [Proving the power of a point relationships](https://math.wiki/prove-power-of-a-point/) **Summary:** Similar triangles formed by the chords or secants give the equal products. **Question:** Draw any line through a fixed point and multiply the two distances from the point out to where the line meets a circle. The answer never changes. How do you prove that, for a point inside the circle and for a point outside it, and what is the number it keeps giving? ## When you reach for it Two chords of a circle cross, and each is cut into two pieces. Measure the pieces and multiply each chord's pair: the two products come out equal. Move outside the circle and send two lines through it from one point, and the same thing happens with the distances from that point. Replace one of the lines by a tangent, and the square of the tangent's length joins in. Power of a point states all this as one fact. This lesson proves it. Reach for it whenever a proof about a circle asks for a length, or for a product or square of lengths, along lines through one point. Reach for its converse when you have lengths and want a circle: equal products put four points on one circle. Every case is proved the same way, by the pattern of proving two triangles similar: name the correspondence, find two pairs of angles, cite AA, read off a proportion, clear the fractions. What is new here is where the angles come from. The circle supplies them, through inscribed angles that open onto the same arc, so the Inscribed Angle Theorem, proved in proving the inscribed angle theorem, does the real work. ## The procedure **The words.** A secant is a line that meets a circle at two points. All through this lesson $P$ is a point **not on the circle**, and every line through $P$ that we use meets the circle. $PA$ is the length of $\overline{PA}$, a distance, so it is never negative. Two chords **cross** at $P$ when $P$ is between the ends of each. ### [Proving the Pythagorean theorem by rearrangement](https://math.wiki/prove-the-pythagorean-theorem-by-area/) **Summary:** Fitting four copies of one right triangle into a square in two different ways. **Question:** How can four copies of one right triangle, moved around inside a square, prove something true of every right triangle there is? ## When you reach for it The Pythagorean theorem says that in a right triangle with legs $a$ and $b$ and hypotenuse $c$, $$a^2 + b^2 = c^2$$ Checking it on a few triangles, such as $3$, $4$, $5$, shows that it is plausible, but a pattern is not a proof: there are infinitely many right triangles, long and flat, tall and thin, and no one can check them all. A proof has to work for a triangle whose sides are only letters. This skill does it with area: work out one region's area two ways, one giving $c^2$ and one giving $a^2 + b^2$, and set them equal. Four copies of the triangle and a square of side $a + b$ are all it takes. Reach for it when you are asked why the theorem is true rather than how to use it, when you want a proof built from nothing more than areas of squares and triangles and the angles of a triangle, and, more widely, whenever an equation about squares can be drawn as a statement about area. ## The procedure **What the proof may use.** Five facts, all settled before this lesson. - A square of side $s$ has area $s^2$ (area of a square). - A right triangle with legs $a$ and $b$ has area $\tfrac{1}{2}ab$, half of an $a$ by $b$ rectangle (area of a triangle). - If a region is cut into pieces that do not overlap, its area is the sum of the pieces' areas, and moving the pieces does not change them (area stays the same when the pieces move). - A copy of a figure, moved by sliding, turning or flipping, has the same side lengths, angles and area as the original (congruence by rigid motion). - The angles of a triangle add to $180^\circ$ (the triangle angle sum). In a right triangle the right angle takes $90^\circ$ of that, so **the two acute angles add to $90^\circ$**. This one fact does most of the work. ### [Proving the Pythagorean theorem with similar triangles](https://math.wiki/prove-the-pythagorean-theorem-with-similarity/) **Summary:** The altitude to the hypotenuse makes two similar copies whose ratios combine into the theorem. **Question:** One segment dropped from the square corner cuts a right triangle into two smaller copies of itself. How do those copies prove the Pythagorean theorem? ## When you reach for it The Pythagorean theorem says that in a right triangle with legs $a$ and $b$ and hypotenuse $c$, $$a^2 + b^2 = c^2$$ The rearrangement proof gets there by counting areas. This proof uses no area at all. It uses one segment and the idea of proving two triangles similar: drop the altitude from the right angle to the hypotenuse, notice that it cuts the triangle into two smaller triangles with the same angles as the whole, and read proportions off those similarities. Two of the proportions, added together, are the theorem. Reach for it when a proof about a right triangle has to be made with lengths and ratios, when a figure shows a right triangle with the altitude to its hypotenuse drawn in, and whenever a problem gives you some of the six lengths in that figure (two legs, the hypotenuse, the altitude and the two pieces of the hypotenuse) and asks for the rest. Along the way the proof gives two more facts about that figure that are worth knowing for their own sake. ## The procedure **The figure and its names.** Start with $\triangle ABC$ with its right angle at $C$, so that $\overline{AB}$ is the hypotenuse. Draw the altitude from $C$: the segment from $C$ perpendicular to $\overline{AB}$, meeting it at a point $D$, called the **foot** of the altitude. Name the six lengths as in Figure 1: $$a = BC \qquad b = AC \qquad c = AB$$ $$h = CD \qquad p = AD \qquad q = DB$$ Each piece of the hypotenuse touches one leg: $p$ touches $b$ at $A$, and $q$ touches $a$ at $B$. ### [Proving the triangle angle sum](https://math.wiki/prove-the-triangle-angle-sum/) **Summary:** Drawing a line through one vertex parallel to the opposite side and reading the angles off it. **Question:** The angles of every triangle make a straight angle, thanks to one added line, and an angle outside a triangle equals the two far angles inside it. What do those arguments look like when every step, the added line included, has to name the fact that allows it? ## When you reach for it Once for each theorem, to earn it, and then every time after, to use it. The triangle angle sum lesson states two theorems, uses them on numbers and gives the reason for each in prose: a line through one vertex, parallel to the opposite side, lays the three angles side by side along a straight line; and an angle outside a corner is whatever the inside angle there leaves of $180^\circ$, which is exactly what the other two angles add to. Its exercises are the place to practice finding angles. Here those two reasons become two-column proofs, set up as in Given, prove and the diagram, with a named reason on every line as in giving a reason for every step. An added line gets a reason too, as every auxiliary line must. On this map the order is fixed: the parallel postulate, then the Corresponding Angles Theorem, then the Alternate Interior Angles Theorem, then the Triangle Angle Sum Theorem, then the Exterior Angle Theorem, with the Exterior Angle Inequality a few lines behind it. Each may cite only what comes before it. Once proved, each theorem is a one-line reason. Whenever a proof meets a triangle, "Triangle Angle Sum Theorem" turns its three angles into an equation. Whenever a triangle has a side running on past a corner, "Exterior Angle Theorem" trades the outside angle for two inside ones. Most of the worked examples are about writing those single lines so that they are justified, and about the three facts that follow from the angle sum and are quoted just as often. ### [Proving the triangle proportionality theorem](https://math.wiki/prove-the-triangle-proportionality-theorem/) **Summary:** A line parallel to one side makes a smaller similar triangle, and that similarity gives the proportion. **Question:** A line runs across a triangle parallel to one side. Why must it cut the other two sides in the same ratio, and why does a cut in the same ratio have to be parallel? ## When you reach for it Similar triangles give proportions between **sides of triangles**. Very often the lengths you want are not sides of any one triangle but pieces of sides: a triangle is cut by a line, and you know the pieces on one side and want the pieces on the other. The triangle proportionality theorem, also called the side splitter theorem, is the bridge. Reach for it when a figure has a line parallel to one side of a triangle and crossing the other two, and the question is about the pieces those two sides are cut into. Reach for its converse in the other direction: when the Given talks about lengths, and the Prove says two lines are parallel. Both are proved from similar triangles. The parallel line cuts off a small triangle that shares an angle with the whole triangle, and that similarity does all the work. What this lesson adds is the careful bookkeeping: which triangle is similar to which, under which correspondence, and the short piece of algebra that turns whole sides into pieces. ## The procedure **The statement.** Here is the theorem with every hypothesis written out. **Triangle proportionality theorem.** In $\triangle ABC$, let $D$ lie on side $\overline{AB}$ and $E$ on side $\overline{AC}$, each strictly between the two vertices. If $\overline{DE} \parallel \overline{BC}$, then $$\frac{AD}{DB} = \frac{AE}{EC}$$ and also, for whole sides, $$\frac{AD}{AB} = \frac{AE}{AC} = \frac{DE}{BC}$$ ### [Proving triangles congruent](https://math.wiki/proving-triangles-congruent/) **Summary:** Pick the criterion, line up the parts, conclude the rest match. **Question:** The Given hands you a midpoint, a bisector or two parallel lines. How do you turn that into a proof that two triangles, and then their parts, are congruent? ## What this area is about A congruence proof is a two-column proof that ends at a line such as $\triangle ABC \cong \triangle DEF$, or passes through one to reach a side or an angle. The tools are fixed: the SSS, SAS and ASA postulates, the AAS theorem, and HL for right triangles, each needing three matching pairs. The Given almost never lists three pairs, and this area is the craft of finding them. ## How these fit together **The core procedure.** Planning a congruence proof: start from the Prove, unpack each given into a pair of parts, find the pairs nobody states (a shared side, vertical angles), and choose the criterion they fit. It also separates overlapping triangles sharing a side, an angle or a piece of one, using the reflexive property and segment or angle addition. **Going past the triangles.** Proving parts congruent after the triangles: prove the triangles congruent, then claim a matching side or angle, because corresponding parts of congruent triangles are congruent. Its two stage proofs use one congruent pair to supply the missing part for another. **A classic, worked in full.** Proving the base angles theorem adds one line from the vertex to split an isosceles triangle into two congruent halves. ## A path through it Planning, then parts after the triangles, then the base angles proof, which uses them. You need the congruence criteria first. After this, congruent triangles serve proofs about other figures, such as why opposite sides of a parallelogram are congruent. ### [Proving two triangles similar](https://math.wiki/prove-triangles-similar/) **Summary:** Finding two congruent pairs of angles, often from parallel lines or from a shared angle. **Question:** Two triangles look like the same shape at different sizes. What is the least you have to prove to be sure, and how do you write the proof? ## When you reach for it A congruence proof ends with equal sides. Many of the results that matter in geometry are about sides that are **not** equal but in proportion: a short post and a tall tree casting shadows, the pieces of two crossing chords, the two legs and the hypotenuse of a right triangle. Those results are proved by showing that two triangles are similar, and then reading a proportion off the similarity. So reach for this whenever the Prove says $\triangle ABC \sim \triangle DEF$, and also whenever it asks for a ratio of lengths, a product of lengths, or a length you cannot measure directly. In the second case the similarity is a step in the middle of the proof rather than its last line. You already have the tools for the proof itself from two-column proofs and planning a congruence proof: unpack the Given into pairs of parts, look for the pairs the figure hands you, then cite a criterion. The criteria change, and the pairs of sides now need to be in proportion instead of equal. Everything else carries over. ## The procedure **What "similar" asks for.** Two triangles are similar when there is a correspondence of their vertices under which **every pair of corresponding angles is congruent and every pair of corresponding sides is in the same ratio**. Both halves are part of the definition. Writing $\triangle ABC \sim \triangle DEF$ claims the correspondence $A \leftrightarrow D$, $B \leftrightarrow E$, $C \leftrightarrow F$, and with it all six of these facts: ### [Pythagorean triples](https://math.wiki/pythagorean-triples/) **Summary:** Whole number side lengths that fit the Pythagorean relationship, such as 3, 4 and 5. **Question:** Most right triangles have a side that is not a whole number. Which ones have three whole-number sides, how many are there, and can you list them all? ## Why this exists Take a right triangle at random and the chances are that one of its sides is a number you cannot write down exactly. Two legs of $1$ give a hypotenuse of $\sqrt{2}$. Legs of $1$ and $2$ give $\sqrt{5}$. Both are decimals that never end and never repeat. A few right triangles behave better. Legs of $3$ and $4$ give a hypotenuse of exactly $5$, with no root and no rounding anywhere. A builder with a tape measure and no calculator can use that: mark $3$ along one wall and $4$ along the other, and if the gap between the marks reads $5$, the corner is square, by the converse of the Pythagorean theorem. That raises three questions. Are there other triangles like it? If so, is there a way to make them on demand, instead of hunting? And does that way reach all of them? The answers are yes, yes and yes, and the method is short enough to write on one line. It also turns up a surprise: every one of these triangles has a side that is a multiple of $3$, a side that is a multiple of $4$, and a side that is a multiple of $5$. ## Definition A **Pythagorean triple** is three positive whole numbers $a$, $b$ and $c$ that satisfy the Pythagorean equation $$a^2 + b^2 = c^2$$ It is written $(a, b, c)$, and by custom $a < b < c$. The number $c$ is always the largest, because $c^2 = a^2 + b^2$ is bigger than either square alone. The two legs can never be equal: $a = b$ would give $c^2 = 2a^2$, so $c \div a = \sqrt{2}$, and the square root of $2$ is not a fraction. ### [Quadrilaterals](https://math.wiki/quadrilaterals/) **Summary:** Squares, rectangles, parallelograms, rhombuses, trapezoids, and how they nest. **Question:** Four straight sides can close up in endless ways. Which four-sided shapes earn a name, and what does each name promise? ## What this area is about A quadrilateral is a flat polygon with four sides. A few conditions on sides, angles or parallel pairs pick out shapes with names, and each name carries facts nobody had to measure. The names are inclusive, so one shape can carry several. ## How these fit together **First names.** Rhombuses, rectangles and squares are sorted with a ruler and the corner of a card, and shapes in more than one category shows that a square is all three, while many quadrilaterals are none. **Definitions and the family tree.** What each special quadrilateral guarantees lists the promises behind each name, and the quadrilateral family tree places square, rectangle, rhombus, parallelogram, trapezoid and kite inside one another. **Results.** Properties of a parallelogram: if both pairs of opposite sides are parallel, then opposite sides and opposite angles are congruent, neighboring angles add to $180^\circ$, and the diagonals bisect each other. Rectangles, rhombuses and squares add more: a rectangle's diagonals are congruent, a rhombus's are perpendicular. In trapezoids and kites, the two angles along each leg of a trapezoid add to $180^\circ$, an isosceles trapezoid has equal base angles and congruent diagonals, and a kite's diagonals are perpendicular. ## A path through it Rhombuses, rectangles and squares, then shapes in more than one category, then what each quadrilateral guarantees. Parallelograms come next, then rectangles and rhombuses, then trapezoids and kites, and the family tree last, since it gathers them all. ### [Radius and diameter](https://math.wiki/radius-and-diameter/) **Summary:** The distance from the center out to the edge, and the distance all the way across through the center. **Question:** A compass, a rope tied to a stake and the hand of a clock all measure a circle from the middle out, but a tape across a table, the size printed on a pizza box and the inside of a tin all measure it straight across. How do you get from one to the other, and what do they tell you about how circles fit in rows and boxes? ## Why this exists A square's size is one number, the length of its side. A circle has no sides, but it too has one number that settles its size: how far its edge is from its center. Know that, and you know how wide the circle is, how far to open a compass to draw it, how long a rope must be to mark it out on the ground, and how big a box it needs. The world hands you that size in two forms. A compass, a rope on a stake or a clock hand gives the distance from the middle out; a tape across a table or the inside of a box gives the distance all the way across. This lesson is about moving between the two, and about fitting circles into rows and boxes. ## Definition The words come from what a circle is. A **radius** is a segment from the center to a point on the circle, and its length is written $r$. A **diameter** is a chord through the center, and its length is written $d$. Each word names the segment and also its length, and the plural of radius is **radii**. A diameter is two radii laid end to end in one straight line, so $$d = 2r \qquad\text{and}\qquad r = \frac{d}{2}$$ In everyday measurements, "across" means the diameter: a pizza $30$ cm across, a cake tin $20$ cm across. "From the center", the opening of a compass and the length of a tether give the radius. ## Intuition Roll a coin between two rulers held parallel on a table. However you turn it, the rulers stay the same distance apart: a circle reaches exactly $r$ beyond its center on each side, whichever way you look, so it is $d$ wide in every direction. That is why a coin in a tray needs $d$ of room each way, and a row of touching coins needs one diameter per coin. ### [Rectangles, rhombuses and squares](https://math.wiki/rectangle-properties/) **Summary:** A parallelogram with four right angles, whose two diagonals are also equal in length. **Question:** A parallelogram already has equal opposite sides and diagonals that cut each other in half. What more do you get when every corner is a right angle, or every side is the same length? ## Why this exists Every parallelogram comes with a set of guarantees: its opposite sides are parallel and congruent, its opposite angles are congruent, its neighboring angles add to $180^\circ$, and its two diagonals bisect each other. Those facts hold for every parallelogram, however slanted it is. Ask the parallelogram for more and it gives more back. Make every corner a right angle, and the two diagonals come out the same length. Make every side the same length, and the diagonals cross at right angles and cut every corner exactly in half. Ask for both at once and you have a square, which has all of these together. These extra facts are what let you find a length or an angle inside a rectangle or a rhombus from very little information: one side and one diagonal, or one angle between a side and a diagonal. They are also the facts that the tests in proving a parallelogram is a rectangle or a rhombus turn round the other way. ## Definition All three shapes are parallelograms with something extra. A **rectangle** is a parallelogram with four right angles. A **rhombus** is a parallelogram whose four sides are all congruent. A **square** is a parallelogram that is both a rectangle and a rhombus: four right angles and four congruent sides. For a quadrilateral $ABCD$, named in order around it, "rectangle" means that each of $\angle A$, $\angle B$, $\angle C$ and $\angle D$ measures $90^\circ$, and "rhombus" means $AB = BC = CD = DA$. ### [Reflections](https://math.wiki/reflect-over-the-axes/) **Summary:** Flipping a figure over the x axis or the y axis by changing the sign of one coordinate. **Question:** A figure on graph paper is flipped over a line. Where exactly does each corner land, and if you see only the two figures, how do you find the line? ## When you reach for it A reflection flips a figure over a line, the **mirror** or **line of reflection**, the way a shape drawn in wet ink prints onto the other half of a folded page. On plain paper you can fold and trace. On the coordinate plane you can do better: name exactly where every point lands, as a pair of numbers, with no folding and no guessing. Reach for this move in two situations. The first is going forwards: a figure and a mirror are given, and you want the image. When the mirror is one of the axes, a vertical or horizontal line, or one of the two diagonals $y = x$ and $y = -x$, the image comes from a one-line rule on the coordinates. The second is going backwards: a figure and its image are given, and you want the mirror, or want to know whether there is one at all. Both rest on the single fact that defines a reflection, which comes from the perpendicular bisector. ## The procedure ### What a reflection does to one point Fix a line $\ell$ as the mirror. The reflection in $\ell$ sends a point $P$ to its image $P'$ like this. - If $P$ is on $\ell$, it stays where it is: $P' = P$. - If $P$ is not on $\ell$, then $P'$ is the point that makes $\ell$ the perpendicular bisector of $\overline{PP'}$. So the segment $\overline{PP'}$ crosses the mirror at a right angle, and the mirror cuts it at its midpoint. Put another way: drop the perpendicular from $P$ to the mirror, and carry on past the mirror by the same distance again. That is the whole definition, and every rule below is this definition worked out for one particular mirror. The precise treatment, and the proof that a reflection keeps every length and every angle, is in rigid motions defined precisely. ### [Regular polygons and their angles](https://math.wiki/interior-angle-of-a-regular-polygon/) **Summary:** Sharing the interior angle sum equally among angles that are all congruent. **Question:** When every side and every angle of a polygon is the same, how big is each angle, and which sizes are possible at all? ## When you reach for it A **regular polygon** is a polygon whose sides all have the same length and whose interior angles all have the same size. A polygon with all its sides equal is called **equilateral**, and one with all its angles equal is called **equiangular**. Regular means both at once. Both conditions are needed, because from four sides on neither one forces the other. The rhombus in Figure 1 has four equal sides, but two of its angles are $60^\circ$ and two are $120^\circ$. The rectangle has four equal angles, all $90^\circ$, but its long sides are not the same length as its short ones. Only the hexagon has both, so only the hexagon is regular. Triangles are the one exception. A triangle with three equal sides has three equal angles, and the other way round, by the facts about isosceles and equilateral triangles. So a regular triangle is just an equilateral triangle, and a regular quadrilateral is a square. Reach for this lesson when you know a polygon is regular, or only that its angles are all equal, and you want the size of one angle. Reach for it the other way round when you know one angle and want the number of sides, or want to know whether a regular polygon with that angle exists at all. Both rest on two facts proved in angle sums of polygons: the interior angles of an $n$-gon add to $(n - 2) \cdot 180^\circ$, and the exterior angles, one at each vertex of a convex polygon, add to $360^\circ$. This lesson shares those totals out equally. ### [Regular polygons and their circles](https://math.wiki/central-angle-of-a-regular-polygon/) **Summary:** The angle at the center between two neighboring vertices, equal to 360 degrees divided by the side count. **Question:** Every regular polygon fits exactly inside one circle and exactly around another. Why, and what does the angle at the center between two neighboring corners tell you? ## Why this exists The angles of a regular polygon are found by working along its edge, one corner at a time. That says how sharply each corner turns, but not how the polygon sits as a whole. A regular polygon also has a middle: one point that is the same distance from every corner and the same distance from every side. Join that point to the corners and the polygon falls apart into identical triangles, all meeting in the middle like slices of a pie. Those triangles are how almost everything else about a regular polygon is worked out. Its perimeter and its area are $n$ copies of one triangle's. Its turns onto itself are turns about the middle point. And the idea runs the other way too. To lay out a regular hexagon for the floor of a garden gazebo, or a twelve-sided table top, nobody measures $150^\circ$ angles along the edge. You draw a circle, share the full turn at its center into equal parts, and join the marks. The angle that does the sharing is the central angle, and this lesson is about it and the two circles that come with it. ## Definition Take a regular polygon with $n$ sides: all its sides are equal and all its angles are equal. - Its **center** is the point $O$ that is the same distance from all of its vertices. Every regular polygon has exactly one such point; "Why it is true" shows it. - The circle with center $O$ through all the vertices is the **circumscribed circle**. Its radius is the **radius** of the polygon, and the polygon is **inscribed** in the circle. - A **central angle** is the angle at $O$ between the radii to two **neighboring** vertices, such as $\angle AOB$ in Figure 1. - The radii to the vertices cut the polygon into $n$ triangles, one standing on each side, such as triangle $OAB$. This lesson calls them the **triangles at the center**; the name is a label for these pages, not a standard term. - The circle with center $O$ that touches every side is the **inscribed circle**. "Touches" means it meets each side at exactly one point and never crosses it, so each side is tangent to the circle. Its radius is the distance from $O$ to a side, which is called the apothem of the polygon. ### [Rhombuses, rectangles and squares](https://math.wiki/rhombuses-rectangles-and-squares/) **Summary:** A rhombus has four equal sides, a rectangle four right angles, and a square has both. **Question:** Every four-sided shape is a quadrilateral. Which ones earn a name of their own, and how can a ruler and the corner of a card tell them apart? ## Why this exists A quadrilateral is any closed shape with four straight sides. Shapes are named by what they must have, whatever their size, as in draw-shapes-with-given-attributes. But "quadrilateral" covers a great deal: a door, a kite in the sky, a slice of a pizza box, a lopsided scrap of paper. The word alone says very little. Two questions about a quadrilateral pick out the shapes people use most: - Are all four **sides** the same length? - Are all four **corners** square corners? Each question has a simple tool. A ruler answers the first. The corner of an index card answers the second. The two answers give three names: **rhombus**, **rectangle** and **square**. A name is a promise. If a carpenter is told a window is a rectangle, the carpenter knows every corner of the frame is square without seeing it. If a quilter is told a patch is a rhombus, the quilter knows all four edges can be cut to one length. ## Definition A **square corner** is a corner that the corner of an index card fits exactly: one edge of the card lies along each side, with no gap and nothing sticking out. Later lessons call a square corner a right angle. - A **rhombus** is a quadrilateral whose four sides are all the same length. - A **rectangle** is a quadrilateral whose four corners are all square corners. - A **square** is a quadrilateral whose four sides are all the same length **and** whose four corners are all square corners. ### [Rigid motions and congruence](https://math.wiki/congruence-by-rigid-motion/) **Summary:** Two figures are congruent exactly when some sequence of slides, flips and turns carries one onto the other. **Question:** Two shapes look like copies of each other. What exactly does "the same shape and the same size" mean, and how could you ever be sure? ## Why this exists "Congruent" is meant to say "the same shape and the same size". That sounds clear until you try to check it. Two triangles on a page sit in different places and point different ways, and one may be a mirror image of the other. Which features have to match? The sides? The angles as well? Is the same area enough? A definition has to say exactly. The answer here uses motion. Trace one figure on tracing paper, then slide the paper, turn it, or flip it over. If some run of those moves lays the tracing exactly on the other figure, the two are congruent. Slides, turns and flips are the rigid motions: they move a figure without stretching, shrinking or bending it. So "same shape and size" becomes something you can do and check. You name the moves and follow every point. The definition pays off twice. Because a rigid motion keeps every length and every angle, congruent figures have matching sides and matching angles. And it explains the familiar shortcuts for triangles, SSS, SAS and ASA, which say that three matching parts are enough. A course can take those three as starting points, and the triangle congruence criteria does. This lesson gives a **second route to the same three facts**: each one can be proved by building the motion that carries one triangle onto the other. ## Definition A **rigid motion** is a translation, a rotation, a reflection, or a sequence of them done one after another. Composing rigid motions gives another rigid motion. Rigid motions defined precisely starts from the other end, calling any move that keeps every distance a rigid motion; in the plane the two descriptions pick out the same moves. ### [Rigid motions defined precisely, and what they preserve](https://math.wiki/rigid-motion-definitions/) **Summary:** Translations, reflections and rotations defined from segments, perpendicular bisectors and circles, and why each keeps distances and angles. **Question:** Slide, flip and turn are easy to picture. Where exactly does each one send a single point, and how can we be sure nothing gets stretched? ## Why this exists "Slide it", "flip it" and "turn it" are good enough for moving a paper triangle across a desk. They are not good enough for a proof. A proof needs to know, for every point $P$ of the plane, exactly which point $P$ lands on, and that answer has to be built from objects geometry has already pinned down: segments, parallel and perpendicular lines, and circles. What rigid motions preserve listed what survives a slide, a flip or a turn: lengths, angle measures, betweenness and parallel lines. Those facts were read off the picture. This lesson defines each motion precisely enough that the facts can be **proved**, and precisely enough that a claim like "this rule is a rotation" can be checked instead of eyeballed. Exact definitions are what congruence by rigid motion and transformational proof are built on. ## Definition A **transformation** of the plane is a rule that assigns to every point $P$ of the plane exactly one point $P'$, called the **image** of $P$. It is a function whose inputs and outputs are points. If the transformation is called $T$, we write $T(P) = P'$. A point with $T(P) = P$ is a **fixed point** of $T$. The rule acts on the whole plane, not only on the figure in front of you. The image of a triangle is just the set of images of its points. A transformation $T$ is a **rigid motion**, also called an **isometry**, when it keeps every distance: $$P'Q' = PQ \quad\text{for every pair of points } P,\ Q$$ ### [Rotating about a point other than the origin](https://math.wiki/rotate-about-any-point/) **Summary:** Turning a figure around a chosen center while every distance to that center stays the same. **Question:** Turning a point about the origin has a neat coordinate rule. What happens when the turn is about some other point? ## When you reach for it A rotation is fixed by three things: the point it turns about, the angle, and the direction, as the lesson on rotations sets out. When the center is the origin, a quarter turn counterclockwise has a one-line rule, $(x, y) \mapsto (-y, x)$. That rule is about the origin and nothing else. Apply it to a turn about $(2, 1)$ and the answer is wrong, because the origin is not where the turning happens. Most turns in the world are not about the origin. A door swings about its hinge, a wheel turns about its hub, and a figure on a grid is often turned about one of its own corners. Reach for this skill whenever the center of a rotation is a point $C = (a, b)$ other than $(0, 0)$ and you want the exact coordinates of the image. Reach for it backwards too, when you can see a point and where it landed and want to know what turn did it. ## The procedure ### What a rotation keeps A rotation about $C$ through an angle $\theta$ sends a point $P$ to the point $P'$ with two properties: - $CP' = CP$: the image is the same distance from the center as the point was, so $P$ travels along a circle centered at $C$; - $\angle PCP' = \theta$, measured in the stated direction: counterclockwise when $\theta$ is positive, clockwise when it is negative. The center itself does not move. Everything below is a way of finding the one point with those two properties. ### Measure from the center ### [Rotational and point symmetry](https://math.wiki/rotational-symmetry/) **Summary:** A figure has rotational symmetry when a turn of less than a full circle maps it onto itself. **Question:** Some shapes look exactly the same after you turn them part of the way around. Which ones, and by how much can you turn them? ## Why this exists Turn a square a quarter of the way around its center and it lands exactly on its own outline. If you looked away while it turned, you could not tell that anything had happened. A pinwheel does the same, and so does a regular hexagon, after a sixth of a turn. Line symmetry does not describe this. A mirror line catches shapes whose two halves match, but a pinwheel has no mirror line at all, and it is still plainly symmetric. What it has is a turn that leaves it unchanged. This lesson names that kind of symmetry, counts it, and finds the one special turn, the half turn, that shows up in every parallelogram and in the graph of every odd function. Symmetry also pays. A turn is a rigid motion, so it keeps every length and angle. When a turn carries a figure onto itself, each side is equal to the side it lands on and each angle to the angle it lands on. A symmetry hands you equal parts without measuring anything. ## Definition A rotation turns every point of the plane about one fixed point $O$ through the same angle. A figure has **rotational symmetry** when some rotation about a point $O$, through an angle strictly between $0^\circ$ and $360^\circ$, maps the figure exactly onto itself: every point of the figure lands on a point of the figure, and every point of the figure is landed on. The point $O$ is the **center** of the symmetry, and each angle that works is an **angle of rotational symmetry**. A turn through $360^\circ$ puts every figure back where it started, so it is left out of the definition: it says nothing about the shape. ### [Rotations](https://math.wiki/center-and-angle-of-rotation/) **Summary:** A rotation is fixed by the point it turns about, the number of degrees, and the direction. **Question:** Turn it a quarter turn" sounds like a complete instruction. What else has to be said before everyone who follows it ends up with the same picture? ## Why this exists A slide is fixed by how far and which way it goes. A flip is fixed by its mirror line. A turn needs more than either. Put a square card on a table and say "turn it a quarter turn". Pin the card at its center and it spins in place. Pin it at a corner and it swings out to a new spot. Turn it to the left or to the right and it lands in two different places again. Three people following the same words can end up with three different pictures. So a rotation is pinned down by three facts: the point it turns about, how far it turns, and which way. Once all three are named, every point of the plane has exactly one place to go, and that is what lets a rotation be drawn, computed, checked and used in a proof. On a coordinate grid, with the turn about the origin and a quarter or half turn, the whole rotation shrinks to a rule as short as swapping two numbers and changing a sign. ## Definition A **rotation** turns the whole plane about one fixed point. It is described by three things: - the **center** $C$, the one point that stays where it is; - the **angle of rotation** $\theta$, how far the plane turns, measured in degrees; - the **direction**, counterclockwise or clockwise. The rotation about $C$ through $\theta$ sends $C$ to itself. It sends every other point $P$ to the point $P'$ for which - $CP' = CP$: the image is exactly as far from the center as the point was; and - the ray $CP'$ is the ray $CP$ turned through $\theta$ in the stated direction. For $\theta$ up to $180^\circ$ this says $m\angle PCP' = \theta$. ### [Scale factor and perimeter, area and volume](https://math.wiki/ratio-of-perimeters-and-areas/) **Summary:** In similar figures perimeters scale like the scale factor and areas scale like its square. **Question:** A model is built exactly like the real thing, one tenth of the size in every direction. Does it need one tenth of the paint, and hold one tenth of the water? ## Why this exists A scale factor tells you what happens to lengths: make a copy with every length three times as long, and every side, every diagonal and every height is three times as long. But most questions people ask about a shape are not about one length. How much paint covers it? How much land is inside the fence? How much water does the tank hold? How heavy is the statue? Those are areas and volumes, and they do not follow the scale factor. A model one tenth of the size needs far less than one tenth of the paint, and holds far less than one tenth of the water. This lesson finds the exact rule. It is short, it holds for every pair of similar figures and every pair of similar solids, whatever their shape, and it explains effects that look surprising until you know it: a large pizza that is only a little wider than a small one but has much more pizza, and a glass filled to half its height that is nowhere near half full. ## Definition Two figures are **similar** when one is a scaled copy of the other: every length in one is the same number $k$ times the matching length in the other. That number is the **scale factor**, $$k = \frac{\text{a length in the copy}}{\text{the matching length in the original}}$$ and $k > 0$. A copy with $k > 1$ is an enlargement, a copy with $k < 1$ is a reduction, and $k = 1$ gives a congruent copy. "Every length" means every one, not only the sides. Heights, diagonals, radii, the distance between two marked points, and the perimeter, which is a sum of lengths, all change by the same factor $k$. The same definition covers solids: two solids are **similar** when every length in one is $k$ times the matching length in the other. Two cylinders are similar exactly when their radii and their heights are in the same ratio; two boxes are similar exactly when all three edges are in the same ratio. ### [Sectors, segments and their areas](https://math.wiki/sector-area-formula/) **Summary:** Taking that same fraction of the full turn and applying it to the area of the whole circle. **Question:** A slice of pie takes some fraction of the full turn. How much of the pie's area does it take, and what is left when the slice is cut straight across instead of to the center? ## When you reach for it The area of a whole circle is $\pi r^2$. Most circular things you meet are not whole circles: a slice of pie, the patch of lawn a sprinkler reaches as it turns through part of a turn, the band a windscreen wiper clears, a shaded piece of a diagram. Each is a **piece** of a circle, and the question is how much of the circle's area that piece holds. Two kinds of piece come up again and again. A **sector** is the region bounded by two radii $OA$ and $OB$ and one of the arcs between $A$ and $B$. It is the shape of a slice of pie. The angle $\angle AOB$ at the center is the sector's **central angle**, written $\theta$. Two radii always cut a circle into two sectors: the **minor sector**, whose angle is less than $180^\circ$, and the **major sector**, whose angle is the rest of the turn, $360^\circ - \theta$. They sit on the minor and major arcs $AB$. A **segment** is the region bounded by a chord $AB$ and one of the two arcs between $A$ and $B$. It is what you get by cutting a pie straight across instead of to the center. Again there are two: the **minor segment** on the minor arc and the **major segment** on the major arc. Reach for this lesson whenever a region is bounded partly by an arc. Every such region is built from sectors, triangles and whole circles, added or taken away. ## The procedure ### The area of a sector A sector takes the same fraction of the circle's area as its central angle takes of the full turn. With $\theta$ measured in degrees, that fraction is $\frac{\theta}{360}$, so ### [Segment length, addition and congruence](https://math.wiki/congruent-segments/) **Summary:** Two segments are congruent when they have the same length. **Question:** A point cuts a segment in two, and somewhere else two segments sit apart, pointing different ways. How do the lengths of the pieces add up to the whole, and when are two separate segments the same size? ## Why this exists Lay two sticks end to end in a straight line and the total length is the sum of the two; that is how you measured length in the first place, units laid end to end. In a geometry proof, though, "it looks like they add" is not a reason. Every step has to point at something already agreed. The **segment addition postulate** is that agreed thing: it says exactly when two lengths add to a third, and it is the fact you cite whenever you split a segment, join two segments into one, or find a piece from the whole. The second job is comparing segments that do not touch. Look at the two segments in Figure 2 that carry one tick each. One runs gently uphill to the right. The other climbs steeply to the left, from a different starting point. They are not the *same segment*: their endpoints are four different points. But a ruler gives the same reading on each, and in almost every problem that is all that matters. A builder cutting two table legs, or a proof about the two equal sides of a triangle, needs to say "these two have the same length" while keeping clear that there are two of them. Geometry's word for that is **congruent**. Both ideas have twins for angles: angle addition and congruent angles. There a ray inside an angle splits a measure; here a point between two others splits a length. ## Definition ### Segments and their lengths A segment $\overline{AB}$ is the endpoints $A$ and $B$ and every point of the line between them: a figure. $AB$, with no bar, is its **length**, a positive number such as $7$ or $4.5$ cm. Some books write the length as $m\overline{AB}$, "the measure of segment $AB$"; it means the same as $AB$. Naming the ends the other way around changes nothing: $\overline{AB}$ and $\overline{BA}$ are one segment, and $AB = BA$. ### [Segment lengths in circles](https://math.wiki/chord-chord-product/) **Summary:** The products of the two pieces of each chord are equal. **Question:** Two chords cross inside a circle and cut each other into four pieces. Is there a rule tying the four lengths together, and does it survive when the crossing point moves outside the circle? ## Why this exists The chord theorems say where a chord sits: how far it is from the center, and that the perpendicular from the center cuts it in half. They say nothing about a chord that is cut somewhere else, by another chord crossing it at a point that is not its midpoint. Draw two chords that cross, and you get four pieces of four different lengths with no obvious relation among them. There is one, and it is about products, not sums. Multiply the two pieces of one chord, multiply the two pieces of the other, and the answers agree, however the chords are drawn. Even more holds: the same rule works when the point is outside the circle and the lines are secants, and when one line is a tangent. All of it comes from a single number attached to the point, its **power**, which depends only on how far the point is from the center. This turns a circle into a tool for finding lengths. Three pieces give the fourth; a tangent's length follows from one secant; and a circle drawn in coordinates hands over the product without finding a single intersection point. ## Definition A **chord** is a segment with both ends on the circle. A **secant** is a line that meets the circle at two points, and a **tangent** meets it at exactly one; all three are set out in chords, secants and tangents. In the three results below, $P$ is a point that is not on the circle, and $PA$ means the length of $\overline{PA}$, a distance, so it is never negative. ### [Segments, rays and length](https://math.wiki/segments-and-rays/) **Summary:** Pieces of a line, and adding lengths along one. **Question:** A line goes on forever both ways. What do you get when you cut a piece off it, and how do lengths behave along it? ## What this area is about A line never ends, so it has no length. Stop it at one point and you get a **ray**, $\overrightarrow{AB}$, which starts at $A$ and runs on forever through $B$. Stop it at two points and you get a **segment**, $\overline{AB}$, which does have a length, written $AB$ with no bar. This area is about that length: how to find it, when two segments match, and how lengths add along one line. ## How these fit together **Measuring.** Distance on a number line is a procedure: it finds a length from the coordinates $a$ and $b$ of the two ends, $AB = |b - a|$, even when they are negative. **Comparing and adding.** Segment length, addition and congruence holds a definition and a rule. Two segments with the same length are congruent, $\overline{AB} \cong \overline{CD}$. The segment addition postulate says that when $B$ is between $A$ and $C$, $AB + BC = AC$. **The middle.** Midpoints and segment bisectors: the midpoint splits a segment into two congruent halves, and any line, ray or segment through it is a segment bisector. The perpendicular bisector of a segment is the line through its midpoint at a right angle to it. ## A path through it Distance on a number line and segment length, addition and congruence can come in either order. Then midpoints and segment bisectors, then the perpendicular bisector, each built on the one before. Finish here and you can measure, compare and split lengths. That is where medians of a triangle, compass and straightedge constructions and distances on a coordinate grid start. ### [Shapes in more than one category, and quadrilaterals in none](https://math.wiki/shapes-in-more-than-one-category/) **Summary:** A square is also a rectangle and a rhombus, while some four-sided shapes belong to none of those groups. **Question:** A square has four equal sides and four square corners. So which group does it belong in, and where does a four-sided shape go when it fits none of the groups? ## Why this exists The lesson rhombuses-rectangles-and-squares gave three names, each with a test: - a **rhombus** passes the **ruler test**: all four sides are the same length; - a **rectangle** passes the **card test**: the corner of an index card fits all four corners; - a **square** passes both. That last line has a surprise in it. A square passes the ruler test, so it is a rhombus. It passes the card test, so it is a rectangle. One shape, three names, and every one of them is right. People are like this too: one person can be a sister, a soccer player and a drummer all at once. Each name says one true thing about her. The other surprise goes the opposite way. Plenty of four-sided shapes pass neither test. They are still quadrilaterals, but none of the three names fits them. Sorting shapes on a mat makes both surprises easy to see. This lesson builds that mat, reads it, and then hunts for the shapes that land outside everything. ## Definition A **category** is a group of shapes that all have some **attribute**: something the shape must have, checked with a tool or by counting. Size, color and which way the shape is turned are not attributes here, because none of them changes what the ruler and the card say. The categories in this lesson: | Category | Attribute every shape in it has | |---|---| | quadrilateral | four straight sides (and so four corners), closed up | | rhombus | a quadrilateral with four sides of the same length | | rectangle | a quadrilateral with four square corners | | square | a quadrilateral with four equal sides **and** four square corners | ### [Sides and angles in order](https://math.wiki/side-angle-inequality-in-a-triangle/) **Summary:** The longest side of a triangle lies opposite the largest angle and the shortest lies opposite the smallest. **Question:** The three angles of a triangle are all different sizes. Without measuring, which side is the longest, and why can the answer never come out differently? ## Why this exists Two facts about triangles are already in hand. Equal sides lie opposite equal angles, the base angles theorem, and equal angles lie opposite equal sides, its converse. Between them they settle every question of the form "are these two equal?" Most triangles have no equal sides at all, and for them those two theorems only say that the three angles are all different too. They do not say which angle is the big one. Given a triangle with angles $40^\circ$, $60^\circ$ and $80^\circ$, which side is longest? Given sides $7$, $9$ and $12$, where is the widest corner? The answer is as simple as it could be: sides and angles come in the same order. The longest side lies opposite the largest angle, the shortest side opposite the smallest, and the middle one opposite the middle one. This lesson states that exactly, in both directions, and proves it. Euclid builds both the triangle inequality and the hinge theorem on it. ## Definition The side **opposite** an angle of a triangle is the one side that does not touch that angle's vertex (parts of a triangle). In $\triangle ABC$ the side opposite $\angle A$ is $\overline{BC}$, and its length is written $a$. In the same way $b = CA$ is opposite $\angle B$ and $c = AB$ is opposite $\angle C$. In words: **within one triangle, the longer of two sides lies opposite the larger of the two angles, and the larger of two angles lies opposite the longer of the two sides.** ### [Sides, corners and sorting shapes](https://math.wiki/sort-shapes-by-attributes/) **Summary:** Grouping flat shapes by number of sides, number of corners, or whether the sides are equal. **Question:** What can a child count and check on a flat shape, and how do those counts let the child put shapes into groups? ## When you reach for it A flat shape has parts a child can count and check: its **sides** (the straight edges) and its **corners** (the points where two sides meet). A child can also check whether the outline is **closed**, with no gap, and whether all its sides are the **same length**. Each of these is a rule for sorting a pile of shapes into groups, and each can be checked with a finger, a sticker or a paper strip. Reach for this once a child can name circles, triangles, squares and rectangles. The names stay in the background this week: the work is counting and checking, so ask "how many sides?" more than "what is it called?" Each day takes about ten to fifteen minutes with paper cutouts on a table. ## Day 1: Sides and corners **You need:** paper cutouts of a few straight-sided shapes, like the ones below, and a sheet of small round stickers. Hold up one cutout. Run a finger slowly along one straight edge and say, "This is a side." Have the child trace each side and count out loud, one number for each side. Then point to a spot where two sides meet: "This is a corner. Is it pointy?" The child puts one sticker on each corner and counts the stickers. After two or three shapes, ask, "What do you notice about the sides and the corners?" On every one of these shapes the two counts match, because each corner sits between two sides as you go around. Then hand over a paper circle and ask the child to trace it. There is no straight side to count and nowhere to put a corner sticker. ### [Similar figures](https://math.wiki/similar-figures/) **Summary:** Matching angles equal, matching sides in one ratio. **Question:** One shape looks like a bigger, turned-round copy of another. What has to be true of their angles and sides for that to be exactly so? ## Why this exists Similarity transformations say what "the same shape" means: two figures are similar when rigid motions and a dilation carry one exactly onto the other. That is the right meaning, but two quadrilaterals printed with their sides and angles labeled do not come with a list of moves. To decide whether they are the same shape, and how much bigger one is, we need a test that works from the measurements themselves. For polygons there is one: two polygons are similar when their angles match and their sides all grow by the same factor. A missing side of a similar figure then comes from a proportion, with no moves at all. ## Definition Two figures are **similar** when one can be carried exactly onto the other by rigid motions (slides, turns and flips) and a dilation. We write $F \sim G$, read "$F$ is similar to $G$". For polygons this means the same as the following, which is the form used in practice. Two polygons are similar when their vertices can be matched up, one to one and in order around each polygon, so that - every pair of **matching angles** is equal, and - every pair of **matching sides** is in the same ratio. That common ratio, a side of the second figure divided by the matching side of the first, is the **scale factor** $k$ from the first figure to the second. It is always positive: $k > 1$ for an enlargement, $k < 1$ for a reduction, and $k = 1$ when the figures are congruent. ### [Similarity](https://math.wiki/similarity/) **Summary:** Same shape, different size. **Question:** A photo and its enlargement have the same shape at different sizes. What exactly has to match, and what does matching let you work out? ## What this area is about Two figures are similar when one is a scaled copy of the other: matching angles equal and every matching length multiplied by one scale factor $k > 0$. Fixed ratios let a length you cannot reach, like a tree's height, come from lengths you can. ## How these fit together **Definitions.** Similar figures gives the working test for polygons. Dilations make a scaled copy from a center, and similarity transformations define similarity as a dilation plus rigid motions. **Tests for triangles.** AA similarity needs two pairs of equal angles; SSS and SAS similarity work from side ratios, SAS with the included angle. **Results.** What a dilation does to a line: a line through the center stays put, and when $k \ne 1$ any other line goes to a parallel one. The triangle proportionality theorem says a line parallel to one side splits the other two sides in one ratio, and the triangle angle bisector theorem says an angle's bisector splits the opposite side in the ratio of the two sides at that angle. Scale factor and perimeter, area and volume: perimeters scale by $k$, areas by $k^2$, and volumes of similar solids by $k^3$. **Procedure.** Proportions in similar triangles finds a missing side. ## A path through it Similar figures first. Dilations come next, before similarity transformations and dilated lines. AA similarity also follows similar figures and precedes SSS and SAS, triangle proportionality and proportions. The angle bisector theorem follows triangle proportionality. Perimeter, area and volume have no fixed place. ### [Similarity transformations](https://math.wiki/similarity-transformations/) **Summary:** A dilation followed by rigid motions, which is what it means for two figures to be similar. **Question:** When are two figures the same shape, even if one is bigger, turned or flipped over, and how do you show it? ## Why this exists Congruence made "the same shape and the same size" exact: two figures are congruent when slides, turns and flips carry one exactly onto the other. Everyday life is full of a looser relation, the same shape at a different size: a photo and its enlargement, a map and the town it shows, a model car and the car. "The same shape" by eye is not a test, though. Are all rectangles the same shape? In ordinary speech they are, since they are all rectangles. Yet a $1$ by $1$ square and a $1$ by $10$ strip are both rectangles, and nobody would call them the same shape. A dilation changes size and keeps shape, but on its own it cannot turn a figure or flip it over: it sends every side to a parallel side. A copy that is both larger and turned is out of its reach. Putting dilations together with the rigid motions closes that gap. The result is a meaning for "the same shape" that works for every kind of figure, polygons, circles, curves and letters of the alphabet alike. It says what to do to show that two figures are similar, and it gives a way to show that they are not. This lesson takes two days. Part 1 is the definition and what a similarity transformation keeps and changes, used to decide whether two figures are similar. Part 2 carries out similarity transformations on the coordinate plane: following a sequence of moves, finding one that carries a figure onto another, undoing one, and seeing why the order of the moves matters. ### [Solids](https://math.wiki/solids/) **Summary:** Three-dimensional shapes, their surfaces and their insides. **Question:** A box, a can and a ball all take up room. How do you describe a solid, unfold it, slice it and measure what it covers and holds? ## What this area is about A solid is a shape in three dimensions. This area names solids and their parts, then measures them: a net for the outside, a slice for the inside, coordinates for position, and formulas for surface area and volume. ## How these fit together **Naming.** Three-dimensional shapes covers faces, edges and vertices, polyhedra, prisms and pyramids, cylinders, cones and spheres, and the five regular solids. **Position.** Coordinates in space pins a point with three numbers $(x, y, z)$ and measures the distance between two points. **Flattening.** Nets unfold a prism, pyramid or cylinder into one flat piece. Cross sections name the flat shape a plane cut leaves. Solids of revolution describe a round solid by the flat region spun about a line to make it. **Measuring.** Surface area totals the outside: the face areas, a curved side unrolled flat, or for a sphere a formula of its own. Volume stacks copies of a base, then finds a pyramid, cone or sphere as a fraction of a matching prism or cylinder. **Using it.** Modeling with solids treats real objects as solids to judge weight, material and design. ## A path through it Three-dimensional shapes first, except that its five regular solids wait until after coordinates in space, which they use. Then nets before surface area, and cross sections before volume. Solids of revolution uses the volume formulas and is cited by cross sections, so read it beside those two. Modeling with solids comes last: it needs surface area and volume. ### [Solids of revolution](https://math.wiki/solids-of-revolution/) **Summary:** The solid swept out when a flat region is spun about a line. **Question:** A vase, a bowl, a table leg and a ball are all round in the same way. Is there one flat shape that describes each of them completely? ## Why this exists A potter's wheel and a woodworker's lathe make round things in the same way. The material spins about a fixed line, and the tool, held at one place, cuts the same shape all the way around. The finished vase or table leg is described completely by one flat outline: the shape of a single slice taken through its middle. That is a large saving. A solid is hard to draw, hard to measure and hard to describe in words. A flat region and a line are easy. If every round solid comes from spinning a flat region about a line, then questions about the solid (what shape is it, what does a slice look like, how much does it hold) can be answered by looking at the flat region instead. The cylinder, the cone and the sphere all turn out to be made this way, and so do many solids that have no name of their own. ## Definition Start with a flat region $R$ and a straight line $\ell$ lying in the same plane, with $R$ entirely on one side of $\ell$. The region may touch $\ell$, but it may not cross it. Turn $R$ one full turn about $\ell$, the way a door turns on its hinges. Every point $P$ of $R$ travels around a circle. The circle lies in the plane through $P$ perpendicular to $\ell$, its center is the point of $\ell$ nearest to $P$, and its radius is the distance $d$ from $P$ to $\ell$. The **solid of revolution** swept out by $R$ about $\ell$ is the set of every point on every one of these circles. The line $\ell$ is the **axis of revolution**. A point of $R$ that lies on the axis has distance $0$ from it, so it does not move at all: its circle is a single point. ### [Special right triangles](https://math.wiki/45-45-90-triangles/) **Summary:** The isosceles right triangle, whose hypotenuse is a leg times the square root of two. **Question:** Cut a square along its diagonal, or an equilateral triangle straight down its middle. Is one side of either half enough to find the other two exactly? ## Why this exists The Pythagorean theorem finds the third side of a right triangle from the other two. Knowing one side and the angles is not enough for it. For most right triangles, one side and an acute angle do fix the other two sides, but finding them takes trigonometry and a calculator, and the answers come out as decimals. Two right triangles give more. Each has a shape so simple that the squaring and adding comes out the same way every time, so it can be done once, with a letter, and read off from then on. - The right triangle with two equal legs is half of a square. Its hypotenuse is a leg times $\sqrt{2}$. - The right triangle with angles of $30^\circ$ and $60^\circ$ is half of an equilateral triangle. Its hypotenuse is twice its shortest side, and its other leg is that side times $\sqrt{3}$. For these two, one measurement gives all three sides, exactly: a whole number, or a whole number times a root. Every square's diagonal and every equilateral triangle's height come with them. ## Definition The **special right triangles** are two right triangles, each named by its three angles: the 45-45-90 triangle and the 30-60-90 triangle. In each, the sides are in a fixed exact ratio, so any one side gives the other two without trigonometry. As in every right triangle, the two sides that form the right angle are the **legs**, and the side opposite the right angle is the **hypotenuse** (see opposite, adjacent and hypotenuse). ### [Special segments in a triangle](https://math.wiki/triangle-segments/) **Summary:** Medians, altitudes, angle bisectors and perpendicular bisectors, the four families of segments drawn in a triangle. **Question:** Of all the segments you could draw inside a triangle, which ones are worth a name, and what does each one know? ## What this area is about A few segments in a triangle recur everywhere: ones that halve a side, halve an angle, or meet a side at a right angle. Each carries a fact about lengths, distances or parallels. ## How these fit together **Definitions.** Medians and altitudes: a median runs from a vertex to the midpoint of the opposite side; an altitude runs from a vertex perpendicular to the line containing the opposite side, so in an obtuse triangle two altitudes fall outside it. **Results about equal distances.** Perpendicular bisectors of a triangle: in a plane, a point is equidistant from $A$ and $B$ if and only if it lies on the perpendicular bisector of $\overline{AB}$, so the three in a triangle meet at one point. Angle bisectors of a triangle: a point inside an angle is equidistant from its two sides if and only if it lies on the bisector, so a triangle's three angle bisectors meet at one point too. **A result about parallels.** Midsegments of triangles and trapezoids: in a triangle, the segment joining the midpoints of two sides is parallel to the third side and half as long; in a trapezoid, the segment joining the midpoints of the legs is parallel to the bases and its length is their average. ## A path through it Medians and altitudes come before midsegments. Perpendicular bisectors and angle bisectors fit anywhere. The points where each family of three meets are the subject of triangle centers. Also next: the triangle angle bisector theorem, geometric means in a right triangle and coordinate proofs. ### [Spheres](https://math.wiki/spheres/) **Summary:** A surface every point of which is the same distance from the center. **Question:** A circle is every point in a flat plane at one distance from a center. What shape do you get when the points may lie anywhere in space? ## Why this exists A circle answers a flat question: which points on a page sit at one distance from a center? Most round things are not flat. A soap bubble, an orange and, very nearly, the Earth are round in every direction at once. To reason about them (how large a box must be to hold one, how wide the face of a cut fruit is, whether a point is inside or outside) they need a description as exact as the circle's. It turns out to be the circle's own, with the words "in a plane" taken away. ## Definition A **sphere** is the set of all points in space that are the same distance from one fixed point. - The fixed point is the **center**, often called $O$. - The fixed distance is the **radius**, $r$. The word also names any segment from the center to a point on the sphere; every such segment has length $r$. - A **chord** is a segment with both ends on the sphere. A chord through the center is a **diameter**. It is two radii laid end to end, so its length is $$d = 2r$$ - A **great circle** is the circle where a plane through the center cuts the sphere. It has the sphere's own center and radius. A globe's equator is one. - A **hemisphere** is half a sphere: either of the two pieces a great circle cuts it into. A sphere is only the surface, the skin of points at distance $r$. The solid it encloses, skin and inside together, is a **ball**: a soap bubble is very nearly a sphere, an orange very nearly a ball. Everyday speech uses "sphere" for both, and this lesson says "ball" whenever it means the solid. The center itself is not on the sphere; it is at distance $0$, not $r$. ### [SSS and SAS similarity](https://math.wiki/sss-similarity/) **Summary:** Triangles are similar when all three pairs of sides are in the same ratio, or when two pairs are and the angles between them are equal. **Question:** You can measure the three sides of two triangles but none of their angles. Is that enough to tell whether they have the same shape? ## Why this exists AA similarity settles the shape of a triangle from its angles: find two pairs of equal angles and the triangles are similar. But angles are often not what you have. A frame cut from timber, a field paced out on foot, a triangle drawn on a coordinate grid: each of these hands you lengths. To use AA you would first have to work the angles out, which is a long way round. Lengths can do the job on their own, because a triangle is rigid. Pin three sticks together at their ends and the shape cannot flex, while four sticks joined in a loop can be pushed out of square. Three lengths fix a triangle completely, which is why SSS congruence works. This lesson is the version of that fact for shape rather than size: if three lengths fix a triangle, then three lengths in a fixed ratio fix its shape. There is a second test with lengths in it. Two sticks hinged at a fixed angle also leave a triangle no freedom, since the third side only has to close the gap. That gives SAS similarity, the other half of this lesson. ## Definition Two triangles are similar when one is a scaled copy of the other: matching angles equal, and matching sides all in one ratio. That ratio is the **scale factor** $k$. This lesson gives two tests that prove triangles similar using their sides. **SSS similarity** (side, side, side). If each side of one triangle is the same multiple $k$ of the matching side of another, ### [SSS, SAS, ASA, AAS](https://math.wiki/congruence-criteria/) **Summary:** The short lists of matching parts that force two triangles to be congruent. **Question:** Two triangles look like copies of each other. How few matching parts does it take to be sure they are? ## Why this exists Two triangles are congruent when all six pairs of corresponding parts match: three pairs of sides and three pairs of angles. Checking six facts is slow, in a proof it is six lines, and in most problems you are handed only two or three of them. A triangle, though, is rigid. Three rods pinned at their ends make a frame that cannot be pushed out of shape, which is why bridges and roofs are built from triangles. Once a few parts of a triangle are fixed, the rest have no freedom left. The four criteria in this lesson say exactly which three parts are enough, so that three matching pairs prove all six. ## Definition Two triangles are **congruent** when their vertices can be paired so that all six pairs of parts are congruent; $\triangle ABC \cong \triangle DEF$ pairs them in letter order, $A$ with $D$, $B$ with $E$, $C$ with $F$ (congruence statements and correspondence). A **congruence criterion** is a list of three pairs of corresponding parts which, once they match, force the other three pairs to match as well. Two words do most of the work. In a triangle, the **included angle** of two sides is the angle between them, at the vertex they share. The **included side** of two angles is the side that joins their vertices. In Figure 1, $\angle Q$ is included between $\overline{PQ}$ and $\overline{QR}$, and $\overline{PR}$ is included between $\angle P$ and $\angle R$. Each criterion is named by its parts, S for a side and A for an angle, read in order around the triangle. **SSS** is three sides. **SAS** is two sides and their included angle. **ASA** is two angles and their included side. **AAS** is two angles and a side that is **not** between them. Figure 2 marks the three parts of each; as in congruent segments, matching ticks show congruent sides and matching arcs show congruent angles. ### [Surface area](https://math.wiki/surface-area/) **Summary:** Add up the area of every face. **Question:** How much material covers a solid exactly once, with nothing left bare and nothing counted twice? ## What this area is about The surface area of a solid is the total area of its outside, measured in square units such as $\text{cm}^2$. For a solid with flat faces it is the sum of the face areas, and a net lays those faces flat so that each is counted once. Curved solids need one more idea each: a curved side that unrolls, or a formula of its own. ## How these fit together All three children are procedures, one per family of solids. **Wrapping straight sides.** Prisms and cylinders: for a right prism or right cylinder, the two congruent bases plus the side faces, which unroll into one rectangle whose length is the perimeter of the base and whose width is the height. **Narrowing to a point.** Pyramids and cones: the base plus the lateral faces, measured along the slant height $\ell$, not the vertical height. For a regular pyramid or a right cone, the Pythagorean theorem links $\ell$ to the height. **Nothing to unfold.** The sphere cannot be flattened without stretching, so it has its own result: $4\pi r^2$, four times the area of a circle of the same radius $r$. ## A path through it None of the three needs another, so any order works: prisms and cylinders, pyramids and cones and the sphere each lean on earlier ideas (nets, circle area, the Pythagorean theorem, the sphere itself), not on one another. After this, modeling with solids uses surface area to design containers and estimate material. ### [Surface area of a sphere](https://math.wiki/surface-area-of-a-sphere/) **Summary:** A sphere has four times the surface area of a circle with the same radius. **Question:** A ball cannot be unrolled flat the way a box can. So how do you measure how much material covers it? ## When you reach for it You want to know how much material covers something round: the leather on a ball, the paint on a dome, the foil around a sweet, the skin of a tank that loses heat through its surface. For a box or a prism, surface-area means unfolding the solid into flat faces and adding up their areas. A sphere has no faces to unfold. It cannot even be flattened: any piece of a sphere pressed onto a table has to stretch or tear, which is why every flat map of the Earth distorts something. So a sphere needs a formula of its own, and it is a short one: $$S = 4\pi r^2$$ where $r$ is the radius of the sphere. Read it as a comparison with something you already know. Cut a sphere through its center and the cut face is a circle of radius $r$, called a **great circle**, with area $\pi r^2$ by the area of a circle. The surface of the sphere is exactly **four** of those circles. Reach for it whenever the surface is a whole sphere, half of one, or a slice of one, and whenever a sphere sits inside a larger solid, such as a capsule or a silo with a rounded roof. ## The procedure ### Why it is four circles The formula comes from comparing the sphere with the cylinder that just fits around it. That cylinder has the same radius $r$ as the sphere, and its height is the sphere's diameter, $2r$. **The claim:** the sphere's surface has the same area as the **curved side** of this cylinder, leaving out its top and bottom. That side unrolls into a flat rectangle, as for any cylinder. Its length is the distance round the cylinder, the circumference $2\pi r$, and its height is $2r$, so its area is ### [Surface area of prisms and cylinders](https://math.wiki/surface-area-of-a-prism/) **Summary:** Adding the two bases to the rectangles that wrap around them. **Question:** How much material does it take to cover a box, a tent or a can exactly once, with nothing left bare and nothing doubled? ## When you reach for it Paper to wrap a present, wood for a planter, metal for a can, cardboard for a tube: each asks how much **area** covers the outside of a solid. That number is the solid's **surface area**, and this lesson finds it for prisms and cylinders, the two shapes behind most boxes, cans and buildings. There is one idea underneath. Unfold the solid flat, and its surface becomes flat shapes whose areas you already know. For a prism or a cylinder that pattern is always the same: two bases, and one rectangle that wraps around them. Real problems also ask **which** faces to cover. A room's walls are painted but not its floor; a tube is open at both ends. So the lesson separates the **lateral area**, the sides alone, from the **total surface area**, everything. This lesson is taught over two days. Part 1 is prisms: the net, boxes and triangular prisms, and lateral against total area. Part 2 is cylinders: the net, the formula, and problems with $\pi$ in them. ## The procedure ### Part 1: prisms **What is being measured.** The surface area of a solid is the total area of all its faces: the material that covers it once, with no gaps and no overlaps. It is an area, so it is measured in square units, such as $\text{cm}^2$ or $\text{m}^2$. For a solid with flat faces, find the area of each face and add. The real work is making sure every face is counted, and counted once. **The faces of a prism.** A prism has two **bases**: polygons that are congruent (the same shape and size) and lie in parallel planes, with each corner of one joined by an edge to the matching corner of the other. In a **right prism** those joining edges are perpendicular to the bases, and every prism in this lesson is a right prism. Each of its other faces is then a rectangle, one for each edge of the base: three for a triangular prism, four for a box, six for a hexagonal prism. The length of the joining edges, the distance between the bases, is the prism's **height** $h$. ### [Surface area of pyramids and cones](https://math.wiki/surface-area-of-a-pyramid/) **Summary:** Adding the base to the triangular faces, which use the slant height rather than the vertical height. **Question:** A pyramid or a cone narrows to a point. How much material does it take to cover one? ## When you reach for it Canvas for a tent, tiles for a pointed roof, card for a party hat, paint for a spire: each asks for the area of the outside of a solid that rises from a flat base to a single point. The surface area of a solid with flat faces is the sum of the areas of its faces, and a net lays those faces out flat so that none is missed. For a prism the faces around the side are rectangles whose height is the height of the prism, which is how prisms and cylinders work. A pyramid is different in one way that matters. Its side faces are triangles that lean inward to meet at the top, so the height of each triangle is measured **along the sloping face**, not straight up. That length is the **slant height**, and it is longer than the height of the solid. When a problem gives you a different length, the real work is finding the slant height, and that is a job for the Pythagorean theorem. A cone is the round version of the same idea: its curved side unrolls into a piece of a circle, and the slant height is the radius of that piece. Reach for this when a solid has a flat base and one point, the apex, joined to every point of the base's edge: a pyramid on any polygon, or a cone on a circle. ## The procedure ### The parts of a pyramid A **pyramid** has a polygon for its base and one point off the base, the **apex**, joined by a straight edge to every vertex of the base. Its faces are the base and one triangle for each edge of the base, the **lateral faces**. The total area of the lateral faces is the **lateral area**, and ### [Tangent properties](https://math.wiki/tangent-radius-perpendicular/) **Summary:** A tangent line is perpendicular to the radius drawn to the point where it touches. **Question:** A line touches a circle at a single point. What does that one touch force on the radius, on a second tangent from the same point, and on a polygon or a second circle that the line also touches? ## Why this exists A tangent is a line that touches a circle and does not cut it; chords, secants and tangents sets out the words. Touching at one point sounds like too weak a condition to say much. It turns out to fix a great deal. The radius to the point of contact always meets the tangent at a right angle, and once there is a right angle there is a right triangle, so the Pythagorean theorem and the angle sum of a triangle take over. That one right angle is behind every result in this lesson. Two tangents drawn from one outside point are equally long. A polygon drawn around a circle has its sides cut into matching pairs of pieces. Two circles can share up to four tangent lines, and their lengths follow from the two radii and the distance between the centers. A belt running round two wheels, a pipe lying in a V-shaped cradle and a coin pushed into the corner of a frame are all these facts at work. ## Definition A line is **tangent** to a circle when it meets the circle at exactly one point, the **point of tangency** or **point of contact**. From a point $P$ outside a circle, a **tangent segment** is the piece $\overline{PA}$ of a tangent line running from $P$ to its point of contact $A$. Its length $PA$ is the **tangent length** from $P$. Two tangent segments from the same point, $\overline{PA}$ and $\overline{PB}$, make the angle $\angle APB$, called a **circumscribed angle** of the circle. ### [Tessellations](https://math.wiki/tessellations/) **Summary:** A tiling that covers the plane with copies of a shape leaving no gaps and no overlaps. **Question:** Which shapes can cover a floor with copies of themselves, leaving no gaps and no overlaps, and what decides it? ## Why this exists A tiled floor, a brick wall, the cells of a honeycomb and a chessboard all cover a flat surface with copies of one shape, and each shape seems to fit against itself without any effort. Cut a few regular pentagons out of card and try the same thing, and it fails at once: three of them around a point leave a thin gap, and a fourth will not go in. So which shapes can do it? The question sounds like one about shapes, but it is mostly a question about angles. Wherever tiles meet at a point, their corners have to fill the space around that point exactly, and the angle sum of a polygon says what each tile's corners measure. That one requirement decides which regular polygons tile, shows that every triangle and every quadrilateral tiles, and rules out mixtures of shapes that look as if they ought to work. ## Definition A **tessellation**, or **tiling**, of the plane is a collection of shapes, called **tiles**, that cover the whole plane with **no gaps** and **no overlaps**: every point of the plane lies in at least one tile, and two tiles share at most points on their boundaries. In this lesson every tile is a polygon. A shape **tiles the plane** when there is a tessellation in which every tile is a copy of it. A copy may be slid, turned or flipped over, but not stretched, so all the tiles are congruent. A **vertex** of a tiling is a point that is a corner of at least one tile. A tiling is **edge-to-edge** when no corner of a tile lies partway along a side of another tile; then two tiles that touch along a side share the whole of it. The square grid on the left of Figure 1 is edge-to-edge. The brick wall on the right is not, because two bricks have their corners in the middle of the long side of the brick above them. ### [The apothem of a regular polygon](https://math.wiki/apothem/) **Summary:** The perpendicular distance from the center of a regular polygon out to one of its sides. **Question:** How far is the middle of a regular polygon from its sides, and how does that distance tie together the side, the radius and the number of sides? ## Why this exists A regular polygon has a center: one point the same distance from every corner and the same distance from every side, as regular polygons and their circles shows. The distance out to a corner is the polygon's radius. The distance out to a side is just as useful, and it gets its own name, the apothem. You meet it whenever a regular polygon is gripped, measured or stacked by its flat sides. A spanner for a hexagonal bolt head closes on two opposite flat sides, so the size stamped on the spanner is the distance across the flats, which is twice the apothem. A hexagonal pencil lies on a desk on one flat side, and its height off the desk is again twice the apothem. And when a regular polygon is cut into its triangles at the center, the apothem is the height of every one of those triangles, which is what makes the area of a regular polygon a short calculation. The apothem, the radius and the side are three lengths of one right triangle. Know the number of sides and any one of them, and the other two follow. This lesson is about that triangle. ## Definition Let a regular polygon have $n$ sides, each of length $s$, and center $O$. Its **radius** $R$ is the distance from $O$ to each vertex. The **apothem** of the polygon is the segment from the center $O$ to a side that meets the side at a right angle. Like the word radius, the word apothem names both the segment and its length, and the length is written $a$. A polygon with $n$ sides has $n$ apothems, one to each side, and they are all the same length. ### [The coordinate plane](https://math.wiki/coordinate-plane/) **Summary:** Two number lines at right angles; every point gets a pair of numbers. **Question:** One number says where a point sits on a line. How do you say where a point sits on a flat sheet? ## What this area is about Cross two number lines at right angles, meeting at the $0$ of each, and every point of the flat plane gets a pair of numbers $(x, y)$: how far across, then how far up. Once points have numbers, a picture can be described in numbers, and a list of number pairs can be drawn as a picture. ## How these fit together **The definition.** Ordered pairs and the first quadrant sets up the axes, the origin and the pair $(x, y)$, using only numbers that are positive or $0$, and shows why the order of the two numbers matters. **Extending it.** Plotting in all four quadrants lets either number be negative, so the pairs reach every side of the origin. **A result.** Reflections across the axes shows that changing the sign of one coordinate mirrors a point across an axis, and changing both turns it a half turn about the origin. **A use.** Two patterns and their graph pairs the matching terms of two number patterns and plots the pairs to see how the patterns relate. ## A path through it Start with ordered pairs. Two patterns and their graph needs only the first quadrant, so it can come any time after. Plotting in all four quadrants comes before reflections across the axes. Finish here and you can read and plot any point. The distance formula, graphing a line from a table, scatter plots and reflections all build on it. ### [The distance formula](https://math.wiki/distance-formula/) **Summary:** The Pythagorean theorem between two points. **Question:** Two points sit on a grid, but neither is straight across from the other nor straight above it. How far apart are they? ## Why this exists On a number line, distance is a subtraction. Points at $2$ and $9$ are $9 - 2 = 7$ apart, and taking the absolute value of the difference handles any order and any signs (distance-on-a-number-line). On the coordinate-plane the same subtraction still works whenever two points share a row or share a column, because then the segment between them lies along a grid line. Most pairs of points do neither. The points $(1, 3)$ and $(13, 8)$ are joined by a slanted segment that cuts through grid squares at awkward places, and no row of unit squares lies along it to be counted. A ruler on paper would give an estimate, but not an exact length, and not one that works for points too far apart to draw. The way through is a right triangle. Go from the first point straight across, then straight up, to the second. Both of those moves run along grid lines, so their lengths can be counted. Together with the slanted segment they make a right triangle, and the pythagorean-theorem turns the two legs into the third side. The distance formula is nothing more than that triangle written once in letters, so that it works for any two points without drawing anything. ## Definition The **distance** between two points $P$ and $Q$ is the length of the segment $\overline{PQ}$ that joins them, written $PQ$. A distance is never negative, and it is $0$ only when $P$ and $Q$ are the same point. Two cases need only subtraction, as on a number line (horizontal-and-vertical-distance): ### [The ellipse](https://math.wiki/foci-of-an-ellipse/) **Summary:** An ellipse is every point whose distances to two fixed points add to the same total. **Question:** Pin the two ends of a loop of string, pull it tight with a pencil, and drag the pencil all the way around. What curve comes out, and what equation does it obey? ## Why this exists Cutting a double cone with a tilted plane gives a closed, stretched curve, the ellipse. That description is a picture, not something you can compute with. A circle became computable once it was defined by distance: every point at distance $r$ from a center, which turned into the equation of a circle. The ellipse needs the same treatment. The idea that works uses **two** fixed points instead of one, and it turns the stretched oval into an equation from which you can read its center, its size and its shape. The lesson runs over two days. Day 1 is the definition, the equation of an ellipse centered at the origin, and its parts. Day 2 moves the center to $(h, k)$, graphs an ellipse from its equation, writes the equation from its features, measures flatness, and uses the result in a room built on an ellipse. ## Definition Fix two points $F_1$ and $F_2$, the **foci** (one is a **focus**), and a length $2a$ greater than the distance $F_1F_2$. An **ellipse** is the set of all points $P$ in the plane with $$PF_1 + PF_2 = 2a.$$ The parts that go with it: - The **center** is the midpoint of $F_1F_2$. Write $c$ for the distance from the center to each focus, so $F_1F_2 = 2c$ and the condition says $a > c \ge 0$. - The **major axis** is the chord through both foci. Its ends are the **vertices**, and its length is $2a$. - The **minor axis** is the chord through the center perpendicular to the major axis. Its ends are the **co-vertices**, and its length is $2b$, where $b^2 = a^2 - c^2$. ### [The five regular solids](https://math.wiki/platonic-solids/) **Summary:** Only five solids have identical regular faces meeting the same way at every vertex. **Question:** There is a regular polygon with any number of sides you like. How many solids can be built from copies of one regular polygon, with every corner exactly alike? ## Why this exists In a flat plane, regular polygons never run out. There is an equilateral triangle, a square, a regular pentagon, a regular hexagon, and one for every number of sides after that, each with its own interior angle. Go up one dimension and ask the same question of solids. Take copies of a single regular polygon and fit them together into a closed solid, so that every face is like every other face and every corner is like every other corner. A fair die is made this way: no face and no corner is special, so no side is favored. You might expect one such solid for each polygon, an endless list again. There are exactly five. Not five that people happened to find: five, and a proof that a sixth cannot exist. The proof needs only the angle of a regular polygon and one fact about corners, and it is one of the oldest complete classifications in mathematics. The last book of Euclid's *Elements* builds all five and ends by showing that no other can be built. They are often called the **Platonic solids**, after Plato, who wrote about them in his dialogue the *Timaeus*. ## Definition A polyhedron is **convex** when the segment joining any two of its points stays inside it: no dents, no holes, no faces folded inward. A **regular polyhedron**, or **Platonic solid**, is a convex polyhedron in which - every face is a regular polygon, and all the faces are congruent; and - the same number of faces meet at every vertex. ### [The hyperbola](https://math.wiki/foci-of-a-hyperbola/) **Summary:** A hyperbola is every point whose distances to two fixed points differ by the same amount. **Question:** The ellipse keeps the sum of two distances fixed. What curve keeps their difference fixed instead, and why does it run off along two straight lines? ## Why this exists Cutting a double cone with a plane steeper than its sides gives a curve in two pieces, the hyperbola. As with the ellipse, a slice is a picture, not something to compute with. The ellipse became an equation once it was described by distances to two fixed points: their **sum** stays the same. Keep the two points and fix the **difference** instead, and the result is the hyperbola. That description gives its equation, its two straight guide lines, and a way to locate a sound from the moment it arrives at two places. The lesson runs over two days. Day 1 is the definition, the equation centered at the origin, the parts, and the asymptotes. Day 2 moves the center to $(h, k)$, graphs with the central box, writes equations from features, completes the square, and locates a sound from a time difference. ## Definition Fix two points $F_1$ and $F_2$, the **foci**, and a positive length $2a$ smaller than the distance $F_1F_2$. A **hyperbola** is the set of all points $P$ in the plane with $$|PF_1 - PF_2| = 2a.$$ The parts that go with it: - The **center** is the midpoint of $F_1F_2$. Write $c$ for the distance from the center to each focus, so $F_1F_2 = 2c$ and the condition says $0 < a < c$. - The curve has two separate pieces, the **branches**: the points with $PF_1 - PF_2 = 2a$ (nearer $F_2$) and those with $PF_2 - PF_1 = 2a$ (nearer $F_1$). - The line through the foci meets the hyperbola at two points, the **vertices**. The segment joining them is the **transverse axis**, of length $2a$. - The **conjugate axis** is the segment through the center, perpendicular to the transverse axis, of length $2b$, where $b^2 = c^2 - a^2$. ### [The inscribed angle theorem](https://math.wiki/inscribed-angles-on-the-same-arc/) **Summary:** Any two inscribed angles opening onto the same arc are congruent. **Question:** Stand anywhere on a circle and look at two fixed points on it. Why is the angle you see the same wherever you stand? ## Why this exists A central angle has its vertex at the center of the circle, and its measure is the measure of its arc by definition. But most angles that turn up in a circle do not sit at the center. They sit on the circle itself: the corners of a triangle whose three vertices all lie on a circle, or the angle where two chords meet at the rim. Such an angle cuts off an arc just as a central angle does, and until now nothing has said how big it is. The answer is as simple as it could be: the angle is exactly half its arc. Two facts fall straight out of that and are used constantly. Every angle whose vertex is on one arc and whose sides pass through the same two points has the same size, wherever on the arc the vertex sits. And every angle with its vertex on the circle and its sides through the two ends of a diameter is a right angle. This lesson takes two days. Part 1 states the theorem, proves it, and uses it to find angles and arcs. Part 2 is the special case of a diameter, the angle in a semicircle, together with its converse, which finds circles from right angles. ## Definition An **inscribed angle** is an angle whose vertex lies on a circle and whose two sides each cross the circle again, so that each side contains a chord from the vertex. If $A$, $B$ and $C$ are three points on a circle, then $\angle ABC$ is an inscribed angle with vertex $B$, and its sides contain the chords $\overline{BA}$ and $\overline{BC}$. ### [The midpoint formula](https://math.wiki/endpoint-from-a-midpoint/) **Summary:** Working backwards from a midpoint and one endpoint to the other endpoint. **Question:** Two points are given by their coordinates. What are the coordinates of the point halfway between them, and how do you run that backwards, or write down every point that is equally far from both? ## When you reach for it The midpoint of a segment is the point on it that is equally far from both ends. On a number line it sits at the average of the two ends. On a coordinate grid the two ends are pairs of numbers, and you want the midpoint as a pair of numbers too. Reach for the midpoint formula whenever a question names two points by their coordinates and asks for the point halfway between them: the center of a circle from the two ends of a diameter, the place a median of a triangle lands on the opposite side, the crossing point of the diagonals of a parallelogram. Reach for it **backwards** when you know the midpoint and one end and need the other end. That turns up as often as the forward question: the far end of a diameter, the fourth corner of a parallelogram, the mirror image of a point. And reach for it as the first step whenever you need the **equation** of the perpendicular bisector of a segment, the line through the midpoint at right angles to the segment. That line is exactly the set of points equally far from the two ends, which is why it is the tool for finding the center of a circle through given points. ## The procedure Every step below is shown on one running example: $A = (3, 2)$ and $B = (9, 6)$. ### The midpoint formula Walk from $A$ to $B$. Along the way you move $9 - 3 = 6$ units right and $6 - 2 = 4$ units up. The midpoint should be halfway along the walk, so make **half of each move**: $3$ right and $2$ up. That lands on ### [The parabola as a locus](https://math.wiki/parabola-focus-and-directrix/) **Summary:** A parabola is every point that is the same distance from a fixed point as from a fixed line. **Question:** A satellite dish catches a faint signal over its whole face and brings all of it to one small receiver. What shape does that, and where exactly does the receiver go? ## Why this exists Vertex form describes a parabola as the graph of $y = a(x - h)^2 + k$: a vertex, a stretch, and a curve that opens up or down. That says what the curve looks like, but not what makes it special, and it cannot describe a parabola that opens sideways. Focus and directrix gave the deeper description: the conic with eccentricity $e = 1$ is every point as far from a fixed point as from a fixed line. This lesson works out that case in full: an equation for each of the four directions, the focus read from the equation, and why dishes, solar cookers and flashlight reflectors are all cut in this one shape. ## Definition Fix a point $F$, the **focus**, and a line $L$ that does not pass through $F$, the **directrix**. The **parabola** with this focus and directrix is the set of all points $P$ in the plane with $$PF = PL,$$ where $PF$ is the distance from $P$ to $F$ (the distance formula) and $PL$ is the distance from $P$ to the line $L$, measured along the perpendicular. The parts of a parabola: - The **axis** is the line through $F$ perpendicular to $L$. The parabola is symmetric about it. - The **vertex** is the point of the parabola on the axis. It is halfway between $F$ and $L$. - The number $p$ is the distance from the vertex to the focus, given a sign: positive when the focus lies in the positive direction ($+y$ or $+x$) from the vertex, negative otherwise. The directrix is the same distance $|p|$ on the other side of the vertex. - The **focal width** (or latus rectum) is the chord through the focus parallel to the directrix. Its length is $|4p|$. ### [The parallel postulate](https://math.wiki/the-parallel-postulate/) **Summary:** Through a point not on a line there is exactly one parallel line, the assumption Euclidean geometry rests on. **Question:** A line and a point off it. How many lines through the point never meet the line, and why could nobody prove the answer? ## Why this exists Every proof has to start somewhere. Euclid, around 300 BC, opened his *Elements* with five postulates: statements accepted without proof, from which everything else is built. Four are short and obvious: you can join two points, extend a line, draw a circle, and all right angles are equal. The fifth is long, and it is about something nobody can check by drawing: what two lines do far away, off the edge of every page. Yet without it you cannot show that a triangle's angles add to $180^\circ$, that alternate interior angles at parallel lines are equal, or even that two lines parallel to a third are parallel to each other. For two thousand years mathematicians tried to prove it from the other four, and failed. The failures were the discovery: it cannot be proved, because there are consistent geometries where it is false. It is a choice of plane, and choosing it is what makes school geometry **Euclidean geometry**. ## Definition Two lines in a plane are **parallel** when they never meet, however far they are extended. We write $m \parallel \ell$, and $m \nparallel \ell$ when they do meet. **The parallel postulate.** Given a line $\ell$ and a point $P$ not on $\ell$, there is **exactly one** line through $P$ parallel to $\ell$. This is the form most modern books use, often called Playfair's axiom after John Playfair, who used it in his 1795 edition of Euclid. Euclid's own wording talks about a transversal, a line crossing two others, and the two **interior angles** it makes on one side, the ones between the two lines: ### [The Pythagorean theorem](https://math.wiki/pythagorean-theorem/) **Summary:** In a right triangle, the squares on the legs add to the square on the hypotenuse. **Question:** You know two sides of a right triangle. Is the third side already decided, and if so, how long is it? ## Why this exists Tell someone that a triangle has sides of $6$ and $8$ and ask for the third side, and they cannot answer. The two sides could meet at any angle, and the third side could be anything from just over $2$ to just under $14$. Now add one fact: the two given sides meet at a **right angle**. Suddenly there is no freedom left. The triangle is pinned down, and the third side is exactly $10$. The Pythagorean theorem is the rule that says how to get that number from the other two, for every right triangle there is. That turns a length you cannot reach into one you can calculate: the diagonal of a screen, the slant of a roof, the straight-line distance across a field you would rather not walk. Any time a right angle hides inside a figure, two lengths you know give you the third. Finding that hidden right angle is half the work, and it has its own part of the results below, "Using the Pythagorean theorem". ## Definition A **right triangle** is a triangle with one right angle, a $90^\circ$ angle, usually marked with a small square in its corner. Its sides have their own names. - The two sides that meet at the right angle are the **legs**. - The side opposite the right angle is the **hypotenuse**. It never touches the square corner. It is the custom to call the legs $a$ and $b$ and the hypotenuse $c$, as in Figure 1. Which leg is $a$ and which is $b$ does not matter. Which side is $c$ matters a great deal. ### [The Pythagorean theorem in three dimensions](https://math.wiki/pythagorean-theorem-in-three-dimensions/) **Summary:** Use the Pythagorean theorem twice to find the space diagonal of a box and other lengths in solid figures. **Question:** A rod has to go into a box corner to corner, rising as it crosses. How long can it be? ## When you reach for it The Pythagorean theorem is about a flat triangle with a right angle. In using the theorem the right triangle always sat in one flat picture: a ladder against a wall, a path across a field. Many lengths in the world do not lie flat. A rod laid corner to corner in a box rises as it crosses the box. The point of a pyramid sits above the middle of its base. No face of the solid contains these lengths, so no right triangle with them as a side is drawn for you. The move is to **find a right triangle that stands up inside the solid**. Usually one of its legs is not given either, so you first find that leg from a second right triangle lying flat in a face. The theorem is used twice: once across a face, then once more through the inside. Reach for it when: - the length you want joins two corners of a box that are not on the same face; - something long has to fit inside a box, a tank or a crate; - a pyramid gives you one of its height, its slant height or its sloping edge, and you need another. ## The procedure ### Naming the parts of a box A box, called a **rectangular prism** in prisms and pyramids, has a length $l$, a width $w$ and a height $h$. Every face is a rectangle. A **face diagonal** joins opposite corners of one face. A **space diagonal** joins two corners that do not share a face, so it runs through the inside of the box. A box has four space diagonals, one from each bottom corner to the top corner furthest from it, and they all have the same length. ### [The quadrilateral family tree](https://math.wiki/quadrilateral-hierarchy/) **Summary:** How square, rectangle, rhombus, parallelogram, trapezoid and kite sit inside one another. **Question:** A square has four right angles, so is it a rectangle? It has four equal sides, so is it a rhombus too? How do all the named four-sided shapes fit together, and what does a shape inherit from each name it carries? ## Why this exists Four-sided shapes come with a crowd of names: trapezoid, parallelogram, rectangle, rhombus, square, kite. Learned one at a time, they look like separate boxes, and a question such as "is a square a rectangle?" turns into an argument about what the shape looks like. The definitions settle it. Each name is a short list of demands, and many of the lists contain one another. A square meets every demand a rectangle makes, so it is a rectangle, in the same way that every right triangle is a triangle. Put every name in one picture, with each shape placed under all the names whose demands it meets, and the arguments stop. The picture also saves work. A fact proved once for every parallelogram is true for every rectangle, rhombus and square without being proved again, because each of them is a parallelogram. Knowing where a shape sits tells you, at a glance, every fact it is owed. ## Definition The names, each one built from an earlier one by adding a demand. They are the definitions used in parallelogram-properties, rectangle-properties and trapezoid-definition, collected in one place. A quadrilateral is a polygon with four sides; for quadrilateral $ABCD$ the sides are $\overline{AB}$, $\overline{BC}$, $\overline{CD}$ and $\overline{DA}$, and two sides that share a vertex are **neighbors**. - A **trapezoid** is a quadrilateral with **at least one** pair of parallel sides. - An **isosceles trapezoid** is a trapezoid whose legs (the two sides that are not the chosen parallel pair) are congruent and **not** parallel. - A **parallelogram** is a quadrilateral with both pairs of opposite sides parallel. - A **rectangle** is a parallelogram with four right angles. - A **rhombus** is a parallelogram with four congruent sides. - A **square** is a parallelogram that is both a rectangle and a rhombus. - A **kite** is a quadrilateral with two pairs of congruent sides, where the two sides of each pair are neighbors, for example $AB = AD$ and $CB = CD$. ### [Three-dimensional shapes](https://math.wiki/three-dimensional-shapes/) **Summary:** Prisms, pyramids, cylinders, cones, spheres: faces, edges, vertices. **Question:** A box, a can, a ball and a pyramid all take up room. What tells them apart, and what words describe a solid to someone who cannot see it? ## What this area is about A solid takes up space: it can be picked up, turned over and seen from every side. Solids split into two kinds, those made only of flat polygons and those with a curved surface, and each kind has a few families worth naming part by part. ## How these fit together **First words.** Naming and describing solid shapes tells a cube, sphere, cone and cylinder apart, and faces, edges and vertices names the parts: flat surfaces, the lines where two faces meet, the corners where edges meet. **Flat-sided solids.** Polyhedra are solids whose faces are all polygons; for every polyhedron without a hole through it, $F + V - E = 2$. Prisms and pyramids are the two families met most: two congruent bases in parallel planes, or one base and an apex. The five regular solids answer a sharper question: a convex polyhedron with congruent regular faces and the same number meeting at every vertex comes in exactly five kinds. **Curved solids.** Cylinders and cones are built on circles, and a sphere is every point in space at one distance $r$ from a center. ## A path through it Naming solid shapes comes first. Faces, edges and vertices and polyhedra both follow it, and prisms and pyramids and the five regular solids both come after polyhedra. Cylinders, cones and spheres need only what a circle is, so they fit anywhere. After this come nets, cross sections, volume of prisms and cylinders and surface area of a sphere. ### [Transformational proof](https://math.wiki/transformational-proof/) **Summary:** Proving a claim by exhibiting the rigid motion or dilation that carries one figure onto another. **Question:** Two figures look like copies of each other. How do you prove it, rather than just say it? ## When you reach for it Congruence has a definition in terms of motion: two figures are congruent exactly when some sequence of rigid motions (translations, rotations and reflections) carries one onto the other. That is congruence as a sequence of rigid motions. Similarity has the same kind of definition with a dilation allowed as well, which is similarity by transformations. So the most direct way to prove two figures congruent is to **produce the motion**: say exactly which motion it is, and show that it carries every point of one figure to the matching point of the other. That is a transformational proof. It is usually written as a paragraph, but like any two-column or paragraph proof, every sentence carries its reason. Reach for it in three situations. - **The figures sit on a coordinate grid.** Each motion has a rule, so where each point lands is a calculation, and the proof is short and certain. - **A figure has a symmetry.** An isosceles triangle, a kite or a parallelogram can be carried onto itself by a reflection or a half turn that swaps two of its parts. Whatever a motion swaps is congruent, with no pair of triangles and no congruence criterion needed. - **You are proving a criterion itself.** SAS, ASA and SSS are theorems only because, for any two triangles that meet them, a motion can be built. Building it is a transformational proof. ## The procedure A transformational proof has three parts: say which motion, show where each point lands and why, and read off the conclusion. ### [Transformations](https://math.wiki/transformations/) **Summary:** Moving figures: slide, flip, turn, and what stays the same. **Question:** A figure is slid, flipped or turned. Exactly where does each point go, and what about the figure is still the same afterwards? ## What this area is about A transformation sends every point $P$ of the plane to exactly one image point $P'$. Slides, flips and turns are the rigid motions: they move a figure without changing any length or angle. That turns "the same shape and the same size" into something you can check, and it explains symmetry. ## How these fit together **The three motions.** Translations, reflections and rotations say where each point goes. Rotating about a point other than the origin is the procedure for a turn whose center is not $(0, 0)$. **Combining them.** Composing transformations does one motion after another, and order can matter. Two reflections in a row give a translation when the mirrors are parallel and a rotation when they cross. **What they keep.** What rigid motions preserve lists lengths, angle measures, betweenness and parallel lines. Rigid motions defined precisely proves those facts. Rigid motions and congruence defines congruent figures by them. **Symmetry.** Line symmetry and rotational and point symmetry are a figure mapped onto itself by a reflection, or by a turn strictly between $0^\circ$ and $360^\circ$. ## A path through it Translations, then reflections, then rotations. After rotations come rotating about another point, composing, and what rigid motions preserve. Composing comes before congruence and two reflections; what rigid motions preserve comes before the precise definitions. The two symmetry lessons have no fixed place. ### [Translations](https://math.wiki/translation-with-a-vector/) **Summary:** Describing a slide by how far the figure moves across and how far it moves up or down. **Question:** You slide a shape across the page without turning it. How do you say exactly where every point of it ends up? ## Why this exists Push a paper triangle across a desk without letting it turn. Every corner moves, every edge moves, and yet nothing about the triangle has changed except where it is. Saying "it moved a bit to the right" is not enough to draw the new position. Two people following that sentence would draw two different triangles. What is needed is a description that settles where **every** point goes, and words for talking about the shape before and after the move. It turns out two numbers are enough: how far across and how far up. That pair of numbers is a translation. Translations are the simplest of the motions that move a figure without changing its size or shape. The others are reflections and rotations, and together they are how geometry says two figures are congruent. A translation is also what makes a wallpaper pattern repeat, and what moves a graph to a new place on the coordinate plane. This lesson runs over two parts. Part 1 sets up the language (preimage, image, mapping) and the coordinate rule for a translation, and uses it in both directions. Part 2 shows what a translation keeps, how to tell whether a copy really is a translation of a figure, how to undo and combine translations, and how a translation moves a whole line. ## Definition **Transformation, preimage and image.** A **transformation** of the plane is a rule that sends every point $P$ of the plane to exactly one point $P'$. The point you start from is the **preimage**, and the point it is sent to is its **image**. We say the transformation **maps** $P$ to $P'$ and write ### [Trapezoids and kites](https://math.wiki/trapezoid-definition/) **Summary:** A quadrilateral with at least one pair of parallel sides, which are called its bases. **Question:** A four-sided shape with only one pair of parallel sides, or with two pairs of equal sides sitting side by side, looks lopsided. What do those few conditions force to be true about its angles and its diagonals? ## Why this exists A parallelogram is generous: two pairs of parallel sides, and in return its opposite sides and opposite angles all match. Plenty of four-sided shapes are not that tidy. A roof seen from the side, a slice of a triangle, a table top that narrows toward one end: each has one pair of parallel sides and no more. A kite flown on a string has no parallel sides at all, only two pairs of equal edges meeting at the top and at the bottom. It is fair to ask whether shapes with so little given have any properties worth knowing. They do. One pair of parallel sides is enough to tie the angles together in pairs. Add equal legs and the shape gains a mirror line, equal angles at each base and equal diagonals. Two neighboring pairs of equal sides give a kite, and its diagonals always cross at a right angle. This lesson names these shapes exactly and shows what each name guarantees. ## Definition A **trapezoid** is a quadrilateral with **at least one** pair of parallel sides. The parallel sides are its **bases**, and the other two sides are its **legs**. The two angles at the ends of one base are a pair of **base angles**, so a trapezoid has two pairs of base angles, one pair for each base. The **height** is the distance between the bases, measured along a segment perpendicular to both. In Figure 1, $\overline{AB} \parallel \overline{DC}$. The bases are $\overline{AB}$ and $\overline{DC}$, the legs are $\overline{AD}$ and $\overline{BC}$, and the base angles are $\angle A$ and $\angle B$ at one base, $\angle D$ and $\angle C$ at the other. The matching arrow marks on the two bases say they are parallel; a figure must mark it, since a drawing that only looks parallel proves nothing. ### [Triangle angle bisector theorem](https://math.wiki/triangle-angle-bisector-proportion/) **Summary:** An angle bisector splits the opposite side in the same ratio as the two sides that form the angle. **Question:** A segment cuts one angle of a triangle exactly in half and runs on to the opposite side. Where along that side does it land? ## Why this exists The bisector of an angle is defined by angles: it cuts the angle into two equal parts. Angle bisectors found what that means for the points on it, which are the points equally far from the two sides. Neither fact says where the bisector of an angle of a triangle meets the side across from that angle. Symmetry settles one case. When the two sides at the angle are equal, the triangle is its own mirror image across the bisector, and the bisector lands at the midpoint of the opposite side. When the two sides differ, the bisector still lands somewhere, and this theorem says exactly where: it splits the opposite side in the same ratio as the two sides that form the angle. A fact about angles becomes a fact about lengths, which can be computed. It places the meeting point of the three bisectors, the incenter, along each bisector, and it is the reason the bisector and the midpoint of the opposite side part company in every triangle that is not isosceles. ## Definition Let $\triangle ABC$ be a triangle. The **bisector from $A$** is the segment that starts at $A$, cuts $\angle BAC$ into two congruent angles, and ends where it meets $\overline{BC}$, at a point $D$. In Figure 1 the two arcs mark the two equal halves. **Triangle angle bisector theorem.** If $\overline{AD}$ bisects $\angle BAC$ and $D$ lies on $\overline{BC}$, then $$\frac{BD}{DC} = \frac{AB}{AC}$$ Read it as: the piece of $\overline{BC}$ next to $B$, over the piece next to $C$, equals the side through $B$, over the side through $C$. Each piece is paired with the side it touches: $\overline{BD}$ and $\overline{AB}$ both end at $B$, and $\overline{DC}$ and $\overline{AC}$ both end at $C$. The third side, $\overline{BC}$, is the side being cut, and it does not appear on the right. ### [Triangle angle sum](https://math.wiki/triangle-angle-sum/) **Summary:** The three angles add to a straight line, and why. **Question:** Triangles come in every shape, long and thin or short and wide. Is there anything about their angles, inside the corners or just outside them, that every one of them shares? ## Why this exists A triangle has three sides and three angles, and it can be almost any shape. A tall thin triangle has one tiny angle at the top and two steep ones at the bottom. A flat wide triangle has one huge angle and two small ones. It looks as if the three angles could be anything at all. They cannot. Whatever the shape and whatever the size, the three angles of a triangle always add to the same total, $180^\circ$, the size of a straight angle. So two angles of a triangle always tell you the third, without a protractor. That one fact is behind a large share of the angle work in geometry, from isosceles triangles to the angles of any polygon. Many of the angles you meet in a figure are not inside the triangle at all. Whenever a triangle sits on a longer line, one of its sides runs on past a corner, and a new angle opens up outside. That outside angle looks like new information. It is not. It is fixed by the two corners at the far end of the triangle, by a rule you can use at a glance: add them. The rule skips a step in every problem, and, more usefully, it lets an angle belonging to one triangle be read straight off another triangle that shares a side with it. That is how long angle chases become short ones. ## Definition ### Interior angles and the angle sum The **angles of a triangle** are its three **interior angles**: the angle at each corner, between the two sides that meet there, on the inside. In $\triangle ABC$ they are $\angle A$, $\angle B$ and $\angle C$. ### [Triangle centers](https://math.wiki/triangle-centers/) **Summary:** Each family of special segments meets at one point, giving the triangle four classic centers. **Question:** Three lines drawn in a triangle almost never pass through one point. Why do these four families always do, and what is special about each meeting point? ## What this area is about Every triangle has four families of special lines: perpendicular bisectors, angle bisectors, medians and altitude lines. In each family the three lines are concurrent, meeting at one point, and that is a theorem, not a lucky drawing. Each meeting point is a center with a job of its own. ## How these fit together All four are concepts, each a point of concurrency plus what that point does. **Centers from equal distances.** The circumcenter $O$, where the perpendicular bisectors meet, is equally far from all three vertices and centers the circle through them; it is inside an acute triangle, at the midpoint of the hypotenuse of a right one, outside an obtuse one. The incenter $I$, where the angle bisectors meet, is equally far from all three sides and centers the circle inside tangent to them. **Centers from segments.** The centroid $G$, where the medians meet, lies two thirds of the way along each median from its vertex and is the balance point. The orthocenter and the Euler line: the altitude lines meet at $H$, outside the triangle when it is obtuse, and in every triangle that is not equilateral, $O$, $G$ and $H$ are three different points on one line. ## A path through it The circumcenter first. The incenter and the centroid both follow it, and the orthocenter comes after the centroid. After this, inscribed polygons and cyclic quadrilaterals build on the circumcircle, and tangent properties on the incircle. ### [Triangle inequality](https://math.wiki/triangle-inequality/) **Summary:** Any two sides together are longer than the third. **Question:** You have three sticks. Will they close into a triangle, and if two of them are fixed, how long can the third be? ## Why this exists Take three sticks of length $3$, $4$ and $8$. Lay the long one down, and start the $3$ stick at one end and the $4$ stick at the other, both pointing inward. Together they reach only $3 + 4 = 7$, one short of $8$. However you turn them, their free ends cannot meet, so the three sticks cannot close into a triangle. Swap the two short sticks for a $5$ and a $6$. Together they reach $5 + 6 = 11$, more than $8$, and sticks of $5$, $6$ and $8$ close up at once. Whether three lengths make a triangle is not a matter of luck or of how carefully you draw. It is decided by the three numbers alone, and the decision takes one addition. That is worth having. A builder cutting three beams, a designer choosing the sides of a frame, and a surveyor checking three measured distances all ask the same question: *can these three lengths be the sides of a triangle?* And the question has a partner. If two sides are already fixed, how long can the third one be? Both questions are answered by one rule, the triangle inequality. ## Definition Write $AB$ for the length of the segment $\overline{AB}$. In a triangle $ABC$ name the sides by the corner they face, $a = BC$, $b = CA$ and $c = AB$. The **triangle inequality** says that in every triangle, any two sides together are longer than the third: $$a < b + c \qquad b < a + c \qquad c < a + b$$ That is three statements, one for each side. "Longer than" means strictly longer: the sum is never equal to the third side in a real triangle. ### [Triangle proportionality theorem](https://math.wiki/triangle-proportionality/) **Summary:** A line parallel to one side splits the other two in the same ratio. **Question:** A line crosses a triangle parallel to one of its sides. How does it share out the other two sides? ## Why this exists Similar triangles compare a whole side with a whole side. Often the lengths you can actually get at are not whole sides. A fence runs across a triangular field parallel to one edge, and what you can pace out is the stretch from the corner to the fence and the stretch from the fence to the far edge. The lines of ruled paper cross a slanting pencil line, and what you see is the run of pieces between them. This theorem is about the pieces. A line parallel to one side of a triangle cuts the other two sides in the same ratio, and the converse says that cutting them in the same ratio is exactly what makes a line parallel. Together they turn "parallel" into a statement about lengths, and a statement about lengths back into "parallel". Take the triangle away and the same fact says that a family of parallel lines cuts every line across it in the same ratios, which is how a segment can be split into equal parts without measuring it. ## Definition Let $\triangle ABC$ be a triangle, and let a line meet side $\overline{AB}$ at $D$ and side $\overline{AC}$ at $E$, each strictly between the vertices. The line **divides the two sides proportionally** when the piece at $A$ compares with the piece at the base in the same ratio on both sides: $$\frac{AD}{DB} = \frac{AE}{EC}$$ **Triangle proportionality theorem.** If $\overline{DE} \parallel \overline{BC}$, then $\overline{DE}$ divides $\overline{AB}$ and $\overline{AC}$ proportionally. ### [Triangles](https://math.wiki/triangles/) **Summary:** The simplest polygon, and most of geometry's theorems. **Question:** A triangle is the simplest polygon. Why do so many of geometry's theorems turn out to be about it? ## What this area is about Three lengths fix a triangle, and every polygon can be cut into triangles. So most of plane geometry comes back to them: angles, sides, when two match, and the right triangle. ## How these fit together **Parts.** Parts of a triangle names vertices, sides and angles. Special segments such as medians meet at the triangle centers. **Angles and sides.** The angle sum is $180^\circ$. By the triangle inequality, any two sides together are longer than the third. Drawing triangles from conditions asks whether given sides and angles fit one triangle, many or none. Sides and angles in order puts the longest side opposite the largest angle; the hinge theorem compares two triangles. **Congruence.** Congruent triangles and SSS, SAS, ASA, AAS prove when two triangles match. Isosceles triangles turn equal sides into equal angles, and back. **Right triangles.** The Pythagorean theorem: $a^2 + b^2 = c^2$ for legs $a$, $b$ and hypotenuse $c$. Using it means spotting the right triangle in a real figure; in a solid it is used twice. Its converse tests for a right angle; triples fit it in whole numbers. Special right triangles have fixed side ratios. Geometric means come from the altitude to the hypotenuse. ## A path through it Parts, angle sum, triangle inequality, drawing from conditions. Then the Pythagorean theorem, its uses and solids, converse, triples. Then congruence, isosceles triangles, sides and angles, hinge, special right triangles. Segments and centers last; geometric means wait for similarity. ### [Triangles, quadrilaterals, pentagons and hexagons from their sides and angles](https://math.wiki/draw-shapes-with-given-attributes/) **Summary:** Shapes are named and drawn from what they must have, such as five sides or three corners, whatever their size. **Question:** If all you are told is how many sides a shape has, or that one of its corners is square, can you name it and draw it, and is there only one way to draw it? ## When you reach for it A child who has met defining attributes knows that a triangle is a closed shape with $3$ straight sides, and that its size, color and the way it is turned do not matter. This lesson takes that habit to four families of shapes and runs it in both directions. **Naming** starts from a shape and asks what it must be called. Count the straight sides, check that the shape is closed, and the count gives the name: $3$ sides a triangle, $4$ a quadrilateral, $5$ a pentagon, $6$ a hexagon. **Drawing** starts from a rule and asks for a shape that fits it. "Draw a pentagon." "Draw a shape with four sides and exactly one square corner." A rule like that almost never has just one answer. Two children can draw quite different shapes and both be right, because the rule says only what the shape must have and leaves everything else free. The skill matters because every later lesson on shapes talks this way. A rectangle is defined by its square corners, a rhombus by its equal sides, and a child who can read a rule and test a shape against it, or build a shape to fit it, can follow all of them. The lesson runs over four class days, one part each. | Part | What it teaches | Examples | Exercises | |:-:|---|:-:|:-:| | $1$ | Naming by sides and corners | $1$, $2$ | $1$ to $7$ | | $2$ | Drawing from a side count | $3$, $4$ | $8$ to $12$ | | $3$ | Rules with square corners | $5$, $6$ | $13$ to $18$ | | $4$ | Shapes that do not belong | $7$, $8$ | $19$ to $22$ | ### [Two number patterns side by side and graphing their pairs](https://math.wiki/two-patterns-and-their-graph/) **Summary:** Two patterns are built from two rules, their matching terms are paired, and the pairs are graphed to see how the patterns relate. **Question:** Two patterns grow side by side, each from its own rule. How are their matching terms related, and what does that look like when the pairs are drawn on a grid? ## When you reach for it A rule such as "start at $0$, add $3$" builds a number pattern one term at a time; that is patterns-from-a-rule. Often two things grow at the same time, each by its own rule. Takumi and Yifan each keep a caterpillar in a jar. Takumi's caterpillar eats $3$ leaves a day and Yifan's eats $6$ leaves a day. Each of them counts the total number of leaves eaten so far, starting from $0$. That gives two patterns, one per caterpillar, growing day by day. You reach for this skill when you want to know how two such patterns are **related**. Is one always bigger? By how much? Is one always a multiple of the other? Writing the two patterns side by side, pairing their matching terms, and plotting the pairs as points on a grid answers these questions, and the picture often makes the answer plain at a glance. The lesson takes three days. Part 1 builds two patterns from two rules and pairs their matching terms. Part 2 finds the relationship between matching terms and explains why it holds at every step. Part 3 graphs the pairs as ordered-pairs, reads the graph, and looks at patterns that do not start at $0$ or whose relationship is not a plain multiple. ## The procedure ### Part 1: two rules, two patterns, matching terms A rule names a starting number and what to do to get from each term to the next. The two caterpillar rules are: - Takumi's pattern: start at $0$, add $3$. - Yifan's pattern: start at $0$, add $6$. ### [Two reflections in a row](https://math.wiki/two-reflections-make-a-rotation/) **Summary:** Reflecting twice gives a translation when the mirrors are parallel and a rotation when they cross. **Question:** You reflect a shape in one mirror line, then reflect the result in a second. Is there one single motion that does the same job? ## Why this exists A reflection is the plainest motion there is: one line, and every point jumps straight across it to the same distance on the other side. Slides and turns look like different kinds of motion altogether. A slide has a direction and a length, a turn has a center and an angle, and neither seems to have a mirror anywhere in it. They are made of mirrors all the same. Reflect a figure in one line and then in a second, and the figure has been slid or turned, never left flipped. Which of the two you get depends only on whether the mirrors are parallel or cross, and how far the figure moves depends only on how far apart the mirrors are, or at what angle they meet. That gives two things at once. You can say where two reflections leave a figure without tracking a single point through both. And since every slide and every turn is two reflections in a row, any fact proved for reflections, such as that they keep lengths and angles, holds at once for every slide and every turn too. ## Definition The **reflection in a line** $\ell$, written $r_\ell$, leaves every point of $\ell$ where it is and sends every other point $P$ to the point $P'$ for which $\ell$ is the perpendicular bisector of $\overline{PP'}$. The line $\ell$ is called the **mirror**. Rigid motions defined precisely sets this out beside the other rigid motions. **Two reflections in a row** means: reflect in $\ell$, then reflect the result in $m$. As with any composition of transformations, it is written with the first motion on the right. So $r_m \circ r_\ell$ means first $r_\ell$, then $r_m$: ### [Two-column and paragraph proofs](https://math.wiki/justifying-each-step/) **Summary:** Matching each statement to a definition, postulate, theorem or property that is already accepted. **Question:** A proof is a list of true statements. What gives each one the right to be on the list? ## When you reach for it Every time you write a proof. A two-column proof lists **statements** down the left and, beside each one, the **reason** it is allowed. The statements carry the argument; the reasons make it a proof instead of a list of things that happen to be true. Anyone should be able to check a single line by reading its reason and the lines above it, without trusting the writer and without measuring the picture. A **paragraph proof** is the same chain of statements and reasons written out as sentences, each reason riding inside the sentence that needs it (Example 1 shows both). The first lines are usually the Given, with the reason "given", and the last line is the Prove, both set up as in Given, prove and the diagram. Everything between them is this skill. ## The procedure **Only five kinds of reason are allowed.** A statement may stand because of - **a given**: a fact stated in the Given, and nothing more; - **a definition**: what a word means, such as "a midpoint cuts a segment into two equal parts"; - **a postulate**: a fact accepted without proof, such as the segment addition postulate or the angle addition postulate; - **a property of equality or congruence**: the rules for trading equal things, listed below; - **a theorem already proved**, such as the Linear Pair Theorem or the Vertical Angles Theorem. "It looks that way", "I measured it" and "obviously" are not on the list. Neither is the Prove itself, and neither is any theorem that was proved using the one you are proving. How facts are sorted into these kinds is definitions, postulates and theorems; why a chain of such steps settles a claim is deductive reasoning. ### [Two-dimensional shapes](https://math.wiki/shapes-2d/) **Summary:** Naming and sorting flat shapes by their sides and corners. **Question:** A huge triangle and a tiny turned one get the same name. What decides a flat shape's name, and what does the name promise? ## What this area is about Flat shapes are named by rules about their parts, such as sides and corners, never by size, color or which way they are turned. This area runs from a young child's first shape words to the angles and distances inside a regular polygon. ## How these fit together **First words.** Position words say where one thing is compared with another. Naming basic shapes gives circles, triangles, squares and rectangles their names, and circles says what makes a curve a circle. **Hands-on parts.** Sorting shapes counts sides and corners; building with sticks and clay makes those parts; composing and decomposing joins and cuts whole shapes. **Definitions.** Defining attributes separates what a name needs from what does not matter, and drawing from attributes runs that rule backwards. Polygons, quadrilaterals and classifying triangles name the families, and the shape hierarchy nests them. **Regular polygons.** The interior angle shares the angle sum equally; the central angle is $360^\circ$ divided by the number of sides $n$; the apothem is the distance from the center to a side. ## A path through it Position words and shape names, then sorting, building, composing and the circle. Next defining attributes and drawing, then polygons and quadrilaterals, which each draw on the other, then classifying triangles and the hierarchy. Last the interior angle, the central angle and the apothem. ### [Using the Pythagorean theorem](https://math.wiki/pythagorean-theorem-applications/) **Summary:** Finding a missing distance in a real figure by spotting a right triangle inside it. **Question:** A ramp, a television, a kite on its string. Nobody handed you a right triangle, so where is it hiding, and what can it tell you? ## When you reach for it You want a length that nobody measured: how far out a ramp starts, how tall a screen is, how long a brace must be, how high a kite is flying. The Pythagorean theorem can find it, but only after the length has been seen as one side of a right triangle whose other two sides are known, or can be worked out. So this skill is mostly about seeing. Right angles hide in ordinary things: a wall standing on level ground, every corner of anything rectangular, a pole standing straight up, the line straight down from something in the air. Find the right angle and the reason it is there, and the theorem does the rest. ## The procedure ### Part 1: one right triangle in a real figure **Find the right angle first, and know why it is there.** The theorem is true only for right triangles, so the right angle is the one thing you may not guess. In a real problem it comes from how things are built or placed: vertical meets level at a right angle, so a wall on flat ground makes one, and so does a mast. Every corner of a rectangle is one. A corner that merely looks square in a sketch is not a reason. **Draw just the triangle and label it.** Sketch the triangle on its own, with the known lengths on it and a letter on the unknown. Often one of its sides is not drawn in the real thing at all: the diagonal of a screen is not a line on the screen. **Name the hypotenuse before writing the equation.** The hypotenuse is the side across from the right angle, and it is always the longest side. In real figures it is usually the slanting thing: the ramp, the diagonal, the wire from a high point down to the ground, the kite string. If the unknown is the hypotenuse, add the squares of the two legs. If it is a leg, take the square of the known leg away from the square of the hypotenuse: ### [Vertical angles](https://math.wiki/vertical-angles/) **Summary:** The opposite pair of angles formed when two lines cross, always equal in measure. **Question:** Two lines cross. What does that force to be true about the four angles? ## Why this exists Draw two straight lines through one point and you have made four angles without measuring anything. It looks like four separate facts to find out. It is two. The angles come in matching pairs, and once you know one of them you know all four. That makes a crossing the cheapest information in geometry: it turns up inside triangles, inside circles, and under every pair of parallel lines cut by a transversal, and each time it hands you three angles for the price of one. Most of the time that is how the fact gets used. A figure hands you one or two angle measures and asks for an angle that was never measured. You could put a protractor on the page, but a drawing is only a sketch, and the question is what the figure **forces** to be true, not what it happens to look like. Vertical angles are one link in that chain, and this lesson also covers the whole job of finding unknown angle measures, with every link named. ## Definition When two lines cross, they make four angles around the crossing point. Two of those angles are **vertical angles** when they sit **opposite each other**: they share the vertex and nothing else, and each side of one is the continuation of a side of the other. In Figure 1, $\angle 1$ and $\angle 3$ are one pair of vertical angles, and $\angle 2$ and $\angle 4$ are the other. Two angles that sit side by side at a crossing, such as $\angle 1$ and $\angle 2$, are not vertical. They share a whole side, and together they make a straight line. A pair like that is a linear pair, and its two measures add to $180^\circ$. ### [Volume of a cylinder](https://math.wiki/volume-of-a-cylinder/) **Summary:** The same stacking idea carried out with a circular base. **Question:** A can is round, so no stack of unit cubes fits inside it exactly. How do you find how much it holds? ## When you reach for it A box is easy to measure because unit cubes fit it exactly: so many in a row, so many rows, so many layers. A can, a pipe, a water tank, a coin and a pillar are all round, and no arrangement of cubes fills a round shape without gaps. Yet each of these is a **cylinder**, a solid with two congruent, parallel circular ends joined by a curved side (see cylinders), and its volume comes from the same idea that measures a box. Reach for this skill whenever the question is how much a round container holds, how much material a round solid or a tube contains, how tall or how wide a cylinder must be to hold a given amount, or how much a water level rises when something is dropped in. It rests on two things you already have: the volume of a prism, base area times height, and the area of a circle, $\pi r^2$. This lesson takes two days. **Part 1** builds the formula and uses it to find volumes from a radius, a diameter or a circumference, in any units, and turns cubic units into liters. **Part 2** runs the formula backwards to find a height or a radius, compares cylinders of different sizes, and measures tubes and the volume of water pushed up by an object. Examples 1 to 4 and Exercises 1 to 11 belong to Part 1; Examples 5 to 9 and Exercises 12 to 25 belong to Part 2. ## The procedure ### Part 1: the formula and how to use it **Why base area times height.** Stand a cylinder of radius $r$ and height $h$ on one of its ends, and slice it into layers one unit thick. Each layer is a flat round disc, a circle of radius $r$ that is $1$ unit deep. A layer with base area $A$ and depth $1$ holds $A \cdot 1$ cubic units, for the same reason a single layer of a box holds as many unit cubes as there are unit squares on its floor. So each layer holds $\pi r^2$ cubic units, and there are $h$ of them. ### [Volume of a sphere](https://math.wiki/volume-of-a-sphere/) **Summary:** A sphere holds two thirds of the cylinder that just contains it. **Question:** A ball has no flat face to stand on and no straight edge to measure. How can its volume be found from its radius alone? ## When you reach for it A **sphere** is every point at one fixed distance, the radius, from a center (see spheres). The solid it encloses is a **ball**, and "the volume of a sphere" always means the volume of that ball: how much a round tank holds, how much rubber or metal a ball contains, how much air fills a balloon. A ball is harder to measure than a box or a can. A box is so many layers of unit cubes, and a can is the same circle stacked through its height (volume-of-a-cylinder). A ball's slices are circles too, but they grow from nothing at the bottom to the widest circle at the middle and shrink back to nothing at the top. Yet the answer turns out to need only the radius, and it is tied exactly to the cylinder around the ball. Reach for this skill whenever the solid is round in every direction: a ball, a bubble, a planet treated as a ball, a dome or a bowl that is half a ball, a hollow ball such as a shell, or a liquid level that rises when balls are dropped in. It rests on the volume of a cylinder, $\pi r^2 h$, and on cube-roots when the formula is run backwards. This lesson takes two days. **Part 1** gives the formula, where it comes from, and how to use it from a radius, a diameter or a circumference, for whole balls and half balls, in any unit. **Part 2** runs the formula backwards to find a radius, compares balls of different sizes, and measures hollow balls and the water that balls push up. Examples 1 to 4 and Exercises 1 to 12 belong to Part 1; Examples 5 to 9 and Exercises 13 to 25 belong to Part 2. ### [Volume of prisms and cylinders](https://math.wiki/volume-of-a-prism/) **Summary:** Stacking copies of the base area all the way up to the height. **Question:** How much does a box, a trough or a can hold, and why is it the area of the bottom times the height? ## When you reach for it Sand for a sandbox, rubble in a skip, juice in a carton, batter in a cake tin: each asks how much space is inside a solid, which is how much it takes to fill it. That amount is the **volume**. It is measured by counting cubes. A cubic centimeter, $1\ \text{cm}^3$, is the space inside a cube $1$ cm along each edge, and a cubic meter, $1\ \text{m}^3$, the space inside a cube $1$ m along each edge. Volume is a different question from surface area. Surface area is the wrapping, the material that covers the outside once; volume is the filling. This lesson handles the solids in which every slice parallel to the bottom is the same shape and size as the bottom: prisms, whose base is a polygon, and cylinders, whose base is a circle. One rule works for all of them, the area of the base times the height. Part 1 builds it for prisms and Part 2 carries it to cylinders. ## The procedure ### Part 1: prisms **What a prism is.** A prism has two faces that are the same polygon, lying in parallel planes, one directly over the other. These are its two **bases**, and the faces joining them are rectangles. Think of the base slid straight up through space, leaving a solid behind it. Because the base slides without turning or shrinking, every slice parallel to the base is an exact copy of the base. three-dimensional-shapes names the faces, edges and vertices of these solids. **Layers of cubes.** Take a box whose base is $5$ by $3$ and whose height is $4$, in some unit. Set unit cubes on the base: a row of $5$, and $3$ rows, so the bottom layer holds $5 \times 3 = 15$ cubes, one on each unit square of the base. This is the count from volume-with-unit-cubes. Every layer above is the same, because every slice is a copy of the base, and there are $4$ layers. The box holds $15 \times 4 = 60$ cubes. ### [Volume of pyramids and cones](https://math.wiki/volume-of-a-pyramid/) **Summary:** A pyramid holds one third of the prism with the same base and height. **Question:** A pyramid and a box can stand on the same floor and reach the same height, but the pyramid narrows to a point. How much of the box does it fill? ## When you reach for it A **pyramid** is the solid you get from a flat polygon, its base, and one point not in the plane of the base, its apex, by joining every point of the base to the apex with a straight segment. A **cone** is built the same way on a circular base (see prisms and pyramids and cylinders and cones). Spires, tent roofs, funnels, piles of sand or grain, the tip of a pencil and a paper drinking cup are all pyramids or cones, or pieces of them. Reach for this skill whenever the question is how much such a solid holds or contains: the air under a pyramid roof, the sand in a heap, the water in a cone-shaped cup or tank. It also runs backwards, to find how tall or how wide a cone must be to hold a given amount, and it measures shapes such as a bucket or a lampshade, which are cones with the tip cut off. It rests on the volume of a prism, base area times height, and for cones on the area of a circle, $\pi r^2$. This lesson takes two days. **Part 1** gives the one-third rule, says why it holds, and uses it to find volumes, including when only the slant height is known. **Part 2** runs the rule backwards, compares solids of different sizes, and measures solids with the tip cut off and solids built from a cone and a cylinder. Examples 1 to 5 and Exercises 1 to 13 belong to Part 1; Examples 6 to 9 and Exercises 14 to 25 belong to Part 2. ## The procedure ### Part 1: one third of the prism ### [Volume of solids](https://math.wiki/volume-of-pyramids-cones-spheres/) **Summary:** One third of the matching prism or cylinder, and the sphere formula. **Question:** A box can be filled with unit cubes, but a can, a cone and a ball cannot. How do you find how much any of them holds? ## What this area is about Volume counts the unit cubes of space a solid fills, in units such as $\text{cm}^3$. Stacking copies of a base gives the volume of a prism or cylinder, and the other formulas are tied back to it. ## How these fit together **Stacking.** Volume of prisms and cylinders: $V = Bh$, base area $B$ times height $h$. Volume of a cylinder carries it out on a circular base, $V = \pi r^2 h$. **Fractions of a stack.** Pyramids and cones hold one third of the prism or cylinder with the same base and the same height, $V = \tfrac13 Bh$, upright or leaning, as long as $h$ is the perpendicular height. A sphere of radius $r$ encloses $\tfrac43 \pi r^3$, two thirds of the cylinder that just contains it. **Why they hold.** Cavalieri's principle: if two solids lie between the same two parallel planes and every plane parallel to them cuts cross sections of equal area, the volumes are equal. Why the volume formulas hold uses it, with dissection and a limit, to justify the cylinder, pyramid, cone and sphere results. **Using them.** Composite solids and density add or subtract pieces and turn volume into mass. ## A path through it Prisms and cylinders first. After them come a cylinder alone, pyramids and cones, and Cavalieri's principle. Composite solids follow pyramids and cones; the derivations follow Cavalieri, the cylinder and pyramids and cones. The sphere needs only what a sphere is, but it leans on the cylinder formula, so it reads best after the cylinder. ### [What a circle is](https://math.wiki/circles-concept/) **Summary:** Every point the same distance from the center; radius, diameter, chord. **Question:** A triangle has three sides and a square has four corners, but a circle has no sides and no corners at all. So what exactly makes a shape a circle? ## Why this exists Shapes like triangles and squares are named by counting: how many straight sides, how many corners. A circle gives you nothing to count. Its edge is one smooth curve that never turns a corner, so the usual way of sorting shapes has nothing to hold on to. Yet everyone knows a circle when they see one, and a wheel shows why the idea matters. A wheel rolls smoothly because the axle is the same distance from every part of the rim. Make the rim even a little lopsided and the cart bumps up and down as it rolls. That one fact is the whole idea. Tie a string to a peg, pull it tight, and walk around the peg: the path you trace keeps exactly the length of the string between you and the peg. A compass draws a circle the same way, with a fixed gap between its point and its pencil. So the honest description of a circle is not about sides at all. It is about **distance from one point**. ## Definition A **circle** is the set of all points in a flat plane that are the same distance from one fixed point. The fixed point is the **center**, and the fixed distance is the **radius**. A circle is named by its center. The circle with center $O$ is written $\odot O$ and read "circle $O$". Three kinds of segment come with every circle (Figure 1). - A **radius** is a segment from the center to a point on the circle, such as $\overline{OA}$. The word also means its length, written $r$. - A **chord** is a segment whose two endpoints are both on the circle, such as $\overline{DE}$. - A **diameter** is a chord that passes through the center, such as $\overline{BC}$. The word also means its length, written $d$. ### [What a dilation does to a line](https://math.wiki/dilations-of-lines/) **Summary:** A line through the center stays exactly where it is and every other line maps to a line parallel to it. **Question:** A dilation moves every point along its own ray from the center. What happens to a straight line when all of its points move at once? ## Why this exists A dilation is described one point at a time. The center $O$ stays put, and every other point $P$ slides along the ray from $O$ through $P$ until it is $k$ times as far from $O$ as it was. That tells you where any single point goes. It says nothing directly about a figure, and figures are made of lines and segments. Dilate the three corners of a triangle and you get three new points. To join them with straight sides and call the result "the image of the triangle", you need to know that the points of each side, all moving at once, land on the straight segment between the new corners and not on some curve. You would also like to know how each new side sits compared with the old one. This lesson answers both questions, and the answer is short. A line through the center stays exactly where it is, and every other line is carried to a line parallel to it. That one fact is why a dilated figure has the same shape as the original, and it is the engine behind similar figures. ## Definition Fix a dilation with center $O$ and scale factor $k > 0$. The image of a point $P \neq O$ is the point $P'$ on ray $OP$ with $OP' = k \cdot OP$, and the image of $O$ is $O$ itself. The **image of a line** $l$ is the set of the images of all the points on $l$. It is written $l'$ and read "$l$ prime". The image of a segment $\overline{PQ}$ is defined the same way, as the set of the images of its points. ### [What rigid motions preserve](https://math.wiki/invariance-under-rigid-motion/) **Summary:** Lengths, angle measures, betweenness and parallelism all survive a rigid motion unchanged. **Question:** A figure has been slid, flipped or turned. Which of its measurements can you still trust without measuring again, and which might have changed? ## Why this exists Moving a figure is useful only if you know what survives the move. A tiler who has measured one tile wants every copy laid by sliding, flipping or turning to have the same edges and corners, so the pattern closes up with no gaps. The lessons on translations, reflections and rotations showed where each motion sends a point. This lesson asks what all three keep. ## Definition A **rigid motion** is a translation, a reflection, a rotation, or a sequence of these. If a motion sends the point $A$ to $A'$, then $A'$ is the **image** of $A$, written $A \mapsto A'$. A property is **preserved** when every figure and its image share it. Rigid motions preserve: - **lengths:** a segment goes to a segment of the same length, $A'B' = AB$; - **lines:** a line goes to a line, and a ray to a ray; - **angle measures:** $m\angle A'B'C' = m\angle ABC$; - **parallel lines:** if $\ell \parallel m$, then $\ell' \parallel m'$; - **betweenness:** if $Q$ lies on $\overline{PR}$, then $Q'$ lies on $\overline{P'R'}$. What can change is where the figure sits, the direction each side points, and its **orientation**: whether going round triangle $ABC$ in the order $A$, $B$, $C$ runs counterclockwise or clockwise. To test it, walk from $A$ to $B$; if $C$ is on your left, the order is counterclockwise. ## Intuition Trace a figure on tracing paper. A translation slides the paper, a rotation turns it, and a reflection turns it face down. The paper never stretches or creases, so lengths, corners, parallel edges and midpoints arrive as they left. What can change is where it lands, which way it points, and which face is up. In Figure 1 the turned and slid copies still run counterclockwise; the reflected copy, the tracing seen from the back, runs clockwise. ### [Why a construction works](https://math.wiki/justify-a-construction/) **Summary:** Every compass and straightedge step can be defended by the congruent triangles or circles it creates. **Question:** A compass and a straightedge have produced a line that looks perpendicular, or a ray that looks as if it cuts an angle in half. How do you know it really does, every time, and not just in this one drawing? ## Why this exists A drawing shows one case. Bisect a segment with compass and straightedge, measure the two halves with a ruler, and they agree to the width of a pencil line. That is evidence about one segment of one length, and it is no more exact than the ruler that read it. A construction promises more: that its steps give the midpoint of every segment, exactly. A promise about every case needs a proof. The steps of a construction create facts about distances and lines, and the result is a claim about the finished figure, such as "this line is perpendicular to $\overline{AB}$ and passes through its midpoint". Justifying the construction means connecting the two with the tools of proof: each fact the drawing creates becomes a given, and congruent triangles, the perpendicular bisector or a reflection carry the givens to the conclusion. The payoff is trust. Once a construction has been justified, later constructions and later proofs can use "draw the perpendicular bisector" as a single step, because it has been shown to work in every case. ## Definition A **justification** of a construction is a proof that the figure its steps produce has the property the construction claims, for every allowed starting figure. Its givens are the facts the two tools create, and nothing else. **An arc gives equal distances.** Every point of an arc drawn about a center $C$ with compass opening $r$ is at distance $r$ from $C$, because a circle is exactly the set of points at distance $r$ from its center. Two arcs drawn without changing the opening have the same radius. ### [Why AAA and SSA fail](https://math.wiki/why-ssa-is-not-a-criterion/) **Summary:** Two sides and an angle that is not between them can build two different triangles, so they do not force congruence. **Question:** Three matching parts make two triangles congruent in SSS, SAS, ASA and AAS. Why not three matching angles, or two sides and an angle that is not between them? ## Why this exists A triangle has six parts, three sides and three angles. SSS, SAS, ASA and AAS each name three matching parts that force two triangles to be congruent. SSS, SAS and ASA are taken as postulates on this map, and AAS is proved from ASA. (The three postulates can also be derived from rigid motions.) Three parts of a triangle can be arranged in six ways: SSS, SAS, ASA, AAS, AAA and SSA. Four are criteria. This lesson is about the other two. **AAA**, the three angles and no side at all, sounds like the most natural test of all. It fails for a reason you can see at once: a photograph and its enlargement have the same angles. **SSA**, two sides and an angle that is not between them, is the one that fools people. It uses the same ingredients as SAS, and AAS shows that a part off to one side can still do its job, so it seems it ought to work. Neither does. This lesson builds the triangles that break each one, shows the one extra fact that turns AAA into a criterion, and says exactly when SSA data does pin a triangle down after all. ## Definition A **criterion** for congruence is a list of matching parts that forces two triangles to be congruent **every time**, whatever the triangles look like. A general rule is broken by a single case where it fails: one pair of triangles that matches the list and still is not congruent shows the list is not a criterion. Such a pair is a counterexample. ### [Why the volume formulas hold: dissection, slicing and Cavalieri](https://math.wiki/derive-volume-formulas/) **Summary:** Informal arguments for the cylinder, pyramid, cone and sphere volume formulas, by dissection, by slicing and by Cavalieri's principle. **Question:** No box of unit cubes fits a cone or a ball. So why does a cone hold exactly one third of the cylinder around it, and a ball exactly two thirds? ## Why this exists The volume of a box comes from counting. A box $4$ by $3$ by $2$ holds $4 \cdot 3 \cdot 2 = 24$ unit cubes, laid in $2$ layers of $12$. Stacking carries the same idea to every prism and cylinder: one layer of the base, repeated all the way up, gives $V = Bh$ (volume-of-a-prism, volume-of-a-cylinder). Pyramids, cones and balls do not stack that way. A pyramid narrows to a point, so no two of its layers are the same, and a ball curves everywhere, so no unit cube sits flush against it. Yet their formulas are short: $$\begin{aligned} V_{\text{pyramid}} &= \frac{1}{3}Bh \\ V_{\text{ball}} &= \frac{4}{3}\pi r^3 \end{aligned}$$ Nobody would guess a $\frac{1}{3}$ or a $\frac{4}{3}$ from counting cubes. This lesson shows where those fractions come from, using three ways of arguing about volume: cutting a solid into pieces whose volumes are known (dissection), comparing two solids slice by slice (Cavalieri's principle), and trapping a solid between simpler solids that close in on it (a limit). This lesson takes two days. Part 1 derives the formulas for prisms, cylinders, pyramids and cones. Part 2 derives the volume of a ball, and from it the area of the sphere's surface. ## Definition A **pyramid** has a flat polygon as its base and a point, the **apex**, outside the plane of the base; the pyramid is made of all the segments from the apex to points of the base. A **cone** is built the same way on a circle. The **height** $h$ of either is the perpendicular distance from the apex to the plane of the base. The apex does not have to sit above the center of the base; when it does not, the solid leans and is called **oblique**. ## Linear algebra ### [Adding and scaling vectors in n dimensions](https://math.wiki/vector-addition-in-rn/) **Summary:** Adding two lists of numbers one position at a time. **Question:** A day at a creamery produces five numbers, not one. How do you add, compare and scale whole lists like that in one move? ## When you reach for it Anela, Kailani and Brynjar run a creamery. Each evening they record the day's output as five numbers in a fixed order: gallons of milk, pounds of butter, quarts of yogurt, pints of ice cream and pounds of cheese. Monday might be $(120, 35, 60, 80, 18)$. That list is one object. The total for two days, the change from one day to the next, and the output of five days at Monday's rate all come from adding, subtracting and scaling whole lists, one position at a time. In the plane and in space these moves were arrows, tip to tail and stretched (vector-operations). Nothing in the rules depends on there being two or three entries. This lesson sets up the arithmetic of lists of any length $n$: equality, adding, scaling, the zero vector and negatives, and the eight properties that let vector expressions be rearranged as freely as number expressions. A list written as a column is an $n \times 1$ matrix, so the arithmetic is that of adding and scaling matrices. What is new is treating the lists as the points of one space, $\mathbb{R}^n$, and proving the properties for every $n$ at once. Reach for it whenever several numbers belong together and are always combined position by position. Every list must use one order and one unit per position; otherwise a sum is defined and means nothing. ## The procedure ### Vectors in $\mathbb{R}^n$ For a positive integer $n$, $\mathbb{R}^n$ is the set of all ordered lists of $n$ real numbers. An element of $\mathbb{R}^n$ is a **vector**, and its numbers are its **entries**, or components. We write a vector as a column, ### [Adding, subtracting and scaling matrices](https://math.wiki/add-subtract-matrices/) **Summary:** Entry by entry, and only when the sizes match. **Question:** Two tables of numbers have the same rows and the same columns. How do you combine them into one table, or scale one by a single factor, and what rules does that arithmetic obey? ## When you reach for it Ardak runs the north branch of an ice cream shop and Bolatbek runs the south branch. Each keeps the weekend's cone sales in the same layout: one row per flavor (vanilla, chocolate, mango) and one column per day (Saturday, Sunday). Each table is a $3 \times 2$ matrix. To get the weekend's sales for the whole shop, you add the two vanilla-on-Saturday numbers, the two vanilla-on-Sunday numbers, and so on, six additions in all. To see where the north branch sold more, you subtract in the same pattern. That is the whole idea of matrix addition: matching positions combine, and nothing else does. Scaling works the same way. If every price on a menu goes up by the same factor, every entry of the price table is multiplied by that factor. Reach for these moves whenever data comes in tables of one shape, and whenever a later topic (systems of equations, transformations, vectors) asks you to combine matrices. ## The procedure ### Adding and subtracting Let $A = [a_{ij}]$ and $B = [b_{ij}]$ both be $m \times n$. Their **sum** $A + B$ and **difference** $A - B$ are the $m \times n$ matrices with entries $$\begin{aligned} (A + B)_{ij} &= a_{ij} + b_{ij} \\ (A - B)_{ij} &= a_{ij} - b_{ij} \end{aligned}$$ for every row $i$ and column $j$. The entry in a position of the answer comes from the two entries in **the same position**, and from nothing else. The dimensions must match. If $A$ is $2 \times 3$ and $B$ is $3 \times 2$, some position of one matrix has no partner in the other, so the rule has nothing to add there. The sum is then **undefined**: not zero, not some partial matrix, simply not defined. Having the same number of entries is not enough; the shapes must be the same. ### [Augmented matrices and row operations](https://math.wiki/elementary-row-operations/) **Summary:** Swapping rows, scaling a row, or adding a multiple of one row to another, none of which change the solutions. **Question:** Elimination keeps copying the same letters while only the numbers change. Can the letters be dropped, and which moves on the numbers alone are guaranteed never to change the solutions? ## When you reach for it In elimination-by-adding a system is solved by adding a multiple of one equation to another until a variable cancels, copying $x$ and $y$ at every step although only the numbers change. Writing the system as $AX = B$ already sorted those numbers into a coefficient matrix $A$ and a column of constants $B$. Setting $B$ beside $A$ and dropping the letters gives the **augmented matrix**, and the moves of elimination become three **row operations** on it. You reach for this whenever a linear system is to be solved with matrices, by hand or by a calculator: it is the format of Gaussian elimination. ## The procedure ### The augmented matrix Take a linear system with every equation lined up: variable terms on the left in the same order, the constant on the right. Its augmented matrix $[A \mid B]$ has one row per equation. The columns left of the bar hold the coefficients, one column per variable, and the column right of the bar holds the constants. For the system $x + 3y = 5$, $2x - y = 3$, $$[A \mid B] = \left[\begin{array}{cc|c} 1 & 3 & 5 \\ 2 & -1 & 3 \end{array}\right]$$ The bar marks where each equals sign stood. A number's column says which variable it multiplies, which is why the equations must be lined up first and a missing variable still gets its entry, $0$. ### The three row operations | Operation | Notation | The same move on the equations | |---|:-:|---| | Swap two rows | $R_i \leftrightarrow R_j$ | write the equations in another order | | Multiply a row by a number $c \ne 0$ | $cR_i \to R_i$ | multiply both sides of one equation by $c$ | | Add a multiple of another row ($j \ne i$) | $R_i + cR_j \to R_i$ | add $c$ times one equation to another | ### [Basis and dimension](https://math.wiki/dimension-of-a-space/) **Summary:** The number of vectors in a basis, which comes out the same for every basis of the same space. **Question:** How few vectors are enough to build every vector of a subspace, and why does every smallest building list come out the same size? ## Why this exists Zilola, Dildora and Umida run a chocolatier. A recipe is a list $(x_1, x_2, x_3, x_4)$ of grams of cocoa mass, cocoa butter, sugar and milk powder, and a change to a recipe is a list of grams added to or taken from each. The changes they allow keep the weight of a bar fixed, $$x_1 + x_2 + x_3 + x_4 = 0,$$ so they form a subspace $W$ of $\mathbb{R}^4$ (subspaces-of-rn). Zilola wants a short list of basic changes that builds every allowed one, each in only one way. The swaps $\mathbf{s}_1 = (1, -1, 0, 0)$, $\mathbf{s}_2 = (0, 1, -1, 0)$ and $\mathbf{s}_3 = (0, 0, 1, -1)$ do it (Example 1). Could two changes do it? A different trio, or four? Every such list turns out to have exactly three vectors. That number is the dimension of $W$, the exact sense in which $W$ is three-dimensional though it sits in $\mathbb{R}^4$. ## Definition Let $V$ be a subspace of $\mathbb{R}^n$. A list $\mathbf{v}_1, \dots, \mathbf{v}_k$ of vectors in $V$ is a **basis** of $V$ when it is linearly independent (linear-independence) and spans $V$, that is, $\operatorname{span}\{\mathbf{v}_1, \dots, \mathbf{v}_k\} = V$ (span-of-a-set-of-vectors). Spanning says every vector of $V$ can be built; independence says nothing in the list is wasted. If $\mathbf{x} = c_1\mathbf{v}_1 + \dots + c_k\mathbf{v}_k$, the weights $(c_1, \dots, c_k)$ are the **coordinates** of $\mathbf{x}$ relative to the basis, taken in the order of the list. ### [Complex eigenvalues and rotation-scaling matrices](https://math.wiki/complex-eigenvalues/) **Summary:** A real matrix with no real eigenvectors rotates and scales, and its complex eigenvalues give the angle and the scale factor. **Question:** A real matrix can turn every arrow in the plane, so that no real direction is kept and the characteristic equation has no real root. What do its complex roots say about the motion? ## Why this exists A leaf floats on a whirlpool in a river. Sangmin records its position $\mathbf{x}_k$, in meters from the center of the whirlpool, once a second, and finds that one matrix carries each position to the next: $$\mathbf{x}_{k+1} = W\mathbf{x}_k, \qquad W = \begin{bmatrix} 1 & -1 \\ \tfrac12 & 0 \end{bmatrix}.$$ The leaf circles as it drifts inward, so no line through the center is carried to itself, and $W$ has no real eigenvector. The characteristic-equation agrees: $\lambda^2 - \lambda + \tfrac12 = 0$ has discriminant $1 - 2 = -1 < 0$. With real numbers only, that ends the analysis. With complex numbers it does not: the roots $\tfrac12 \pm \tfrac12 i$ carry the whole motion. Their modulus $\tfrac{\sqrt2}{2}$ is the factor by which the leaf closes in each second, and their argument $45^\circ$ is how far it turns, once the plane is measured in axes suited to the whirlpool. This lesson finds those axes. ## Definition Let $A$ be a real $n \times n$ matrix. A complex number $\lambda$ is an **eigenvalue** of $A$ when $A\mathbf{v} = \lambda\mathbf{v}$ for some nonzero vector $\mathbf{v}$ with complex entries; $\mathbf{v}$ is a **complex eigenvector**. The eigenvalues are exactly the complex roots of $\det(A - \lambda I)$, and the eigenvectors are found by row reducing $A - \lambda I$ as before, with complex arithmetic. A **complex eigenvalue** here means a non-real one, $a + bi$ with $b \neq 0$. ### [Composing transformations as a matrix product](https://math.wiki/composition-as-matrix-product/) **Summary:** Doing one transformation and then another is the same as multiplying their two matrices. **Question:** A choreographer moves a whole formation one way and then another. Is there a single matrix for the combined move, and how is it built from the matrices of the two moves? ## Why this exists Txia is a choreographer. She records each dancer's spot as a vector from center stage, in meters: the first entry runs across the stage, positive to the audience's right, and the second runs upstage, away from the audience. A move of the whole formation is then a function from $\mathbb{R}^2$ to $\mathbb{R}^2$, and her usual moves (slants, turns about center stage, mirror flips) are linear transformations, each with a matrix. A piece is one move after another, and Txia wants one matrix for the combined move, rather than pushing every dancer through every matrix in turn. This lesson shows it is the product of the matrices, with the move done first on the right, and that this is why matrix multiplication is defined the way it is. ## Definition To compose two functions is to feed the output of one into the other. If $S: \mathbb{R}^n \to \mathbb{R}^m$ and $T: \mathbb{R}^m \to \mathbb{R}^p$, the **composition** $T \circ S: \mathbb{R}^n \to \mathbb{R}^p$ is $$(T \circ S)(\mathbf{x}) = T(S(\mathbf{x})).$$ It is read "$T$ after $S$": $S$ acts first though it is written second, because it sits next to $\mathbf{x}$. It makes sense only when the codomain of $S$ is the domain of $T$. The product $BA$ of a $p \times m$ matrix $B$ and an $m \times n$ matrix $A$ is the $p \times n$ matrix of matrix-multiplication-rule. Read a column at a time, its row-by-column rule says **column $j$ of $BA$ is $B\mathbf{a}_j$**, where $\mathbf{a}_j$ is column $j$ of $A$, since entry $i$ of $B\mathbf{a}_j$ is row $i$ of $B$ times $\mathbf{a}_j$. ### [Computing a determinant](https://math.wiki/determinant-2x2/) **Summary:** Multiply the diagonals and subtract. **Question:** A square array of numbers holds a whole matrix. Is there one number that says whether its columns point in genuinely different directions, and how much room they span? ## When you reach for it Every square matrix carries one number, its **determinant**. You compute it when you need to know whether the columns of the matrix point in genuinely different directions or collapse onto one line (for a $3 \times 3$ matrix, onto one plane). That question decides whether a system with that coefficient matrix has exactly one solution, which you have met through row reduction; whether the matrix can be undone by an inverse; and whether Cramer's rule applies. The determinant also measures something you can see: the area of the parallelogram the columns span. The words come from matrix-vocabulary: rows, columns, entries $a_{ij}$, square matrices and $I_2$. Unlike adding matrices, which returns a matrix, a determinant returns a single number, and only a square matrix has one. ## The procedure ### The 2 by 2 determinant For $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, $$\det A = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$$ Multiply the entries on the main diagonal, multiply the entries on the other diagonal, and subtract the second product from the first. Straight bars around the entries mean the determinant. Some books write $|A|$; those bars are not an absolute value, since a determinant can be negative. Why this combination of the four entries? Read the columns of $A$ as two arrows from the origin, $(a, c)$ and $(b, d)$. ### What the number measures ### [Coordinates in a basis and change of basis](https://math.wiki/coordinates-and-change-of-basis/) **Summary:** The same vector has different coordinates in different bases, and an invertible matrix converts one set of coordinates into the other. **Question:** A point on a slanted tile grid can be named by its screen position or by its tile steps. How do you turn one description into the other, and between two different grids? ## Why this exists Pehuen is drawing a pixel-art game on a slanted tile grid, in the spirit of isometric art. The engine stores every position as a screen vector, so many units right and so many units up. Amancay, who designs the levels, never thinks that way: a gem sits "two tiles along one edge, three along the other". One tile edge is $\mathbf{b}_1 = (2, 1)$ on screen and the other is $\mathbf{b}_2 = (1, 3)$, so her gem is at $$2\mathbf{b}_1 + 3\mathbf{b}_2 = (4, 2) + (3, 9) = (7, 11).$$ The game needs both directions of translation: tile steps to screen position for drawing, and screen position to tile steps when a player clicks. Then Kuyen brings in a second tile set with different edges, and every level has to be converted from one grid to the other. All three jobs are the same idea. A basis is a coordinate grid, the coordinates of a vector are its steps on that grid, and an invertible matrix carries one set of coordinates to another. ## Definition Let $V$ be a subspace of $\mathbb{R}^n$ and $\mathcal{B} = \{\mathbf{b}_1, \dots, \mathbf{b}_k\}$ a basis of $V$, taken in this order (dimension-of-a-space). For $\mathbf{x}$ in $V$, the **coordinate vector** of $\mathbf{x}$ relative to $\mathcal{B}$ is the column of weights $$[\mathbf{x}]_{\mathcal{B}} = \begin{bmatrix} c_1 \\ \vdots \\ c_k \end{bmatrix} \quad \text{where} \quad \mathbf{x} = c_1\mathbf{b}_1 + \dots + c_k\mathbf{b}_k.$$ ### [Cramer's rule](https://math.wiki/cramers-rule/) **Summary:** Solve a system with a ratio of determinants. **Question:** The inverse of a matrix solves a whole system at once. Is there a formula that gives one unknown by itself, straight from the coefficients? ## When you reach for it A square linear system with exactly one solution can be solved by row reduction or, as in inverse-2x2, by $X = A^{-1}B$. **Cramer's rule** writes each unknown on its own as a ratio of two determinants. You reach for it when you want one unknown and not the others, when a coefficient is a letter such as $k$ and you want the answer as a formula in $k$, and for small systems with awkward numbers, where it avoids fractions until the last step. ## The procedure ### Two equations Take the system $$\begin{aligned} ax + by &= e \\ cx + dy &= f \end{aligned}$$ and form three determinants. $D$ is the determinant of the coefficient matrix. $D_x$ is the same determinant with the $x$ column replaced by the constants $e, f$, and $D_y$ has the $y$ column replaced instead: $$D = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$$ $$D_x = \begin{vmatrix} e & b \\ f & d \end{vmatrix}$$ $$D_y = \begin{vmatrix} a & e \\ c & f \end{vmatrix}$$ If $D \neq 0$, the system has exactly one solution, and $$x = \frac{D_x}{D}, \qquad y = \frac{D_y}{D}$$ **Why it is true.** With $D \neq 0$ the coefficient matrix $A$ has the inverse $\frac{1}{D}\begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$, so the only solution is $$\begin{aligned} \begin{bmatrix} x \\ y \end{bmatrix} &= A^{-1}\begin{bmatrix} e \\ f \end{bmatrix} \\ &= \frac{1}{D}\begin{bmatrix} de - bf \\ af - ce \end{bmatrix} \end{aligned}$$ and $de - bf$ is exactly $D_x$, while $af - ce$ is exactly $D_y$. ### [Determinants by cofactor expansion](https://math.wiki/cofactor-expansion/) **Summary:** A determinant of any size breaks into smaller determinants along any row or column, with alternating signs. **Question:** The determinant of a 2 by 2 or 3 by 3 matrix has a formula you can write on one line. How do you find it for a 4 by 4, a 5 by 5, or any size at all? ## When you reach for it Oriwa, Waimarie and Ruiha are in an engineering class sizing a small footbridge. Balancing the forces at the joints gives four equations in four unknown forces, and they want to know whether the system has exactly one solution. By determinant-meaning the determinant is the factor by which a matrix scales volume, and nonzero means nothing is flattened and the system has exactly one solution; for $4 \times 4$ that is stated below, proved later. They need a $4 \times 4$ determinant. determinant-2x2 expanded $3 \times 3$ determinants along a row: each entry times its **minor** $M_{ij}$ (the determinant left after crossing out row $i$ and column $j$), with the checkerboard sign $(-1)^{i+j}$. Here that move becomes the definition for every size, works along columns too, and meets its best uses and its limit. ## The procedure ### Day 1: the definition for any size The **cofactor** of the entry in position $(i, j)$ of an $n \times n$ matrix $A$ is its minor with the checkerboard sign attached: $$C_{ij} = (-1)^{i+j} M_{ij}$$ The minor of an $n \times n$ matrix is an $(n - 1) \times (n - 1)$ determinant. For $n = 4$ the signs $(-1)^{i+j}$ are $$\begin{bmatrix} + & - & + & - \\ - & + & - & + \\ + & - & + & - \\ - & + & - & + \end{bmatrix}$$ Every diagonal position is $+$, and the sign changes with each step along a row or down a column. **Definition.** For a $1 \times 1$ matrix, $\det [a] = a$. For $n \geq 2$, ### [Diagonalization](https://math.wiki/diagonalization/) **Summary:** Using a basis of eigenvectors, a transformation turns into plain scaling along each direction. **Question:** Most matrices turn almost every arrow they act on. When can a matrix be redescribed, in a well-chosen basis, as nothing but a separate stretch along each axis, and what does that make easy? ## Why this exists Malika, Bakyt and Mihi make mosaics. A border strip one unit wide and $n$ units long is laid from white square tiles ($1$ by $1$) and dominoes ($1$ by $2$) that come in two colors, blue and gold. Malika wants to know how many different strips of each length there are. Sorting the strips by their last piece gives a rule: a strip ending in a square is a strip of length $n - 1$ plus that square, and a strip ending in a domino is a strip of length $n - 2$ plus one of two dominoes (recurrences-from-counting-problems). So $$a_n = a_{n-1} + 2a_{n-2}, \qquad a_1 = 1, \quad a_2 = 3.$$ The rule is linear, so one step is a matrix: $(a_{n+1}, a_n) = M(a_n, a_{n-1})$ with $M = \begin{bmatrix} 1 & 2 \\ 1 & 0 \end{bmatrix}$, and a strip of length $30$ needs $M^{28}$ applied to $(a_2, a_1)$. Powers of a general matrix show no pattern; powers of a diagonal matrix are the powers of its diagonal entries. This lesson finds when a matrix is a diagonal matrix in disguise, seen through a basis of eigenvectors. Day 1 proves the diagonalization theorem and reads it as a change of basis; day 2 handles repeated eigenvalues, powers, and the strip count. ## Definition A square matrix $D$ is **diagonal** when every entry off its main diagonal is $0$; write $D = \operatorname{diag}(\lambda_1, \dots, \lambda_n)$. An $n \times n$ matrix $A$ is **diagonalizable** when there are an invertible $n \times n$ matrix $P$ and a diagonal matrix $D$ with ### [Dot product, length and orthogonality in n dimensions](https://math.wiki/dot-product-and-length-in-rn/) **Summary:** Multiplying matching entries and adding gives the dot product, which measures length, angle and perpendicularity in any number of dimensions. **Question:** With no picture to measure, how long is a list of six numbers, and how alike are two such lists? ## Why this exists Dritan, Valbona and Ilir run a perfumer's workshop. They describe each scent by the intensities of six notes, citrus, rose, jasmine, cedar, vanilla and musk, so a scent profile is a vector in $\mathbb{R}^6$ (vector-addition-in-rn). How strong is a profile overall, and how alike are two profiles? In the plane and in space, geometry answered both first. Length came from the Pythagorean theorem, the angle between two arrows could be measured, and $\mathbf{u} \cdot \mathbf{v} = \lVert \mathbf{u} \rVert \lVert \mathbf{v} \rVert \cos\theta$ was proved from a drawn triangle (dot-product-as-magnitudes-and-cosine). In $\mathbb{R}^6$ there is nothing to draw. So the order turns around: define the dot product by arithmetic, define length from it, and prove the Cauchy-Schwarz inequality, which is what makes "the angle between two vectors" meaningful in any number of dimensions. ## Definition The **dot product** of two vectors in the same $\mathbb{R}^n$ multiplies matching entries and adds: $$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + \dots + u_n v_n.$$ The result is a number, not a vector. Vectors with different numbers of entries have no dot product. The **length**, or **norm**, of $\mathbf{v}$ is $$\lVert \mathbf{v} \rVert = \sqrt{\mathbf{v} \cdot \mathbf{v}} = \sqrt{v_1^2 + v_2^2 + \dots + v_n^2},$$ a square root of a sum of squares, so always defined. The **distance** between $\mathbf{u}$ and $\mathbf{v}$ is $\operatorname{dist}(\mathbf{u}, \mathbf{v}) = \lVert \mathbf{u} - \mathbf{v} \rVert$. ### [Echelon forms and Gaussian elimination](https://math.wiki/row-echelon-form/) **Summary:** A staircase shape where each row starts further right than the one above and zero rows sit at the bottom. **Question:** Row operations never change the solutions, but which shape should they aim for, so that the solution can simply be read off? ## Why this exists Row operations change an augmented matrix without changing the solutions of its system, but there are endless ways to apply them and most of them wander. What was missing is a target, a shape from which the solution can be read with almost no work, and a strategy that reaches it every time. The target is a staircase of zeros in the lower left: a last equation in one variable, the one above it in two, and so on. ## Definition In a nonzero row of a matrix, the **leading entry** is the first entry, reading from the left, that is not $0$. A row of all zeros has no leading entry. A matrix is in **row echelon form** (echelon form, for short) when - every row of all zeros sits below every nonzero row, and - each leading entry lies in a column strictly to the right of the leading entry of the row above it. It follows that every entry below a leading entry is $0$. The position of a leading entry is a **pivot**, and its column a pivot column. A matrix is in **reduced row echelon form** when, in addition, - every leading entry is $1$, and - each leading $1$ is the only nonzero entry in its column. For an augmented matrix, the whole row, constant included, is read from left to right. **Gaussian elimination** brings a matrix to echelon form: work through the columns from left to right, choose a pivot in each, and clear every entry below it with replacements $R_i + cR_j \to R_i$. **Back-substitution** then solves the system from the bottom row up. Going on to scale each pivot to $1$ and clear above it as well is **Gauss-Jordan elimination**, and it ends in the reduced form. ### [Eigenvalues and eigenvectors](https://math.wiki/eigenvalues-and-eigenvectors/) **Summary:** A vector that a transformation only stretches, together with the number by which it stretches. **Question:** A matrix turns almost every arrow it acts on. Are there directions it only stretches, and what do they reveal about the matrix? ## What this area is about For a square matrix $A$, an eigenvector is a nonzero vector $\mathbf{v}$ with $A\mathbf{v} = \lambda\mathbf{v}$: along it, $A$ does nothing but multiply by the number $\lambda$, the eigenvalue. Those directions and numbers describe what $A$ does far better than its entries, and they make repeated application of $A$ easy to predict. ## How these fit together **Procedures.** The characteristic equation finds the eigenvalues first, as the roots of $\det(A - \lambda I) = 0$, the numbers that make $A - \lambda I$ fail to be invertible. Finding an eigenvector puts each eigenvalue back and solves $(A - \lambda I)\mathbf{x} = \mathbf{0}$ for its eigenspace. **Results.** Diagonalization shows that an $n \times n$ matrix with $n$ linearly independent eigenvectors is $PDP^{-1}$ with $D$ diagonal. Complex eigenvalues covers a real matrix whose characteristic equation has non-real roots: for a $2 \times 2$ one, in suitable axes it rotates and scales. **Application.** Matrix powers and dynamical systems reads the long run of $\mathbf{x}_{k+1} = A\mathbf{x}_k$ off the eigenvalues, including the steady state of a Markov matrix. ## A path through it Start with the characteristic equation, then finding eigenvectors, then diagonalization, then dynamical systems. Complex eigenvalues branches off after the characteristic equation and also needs the polar form of a complex number; dynamical systems sends its spiral cases there. ### [Finding an eigenvector](https://math.wiki/find-an-eigenvector/) **Summary:** Putting an eigenvalue back in and solving the system for the vectors it leaves in place. **Question:** Once the stretch factors of a matrix are known, how do you find every direction that each one stretches, and how many independent directions does each factor bring? ## When you reach for it Javlon, Shoira and Kamila clamp a thin steel plate in a loading rig and mark arrows on it from a center point. Under the load, the point at $\mathbf{x}$ moves to $A\mathbf{x}$, a linear-transformation. Most arrows come out turned. They want the ones that come out only stretched, shrunk or reversed: along those directions the plate is purely stretched. The characteristic-equation has already given the stretch factors, the eigenvalues $\lambda$, as the roots of $\det(A - \lambda I) = 0$. A number is not a direction, though. This lesson takes each eigenvalue, finds **all** the vectors it stretches, describes them with a basis, and counts the independent directions each eigenvalue brings. That count decides whether there are enough pure-stretch directions to describe every arrow, the question diagonalization answers. Day 1 finds eigenspaces of $2 \times 2$ matrices and checks eigenvectors by one multiplication. Day 2 moves to $3 \times 3$ matrices, compares the dimension of an eigenspace with the multiplicity of its eigenvalue, and uses the independence of eigenvectors for different eigenvalues. An inline $(a, b)$ means the column vector $\begin{bmatrix} a \\ b \end{bmatrix}$. ## The procedure ### The eigenspace is a kernel A nonzero $\mathbf{v}$ is an eigenvector of the $n \times n$ matrix $A$ for $\lambda$ when $A\mathbf{v} = \lambda\mathbf{v}$. Since $\lambda\mathbf{v} = \lambda I\mathbf{v}$ and matrix products distribute over subtraction, ### [Finding an inverse by row reducing A next to I](https://math.wiki/inverse-by-row-reduction/) **Summary:** Row reduce A with the identity beside it; when A has become I, the identity has become the inverse of A. **Question:** The swap-and-divide formula only covers 2 by 2 matrices. How do you find the inverse of a larger square matrix, or find out that it has none, and why does the same row reduction answer both? ## When you reach for it A square matrix $A$ sends an input $\mathbf{x}$ to an output $A\mathbf{x}$, and its inverse $A^{-1}$ returns the input from the output. For a $2 \times 2$ matrix the swap, negate and divide formula gives $A^{-1}$ at a glance. From $3 \times 3$ on, the tool is row reduction. Ketsana runs a letterpress print shop. Every shade is mixed from three stock inks, each a fixed blend of three pigments. If $\mathbf{x}$ lists the spoons of each stock ink, then $A\mathbf{x}$ lists the grams of each pigment, where column $j$ of $A$ is the pigment content of one spoon of ink $j$. Mixing is multiplying by $A$. A customer brings a wanted shade $\mathbf{b}$; undoing the mix is multiplying by $A^{-1}$, and one inverse serves every customer. Reach for it when you need $A^{-1}$ itself, or to decide whether it exists. For one system, reducing $[A \mid \mathbf{b}]$ as in solve-a-system-by-row-reduction is less work. ## The procedure ### Reduce A with the identity beside it Set the $n \times n$ identity beside $A$ and carry $[A \mid I]$ toward reduced-row-echelon-form with the elementary-row-operations. Each operation is chosen by the left half and applied to the whole row. There are two possible endings. - The left half becomes $I$. Then the right half is $A^{-1}$: $$[A \mid I] \;\longrightarrow\; [I \mid A^{-1}]$$ - The left half produces a row of zeros (some column of $A$ gets no pivot). Then $A$ has no inverse, and you can stop. ### [Finding the matrix of a transformation](https://math.wiki/matrix-of-a-rotation/) **Summary:** Where a transformation sends each unit vector gives the columns of the matrix that performs it. **Question:** A printmaker turns or flips a design about a fixed pin. Where does every point of the design land, and which matrix does the moving? ## Why this exists Nenad cuts a woodcut block and pins it to the press bed through a hole at the origin. Between pulls he turns the block about the pin, or flips it face down across a guide line through the pin. Libuse plans each sheet on graph paper and wants, for each move, one rule saying where every point lands. linear-transformation proved that a linear rule $T: \mathbb{R}^n \to \mathbb{R}^m$ is $\mathbf{x} \mapsto A\mathbf{x}$, where column $j$ of $A$ is $T(\mathbf{e}_j)$. So Libuse only has to follow $\mathbf{e}_1$ and $\mathbf{e}_2$ through the move. This lesson does that geometry for rotations, reflections and projections that fix the origin, and for turns of space about a coordinate axis. ## Definition Let $L$ be a line through the origin with a unit direction vector $\mathbf{u}$, so $\lVert \mathbf{u} \rVert = 1$. - The **rotation by $\theta$**, written $R_\theta$, turns every vector counterclockwise about the origin through the angle $\theta$, keeping its length (center-and-angle-of-rotation, with the center at the origin). A negative $\theta$ turns clockwise. - The **projection onto $L$**, written $P$, sends $\mathbf{x}$ to the foot of the perpendicular from $\mathbf{x}$ to $L$, the point of $L$ closest to $\mathbf{x}$. - The **reflection across $L$**, written $F$, sends $\mathbf{x}$ to its mirror image: $L$ is the perpendicular bisector of the segment from $\mathbf{x}$ to $F(\mathbf{x})$, and $F(\mathbf{x}) = \mathbf{x}$ for $\mathbf{x}$ on $L$. ### [Homogeneous systems and solutions in parametric vector form](https://math.wiki/solution-sets-of-linear-systems/) **Summary:** The solutions of Ax = 0 form a span, and every solution of Ax = b is one particular solution plus a solution of Ax = 0. **Question:** A linear system has infinitely many solutions. How do you write the whole set down at once, and what shape does it have? ## When you reach for it solve-a-system-by-row-reduction ends with a solution set in coordinates, such as $(x, y, z) = (2 - t, 1 + 3t, t)$, one parameter per free variable. That is correct but hides the shape of the set, and says nothing about how it compares with the solutions for a different right side. This lesson writes the same answer as $$\mathbf{x} = \mathbf{p} + s\mathbf{v}_1 + t\mathbf{v}_2,$$ where $\mathbf{p}$ is one solution and $s\mathbf{v}_1 + t\mathbf{v}_2$ runs over every solution of the system with its right sides set to $0$. Reach for it to describe, compare or picture a solution set, to decide uniqueness without solving, and to understand problems with many acceptable answers, such as mixing materials to a target. The tool underneath is matrix-times-a-vector: $A\mathbf{x} = x_1\mathbf{a}_1 + \dots + x_n\mathbf{a}_n$, and $A(c\mathbf{u} + d\mathbf{v}) = cA\mathbf{u} + dA\mathbf{v}$. ## The procedure ### Day 1: homogeneous systems, and their solutions as a span A system is **homogeneous** when every right side is $0$: $A\mathbf{x} = \mathbf{0}$. Since $A\mathbf{0} = \mathbf{0}$, it is **always consistent**, with the **trivial solution** $\mathbf{x} = \mathbf{0}$. A solution $\mathbf{x} \neq \mathbf{0}$ is **nontrivial**, and the question is whether any exist. Row reduce $A$ alone: every row operation turns a column of zeros into a column of zeros, so the augmented column can be left off. ### [Image](https://math.wiki/image-of-a-matrix/) **Summary:** All the vectors a transformation can produce, which is the span of its columns. **Question:** A linear transformation sends every input somewhere. Which outputs actually occur, what shape does that set have, and when is every vector of the target space reached? ## Why this exists In a recording studio, Ekaterine runs a mixing desk with three input channels and three speakers: left, center and right. Channel $j$ with its fader at level $1$ sends to the speakers the levels in column $j$ of $$B = \begin{bmatrix} 1 & 2 & 0 \\ 0 & 1 & 1 \\ 1 & 3 & 1 \end{bmatrix},$$ row $1$ for the left speaker, row $2$ for the center, row $3$ for the right. Signals add, and a negative setting sends a channel with its polarity flipped, so a setting of the three faders is a vector $\mathbf{x}$ in $\mathbb{R}^3$ and the speaker levels are $B\mathbf{x}$. Irakli wants the same level in all three speakers. Before anyone turns a fader, the whole question can be asked at once: which speaker patterns can this desk produce at all? That set of outputs is the **image**. It is always a subspace, it can be read from the columns, and when it is all of $\mathbb{R}^3$ every request can be met. Day 1 is the definition, the proof that the image of a transformation is the span of its matrix's columns, and a basis for it. Day 2 is whether every output is reached, and images read geometrically. ## Definition Let $T: \mathbb{R}^n \to \mathbb{R}^m$ be a linear-transformation. The **image** of $T$ is the set of all its outputs: $$\operatorname{im} T = \{\, T(\mathbf{x}) : \mathbf{x} \in \mathbb{R}^n \,\}.$$ So $\mathbf{b}$ is in $\operatorname{im} T$ exactly when $T(\mathbf{x}) = \mathbf{b}$ has at least one solution. The codomain $\mathbb{R}^m$ is where outputs may land; the image is where they do land. Many books say **range**, as for the functions of domain-and-range. ### [Inverse of a 2 by 2 matrix, and solving with it](https://math.wiki/inverse-2x2/) **Summary:** Swap, negate, divide by the determinant. **Question:** Dividing by a number undoes multiplying by it. Matrices have no division, so is there a matrix that undoes multiplying by a given one, and how do you find it? ## When you reach for it For numbers, the equation $3x = 12$ is solved by multiplying both sides by $\frac{1}{3}$, the number that undoes multiplying by $3$, because $\frac{1}{3} \cdot 3 = 1$. A linear system written as a matrix equation $AX = B$ looks just as simple, but there is no dividing by a matrix. What can exist is a matrix that undoes $A$: its **inverse**, $A^{-1}$. You reach for the inverse when a square system $AX = B$ has to be solved, especially when the same coefficient matrix $A$ meets many right sides $B$, as in a code where every block of a message is multiplied by one key matrix. For a $2 \times 2$ matrix the inverse comes from the determinant in one line; for a larger one it comes from row reduction. The lesson runs over two days. Day 1 is the inverse itself: what it is, the $2 \times 2$ formula and why it works, and when there is none. Day 2 puts it to work: solving $AX = B$, a $3 \times 3$ inverse by row reduction, and the inverse of a product. ## The procedure ### Day 1: what an inverse is The **identity matrix** $I$ of size $n$ has $1$s on the main diagonal and $0$s elsewhere, as in $I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$. By the row-by-column rule, $AI = IA = A$ for every $n \times n$ matrix $A$, so $I$ plays the part of the number $1$. A square matrix $A$ is **invertible** when there is a matrix $A^{-1}$ of the same size with $$AA^{-1} = A^{-1}A = I$$ ### [Kernel](https://math.wiki/kernel-of-a-matrix/) **Summary:** All the vectors that a transformation sends to zero. **Question:** Some inputs to a linear transformation come out as the zero vector. Which ones, what shape does that set have, and what does it say about whether the transformation can tell its inputs apart? ## Why this exists At a village market, goods change hands by barter. Wara, Asiri and Achik keep a tally of how many baskets of goods each of them holds, and three standing deals run between them and the farms of the valley. Doing deal $j$ once changes the three tallies by the numbers in column $j$ of $$A = \begin{bmatrix} 1 & -1 & 2 \\ 2 & 1 & 1 \\ 1 & 2 & -1 \end{bmatrix},$$ row $1$ for Wara, row $2$ for Asiri, row $3$ for Achik. A **bundle** of trades says how many times each deal is done, a negative count meaning the deal runs in reverse, so a bundle is a vector $\mathbf{x}$ in $\mathbb{R}^3$ and its effect on the tallies is $A\mathbf{x}$. Some bundles leave every tally unchanged: goods move, and nobody ends up with more or fewer baskets. They also answer a second question, when two different bundles have the same effect: exactly when their difference is a bundle of the first kind. The set of inputs a linear transformation sends to zero is its **kernel**, and it measures what the transformation throws away. Day 1 is the definition, the proof that a kernel is a subspace, and how to find a basis for it. Day 2 is what the kernel says about the transformation: when it is one-to-one, how the kernel looks geometrically, and why a matrix with more columns than rows always has a nonzero kernel vector. ## Definition Let $T: \mathbb{R}^n \to \mathbb{R}^m$ be a linear-transformation. The **kernel** of $T$ is the set of all inputs that $T$ sends to the zero vector: ### [Least squares: the best approximate solution of Ax = b](https://math.wiki/least-squares-solutions/) **Summary:** When Ax = b has no solution, the normal equations give the x that brings Ax as close to b as possible. **Question:** A system Ax = b has no solution, because the data that built it carry measurement error. Which x comes closest, and how do you find it without trying every x? ## When you reach for it Corsin, Seraina and Flurin run an alpine cheese cellar. A wheel loses water as it ages, so it loses weight, and once a week they weigh one wheel and record $y$, the weight it has lost since it left the brine bath, in hundreds of grams. At weeks $t = 0, 1, 2, 3$ they record $y = 1, 3, 4, 4$. They want a trend line $y = c_0 + c_1 t$. Asking the line to pass through every point gives one equation per weighing: $$\begin{aligned} c_0 + 0c_1 &= 1 \\ c_0 + 1c_1 &= 3 \\ c_0 + 2c_1 &= 4 \\ c_0 + 3c_1 &= 4 \end{aligned}$$ or $A\mathbf{x} = \mathbf{b}$ with $$A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ 1 & 2 \\ 1 & 3 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} c_0 \\ c_1 \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 1 \\ 3 \\ 4 \\ 4 \end{bmatrix}.$$ The first two equations force $c_0 = 1$ and $c_1 = 2$, and then the third says $5 = 4$. The system is inconsistent (solution-sets-of-linear-systems), as four measured points with real error almost always are. Row reduction can only report "no solution". The useful question is which $\mathbf{x}$ makes $A\mathbf{x}$ as close to $\mathbf{b}$ as possible. You reach for least squares whenever $A\mathbf{x} = \mathbf{b}$ has no solution, typically more equations than unknowns built from measurements, and an approximate one is wanted. ## The procedure Let $A$ be an $m \times n$ matrix and $\mathbf{b}$ a vector in $\mathbb{R}^m$. A **least-squares solution** of $A\mathbf{x} = \mathbf{b}$ is a vector $\hat{\mathbf{x}}$ in $\mathbb{R}^n$ with ### [Linear algebra](https://math.wiki/linear-algebra/) **Summary:** Vectors, matrices and the linear transformations they describe: solving many equations at once, and what a matrix does to space. **Question:** Many equations share the same unknowns, and many quantities change together. Is there one language for solving them all at once and for seeing what a matrix does to space? ## What this area is about Linear algebra studies vectors, matrices and the maps between them that respect adding and scaling. It solves large linear systems by routine, and reads a matrix as a motion of space: what it reaches, what it crushes, and which directions it only stretches. ## How these fit together **Tools.** Matrices give the arithmetic and the determinant; row reduction solves any linear system; vectors in n dimensions bring span, independence and dimension. **The geometric reading.** Matrices as transformations covers image, kernel and invertibility. Orthogonality and least squares finds closest points and best approximate solutions. Eigenvalues and eigenvectors finds the directions a square matrix only stretches. **Proof-based courses, not yet written.** Abstract vector spaces: vector spaces from axioms. Linear maps: maps between vector spaces and their matrices. Eigenvalues, proved: invariant subspaces and diagonalizability. Inner product spaces: length and angle in any vector space. Structure of operators: generalized eigenvectors and Jordan form. ## A path through it Matrices, row reduction and matrices as transformations lean on one another, so they are taken together, matrices first. Vectors in $\mathbb{R}^n$ follow matrix addition and come before transformations. Orthogonality and eigenvalues come after; eigenvalues needs transformations. The proof-based courses follow in order: abstract vector spaces, linear maps, eigenvalues proved, then inner product spaces and structure of operators. ### [Linear combination](https://math.wiki/linear-combination-of-vectors/) **Summary:** A sum of scaled vectors, which is the single move the whole of linear algebra is built on. **Question:** A few vectors are given. Which new vectors can be built from them by scaling and adding, and how do you find the amounts? ## Why this exists Arnav, Ishani and Saanvi mix paint from three base paints. Each liter of a base carries a fixed amount of each pigment, so a base is a vector: $(3, 1, 0)$ means $3$ grams of red, $1$ of yellow and none of white per liter. A batch of $c_1$ liters of base $\mathbf{s}_1$, $c_2$ of $\mathbf{s}_2$ and $c_3$ of $\mathbf{s}_3$ holds $$c_1\mathbf{s}_1 + c_2\mathbf{s}_2 + c_3\mathbf{s}_3$$ grams of pigment: the adding and scaling of vector-addition-in-rn, used at once. Forward, finding the batch from the liters is arithmetic. Backward, finding the liters that give the pigment a color needs, if any exist, means solving equations whose unknowns are the amounts. That backward question is the one linear algebra asks more than any other. ## Definition Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $\mathbb{R}^n$, all with the same $n$, and let $c_1, c_2, \dots, c_k$ be scalars. The vector $$c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \dots + c_k\mathbf{v}_k$$ is the **linear combination** of $\mathbf{v}_1, \dots, \mathbf{v}_k$ with **weights** $c_1, \dots, c_k$. A vector $\mathbf{b}$ **is a linear combination of** $\mathbf{v}_1, \dots, \mathbf{v}_k$ when some choice of weights makes the combination equal to $\mathbf{b}$. The weights can be any real numbers, including negatives, fractions and zero; the list is finite; the result is in $\mathbb{R}^n$. A multiple $c\mathbf{v}$, a sum $\mathbf{u} + \mathbf{v}$ (weights $1, 1$) and a difference $\mathbf{u} - \mathbf{v}$ (weights $1, -1$) are all combinations. Weights all zero give $\mathbf{0}$, so $\mathbf{0}$ is a combination of any list. In the plane, $3\mathbf{i} - 5\mathbf{j}$ (i-and-j-notation) is a combination of $\mathbf{i}$ and $\mathbf{j}$. ### [Linear independence](https://math.wiki/linear-independence/) **Summary:** A set is independent when no vector in it can be built from the others. **Question:** Given a list of vectors, can any of them be built from the others, and how can you tell without testing each one in turn? ## Why this exists Rokeya, Nusrat and Imtiaz sell spice blends. A blend is a vector: $(3, 1, 0, 0)$ means $3$ grams of cumin, $1$ of coriander and none of turmeric or chili per scoop, and mixing scoops forms a linear combination. A blend is **redundant** when it can be mixed from the others. Keeping it adds nothing new to what the shelf can make, and it makes recipes ambiguous: the same mixture can then be reached in more than one way. Checking each blend against the rest means one system per blend. This lesson replaces them with a single system whose right side is $\mathbf{0}$, and shows that its answer also decides whether recipes are unique. ## Definition Let $\mathbf{v}_1, \dots, \mathbf{v}_k$ be vectors in $\mathbb{R}^n$. Weights all zero always give $0\mathbf{v}_1 + \dots + 0\mathbf{v}_k = \mathbf{0}$, the **trivial combination**. The list is **linearly independent** when that is the only way to reach $\mathbf{0}$: $$c_1\mathbf{v}_1 + \dots + c_k\mathbf{v}_k = \mathbf{0} \ \text{ only when } \ c_1 = \dots = c_k = 0.$$ The list is **linearly dependent** when some weights, not all zero, give $\mathbf{0}$; such an equation is a **dependence relation**. Independence belongs to the whole list, not to one vector in it. A list with a vector repeated is dependent, since $\mathbf{v} - \mathbf{v} = \mathbf{0}$, which is why this lesson says "list"; "set" is the common word when the vectors are distinct. ### [Linear systems in use: networks, chemical equations, fitting curves](https://math.wiki/applications-of-linear-systems/) **Summary:** Balancing a chemical equation, the flows through a network and a polynomial through given points each become one linear system. **Question:** Traffic around a block, atoms in a reaction, a curve through measured points. Why does each one come down to a single linear system, and what does the solution set say about it? ## When you reach for it Reach for a linear system whenever a situation is described by **quantities that must add up**: cars into a corner equal cars out, atoms of each element before a reaction equal atoms after, and the height of a curve at each measured point equals the measurement. Each such requirement is one linear equation, and the whole situation is one system $A\mathbf{x} = \mathbf{b}$. The work is in setting it up. Solving is already known: row reduce and write the answer as in solution-sets-of-linear-systems, one particular solution plus the homogeneous ones, $\mathbf{x} = \mathbf{p} + \mathbf{v}_h$. What is new is reading that answer back: a free variable is a real freedom in the network, the null space of a reaction is its set of balanced forms, and a unique solution is a curve the points pin down. ## The procedure ### Day 1: networks and chemical equations **Networks.** A **network** is a set of junctions (nodes) joined by branches, with something flowing along each branch: cars, water, current. Give each branch an unknown flow $x_1, x_2, \dots$ and a direction. The one law is **conservation at each node**: nothing is stored or created there, so $$\text{flow in} = \text{flow out}.$$ That is one linear equation per node, with the flows that enter or leave the network from outside as constants. Two features of the answer recur. First, in a connected network every branch leaves one node and enters another, so adding all the node equations cancels every unknown and leaves "total entering the network = total leaving it". One node equation is therefore always a combination of the others: if the totals agree it reduces to $0 = 0$, and if they do not, the system is inconsistent. Second, flow that runs around a closed loop enters and leaves every node equally, so it is invisible to the equations. Each independent loop shows up as a free variable, and the homogeneous solutions are exactly these circulations. ### [Linear transformation](https://math.wiki/linear-transformation/) **Summary:** A function on vectors that respects addition and scaling, which is exactly what a matrix does. **Question:** A rule moves every vector to a new vector. When can that rule be carried out by a single matrix, and how do you find the matrix? ## Why this exists Leyli is a type designer. Her letters are drawn as points and strokes on a grid, and she wants an italic version of a whole alphabet: every letter slanted the same way. Moving each point by hand would be endless, so she wants one rule that moves every point of the plane at once. Her slant moves $(x, y)$ to $(x + y/4, y)$: the higher a point sits, the farther right it slides, a quarter unit for each unit of height, and the baseline stays put. A rule like that is a function whose inputs and outputs are vectors. Most such functions are useless to a designer, because they bend straight strokes into curves. The ones that keep the drawing intact, straight strokes to straight strokes, parallel strokes to parallel strokes, the midpoint of a stroke to the midpoint of its image, are the ones that respect the two operations on vectors: adding and scaling. Those are the **linear transformations**. They are worth naming because, as this lesson proves, they are exactly the functions a matrix can carry out, so every question about them becomes a question about a matrix. This lesson runs over two days. Day 1 is the definition and how to test a rule against it. Day 2 is the standard matrix: every linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ is multiplication by one matrix, and its columns are where the standard vectors go. ## Definition A function $T: \mathbb{R}^n \to \mathbb{R}^m$ assigns to each vector $\mathbf{x}$ in $\mathbb{R}^n$ exactly one vector $T(\mathbf{x})$ in $\mathbb{R}^m$, called the **image** of $\mathbf{x}$. $\mathbb{R}^n$ is the **domain** and $\mathbb{R}^m$ the **codomain**. The two may be different spaces. ### [Matrices](https://math.wiki/matrices/) **Summary:** Rectangular arrays of numbers, with their own arithmetic. **Question:** A system of equations is mostly repeated letters around a block of numbers. What can be done with the block alone? ## What this area is about A matrix is a rectangle of numbers with its own arithmetic. It packs a whole linear system into one equation $AX = B$, and for a square $A$ one number, the determinant, says whether the system has exactly one solution. ## How these fit together **Definitions.** Rows, columns and dimensions names sizes and entries; the transpose swaps rows with columns. **Arithmetic.** Adding and scaling works entry by entry on matrices of one size. Matrix multiplication pairs rows with columns, so $AB$ needs as many columns in $A$ as rows in $B$. Writing a system as a matrix equation puts it to work. **Determinants.** Computing a determinant and cofactor expansion are procedures; what the determinant means (area or volume scale) and its properties (row operations, $\det(AB) = \det A \det B$) are results. **Solving.** A square $A$ is invertible exactly when $\det A \ne 0$. The 2 by 2 inverse, row reducing A next to I and Cramer's rule then solve $AX = B$. ## A path through it Vocabulary first. Then adding, multiplying, and the transpose or the matrix equation. Separately, compute a determinant. The $2 \times 2$ inverse needs both strands; Cramer's rule and the row-reduction inverse follow it, the second also needing row reduction. The determinant's meaning (which needs linear transformations) comes before cofactor expansion, then properties, which need row operations. Next: matrices as transformations and row reduction. ### [Matrices as linear transformations](https://math.wiki/matrices-as-transformations/) **Summary:** A matrix moves every vector in the plane; columns show where the basis goes. **Question:** A matrix can be read as a machine that moves every vector at once. What does the machine do to space, and when can it be run backward? ## What this area is about An $m \times n$ matrix $A$ defines a function $\mathbf{x} \mapsto A\mathbf{x}$ from $\mathbb{R}^n$ to $\mathbb{R}^m$, and every linear transformation between these spaces is one of these. This area reads matrices as such maps: what they reach, what they crush to $\mathbf{0}$, and when they can be undone. ## How these fit together **Definitions.** Linear transformations are the maps that respect addition and scaling. The image is the set of outputs, the span of the columns; the kernel is the set of inputs sent to $\mathbf{0}$. **Procedures.** Multiplying a matrix by a vector reads $A\mathbf{x}$ as a combination of the columns. Finding the matrix of a transformation builds $A$ from where each $\mathbf{e}_j$ lands; composing transformations multiplies their matrices. **Results.** Rank and nullity: for an $m \times n$ matrix, the dimensions of image and kernel add to $n$. Invertible transformations need a square matrix whose kernel is $\{\mathbf{0}\}$, equivalently $\det A \ne 0$. The invertible matrix theorem gathers a dozen equivalent tests for a square matrix. ## A path through it Matrix times a vector, then linear transformations. After that, finding a transformation's matrix, composition, the image and the kernel. Rank and nullity needs both image and kernel; invertible transformations needs the kernel. The invertible matrix theorem comes last. Next come eigenvalues and eigenvectors, the directions a transformation only stretches. ### [Matrix multiplication](https://math.wiki/matrix-multiplication-rule/) **Summary:** Each entry is a row of the first dotted with a column of the second. **Question:** One table says how many of each flower goes into each arrangement, and another says what each flower costs at each supplier. How do you combine the two into one table of totals, and what kind of multiplication does that turn out to be? ## Why this exists Reiko runs a flower stall and makes two table centerpieces. A small one takes $3$ tulips, $5$ lilies and $2$ carnations; a large one takes $7$ tulips, $9$ lilies and $3$ carnations. Two suppliers charge different prices per stem. To cost a small centerpiece from the first supplier, multiply each stem count by that supplier's price for the flower, and add: $$3(1.20) + 5(2.50) + 2(0.90) = 17.90$$ That is one of four totals, all of the same shape: walk along a recipe, walk down a price list, multiply matching items, add. Matrix multiplication does this for every recipe and price list at once. It is not entry by entry, as adding matrices is: it pairs a **row** of the first table with a **column** of the second. ## Definition In plain words: the entry in row $i$, column $j$ of the product $AB$ comes from row $i$ of $A$ and column $j$ of $B$. Multiply first entry by first entry, second by second, and so on to the end, and add the products. In symbols, let $A = [a_{ij}]$ be $m \times n$ and $B = [b_{ij}]$ be $n \times p$ (sizes and entry names as in matrix-vocabulary). The **product** $AB$ is the $m \times p$ matrix with entries $$\begin{aligned} (AB)_{ij} &= a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{in}b_{nj} \\ &= \sum_{k=1}^{n} a_{ik}\,b_{kj} \end{aligned}$$ **When the product exists.** $AB$ is defined only when the number of columns of $A$ equals the number of rows of $B$. Written side by side, the inner sizes must match, and the outer sizes give the size of the answer: ### [Matrix powers, dynamical systems and steady states](https://math.wiki/dynamical-systems-and-steady-states/) **Summary:** Applying a matrix again and again is simple in a basis of eigenvectors, which shows where a system settles in the long run. **Question:** A matrix carries this year's state of a system to next year's. Where is the system after fifty years, and does it settle down, die out or blow up? ## Why this exists Two mountain lakes are joined by a stream. Each year Joost, who studies the trout, finds that $75\%$ of the trout in the upper lake stay there and $25\%$ swim down, while $80\%$ of those in the lower lake stay and $20\%$ swim up. Writing $u_k$ and $\ell_k$ for the counts after $k$ years, $$\begin{bmatrix} u_{k+1} \\ \ell_{k+1} \end{bmatrix} = \begin{bmatrix} 0.75 & 0.2 \\ 0.25 & 0.8 \end{bmatrix}\begin{bmatrix} u_k \\ \ell_k \end{bmatrix}.$$ Next year is one matrix product away, and year fifty is fifty products away: $\mathbf{x}_{50} = A^{50}\mathbf{x}_0$. Multiplying fifty times answers one question for one start, and says nothing about why the counts behave as they do. diagonalization made $A^k$ easy to write. This lesson reads the long run straight off the eigenvalues: which part of the state survives, where it settles, and how fast. ## Definition A **discrete dynamical system** is a rule $\mathbf{x}_{k+1} = A\mathbf{x}_k$ with $A$ an $n \times n$ matrix. The vector $\mathbf{x}_k$ is the **state** at step $k$, and the sequence $\mathbf{x}_0, \mathbf{x}_1, \mathbf{x}_2, \dots$ is the **trajectory** from $\mathbf{x}_0$. Since each step multiplies by $A$, $\mathbf{x}_k = A^k\mathbf{x}_0$. A **fixed point** is a state with $A\mathbf{x} = \mathbf{x}$: the vector $\mathbf{0}$, or an eigenvector for $\lambda = 1$. An eigenvalue $\lambda_1$ is **dominant** when $|\lambda_1| > |\lambda_i|$ for every other eigenvalue $\lambda_i$. ### [Multiplying a matrix by a vector](https://math.wiki/matrix-times-a-vector/) **Summary:** The answer is a linear combination of the columns, with the vector entries as the weights. **Question:** A matrix sits next to a vector. What single vector does their product stand for, and what does it tell you about which right sides an equation can reach? ## When you reach for it Three earlier lessons meet here. linear-combination-of-vectors scales vectors and adds them, and asks which weights build a given vector. matrix-multiplication-rule gives each entry of a product as a row of the first factor times a column of the second. system-as-a-matrix-equation packs a system into one equation $A\mathbf{x} = \mathbf{b}$. A matrix times a vector can be read row by row, as the multiplication rule computes it, or column by column, as a combination of the columns of $A$ with the entries of $\mathbf{x}$ as weights. The two readings always agree, and the column reading is the one that explains things: it is why $A(\mathbf{u} + \mathbf{v}) = A\mathbf{u} + A\mathbf{v}$, and why $A\mathbf{x} = \mathbf{b}$ has a solution exactly when $\mathbf{b}$ is built from the columns. Reach for it whenever fixed ingredients are mixed in chosen amounts. ## The procedure ### Two readings of one product Let $A$ be an $m \times n$ matrix with columns $\mathbf{a}_1, \dots, \mathbf{a}_n$, each in $\mathbb{R}^m$, and let $\mathbf{x} = (x_1, \dots, x_n)$ be a column vector in $\mathbb{R}^n$. Then $$A\mathbf{x} = x_1\mathbf{a}_1 + x_2\mathbf{a}_2 + \dots + x_n\mathbf{a}_n.$$ This is the **column reading**: entry $j$ of $\mathbf{x}$ is the weight on column $j$. It needs one weight per column, so $\mathbf{x}$ must have $n$ entries, and the result lives where the columns live, in $\mathbb{R}^m$. ### [Orthogonal bases and projection onto a subspace](https://math.wiki/orthogonal-projection/) **Summary:** With an orthogonal basis, the point of a subspace closest to a vector is a sum of simple projections onto the basis vectors. **Question:** A plane through the origin and a point off it. Which point of the plane is closest, and is there a formula for it that needs no calculus? ## Why this exists Pirjo, Tuula and Kaija are surveying a climbing crag. In a frame with the base station at the origin, three perpendicular axes (the third pointing up) and coordinates in tens of meters, they model one steep rock face as the plane $W = \operatorname{span}\{\mathbf{u}_1, \mathbf{u}_2\}$ with $\mathbf{u}_1 = (1, 2, 2)$ and $\mathbf{u}_2 = (2, 1, -2)$. A survey drone hovers at $\mathbf{y} = (6, 3, 3)$. Where on the face is the drone closest, and how far away is it? For a line through the origin the answer is known: drop a perpendicular, and the foot is $\frac{\mathbf{y} \cdot \mathbf{u}}{\mathbf{u} \cdot \mathbf{u}}\mathbf{u}$ (scalar-projection). For a plane, or a subspace of any dimension, minimizing a distance over every point sounds like calculus, but it is not needed. If the basis of $W$ is **orthogonal**, the closest point is the sum of the projections onto the basis vectors, one at a time. ## Definition Throughout, $W$ is a subspace of $\mathbb{R}^n$ (subspaces-of-rn) and the dot product, length and orthogonality are those of dot-product-and-length-in-rn. A set of vectors $\{\mathbf{u}_1, \dots, \mathbf{u}_k\}$ is an **orthogonal set** when every pair of different vectors in it is orthogonal: $\mathbf{u}_i \cdot \mathbf{u}_j = 0$ whenever $i \neq j$. An **orthogonal basis** of $W$ is a basis of $W$ that is also an orthogonal set. An **orthonormal set** is an orthogonal set of unit vectors, and an **orthonormal basis** is a basis that is an orthonormal set. ### [Orthogonality and least squares](https://math.wiki/orthogonality-and-least-squares/) **Summary:** Perpendicular vectors, projection onto a subspace, and the best approximate solution of a system that has no exact one. **Question:** A system built from measured data has no exact solution. Which answer comes closest, and how does perpendicularity find it? ## What this area is about Perpendicular vectors, those with dot product $0$, make distances easy. In $\mathbb{R}^n$, the point of a subspace $W$ closest to a vector $\mathbf{y}$ is the one whose difference from $\mathbf{y}$ is perpendicular to $W$. That one fact turns an inconsistent system $A\mathbf{x} = \mathbf{b}$, with no exact solution, into a best approximate one. ## How these fit together **Result.** Orthogonal bases and projection onto a subspace gives the closest point: with an orthogonal basis of $W$, it is a sum of projections onto the basis vectors, one at a time. **Procedure.** The Gram-Schmidt process turns any basis of a subspace into an orthogonal one, so that projection formula always applies. **Application.** Least squares takes the $\hat{\mathbf{x}}$ that brings $A\hat{\mathbf{x}}$ as close to $\mathbf{b}$ as possible, found from the normal equations $A^T A\hat{\mathbf{x}} = A^T\mathbf{b}$; when the columns of $A$ are linearly independent, $\hat{\mathbf{x}}$ is unique. Fitting a trend line to data is the standard use. ## A path through it Projection first, since both other lessons use it. Then Gram-Schmidt, which least squares relies on to know an orthogonal basis of the column space exists. Least squares comes last. The area assumes dot products and subspaces of $\mathbb{R}^n$, and the transpose from matrices. Later, inner product spaces take length and angle to any vector space. ### [Properties of determinants: row operations, products, transposes](https://math.wiki/properties-of-determinants/) **Summary:** A row swap flips the sign, scaling a row scales the determinant, and the determinant of a product is the product of the determinants. **Question:** Cofactor expansion of a large determinant takes longer than anyone can wait. Row reduction is fast, but each row operation changes the matrix. What does each one do to the determinant, and what else follows? ## Why this exists A full cofactor-expansion has $n!$ products: $24$ for $n = 4$, about $3.6$ million for $n = 10$. Row reduction (elementary-row-operations) reaches a triangular matrix, whose determinant is the product of its diagonal, in about $n^3/3$ multiplications. That works only if we know what each row operation does to the determinant. The same knowledge settles two open questions: is $\det A = 0$ exactly when $A$ has no inverse, and is $\det(AB) = \det A \det B$ for every size? ## Definition $\det A$ is the number defined by cofactor expansion, which by the theorem of cofactor-expansion can run along any row or any column. The three row operations are a **swap** ($R_i \leftrightarrow R_j$), a **scaling** ($R_i \to cR_i$ with $c \neq 0$), and a **replacement** ($R_j \to R_j + kR_i$, $i \neq j$). An **elementary matrix** $E$ is $I$ with one row operation done to it, and doing that operation to $A$ gives $EA$ (inverse-by-row-reduction). A **column operation** is the same three moves applied to columns. ## Intuition For a $2 \times 2$ matrix, $ad - bc$ is the signed area of the parallelogram on its rows, as determinant-meaning reads it on the columns. Swapping the rows flips orientation and sign. Scaling one row by $c$ stretches one side by $c$, so the area scales by $c$. Adding a multiple of row $1$ to row $2$ slides the far side along a line parallel to row $1$: same base, same height, same area. ### [Rank](https://math.wiki/rank-of-a-matrix/) **Summary:** The number of leading entries left after reduction, which says how many rows carry new information. **Question:** A matrix can have many rows and columns and still say very little. How many of its rows carry new information, and why does counting its columns give the same number? ## Why this exists Sokha, Vanna and Phalla run a coffee roastery. Each blend is a row of grams of four beans: Brazilian, Colombian, Ethiopian and Sumatran. Sokha's notebook holds four blends, $$A = \begin{bmatrix} 1 & 3 & 0 & 2 \\ 2 & 6 & 1 & 5 \\ 3 & 9 & 1 & 7 \\ 0 & 0 & 3 & 3 \end{bmatrix}.$$ Blend $3$ is blends $1$ and $2$ poured together. Blend $4$ is $3(\text{blend } 2) - 6(\text{blend } 1)$: no one weighs out a negative amount, but its makeup is still fixed by the first two. Four rows, two pieces of information. That count is the **rank**, and it says, before any solving, whether a system can be solved for every right-hand side and whether a solution can be unique. ## Definition The **rank** of a matrix $A$, written $\operatorname{rank} A$, is the number of pivot positions of $A$: the number of leading $1$s in its reduced row echelon form. Every echelon form of $A$ has its leading entries at those positions, so the rank is also the number of nonzero rows in any echelon form of $A$, however it was reached. Two subspaces come with an $m \times n$ matrix $A$ (subspaces-of-rn): - the **row space** $\operatorname{Row} A$, the span of the rows, in $\mathbb{R}^n$; - the **column space** $\operatorname{Col} A$, the span of the columns, in $\mathbb{R}^m$. The **transpose** $A^{T}$ is the $n \times m$ matrix whose rows are the columns of $A$, so $\operatorname{Row} A^{T} = \operatorname{Col} A$ (transpose-and-symmetric-matrices). ### [Rank and nullity](https://math.wiki/rank-and-nullity/) **Summary:** The dimensions of the image and of the kernel always add up to the number of columns. **Question:** A linear transformation crushes some input directions to zero and carries the rest into its outputs. Is there a fixed budget shared between what is lost and what survives? ## Why this exists A city bike-share system has three stations, North, Market and River, and a depot. Each night a rebalancing truck can run six standard moves, each dropping bikes from the depot at some stations. Siku, who runs the night shift, keeps a count sheet with four lines: the change at North, at Market, at River, and the total change. Running move $j$ once changes the four lines by column $j$ of $$A = \begin{bmatrix} 1 & 2 & 0 & 1 & 3 & 0 \\ 0 & 0 & 1 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 2 & 1 & 3 & 4 & 1 \end{bmatrix}.$$ A **plan** says how many times each move is run, a negative count meaning the move is run in reverse, collecting those bikes. So a plan is a vector $\mathbf{x}$ in $\mathbb{R}^6$ and its effect is $A\mathbf{x}$. Which effects can a plan produce? That is the image of $A$, in $\mathbb{R}^4$. Which plans change nothing? That is the kernel, in $\mathbb{R}^6$. Different spaces, yet their dimensions are tied: they always add up to the number of columns. Day 1 is the theorem and its proof; Day 2, its consequences and the row space. ## Definition Let $A$ be an $m \times n$ matrix. Its **rank** is the dimension of its image, which is its column space, and by rank-of-a-matrix it is also the number of pivot columns. Its **nullity** is the dimension of its kernel: $$\operatorname{rank} A = \dim \operatorname{Col} A, \qquad \operatorname{nullity} A = \dim \ker A.$$ ### [Reduced row echelon form](https://math.wiki/reduced-row-echelon-form/) **Summary:** The staircase taken further, so each leading entry is 1 and is the only non-zero entry in its column. **Question:** Row reduction can be steered many ways and works on a matrix of any shape. Does every route end at the same matrix, and what does that final matrix say about the columns we started with? ## Why this exists Echelon forms solved square systems with one solution, and the reduced form appeared there as the shape the solution is read from. Two things were left open: most matrices in linear algebra are not square, and the claim that every route of row operations ends at the same reduced form was stated, not proved. One fact settles both. Row operations change every column of a matrix but keep every linear combination that ties the columns together, and the reduced form writes those ties down in the plainest way, so it belongs to the matrix. Day 1 reduces matrices of any shape and proves the reduced form unique. Day 2 reads it: consistency, uniqueness, redundant vectors, and several systems at once. ## Definition Recall from row-echelon-form: a matrix is in **reduced row echelon form** when it is in echelon form, every leading entry is $1$, and each leading $1$ is the only nonzero entry in its column. Two matrices are **row equivalent**, $A \sim B$, when row operations turn one into the other; every row operation can be undone (elementary-row-operations), so this runs both ways. **Gauss-Jordan elimination on an $m \times n$ matrix.** Keep a current row, starting at row $1$, and visit the columns from left to right. If a column is $0$ in the current row and every row below, it gets no pivot and the current row stays put. Otherwise swap a row with a nonzero entry there into the current row, scale that entry to $1$, clear every other entry of the column, above and below, with replacements $R_i + cR_j \to R_i$, and move the current row down one. Stop when the rows or the columns run out. A finished column is never disturbed, since every row used later is $0$ in all columns already visited, so the result is in reduced form. ### [Row reduction](https://math.wiki/row-reduction/) **Summary:** Systematic elimination on a matrix to solve any linear system. **Question:** Elimination works on two equations by insight. Is there one routine that settles any linear system, of any size, every time? ## What this area is about Row reduction is elimination made mechanical. Write a linear system as an augmented matrix, apply moves that never change its solutions, and aim for a staircase shape. The last rows then say whether the system has one solution, none (it is inconsistent), or infinitely many, and give them all. ## How these fit together **Definitions and moves.** Augmented matrices and row operations sets up the matrix and the three moves: swap two rows, scale a row by a nonzero number, add a multiple of one row to another. **Procedures.** Echelon forms and Gaussian elimination gives the staircase target and a strategy that reaches it. Solving a system by row reduction reads the three cases off the final rows. **Results.** Reduced row echelon form is unique for each matrix, whatever route reaches it, so the rank, defined as its number of pivots, is a fixed number for each matrix. Solution sets write every solution of a consistent $A\mathbf{x} = \mathbf{b}$ as one particular solution plus the solutions of $A\mathbf{x} = \mathbf{0}$. **Application.** Linear systems in use sets up networks, chemical equations and curve fits as systems. ## A path through it Row operations, then echelon forms. From there two strands run, and they refer to each other: reduced row echelon form, then rank; and solving by row reduction, then solution sets, then applications. Row reduction finds inverses (row reducing A next to I) and kernels (kernel), and rank feeds rank and nullity. ### [Rows, columns and dimensions](https://math.wiki/matrix-vocabulary/) **Summary:** Naming a matrix by its size, and naming an entry by its address. **Question:** You have a rectangle of numbers with no labels on it. How do you say how big it is, and point to exactly one number inside it? ## Why this exists A system of equations such as $$\begin{aligned} 2x + 3y &= 7 \\ x - 4y &= 5 \end{aligned}$$ carries its information in six numbers. The letters $x$ and $y$, the plus signs and the equals signs only say where each number sits. Strip them away, keep the positions, and what is left is a rectangle of numbers. The same rectangle turns up whenever data comes in two directions at once: items down the side and days across the top, or cities down the side and cities across the top. A rectangle of numbers is only useful if you can talk about it precisely. You need to say how big it is, and you need to point at one number in it without drawing an arrow. This lesson sets up exactly that language: a name for the shape of the whole block, and an address for every position inside it. Every later piece of matrix arithmetic, from adding matrices to multiplying them, is written in these words. ## Definition A **matrix** is a rectangular array of numbers, arranged in rows and columns. A **row** is a line of numbers running across, left to right. A **column** is a line of numbers running down, top to bottom. Each number in the array is an **entry** (some books say *element*) of the matrix. The plural is *matrices*. A matrix is enclosed in square brackets, or in some books parentheses, and is named with a capital letter. $$A = \begin{bmatrix} 5 & -2 & 0 & 8 \\ 1 & 4 & 7 & -3 \\ 6 & 9 & -1 & 2 \end{bmatrix}$$ ### [Solving a system by row reduction: one, none or infinitely many solutions](https://math.wiki/solve-a-system-by-row-reduction/) **Summary:** Reducing the augmented matrix and reading the solution off the final rows. **Question:** Row reduction always ends in a staircase, but not every staircase names one point. How do the last rows say whether a system has one solution, none, or infinitely many, and how is an infinite family of solutions written down? ## When you reach for it Any system of linear equations, in any number of variables, can be written as an augmented matrix and reduced to a row echelon form. That lesson kept to systems whose staircase has a pivot in every variable column, which gives exactly one solution. Most systems met in practice are not so tidy: an equation may contradict the others, or repeat what they already say. Row reduction handles these too, and the evidence is in the last rows of the echelon form, even when a coefficient is a letter whose value decides the case. A system is **consistent** when it has at least one solution and **inconsistent** when it has none. Every linear system is in exactly one of three cases: one solution, no solution, or infinitely many. ## The procedure ### Day 1: reading the final rows Reduce the augmented matrix to echelon form with row operations, naming each one ($R_i \leftrightarrow R_j$, $kR_i \to R_i$, $R_i + cR_j \to R_i$). Row operations never change the solution set, so whatever the final matrix says about solutions is true of the original system. Then read the rows from the bottom up. **A row $\left[\begin{array}{ccc|c} 0 & 0 & 0 & c \end{array}\right]$ with $c \ne 0$.** It is the equation $0x + 0y + 0z = c$, that is $0 = c$. No triple of numbers makes it true, so no triple solves the reduced system, and so none solves the original. The system is inconsistent; stop. In the language of pivots, a leading entry has appeared in the constants column, to the right of the bar. ### [Span](https://math.wiki/span-of-a-set-of-vectors/) **Summary:** Every vector that can be reached from a given set using linear combinations. **Question:** A few vectors can be scaled and added. Which vectors can be reached that way at all, and what shape does that collection make? ## Why this exists Nemanja, Branka and Dragica run the camera at a small observatory. The camera rides on a mount driven by two motors. One turn of the first motor moves the camera by a fixed vector $\mathbf{p}$, in centimeters from its home position; one turn of the second moves it by $\mathbf{q}$. Each motor runs forward or in reverse and can stop at any fraction of a turn, so after $c_1$ turns of the first and $c_2$ of the second the camera sits at $c_1\mathbf{p} + c_2\mathbf{q}$. linear-combination-of-vectors answered whether one given position can be reached. The crew needs every position the mount can reach, the shape they make, and whether a third motor would add any. Everything reachable from a list by scaling and adding is its span. ## Definition Let $\mathbf{v}_1, \dots, \mathbf{v}_k$ be vectors in $\mathbb{R}^n$. Their **span** is the set of all their linear combinations: $$\begin{aligned} &\operatorname{span}\{\mathbf{v}_1, \dots, \mathbf{v}_k\} \\ &\quad = \{\, c_1\mathbf{v}_1 + \dots + c_k\mathbf{v}_k : c_1, \dots, c_k \in \mathbb{R} \,\}. \end{aligned}$$ A vector $\mathbf{b}$ is **in the span**, written $\mathbf{b} \in \operatorname{span}\{\mathbf{v}_1, \dots, \mathbf{v}_k\}$, when it is a linear combination of $\mathbf{v}_1, \dots, \mathbf{v}_k$. The vectors **span** a set $W$ (for instance all of $\mathbb{R}^n$) when their span is exactly $W$. The list is finite, but the span is usually an infinite set. It always holds $\mathbf{0}$ (all weights zero) and each $\mathbf{v}_i$ (weight $1$ on $\mathbf{v}_i$, $0$ on the rest). ### [Subspaces](https://math.wiki/subspaces-of-rn/) **Summary:** A set of vectors closed under adding and scaling, such as a line or plane through the origin, is a subspace. **Question:** Lines and planes through the origin keep turning up, as spans and as solution sets of equations. What do they share that a line missing the origin, or a curve, does not? ## Why this exists Carmela, Gennaro and Sokol run the vats at a dyehouse. When a batch comes out slightly off, they adjust the recipe for the next one. An adjustment is a vector $(x, y, z)$ in $\mathbb{R}^3$: the change in grams per liter of red, yellow and blue dye, positive for more and negative for less. Their shade meter turns a recipe into one reading, and an adjustment $(x, y, z)$ changes the reading by $x - 2y + 3z$. An adjustment that leaves the reading unchanged is balanced. Two balanced adjustments made in turn are balanced, and so is one made twice as strongly, or reversed, or no change at all. Adding and scaling never lead out of the set. Sets like this are the subspaces: the flat pieces of $\mathbb{R}^n$ through the origin, where the rest of linear algebra takes place. ## Definition A **subspace** of $\mathbb{R}^n$ is a set $W \subseteq \mathbb{R}^n$ with three properties. - **It contains zero:** $\mathbf{0} \in W$. - **It is closed under addition:** if $\mathbf{u} \in W$ and $\mathbf{v} \in W$, then $\mathbf{u} + \mathbf{v} \in W$. - **It is closed under scalar multiplication:** if $\mathbf{u} \in W$ and $c$ is any real number, then $c\mathbf{u} \in W$. "Closed" means the operation never leads out of the set. $\mathbb{R}^n$ is a subspace of itself, and so is $\{\mathbf{0}\}$, the **zero subspace**, since $\mathbf{0} + \mathbf{0} = \mathbf{0}$ and $c\mathbf{0} = \mathbf{0}$. ### [The characteristic equation and finding eigenvalues](https://math.wiki/characteristic-equation/) **Summary:** Eigenvalues are the numbers that make the shifted matrix fail to be invertible. **Question:** A linear map stretches a rubber sheet and turns almost every arrow drawn on it. Which directions come out only stretched or shrunk, never turned, and how can the stretch factors be found without guessing one direction at a time? ## Why this exists Sarnai, Tuvshin and Odval clamp a square sheet of rubber membrane in an adjustable frame and draw arrows on it from a center point. Pulling the frame moves the point at $(x, y)$ to $A\begin{bmatrix} x \\ y \end{bmatrix}$, a linear-transformation with matrix $$A = \begin{bmatrix} 7 & -4 \\ 2 & 1 \end{bmatrix}.$$ After the pull, almost every arrow points a new way. Odval asks whether any arrow comes out pointing along its old line, only longer, shorter or reversed. Along such a direction the map does the simplest thing a matrix can do: it multiplies by one number. But there are infinitely many directions to try, and testing them one by one never ends. This lesson turns the search into one polynomial equation in one unknown number. Its roots are the stretch factors, called eigenvalues, and they can be found before any direction is known. The lesson runs over two days: day 1 builds the equation and solves it for $2 \times 2$ matrices; day 2 takes it to larger matrices and reads the trace and determinant off the eigenvalues. ## Definition Let $A$ be an $n \times n$ matrix. A number $\lambda$ is an **eigenvalue** of $A$ when some **nonzero** vector $\mathbf{v}$ has $$A\mathbf{v} = \lambda\mathbf{v}.$$ Such a $\mathbf{v}$ is an **eigenvector** of $A$ for $\lambda$. In plain words, $\mathbf{v}$ is a direction the map keeps on its own line, and $\lambda$ is the factor applied along it: a stretch when $\lambda > 1$, a shrink when $0 < \lambda < 1$, a reversal when $\lambda < 0$, and a collapse to $\mathbf{0}$ when $\lambda = 0$. ### [The Gram-Schmidt process](https://math.wiki/gram-schmidt-process/) **Summary:** Any basis becomes an orthogonal one by subtracting from each vector its projections onto the vectors before it. **Question:** A subspace comes with a basis whose vectors lean on one another. How do you trade it for a perpendicular basis of the same subspace, and what does the trade record about the vectors you started with? ## When you reach for it Meseret, Biruk and Kidist run a robotics lab. Their camera rig reports a position as amounts along three sensor axes which, in the lab's calibration frame (units of decimeters), are $$\mathbf{x}_1 = \begin{bmatrix} 2 \\ 2 \\ 1 \end{bmatrix}, \quad \mathbf{x}_2 = \begin{bmatrix} 3 \\ 0 \\ 3 \end{bmatrix}, \quad \mathbf{x}_3 = \begin{bmatrix} 7 \\ 1 \\ 2 \end{bmatrix}.$$ The axes are skewed: $\mathbf{x}_1 \cdot \mathbf{x}_2 = 9 \neq 0$. A reading $(c_1, c_2, c_3)$ means the point $c_1\mathbf{x}_1 + c_2\mathbf{x}_2 + c_3\mathbf{x}_3$, so turning a point into a reading means solving a system. Along perpendicular unit axes, each coordinate would be one dot product. The team wants perpendicular axes that keep the first new axis along $\mathbf{x}_1$ and the first two in the plane of $\mathbf{x}_1$ and $\mathbf{x}_2$. Reach for the Gram-Schmidt process whenever a subspace $W$ of $\mathbb{R}^n$ is given by a basis, or any spanning list, and an orthogonal or orthonormal basis of $W$ is needed: to project onto $W$ (orthogonal-projection), to read off coordinates by dot products, or to factor $A = QR$. That lesson's last example repaired a basis of a plane with one subtraction; here the subtraction is repeated for any number of vectors, with a proof that it always works. ## The procedure ### Subtract the projections onto what is already built Let $\mathbf{x}_1, \dots, \mathbf{x}_p$ be a basis of a subspace $W$ of $\mathbb{R}^n$. Keep the first vector, and from each later vector remove its projections onto the new vectors already built: ### [The invertible matrix theorem](https://math.wiki/invertible-matrix-theorem/) **Summary:** For a square matrix, a dozen conditions, from a nonzero determinant to a kernel holding only zero, all say the same thing. **Question:** For a square matrix, the questions "can it be undone?", "is the only solution of Ax = 0 the zero vector?", "do its columns span the space?" and "is its determinant nonzero?" look like different questions. Are they? ## Why this exists Thuy, Trang and Kioni run an instrument lab. Each instrument reads three true quantities $\mathbf{x}$ in $\mathbb{R}^3$ and reports $A\mathbf{x}$, where $A$ is its $3 \times 3$ calibration matrix. The lab must recover $\mathbf{x}$ from every reading, so for each instrument it asks whether $A$ can be undone. Earlier lessons gave many tests: reduce $A$ beside $I$ (inverse-by-row-reduction), look for a nonzero solution of $A\mathbf{x} = \mathbf{0}$ (kernel-of-a-matrix), ask whether every reading is reachable (image-of-a-matrix), test the columns (linear-independence), compute $\det A$ (properties-of-determinants). invertible-transformations proved that four of these agree. For a square matrix all of them, and more, agree, so the cheapest test decides the rest. Day 1 is the theorem and its proof. Day 2 uses it: many conclusions from one fact, families with a parameter, and what breaks for matrices that are not square. ## Definition The theorem is about an $n \times n$ matrix $A$, its transformation $T(\mathbf{x}) = A\mathbf{x}$ from $\mathbb{R}^n$ to $\mathbb{R}^n$, and the following conditions, each true or false for the given $A$. - (a) $A$ is invertible. - (b) $A$ is row equivalent to $I_n$: its reduced row echelon form is $I_n$. - (c) $A$ has $n$ pivot positions. - (d) $A\mathbf{x} = \mathbf{0}$ has only the solution $\mathbf{x} = \mathbf{0}$: $\ker A = \{\mathbf{0}\}$. - (e) The columns of $A$ are linearly independent. - (f) $T$ is one-to-one. - (g) $A\mathbf{x} = \mathbf{b}$ has at least one solution for every $\mathbf{b}$ in $\mathbb{R}^n$. - (h) The columns of $A$ span $\mathbb{R}^n$. - (i) $T$ is onto: $\operatorname{im} A = \mathbb{R}^n$. - (j) There is an $n \times n$ matrix $C$ with $CA = I_n$. - (k) There is an $n \times n$ matrix $D$ with $AD = I_n$. - (l) $A^{T}$ is invertible. - (m) $\operatorname{rank} A = n$. - (n) The columns of $A$ form a basis of $\mathbb{R}^n$. - (o) $A\mathbf{x} = \mathbf{b}$ has exactly one solution for every $\mathbf{b}$ in $\mathbb{R}^n$. - (p) $\det A \neq 0$. - (q) $0$ is not an eigenvalue of $A$: no $\mathbf{v} \neq \mathbf{0}$ has $A\mathbf{v} = 0\mathbf{v}$. ### [The transpose and symmetric matrices](https://math.wiki/transpose-and-symmetric-matrices/) **Summary:** Flipping a matrix across its diagonal swaps its rows and columns, and a matrix equal to its flip is symmetric. **Question:** A table of distances between huts reads the same down a column as along the matching row. What operation on a matrix captures that, and what does it do to sums, products and dot products? ## Why this exists Gwenaelle keeps the hiking club's table of trail distances, in kilometers, between four mountain huts. Row $i$, column $j$ holds the distance from hut $i$ to hut $j$: $$D = \begin{bmatrix} 0 & 7 & 12 & 9 \\ 7 & 0 & 6 & 11 \\ 12 & 6 & 0 & 8 \\ 9 & 11 & 8 & 0 \end{bmatrix}.$$ The trail from hut $1$ to hut $3$ is the trail from hut $3$ to hut $1$, so $d_{13} = d_{31} = 12$. Row $1$ read across and column $1$ read down hold the same four numbers, so swapping rows with columns changes nothing. Naming that swap pays off well beyond tables: a dot product is a row times a column, so writing $\mathbf{u} \cdot \mathbf{v}$ as a matrix product needs a way to turn a column into a row, and the best approximate solution of an unsolvable system comes from the matrix $A^T A$ (least-squares-solutions). ## Definition An $m \times n$ matrix has $m$ rows and $n$ columns, and $a_{ij}$ is its entry in row $i$, column $j$ (matrix-vocabulary). The **transpose** of an $m \times n$ matrix $A$ is the $n \times m$ matrix $A^T$ whose rows are the columns of $A$, in order. Entry by entry, $$(A^T)_{ij} = a_{ji} \quad \text{for } 1 \le i \le n,\ 1 \le j \le m.$$ The transpose of a column vector in $\mathbb{R}^n$ (an $n \times 1$ matrix) is a $1 \times n$ row, and the transpose of a row is a column. A matrix $S$ is **symmetric** when $S^T = S$. Since $S^T$ is $n \times m$ when $S$ is $m \times n$, a symmetric matrix must be square, and a square matrix is symmetric exactly when ### [Vectors in n dimensions](https://math.wiki/vectors-in-rn/) **Summary:** Lists of numbers with addition and scaling; linear combinations and span. **Question:** Arrows in the plane add tip to tail. What survives when a vector is a list of six numbers, or sixty, with no picture to draw? ## What this area is about A vector in $\mathbb{R}^n$ is a list of $n$ numbers, added and scaled entry by entry. From those two moves come the central questions of linear algebra: which vectors can be built from a given few, whether any of the few is redundant, and how many are needed. ## How these fit together **Operations.** Adding and scaling vectors sets up the arithmetic; the dot product adds length, angle and perpendicularity. **Definitions.** A linear combination is a sum of scaled vectors. The span of a set is every combination of it. Linear independence means no vector in the set is a combination of the others. Subspaces are sets closed under adding and scaling, such as a line or plane through the origin. **Bases.** Basis and dimension defines a basis as an independent spanning list and proves the central result: every basis of a subspace has the same number of vectors. Coordinates and change of basis names a vector by its weights in a basis and converts between bases with an invertible matrix. ## A path through it Adding and scaling first. The dot product can follow at any point after that; nothing else here uses it. Then linear combinations, then span and linear independence; subspaces need span. Basis and dimension need both independence and subspaces, and coordinates come last, also needing a matrix inverse found by row reduction. This area is the input to matrices as transformations, orthogonality and least squares and eigenvalues. ### [What the determinant means](https://math.wiki/determinant-meaning/) **Summary:** It is the area or volume scaling factor, and zero means the matrix flattens everything. **Question:** A matrix moves every point of the plane at once. Is there one number that says how much it stretches or shrinks the area of every shape, and whether it flips shapes over? ## Why this exists A linear-transformation $T(\mathbf{x}) = A\mathbf{x}$ of the plane moves every point at once. Tshepo prints a tailor's pattern pieces on stretchy fabric, and a press distorts the fabric by such a map: a collar piece, a round patch and a sleeve all change shape, and Dineo, who buys the cloth, wants to know how much fabric each piece covers afterward, curved pieces included. determinant-2x2 stated that the columns of $A$ span a parallelogram of area $\lvert \det A \rvert$, proved there for one drawn case. Here it is proved in general and read as a fact about the map: **every** region, square or curved, has its area multiplied by $\lvert \det A \rvert$. The sign says whether the map flips the plane over, and $\det A = 0$ says it flattens the plane, so it cannot be undone. ## Definition The **signed area** $S(\mathbf{u}, \mathbf{v})$ of an ordered pair of vectors in $\mathbb{R}^2$ is the area of the parallelogram they span, with a sign. Stand at the origin facing along $\mathbf{u}$. If $\mathbf{v}$ points to your left (the counterclockwise side of the line through $\mathbf{u}$), the sign is $+$; to your right, $-$. If $\mathbf{u}$ and $\mathbf{v}$ lie on one line through the origin (a zero vector included), $S(\mathbf{u}, \mathbf{v}) = 0$. This is the same sign as the turn rule of determinant-2x2: $\mathbf{v}$ is on the left of $\mathbf{u}$ exactly when the shorter turn from $\mathbf{u}$ to $\mathbf{v}$ is counterclockwise. ### [When a matrix or transformation can be undone](https://math.wiki/invertible-transformations/) **Summary:** A matrix can be undone exactly when nothing but zero lands on zero, which is when its determinant is not zero. **Question:** A linear transformation scrambles every vector of space. When can the scrambling be undone, so that each output tells you exactly which input it came from, and what does the undoing look like? ## Why this exists Paqari, Tayen and Wayta run a secret-message club. Each letter becomes a number, $\text{A} = 1$ up to $\text{Z} = 26$, with $0$ for a space. A message is cut into pairs, and each pair $\mathbf{x}$ is sent as $K\mathbf{x}$ for a $2 \times 2$ key matrix $K$ the members share. Encoding is the linear-transformation $T(\mathbf{x}) = K\mathbf{x}$. A key is useful only if a second transformation takes each sent pair back to its message, and some keys fail. This lesson says exactly which linear transformations can be undone, proves that the undoing is again linear, and shows why the matrix must be square. ## Definition A transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ is **invertible** when some transformation $S: \mathbb{R}^m \to \mathbb{R}^n$ undoes it from both sides: $$\begin{aligned} S(T(\mathbf{x})) &= \mathbf{x} \ \text{ for every } \mathbf{x} \text{ in } \mathbb{R}^n, \\ T(S(\mathbf{y})) &= \mathbf{y} \ \text{ for every } \mathbf{y} \text{ in } \mathbb{R}^m. \end{aligned}$$ Then $S$ is the **inverse** of $T$, written $T^{-1}$. Recall that $T$ is **one-to-one** when different inputs give different outputs, and **onto** when every vector of $\mathbb{R}^m$ is an output, that is, when its image is all of $\mathbb{R}^m$. For matrices the definition is the one from inverse-2x2: an $n \times n$ matrix $A$ is **invertible** when some $n \times n$ matrix $A^{-1}$ has $AA^{-1} = A^{-1}A = I$. A square matrix that is not invertible is **singular**. ### [Writing a system as a matrix equation](https://math.wiki/system-as-a-matrix-equation/) **Summary:** Coefficients, variables and constants, split into three matrices. **Question:** A system of equations is a stack of separate lines, each repeating the same letters. Can the whole stack be written as one equation, so that matrix arithmetic can work on all of it at once? ## When you reach for it A linear system such as $$\begin{aligned} 5x + 2y &= 12 \\ 3x - 4y &= 2 \end{aligned}$$ carries three kinds of information: the numbers multiplying the variables, the variables themselves, and the numbers on the right. Elimination copies the letters $x$ and $y$ over and over, although only the numbers ever change. Writing the system as a **matrix equation** $AX = B$ puts the three kinds of information in three matrices and packs the whole system into one statement. You reach for it whenever a system is to be handled with matrices: by row operations, an inverse matrix, Cramer's rule, or a calculator. ## The procedure ### The three matrices Take a system of $m$ linear equations in $n$ variables, each equation written with the variable terms on the left, **in the same order in every equation**, and the constant on the right. - The **coefficient matrix** $A$ is $m \times n$. Row $i$ holds the coefficients of equation $i$; column $j$ holds every coefficient of variable $j$. A variable that does not appear in an equation has coefficient $0$ there. A variable with no number written in front, as in $x$ or $-y$, has coefficient $1$ or $-1$. - The **variable matrix** $X$ is the $n \times 1$ column of variables, in the chosen order. - The **constant matrix** $B$ is the $m \times 1$ column of right-hand sides. For the system above, $$A = \begin{bmatrix} 5 & 2 \\ 3 & -4 \end{bmatrix} \quad X = \begin{bmatrix} x \\ y \end{bmatrix} \quad B = \begin{bmatrix} 12 \\ 2 \end{bmatrix}$$ ## Logic, proof and discrete math ### [A conditional and its contrapositive](https://math.wiki/contrapositive-equivalence/) **Summary:** Swapping the two parts of an if-then statement and negating both gives a statement with the same meaning. **Question:** An if-then statement can be turned around in three ways. Which of them is guaranteed to say exactly what the original said? ## Why this exists "If $x > 4$, then $x^2 > 16$." That is true for every real number. Now turn it around: "if $x^2 > 16$, then $x > 4$." That is false, since $x = -5$ has $(-5)^2 = 25 > 16$. Turning a true statement around can break it. Yet there is one way of turning it around that never breaks anything: "if $x^2 \le 16$, then $x \le 4$." Swap the two parts **and** negate both, and the new statement is true exactly when the old one is. This is the **contrapositive**, and it matters because it is often far easier to check or prove than the statement you started with. Knowing which turnaround is safe, and why, lets you make that trade without changing the question. ## Definition Start from a conditional $P \to Q$, "if $P$, then $Q$", where $P$ is the hypothesis and $Q$ the conclusion. Three other conditionals are built from the same two parts: | Name | Statement | In words | |---|:-:|---| | original | $P \to Q$ | if $P$, then $Q$ | | converse | $Q \to P$ | swap the parts | | inverse | $\neg P \to \neg Q$ | negate both parts | | contrapositive | $\neg Q \to \neg P$ | swap the parts and negate both | Two statements are logically equivalent, written $\equiv$, when their truth tables agree on every row: whatever $P$ and $Q$ are, the two statements are true together and false together. The claim of this lesson is that a conditional is logically equivalent to its contrapositive, and the converse is logically equivalent to the inverse, but the original is not equivalent to the converse. ### [Adjacency matrices and their powers](https://math.wiki/adjacency-matrix-powers/) **Summary:** Each entry of the nth power counts the walks of length n between one vertex and another. **Question:** A network can be stored as a square table of 0s and 1s. What does multiplying that table by itself count, and which questions about the network does that arithmetic answer? ## Why this exists A drawing of a graph suits the eye, not a computer. A table with a row and a column for each vertex, holding $1$ where two vertices are adjacent and $0$ elsewhere, stores the same graph exactly. The surprise is what happens when that table is multiplied by itself with the ordinary rule for matrices: the entries of the powers count walks. Questions about reach, distance and triangles become arithmetic. ## Definition Graphs are as in degree-of-a-vertex: $G = (V, E)$ is simple, $uv$ is an edge and $\deg(v)$ a degree. Walks, their length, "joined", "connected" and the distance $d(u, v)$ are those of walks-paths-and-cycles; directed graphs $D = (V, A)$, arcs $(u, v)$, $\deg^+(v)$ and $\deg^-(v)$ are those of weighted-graphs. List the vertices of $G$ in an order $v_1, v_2, \ldots, v_n$. The **adjacency matrix** of $G$ is the $n \times n$ matrix $A = [a_{ij}]$ with $$a_{ij} = \begin{cases} 1 & \text{if } v_iv_j \in E, \\ 0 & \text{otherwise.} \end{cases}$$ For a directed graph $D = (V, A)$ the adjacency matrix is written $M = [m_{ij}]$ here, so that the letter $A$ keeps naming the arc set: $m_{ij} = 1$ if $(v_i, v_j)$ is an arc and $0$ otherwise. Row $i$ belongs to the tail, column $j$ to the head. Products are those of matrix-multiplication-rule: $(XY)_{ij} = \sum_{m} x_{im}y_{mj}$, row $i$ of $X$ times column $j$ of $Y$. $A^k$ is the product of $k$ copies of $A$, and $A^0 = I_n$, the identity matrix. The **trace** $\operatorname{tr}(X)$ of a square matrix is the sum of its diagonal entries. A **triangle** in $G$ is a cycle of length $3$: three vertices that are pairwise adjacent. ### [And and or: conjunction and disjunction](https://math.wiki/conjunction-and/) **Summary:** A statement joined by and, true only when both of its parts are true. **Question:** Two claims are joined into one sentence, by "and" or by "or". When does the whole sentence count as true, and how do you list every case at once so nothing is missed? ## Why this exists A single statement is true or false on its own: "$12$ is even" needs no help from anywhere else to be decided. Most of the claims worth making, though, are built out of smaller ones. "The number is even and it is greater than $10$" is really two statements riding on one sentence. And a great deal of mathematics does not pin down a single answer at all: solve $x^2 = 9$ and the honest answer is "$x = 3$ or $x = -3$"; divide a whole number by $2$ and the remainder is $0$ or $1$. English already has the two joining words, "and" and "or", but it is loose about both. "She opened the door and walked in" quietly means first one, then the other, while "$2$ is even and $3$ is odd" carries no order at all. "You can have cake or ice cream" usually means pick one, while "is anyone here hungry or thirsty?" plainly includes the person who is both. A proof that leans on these words has to know exactly what it is allowed to conclude, so mathematics fixes each of them to one meaning. "And" keeps its demand that both halves hold and loses any sense of order or cause. "Or" means at least one, both included. The joined statements are the **conjunction** and the **disjunction**, two of the basic logical connectives, and the clearest way to see everything they do is to list every case in a **truth table**. ## Definition ### Conjunction: and Let $P$ and $Q$ be statements. Their **conjunction**, written $P \land Q$ and read "$P$ and $Q$", is the statement that is true when $P$ and $Q$ are both true, and false in every other case. ### [Axioms, definitions and theorems](https://math.wiki/what-an-axiom-is/) **Summary:** A statement accepted without proof, chosen as a starting point for everything built after it. **Question:** Every proof answers "why?" with an earlier fact, in words that must mean one exact thing. Where do the first facts and the exact meanings come from? ## Why this exists A proof gives a reason for a statement, and the reason is an earlier statement. Ask "why?" of that one, and you are handed an even earlier one. The questions cannot run backward forever, and they cannot go around in a circle, because a circle proves nothing. Somewhere the answers have to stop. The statements where they stop are called **axioms**. Choosing them openly is what lets mathematics say exactly what it is talking about. Euclid built the geometry of the plane on a short list of starting statements, and every theorem in it can be traced back to that list. In the nineteenth century people changed one item on the list and found a second, different geometry that works just as well as the first. Around 1900 a harsher lesson arrived in the theory of sets: a rule that sounded perfectly obvious, that any description picks out a set, turned out to contradict itself. Both stories say the same thing. Whatever you start from has to be stated and examined, because you cannot see what a proof depends on unless you can see where it starts. The words in a proof need the same care. Two people argue about whether $0$ is even. One says it cannot be, because even numbers are the ones you can share out in pairs, and there is nothing to share. The other says it must be, because it sits between $-1$ and $1$, and both of those are odd. They can go on all evening. Nothing either of them says can settle it, because each is using the word "even" to mean a slightly different thing. The cure is to agree, in words that leave no room, on exactly what the word means. After that the argument is over, because it is no longer an argument about words. You hold the agreed meaning up against the number and look. That is what a **definition** is for. It turns a word that has a feel into a rule that has a verdict. "Rectangle", "prime", "continuous" and "derivative" are each, before anything is proved about them, one exact sentence, and every proof about them begins from that sentence. A proof has no way of knowing what a word "really" means. It knows only what the word was defined to mean, and that is all it needs. ### [Bezout's identity and the extended Euclidean algorithm](https://math.wiki/extended-euclidean-algorithm/) **Summary:** Running the Euclidean algorithm backwards to find the combination that actually gives the divisor. **Question:** The greatest common divisor of two integers divides every combination of them. Is the greatest common divisor itself one of those combinations, and how would you find the multipliers? ## When you reach for it The euclidean-algorithm finds $\gcd(527, 403) = 31$ in three divisions. Every common divisor of $527$ and $403$ divides every combination $527x + 403y$ with $x, y \in \mathbb{Z}$ (proof-with-divisibility), so every such combination is a multiple of $31$. A sharper question is whether $31$ itself is one of them. It is: $31 = 4 \cdot 403 - 3 \cdot 527$ (Example 1), and the multipliers come from the same divisions, read backward. Reach for this move when a proof needs the greatest common divisor written as a combination: to show two integers are coprime by producing $1$, to prove **Euclid's lemma** (a prime dividing a product divides a factor), and to solve an equation $ax + by = c$ in integers, such as finding how many $\$7$ and $\$11$ items cost exactly $\$100$. The lesson runs over two days. Day 1 finds the combination and proves it always exists (Bezout's identity). Day 2 uses it: Euclid's lemma and equations $ax + by = c$. ## The procedure ### Day 1: running the divisions backward An **integer combination** (or linear combination) of $a$ and $b$ is a number $ax + by$ with $x, y \in \mathbb{Z}$; the integers $x, y$ are its **coefficients**. Run the Euclidean algorithm on $a$ and $b > 0$: $$\begin{aligned} a &= bq_1 + r_1, \\ b &= r_1 q_2 + r_2, \\ &\ \ \vdots \\ r_{n-2} &= r_{n-1} q_n + r_n, \end{aligned}$$ and then $r_{n-1} = r_n q_{n+1}$, with $r_n = \gcd(a, b)$. **Back substitution** solves each line for its remainder, $r_k = r_{k-2} - q_k r_{k-1}$ (with $r_{-1} = a$, $r_0 = b$), and works upward. The last line writes $r_n$ as a combination of $r_{n-2}$ and $r_{n-1}$. Replace $r_{n-1}$ by its own expression in $r_{n-3}$ and $r_{n-2}$, collect terms, and the result is a combination of $r_{n-3}$ and $r_{n-2}$. Each substitution moves one line up, and at the top only $a$ and $b$ are left. Every step is the distributive property and collecting like terms, so the final equation is true because each one before it was. ### [Biconditional](https://math.wiki/biconditional/) **Summary:** A statement saying two things hold together and fail together, written as if and only if. **Question:** Two statements always turn out true together and false together. What is the one connective that says exactly that, no more and no less? ## Why this exists And and or join two statements into one. If-then joins them a third way, but only in one direction, and the two ends of an if-then do not play the same part. "If a number is divisible by $6$, then it is even." Learn that a number is divisible by $6$ and you know for certain that it is even. Learn that a number is even and you know nothing certain about $6$: the number $4$ is even and is not a multiple of $6$. Two words name the two ends: divisibility by $6$ is **sufficient** for being even, since it is enough on its own, and being even is **necessary** for divisibility by $6$, since no number manages that without it. Often one direction is not what you mean. "A number is divisible by $10$ if and only if it ends in $0$" is not claiming a one-way link; it claims that the two facts always travel together, so that knowing either one tells you the other. Each condition is both necessary and sufficient for the other. Writing that out as two separate if-then sentences works, but it is clumsy, and it hides the fact that the two directions are being asserted as a single package. The **biconditional** is the connective that says it in one piece, and its four-row truth table is a small tool for testing when two statements say the same thing. ## Definition For statements $P$ and $Q$, the biconditional $P \leftrightarrow Q$ is read "$P$ if and only if $Q$," often shortened to "$P$ iff $Q$." It is true exactly when $P$ and $Q$ have the **same truth value**: both true, or both false. It is false exactly when they disagree. ### [Bipartite graphs](https://math.wiki/bipartite-graphs/) **Summary:** A graph whose vertices split into two groups with every edge running between the groups. **Question:** When every edge of a network runs between two kinds of things, students and projects, applicants and jobs, what does that force on the network, and when can every student get a project of their own? ## Why this exists Many networks join two kinds of things and never two of one kind: students to the projects they would accept, applicants to the jobs they qualify for. Such a graph can be drawn with one kind in a top row and the other below, every edge running between the rows. That one feature decides a great deal: two colors suffice, there is no cycle of odd length, there are at most $\frac{n^2}{4}$ edges, and there is the question of matching: can each student get a project of their own? ## Definition Graphs are finite and simple, in the notation of degree-of-a-vertex and walks-paths-and-cycles (walks, cycles, components, the distance $d(u, v)$, the cycle graph $C_n$). Colorings and $\chi(G)$ are as in chromatic-number, and the grids and cube graphs as in hamiltonian-paths-and-cycles. - $G = (V, E)$ is **bipartite** if $V$ is the union of two disjoint sets $X$ and $Y$, either of which may be empty, such that every edge has one end in $X$ and the other in $Y$. The pair $(X, Y)$ is a **bipartition** and $X$, $Y$ are its **parts**. - For $m, n \ge 1$ the **complete bipartite graph** $K_{m,n}$ has parts of sizes $m$ and $n$, every vertex of one part joined to every vertex of the other, and no other edges. - The **$m \times n$ grid** has vertices $(i, j)$ with $1 \le i \le m$, $1 \le j \le n$, two adjacent when they agree in one coordinate and differ by $1$ in the other. For $n \ge 1$ the **cube graph** $Q_n$ has the $2^n$ strings of $0$s and $1$s of length $n$ as vertices, two adjacent when they differ in exactly one place. - A **matching** is a set $M$ of edges no two of which share an end. $M$ **covers** each end of its edges. A matching is **perfect** if it covers every vertex. - For a set $S$ of vertices, $N(S)$ is the set of vertices adjacent to at least one vertex of $S$. In a bipartite graph with parts $X$ and $Y$, **Hall's condition** for $X$ is ### [Building a truth table for a compound statement](https://math.wiki/build-a-compound-truth-table/) **Summary:** Working outward from the small parts of a statement, one column at a time, to reach its whole value. **Question:** A statement mixes several letters and several connectives at once. How do you find its truth value in every possible case, without getting lost? ## When you reach for it And, or and not each have a table of their own, four rows or two, small enough to hold in your head. A real statement is rarely just one of them. It joins several smaller statements with several connectives at once, sometimes the same connective more than once, sometimes one connective sitting inside another. There is no table to memorize for a statement like that, because there are infinitely many such statements. What there is instead is a way to build the table for any one of them, out of the small tables you already have, one column at a time. Reach for this whenever a statement mixes more than one connective, or uses more than two letters, and you want to know exactly when it is true: not by guessing, and not by working it out fresh for every situation, but by listing every situation once and filling it in. ## The procedure **Count the letters, and that fixes the number of rows.** Call the number of distinct simple statements in the formula $n$. Each one can independently be true or false, so there are $2^n$ combinations to account for, and the table needs exactly $2^n$ rows: no more, since nothing rules a combination out in advance, and no fewer, since leaving one out is a case the table never covers. **Generate the rows in a fixed order, the same way every time.** With two letters, $P$ and $Q$, the four rows are $TT, TF, FT, FF$: $P$ is true for the top half and false for the bottom half, and within each half $Q$ alternates. A third letter doubles that to eight rows by splitting every existing row into two, once with the new letter true and once false, so $P$ is true for the top four rows and false for the bottom four, $Q$ alternates in blocks of two, and $R$ alternates every single row: ### [Cantor's theorem: no set matches its power set](https://math.wiki/cantors-theorem/) **Summary:** No set can be matched one to one with its power set, so there is no largest infinity. **Question:** The real numbers are a bigger infinity than the natural numbers. Is there a biggest infinity of all, or can every set be outgrown? ## Why this exists cantor-diagonal-argument showed that $\mathbb{R}$ cannot be listed: infinite sets come in at least two sizes. Is $\mathbb{R}$ as big as a set can get, or is there something bigger still, and something bigger than that? Cantor's theorem answers it for every set at once. No set $A$ can be matched one to one with its power-set $\mathcal{P}(A)$, and $A$ always fits inside $\mathcal{P}(A)$, so $\mathcal{P}(A)$ is strictly bigger. Applying this over and over gives a ladder of larger and larger infinities that never stops. In a paper of 1891 Cantor gave this argument in the form of functions taking only the values $0$ and $1$, which (Example 5) is the same thing as subsets. Throughout, $\mathbb{N} = \{0, 1, 2, \dots\}$. ## Definition Recall from matching-sets-one-to-one that $\lvert A \rvert = \lvert B \rvert$ means there is a bijection $A \to B$. Comparing sizes uses injections: - $\lvert A \rvert \le \lvert B \rvert$ means there is an injection $A \to B$; - $\lvert A \rvert < \lvert B \rvert$ means there is an injection $A \to B$ and no bijection $A \to B$. As with $=$, these compare a pair of sets; no number $\lvert A \rvert$ is defined. A function $f : A \to \mathcal{P}(A)$ assigns to each member $a$ of $A$ a subset $f(a)$ of $A$, and $a$ itself either belongs to that subset or does not. The **diagonal set** of $f$ collects the members that do not: $$D_f = \{a \in A : a \notin f(a)\}.$$ ### [Cardinality of a finite set](https://math.wiki/cardinality-of-a-finite-set/) **Summary:** How many members a set has, counting each distinct member exactly once. **Question:** Two descriptions of a set can look nothing alike. How do you say exactly how big a set is, and when can you tell that two sets are the same size without counting either one? ## Why this exists A set is decided by what belongs to it (elements and membership), and roster notation writes those elements down. Once a set is written down, the first question anyone asks about it is how big it is. For a short roster the answer looks obvious: count the entries. But the entries of a roster are not always the elements. The roster $\{3, 3, 7\}$ has three entries and only two elements, and $\{\tfrac{1}{2}, 0.5\}$ has two entries naming one number. Many sets are not given by a roster at all but by a rule, such as "the integers whose square is less than $20$" or "the solutions of $x^3 = 16x$", and some are too long to list, such as the two-digit numbers that leave remainder $5$ on division by $7$. So the size of a set needs a precise meaning: count each **distinct** element once, whatever the description. With that meaning settled, sizes can be compared, added and, in the lessons that follow, computed for sets far too large to list. ## Definition The **cardinality** of a finite set $A$ is the number of distinct elements of $A$. It is written $\lvert A\rvert$, read "the cardinality of $A$" or "the size of $A$". Some books write $n(A)$ for the same number. Counting a set means pairing its elements with $1, 2, 3, \dots$ in turn, using each element exactly once, until none are left. If the last number used is $n$, then $\lvert A\rvert = n$. The empty set has nothing to pair, so $\lvert\varnothing\rvert = 0$. ### [Cartesian product](https://math.wiki/cartesian-product/) **Summary:** The set of all ordered pairs made by taking one member from each of two sets. **Question:** A point in the plane needs two numbers in a fixed order, and so does a crepe order of one batter and one filling. What single set holds every such pair at once? ## Why this exists A point of the plane is two numbers playing different parts: $(3, 1)$ and $(1, 3)$ are different points (ordered-pairs). A crepe order at Nayeli's stand is also two things, a batter and a filling. The Cartesian product gathers every such pair into one set. The plane is a product, a relation is a subset of one, and counting choices made in stages (counting-choices-in-stages) is counting a product. ## Definition **Ordered pair.** An **ordered pair** $(a, b)$ is two objects in a fixed order: $a$ is the **first coordinate**, $b$ the **second coordinate**. Everything about pairs used here comes from one rule: $$(a, b) = (c, d) \iff a = c \text{ and } b = d$$ So $(1, 2) \neq (2, 1)$, and $(3, 3)$ is a genuine pair whose coordinates happen to agree. **Why a set of two will not do.** Since $\{1, 2\} = \{2, 1\}$, a set loses order, and since $\{3, 3\} = \{3\}$, a repeated entry collapses (roster-notation). An ordered pair is a new kind of object. It can be built from sets (one standard choice is $\{\{a\}, \{a, b\}\}$), but nothing here depends on how, only on the rule above. **Cartesian product.** For sets $A$ and $B$, the **Cartesian product** is the set of all ordered pairs with first coordinate in $A$ and second coordinate in $B$: $$A \times B = \{(a, b) : a \in A,\ b \in B\}$$ It is read "$A$ cross $B$". We write $A^2 = A \times A$; it holds every pair of members of $A$, including each $(a, a)$, and both $(a, b)$ and $(b, a)$. ### [Closed form versus recursive form](https://math.wiki/closed-form-versus-recursive/) **Summary:** A recursive rule needs every earlier term, while a closed formula gives any term straight away. **Question:** To find the 200th term of a sequence, do you have to work out the 199 terms before it? ## Why this exists A sequence can be told in two ways. "Start at $4$ and add $7$ each time" says how to get from one term to the next. "The term in position $n$ is $7n - 3$" says how to get any term straight from its position. Both describe the same list, $4, 11, 18, 25, \dots$ The first way is how rules usually arrive. A reading challenge says "each day, read $5$ more pages than the day before"; a pendulum loses the same fraction of its reach on every swing; a savings plan says "each week, put in $2$ dollars more". Each describes a change from one term to the next. A step rule has a cost, though. To know the $200$th term you need the $199$th, and for that the $198$th, and so on back to the start: $199$ uses of the step. A formula in $n$ answers in one line: $7(200) - 3 = 1397$. This lesson is about moving between the two kinds of rule, and about telling when two rules, written differently, describe the same sequence. ## Definition A sequence is a function whose inputs are the counting numbers $1, 2, 3, \dots$, and the output for input $n$ is the term $a_n$ (sequence-basics). There are two common ways to give that function. A **recursive rule** gives the first term and a step that makes each term from the one before it (recursive-formula): $$a_1 = 4, \qquad a_n = a_{n-1} + 7 \ \text{ for } n \ge 2$$ An **explicit formula**, also called a **closed form**, gives each term directly from its position $n$, with no earlier term needed: ### [Combinatorial proof: counting one set two ways](https://math.wiki/combinatorial-proof/) **Summary:** An identity is proved by showing that both sides count the same collection in two different ways. **Question:** Two expressions give the same number every time you test them. How can you see why they must agree, without pushing symbols around? ## Why this exists An identity among binomial coefficients can be checked by algebra, writing each side with factorials and simplifying. That proves it but rarely shows why it holds, and for a sum such as $\sum_k k(n - k)\binom{n}{k}^2$ the algebra is long. If one collection of objects is counted in two ways, the two answers are the same number, however different they look on paper. Pascal's rule was proved this way, and so were the row sum $2^n$ and the diagonal sum. This lesson makes the method explicit and adds its second form: matching two collections one to one. ## Definition This lesson takes two days. Day 1 is double counting (Examples 1 to 4); Day 2 is bijective proof (Examples 5 to 7). Throughout, $\lvert S \rvert$ is the number of members of a finite set $S$, and $\binom{n}{k}$ is the number of $k$-element subsets of an $n$-element set (combination-notation), taken as $0$ when $k < 0$ or $k > n$. A **combinatorial proof** of an identity $A = B$, where both sides are counts (nonnegative integers) for every allowed value of the variables, is a proof of one of two kinds. - **Double counting.** Name one finite set $S$. Show by one way of counting that $\lvert S \rvert = A$, and by another that $\lvert S \rvert = B$. - **Bijective proof.** Name finite sets $S$ and $T$ with $\lvert S \rvert = A$ and $\lvert T \rvert = B$, and give a bijection $f : S \to T$. Each count must count every member exactly once. Usually this means sorting $S$ into cases that do not overlap and together cover $S$, then counting each case in stages by the multiplication principle. ### [Composing functions between sets, and when an inverse exists](https://math.wiki/composition-and-inverses-on-sets/) **Summary:** Following one function with another gives a composite, and a function has an inverse exactly when it is a bijection. **Question:** When one function is followed by another, which of one to one and onto survive the chaining, and when can a function be run backward? ## Why this exists A function can be fed into another: the outputs of the first become the inputs of the second. With real-number formulas this is composing algebraically, and undoing a formula is finding an inverse by swapping and solving. Between arbitrary sets (strings, subsets, finite sets of labels) there is no formula to solve, and two questions take its place. When one function follows another, does the chain keep the properties of one to one and onto? And exactly when can a function be run backward, so that a second function undoes it? The answers are short theorems, used constantly afterward: to prove a function is a bijection by writing down its inverse, and to compare sizes of sets by chaining matchings. ## Definition ### Day 1: composition and identity Let $f : A \to B$ and $g : B \to C$. The **composite** of $f$ and $g$ is the function $g \circ f : A \to C$ given by $$(g \circ f)(a) = g\big(f(a)\big) \quad \text{for every } a \in A.$$ It is read right to left: $f$ acts first, then $g$. The composite is defined only when the codomain of $f$ is the domain of $g$, so that every output of $f$ is something $g$ accepts. Two functions $f, f' : A \to B$ are **equal** when they have the same domain, the same codomain, and $f(a) = f'(a)$ for every $a \in A$. Proving two functions equal means checking every input. For any set $A$, the **identity function** $\mathrm{id}_A : A \to A$ is $\mathrm{id}_A(a) = a$. It is a bijection. ### [Conditionals, converse, inverse, contrapositive](https://math.wiki/conditionals-and-converse/) **Summary:** If p then q, and the three ways to turn it around; only one keeps the truth. **Question:** Almost every theorem says \"if this, then that\". What exactly does such a sentence promise, and which of the three ways of turning it around still makes the same promise? ## Why this exists Open any mathematics book at a theorem and you will almost always find the same shape underneath: *if* something holds, *then* something else holds. "If two angles are vertical angles, then they are equal." "If a triangle has a right angle, then the squares on its two shorter sides add up to the square on the longest." Even a sentence that does not look like one usually hides one: "every multiple of $4$ is even" means *if* a whole number is a multiple of $4$, *then* it is even. Such a sentence is called a **conditional**, and two things have to be settled before it can be used safely. First, exactly what does it promise, and in which situations is it broken? Second, people turn conditionals around all the time, often without noticing. From "every multiple of $4$ is even" it is tempting to slide to "every even number is a multiple of $4$", which is false, since $6$ is even and not a multiple of $4$. There are three natural ways to turn a conditional around, and only one of them is guaranteed to say the same thing as the original. Knowing which one is the difference between a correct proof and a convincing-looking wrong one. ## Definition For statements $P$ and $Q$, the **conditional** "if $P$, then $Q$" is written $P \to Q$. The statement $P$ after "if" is its **hypothesis** (also called the antecedent), and the statement $Q$ after "then" is its **conclusion** (the consequent). Many books, especially in geometry, write the same arrow as $P \Rightarrow Q$. ### [Congruence modulo n and its arithmetic](https://math.wiki/congruence-notation/) **Summary:** Two numbers are congruent when they leave the same remainder on division by a fixed number. **Question:** A clock forgets how many times its hand has gone around, and a weekday forgets how many weeks have passed. Can we do arithmetic that keeps only the remainder, and trust every step of it? ## Why this exists Ask what day of the week it will be $260$ days from now, or what the last two digits of $3{,}147 \cdot 5{,}286$ are, and most of each number is irrelevant. Only a remainder matters: on division by $7$ for the weekday, by $100$ for the digits. Congruence is the language for "equal as far as the remainder can tell." Its value is that it behaves like equality under addition, subtraction and multiplication, so a large number may be swapped for a small one partway through a computation without changing the remainder at the end. This lesson sets up that language on the first day and proves the arithmetic rules on the second, together with the one place where the analogy with equality breaks: division. ## Definition ### Day 1: congruence and remainders Fix a positive integer $n$, the **modulus**. For integers $a$ and $b$, $$ a \equiv b \pmod{n} \quad \text{means} \quad n \mid (a - b). $$ It is read "$a$ is congruent to $b$ modulo $n$." When $n \nmid (a - b)$ we write $a \not\equiv b \pmod{n}$. Since $n \mid (a - b)$ means $a - b = kn$ for some integer $k$ (divisibility-as-a-relation), an equivalent form is $$ \begin{aligned} a \equiv b \pmod{n} &\iff a = b + kn \\ &\phantom{\iff} \text{ for some integer } k. \end{aligned} $$ So the integers congruent to $b$ are $b$ and everything reached from it by whole steps of $n$, forward or backward. The examples use $n \ge 2$; with $n = 1$ every pair of integers is congruent, since $1$ divides everything. ### [Conjecture, and why a pattern is not a proof](https://math.wiki/conjecture/) **Summary:** A statement someone believes is true but that nobody has proved yet. **Question:** Something has held every time you tried it. How do you state it so that it can be tested, and what has to happen before it stops being a guess? ## Why this exists Mathematics is found before it is proved. Someone adds up the first few odd numbers and gets square numbers, or colors a map and never needs a fifth color, and a guess forms. Most of the time the guess is right. Sometimes it survives for centuries with no proof, and sometimes it breaks on the very next case. A guess like that needs a status. It is not a fact, because nothing has shown it always holds, so nothing safe can be built on it. It is not trash either, because it points at something real. **Conjecture** is the name for the middle state: a claim that is worth working on and is not yet safe to use. The word keeps everyone honest. Whoever hands one over says plainly that it is unproved, and whoever builds on one says plainly that the building is only as sure as the guess beneath it. History shows how long the middle state can last. The claim that four colors always suffice for a map was made in 1852 and proved in 1976. Fermat wrote his famous claim in a margin around 1637 and it was proved in 1994. Goldbach set out his in a letter in 1742, and it is still open today. It is tempting to shorten the wait by checking more cases. Draw a circle and mark one point on it. The inside is one region. Mark a second point and join the two with a straight chord: two regions. Three points, with all three chords between them, give four regions. Four points give eight, and five points give sixteen. ### [Countable sets](https://math.wiki/countable-sets/) **Summary:** An infinite set is countable when its members can be lined up in a list that goes on forever. **Question:** An infinite set can never be counted to the end. When can its members still be put in one list, first, second, third and on forever, so that each of them has a place? ## Why this exists matching-sets-one-to-one defined "same size" by a bijection and found that $\mathbb{N}$ is the same size as the even natural numbers, and that a full hotel with one room for each natural number can take one more guest. Throughout, $\mathbb{N} = \{0, 1, 2, \dots\}$. Matching a set with $\mathbb{N}$ is the same as writing its members in one list that never ends, each member at some numbered position. The surprise is how much can be listed: the integers, the pairs of natural numbers, every fraction, every finite piece of text. ## Definition ### Day 1: countable and countably infinite A set $A$ is **countably infinite** when it is the same size as $\mathbb{N}$, that is, when there is a bijection $f : \mathbb{N} \to A$. Some books write $\lvert A \rvert = \aleph_0$. A set is **countable** when it is finite or countably infinite. Some books use "countable" for countably infinite only (and call such a set "denumerable"); check which meaning a book uses before quoting one of its theorems. A **list** of $A$ is a sequence $a_0, a_1, a_2, \dots$, one term for each $n \in \mathbb{N}$, in which every member of $A$ appears exactly once. Writing $a_n = f(n)$ turns a bijection $f : \mathbb{N} \to A$ into a list and back. ### Day 2: building countable sets Day 2 needs two words. A **string** over a finite set of symbols (an **alphabet**) is a finite sequence of symbols, such as $\texttt{cab}$ over $\{\texttt{a}, \texttt{b}, \texttt{c}\}$; the empty string has no symbols. A number is **algebraic** when it is a root of a nonzero polynomial with integer coefficients. ### [Counterexample](https://math.wiki/counterexample/) **Summary:** One case that fails kills a universal claim. **Question:** Someone claims a rule works for every number. What does it take to prove them wrong, and what can never do it? ## When you reach for it A claim that something holds for **every** member of a collection is a universal statement, a "for all". Proving one takes an argument that covers every member at once. Refuting one takes much less: a single member for which it fails. That member is a **counterexample**. Reach for one whenever a claim says "every", "all", "always" or "never", or hides that word inside an "if ... then" ("if a number is prime, it is odd" means every prime is odd). It is the first test to run on a guess of your own, before spending an hour trying to prove it, and the fastest answer to someone else's. ## The procedure **What counts.** For a claim $\forall x \in D,\ P(x)$, a counterexample is a member $a$ of the domain $D$ with $P(a)$ false. Most claims have the shape "for every $x$, if $H(x)$ then $C(x)$", and then a counterexample is a member that makes the hypothesis $H$ **true** and the conclusion $C$ **false**. A member that makes $H$ false cannot break the claim, because an if-then with a false "if" part is true (the truth table for the conditional). **Why one is enough.** Negating the claim gives $$\neg\big(\forall x \in D,\ P(x)\big) \;\equiv\; \exists x \in D,\ \neg P(x)$$ and a "there exists" is proved by showing one object that does the job. The counterexample is that object. **Where to look.** A claim is usually a generalization from the cases its author had in mind, so its failures hide in the cases they did not picture: $0$, $1$, negative numbers, fractions between $0$ and $1$, two values that are equal, the smallest allowed case, a square counted as a rectangle. A second place to look is the claim's structure: factor it, and ask which inputs would make every term share a factor, or make two terms cancel. A third is a failed proof. Try to prove the claim, and note the step that needs something extra, such as "$x$ is positive". Inputs without that extra property are where to search. ### [De Morgan's laws for statements](https://math.wiki/de-morgans-laws-logic/) **Summary:** Negating an and turns it into an or, and negating an or turns it into an and. **Question:** Negating \"P and Q\" looks like it should give \"not P and not Q.\" It does not. What is the actual negation, and why does the connective itself have to change? ## Why this exists You already know how to negate a single statement, and you know exactly what $\land$ and $\lor$ each demand: an and is hard to satisfy, since both halves must hold, and an or is easy, since one is enough. So a natural guess for negating a compound statement is to leave the connective alone and just negate each piece: surely "not (raining and cold)" is "not raining and not cold"? It is not, and a quick check shows why. Suppose it is raining but not cold. "Raining and cold" is false, so its negation should be true. But "not raining and not cold" is also false: it is raining, so "not raining" already fails. The guess and the true negation disagree here, and again in the mirror case, cold but not raining. A conjunction is easy to break, one failing part is enough, so its negation, the statement that is true whenever the conjunction breaks, has to be just as easy to satisfy. That means the negation of an "and" must itself be an "or". Getting this exactly right matters whenever an argument has to say precisely what it is ruling out, which is most of the time in a proof. ## Definition For statements $P$ and $Q$, $$\neg(P \land Q) \equiv \neg P \lor \neg Q$$ $$\neg(P \lor Q) \equiv \neg P \land \neg Q$$ where $\equiv$ reads "is logically equivalent to": the two sides are true in exactly the same rows of a truth table. These are **De Morgan's laws**, named for the nineteenth-century mathematician Augustus De Morgan. In words: the negation of an "and" is the "or" of the negations, and the negation of an "or" is the "and" of the negations. Two things happen at once. The connective flips, $\land$ becomes $\lor$ and $\lor$ becomes $\land$, and the $\neg$ that sat outside the whole statement moves inside, landing separately on $P$ and on $Q$. ### [Degree, the handshake lemma and complete graphs](https://math.wiki/degree-of-a-vertex/) **Summary:** How many edge ends meet at a vertex. **Question:** Every link in a network has two ends. What does that one fact force on the number of links at each point, and which lists of link counts can a network actually have? ## Why this exists Computers joined by cables, people who know each other, cities joined by flights: each is a set of points with some pairs linked. Graphs are the language for all of them. The first number to ask about a point is how many links meet there, its degree. Degrees are local, one point at a time, yet they add up to a fact about the whole network: every link has two ends, so the degrees always sum to twice the number of links. That one count rules out plans before anyone builds them, and it starts the question of which lists of degrees a network can have at all. ## Definition This lesson takes two days. Day 1 is degree, the handshake lemma, complete graphs and regular graphs (Examples 1 to 4). Day 2 is degree sequences, complements and degree bounds (Examples 5 to 7). A **graph** $G = (V, E)$ is a finite nonempty set $V$ of **vertices** together with a set $E$ of **edges**, each edge an unordered pair $\{u, v\}$ of two different vertices. The edge $\{u, v\}$ is written $uv$; $uv$ and $vu$ are the same edge. Its two vertices are its **endpoints**, and they are **adjacent**, or **neighbors**. A drawing shows each vertex as a dot and each edge as a line between its endpoints; where two lines cross in a drawing there is no vertex unless a dot is drawn. The number of vertices is $\lvert V \rvert$ and the number of edges $\lvert E \rvert$. Because an edge is a set of two different vertices, no vertex is joined to itself (no **loops**), and because $E$ is a set, no pair is joined twice (no **repeated edges**). Graphs with both properties are called **simple**, and every graph in this lesson is simple. A **multigraph** allows loops and repeated edges; some later problems need one. ### [Direct proof](https://math.wiki/direct-proof/) **Summary:** Start from what is given and reason step by step to the claim. **Question:** You are given a hypothesis and asked for a conclusion. How do you get from one to the other with every step justified? ## What this area is about A direct proof of "if $p$, then $q$" assumes $p$ and reasons forward, one justified step at a time, until it reaches $q$. Each step uses a definition, an earlier result or a rule of algebra or logic, and nothing else. The two lessons here practice it on integers, where the key move is the same both times: replace a word such as "even" or "divides" by the equation that defines it, do algebra on the equation, then read the word back off the result. ## How these fit together Both children are procedures built on a definition. Proving statements about even and odd numbers uses $n = 2k$ for even and $n = 2k + 1$ for odd, with $k$ an integer, to prove rules for sums, products and powers, and to show some equations have no integer solution. Proving statements about divisibility uses the definition that, for integers $a$ and $b$, $a \mid b$ means $b = ak$ for some integer $k$. It proves the divisibility rules that later proofs cite and applies them to expressions. ## A path through it Even and odd first. The divisibility lesson reuses its shape, with the divisor $2$ replaced by any integer, so it reads best second. Direct proof is the base the other methods start from: proof by contradiction, proof by contraposition, proof by cases and proving a set identity all assume it. ### [Divisibility as a relation](https://math.wiki/divisibility-as-a-relation/) **Summary:** One integer divides another when the second is an integer multiple of the first. **Question:** Inside the integers, where dividing does not always work, how do we say exactly that one number goes into another, and what kind of comparison between two numbers is that? ## Why this exists Add, subtract or multiply two integers and you get an integer. Divide them and you often do not: $12 \div 5 = 2.4$ has left the integers. Number theory needs a way to say "$3$ goes into $12$ evenly" that never leaves them. The way out is to ask about multiplication instead: $12$ is $3$ times some integer. Written this way, divisibility becomes a *relation*: a yes-or-no comparison between two integers, like $a \le b$, with rules of its own that can be proved. The earlier work on factors (divisibility-and-primes, and listing factors in find-all-factors-of-a-number) used the idea for positive whole numbers; here it is made exact for every integer. ## Definition For integers $a$ and $b$, we say **$a$ divides $b$**, and write $a \mid b$, when there is an integer $k$ with $$b = ak.$$ Then $a$ is a **divisor** (or **factor**) of $b$, and $b$ is a **multiple** of $a$. When no such integer exists we write $a \nmid b$, "$a$ does not divide $b$". So $3 \mid 12$ because $12 = 3 \cdot 4$, and $5 \nmid 12$ because $5 \cdot 2 = 10$ and $5 \cdot 3 = 15$ step over $12$. Nothing in the definition excludes $a = 0$ or negative numbers; those cases are settled below. An integer $n$ is even exactly when $2 \mid n$, which is the definition used in proof-with-even-and-odd. **A statement, not a number.** $a \mid b$ is a sentence. It is either true or false, the way $7 < 2$ is false. The fraction $\frac{b}{a}$ is a number. They answer different questions: $3 \mid 12$ says "yes, $3$ goes in evenly", and $\frac{12}{3}$ says "it goes in $4$ times". So "$3 \mid 12 = 4$" means nothing, and $3 \mid 12$ is not "$\frac{3}{12}$" either. Notice the order: in $a \mid b$ the divisor comes first, in $\frac{b}{a}$ it comes second. ### [Elements and membership](https://math.wiki/element-and-membership/) **Summary:** An object either belongs to a set or does not, and a set is decided entirely by what belongs to it. **Question:** When you gather things into one collection, what does it mean for a thing to belong to it, and when are two collections the same? ## Why this exists Mathematics is full of collections: the whole numbers from $1$ to $10$, the solutions of an equation, the points on a line, the letters of a word. Before anyone can say something true about a collection, two plain questions need settled answers. Is this particular thing in it, or not? And when are two collections the same collection? Without answers, arguments about collections fall apart. Is the collection of letters in "LOOT" the same as the collection of letters in "TOOL"? Is a collection that lists the number $7$ twice different from one that lists it once? A collection of solutions found by factoring and a collection of solutions found by a graph: are they the same? The idea of a **set** gives firm answers to both. A thing is either in a set or it is not, with nothing in between. And a set is decided **entirely by what is in it**: nothing else about how it was described or written down matters. Every later idea about sets, subsets, unions, counting and proof, rests on these two sentences. ## Definition A **set** is a collection of objects. The objects in it are called its **elements**, or its **members**. For this lesson, numbers and letters count as plain objects, not as collections. The simplest way to name a set is to list its elements between braces and give the set a capital letter: $$A = \{2, 4, 6, 8\}$$ This is read "$A$ is the set whose elements are $2$, $4$, $6$ and $8$". ### [Euler paths and circuits](https://math.wiki/condition-for-an-euler-circuit/) **Summary:** A connected graph has a closed route using every edge once exactly when every vertex has even degree. **Question:** Can a walk cross every bridge of a city exactly once, and how could anyone tell without trying every route? ## Why this exists The old city of Königsberg, today Kaliningrad, was built where its river forks around an island, so four pieces of land faced each other across the water, joined by seven bridges. The puzzle, as it is usually told, asked for a walk crossing each bridge exactly once. Trying routes is slow, and failing to find one proves nothing. Euler settled the question in 1736 with an argument that forgets the shape of the map and keeps only which land each bridge touches. The same question comes back whenever every link of a network must be covered once: a mail carrier's streets, a snowplow's route, a drawing made without lifting the pen. One count, the degree of each vertex, answers all of them. ## Definition This lesson takes two days. Day 1 is Euler circuits and trails and the theorem that decides them (Examples 1 to 4). Day 2 is building a circuit, adding edges, and route problems (Examples 5 to 7). The words are those of walks-paths-and-cycles: a walk $(v_0, v_1, \ldots, v_k)$ is closed when $k \ge 1$ and $v_k = v_0$, a trail is a walk using no edge twice, and a graph is connected when every two vertices are joined by a walk. Degree is as in degree-of-a-vertex. The bridge map has two bridges between the same pair of land areas, so this lesson allows **repeated edges**. A **multigraph** here is a graph in which two vertices may be joined by more than one edge (but no edge joins a vertex to itself). Repeated edges are told apart by names, and a walk lists them, $(v_0, e_1, v_1, \ldots, e_k, v_k)$ with $e_i$ joining $v_{i-1}$ and $v_i$. A list of vertices alone is enough when no pair is passed more often than it has edges: each pass takes a different one. The degree $\deg(v)$ counts edge ends at $v$, each repeated edge separately, and the handshake lemma $\sum_v \deg(v) = 2\lvert E \rvert$ holds with the same proof. A vertex is **odd** or **even** as its degree is. Every simple graph is a multigraph. ### [Exclusive or and logic circuits](https://math.wiki/exclusive-or-and-logic-circuits/) **Summary:** Exclusive or is true when exactly one part is true, and gates for and, or and not wire statements into circuits that compute. **Question:** How can one extra bit reveal that a message was changed on the way, and how can parts that only compute and, or and not end up adding numbers? ## Why this exists And and or gave exclusive or, $P \oplus Q$, "$P$ or $Q$ but not both", its table, and the biconditional showed it is $\neg(P \leftrightarrow Q)$: true exactly when the parts differ. It earns a lesson of its own for two reasons. First, it has an algebra that and and or lack. A statement exclusive-or'ed with itself vanishes, and exclusive or with T is negation. So a long chain of exclusive ors is easy to evaluate: it is true exactly when an odd number of its parts are true. One extra bit sent with a message, a **parity bit**, uses this to let the receiver notice that a bit was changed on the way. Second, a connective can be built. A **logic gate** is a small electronic part whose output is one connective of its inputs. Wired together, gates compute any compound statement, and an XOR gate with an AND gate adds two one-bit numbers. ## Definition **Exclusive or.** $P \oplus Q$ is true when exactly one of $P$ and $Q$ is true: when $P$ and $Q$ have different truth values. In the usual row order TT, TF, FT, FF its column is F, T, T, F. **Bits and parity.** In computing a truth value is a **bit**: $1$ for T and $0$ for F. A message is a string of bits $b_1 b_2 \ldots b_n$. Its **even parity bit** is $$p = b_1 \oplus b_2 \oplus \cdots \oplus b_n,$$ and the sender transmits the **word** $b_1 b_2 \ldots b_n p$. (No parentheses are needed in the chain, by the associative law below.) ### [Existence proof](https://math.wiki/existence-proof/) **Summary:** A proof that at least one object with a stated property exists. **Question:** Is there one? How do you show that something exists without searching forever? ## Why this exists Many questions in mathematics begin with "is there one?" Is there a real number whose cube plus itself equals $1$? Is there a whole number that is a perfect square and a perfect cube? Is there, among any five whole numbers, a pair with some special property? Searching is the obvious way to answer, and it has two weaknesses. When the collection is infinite the search never ends, and when a search comes up empty it proves nothing, because the thing may be just past where you stopped looking. An **existence proof** is an argument that settles the question in finitely many steps, whatever the size of the collection. There is also a lopsidedness that makes existence the friendliest kind of claim to prove. To show that *every* whole number has a property, no list of examples is ever enough. To show that *some* whole number has it, a single example is the entire proof. ## Definition A statement that something exists has the form $$\exists x \in S\;\; P(x)$$ read "there is an $x$ in $S$ such that $P(x)$". Here $S$ is the collection being searched and $P$ is the property wanted. Every existence claim has a collection, stated or understood: "there is a number with $x^2 = 2$" means one thing among the real numbers and another among the fractions. The symbol $\exists$ is the existential quantifier, and "exists" always means **at least one**: one object, ten, or infinitely many all make the statement true. ### [Fermat's little theorem](https://math.wiki/fermats-little-theorem/) **Summary:** Raising a number to the power one less than a prime gives 1 modulo that prime, unless the prime divides it. **Question:** For a prime p and a number a that p does not divide, the powers of a modulo p always come back to 1. Can we say in advance which power does it, and does the same thing tell a prime from a composite number? ## Why this exists In modular-exponentiation the powers of a number modulo $n$ were found to repeat in a cycle, and when a power reached $1$ the exponent could be cut down to its remainder. But finding where the cycle closes meant listing powers until $1$ appeared, and that lesson only stated, without proof, that $1$ does appear when $a$ and $n$ have no common factor. For a prime modulus Fermat's little theorem settles it in advance: the power $a^{p-1}$ is always $1$ modulo $p$. That turns exponents in the billions into small ones, gives a formula for inverses, and gives a way to prove a number composite without finding a single factor of it. ## Definition ### Day 1: the theorem and two proofs Let $p$ be a prime and $a$ an integer. **Fermat's little theorem** says two equivalent things. - If $p \nmid a$, then $a^{p-1} \equiv 1 \pmod{p}$. - For every integer $a$, $a^p \equiv a \pmod{p}$. The first form is the useful one for computing; the second needs no condition on $a$. They say the same thing: when $p \mid a$ the second form reads $0 \equiv 0$, and when $p \nmid a$ the two differ by a factor of $a$, which may be multiplied in (Theorem 3 of congruence-notation) or canceled (Lemma 2 below). ### Day 2: what the theorem is used for Day 2 keeps the theorem and changes the question. Read forward, it computes inverses modulo $p$. Read backward, it is a test: a number $n$ with $a^{n-1} \not\equiv 1 \pmod{n}$ for some $a$ that $n$ does not divide cannot be prime. For a modulus that is not prime, **Euler's phi function** $\varphi(n)$ counts the integers from $1$ to $n$ that share no factor greater than $1$ with $n$ (inclusion-exclusion-principle computed it), and $\varphi(n)$ takes the place of $p - 1$. ### [Finding a shortest path](https://math.wiki/shortest-path-in-a-weighted-graph/) **Summary:** Growing a set of settled vertices outward, each time taking the cheapest edge that reaches something new. **Question:** A map gives the miles on every road. From one town, what is the shortest drive to every other town, and how can it be found without listing every possible route? ## When you reach for it You have a weighted graph, or a weighted directed graph, whose weights are all $\ge 0$: miles, minutes or dollars. You have a start $s$, and you want the weighted distance $d_w(s, v)$ of weighted-graphs to every vertex $v$, with a cheapest route. Listing every path works for five vertices, but in a network where every two of $10$ places are joined there are $109{,}601$ paths between two of them (Example 7). **Dijkstra's algorithm** finds every distance from $s$ in about $v$ rounds for $v$ vertices. This lesson takes two days. Day 1 is the method, the route it records, and the proof that it is right (Theorem 1, Examples 1 to 3). Day 2 is the tree of routes, the subpath property, directed graphs, unweighted graphs, negative weights and the count of the work (Theorems 2 and 3, Examples 4 to 7). Walks, paths and the plain distance $d$ are those of walks-paths-and-cycles. Weights here may be $0$. A cheapest walk may then repeat a vertex, but cutting out the repeat removes a stretch of weight $\ge 0$, so $d_w(s, v)$ is still the least weight of a path from $s$ to $v$. ## The procedure ### Day 1: settling one vertex a round The method keeps a set $S$ of **settled** vertices, whose distances from $s$ are known. Every vertex $v$ carries a **label** $\ell(v)$, the weight of the cheapest route to $v$ found so far, and a **predecessor** $p(v)$, the vertex before $v$ on that route. At the start $S$ is empty, $\ell(s) = 0$, and $\ell(v) = \infty$ for every other $v$. ### [For all](https://math.wiki/universal-quantifier/) **Summary:** A claim that a statement holds for every object in a stated collection. **Question:** A rule has worked in every case anyone has tried. What would it mean for it to hold in every case, and how could anyone check infinitely many? ## Why this exists In letters of 1742, Christian Goldbach and Leonhard Euler discussed a guess that now carries Goldbach's name: every even number greater than $2$ is the sum of two primes. $4 = 2 + 2$, $6 = 3 + 3$, $8 = 3 + 5$, $10 = 5 + 5$, and it keeps working. Computers have checked every even number up to $4 \times 10^{18}$ and found no exception. The guess is still not a theorem. It is a claim about **every** even number, and past any point that has been checked there are infinitely many more. In 1769 Euler made a guess of the same shape: among other things, that four fifth powers of positive whole numbers can never add up to a fifth power. It stood for almost two hundred years, until in 1966 L. J. Lander and T. R. Parkin, searching by computer, found $$27^5 + 84^5 + 110^5 + 133^5 = 144^5$$ One equation, and the claim was dead. Those are the two faces of a claim about everything: no pile of successes proves it, and one failure destroys it. Most theorems have this shape ("the angles of **every** triangle add to $180^\circ$"), and English says it with "all", "every", "each", "any" and "always", not always in the same way. A proof needs one exact meaning and a symbol for it. ## Definition As for "there exists", start with a collection of objects $D$, the **domain**, and a **predicate** $P(x)$: a sentence about $x$ that becomes true or false once $x$ is replaced by a member of $D$. The statement **"for all $x$ in $D$, $P(x)$"** is true when **every** member of $D$ makes $P(x)$ true, and false when **at least one** member makes it false. In symbols: ### [Generating a sequence from a rule](https://math.wiki/sequence-from-a-rule/) **Summary:** Starting from the given first terms and applying the rule again and again to get the next ones. **Question:** You know how a sequence starts and how each term makes the next. What does the whole sequence look like, and what does it do that the rule never said? ## When you reach for it A rule such as "double the term and subtract $1$" says nothing on its own. It only turns one number into the next. Give it a first term and it produces a whole sequence, one term at a time: from $4$ it gives $7$, from $7$ it gives $13$, and so on for as long as you care to go. You reach for this move whenever a sequence is described by **how it grows** rather than by where each term sits: a count that builds on the counts before it, a distance logged hour by hour, a game counter whose next square depends on the square it is on. Rules like "add $3$" and "multiply by $2$" appeared in patterns from a rule and the basics of sequences. Here the rules square, mix two operations, look back two terms, or treat even and odd terms differently. The second half of the skill is to **look at what comes out**. A rule often produces features it never states: every term odd, ones digits that cycle, two sequences in a fixed ratio. Spotting a feature is useful; explaining it from the rule is what makes it certain. ## The procedure **Starting terms.** A rule names the terms it uses. "Double and subtract $1$" uses only the term just before, so one starting term settles everything. "Add the two terms before" uses two, so it needs two starting terms: a single $1$ has nothing to be added to. The number of starting terms matches how far back the rule looks. Without them the rule makes no sequence; with them it makes exactly one, because each new term is forced by terms already known. ### [Graph coloring](https://math.wiki/chromatic-number/) **Summary:** The smallest number of colors a given graph can manage with. **Question:** Some pairs of meetings cannot happen at the same time. What is the fewest number of time slots that works, and why can every map, however tangled, be colored with only four colors so that no two neighboring countries match? ## Why this exists Exams that share a student cannot be given in the same period. Rehearsals that share a player cannot be held in the same hour. Neighboring countries on a map should not get the same color. Each time things in conflict must be kept apart in as few groups as possible. Join two things that conflict, and a plan is a coloring of the graph in which joined vertices differ; the fewest groups is the chromatic number. For maps the graph is planar, and eulers-formula-for-planar-graphs gives the fact that shows six colors always suffice. ## Definition This lesson takes two days. Day 1 is colorings, the chromatic number and its bounds, with scheduling (Results 1 to 5, Examples 1 to 3). Day 2 is maps and planar graphs, and a bound linking a graph to its complement (Results 6 to 10, Examples 4 to 6). Graphs are finite and simple, in the notation of degree-of-a-vertex ($\deg(v)$, $\Delta(G)$, $K_n$, the complement $\overline{G}$) and walks-paths-and-cycles (the cycle graph $C_n$ on the vertices $0, 1, \ldots, n - 1$). $H$ is a **subgraph** of $G$ when every vertex and every edge of $H$ is one of $G$, and $G - u$ is $G$ with the vertex $u$ and its edges erased. - A **coloring** of $G$ with $k$ colors is a function $\varphi : V \to \{1, 2, \ldots, k\}$. It is **proper** when $\varphi(u) \ne \varphi(v)$ for every edge $uv$. The numbers are only names for colors. $G$ is **$k$-colorable** when it has a proper coloring with $k$ colors. - The **chromatic number** $\chi(G)$ is the smallest $k$ for which $G$ is $k$-colorable. Giving each vertex its own color is proper, so $\chi(G) \le \lvert V \rvert$ and the smallest $k$ exists. - The vertices of one color form a **color class**. A set of vertices no two of which are adjacent is **independent**, so a coloring is proper exactly when every color class is independent. - A **clique** is a set of vertices every two of which are adjacent. The **clique number** $\omega(G)$ is the size of a largest clique. - **Greedy coloring** in the order $v_1, v_2, \ldots, v_n$ gives each $v_i$ in turn the smallest positive integer not already given to a neighbor of $v_i$ among $v_1, \ldots, v_{i-1}$. - (Day 2) A **map** divides part of the plane into countries, each one connected piece. Two countries are **neighbors** when their borders share a stretch of positive length; meeting at a single point does not count. A map is colored properly when neighbors get different colors. ### [Graph isomorphism: when two drawings are the same graph](https://math.wiki/graph-isomorphism/) **Summary:** Two graphs are the same when some matching of their vertices keeps every edge, and a difference in degrees or cycles proves they are not. **Question:** Two drawings of a network look nothing alike. How do we prove they show the same network, and how do we prove they do not? ## Why this exists A drawing is only one way to show a graph, and two drawings of one graph can share nothing visible. Before anyone can say "there are $11$ graphs on four vertices" or "this network is a cycle", "the same graph" needs a precise meaning: a renaming of the vertices that carries edges to edges. The harder half is proving that no such renaming exists. ## Definition This lesson takes two days. Day 1 is the definition, proving graphs isomorphic, and invariants that prove they are not (Examples 1 to 3). Day 2 is isomorphism as an equivalence relation, counting graphs up to isomorphism, complements and automorphisms (Examples 4 to 6). The notation is that of degree-of-a-vertex and walks-paths-and-cycles: graphs are simple, $G = (V, E)$, $uv$ is the edge joining $u$ and $v$, and $\overline{G}$ is the complement. Let $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$ be graphs. An **isomorphism** from $G_1$ to $G_2$ is a bijection $f : V_1 \to V_2$ (one-to-one and onto, injective-surjective-bijective) such that for all $u, v \in V_1$, $$uv \in E_1 \quad \text{exactly when} \quad f(u)f(v) \in E_2.$$ - $G_1$ and $G_2$ are **isomorphic**, written $G_1 \cong G_2$, if there is an isomorphism from $G_1$ to $G_2$. - An **automorphism** of $G$ is an isomorphism from $G$ to itself; $\operatorname{Aut}(G)$ is the set of them. The identity map is always one. - An **invariant** is a number or property that is the same for any two isomorphic graphs. - The graphs isomorphic to $G$ form its **isomorphism class**; counting graphs **up to isomorphism** counts classes. A **labeled graph** on a set $S$ is a graph whose vertex set is $S$; two labeled graphs on $S$ are equal only when their edge sets are equal. - $G$ is **self-complementary** if $G \cong \overline{G}$. ### [Graphs, vertices and edges](https://math.wiki/graphs-vertices-and-edges/) **Summary:** A network as a set of dots joined by lines. **Question:** A network is points with some pairs linked. What can be said about it before anyone asks for a route through it? ## What this area is about A graph is a finite nonempty set of vertices together with edges, each edge joining two different vertices. That bare picture models cables, friendships and flights alike. This area sets up the vocabulary and first facts: how many edges meet at a vertex, what numbers or directions an edge can carry, how a graph is stored as a table, and when two drawings show the same graph. ## How these fit together **Definitions and a first result.** Degree, the handshake lemma and complete graphs defines simple graphs and degree, and proves that in every graph the degrees sum to $2|E|$. **Extra structure.** Weighted and directed graphs put a number on every edge, or an arrow on it, so that a route has a weight and degree splits into in and out. **Comparing and computing.** Graph isomorphism makes "the same graph" precise as a bijection of vertices that keeps every edge, and uses invariants such as degrees to prove two graphs differ. Adjacency matrices and their powers store a graph as a $0$ and $1$ matrix $A$; the entry in row $i$, column $j$ of $A^k$ counts the walks of length $k$ from $v_i$ to $v_j$. ## A path through it Degree first: the other three lessons borrow its definitions, and isomorphism needs it. Then take walks, paths and cycles: weighted graphs price its walks, and graph isomorphism and adjacency matrices both require it. Finish with those three, weighted graphs first. ### [Hamiltonian paths and cycles](https://math.wiki/hamiltonian-paths-and-cycles/) **Summary:** Routes that visit every vertex once, which unlike Euler routes have no simple test. **Question:** A route that uses every road exactly once has a simple test. What about a round trip that stops at every place exactly once? ## Why this exists Kepa services weather beacons on hilltops joined by service roads. He wants a round trip that stops at every beacon exactly once and ends where it began. For a route using every road exactly once, condition-for-an-euler-circuit gives a complete answer read off the degrees. A route through every vertex looks like its twin, yet no such test is known: only tests that rule a tour out and conditions that guarantee one. ## Definition The notation is that of degree-of-a-vertex and walks-paths-and-cycles: $G = (V, E)$ is a simple graph with $n = \lvert V \rvert$ vertices and smallest degree $\delta(G)$, and $K_n$ and $C_n$ are the complete graph and the cycle graph. - A **Hamiltonian path** in $G$ is a path containing every vertex of $G$, $(v_0, v_1, \ldots, v_{n-1})$, each vertex appearing once. Its length is $n - 1$. - A **Hamiltonian cycle** is a cycle containing every vertex, $(v_0, v_1, \ldots, v_{n-1}, v_0)$ with $v_0, \ldots, v_{n-1}$ all the vertices. Its length is $n$, so $n \ge 3$. Two Hamiltonian cycles are the same exactly when they use the same edges. - $G$ is **Hamiltonian** if it has a Hamiltonian cycle. - For a set $S$ of vertices, $G - S$ is what is left after deleting the vertices of $S$ and every edge with an endpoint in $S$. - The **$m \times n$ grid** has the $mn$ vertices $(i, j)$ with $1 \le i \le m$ (the row) and $1 \le j \le n$ (the column); two are adjacent when they agree in one coordinate and differ by $1$ in the other. Here $n$ counts columns, not vertices. - The **cube graph** $Q_3$ has the $8$ binary strings of length $3$ as vertices, two adjacent when they differ in exactly one place; $Q_d$ uses the $2^d$ strings of length $d$. - A **Gray code** of length $d$ lists all $2^d$ binary strings of length $d$ so that each differs in exactly one place from the next, and the last from the first: a Hamiltonian cycle in $Q_d$. ### [How induction works: base case, hypothesis and step](https://math.wiki/inductive-step/) **Summary:** The part of an induction that carries the statement from one value to the next one. **Question:** A claim about every whole number is an endless list of claims. How can two finite pieces of work, one of which assumes a case it has not proved, settle all of them? ## Why this exists A claim about every whole number is an endless list of claims in disguise. "The sum of the first $n$ positive whole numbers is $\frac{n(n+1)}{2}$" is really $P(1)$, $P(2)$, $P(3)$, and so on without end, and no amount of checking reaches the end of a list like that. Checking cases is how such a claim is usually found, which is inductive reasoning, and a claim that has held for every case checked so far can still fail at the next one, as why a pattern is not a proof shows. Something other than checking has to finish the job. What makes it possible is that the cases often come in a chain, each built from the one before. The total of ${k+1}$ terms is the total of $k$ terms with one more added. A collection of ${k+1}$ objects is a collection of $k$ objects with one more thrown in. If something is already known about the case for $k$, the case for ${k+1}$ may need only a small piece of extra work. Proof by induction turns that into a method with three named parts, and this lesson is about all three. - The **base case** proves the first statement on the list outright. - The **inductive hypothesis** is the assumption that one statement, $P(k)$, is true. - The **inductive step** is the argument that leads from $P(k)$ to the next statement, ${P(k+1)}$. The step is written once and used forever. Take the claim that the first $n$ even numbers add up to $n(n+1)$. To go from six of them to seven: the six add to $6 \cdot 7 = 42$, the seventh even number is $14$, and $42 + 14 = 56 = 7 \cdot 8$. To go from seven to eight: $56 + 16 = 72 = 8 \cdot 9$. These are the same argument with different numbers in it, and so would be the argument from $99$ to $100$. Nobody can write out all of them, so the argument is written once, with a letter in place of the number: ### [Infinite sets](https://math.wiki/infinite-sets/) **Summary:** A set whose members never run out, so its size cannot be given by a counting number. **Question:** When a set never runs out, what can "how many" still mean, and are all infinite sets the same size? ## What this area is about A finite set is counted by matching it with $\{1, 2, \ldots, n\}$. An infinite set has no last number to count to, but matching still works: two sets have the same size when there is a bijection between them. Taking that as the definition gives surprises. An infinite set can be the same size as part of itself, and not every infinite set is the same size as $\mathbb{N} = \{0, 1, 2, \ldots\}$. ## How these fit together **The definition.** Matching two sets one to one defines "same size" by a bijection. **The first size of infinity.** Countable sets are those that are finite or can be put in one endless list. The integers and the rationals are countable. **Results that go beyond it.** The diagonal argument proves that $\mathbb{R}$ is uncountable: any list of real numbers misses one. Cantor's theorem proves that no set $A$ is the same size as its power set $\mathcal{P}(A)$, so there is no largest infinity. ## A path through it The lessons form a chain, each using the one before: matching sets one to one, then countable sets, then the diagonal argument, then Cantor's theorem. Before starting, the area needs bijections and the cardinality of a finite set; the diagonal argument also uses proof by contradiction, and Cantor's theorem uses power sets. ### [Introductory graph theory](https://math.wiki/graph-theory-intro/) **Summary:** Dots and lines, and the puzzles they solve. **Question:** Bridges, cables, schedules and maps all reduce to dots and lines. Which questions about them can be settled by a count or a method instead of trying every possibility? ## What this area is about A graph is a set of vertices with edges joining some pairs. Forgetting everything else about a network, its shape and its scale, often makes a hard question exact: can a route use every link once, how many links must a connected network have, how few colors keep neighbors apart. The graphs here are finite and, unless a lesson says otherwise, simple. ## How these fit together **Foundations.** Graphs, vertices and edges gives degree, weighted and directed graphs, isomorphism and adjacency matrices. Walks, paths, cycles and connected graphs defines routes and components. **Routes.** Euler circuits: a connected graph with at least one edge has a closed route using every edge once exactly when every degree is even. Hamiltonian paths and cycles visit every vertex once, and no simple test for them is known. **Trees and algorithms.** Trees: a tree on $n$ vertices has $n - 1$ edges. Shortest paths (Dijkstra's algorithm, weights $\ge 0$) and minimum spanning trees are procedures on weighted graphs. **Drawing and coloring.** Euler's formula: a connected plane drawing has $V - E + F = 2$. Graph coloring and bipartite graphs, the graphs with a proper 2-coloring, follow. ## A path through it Start with degree, then walks, paths and cycles, then weighted graphs, isomorphism and adjacency matrices. After walks come Euler circuits, Hamiltonian cycles, shortest paths and trees (which also need induction); minimum spanning trees need weighted graphs. Euler's formula comes before coloring, and coloring before bipartite graphs. ### [Introductory number theory](https://math.wiki/number-theory-intro/) **Summary:** Properties of the integers, proved. **Question:** The integers can be added, subtracted and multiplied, but not always divided. What can be proved about them when every claim must hold for every integer? ## What this area is about Number theory studies the integers $\mathbb{Z}$, and here every fact is proved, not checked on examples. The central ideas are divisibility, remainders and common divisors. From them come working tools, such as the Euclidean algorithm and arithmetic modulo $n$, and classic results, such as the fact that the primes never run out. ## How these fit together **Definitions.** Divisibility as a relation: for integers $a$ and $b$, $a \mid b$ when $b = ak$ for some integer $k$. **Results that make remainders exact.** The division algorithm: for any integer $a$ and any integer $b > 0$ there are unique integers $q$ and $r$ with $a = bq + r$ and $0 \le r < b$. Numbers in other bases proves that every positive integer has exactly one numeral in each base $b \ge 2$. **Procedures.** The Euclidean algorithm finds $\gcd(a, b)$ by repeated division, and the extended Euclidean algorithm writes it as $ax + by$ with integers $x$ and $y$. Modular arithmetic works with remainders alone, up to inverses, Fermat's little theorem and RSA. **Classic results.** There are infinitely many primes, by contradiction, and triangular numbers, the totals $1 + 2 + \cdots + n$. ## A path through it Divisibility, then the division algorithm; base conversion follows it. Modular arithmetic is split: congruence notation comes after the division algorithm and before the Euclidean algorithm, then the extended Euclidean algorithm, and only then modular inverses and what follows them. The primes and triangular numbers stand apart from that chain: the first needs proof by contradiction, the second partial sums. ### [Logic](https://math.wiki/logic/) **Summary:** Statements, how they combine, and what follows from what. **Question:** Mathematics argues only about sentences that are true or false. How are they built, and when does one force another? ## What this area is about Logic is the grammar underneath every proof. It starts from statements, sentences that are true or false, asks how the truth of a sentence built from them depends on its parts, and ends with the question a proof has to answer: when do some statements force another to be true? ## How these fit together **The starting point.** Statements and negation: what counts as a statement, and $\neg P$, its opposite. **Building statements.** A connective joins statements into one whose truth depends only on the truth of its parts. Conjunction and disjunction: $P \land Q$ is true only when both parts are true, and $P \lor Q$ is false only when both are false. Conditionals cover "if ... then" and its turn-arounds. The biconditional, $P \leftrightarrow Q$, is true when the parts agree. Quantifiers, not connectives, add "for all" and "there exists". **Deciding statements.** Truth tables list every case. Logical equivalence: statements whose tables agree, as in De Morgan's laws. Exclusive or, $P \oplus Q$, is true when exactly one part is; logic gates wired together compute any compound statement. **Drawing conclusions.** What an argument is sorts valid forms from fallacies. ## A path through it Negation, then and, or, the conditional, if and only if. Then truth tables; after them equivalence, exclusive or and circuits; quantifiers and arguments in either order. Then you can say what a claim asserts, what would break it, and whether a step follows, which is what direct proof, proof by contraposition and proof by contradiction are built on. ### [Logic, proof and discrete math](https://math.wiki/logic-and-proof/) **Summary:** How mathematical truth is argued, plus sets, counting structures and a first look at linear algebra. **Question:** How does mathematics argue that something is true for every case, and which structures does it argue about first? ## What this area is about This branch is about how mathematical truth is argued, and about the discrete structures where that arguing is first practiced: the integers, sets, sequences defined by rules, and networks of dots and lines. Here a claim is accepted only when it is proved from definitions and earlier results, never because examples agree. ## How these fit together **The tools of argument.** Logic covers statements, connectives, quantifiers and valid arguments. Proof techniques turns them into methods: direct proof, contraposition, contradiction, cases, induction and more. **The language.** Sets gives the vocabulary of collections, functions and relations, up to the sizes of infinite sets. **Structures to prove things about.** Introductory number theory studies divisibility, remainders and common divisors in $\mathbb{Z}$, through modular arithmetic. Recurrence relations build each term of a sequence from earlier ones and solve some of them in closed form. Introductory graph theory studies vertices and edges: routes, trees, planar drawings and colorings. ## A path through it Logic and proof techniques come first and interleave: most of logic comes before proof, but the existential quantifier, and the quantifier lessons built on it, follow direct proof. Proof techniques come before number theory, sets, recurrences and graph theory. Number theory comes before the equivalence classes in sets, since those use congruence. Sets, which also use De Morgan's laws from logic, come before graph theory. Recurrences need induction, and can follow proof techniques. ### [Logical equivalence](https://math.wiki/logical-equivalence/) **Summary:** Two statements are equivalent when their truth tables agree on every single row. **Question:** Two statements are worded differently. How can you tell they say exactly the same thing? ## What this area is about Two statements are logically equivalent, written $P \equiv Q$, when their truth tables agree on every row: whatever truth values the letters take, the two are true together and false together. Equivalent statements can be swapped for each other anywhere, so an awkward statement can be traded for an easier one without changing what is claimed. ## How these fit together **The test.** Equivalence is checked by building both truth tables and comparing the final columns. A single row where they differ is enough to show two statements are not equivalent. **Three results.** Each is one equivalence worth knowing by heart. De Morgan's laws negate an "and" or an "or": the connective flips and the "not" moves inside, as in $\neg(P \land Q) \equiv \neg P \lor \neg Q$. A conditional and its contrapositive shows $p \to q$ says the same as $\neg q \to \neg p$, while the converse does not. Saying a conditional without if-then rewrites $P \to Q$ as $\neg P \lor Q$, which makes a conditional's negation routine. ## A path through it De Morgan's laws first, then the contrapositive, then the conditional as an "or", which leans on De Morgan to negate it. These equivalences are what justify proof by contraposition, they return as the rule for negating a quantified statement, and De Morgan's laws reappear almost word for word as De Morgan's laws for sets. ### [Matching two sets one to one](https://math.wiki/matching-sets-one-to-one/) **Summary:** Two sets are the same size when their members pair off with none left over on either side. **Question:** Without counting, how can you tell that two collections are the same size, and what does "same size" mean when neither collection ever runs out? ## Why this exists A child who sets one cup in front of each chair knows there are as many cups as chairs without knowing how many of either (one-to-one-correspondence). Counting is a special case of the same move: to count a set is to match it with $\{1, 2, \dots, n\}$ (cardinality-of-a-finite-set). A set that never runs out, such as $\mathbb{N} = \{0, 1, 2, \dots\}$, has no last number to count to, but matching still makes sense. Taken as the definition of "same size", it leads at once to a surprise: an infinite set can be the same size as a part of itself. Galileo, in his *Two New Sciences* (1638), noticed that the perfect squares pair off with all the positive integers though most positive integers are not squares, and concluded that "equal", "greater" and "less" should not be applied to infinite collections. The definition below keeps "equal" and accepts the surprise. ## Definition Sets $A$ and $B$ have the **same size** (the same **cardinality**) when there is a bijection $f : A \to B$. We write $$\lvert A \rvert = \lvert B \rvert,$$ and $\lvert A \rvert \neq \lvert B \rvert$ when there is no bijection at all. For infinite sets, read $\lvert A \rvert = \lvert B \rvert$ as one statement about the pair; no number called $\lvert A \rvert$ has been defined. A bijection is exactly a matching: no member of $B$ is used twice (injective) and none is left out (surjective). ## Intuition Figure 1 matches $\mathbb{N}$ with the even natural numbers $E = \{0, 2, 4, \dots\}$. Every number in the top row has its partner below, and every number in the bottom row receives one arrow. Yet $E$ is a proper subset of $\mathbb{N}$, missing $1, 3, 5, \dots$. For a finite set that cannot happen (Result 3). ### [Modular arithmetic](https://math.wiki/modular-arithmetic/) **Summary:** Clock arithmetic: only the remainder matters. **Question:** If only the remainder matters, which parts of ordinary arithmetic still work, and what can that buy you? ## What this area is about Fix a positive integer modulus $n$. Two integers are congruent, $a \equiv b \pmod{n}$, when $n \mid (a - b)$: they leave the same remainder on division by $n$. Congruence behaves like equality under addition, subtraction and multiplication, so large numbers can be swapped for small ones mid-calculation. Division is the exception, and much of this area is about when it can be repaired. ## How these fit together **The language.** Congruence modulo n and its arithmetic defines congruence and proves the rules for $+$, $-$ and $\times$. **Procedures.** Divisibility tests from congruences reads remainders off the decimal digits. Powers modulo n finds $a^k \bmod n$ by reducing after each step and by repeated squaring. Modular inverses, for $n \ge 2$, finds a $b$ with $ab \equiv 1 \pmod{n}$, which exists exactly when $\gcd(a, n) = 1$, and solves $ax \equiv c \pmod{n}$. **Results.** Fermat's little theorem: for a prime $p$ with $p \nmid a$, $a^{p-1} \equiv 1 \pmod{p}$. The Chinese remainder theorem: for pairwise coprime moduli, every choice of remainders is met by a number, unique modulo the product of the moduli. **An application.** RSA public-key encryption combines powers, Fermat and inverses. ## A path through it Congruence notation first: the divisibility tests, powers and inverses all need it. Powers come before Fermat's little theorem. Inverses also need the extended Euclidean algorithm, and come before the Chinese remainder theorem. RSA comes last, after both Fermat and inverses. ### [Modular inverses and linear congruences](https://math.wiki/modular-inverse/) **Summary:** A number that multiplies with a given one to give 1 modulo n, which exists only when the two are coprime. **Question:** Modulo n we can add, subtract and multiply freely, but we cannot simply divide. When can a multiplication be undone, and how do we solve an equation like 23x = 7 when only remainders count? ## Why this exists Congruences modulo $n$ may be added, subtracted and multiplied like equations (congruence-notation), but division breaks: $2 \cdot 3 \equiv 2 \cdot 8 \pmod{10}$ while $3 \not\equiv 8 \pmod{10}$. With rational numbers, dividing by $a$ is multiplying by $\tfrac{1}{a}$, the number whose product with $a$ is $1$ (properties-of-equality). Modulo $n$ there are no fractions, but there may be an integer $b$ with $ab \equiv 1 \pmod{n}$, and multiplying by $b$ undoes multiplying by $a$. Day 1 decides when such a $b$ exists and finds it; day 2 uses it to solve $ax \equiv b \pmod{n}$ in every case. ## Definition Throughout, $n \ge 2$ is the modulus (modulo $1$ everything is congruent). The lesson writes plain integers; in $\mathbb{Z}_n$ (equivalence-classes) each statement is about classes, and "unique modulo $n$" means "one class." ### Day 1: inverses An **inverse of $a$ modulo $n$** is an integer $b$ with $$ ab \equiv 1 \pmod{n}. $$ If $a$ has an inverse modulo $n$, then $a$ is a **unit** modulo $n$. The inverse in $\{1, 2, \dots, n - 1\}$ is written $a^{-1} \bmod n$. For example $23 \cdot 47 = 1{,}081 = 18 \cdot 60 + 1$, so $47$ is an inverse of $23$ modulo $60$. The symbol $a^{-1}$ here names an integer, never the fraction $\tfrac{1}{a}$. ### Day 2: linear congruences A **linear congruence** is $ax \equiv b \pmod{n}$ with integers $a$, $b$ and unknown $x$. A **solution** is an integer $x$ that makes it true. If $x$ is one, so is every $x' \equiv x$, by the multiplication rule, so the **number of solutions modulo $n$** means the number among $0, 1, \dots, n - 1$. ### [Modus ponens, chaining, and a common fallacy](https://math.wiki/modus-ponens/) **Summary:** Given if p then q, and given p, the conclusion q follows. **Question:** One fact is known to bring a second with it. If the first holds, is the second now certain? If the second holds, does that bring back the first? And when two such links meet end to end, what do they give? ## Why this exists Mathematics stores most of what it knows as if-then statements. "If a triangle has a right angle, then the square of its longest side is the sum of the squares of the other two." "If a whole number ends in $0$, then it is divisible by $10$." A statement like that, on its own, tells you nothing about any particular triangle or number. It is a promise waiting for a case. The step that turns the promise into a fact is so natural it is easy to miss: you check that this triangle does have a right angle, and conclude that its sides obey the rule. That step is **modus ponens**, Latin for roughly "the way of affirming": affirm the if-part, and the then-part comes with it. Every time a theorem is applied, and every time a line of algebra follows from the line above it, modus ponens is doing the work. Two more moves sit right beside it. The first joins promises together. Suppose you know two separate facts: if the ferry is canceled, the team drives instead, and if the team drives, they arrive after dark. Neither fact alone links the ferry to the time they arrive. Put together, they say something neither one said by itself: if the ferry is canceled, the team arrives after dark. That new fact came out without checking whether the ferry actually was canceled, and without restating the driving step. Most real reasoning is built out of chains like this one: several "if... then..." statements laid end to end, each picking up where the last left off, and a direct proof often strings together several such steps. Rather than re-justify the whole chain every time, it helps to have one rule that says: whenever two conditionals share a middle statement this way, they collapse into a single conditional connecting the two ends. That rule is **hypothetical syllogism**. ### [Modus tollens, ruling out, and a common fallacy](https://math.wiki/modus-tollens/) **Summary:** Given if p then q, and given that q is false, it follows that p is false. **Question:** You know that if one thing is true, a second thing is true, or that at least one of two things is true. Then something gets ruled out. What are you entitled to conclude, and what only looks as if it follows? ## Why this exists Picture a claim you already believe: *if the alarm goes off, the building empties.* You walk past the building and it is nowhere near empty; people are still at their desks. You did not hear an alarm, and you do not need to have heard one to know it did not go off. If it had, the building would be empty, and it is not. That step, going from "if $P$ then $Q$" together with "$Q$ is false" to "$P$ is false", is used every time something gets ruled out because what it would force to happen did not happen. A doctor rules out a diagnosis because a symptom it demands is missing. A number gets ruled out as prime because it has a divisor it should not have. A whole proof technique is built from this one move, used once, deliberately. The move has a name, **modus tollens**, and together with modus ponens it is one of the two rules of deduction that proofs use most. Ruling things out has a second shape, built on "or" instead of "if". Your keys are in your coat or in your bag; they are not in the coat; so they are in the bag. An equation has two solutions, $x = 5$ or $x = -2$; the problem asks for a length, and a length is not negative; so $x = 5$. Here one fact says at least one of the possibilities holds and the other crosses one possibility off. Neither settles anything alone, since an "or" statement, a disjunction, never says which part is true, and ruling one possibility out says nothing about the rest; together they force a conclusion. That move is the **disjunctive syllogism**. ### [Negating a quantified statement](https://math.wiki/negate-a-quantified-statement/) **Summary:** The opposite of a for-all claim is a there-exists claim about the opposite, and the other way round. **Question:** \"Not every prime is odd.\" Rewritten as its own quantified claim, what exactly does that sentence say? ## When you reach for it "Not every prime is odd" and "every prime is not odd" sound like small rewordings of the same idea. They are not. The first says one prime fails to be odd (true, witness $2$). The second says every prime fails to be odd (false, since $3$ is a counterexample to that). Only one of these is the actual opposite of "every prime is odd," and getting it right is not a matter of taste: a proof that sets out to disprove a for-all claim has to know exactly what statement it is trying to establish instead. That is what this skill is for. Given a statement built from $\forall$ and $\exists$, it produces the one statement that is true exactly when the original is false, and false exactly when the original is true. Reach for it whenever a claim starts "every..." or "there is..." and you need its precise opposite: to test a conjecture, to know what a counterexample would even have to look like, or to read a definition backwards, the way "the sequence does not converge" has to be unpacked into its own $\forall\exists$ statement before it can be used. ## The procedure Everything rests on two swaps, and each one is forced by what the quantifiers already mean. $$\neg\big(\forall x \in D,\ P(x)\big) \;\equiv\; \exists x \in D,\ \neg P(x)$$ $$\neg\big(\exists x \in D,\ P(x)\big) \;\equiv\; \forall x \in D,\ \neg P(x)$$ The first is allowed because $\forall x \in D,\ P(x)$ is false exactly when $P$ fails somewhere in $D$, and "$P$ fails somewhere" is precisely what $\exists x \in D,\ \neg P(x)$ says. The second is its mirror: $\exists x \in D,\ P(x)$ is false exactly when $P$ fails everywhere in $D$, which is $\forall x \in D,\ \neg P(x)$. Neither swap touches the domain $D$; only the quantifier's type flips, from $\forall$ to $\exists$ or back, and the negation lands on the predicate instead of on the quantifier. ### [Nested quantifiers](https://math.wiki/nested-quantifiers/) **Summary:** Statements with one quantifier inside another, where changing their order changes what is being claimed. **Question:** Every whole number has a bigger whole number. Does that mean some whole number is bigger than all of them? How do you say exactly which of the two you mean? ## Why this exists A single quantifier, for all or there exists, is enough for a claim about one variable: every real number has a square of at least $0$, some integer is even. But most claims worth making tie two things together. "Every integer has an integer bigger than it" is about a pair of numbers, the one you are given and the one you answer with, and it quantifies each of them. Once there are two quantifiers, their order starts to matter, and English is careless about order. Compare - for every integer there is a bigger integer, and - there is an integer bigger than every integer. The first is true: whatever integer you name, add $1$. The second is false: no single integer beats all the others, since it cannot beat itself. The words are almost the same and the claims are opposite. Mathematics needs a way to write each one so that nobody can mistake it for the other, and a way to decide which one a proof has actually shown. ## Definition A **predicate in two variables**, written $P(x, y)$, is a sentence about $x$ and $y$ that becomes a statement, true or false, as soon as both $x$ and $y$ are replaced by members of their domains. "$x + y = 0$" and "$x$ divides $y$" are examples. A **nested quantifier** statement puts one quantifier inside the reach of another: $$\forall x \in D,\ \exists y \in E,\ P(x, y)$$ It is read from the left, and each quantifier governs everything to its right. So this one says: **for every** $x$ in $D$, the statement $\exists y \in E,\ P(x, y)$ is true. In words, whatever $x$ is chosen, some $y$ can be found to go with it. The $y$ is found *after* $x$ is known, so it may be a different $y$ for each $x$. ### [Numbers in other bases](https://math.wiki/convert-to-base-ten/) **Summary:** Multiplying each digit by the value of its place and adding the results. **Question:** Ten is only one choice of base. Does every positive integer have a numeral in base seven or base two, and could one number ever have two different numerals? ## When you reach for it other-number-bases bundled straws in fives and in twos: a digit counts how many of a place, and each place is worth $b$ of the place to its right. This lesson states that as a theorem and proves it: every positive integer has a base-$b$ numeral, and only one. Reach for it to read or write a number in another base (binary and base sixteen in computing), and when a proof works with a number through its digits. ## The procedure **Digits and expansions.** Fix an integer $b \ge 2$, the **base**. The base-$b$ **digits** are the integers $0, 1, \dots, b - 1$; above base ten, letters stand in for digits from ten up, so base sixteen uses $\mathrm{A} = 10$ through $\mathrm{F} = 15$. For digits $d_0, \dots, d_k$, write $$\begin{aligned}(d_k d_{k-1} \cdots d_1 d_0)_b &= d_k b^k + d_{k-1} b^{k-1} + \cdots + d_1 b + d_0 \\ &= \sum_{i=0}^{k} d_i b^i.\end{aligned}$$ The digit $d_i$ sits in the $b^i$ place. The string is a **base-$b$ expansion** of $n$ when its value is $n$ and its leading digit $d_k$ is not $0$. A numeral with no subscript is in base ten. **The theorem.** Let $b \ge 2$ be an integer. Every positive integer $n$ has exactly one base-$b$ expansion: exactly one $k \ge 0$ and one list of digits $d_0, \dots, d_k$ in $\{0, 1, \dots, b - 1\}$ with $d_k \ne 0$ and $n = \sum_{i=0}^{k} d_i b^i$. The proof rests on the-division-algorithm, used here and proved in its own lesson: for integers $a \ge 0$ and $b \ge 1$ there is exactly one pair of integers $q, r$ with $a = qb + r$ and $0 \le r < b$. If $k \ge 1$, factoring $b$ out of every term but the last (the distributive property) gives ### [One-to-one, onto and bijective functions](https://math.wiki/injective-surjective-bijective/) **Summary:** A function is one to one when no two inputs share an output, onto when every target is reached, and bijective when it is both. **Question:** A rule sends every member of one set to a member of another. Does it ever send two members to the same place, and does it reach every member of the target? ## Why this exists A function from one set to another can fail to be a perfect matching in two separate ways. It can send two different inputs to the same output, so the output no longer tells you which input it came from. Or it can leave some members of the target set unreached, so some outputs are never produced. These are the two questions behind almost everything done with functions between sets. Undoing a function needs the first to never happen and the second to never happen. Comparing the sizes of two sets, including infinite ones, is done by finding a function that avoids both. This lesson names the two properties, states each with quantifiers, and proves and disproves them. ## Definition ### Day 1: functions between sets, and injective A **function** $f : A \to B$ assigns to every member $a$ of $A$ exactly one member $f(a)$ of $B$ (recap of function-definition; as a set of pairs it is a relation in which every member of $A$ is a first coordinate exactly once). $A$ is the **domain** and $B$ the **codomain**. Both are part of the function: the same rule with a different domain or codomain is a different function. The **image** of $f$ is the set of outputs actually produced, $$f(A) = \{\, f(a) : a \in A \,\}.$$ It is always a subset of $B$, and it may be a proper one. Many books call $f(A)$ the **range**; since others use "range" for the codomain, this lesson says image. Here $\mathbb{N} = \{0, 1, 2, 3, \dots\}$; some books start the natural numbers at $1$. ### [Planar graphs and Euler's formula](https://math.wiki/eulers-formula-for-planar-graphs/) **Summary:** For any connected drawing with no crossings, vertices minus edges plus faces always equals 2. **Question:** Some networks can be drawn flat with no two links crossing and some cannot. When a flat drawing exists, what does it force on the numbers of points, links and regions, and how does that prove some networks can never be drawn flat? ## Why this exists A circuit board with copper on one layer, a subway map with no bridges, a puzzle about pipes: each asks whether a network can be drawn flat with no two links crossing. Trying drawings settles the question only when one works. Euler's formula turns it into arithmetic: a flat drawing of a connected network always has vertices minus edges plus regions equal to $2$. Since every region needs at least three edges around it, the formula caps the number of edges, and that cap proves that $K_5$ and the three-utilities graph can never be drawn flat. ## Definition This lesson takes two days. Day 1 is plane drawings, faces and Euler's formula (Results 1 and 2, Examples 1 to 4). Day 2 is the edge bounds, the two smallest graphs that cannot be drawn flat, and Kuratowski's theorem (Results 3 to 7, Examples 5 to 7). The notation is that of degree-of-a-vertex and walks-paths-and-cycles: $G = (V, E)$ is a finite simple graph, and paths, cycles, connected graphs and components are as defined there. Trees and forests are as in tree-edge-count, and $G - xy$ is $G$ with the edge $xy$ erased. - A **plane drawing** of $G$ puts each vertex at its own point and draws each edge $xy$ as a curve from $x$ to $y$ that does not cross itself and passes through no other vertex, so that two edges meet only at a shared endpoint. $G$ is **planar** if it has a plane drawing; a **plane graph** is a graph together with one plane drawing of it. - Erasing the drawing from the plane leaves separate regions, the **faces**. Exactly one face is unbounded, the **outer face**. - Here $v = \lvert V \rvert$, $e = \lvert E \rvert$, $f$ is the number of faces and $c$ the number of components. A single edge is always named by its endpoints, as $xy$. - (Day 2) Each edge has two sides, and each side borders exactly one face. The **degree** $\deg(F)$ of a face $F$ is the number of edge sides bordering it, so an edge with $F$ on both sides counts twice. - (Day 2) The **complete bipartite graph** $K_{m,n}$ has a group of $m$ vertices and a group of $n$, every vertex of one group joined to every vertex of the other, and no edge inside a group (more in bipartite-graphs). - (Day 2) **Subdividing** the edge $xy$ replaces it by a new vertex $w$ and edges $xw$, $wy$. A **subdivision** of $H$ is a graph made from $H$ by subdividing edges any number of times. ### [Power set](https://math.wiki/power-set/) **Summary:** The set whose members are all the subsets of a given set, the empty set and the whole set included. **Question:** A set has a handful of subsets. What happens if we gather all of them into one set, and how large is that set? ## Why this exists Subsets answer questions one at a time: is this set inside that one? Many questions are about all the subsets at once. How many different ways can a committee be chosen from a club? Which settings can a board of switches be in? Which subsets of $A$ are also subsets of $B$? To talk about all the subsets of a set as a single object, we collect them into one set. That set is the **power set**. Its members are themselves sets, so it is the first place where a set of sets is not a curiosity but the natural thing to write down. Counting it gives one of the most used formulas in discrete mathematics, $2^n$, and comparing power sets turns facts about subsets into facts about membership. ## Definition The **power set** of a set $A$ is the set of all subsets of $A$. It is written $\mathcal{P}(A)$: $$\mathcal{P}(A) = \{X : X \subseteq A\}$$ So for any object $X$, $$X \in \mathcal{P}(A) \iff X \subseteq A$$ That one line is the whole definition at work: a question about membership in $\mathcal{P}(A)$ is a question about being a subset of $A$, and the other way round. For $S = \{r, s, t\}$ there are eight subsets, so $$\begin{aligned} \mathcal{P}(S) = \{\, &\varnothing, \{r\}, \{s\}, \{t\}, \\ &\{r, s\}, \{r, t\}, \{s, t\}, \{r, s, t\} \,\} \end{aligned}$$ Three things are worth seeing in this list. - The empty set and $S$ itself are always members, because $\varnothing \subseteq A$ and $A \subseteq A$ for every set $A$. - Every member of $\mathcal{P}(S)$ is a set. The letter $r$ is a member of $S$ but **not** of $\mathcal{P}(S)$; the set $\{r\}$ is a member of $\mathcal{P}(S)$ but not of $S$ (elements and membership). - $\lvert \mathcal{P}(S)\rvert = 8$, counting each subset once, as in the cardinality of a finite set. ### [Powers modulo n](https://math.wiki/modular-exponentiation/) **Summary:** Reducing at every step, and squaring repeatedly, keeps a huge power manageable. **Question:** The number 11 to the 53rd power has 56 digits, yet its remainder on division by 97 is a number below 97. Can we find that remainder without ever writing the power down? ## When you reach for it Reach for this move whenever a question asks for a power $a^k$ only up to a remainder: the last digit of $3^{2026}$, the weekday a huge number of days from now, whether $23$ divides $2^k - 1$. The power itself is usually far too large to write. The remainder is small, and two facts get to it in a few lines. First, by congruence-notation a number may be replaced by its remainder after every multiplication, so no number in the work ever exceeds $(n - 1)^2$. Second, squaring over and over, guided by the binary digits of $k$, reaches $a^k$ in about $2 \log_2 k$ multiplications instead of $k - 1$. A third idea often finishes the job by hand: the remainders of $a^0, a^1, a^2, \dots$ eventually repeat in a cycle, and a cycle turns a huge exponent into a small one. ## The procedure **Reducing after every multiplication.** Since $a^{k+1} = a^k \cdot a$, suppose $a^k \equiv r \pmod{n}$. The multiplication rule (Theorem 3 of congruence-notation), applied to $a^k \equiv r$ and $a \equiv a$, gives $$ a^{k+1} \equiv r \cdot a \pmod{n}. $$ So each new power comes from the previous remainder, and $r \cdot a$ may be reduced at once. If $a$ is first replaced by its remainder too (Corollary 4 there), every product in the work is of two numbers below $n$. **Repeated squaring.** One step at a time still takes $k - 1$ multiplications for $a^k$. Squaring is faster: by the power-of-a-power law (laws-of-exponents), $(a^{2^j})^2 = a^{2^{j+1}}$, so the list ### [Proof by cases](https://math.wiki/cases-from-parity/) **Summary:** Using the fact that every integer is either even or odd to cover everything with two cases. **Question:** Some claims are easy once you know whether $n$ is even or odd, but you are not told which. How can a proof use a fact it does not have? ## When you reach for it You reach for cases when a claim would be easy if you knew one more fact, and every possible value of that fact can be handled. For an integer $n$ you do not know whether $n$ is even or odd, but you know it is one of the two, so you prove the claim for even $n$, then for odd $n$, and you are done. In logical form: if at least one of $Q_1, Q_2, \ldots, Q_m$ is true, and each $Q_i \to P$ is proved, then $P$ is true. This lesson takes two days. **Part 1** splits by parity, for one integer and for two, and shows how symmetry shortens a split. **Part 2** splits by remainders on division by $3$ and by the sign of a real number. ## The procedure ### Part 1: splitting by parity A proof by cases makes three promises, and each one has a reason behind it. **The cases cover every possibility.** This is the step that makes the method valid. If some integer fell in no case, nothing would have been proved about it. Parity is the standard split because proof-with-even-and-odd proved that every integer is even or odd. Coverage is all that is needed: cases may overlap, and that does no harm, as long as nothing falls outside all of them. **Each case is a full proof.** Inside a case you gain one assumption, such as "$n$ is even", and you unpack it exactly as in a direct proof: $n = 2k$ for some integer $k$. A case may cite the parity rules for sums and products from proof-with-even-and-odd. It may not borrow anything from the other case. ### [Proof by contradiction](https://math.wiki/proof-by-contradiction/) **Summary:** Assume the opposite and reach something impossible. **Question:** How can anyone prove that $\sqrt{2}$ is not a fraction, when there are infinitely many fractions to rule out and none of them is there to calculate with? ## When you reach for it Some statements give a direct proof nothing to start from. "$\sqrt{2}$ is irrational" says that **no** fraction squares to $2$. There are infinitely many fractions, so checking them one at a time never finishes, and there is no particular fraction on the table to calculate with. The same is true of "there are no integers $x$ and $y$ with ...", "there is no largest ...", and "at most one ...". A **proof by contradiction** turns the problem around. It assumes, for the length of the argument, that the statement is false. That negation is usually something concrete: "$\sqrt{2} = \frac{a}{b}$ for some integers $a$ and $b$" hands over two integers to work with. The proof reasons from the assumption, using only facts already known, until two statements appear that cannot both be true. Every step was valid, so the fault lies with the one thing that was not known to be true: the assumption. It is false, and so the statement is true. Reach for it when the statement says that something is impossible, does not exist, or is not a member of some set (irrational means *not* rational), and when its negation gives you an object to name. A direct proof goes forward from what you know. A proof by contradiction adds one more thing to know, for a while, and then shows that it could never have been true. ## The procedure ### Why assuming the opposite proves anything In mathematics a statement is either true or false, so exactly one of $P$ and $\neg P$ holds. A statement of the form $Q \wedge \neg Q$, "$Q$ and not $Q$", is false whatever $Q$ says: it is a contradiction. Suppose you have proved ### [Proof by contraposition](https://math.wiki/proof-by-contraposition/) **Summary:** Proving if p then q by instead proving that q being false forces p to be false. **Question:** How do you prove "if $n^2 - 6n + 5$ is even, then $n$ is odd", when knowing that $n^2 - 6n + 5$ is even says almost nothing you can use about $n$ itself? ## When you reach for it To prove "if $P$, then $Q$" directly, you assume $P$ and work toward $Q$ (direct proof). That works when $P$ hands you something to calculate with. Often it does not. "If $n^2 - 6n + 5$ is even, then $n$ is odd" gives $n^2 - 6n + 5 = 2k$, an equation with $n$ buried inside a quadratic, and nothing in it says how to write $n$ as $2m + 1$. Run the other way and the work is easy: start from $n$, a single letter with a definition, and compute. A conditional and its contrapositive $\neg Q \to \neg P$ are logically equivalent (a conditional and its contrapositive), so a direct proof of the contrapositive is a proof of the original. That is **proof by contraposition**. Reach for it when - the conclusion is about a simple object, such as one variable, and the hypothesis is about a combination of it, so the reasoning wants to run from the simple object to the combination; - the hypothesis is negative ("is not divisible by", "is irrational", $\ne$), which gives nothing to compute with, while its negation is positive and does; - the conclusion is an "or", which is hard to aim at, while its negation is an "and", which hands over two facts at once. It is the proof-writing form of modus tollens: whenever $Q$ fails, so does $P$. ## The procedure ### The write-up A proof by contraposition has four parts, and the reader must be able to see each one. **Say what you are doing.** Open with "We prove the contrapositive", with the contrapositive written out in full. The reader then knows that the first assumption is $\neg Q$, not $P$, and can check that the statement proved really is the contrapositive. ### [Proof by induction](https://math.wiki/proof-by-induction/) **Summary:** Prove the first case, then that each case forces the next. **Question:** A claim about every whole number is an endless list of claims. How can a finite amount of work prove all of them? ## What this area is about Proof by induction proves a statement $P(n)$ for every integer $n \ge n_0$ in two parts. The base case proves $P(n_0)$. The inductive step proves that, for every $k \ge n_0$, if $P(k)$ holds then $P(k+1)$ holds. Both parts are needed: a step without a base case proves nothing, and checking many cases is not a proof. Together they reach every $n \ge n_0$, one case after another. ## How these fit together **The method.** How induction works: base case, hypothesis and step names the three parts, explains why they settle every case, and shows how a proof can go wrong. **A variant.** Strong induction and well-ordering lets the step assume $P(n_0), \ldots, P(k)$, every earlier case, to prove $P(k+1)$. It fits when breaking a case apart gives smaller pieces whose sizes are not known in advance. The well-ordering principle runs the same idea as a search for a smallest counterexample. **A standard use, not yet written.** Induction on a sum formula will prove a closed formula for a running total by adding one more term to the assumed case. Until it exists, the first lesson's own section on sum formulas covers this use. ## A path through it How induction works comes first; strong induction builds on it. Induction is the tool behind trees, where a tree on $n$ vertices has $n - 1$ edges, and behind checking a closed formula in solving a two-term recurrence. ### [Proof techniques](https://math.wiki/proof-techniques/) **Summary:** The standard ways of showing something must be true. **Question:** A claim holds in every case you have tried. What does it take to show it holds in every case, and which way of arguing fits which claim? ## What this area is about A proof is a chain of justified steps from accepted starting points to the exact claim; examples are evidence, not proof. This area covers what a proof may assume and must reach, then the standard methods, one per shape of claim. ## How these fit together **What a proof is.** Axioms, definitions and theorems says where reasons stop. What may be assumed and what must be shown and writing a proof as prose fix the start and the finish. A conjecture is a claim not yet proved; a counterexample refutes a "for all" claim by one failing case. **Proving "if $P$, then $Q$".** Direct proof assumes $P$ and reaches $Q$. Contraposition proves $\neg Q \to \neg P$ instead. Contradiction assumes the claim false and reaches something impossible. Proof by cases covers every possibility, such as $n$ even or odd. An if and only if is two conditionals, proved separately. **Special tools.** Existence proofs show an object exists; the pigeonhole principle is one way to do it. Induction proves $P(n)$ for all integers $n \ge n_0$ from a base case and a step. Combinatorial proof counts one set two ways. ## A path through it Logic comes first, except the existential quantifier and the quantifier lessons built on it, which follow direct proof. What may be assumed comes before writing a proof as prose. Direct proof comes before contraposition, contradiction, cases, if and only if, induction and pigeonhole, which also needs existence proofs. Counterexamples need quantifiers, existence proofs the existential quantifier, combinatorial proof Pascal's triangle. Axioms and conjectures need nothing. ### [Proving a set identity](https://math.wiki/prove-a-set-identity/) **Summary:** Showing each side is a subset of the other by following one element through the definitions. **Question:** Two descriptions of a set look different. How can anyone show that they pick out exactly the same elements for every choice of sets, and how can anyone show that they do not? ## When you reach for it A set identity claims that two expressions name the same set whatever sets are put in. set-equality showed what that means, each side a subset of the other, and how to prove a subset with an arbitrary member. distributive-set-laws tested identities with membership tables and listed the standard laws. This lesson writes the proof as mathematicians write it: follow one element through the definitions, in sentences. It works for any identity, including ones about sets given by a property or about infinitely many sets, where a table cannot be drawn. This lesson takes two days. **Part 1** is the element-chasing proof, with two new operations, set difference and symmetric difference. **Part 2** proves identities by algebra from the standard laws, proves conditional statements about sets, and disproves a proposed identity, including the case where one inclusion holds and the other fails. ## The procedure ### Part 1: following one element **Two new operations.** For sets $A$ and $B$, the **difference** $A \setminus B$ is the set of elements of $A$ that are not in $B$: $$A \setminus B = \{x : x \in A \text{ and } x \notin B\}$$ When $A$ and $B$ lie in a universal set $U$, this is $A \cap B^{c}$, since for $x \in U$, $x \notin B$ means $x \in B^{c}$. The **symmetric difference** is what lies in exactly one of the two sets: $$A \mathbin{\triangle} B = (A \setminus B) \cup (B \setminus A)$$ ### [Proving an if and only if](https://math.wiki/if-and-only-if-proof/) **Summary:** Splitting the claim into two conditionals and proving each direction on its own. **Question:** How do you prove that two statements always stand or fall together, and not just that one brings the other? ## When you reach for it A claim of the form "$P$ if and only if $Q$", written $P \leftrightarrow Q$, says that $P$ and $Q$ are true together and false together. Its meaning, and the words "necessary and sufficient", are the subject of the biconditional. This lesson is about proving one. You reach for this move whenever a claim promises two things at once: "if and only if", "iff", "exactly when", "is necessary and sufficient for", "the following are equivalent", and also "find every $x$ with...", since "the solutions are exactly these" says that being a solution and being on the list are equivalent. ## The procedure ### Day 1: two directions **Split the claim.** The biconditional $P \leftrightarrow Q$ has the same truth table as $(P \to Q) \land (Q \to P)$, so it is true exactly when both conditionals are true. A proof of $P \leftrightarrow Q$ is therefore two proofs, one of each conditional. Neither alone is enough: "if an integer is divisible by $6$, then it is even" is true, and "an integer is divisible by $6$ if and only if it is even" is false. **Label each direction.** The forward direction, marked ($\Rightarrow$), assumes $P$ and must show $Q$. The backward direction, marked ($\Leftarrow$), assumes $Q$ and must show $P$. The labels matter because the two halves start from different givens: inside ($\Rightarrow$) you may use $P$ and never $Q$, and inside ($\Leftarrow$) the reverse. Open each half with a sentence that says what it assumes and what it must show, as in what may be assumed and what must be shown. Words work as labels too: "Suppose $P$..." and "Conversely, suppose $Q$...". ### [Proving statements about divisibility](https://math.wiki/proof-with-divisibility/) **Summary:** Replacing the words a divides b by the equation b equals a times an integer, then working with that. **Question:** You know that some integers divide some others. How do you prove that a different integer divides a different number? ## When you reach for it A claim says that one integer divides another: $12 \mid n^2 + 2n$, or $7 \mid ab - 1$, or "$d$ divides $1$". The statement $a \mid b$ is true or false; there is nothing inside it to calculate with. The move of this lesson is to trade it for an equation. By the definition of divides, $a \mid b$ means $b = ak$ for some integer $k$, and an equation can be substituted, expanded and factored. At the end the same definition is used in the other direction, to trade an equation back for the statement. Reach for it whenever the hypothesis or the conclusion says "divides", "is a multiple of" or "is divisible by". The words come from divisibility-and-primes: a factor of $b$ goes into $b$ with nothing left over. The shape is the one used in proof-with-even-and-odd, where "even" was traded for $n = 2k$. That was the divisor $2$; this lesson is every other divisor. The lesson takes two days. Day 1 is the unpacking and packing, and the rules it proves, which later proofs cite instead of redoing. Day 2 uses them on expressions: factoring, numbers that sit a fixed amount past a multiple, and claims that turn out to be false. ## The procedure ### Day 1: unpack, work, pack **Unpack each hypothesis.** A hypothesis $a \mid b$ becomes $b = ar$ for some integer $r$, and each hypothesis gets its own new letter. Writing $b = ak$ and $c = ak$ with the same $k$ would claim $b = c$, which nobody said. ### [Proving statements about even and odd numbers](https://math.wiki/proof-with-even-and-odd/) **Summary:** Writing an even number as two times an integer and an odd number as two times an integer plus one. **Question:** Even and odd are about pairing things up. How do you turn that into a statement precise enough to prove things about every integer, including ones you could never pair by hand? ## When you reach for it Whenever a claim mentions even or odd integers, or can be settled by asking whether some quantity is even or odd. Pairing objects (even and odd numbers) decides the parity of one number you can see. A proof has to decide it for a number written as a letter, for every integer at once, negative ones and $0$ included. The move is always the same: replace the word "even" or "odd" by an equation, do algebra, and read the word back off the result. This lesson takes two days. **Part 1** sets out the definitions and proves the rules for sums, differences, products and powers. **Part 2** uses parity as a tool: to show an equation has no integer solution, and to show a puzzle position can never be reached. ## The procedure ### Part 1: the definitions and the parity rules An integer $n$ is **even** if $n = 2k$ for some integer $k$. It is **odd** if $n = 2k + 1$ for some integer $k$. Whether an integer is even or odd is its **parity**. The definitions cover all of $\mathbb{Z}$. Zero is even, since $0 = 2 \cdot 0$. Negative integers have a parity too: $-6 = 2(-3)$ is even and $-1 = 2(-1) + 1$ is odd. For a positive integer, $k$ is the number of pairs and the $+1$ is the one left over, which is exactly the pairing picture. The word "integer" on $k$ is not decoration: $7 = 2 \cdot \tfrac{7}{2}$ does not make $7$ even. **Theorem 1.** Every integer is even or odd, and no integer is both. ### [Quantifiers](https://math.wiki/quantifiers/) **Summary:** For all, there exists, and how to negate each. **Question:** A rule holds for every number, or for at least one. How do you say which, exactly, and what does each take to prove? ## What this area is about A sentence like $x^2 = 2$ is not true or false until you know what $x$ is. A quantifier settles that by saying how many members of a domain make it true: all of them, at least one, or exactly one. The same sentence can flip from true to false when only the domain changes. ## How these fit together **The two quantifiers.** For all, $\forall$, and there exists, and there is exactly one, $\exists$ and $\exists!$, are the definitions everything else is built from. "For all" and "there exists" are lopsided mirror images: one counterexample breaks a "for all", and one witness proves a "there exists". "Exactly one" is "there exists" plus the claim that no second witness exists, so it shares that page. **The procedure.** Negating a quantified statement turns each $\forall$ into $\exists$ and each $\exists$ into $\forall$, and moves the "not" onto the sentence inside. **Combining them.** Nested quantifiers put one inside another, and there the order matters: $\exists y\ \forall x$ always gives $\forall x\ \exists y$, but not the other way round. ## A path through it For all, then there exists and exactly one, then negation, then nesting. Finish here and you can read any claim shaped "every ... has some ..." and say what would prove it and what would break it. That is what a counterexample needs, and it is where existence proofs and uniqueness proofs start. ### [Recurrence relations](https://math.wiki/recurrence-relations/) **Summary:** A rule that builds each term of a sequence out of the terms that came before it. **Question:** A rule tells you how each term grows out of the ones before it. When can you skip the steps and name the hundredth term directly? ## What this area is about A recurrence relation is a rule that builds each term of a sequence from earlier terms, together with enough starting values to begin. It fixes every term, but only one at a time. This area is about running such rules, finding them in counting problems, and, for one well-behaved family, replacing them with a closed formula proved correct for every $n$. ## How these fit together **Running a rule.** Generating a sequence from a rule produces terms from the starting values and looks at what comes out. Closed form versus recursive form compares a step rule with a formula in $n$, for rules that look back one term. **The model example.** The Fibonacci sequence, $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \ge 3$, collects its identities and proves them by induction. **Building rules.** Setting up a recurrence from a counting problem sorts objects of size $n$ by their first or last piece. **Solving rules.** For $a_n = c_1 a_{n-1} + c_2 a_{n-2}$ with constants $c_2 \ne 0$, solving a two-term recurrence handles two distinct characteristic roots, and repeated roots handles the double root, when $c_1^2 + 4c_2 = 0$. ## A path through it Generating a sequence from a rule comes before Fibonacci, and Fibonacci before setting up recurrences from counting. Closed versus recursive form comes before solving a two-term recurrence, which also needs proof by induction; repeated roots come last. ### [Relations and their properties: reflexive, symmetric, antisymmetric, transitive](https://math.wiki/properties-of-relations/) **Summary:** A relation is a set of ordered pairs, and four properties sort relations into equivalences and orders. **Question:** Equal to, at most, divides, has the same parity as. Each compares two things in order. What do all such comparisons have in common, and which habits decide whether one sorts things into groups or lines them up? ## Why this exists The statements $3 \le 5$, $4 \mid 12$ and $\{1\} \subseteq \{1, 2\}$ each answer yes or no about two objects in order. A **relation** gathers every yes answer into one set of ordered pairs (relation-concept). Four properties separate the comparisons that behave like equality, sorting a set into groups, from those that behave like $\le$, arranging it in layers, and say exactly where any other relation falls short. ## Definition ### Day 1: relations and their four properties **Relation.** A **relation from $A$ to $B$** is any subset $R$ of the product $A \times B$ (cartesian-product). A **relation on $A$** is a subset of $A \times A$. When $(a, b) \in R$ we write $a \mathrel{R} b$ and say $a$ is related to $b$; when $(a, b) \notin R$, it is not. Order matters: $a \mathrel{R} b$ and $b \mathrel{R} a$ are different claims. The relation $\le$ on $\mathbb{R}$ is the set $\{(x, y) \in \mathbb{R}^2 : x \le y\}$, and $3 \le 5$ says $(3, 5)$ is in it. **Three pictures of a finite relation.** A relation $R$ on a finite set $A$ can be given as - the **list** of its pairs; - an **arrow diagram**: a dot for each member of $A$, an arrow from $a$ to $b$ whenever $a \mathrel{R} b$, and a **loop** at $a$ when $a \mathrel{R} a$; - a **matrix**: a square table with one row and one column for each member of $A$, in the same order, holding $1$ in row $a$, column $b$ when $a \mathrel{R} b$ and $0$ otherwise. ### [Relations, equivalence classes and partitions](https://math.wiki/equivalence-classes/) **Summary:** The groups an equivalence relation sorts a set into, where everything inside a group counts as the same. **Question:** When a relation says which things count as "the same," what are the groups it sorts a set into, and can we compute with a whole group as if it were a single object? ## Why this exists Mathematics often declares different things "the same for present purposes": integers with the same remainder, the fractions $\tfrac{1}{2}$ and $\tfrac{3}{6}$, seatings that differ by a turn of the table. The useful step is to collect everything that counts as the same into one group and treat the group as a single object. That needs a guarantee: the groups never overlap, and every element lands in one. Day 1 proves the guarantee for every equivalence relation. Day 2 uses it to build the integers modulo $n$, to test when arithmetic on groups makes sense, and to count objects up to a symmetry. ## Definition ### Day 1: classes and partitions Recall from properties-of-relations: a **relation on a set** $A$ is a subset of $A \times A$, and an **equivalence relation** is one that is reflexive, symmetric and transitive. Write $a \sim b$ when $a$ is related to $b$. **Equivalence class.** Let $\sim$ be an equivalence relation on $A$ and $a \in A$. The **class of $a$** is $$ [a] = \{x \in A : x \sim a\}. $$ Any member of a class is a **representative** of it. The set of all classes is written $A / {\sim}$, read "$A$ modulo $\sim$." **Partition.** A **partition** of $A$ is a collection of subsets of $A$, called **blocks**, such that every block is nonempty, any two different blocks are disjoint, and the union of all the blocks is $A$. Equivalently, every element of $A$ lies in exactly one block. Given a partition $P$, write $a \sim_P b$ when $a$ and $b$ lie in the same block. ### [RSA public-key encryption](https://math.wiki/rsa-encryption/) **Summary:** Anyone can lock a message with a public power mod n, but unlocking it needs a private exponent that only the prime factors of n reveal. **Question:** Can someone publish the instructions for locking a message to them, so that anyone at all can send them a secret, and still be the only person able to unlock it? ## Why this exists A classical cipher uses one key to lock and the same key to unlock. Two people who want to write in secret must first agree on that key, and agreeing on it in secret is the very problem the cipher was meant to solve. A **public-key** system splits the key in two. The locking key is published, so anyone can lock a message; the unlocking key is held by one person, and knowing how to lock does not reveal how to unlock. Number theory supplies such a pair. Raising to a power modulo $n$ is fast even for numbers hundreds of digits long (modular-exponentiation). Undoing the power is easy for whoever knows the prime factors of $n$, and as far as anyone knows it is hopelessly slow for everyone else. The scheme built on this is named RSA, after Rivest, Shamir and Adleman, who published it in $1977$. ## Definition ### Day 1: the keys, locking and unlocking **Making the keys.** Choose two different primes $p$ and $q$ and put $$ n = pq, \qquad \varphi(n) = (p - 1)(q - 1). $$ Choose an integer $e$ with $1 < e < \varphi(n)$ and $\gcd(e, \varphi(n)) = 1$. Let $d$ be the inverse of $e$ modulo $\varphi(n)$ with $1 < d < \varphi(n)$, so $$ ed \equiv 1 \pmod{\varphi(n)}. $$ It exists because $\gcd(e, \varphi(n)) = 1$ (Theorem 1 of modular-inverse) and the extended-euclidean-algorithm finds it. The **public key** is the pair $(n, e)$. The **private key** is $d$; the numbers $p$, $q$ and $\varphi(n)$ are kept secret too. ### [Saying a conditional without if-then](https://math.wiki/conditional-as-or/) **Summary:** If p then q says the same thing as not p or q, which is what makes its negation easy to write. **Question:** You have rules for "and", "or" and "not". Is "if ... then ..." really a fourth kind of statement, or can it be built out of the other three? ## Why this exists "And", "or" and "not" come with an algebra: De Morgan's laws push a negation through them, and the disjunctive syllogism lets you eliminate part of an "or". A conditional, "if $P$ then $Q$", looks like it stands outside that algebra entirely, a fourth and stranger kind of statement with its own private rules. It is not. "If $P$ then $Q$" says exactly what "$P$ is false, or $Q$ is true" says, no more and no less, so every tool already built for "or" and "not" works on a conditional as soon as it is written that way. The single biggest payoff is negation. Negating an "if ... then ..." head-on is confusing, since it is not obvious what the opposite of a promise even means. Negating an "or" is routine: push the negation through with De Morgan's laws. Rewrite the conditional first, and its negation stops being a special case. ## Definition Let $P$ and $Q$ be statements. The conditional $P \to Q$ ("if $P$ then $Q$") is **logically equivalent** to the disjunction $\neg P \lor Q$: $$P \to Q \ \equiv\ \neg P \lor Q$$ Two statements are logically equivalent, the subject of logical equivalence, when they have the same truth value for every assignment of true and false to $P$ and $Q$. $\neg P \lor Q$ reads two ways that both match "if $P$ then $Q$" in plain English: - "$P$ is false, or $Q$ is true." - "Either not $P$, or $Q$." Both readings use only "or" and negation, the two connectives this rewriting needs. The conditional itself and its truth table are covered in the truth table for the conditional, and the words converse, inverse and contrapositive that go with it in contrapositive equivalence; this lesson does not repeat them, only puts the conditional in a new set of clothes. ### [Set identities: distributive and De Morgan laws](https://math.wiki/distributive-set-laws/) **Summary:** An intersection spread across a union, and a union spread across an intersection. **Question:** Intersection spreads over union the way multiplication spreads over addition. Which rules like that hold for sets, and how can anyone be sure a proposed rule holds for every choice of sets? ## Why this exists In arithmetic, $a(b + c) = ab + ac$ lets one expression be rewritten as another. Sets built with $\cup$, $\cap$ and complement need rules of the same kind. "Not in either list" and "not in the first and not in the second" describe the same things; "in $A$, and in $B$ or $C$" is the same as "in $A$ and $B$, or in $A$ and $C$". Rules like these, true for every choice of sets, are **set identities**, and each one turns out to be a law of logic in disguise. ## Definition Throughout, $A$, $B$, $C$ are subsets of a stated universal set $U$, and $A^{c}$ is the complement of $A$ in $U$, as in union-and-intersection. A **set identity** is an equation between two expressions built from set letters, $U$, $\varnothing$, $\cup$, $\cap$ and complement, that holds for every choice of the sets. Since sets are equal when they have the same elements (set-equality), it says both sides always contain the same elements. **The distributive laws.** Intersection distributes over union, and union over intersection: $$A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$ $$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$$ **De Morgan's laws for sets.** The complement of a union is the intersection of the complements, and the complement of an intersection is the union of the complements: $$(A \cup B)^{c} = A^{c} \cap B^{c}$$ $$(A \cap B)^{c} = A^{c} \cup B^{c}$$ **Membership table.** For an expression in $n$ set letters, a membership table has one row for each way an element can sit relative to the letters, $2^{n}$ rows in all, with $1$ for "in" and $0$ for "out". Each further column records whether an element of that row is in the set heading the column. It is a truth table with $1$ and $0$ in place of T and F. ### [Sets](https://math.wiki/sets/) **Summary:** Collections of things, and the ways they combine. **Question:** Almost every object in mathematics is a collection of other objects. What exactly is a collection, and what can be built from collections? ## What this area is about A set is decided entirely by its members: an object is in it or not. From that one idea come operations that build new sets, ways to compare and count sets, and the functions and relations that connect one set to another. Much of later mathematics is written in this language. ## How these fit together **Definitions.** Elements and membership, roster and rule notation, subsets and equal sets, cardinality $|A|$ of a finite set, and the power set $\mathcal{P}(A)$. **Building sets.** Union, intersection and complement and the Cartesian product $A \times B$. **Results and proofs.** The distributive and De Morgan laws, proving a set identity by showing each side is a subset of the other, and counting a union: for finite sets, $|A \cup B| = |A| + |B| - |A \cap B|$. **Relations and functions.** Properties of relations lead to equivalence classes. One-to-one and onto functions lead to composition and inverses. Infinite sets compare sizes by bijections. ## A path through it Elements first, then roster notation and subsets. Roster notation opens cardinality, power sets, products and union and intersection. Union leads to the set laws (with De Morgan's laws from logic) and to counting a union. Proving a set identity needs subsets and direct proof. Products lead to relations; equivalence classes come after relations, counting a union and congruence from number theory. Subsets lead to one-to-one and onto functions, then composition. Infinite sets come after cardinality, power sets and bijections. ### [Setting up a recurrence from a counting problem](https://math.wiki/recurrences-from-counting-problems/) **Summary:** Sort the cases by what happens first or last, and the count for n becomes a sum of counts for smaller n. **Question:** A count for size 20 is hard to list, but the objects of size 20 are built from smaller ones. How do you turn that into a rule that computes every count from the ones before it? ## When you reach for it Some counts have no quick formula but have an obvious way to grow: a tiling of a long strip is a tiling of a shorter strip plus one tile, and a string is a shorter string plus one letter. When the objects of size $n$ are built from objects of the same kind and smaller size, the count $a_n$ can be written in terms of $a_{n-1}$, $a_{n-2}$ and so on. That equation, with enough starting values, is a **recurrence**, and it computes every term exactly (sequence-from-a-rule). Reach for it when listing works for small $n$ but the objects are too many to list later, and when the structure of one object can be read off from its first or last piece. The recurrence is also the input to solve-a-linear-recurrence, which turns many such rules into closed formulas. ## The procedure ### Day 1: sort by the last piece Everything rests on the **addition principle** (counting-principles): if a finite set is split into pieces with no element in two pieces and every element in some piece, its size is the sum of the sizes of the pieces. **Name the count.** Say exactly what $a_n$ counts, including what size $0$ would mean. Often there is one object of size $0$, the empty one, and taking $a_0 = 1$ makes the rule work from small $n$. **Sort by what happens at one end.** Look at the last piece (or the first): the last tile, the last letter, the last stride. Each possible last piece gives a case. The cases must be **disjoint** (an object has only one last piece) and must **cover everything** (every object of size $n \ge 1$ has a last piece). Then the addition principle applies. ### [Solving and checking a two-term recurrence](https://math.wiki/solve-a-linear-recurrence/) **Summary:** Trying a power as a guess turns the rule into a quadratic whose roots build the closed formula. **Question:** A rule that builds each term from the two before it gives every term in the end, but only by computing all the earlier ones first. Is there a formula that goes straight to the hundredth term? ## When you reach for it A rule of the form $$a_n = c_1 a_{n-1} + c_2 a_{n-2} \quad (n \ge 2),$$ with fixed numbers $c_1$ and $c_2 \ne 0$ and two starting values $a_0$, $a_1$, fixes every term, but only one at a time. Reach for this move when you want $a_n$ straight from $n$: a far term, the rate of growth, or a fact about every term at once. closed-form-versus-recursive turned one-term rules into formulas by spotting a pattern; a rule that looks back two places shows no pattern, yet its answer is still built from powers. The same method gives the closed formula for the Fibonacci numbers of fibonacci-sequence, with the golden ratio in it. A second move belongs here too: checking a closed formula against the rule by strong-induction proves it for every $n$, however the formula was found. Such a rule is a **linear homogeneous recurrence of order two with constant coefficients**: earlier terms are only multiplied by constants and added, nothing else is added, and it looks back two places. A **solution** is any sequence that obeys the rule for every $n \ge 2$, whatever its starting values. ## The procedure ### Day 1: trying a power A geometric sequence obeys a one-term rule, so try one in the two-term rule. For $r \ne 0$, the sequence $r^n$ is a solution exactly when $$r^n = c_1 r^{n-1} + c_2 r^{n-2} \quad \text{for every } n \ge 2.$$ Dividing by $r^{n-2}$, which is not zero, turns every one of these equations into the same quadratic, ### [Spanning trees and minimum spanning trees](https://math.wiki/minimum-spanning-tree/) **Summary:** Choosing the cheapest set of edges that keeps everything connected, adding safe edges cheapest first. **Question:** Every site must be linked to every other, and every possible link has a price. Which links should be built so that everything is connected at the least total cost? ## When you reach for it A set of sites must all be joined: huts by groomed trails, buildings by cable, tanks by pipe. Each possible link has a weight, in miles or dollars, as in weighted-graphs, and you want the cheapest set of links in which every site can reach every other. With positive weights the cheapest such set has no cycle, since deleting one edge of a cycle keeps everything connected and lowers the total. So it is a tree that reaches every vertex: a minimum spanning tree. Trying every tree is hopeless: by Cayley's formula (tree-edge-count, Result 11) the complete graph on $15$ sites has $15^{13} \approx 1.9 \times 10^{15}$ spanning trees. Kruskal's and Prim's methods look at each edge about once. This is the cheapest *network*; the cheapest *route* is shortest-path-in-a-weighted-graph. ## The procedure Two days. Day 1: spanning trees and Kruskal's method (Examples 1 to 3). Day 2: why it is right, Prim's method, uniqueness and ties (Examples 4 to 6). ### Day 1: spanning trees and Kruskal's method Graphs are finite and simple, with the trees, components and results of tree-edge-count and the weights of weighted-graphs. Let $G = (V, E)$ have $n$ vertices. - A **spanning tree** of $G$ is a set $T \subseteq E$ of edges such that the graph $(V, T)$, which has every vertex of $G$, is a tree. By Result 3 of tree-edge-count, $\lvert T \rvert = n - 1$. - For a set $H$ of edges, $w(H)$ is the sum of $w(e)$ over $e \in H$. A **minimum spanning tree** is a spanning tree $T$ with $w(T) \le w(T')$ for every spanning tree $T'$ of $G$. - For an edge $e \notin T$ and an edge $f \in T$, $T + e - f$ is $T$ with $e$ added and $f$ removed. ### [Statements and negation](https://math.wiki/statements-and-negation/) **Summary:** A sentence that is true or false, and its opposite. **Question:** A sentence is true, or it is false, and never both. What single sentence swaps the two? ## Why this exists Before you can decide whether a sentence is true, you have to decide whether it is the kind of sentence that could be true at all. "Close the door" cannot be true or false; neither can "Is it raining?" Mathematics only argues about sentences that settle one way or the other, so the first job, before any proof or calculation, is picking out which sentences those are. Once you have one, the fastest way to prove you understand it is to say exactly what would make it false. That opposite sentence is the **negation**, and it comes up everywhere a proof needs to rule something out: to show a shape is *not* a square, a number is *not* prime, or a line is *not* parallel to another, you first have to know precisely what sentence you are trying to make true. What negation means takes one sentence to say. Using it inside longer statements needs some bookkeeping as well: a way to record, in one small table, what "not" does to every truth value a statement can have, so that the negation of a statement can be one column of a bigger table and every $\neg$ in a formula is read as covering exactly what it covers and nothing more. That table, the **truth table for not**, is the second half of this lesson. ## Definition A **statement** is a sentence that is either true or false, and not both. It does not have to be about mathematics, and nobody has to know which one it is yet, but there has to be a fact of the matter. ### [Strong induction and well-ordering](https://math.wiki/strong-induction/) **Summary:** Assuming the statement for all earlier values, not only the one just before, to prove the next. **Question:** To prove a claim about $n$, what if the step needs the claim at some smaller value that you cannot name in advance, not at $n - 1$? ## When you reach for it An ordinary induction step (how induction works) proves $P(k+1)$ from $P(k)$. That fits when case $k+1$ is case $k$ with one piece more. Many objects break apart otherwise. A composite $n$ splits as $n = ab$, and $a$ may be $2$, or $\frac{n}{2}$, or anything between. Halving $n$ lands near $\frac{n}{2}$, nowhere near $n - 1$. The step needs the claim at a smaller value, but not one fixed in advance. **Strong induction** hands the step every smaller case at once. Reach for it when breaking case $n$ apart gives pieces of unknown size below $n$. Its partner, the **well-ordering principle**, runs the same idea as a contradiction: if the claim failed anywhere, there would be a smallest failure, and you show that it cannot exist. ## The procedure ### Day 1: the strong hypothesis Let $P(n)$ be a statement about each integer $n \ge n_0$. If the lines above the bar hold, the line below it holds. $$ \begin{array}{c} P(n_0) \text{ is true} \\ P(n_0) \land \cdots \land P(k) \to P(k+1) \\ \text{for every } k \ge n_0 \\ \hline P(n) \text{ is true for every } n \ge n_0 \end{array} $$ The assumption that $P(n_0), \dots, P(k)$ all hold is the **strong hypothesis**, and the step may call on any of these cases, as often as it likes. Proofs often write $n$ for $k + 1$: "Let $n > n_0$, and assume $P(m)$ for every $m$ with $n_0 \le m < n$." Every value the step calls on must be at least $n_0$, since the hypothesis says nothing below $n_0$, and strictly less than the value being proved, or the argument is circular. ### [Subsets, proper subsets and equal sets](https://math.wiki/set-equality/) **Summary:** Two sets are equal when each one is a subset of the other, which is how set identities get proved. **Question:** Every member of one set is also in another. Is the second set bigger, or are the two really one set described in different words? ## Why this exists Once there are sets, the first thing to ask about two of them is how they sit against each other. The simplest question is whether one lies inside the other: is every member of this set also a member of that one? Every student in the chess club is a student at the school. Every multiple of $6$ is a multiple of $3$. Every counting number is an integer. The word for this is **subset**, and it is the relation that the rest of the language of sets is built on. The word subset is generous, though. The statement "$A$ is a subset of $B$" is true when $A$ is a small part of $B$, and it is just as true when $A$ and $B$ are the same set. So on its own it cannot tell you whether $B$ has anything extra. Often the extra is the whole point: the integers have negative numbers that the counting numbers never reach. To say that two sets are nested and **not** the same, mathematics needs a stricter word, **proper subset**. The other end of the question is just as common. Sets are named in many ways. One description lists the members, another gives a rule, and a third builds the set from two others with unions and intersections. Sooner or later two descriptions turn up that look nothing alike, and the question is whether they name one set or two. Listing both and comparing works when the lists are short. It fails for a set such as "the integers that are multiples of both $2$ and $3$", which has infinitely many members, so no list is ever finished. What is needed is a test that never lists anything: a way to settle equality by an argument about one typical member. That test is one of the most used moves in mathematics. Two sets are **equal** exactly when each is a subset of the other. ### [Tautologies and contradictions](https://math.wiki/tautology/) **Summary:** A statement that comes out true on every row of its truth table. **Question:** A compound statement's letters can be true or false in every possible way there is. Can its form alone force it to be true in every case, or false in every case, and what is either one good for? ## Why this exists Building a table for a compound statement usually gives a mixed column: some rows T, some F, depending on which combination of true and false you plugged into the letters. That is the ordinary case. Whether "the shop is open" is true depends on the shop: you have to go and look. Joining it to other statements with "and", "or" and "not" usually keeps that dependence, and the compound is true in some situations and false in others. Every so often the last column comes out T on every single row, however the letters are chosen. That is no longer luck: it means the connectives alone force the truth, with no help at all from what $P$ or $Q$ stand for. You could let $P$ be "the bus is late" and $Q$ be "seven is prime", or any other pair of sentences at all, and the compound statement would still come out true, because nothing about it ever asked whether $P$ or $Q$ were actually true. Such a statement is a tautology, and it turns out to be exactly the fact underneath why some arguments work regardless of their subject matter, and why two differently written statements can be guaranteed to say the same thing. The column can also come out F on every row. "The shop is open and the shop is not open" is false before anyone looks at the shop, and it would be false whatever shop it named. Its falsehood comes from how it is built, not from the world. A statement like that is a contradiction. Recognizing one pays twice. A claim with the form of a contradiction can be rejected without gathering any evidence. And reaching a contradiction is the finishing move of one of the strongest methods of proof: if an assumption leads to a statement that cannot be true, the assumption was false. ### [Testing an argument with a truth table](https://math.wiki/test-an-argument-with-a-truth-table/) **Summary:** Checking every row where the given statements are all true to see whether the conclusion is ever false. **Question:** An argument sounds convincing. How can you be certain that its conclusion really follows from what it assumes, and does not just sound as if it does? ## When you reach for it An argument is a list of premises followed by a conclusion, and it is valid when the conclusion cannot be false while every premise is true. Reading a persuasive argument again rarely tells you whether its conclusion is forced or only sounds nearby. A truth table settles the question mechanically. Give each simple statement a letter, and the table lists every way those letters can be true or false: every situation the argument can tell apart. Validity is a claim about every situation, so a table that lists them all checks it completely. Reach for it when an argument is built from whole statements joined by "and", "or", "not", "if ... then" and "if and only if": to test a pattern of reasoning before relying on it, or to find the concrete situation that breaks one. ## The procedure Take the argument: "If the heater is on, the room is warm. The room is not warm. Therefore the heater is not on." **Give each simple statement a letter.** Let $P$ be "the heater is on" and $Q$ be "the room is warm". The premise "the room is not warm" is the denial of $Q$, so it is written $\neg Q$, not given a new letter. A separate letter would let the table contain rows in which the room is both warm and not warm, situations that cannot happen, and one of them could pass for a counterexample. In symbols the argument is $$\begin{array}{l} P \rightarrow Q \\ \neg Q \\ \hline \therefore\ \neg P \end{array}$$ ### [The Chinese remainder theorem](https://math.wiki/chinese-remainder-theorem/) **Summary:** Remainders on division by numbers with no common factor pin down a number uniquely up to their product. **Question:** A number leaves remainder 2 on division by 5, remainder 3 on division by 7 and remainder 4 on division by 9. Is there such a number, how do we find it, and how much do the three remainders pin it down? ## Why this exists Knowing the remainder of $x$ on division by $5$ pins $x$ down modulo $5$; knowing it on division by $7$ pins it down modulo $7$. Do the two together pin $x$ down modulo $35$, and can any pair of remainders occur? The opening question asks this for three divisors: is there an $x$ with $$ \begin{aligned} x &\equiv 2 \pmod{5}, \\ x &\equiv 3 \pmod{7}, \\ x &\equiv 4 \pmod{9}, \end{aligned} $$ and if so, how many? The answer is yes, exactly one modulo $5 \cdot 7 \cdot 9 = 315$, and the proof builds it with the inverses of modular-inverse. A problem of this kind, a number to be found from its remainders, appears in the *Sunzi Suanjing*, a Chinese arithmetic text from the 3rd to 5th century, hence the name. Day 1 treats coprime moduli; day 2 drops that condition and recasts the theorem as a correspondence between $\mathbb{Z}_{mn}$ and pairs. ## Definition Moduli are integers at least $2$. As in equivalence-classes, $\mathbb{Z}_n$ is written as the remainders $0, 1, \dots, n - 1$, each standing for its class. ### Day 1: systems and pairwise coprime moduli A **system of congruences** in the unknown $x$ is a list $$ \begin{aligned} x &\equiv a_1 \pmod{m_1}, \\ &\ \ \vdots \\ x &\equiv a_k \pmod{m_k}. \end{aligned} $$ A **solution** is an integer $x$ satisfying all of them at once. The moduli are **pairwise coprime** when $\gcd(m_i, m_j) = 1$ for every $i \ne j$. For example $5, 7, 9$ are pairwise coprime, while $6, 10, 15$ are not, although no prime divides all three. ### [The division algorithm](https://math.wiki/the-division-algorithm/) **Summary:** Dividing any integer by a positive one leaves a quotient and a remainder smaller than the divisor. **Question:** Dividing $389$ by $14$ gives $27$ with $11$ left over. Does every integer, negative ones included, split this way against every positive divisor, and in only one way? ## Why this exists Grade-school division already answers "how many times, and how much is left": $389 \div 14$ is $27$ with $11$ left over, because $389 = 14 \cdot 27 + 11$ (interpret-remainders-in-context, standard-long-division-algorithm). A proof needs a statement that holds for every integer, negative ones included, and it needs the remainder to be one definite number, so that "the remainder of $a$ on division by $b$" means something. Divisibility asks a yes or no question: is $a = bk$ for some integer $k$? The division algorithm says what happens when the answer is no. The proofs by cases in cases-from-parity used the forms $3k$, $3k + 1$, $3k + 2$ on the strength of this theorem, stated there without proof. This lesson proves it. ## Definition **The division theorem.** Let $a$ be an integer and $b$ a positive integer. Then there is exactly one pair of integers $q$ and $r$ with $$a = bq + r \quad \text{and} \quad 0 \le r < b.$$ The number $a$ is the **dividend**, $b$ the **divisor**, $q$ the **quotient** and $r$ the **remainder**. The remainder is written $a \bmod b$, read "$a$ mod $b$". So $389 \bmod 14 = 11$. The result is a theorem; the name "division algorithm" is traditional. The **floor** of a real number $x$, written $\lfloor x \rfloor$, is the greatest integer that is at most $x$. So $\lfloor 3.7 \rfloor = 3$, $\lfloor 5 \rfloor = 5$ and $\lfloor -7.8 \rfloor = -8$. ## Intuition ### [The Euclidean algorithm and coprime numbers](https://math.wiki/euclidean-algorithm/) **Summary:** Find a greatest common factor by repeated division, and why it works. **Question:** Factoring a large number can take thousands of trial divisions. Is there a way to find the largest number dividing two integers without ever factoring either one? ## When you reach for it Reach for the Euclidean algorithm whenever you need the largest integer dividing two given integers, or need to know whether two integers share any factor at all. For small numbers you can list the factors (gcf-by-listing-factors) or compare prime factorizations (gcf-and-lcm-from-prime-factors). Both need the factors first, and finding factors is slow. Ksenija wants the greatest common divisor of $4757$ and $3763$. To factor $4757$ she would try the primes $2, 3, 5, 7, \dots$ in turn; the first that works is $67$, the nineteenth prime, giving $4757 = 67 \cdot 71$. A second search gives $3763 = 53 \cdot 71$, so the answer is $71$. The Euclidean algorithm reaches $71$ in six divisions and never learns that $67$ or $53$ exist (Example 1). The gap grows with the numbers: for two $100$-digit numbers the algorithm needs at most a few hundred divisions, while factoring numbers of that size by trial division is out of reach. ## The procedure **Greatest common divisor.** Let $a$ and $b$ be integers, not both zero. A **common divisor** of $a$ and $b$ is an integer $d$ with $d \mid a$ and $d \mid b$. The **greatest common divisor** $\gcd(a, b)$ is the largest of them. It exists: $1$ is always a common divisor, and if, say, $a \ne 0$, every common divisor has $|d| \le |a|$ by the size bound of divisibility-as-a-relation, so there are finitely many. Hence $\gcd(a, b) \ge 1$. Since $d \mid a$ exactly when $d \mid -a$, signs do not matter: $\gcd(a, b) = \gcd(|a|, |b|)$, and clearly $\gcd(a, b) = \gcd(b, a)$. ### [The Fibonacci sequence](https://math.wiki/fibonacci-sequence/) **Summary:** The sequence starting 1, 1 in which each later term is the sum of the two before it. **Question:** A sequence where each term is the sum of the two before it has no obvious formula. How can anything be proved about all of its terms? ## Why this exists A rule that looks back one term, such as $a_n = 2a_{n-1} + 1$, can often be unwound into a formula (closed-form-versus-recursive). The simplest rule that looks back **two** terms, "add the last two", already resists that: its closed formula involves $\sqrt{5}$ and is the subject of solve-a-linear-recurrence. Yet the sequence it builds is full of exact patterns about squares, products, divisors and sizes. Every one of them has to be proved straight from the rule, which makes this sequence the standard place to practice induction with two base cases. Earlier lessons have already met it: sequence-from-a-rule counted its odd terms and found it in a bee family tree (Exercises 7 and 16 there), inductive-step proved $F_1 + F_2 + \cdots + F_n = F_{n+2} - 1$ (Exercise 24 there), and strong-induction bounded other rules that look back two or three terms. This lesson collects the main facts and proves them. ## Definition The **Fibonacci sequence** starts with two ones, and every later term is the sum of the two terms before it: $$\begin{aligned} &F_1 = 1, \qquad F_2 = 1, \\ &F_n = F_{n-1} + F_{n-2} \ \text{ for } n \ge 3. \end{aligned}$$ The terms $F_n$ are the **Fibonacci numbers**. Running the rule backward once, $F_2 = F_1 + F_0$ forces $F_0 = 0$; some results below are neater with $F_0$ allowed. The sequence is named for Leonardo of Pisa, known as Fibonacci, whose book *Liber Abaci* introduced it to European readers. ### [The pigeonhole principle](https://math.wiki/pigeonhole-in-proofs/) **Summary:** Choosing what counts as the items and what counts as the containers so the conclusion falls out. **Question:** More things than places to put them. What does that force, and how do you decide what the things and the places are? ## When you reach for it Some claims say that in *every* arrangement of a collection, two members must share something: two chosen numbers add to a fixed total, two points sit close together, two guests know the same number of people. Listing arrangements cannot prove such a claim. The pigeonhole principle proves it in one stroke. It is a tool of existence proof: it shows that a pair with the property exists, usually without saying which pair. The principle itself is close to obvious. The skill is in the setup: deciding what the objects (the "pigeons") are and what the boxes (the "holes") are, so that two objects in one box gives exactly the conclusion wanted. ## The procedure This lesson takes two days. Day 1 is the principle and the setup; Day 2 is the generalized principle and longer arguments. Examples 1 to 3 go with Day 1, Examples 4 to 7 with Day 2. ### Day 1: the principle and why it is true **The pigeonhole principle.** If more than $k$ objects are placed in $k$ boxes, then some box holds at least two of them. *Proof.* Suppose, for contradiction, that every box holds at most one object. Adding over the $k$ boxes, there are at most $k$ objects in all. But there are more than $k$. This contradiction shows that some box holds at least two objects. $\square$ The same fact in the language of functions: placing each object of a finite set $A$ in a box from a finite set $B$ is a function $f : A \to B$. If $\lvert A \rvert > \lvert B \rvert$, then $f$ is not injective, so there are $a \ne a'$ in $A$ with $f(a) = f(a')$. This is the finite size rule of injective-surjective-bijective read backward. ### [There are infinitely many primes](https://math.wiki/infinitude-of-primes/) **Summary:** Euclid's argument, the first great proof by contradiction. **Question:** The primes thin out as the numbers grow. Do they ever stop? ## Why this exists A prime is an integer greater than $1$ whose only positive divisors are $1$ and itself. The sieve shows them thinning out: $25$ primes below $100$, only $21$ between $100$ and $200$, and stretches with no prime as long as you like (existence-proof builds them). Perhaps the primes run out, and some largest prime exists. No search can settle this, since the question is about what lies beyond any search. It needs an argument about every finite list of primes at once. The standard one is over two thousand years old, it is often a reader's first real proof-by-contradiction, and its idea, "multiply and add one", reaches well past where it started. Day 2 pushes it to primes of a special form and to a bound on the size of the $n$th prime. ## Definition A set is **finite** if its elements can be written as a list $a_1, a_2, \ldots, a_k$ for some whole number $k$; otherwise it is **infinite**. The statement of this lesson is $$\text{The set of primes is infinite.}$$ Equivalently: for every finite list of primes there is a prime that is not on the list. The primes in increasing order are $p_1 = 2$, $p_2 = 3$, $p_3 = 5$, and so on, so $p_n$ is the $n$th prime. For a positive integer $n$, $n! = 1 \cdot 2 \cdot 3 \cdots n$. An integer has the **form $4k + 3$** if it equals $4k + 3$ for some integer $k$, that is, if it leaves remainder $3$ on division by $4$. The forms $4k + 1$, $6k + 1$ and $6k + 5$ read the same way. ### [There exists, and there is exactly one](https://math.wiki/existential-quantifier/) **Summary:** A claim that at least one object in a stated collection makes a statement true. **Question:** You want to say that at least one number works, without saying which, or that exactly one does. How do you say each exactly, and what does it take to prove it? ## Why this exists Pierre de Fermat believed that every number of the form $2^{2^n} + 1$ is prime. The first five, $3$, $5$, $17$, $257$ and $65537$, are. The sixth is $2^{32} + 1 = 4294967297$, and for almost a century nobody knew whether it was prime. In 1732 Leonhard Euler settled the question by showing a single number: $$4294967297 = 641 \times 6700417$$ He did not need to list every divisor, or say anything about the other numbers of that form. One divisor was enough to prove that a divisor exists, and so that the number is not prime. Many questions in mathematics have this shape: is there a solution, is there a case where the rule breaks? English asks them with loose words like "some", which carry hints ("some students passed" suggests that not all did). A proof needs one exact meaning for "at least one", and a symbol. There is a second problem. A sentence such as $x^2 = 2$ is not true and not false: it depends on what $x$ is. It is not yet a statement. Saying "there is an $x$ for which this holds" turns it into one. And there is a stronger claim hiding in a very small word. "The solution of $3x - 6 = 0$ is $x = 2$." The word "the" is a promise: it says a solution exists, and it says there is only one, so that naming it with a single number is not misleading. "There exists" makes only the first half of that promise. ${\exists x \in \mathbb{R},\ x^2 = 4}$ is true, but nobody should write "the solution of $x^2 = 4$", because there are two of them, $2$ and $-2$, and neither is entitled to the definite article. Mathematics is full of objects introduced with "the": the midpoint of a segment, the additive inverse of a number, the remainder when you divide by $7$. Each of those phrases carries a claim that could be false, and before "the" is allowed someone has to check that exactly one object earns it, neither zero nor two. That claim gets its own symbol too. ### [Trees](https://math.wiki/tree-edge-count/) **Summary:** Every tree with n vertices has exactly n minus 1 edges, which induction on the vertex count shows. **Question:** A network hangs together and has no loop. How many links must it have, and what else does that one count force? ## Why this exists Some networks hang together with nothing to spare: the branching of a river, the folders on a computer, the chain of atoms in a fuel molecule. There is one route between any two points and no loop. These are trees. Cut one link and a tree falls apart; add one and it gains a loop. And one count holds for every tree: $n$ points need exactly $n - 1$ links, the fewest any connected network can have. That count lets leaves, degrees and chemical formulas be read off without a drawing. ## Definition This lesson takes two days. Day 1 is trees, forests, leaves, the edge count and unique paths (Results 1 to 7, Examples 1 to 3). Day 2 is the fewest edges of a connected graph, degrees, rooted trees and labeled trees (Results 8 to 11, Examples 4 to 7). Notation is that of degree-of-a-vertex and walks-paths-and-cycles: $G = (V, E)$ is a finite simple graph, with degrees $\deg(v)$, paths, cycles, components and distance $d(u, v)$ as there. - A **tree** is a connected graph with no cycle. A single vertex is a tree with no edges. - A **forest** is a graph with no cycle. Its components are trees. - A **leaf** is a vertex of degree $1$, and $n_i$ is the number of vertices of degree $i$. - $G - e$ removes the edge $e$ and keeps every vertex. $G - v$ removes the vertex $v$ and every edge at $v$. For non-adjacent vertices $u \ne v$, $G + uv$ adds the edge $uv$. - (Day 2) A **rooted tree** is a tree with one vertex $r$ chosen as **root**. The **level** of $v$ is $d(r, v)$ and the **height** is the largest level. For $v \ne r$, the **parent** of $v$ is the vertex just before $v$ on the path from $r$ to $v$ (unique by Result 5), and $v$ is its **child**. - (Day 2) A **labeled tree on $\{1, \ldots, n\}$** is a tree with that vertex set. Two are the same exactly when they have the same edges. ### [Triangular numbers](https://math.wiki/triangular-numbers/) **Summary:** The running totals 1, 3, 6, 10 and onward, each counting the dots in a triangle. **Question:** Stack dots in rows of one, two, three and onward. Which numbers turn up as the totals, and what do they do when you combine them? ## Why this exists The total $1 + 2 + \cdots + n$ has already turned up several times. partial-sum found it by fitting two triangles of dots into a rectangle, inductive-step proved the same formula by induction, conjecture found that two neighbors add to a square, and pascals-triangle-and-combinations found the totals running down a diagonal of Pascal's triangle. Each time the totals were a tool for something else. This lesson treats them as objects in their own right. Is a given number one of them, and which one? What happens when two are combined? Which are odd? What do their reciprocals add to? Each answer is a short proof, and some run on nothing but the rule that builds each total from the one before, which is exactly the shape induction needs. ## Definition The **$n$th triangular number** $T_n$ is the number of dots in a triangle whose rows hold $1, 2, 3, \ldots, n$ dots. Adding a row of $n$ dots to the triangle for $n - 1$ gives the triangle for $n$, and the empty triangle has no dots, so as a recursive rule $$T_0 = 0, \qquad T_n = T_{n-1} + n \quad \text{for } n \ge 1.$$ A whole number $m$ is **triangular** if $m = T_n$ for some whole number $n$, and then $n$ is its **index**. From $T_0$ to $T_{10}$ the values are $0, 1, 3, 6, 10, 15$ and then $21, 28, 36, 45, 55$. Taken from the lessons named above, and used here without being proved again: $$T_n = 1 + 2 + \cdots + n = \frac{n(n+1)}{2} = \binom{n+1}{2}.$$ ### [Truth table for the conditional](https://math.wiki/truth-table-for-the-conditional/) **Summary:** Showing that if-then counts as false only when the first part is true and the second is false. **Question:** An if-then statement makes one promise about two things. When exactly does it break that promise? ## When you reach for it "If $n$ is divisible by $6$, then $n$ is even" is a single statement built out of two smaller ones, joined by *if...then*. With $P$ for the first part and $Q$ for the second, it is written $P \to Q$: a conditional. Like and and or, it has a truth value that depends only on the truth values of $P$ and $Q$, and that dependence can be written down once, as a table. Reach for the table whenever you need to decide, from the truth of $P$ and the truth of $Q$ alone, whether $P \to Q$ holds; whenever a conditional sits inside a longer statement you are building a bigger table for; and whenever you want to see, in black and white, why a conditional with a false hypothesis still counts as true, which is the one part of this table that surprises almost everyone the first time. ## The procedure **Two letters give four rows, in the usual order.** $P$ is true for the top half and false for the bottom half, and $Q$ alternates within each half: $TT$, $TF$, $FT$, $FF$. Here is the finished table; the paragraphs after it say where each entry comes from. | Row | $P$ | $Q$ | $P \to Q$ | |:-:|:-:|:-:|:-:| | 1 | T | T | T | | 2 | T | F | F | | 3 | F | T | T | | 4 | F | F | T | **Rows $1$ and $2$ come straight from the promise.** "If $P$, then $Q$" promises that $Q$ holds whenever $P$ does. In row $1$, $P$ holds and $Q$ holds, exactly as promised, so the statement is true. In row $2$, $P$ holds and $Q$ fails: the promise is broken, so the statement is false. ### [Truth tables](https://math.wiki/truth-tables/) **Summary:** Every combination of true and false, and what the compound statement does. **Question:** A statement built from smaller ones is true in some cases and false in others. How do you list every case and see which is which? ## What this area is about Whether $P \land Q$ is true depends only on whether $P$ and $Q$ are true, not on what they say. So its whole behavior fits in a short list: one row for each way the letters can come out, $2^n$ rows for $n$ letters. That list is a truth table, and this area is about writing them and reading them. ## How these fit together **The connective tables.** Each connective's table is its definition written out case by case: two rows for not, four for the rest. The tables for not, and, or and if and only if are taught with the connectives themselves, in statements and negation, and and or and the biconditional. The one with its own star here is the truth table for the conditional, whose false-hypothesis rows surprise almost everyone. **The procedure.** Building a truth table for a compound statement joins those small tables into one for any statement, one column at a time, from the inside out. **Two special kinds of statement.** Tautologies and contradictions: a tautology is true on every row, a contradiction false on every row. Most statements are neither. ## A path through it The tables for not, and and or come first, since building needs them. Then the conditional's table, then the building method, then tautologies and contradictions. Finish here and you can settle a claim about statements by checking every case. Testing an argument with a truth table does exactly that, and logical equivalence asks when two tables match row for row. ### [Two-term recurrences with a repeated root](https://math.wiki/repeated-characteristic-roots/) **Summary:** When the characteristic equation has a double root r, the solutions are built from r to the n and n times r to the n. **Question:** When the characteristic equation of a two-term rule has only one root, the method of two powers has only one power to work with. What takes the place of the missing one? ## When you reach for it The rule $$a_n = c_1 a_{n-1} + c_2 a_{n-2} \quad (n \ge 2), \qquad c_2 \ne 0,$$ has characteristic equation $x^2 - c_1 x - c_2 = 0$, as in solve-a-linear-recurrence (written here in the letter $x$, so that $r$ can name the root). Its discriminant is $c_1^2 + 4c_2$ (discriminant-and-the-graph). When that is $0$, the equation has only one root, a **double root** (or repeated root) $r = \frac{c_1}{2}$, and the quadratic is $(x - r)^2 = x^2 - 2rx + r^2$. Comparing coefficients, $c_1 = 2r$ and $c_2 = -r^2$. So every rule with a double root has the form $$a_n = 2r\,a_{n-1} - r^2 a_{n-2},$$ with $r \ne 0$ because $c_2 \ne 0$. The distinct-roots theorem of solve-a-linear-recurrence needs two different roots, so it says nothing about these rules. Reach for this move whenever $c_1^2 + 4c_2 = 0$, as in $a_n = 6a_{n-1} - 9a_{n-2}$, where $6^2 + 4(-9) = 0$ and $x^2 - 6x + 9 = (x - 3)^2$. ## The procedure ### One power is not enough The sequence $r^n$ is still a solution, since $r$ is a characteristic root. But the family the distinct-roots method would write, $Ar^n + Br^n = (A + B)r^n$, has only one free constant, $C = A + B$. At $n = 0$ and $n = 1$ it gives $a_0 = C$ and $a_1 = Cr$, so it fits only those starting values with $a_1 = r\,a_0$. For $a_n = 6a_{n-1} - 9a_{n-2}$ with $a_0 = 2$ and $a_1 = 9$ it would need $9 = 3 \cdot 2$, which is false. A second solution, not a multiple of $r^n$, is needed. ### [Uncountable sets and the diagonal argument](https://math.wiki/cantor-diagonal-argument/) **Summary:** Any list of real numbers misses one, built to differ from the first number in the first digit and so on. **Question:** The integers, the fractions and every finite piece of text can all be put in one endless list. Is there any infinite set that no list can ever finish, however cleverly it is arranged? ## Why this exists countable-sets listed the integers, the pairs of natural numbers, the rationals, every finite string and every polynomial with integer coefficients. Each time a clever route reached every member. It is natural to suspect that every infinite set can be listed. Throughout, $\mathbb{N} = \{0, 1, 2, \dots\}$. It cannot. Georg Cantor proved in 1874 that the real numbers cannot be listed, by an argument about nested intervals; the argument taught here, which turns any proposed list against itself, appeared in a later paper of his, from 1891. It shows that infinite sets come in more than one size, and it is one of the most reused ideas in mathematics. ## Definition ### Day 1: uncountable sets A set is **uncountable** when it is not countable, that is, when it is neither finite nor countably infinite (countable-sets). An **infinite binary sequence** $s$ assigns a term $s(k) \in \{0, 1\}$ to each place $k \in \mathbb{N}$, and is written $s = (s(0), s(1), s(2), \dots)$. Two sequences are equal when they agree in every place. The set of all of them is written $\{0, 1\}^{\mathbb{N}}$. A **decimal expansion** of a number $x$ in $[0, 1)$ is a sequence of digits $d_1, d_2, d_3, \dots$, each in $\{0, 1, \dots, 9\}$, with $$x = \sum_{k=1}^{\infty} d_k \, 10^{-k}, \quad \text{written } x = 0.d_1 d_2 d_3 \dots$$ Every number in $[0, 1)$ has at least one decimal expansion. That fact is stated without proof here; it rests on the completeness of the real line (irrational-and-real-numbers). Some numbers have two: $0.4999\ldots$ and $0.5000\ldots$ are the same number, $\frac{1}{2}$. ### [Union, intersection and complement](https://math.wiki/union-and-intersection/) **Summary:** Either, both, or not. **Question:** Two sets are on the table. How do you name everything in either one, only what they share, or everything outside one of them? ## Why this exists Sets are rarely met one at a time. Who is in the drama club and also in the robotics club? Which days of a month are neither weekend days nor rainy? Which numbers $x$ satisfy $x < 3$ and $x \ge -1$ at once? Each question builds a new set from sets that already exist, and the words doing the work are always "or", "and" and "not". This lesson turns those three words into three operations on sets, first for small sets written as a roster, then for stretches of the number line. ## Definition $A$ and $B$ are sets, and membership is read as in element-and-membership: $x \in A$ says $x$ is an element of $A$, and $x \notin A$ says it is not. **Union.** $A \cup B$ is the set of everything that belongs to $A$, or to $B$, or to both. $$x \in A \cup B \iff x \in A \text{ or } x \in B$$ **Intersection.** $A \cap B$ is the set of everything that belongs to both $A$ and $B$. $$x \in A \cap B \iff x \in A \text{ and } x \in B$$ **Universal set.** A universal set $U$ is the set of everything under discussion, fixed before the question is asked: the whole numbers from $1$ to $12$, the students in one class, all real numbers. Every set in the discussion is a subset of $U$. **Complement.** Once a universal set $U$ is stated, $A^{c}$ is the set of everything in $U$ that is not in $A$. A complement is always taken relative to that stated $U$. $$x \in A^{c} \iff x \in U \text{ and } x \notin A$$ ### [Valid and sound arguments](https://math.wiki/valid-and-sound-arguments/) **Summary:** Valid means the conclusion must follow, and sound adds that the given statements are actually true. **Question:** An argument's conclusion follows from its premises by the rules of logic. Does that make the conclusion true? ## Why this exists An argument offers some statements as reasons for a final one. Some arguments force their conclusion; others only look as if they do. And even an argument whose logic is airtight can start from a false statement. "Does the conclusion follow?" and "are the starting statements true?" are two separate questions, and a reader who runs them together can be taken in by either kind of bad argument. The two words in this lesson keep them apart: **valid** answers the first question, **sound** answers both. ## Definition An **argument** is a list of statements, the **premises**, followed by one more statement, the **conclusion**, which the premises are offered to support. Arguments are usually written with the premises stacked, a line, and the conclusion marked $\therefore$ ("therefore"): $$ \begin{array}{l} P \to Q \\ P \\ \hline \therefore\ Q \end{array} $$ Every premise and the conclusion must be a statement: a sentence that is definitely true or definitely false, never both and never neither. **An argument is valid** when there is no way for every premise to be true and the conclusion false at the same time. Equivalently: *whenever* all the premises are true, the conclusion is forced to be true along with them. Validity says nothing about whether the premises actually are true; it only says what would happen if they were. **An argument is sound** when it is valid **and** every one of its premises is actually true. ### [Venn diagrams and counting a union](https://math.wiki/inclusion-exclusion-two-sets/) **Summary:** Adding the sizes of two sets counts the overlap twice, so the overlap is subtracted once. **Question:** You know how many are in one group, how many are in another, and how many are in both. How many are in at least one, and how many in neither? ## When you reach for it A question asks how many things are in one group **or** another: students who take art or music, numbers divisible by $6$ or by $15$, households with a dog or a cat. Often it asks the opposite too, how many are in **neither**. The sizes of the two groups are known, and so is the size of their overlap, or enough to find it. Adding the two sizes is the natural first move, and it is right only when the groups share nothing. The cardinality of a union counts each element once, while $\lvert A\rvert + \lvert B\rvert$ counts every element of the overlap twice: once as a member of $A$ and once as a member of $B$. This lesson removes that second count, and shows how a Venn diagram keeps track of all four regions at once. ## The procedure Throughout, $A$ and $B$ are finite sets inside a universal set $U$, and $\lvert A\rvert$ is the number of elements of $A$. **Four regions.** Two overlapping loops in a Venn diagram cut $U$ into four regions: in $A$ only, in both, in $B$ only, in neither. Every element of $U$ lies in exactly one of them, so the four region counts add up to $\lvert U\rvert$. The regions are linked to the given totals: | Region | Its size | |---|---| | In both, $A \cap B$ | $\lvert A \cap B\rvert$ | | In $A$ only, $A \cap B^{c}$ | $\lvert A\rvert - \lvert A \cap B\rvert$ | | In $B$ only, $B \cap A^{c}$ | $\lvert B\rvert - \lvert A \cap B\rvert$ | | In neither, $(A \cup B)^{c}$ | $\lvert U\rvert - \lvert A \cup B\rvert$ | ### [Walks, paths, cycles and connected graphs](https://math.wiki/walks-paths-and-cycles/) **Summary:** Routes through a graph, a path repeating no vertex and a cycle returning to where it began. **Question:** In a network, when can you get from one point to another by following links, what is the shortest way, and when does the network fall into separate pieces? ## Why this exists A graph records single links, but the questions people ask are about routes: can a visitor get from this pavilion to that one, by how few walkways, is there a loop back to the start, does the whole site hang together? Answering them needs exact words for a route, a way to stop a route revisiting points, and connected graphs and their pieces, the components. ## Definition The notation is that of degree-of-a-vertex: $G = (V, E)$ is a simple graph, with $\lvert V \rvert$ vertices, $\lvert E \rvert$ edges and degrees $\deg(v)$; $\delta(G)$ is the smallest degree. - A **walk of length $k$** from $u$ to $v$ is a sequence of vertices $(v_0, v_1, \ldots, v_k)$ with $v_0 = u$, $v_k = v$, and $v_{i-1}v_i \in E$ for each $i = 1, \ldots, k$. The **length** $k$ counts a repeated edge each time it is used. The single vertex $(u)$ is a walk of length $0$. The **edges of the walk** are the edges $v_{i-1}v_i$. - A walk is **closed** if $k \ge 1$ and $v_k = v_0$. - A **trail** is a walk that uses no edge twice. - A **path** is a walk with no vertex repeated. It uses no edge twice, so every path is a trail. - A **cycle of length $k$** is a closed walk $(v_0, v_1, \ldots, v_k = v_0)$ with $k \ge 3$ in which $v_0, v_1, \ldots, v_{k-1}$ are all different. Starting at another of its vertices or running it backward gives the same cycle. - $u$ and $v$ are **joined** if some walk runs from $u$ to $v$. $G$ is **connected** if every two of its vertices are joined, and **disconnected** otherwise. - A **component** of $G$ is a class of the relation "is joined to" (Result 2) together with every edge of $G$ between vertices of that class. - The **distance** $d(u, v)$ is the length of a shortest walk from $u$ to $v$, and $d(u, v) = \infty$ if $u$ and $v$ are not joined. - For $n \ge 3$ the **cycle graph** $C_n$ has vertices $0, 1, \ldots, n - 1$, with $i$ joined to $i + 1$ for each $i$, taken modulo $n$. ### [Weighted and directed graphs](https://math.wiki/weighted-graphs/) **Summary:** Graphs whose edges carry a number, such as a distance, a cost or a time. **Question:** When every link of a network carries a price, a distance or a travel time, or can be used in one direction only, what is the cheapest way through it, and what can still be counted? ## Why this exists A graph says which pairs are linked and nothing more. A messenger needs the minutes each street takes, the dollars each handoff costs, and which streets run one way. A number on every edge makes a **weighted graph**, where a route's weight is a sum and the direct link is not always the cheapest way. A direction on every edge makes a **directed graph**, the model for one-way streets, for task schedules and for any relation on a finite set. Degree splits into arrows in and arrows out, and the handshake count survives. ## Definition Graphs are written as in degree-of-a-vertex: $G = (V, E)$ is a simple graph with degrees $\deg(v)$. Walks, paths and cycles are those of walks-paths-and-cycles. - A **weighted graph** is a graph $G = (V, E)$ with a **weight function** $w : E \to \mathbb{R}$. The number $w(uv)$ is the **weight** of the edge $uv$: miles, minutes or dollars. Here every weight is positive unless said otherwise. - The **weight of a walk** $W = (v_0, v_1, \ldots, v_k)$ is the sum of $w(v_{i-1}v_i)$ for $i = 1, \ldots, k$, an edge counted each time it is used. The walk $(u)$ has weight $0$. - A **cheapest walk** from $u$ to $v$ is one of least weight; its weight is the **weighted distance** $d_w(u, v)$, and $d_w(u, v) = \infty$ if $u$ and $v$ are not joined. - A **weight table** has a row and a column for each vertex, with $w(uv)$ in row $u$, column $v$, left blank when $uv$ is not an edge. A graph's weight table is symmetric. - A **directed graph** $D = (V, A)$ is a finite nonempty set $V$ of vertices with a set $A$ of **arcs**, each an ordered pair $(u, v)$ of two different vertices, drawn as an arrow from its **tail** $u$ to its **head** $v$. The arcs $(u, v)$ and $(v, u)$ are different; $D$ may have one, both or neither. There are no loops and no repeated arcs. - The **out-degree** $\deg^+(v)$ counts the arcs with tail $v$, the **in-degree** $\deg^-(v)$ those with head $v$. - A **directed walk** is a sequence $(v_0, v_1, \ldots, v_k)$ with $(v_{i-1}, v_i) \in A$ for each $i$. A **directed path** repeats no vertex. A **directed cycle** is a directed walk with $k \ge 2$, $v_k = v_0$ and $v_0, \ldots, v_{k-1}$ all different; $(u, v, u)$ counts, using two arcs. - A **weighted directed graph** carries $w : A \to \mathbb{R}$, with walk weights and $d_w$ as above using directed walks. Its weight table, tail as row and head as column, need not be symmetric. ### [What an argument is](https://math.wiki/logical-argument/) **Summary:** A list of statements taken as given, followed by a conclusion drawn from them. **Question:** Some statements are given, and a conclusion is drawn from them. When does the conclusion really have to follow? ## What this area is about An argument is a list of statements taken as given, the premises, followed by a conclusion drawn from them. Whether the conclusion is forced depends only on the argument's shape, not on what its statements are about. A few shapes always work, a few look almost the same and never do, and this area teaches you to tell them apart. ## How these fit together **The idea.** Valid and sound arguments names the two questions: does the conclusion follow, and are the premises true? **Two valid forms, each with its fallacy.** Modus ponens, chaining, and a common fallacy goes from $p \to q$ and $p$ to $q$, and chains $p \to q$ and $q \to r$ into $p \to r$ (hypothetical syllogism). Its fallacy, affirming the consequent, goes from $p \to q$ and $q$ to $p$. Modus tollens, ruling out, and a common fallacy goes from $p \to q$ and $\neg q$ to $\neg p$, and from $p \lor q$ and $\neg p$ to $q$ (disjunctive syllogism). Its fallacy, denying the antecedent, goes from $p \to q$ and $\neg p$ to $\neg q$. Each fallacy uses the right conditional from the wrong end. **Method.** Testing an argument with a truth table checks any argument built from whole statements for a row where every premise is true and the conclusion false. ## A path through it Valid and sound first, then modus ponens, then modus tollens, each with its fallacy. Then the truth-table test, which confirms each valid form and breaks each fallacy. ### [What may be assumed and what must be shown](https://math.wiki/anatomy-of-a-proof/) **Summary:** A proof starts from stated givens and definitions and ends at the exact claim, with nothing else assumed. **Question:** Someone says a claim is always true. While you argue for it, what may you take for granted, and what is left for you to earn? ## Why this exists A claim can be tested. Try ten numbers, or draw five pictures, and it holds every time. That is evidence, and it is worth having. It is not a proof, because the eleventh number or the sixth picture might break it, and nothing you did rules that out. A proof does something stronger. It is a short chain of statements that makes the claim unavoidable for anyone who accepts where the chain starts. That last part carries all the weight. The chain is only as good as its starting point, so a proof has to be strict about what it starts from. There are two ways to get this wrong. Start from too little, and you cannot take a single step. Start from too much, and the proof looks fine but proves nothing: it may have quietly started from the very thing it was meant to show, or from what a picture seems to say, or from one lucky example. This lesson fixes the rules: exactly what a proof may assume, and exactly what it must show. ## Definition Nearly every claim worth proving is a conditional: it has the shape "if $P$, then $Q$", even when the words hide it. "The sum of two even integers is even" means: if $a$ and $b$ are even integers, then $a + b$ is even. The shape is written $$P \Rightarrow Q$$ - $P$ is the **hypothesis**, also called the **given**. It is what the claim is allowed to start from. - $Q$ is the **conclusion**, what the proof must **prove**. A **proof** of $P \Rightarrow Q$ is a finite list of statements whose last line is $Q$, where every line is there for one of these reasons, and no other. ### [Where the divisibility tests come from](https://math.wiki/divisibility-tests-from-congruences/) **Summary:** Because 10 leaves remainder 1 modulo 9, a number and its digit sum leave the same remainder. **Question:** Adding up the digits of a number tells you whether $9$ divides it. Why should the digits know anything about $9$, and which other divisors can be read off the digits the same way? ## When you reach for it Two sets of tests are already familiar. The tests for 2, 5 and 10 read only the last digit. The tests for 3 and 9 add up all the digits, and showed why with bags of ten: each ten is a group of nine and one over. This lesson proves those tests with congruences, and the same proof produces new tests: an alternating sum for $11$, sums of three-digit blocks for $7$ and $13$, and a check on a long multiplication. Reach for it when a remainder or a divisibility is wanted for a number given by its digits. ## The procedure Every positive integer $n$ has a base-ten expansion $n = d_k 10^k + \cdots + d_1 10 + d_0$ with digits $d_i \in \{0, \dots, 9\}$ (convert-to-base-ten). Write $$ \begin{aligned} S(n) &= d_0 + d_1 + \cdots + d_k, \\ A(n) &= d_0 - d_1 + d_2 - \cdots + (-1)^k d_k \end{aligned} $$ for the **digit sum** and the **alternating digit sum**. In $A(n)$ the ones digit always carries a plus sign. Every test below comes from one move: find a power of $10$ that is congruent to $1$ or to $-1$ modulo $m$, and use the replacement rule (Corollary 4 of congruence-notation) to put that small number in place of the power. **Theorem 1 (nine and three).** For every positive integer $n$, $n \equiv S(n) \pmod{9}$ and $n \equiv S(n) \pmod{3}$. So $n$ and $S(n)$ leave the same remainder on division by $9$, and on division by $3$; in particular $9 \mid n$ exactly when $9 \mid S(n)$, and likewise for $3$. ### [Writing a proof as prose](https://math.wiki/how-a-proof-ends/) **Summary:** A proof stops when the sentence on the page is the claim itself, often marked by a small square or QED. **Question:** You have written a page of correct steps. How do you know the proof is finished? ## Why this exists A proof is an argument with one destination. Before it starts, somebody has named a claim: *the sum of two even numbers is even*, or *vertical angles are congruent*. Every line of the proof is there to get to that claim. That makes the ending the first thing a careful reader checks. They do not have to follow every step to ask two cheap questions: where did this start, and where did it stop? If it starts from what was given and stops at the claim, the steps in between are worth checking. If it stops anywhere else, it does not matter how good those steps are. The page proves something, just not what was promised. It is easy to stop when the algebra looks finished instead of when the claim has been reached. This lesson is about telling those two moments apart. ## Definition A proof **ends** at the line that states the claim itself, with a reason showing that it follows from the lines above. Lines before that one are unfinished. Lines after it are not part of the proof. Most proofs also say that they have ended, in two small ways. - A closing sentence that says the claim aloud, usually beginning with *therefore*, *so* or *hence*. The symbol $\therefore$ stands for "therefore" when space is short. - An end mark: a small square $\square$ or a filled one $\blacksquare$, or the letters $\text{Q.E.D.}$, short for the Latin *quod erat demonstrandum*, "which was to be shown". Euclid closed his theorems with the Greek words for that same phrase. ### [Writing sets: roster, rule and the empty set](https://math.wiki/roster-notation/) **Summary:** Describing a set by listing its members inside braces. **Question:** How do you write down exactly which things belong to a set, when there are a few of them, a great many, endlessly many, or none at all? ## Why this exists A set is decided by what belongs to it (elements and membership). That settles what a set is, but not how to write one down, and the sets mathematics needs come in very different sizes. Some are small with no pattern, like the people in a club, and the only way to say who is in them is to name names. Some are large but regular: there are $142$ positive multiples of $7$ below $1000$, and nobody wants to write them all when the pattern is plain after three. Some never end, like the positive multiples of $5$. Some have nothing in them, like the whole numbers between $7$ and $8$. So mathematics uses a few ways of writing sets: a **list** for the small ones, a **list with an ellipsis** for the regular ones, a **rule in words** for any set at all, and a symbol for the set with nothing in it. ## Definition **Roster notation.** A set is written by listing its elements between braces, separated by commas. The list of entries is called a **roster**. $$\{11, 13, 15, 17, 19\}$$ This is the set whose elements are exactly $11$, $13$, $15$, $17$ and $19$, and nothing else. **Order and repeats do not matter**: $\{13, 11, 19, 17, 15\}$ is the same set, and writing $13$ twice adds nothing. **A rule in words.** A set can also be described by a condition that every object either meets or fails. The set above is "the odd whole numbers between $10$ and $20$". The rule and the roster name the same set, because they let in exactly the same objects. A rule must give a clear yes or no for every object it could be asked about; "the big numbers" is not a rule, because it gives no answer for $60$. ## Measurement ### [An angle is an amount of turn](https://math.wiki/angle-as-an-amount-of-turn/) **Summary:** The size of an angle is how far one arm has to swing round to land on the other **Question:** A weather vane swings from pointing north to pointing east. How much has it moved, and would the answer change if its arrow were twice as long? ## Why this exists An angle is two rays that start at the same point, the vertex, and naming angles gives each angle a name. A name says which angle you mean. It does not say how big the angle is. So what is the size of an angle? It cannot be the length of its arms. A ray goes on forever, so every arm is endlessly long, and when we draw an angle we just stop the arms wherever the paper runs out. Two pictures of the same corner can have short arms or long ones. What does stay the same is how far apart the two arms point. A weather vane on a barn roof, the minute hand of a clock and a revolving door all show this: each one turns about a fixed center, and the question is always "how far did it turn?" This lesson answers that question with fractions of a full turn. Later, degrees gives the same answer as a number of small units. ## Definition Start with a second ray lying right on top of one arm of the angle. Hold the vertex still and swing that ray around the vertex until it lies along the other arm. **The size of the angle is the amount the ray turned.** It is measured the shorter way around, so the size is never more than a half turn. A **full turn** brings the ray all the way around, back to where it started. Smaller turns are fractions of it: - a **half turn** leaves the ray pointing the opposite way, so the two arms make one straight line. That angle is a **straight angle**. - a **quarter turn** leaves the arms making a square corner, like the corner of a page. That angle is a **right angle**. ### [Angle measures add, and finding a missing angle](https://math.wiki/angle-measures-add/) **Summary:** An angle split into parts has a measure equal to the parts added together **Question:** An angle is cut into smaller angles. How is the size of the whole tied to the sizes of the parts, and how do you find a part that nobody measured? ## Why this exists A protractor measures one angle at a time. But angles are often built out of smaller angles. The spray of a lawn sprinkler sweeps across a sidewalk and then across the lawn. The slats of a folding hand fan cut its opening into small gaps. Two pattern blocks fit together at one corner. Each time, two questions come up. If you know the parts, how big is the whole? And if you know the whole and some of the parts, how big is the part nobody measured? Both have short answers, because angle measures add the way lengths do. This lesson takes two days. Part 1 is joining angles: the measure of the whole is the sum of the parts. Part 2 runs that backward to find a missing angle, with an equation and a letter for the unknown, and uses three wholes you already know: a right angle, a straight angle and a full turn. ## Definition An angle is **split into parts** when one or more rays from its vertex run between its two arms. The parts then sit side by side. Each part shares the vertex, neighboring parts share an arm, no part reaches into another (no **overlap**), and together they fill the whole angle (no **gap**). **Angle measures add.** When an angle is split into parts that do not overlap, the measure of the whole angle is the sum of the measures of the parts. In symbols: if ray $BC$ lies inside $\angle ABD$, then $$m\angle ABD = m\angle ABC + m\angle CBD$$ Recall that $m\angle ABD$ means the measure of $\angle ABD$ in degrees (naming angles). ### [Area and perimeter](https://math.wiki/area-and-perimeter/) **Summary:** Around the edge, and how much surface inside. **Question:** A flat shape has an edge and an inside. How do you measure each one, and why does knowing one not tell you the other? ## What this area is about Perimeter is the length all the way around a flat shape, counted in length units. Area is how much flat surface it covers, counted in square units. This area builds both, then finds the area of shapes that have no formula of their own. ## How these fit together **What area and perimeter are.** Area as covering, measuring with unit squares and splitting a rectangle into rows and columns of squares set up area. Perimeter adds the sides. Perimeter versus area shows the two are separate measures. **Formulas.** Rectangles, parallelograms and trapezoids, triangles and regular polygons each get a formula and the reason for it. Why the circle formula works explains $A = \pi r^2$. **Procedures.** Splitting into rectangles, cutting any polygon into pieces and composite figures cut a shape into pieces with formulas, or take a known piece away from a bigger shape. A missing side from the area and from the perimeter run the formulas backward. ## A path through it Area: covering, then unit squares (splitting a rectangle into squares can sit beside them), then rectangles. From rectangles, splitting into rectangles, then parallelograms and trapezoids, then triangles, then regular polygons, any polygon and composite figures. Perimeter: the sum of the sides, then the missing side. Once both are in hand, the missing side from the area (its second half uses perimeter) and perimeter versus area. The circle formula waits for the area of a circle. ### [Area as covering](https://math.wiki/area-as-covering/) **Summary:** How much flat space a shape takes up, found by covering it with copies of a unit shape **Question:** A placemat, a picnic blanket and a puzzle piece each take up some flat space. How do you say how much, and how can you tell which of two shapes takes up more? ## Why this exists Tomoya wants to know how big his placemat is. A ruler tells him how long it is, and how wide, but neither number on its own says how much table it covers. A long thin placemat and a short wide one can have the same length and still cover very different amounts of table. What Tomoya is after is a different kind of size: how much **flat space** the placemat takes up. That kind of size is called **area**. Length already has a way to be measured. To measure a length you pick a unit, such as a paper clip, and lay copies of it end to end along the thing, counting as you go (see iterating-one-unit). Area is measured the same way, one step up. You pick a small flat shape as the unit, lay copies of it over the whole surface so that they cover it, and count the copies. Tomoya has a pad of square sticky notes. He covers the placemat with them, and the count, $12$, is its area: $12$ sticky notes. This one idea answers questions that come up all the time. How much paper covers a desk? Which of two blankets gives more room to sit? If a shape is cut up and its pieces are moved, is there more of it, less, or the same? The lesson takes these in three parts, one for each day: covering and counting, choosing a unit and comparing, and cutting and rearranging. ## Definition The **area** of a flat shape is the amount of flat space inside it. To **measure** an area, choose a **unit**: one small flat shape, such as a sticky note, a paper triangle or a paper card. Then cover the shape with copies of the unit so that ### [Area of a regular polygon](https://math.wiki/area-of-a-regular-polygon/) **Summary:** Splitting a regular polygon into equal triangles from the center and adding their areas **Question:** A regular hexagon has no length and no width to multiply. How do you measure the space inside it? ## When you reach for it A rectangle's area is length times width, and a triangle's is half the base times the height. A regular hexagon has neither a length nor a width, and neither does a regular pentagon or octagon. Yet these shapes are everywhere: floor tiles, the head of a bolt, a road sign, the cells of a honeycomb, the floor of a gazebo. Reach for this move whenever the shape is a **regular polygon**, one whose sides are all equal and whose angles are all equal. The move is to cut the polygon from its center into equal triangles, find the area of one of them, and multiply by how many there are. The height of every one of those triangles is the polygon's apothem, which is the reason the apothem is worth naming at all. Done once with letters, the move gives a single formula for every regular polygon: $$A = \frac{1}{2}aP$$ half the apothem times the perimeter. ## The procedure ### Cutting from the center Every regular polygon has a center: a point equally far from all of its vertices and equally far from all of its sides (see regular polygons and their circles). Draw a segment from the center to every vertex. A polygon with $n$ sides is cut into $n$ triangles, one standing on each side, and they fill the polygon without overlapping. **The triangles are all congruent.** Each has two sides running from the center to a vertex, and these all have the same length $r$, the radius, because the center is equally far from every vertex. Its third side is a side of the polygon, of length $s$, and those are all equal because the polygon is regular. Three sides of one triangle match three sides of another, so the triangles are congruent by side-side-side. Congruent triangles have equal areas, so finding one is enough. ### [Area of any polygon by cutting it into triangles and rectangles](https://math.wiki/area-of-polygons-by-decomposing/) **Summary:** Find the area of an irregular polygon by cutting it into rectangles and triangles, or by subtracting pieces from a surrounding rectangle. **Question:** A camp plot or the side of a bookend has straight edges but no formula of its own. How do you measure the surface inside it? ## When you reach for it Rectangles, triangles, parallelograms and trapezoids each have an area formula (area-of-rectangles, area-of-triangles, area-of-parallelograms-and-trapezoids). Most real shapes with straight edges are none of these: a camp plot with five corners, the side of a wooden bookend with a slanted brace, a marker sign shaped like an arrow. Such a shape is a **polygon**, a flat shape closed in by straight sides (polygons), and any polygon can be broken into pieces that do have formulas. There are two ways in. **Cut and add**: slice the shape into rectangles and triangles and add their areas. **Surround and subtract**: draw the smallest rectangle that holds the shape, then take away the pieces of that rectangle lying outside it. This lesson takes two days: Day 1 (Examples 1 to 3) cuts and adds, and Day 2 (Examples 4 to 6) surrounds and subtracts and asks which way needs the fewest measurements. ## The procedure ### Cut and add (Day 1) Area counts the unit squares a shape covers (area-as-covering). If the shape is cut into pieces that do not overlap and leave no gaps, every unit square of it lies in exactly one piece, so $$\begin{aligned} \text{area of the shape} &= \text{area of piece 1} \\ &\quad + \text{area of piece 2} + \cdots \end{aligned}$$ Pieces may share an edge, because a line covers no surface. Good cuts run along a line that continues one of the shape's own edges, or join two corners. A rectangle piece needs its length and width. A triangle piece needs a base and the height at right angles to it, and nothing else: where its top corner sits along the base does not change $\frac{1}{2}bh$. When the shape is all right angles, cutting into rectangles alone is enough, as in decompose-a-figure-into-rectangles. ### [Area of composite figures and shaded regions](https://math.wiki/area-of-composite-figures/) **Summary:** Split into pieces you know, add or subtract. **Question:** A floor, a flag or a lawn is rarely one plain rectangle or one circle. How do you find the area of a shape that has no formula of its own? ## When you reach for it There is no "area of an L" to look up for an L-shaped room, and none for a flag with a notch cut out or a pool with rounded ends. What such shapes do have is edges that belong to shapes you already know: rectangles, triangles, parallelograms and trapezoids, and the half and quarter circles of area-of-a-circle. A **composite figure** is one that can be cut into pieces that each have a formula, or seen as a bigger known shape with a piece missing. A **shaded region** in a drawing is usually the second kind: what is left of one shape when another is taken out, such as the lawn around a pond. The lesson takes two days. Part 1 (Examples 1 to 4) uses straight-sided pieces. Part 2 (Examples 5 to 7) brings in half and quarter circles and shaded regions. ## The procedure ### Cutting up and adding (Part 1) Area counts the unit squares a shape covers (area-as-covering). Cut a figure into pieces that do not overlap and leave no gaps, and every bit of it lies in exactly one piece, so the areas add: $$\begin{aligned} \text{area of the figure} &= \text{area of piece 1} \\ &\quad + \text{area of piece 2} + \cdots \end{aligned}$$ Pieces may share an edge, since a line covers no surface, but they may not overlap, or the shared part is counted twice. Where to cut is your choice: the figure covers the same surface however it is sliced, so every cut gives the same total. Choose the simplest pieces, and cut a second way when you want a check. Splitting into rectangles alone is decompose-a-figure-into-rectangles; here the pieces may also be triangles, trapezoids and parts of circles. ### [Area of parallelograms and trapezoids](https://math.wiki/area-of-parallelograms-and-trapezoids/) **Summary:** Cut and slide to make a rectangle. **Question:** A shape leans over, so it is no longer a rectangle. How much surface does it cover, and which lengths do you need to find out? ## Why this exists A rectangle's area is easy: length times width, because rows of unit squares fill it exactly. But plenty of real shapes lean. The lines of a slanted parking space make a parallelogram. The side of a ski ramp is wider at the bottom than at the top. A pane of stained glass is often cut on a slant so the pieces fit together. Counting squares no longer works cleanly on a leaning shape, because its slanted edges cut through the squares. What you want is a way to turn a leaning shape back into a rectangle without changing how much surface it covers. That turns out to be easy, and it shows exactly which measurements matter. The surprise is that the slanted sides, the lengths that are easiest to measure with a ruler, are not the ones you need. This lesson runs over two days. Day 1 is the parallelogram and what "height" means. Day 2 is the trapezoid, whose formula comes from the parallelogram's. ## Definition A **parallelogram** is a four-sided shape whose opposite sides are parallel: two pairs of parallel sides. Its opposite sides are also equal in length (properties of parallelograms). A rectangle is a parallelogram whose corners are all right angles. A **trapezoid** is a four-sided shape with at least one pair of parallel sides (trapezoids). The two parallel sides are its **bases**, written $b_1$ and $b_2$. The other two sides are its **legs**. Some books say "exactly one pair" instead; with the "at least one" wording used here, every parallelogram is also a trapezoid, and the formulas below agree about it. ### [Area of rectangles and squares](https://math.wiki/area-of-rectangles/) **Summary:** Rows of unit squares: length times width, and why. **Question:** A game board is ruled into $35$ inch squares. Could you know that from its two side lengths alone, without counting a single square? ## Why this exists Ville is making a game board from a sheet of card. He rules it into squares $1$ inch on a side, and it comes out $7$ squares across and $5$ squares down. To find its area he could count the squares one at a time, as in tiling-with-unit-squares. With $35$ squares that is slow, and the more squares there are, the easier it is to skip one or count one twice. But the squares on a rectangle are not scattered about. They sit in rows, every row is the same length, and the rows stack one on another with no gaps. That is an array, from arrays-and-multiplication, and an array is counted by multiplying. Better still, the two numbers of the array are the side lengths themselves: $7$ inch squares fit along a side $7$ inches long, and $5$ rows fit up a side $5$ inches long. So the area of a rectangle can be found with a ruler and one multiplication, with no squares drawn at all. That is how the area of a classroom floor, a garden bed or a sheet of paper is found in practice. This lesson shows why it works, uses it for squares and for real surfaces, and runs it backward once. ## Definition A **rectangle** is a flat shape with four straight sides and four square corners. Its opposite sides are the same length. The lengths of two sides that meet at a corner are called its **length** and its **width**. Which of the two is called the length does not matter. A **square** is a rectangle whose four sides are all the same length. So every square is a rectangle, and a rectangle is a square only when its length and its width are equal. ### [Area of triangles](https://math.wiki/area-of-triangles/) **Summary:** Half a parallelogram. **Question:** A triangle's slanted sides never line up with rows of squares. How can its area be found exactly? ## Why this exists Area began as counting unit squares (area-as-covering), and for a rectangle the count is quick: rows times columns (area-of-rectangles). A parallelogram followed by cutting off a slanted end and sliding it across, which turns it into a rectangle with the same base and height (area-of-parallelograms-and-trapezoids). A triangle does not give way so easily. Its slanted sides cut the squares they cross into odd pieces, and no single cut and slide makes it a rectangle. Yet triangles are everywhere a flat surface has to be measured: the sail on a boat, the side of a wooden doorstop, the pieces of a wooden puzzle. Every flat shape with straight sides can also be cut into triangles, so a rule for the triangle is a rule for all of them. The way in is a surprise: instead of cutting a triangle up, put two of them together. ## Definition The **base** of a triangle is whichever side is chosen to measure from. Any of the three sides may be the base; it does not have to be the side drawn at the bottom. The **height** that goes with that base is the distance from the opposite corner to the line the base lies on, measured at right angles to that line (perpendicular-lines-geometry). It is a straight distance to the base's line, never the length of a slanted side. The area of the triangle is then $$A = \frac{1}{2}bh$$ where $b$ is the length of the base and $h$ is the height that goes with it, both in the same unit. The area comes out in square units of that unit. ### [Attributes you can measure](https://math.wiki/attributes-you-can-measure/) **Summary:** Length, area, volume, mass, angle and time are separate things about an object that can each get a number **Question:** What things about one object can be measured, and how do you tell them apart? ## Why this exists One object has several separate things about it that can be measured: how long or tall it is, how heavy it is, and how much it holds. Each is its own question with its own words, and one does not decide another: a long scarf can be light, and a small stone can be heavy. Before a child can measure anything, the child has to know *which* thing is being measured, so these three days name each one and practice telling them apart. Color and shape are true things about an object too, but they are not "how much" things, so they are not measured this way. Keep to the three above; comparing two objects side by side comes next, in direct-comparison-of-two-objects. ## Day 1: How long and how tall **You need:** a pencil, a ribbon or piece of string, a cereal box. Hold up the pencil and say, "Tell me about this pencil." Take every answer: yellow, pointy, for writing. Then say, "Here is one thing we can measure about it." Have the child run a finger from the flat end to the tip. "That is *how long* it is. Its length goes all the way from one end to the other." Now stand the pencil on its flat end. "When it stands up, we say *how tall* it is." Tall is the same idea as long, only standing up, from the bottom to the top. Ask, "Is *yellow* about how long it is?" No: yellow is its color, and painting it red would not make it longer or shorter. **Practice.** 1. Show how long the ribbon is with your hands. *Answer:* one hand at each end; the length is the whole way from one end to the other. 2. Stand the cereal box up. Which word fits, long or tall? *Answer:* tall, because it is standing; its height goes from the bottom to the top. 3. Lay the box down on its side. Which word fits now? *Answer:* long; it is the same box, and the edge that went up now runs along the table. 4. "The pencil is pointy." Does that tell how long it is? *Answer:* no, pointy tells its shape; long or short tells its length. 5. Show the length of your own arm. *Answer:* from the shoulder to the tips of the fingers. ### [Balancing to compare mass](https://math.wiki/balance-scales-and-equal-mass/) **Summary:** Deciding which of two objects has more matter by seeing which side of a balance goes down **Question:** How can a child tell which of two things is heavier, and what does it mean when a balance stays level? ## When you reach for it Two things can be compared by how heavy they are, not only by how long or tall they are. A child can feel the difference in two hands, and a two-pan balance shows it plainly: the side that goes down holds the heavier thing, and a level beam means the two are the same. The star's word is **mass**; with a young child, just say "how heavy", "heavier" and "lighter". This builds on comparing two objects side by side. Each day takes about ten to fifteen minutes. Keep it to comparing: no numbers of grams or pounds yet. ## Day 1: Heavier and lighter in two hands **You need:** a shoe and a sock, a book and a sheet of paper, a full water bottle and an empty one, and two crayons that look alike. Put one thing in each of the child's open hands. Ask, "Which one pulls your hand down more? That one is **heavier**. The other one is **lighter**." Let the child say the full sentence: "The book is heavier than the paper." Then swap the hands and ask again, so the child sees the answer stays with the thing, not the hand. End with two things that feel nearly the same, like the two crayons. Hands cannot tell, and that is fine: "My hands can't tell. Tomorrow we'll use a balance to check." **Practice.** Hand the child each pair, one thing in each hand. 1. A shoe and a sock. Which is heavier? **Answer:** the shoe. 2. A book and a sheet of paper. Which is lighter? **Answer:** the paper. 3. A full water bottle and an empty one. They look the same size. Which is heavier? **Answer:** the full one. Same size does not mean same heaviness. 4. Two crayons that look alike. Which is heavier? **Answer:** hands can't tell; they feel the same. ### [Bearings and navigation](https://math.wiki/navigation-with-bearings/) **Summary:** Tracking a position after one or more legs of travel, each given as a bearing and a distance. **Question:** A boat runs one course, turns, and runs another. Where is it now, and which way is home? ## When you reach for it A navigator does not think in coordinates. Sverrir, skippering a search-and-rescue boat, is told "run $8$ nautical miles on a bearing of $040^\circ$, then $5$ on $145^\circ$." An orienteer reads the same kind of instruction off a map: a direction and a distance, then another, then another. Reach for this skill whenever motion comes as **legs**, each a direction and a distance, and you want to know where you end up, how far you are from where you began, and which way leads back. The idea is to break every leg into how far it goes east and how far it goes north. Those amounts simply add, however many legs there are, and at the end a right triangle turns the totals back into a distance and a direction. ## The procedure ### Two ways to write a direction A **three-figure bearing** is the angle measured **clockwise from north**, from $000^\circ$ to just under $360^\circ$, always written with three digits. Due east is $090^\circ$, due south $180^\circ$, due west $270^\circ$. A **compass bearing** starts from north or south, whichever is nearer, and turns an acute angle toward east or west. $\text{N}\,35^\circ\,\text{E}$ reads "$35^\circ$ east of north": face north, turn $35^\circ$ toward east. $\text{S}\,40^\circ\,\text{E}$ means face south and turn $40^\circ$ toward east. Reading each compass bearing off the figure gives its three-figure form. With $\varphi$ the acute angle: ### [Benchmarks, estimating and choosing a unit](https://math.wiki/benchmark-referents-for-units/) **Summary:** Keeping a familiar object in mind, like a doorway for two meters, to feel how big a unit is **Question:** Before you measure something, how can you already know about how long it is, and how do you know which unit and which tool to measure it with? ## Why this exists A child who can measure a length can lay a ruler along a pencil and read a number. That child can still have no feel for what the number means. Ask how long the classroom is and the answer might be $10$ inches or $1{,}000$ feet, because "inch" and "foot" are only words until each one is tied to something the child has seen and held. A **benchmark** is that tie. It is a familiar thing that is about one unit long: the edge of a floor tile for a foot, a crayon's thickness for a centimeter. With a benchmark in mind a child can do three things that a ruler alone does not give. - **Estimate.** Picture the benchmark laid along the object again and again, and count. That gives a length before any tool comes out. - **Check a measurement.** If the estimate says "about $6$ feet" and the ruler work says $60$, a slip has happened somewhere, and the child knows to look again. - **Choose a unit and a tool.** A unit fits a job when a sensible number of copies of it fit along the object. A benchmark shows that at a glance: nobody measures the playground fence in crayon widths. This lesson takes three days. Day 1 builds a benchmark for each unit a child meets now (inch, foot, yard, centimeter, meter). Day 2 estimates a length, then measures it, then compares the two. Day 3 chooses the unit and the tool that fit a job. All lengths here are whole units. ## Definition ### Day 1: a benchmark ### [Calendars and dates](https://math.wiki/calendars-and-dates/) **Summary:** How days are grouped into weeks, months and years and named by a date **Question:** Alper leaves for summer vacation on a Saturday and comes home $30$ days later. What day of the week does he come home, and how do you name that day so nobody mixes it up with another? ## Why this exists Alper is planning his family's summer vacation, and Mirri and Coby are reading the school year calendar that came home in their backpacks. The vacation is "thirty days", but which thirty? School starts again "in August", but on which day, and is it a Monday? Counting days one by one works for a few days and breaks down for a few hundred. A **calendar** fixes this. It groups the days into weeks, months and years, gives each group a name, and so gives every single day its own name: its **date**. With dates, two people can agree on a day months ahead, count how far apart two days are, and work out the day of the week without a long count. ## Definition A **calendar** groups days in three ways at once. - A **week** is $7$ days, named Sunday, Monday, Tuesday, Wednesday, Thursday, Friday and Saturday. The names repeat in the same order forever. - A **month** is $28$, $29$, $30$ or $31$ days. There are $12$ months in a year. - A **year** is $365$ days, or $366$ days in a **leap year**. | Month | Days | Month | Days | |:-:|:-:|:-:|:-:| | January | $31$ | July | $31$ | | February | $28$, or $29$ in a leap year | August | $31$ | | March | $31$ | September | $30$ | | April | $30$ | October | $31$ | | May | $31$ | November | $30$ | | June | $30$ | December | $31$ | A **date** names one day by its month, its day of the month and its year: July $4$, $2027$. In the United States it is written in numbers as month/day/year, so July $4$, $2027$ is $7/4/2027$. Many other countries write day/month/year, where $7/4/2027$ means the $7$th of April. Writing the month as a word removes any doubt. ### [Comparing two objects](https://math.wiki/direct-comparison-of-two-objects/) **Summary:** Deciding which of two things is taller, longer, or holds more by lining them up or pouring from one into the other **Question:** How can a child tell which of two things is longer, taller, or holds more, without measuring with numbers? ## When you reach for it Two things can be compared without any numbers: put them side by side so they start at the same place, and look at which one goes further. That gives *longer* and *shorter*, *taller* and *shorter*, or *the same length*; for containers, pouring one into the other shows which *holds more*. It comes right after a child can name what can be measured about an object, and it is about size, not about how many things are in a group. Each day below takes about ten to fifteen minutes and uses things already in the room. ## Day 1: Taller and shorter **You need:** about $10$ snap cubes or blocks, a flat table, a thick book. Build two towers, one of $5$ cubes and one of $3$, and stand them side by side on the table. Ask, "Which tower is taller? Which one is shorter?" Let the child lower a flat hand slowly from above: the hand lands on the taller one first. Now put the short tower on top of the book and ask again. Its top may now be higher, but the tower did not grow. Take it off the book and stand both towers on the table again. Say, "To compare how tall, both things have to stand on the same floor." Then stand back to back with the child and ask a third person (or a mirror) to say who is taller. **Practice.** 1. A tower of $6$ cubes and a tower of $4$ cubes stand on the table. Which is taller? *Answer:* the tower of $6$, so the tower of $4$ is shorter. 2. A child and a grown-up stand back to back on the floor, and the child's head reaches the grown-up's shoulder. Who is shorter? *Answer:* the child. 3. A milk carton and a toy truck stand on the table, and the carton's top is higher. Which is taller? *Answer:* the milk carton. 4. One doll stands on the floor and another on a chair, and the second doll's head is higher. Can you tell which doll is taller? *Answer:* not yet. Stand both dolls on the floor first, then look. ### [Comparing with a middle object](https://math.wiki/indirect-comparison-by-transitivity/) **Summary:** Using a third object as a go between to compare two things that cannot be put side by side **Question:** A toy boat is at home and the cubby shelf is at school, so they can never be set side by side. How can we still find out whether the boat will fit? ## When you reach for it Two pencils are easy to compare. Stand them side by side on the table with their bottom ends level, and whichever one sticks out past the other at the top is longer. That is comparing two objects: line up the ends and look at the other end. Some things cannot be lined up. A cubby shelf is built into the classroom wall. A poster is pinned high on the wall, out of reach of the floor. A toy boat is at home while the shelf it is meant for is at school. Nobody can hold the shelf against the boat. Looking across the room and guessing is not good enough either, because things that are far apart are hard to judge by eye. The way out is to bring a **third object** that *can* travel: a piece of yarn, a strip of paper, a stick. Compare the first thing with the go-between, carry the go-between over, and compare it with the second thing. Two direct comparisons, each with the ends properly lined up, answer the question that one comparison could not. The same idea answers a second kind of question. If one ribbon is longer than a second, and the second is longer than a third, then the first is longer than the third, even though those two were never held together. That lets a child put three things in order from only two comparisons. This lesson runs over three days. Day 1 is the go-between itself. Day 2 is three objects and their order. Day 3 is what a go-between needs in order to tell the truth, and how it leads to measuring with units. ### [Conversion factors and dimensional analysis](https://math.wiki/conversion-factors-equal-one/) **Summary:** Twelve inches over one foot equals one, so multiplying by it changes the unit and not the amount **Question:** A bag of rice is labeled in pounds and the recipe asks for kilograms. How do you change the unit without changing the amount, and know which way to go? ## Why this exists Every measurement is a number with a unit, and the same amount has many names. A relay leg of $100$ meters is about $109$ yards. A $50$ pound sack of rice is $800$ ounces, or about $23$ kilograms. The amount stays put; only its name changes. Single steps you already know. Customary conversions say a foot is $12$ inches, and metric conversions say a meter is $100$ centimeters. For one step, common sense picks between multiplying and dividing: inches are small, so there are more of them. But a chain of steps, such as yards to inches to centimeters to meters, or a unit made of two units, such as square feet or dollars per pound, leaves too many choices to guess. This lesson writes each fact as a fraction equal to $1$, and then the units themselves show which way the fraction goes. ## Definition A **conversion factor** is a fraction whose top and bottom are the same amount written in two different units. Because the top and the bottom are equal, the fraction is equal to $1$. Each fact gives two conversion factors, one the other turned upside down. From $1\text{ ft} = 12\text{ in}$: $$\frac{12\text{ in}}{1\text{ ft}} = 1 \qquad\qquad \frac{1\text{ ft}}{12\text{ in}} = 1$$ **Dimensional analysis** means carrying the units through a calculation along with the numbers: multiplying by conversion factors, and canceling a unit that sits both on top and on the bottom, just as a number on the top and the bottom of a fraction cancels. ### [Converting between Celsius and Fahrenheit](https://math.wiki/converting-celsius-and-fahrenheit/) **Summary:** Trading one temperature scale for the other with a rule that both stretches and shifts **Question:** A forecast abroad says 25 degrees and a recipe says 425 degrees. How do you turn a reading on one temperature scale into the same temperature on the other? ## When you reach for it Most countries give temperatures in degrees Celsius; a few, the United States among them, use degrees Fahrenheit. A weather report in a city you are traveling to, an oven temperature in a recipe from another country, a reading on a medical thermometer, the setting in a freezer manual: each can arrive in one scale while you think in the other. The two scales measure the same thing but disagree in two ways, both laid out in the lesson on temperature scales. Their zeros are in different places: water freezes at $0^\circ\text{C}$, which is $32^\circ\text{F}$. And their degrees are different sizes: from freezing to boiling there are $100$ Celsius degrees but $212 - 32 = 180$ Fahrenheit degrees. So converting is not a single multiplication, the way changing meters to centimeters is. It needs a **stretch**, for the size of the degree, and a **shift**, for the place of the zero. The rule that does both is a formula, and using it is substituting the reading you have. ## The procedure ### Building the rule from two anchors The freezing and boiling points of water are the two anchors. Between them lie $100$ Celsius degrees and $180$ Fahrenheit degrees, the same stretch of temperature counted in two sizes of step. So one Celsius degree is worth $$\frac{180}{100} = \frac{9}{5} \text{ Fahrenheit degrees,}$$ which is $1.8$ of them. A Celsius degree is the bigger step, which is why the Fahrenheit ticks in the figure are packed more tightly. ### [Converting metric units by powers of ten](https://math.wiki/unit-conversion-metric/) **Summary:** Powers of ten: shift the decimal point. **Question:** A zipline cable is measured in meters, but the rope for its seat is measured in centimeters. How do you change one metric unit into another, and why is it always a matter of tens? ## When you reach for it Burak, Olesya and Urvi are building a zipline in the backyard. They measure the long steel cable with a tape measure, in meters. They measure the rope that holds the seat with a ruler, in centimeters. Indoors, Olesya is making soup. The pot is marked in liters, but her measuring cup is marked in milliliters. Two measures in different units cannot be compared or put together until they are in the same unit. Is a rope of $480$ centimeters longer than $5$ meters? Will $5{,}800$ milliliters of soup fit in a $6$ liter pot? Each question asks you to change one metric unit into another first. In the metric system that change is always a step of $10$, $100$ or $1{,}000$. So every metric conversion is a multiplication or a division by one of those numbers, and you already know that move: the digits slide over by places, as times-ten-shifts-place-value shows. This lesson uses that fact to change units in both directions. ## The procedure This lesson takes three days. Day 1 changes a bigger unit into a smaller one. Day 2 goes the other way and works with measures written in two units, such as $2$ meters $35$ centimeters. Day 3 compares measures given in different units. ### Day 1: steps of ten, and multiplying to a smaller unit The metric system has one basic unit for each kind of measure. Length uses the **meter**. Mass, which tells how heavy something is, uses the **gram**. Capacity, which tells how much a container holds, uses the **liter**. A word in front of the unit, called a **prefix**, tells how big the unit is compared to the basic one. A prefix means the same thing in front of any unit, as metric-prefixes explains. Three prefixes do almost all the work here: ### [Converting square and cubic units](https://math.wiki/converting-square-units/) **Summary:** Changing area units uses the length factor twice, so a hundred per meter becomes ten thousand **Question:** A meter is a hundred centimeters. Why is a square meter not a hundred square centimeters? ## When you reach for it You reach for this move whenever an area or a volume is measured in one unit and needed in another. A room is measured in feet but carpet is sold by the square yard. A field is measured in meters but farmland is sold by the hectare. A fish tank is measured in centimeters but its pump is rated in liters. For a length, the move is the one from conversion factors: since $1\text{ ft} = 12\text{ in}$, a length in feet becomes inches by multiplying by $12$. Areas and volumes need more care, because an area unit is not a length. A **square foot** is the area of a square $1$ ft on each side, and a **cubic yard** is the volume of a cube $1$ yd on each edge. The question is how many of the small units fit inside one big one, and the figures answer it by counting. Each side of a square foot is $12$ inches, so the square holds $12$ rows of $12$ square inches. That is $12 \times 12 = 144$ square inches, not $12$, for the same reason a rectangle's area is length times width (see area of rectangles): the factor $12$ is used once along each side. A cube $1$ yd on each edge is $3$ ft on each edge. Its bottom layer holds $3 \times 3 = 9$ cubes of $1$ cubic foot, and there are $3$ layers, so it holds $27$ cubic feet (see volume with unit cubes). The length factor $3$ is used three times. This is the same rule as for scale copies in scale factor, area and volume: multiply every length by $k$ and the area is multiplied by $k^2$, the volume by $k^3$. Changing the unit is a scaling of that kind. ### [Counting mixed coins](https://math.wiki/count-mixed-coins/) **Summary:** Adding up a handful of different coins by starting with the biggest values **Question:** A handful holds quarters, dimes, nickels and pennies all mixed up. How do you find what it is worth without losing your place? ## When you reach for it A child who has met money knows what each coin is worth and can count a stack of one kind: quarters by twenty-fives, dimes by tens. They can also count two kinds, bigger first. Real money does not come that tidy. A pocket, a purse or a piggy bank gives up a jumble of three or four kinds at once, and the question "how much is this?" has to be answered from the jumble. The move this lesson teaches is to **sort first, then count once, biggest worth first**, switching the skip count each time the kind of coin changes. It is skip-counting put to work, with one new demand: the count does not always start from zero. After two quarters the dimes are counted on from $50$; after one quarter they are counted on from $25$, and "$35$, $45$, $55$" is a skip count by tens that never lands on a ten. The breaking apart and counting on of mental-strategies carry the child through those spots. The lesson runs over four days. - **Day 1: sort and count.** Three or four kinds of coin, sorted into stacks and counted biggest first, switching from twenty-fives to tens to fives to ones. - **Day 2: counting on from an awkward spot.** Coins that arrive one at a time and cannot be sorted, and a quarter or a dime added to a count that is not on a ten. - **Day 3: past a dollar.** A count that goes beyond $100$ cents, said as dollars and cents, and bills and coins counted together. - **Day 4: using the count.** Comparing two handfuls, and deciding whether there is enough to buy something. ### [Currency exchange](https://math.wiki/currency-exchange/) **Summary:** Trading an amount of one country's money for another using a rate between them **Question:** One dollar trades for a fixed number of euros. How do you turn any amount of one country's money into another's, which way around do you multiply, and when does the trade stop being that simple? ## When you reach for it Most countries have their own money, their own **currency**: dollars in the United States, euros across much of Europe, pounds in Britain, yen in Japan, pesos in Mexico and in the Philippines. A traveler landing in Spain with dollars needs euros. Someone in Ohio buying a sketchbook from a shop in Tokyo sees its price in yen but pays in dollars. Someone in the United States who sends money to family in the Philippines hands over dollars, and the family collects pesos. Whenever an amount of one currency has to be stated in another, you reach for an **exchange rate**. An exchange rate says how much of one currency trades for one unit of another, for example $$1 \text{ dollar} = 0.92 \text{ euros}$$ Real rates change from day to day, and from one bank, booth or website to the next. **Every rate in this lesson is an example**, made up but of a realistic size; a real trade uses the rate quoted on the day. Amounts of money are decimals counted in hundredths, as in money written as decimals. ## The procedure **The rate is a constant of proportionality.** At $1$ dollar to $0.92$ euros, every dollar trades for the same $0.92$ euros, whether it is the first dollar handed over or the three-hundredth, and no dollars trade for no euros. So the number of euros is proportional to the number of dollars, and the rate is the constant, $0.92$ euros per dollar (the constant of proportionality): ### [Customary capacity units](https://math.wiki/customary-capacity-units/) **Summary:** Cups, pints, quarts and gallons, the United States units for liquid amounts **Question:** A kiddie pool, a milk jug and a mug of cocoa all hold liquid. Which unit fits each one, and how can you tell whether 3 quarts is more or less than a gallon? ## Why this exists Yash, Yarema and Nayra are filling a kiddie pool in the backyard with a garden hose. Yarema wonders how much water the pool holds. A ruler is no help: water has no length of its own, it takes the shape of whatever it is in. Water in a tall jug and the same water in a flat pool look nothing alike. So liquid is measured a different way. Pick one container as the unit, and count how many times it would take to fill the pool. The answer to "how much does it hold?" is that count, together with the name of the container. How much a container holds is its **capacity**, also called liquid volume. In the United States the everyday capacity units are the cup, the pint, the quart and the gallon, with the fluid ounce for small amounts. They belong to the customary system, met in the-customary-system. This lesson looks closely at these five units: how big each one is, how they fit inside each other, which one suits which job, and how to compare amounts given in different units. This lesson takes two days. Day 1 (Examples 1 to 3) is the units, their sizes, and how each one splits into halves. Day 2 (Examples 4 to 7) is comparing capacities given in different units, and how capacity is related to volume. ## Definition ### Day 1: five units for liquid The **capacity** of a container is how much it holds. A capacity is a number together with a unit: "the jug holds $1$ gallon". | unit | short form | equals | |---|---|---| | fluid ounce | $\text{fl oz}$ | the smallest unit here | | cup | $\text{c}$ | $8$ fluid ounces | | pint | $\text{pt}$ | $2$ cups | | quart | $\text{qt}$ | $2$ pints, which is $4$ cups | | gallon | $\text{gal}$ | $4$ quarts, which is $8$ pints or $16$ cups | ### [Customary conversions](https://math.wiki/unit-conversion-customary/) **Summary:** Inches, feet, yards, miles; ounces, pounds; cups to gallons. **Question:** The bike race is 3 miles long, but the course map is marked in feet. How do you say the same length in another unit, and how do you know whether the number should get bigger or smaller? ## When you reach for it Emre, Mykola and Aminata are helping at a bike race. The race is $3$ miles long, but the course map is marked in feet. Mykola sets out cones by the yard. At the food tent the punch is mixed by the quart and poured by the cup, and at the butcher counter across the street meat is sold by the pound and weighed in ounces. Each time, one amount is named in two different units. The amount itself does not change: the race is just as long whether you count it in miles or in feet. Only the unit you count with changes. You reach for a conversion whenever two measurements of the same kind come in different units and you need them in one unit: to compare them, to add them, or to give an answer in the unit someone asked for. This lesson changes one unit into one other unit, inside one kind of measure: length to length, weight to weight, capacity to capacity. What the units are, and why the steps between them are not all the same size, is in the-customary-system. These are the facts used here. Every one is exact: a foot is defined to be $12$ inches, not "about $12$". | Length | Weight | Capacity | |:-:|:-:|:-:| | $1$ foot $= 12$ inches | $1$ pound $= 16$ ounces | $1$ cup $= 8$ fluid ounces | | $1$ yard $= 3$ feet | $1$ ton $= 2{,}000$ pounds | $1$ pint $= 2$ cups | | $1$ mile $= 5{,}280$ feet | | $1$ quart $= 2$ pints | | | | $1$ gallon $= 4$ quarts | ## The procedure ### Part 1: from a larger unit to a smaller one, by multiplying ### [Customary length units](https://math.wiki/customary-length-units/) **Summary:** Inches, feet, yards and miles, the length units used in the United States **Question:** A dog brush, a picnic table, a snow fort wall and the walk to the park are all lengths. Which named unit fits each one, how big is each unit, and how do the inch, the foot and the yard fit together? ## Why this exists Amarachi and Hohepa help out at a pet shelter. Amarachi measures a shelf for new dog beds with her hand and gets $8$ hand spans. Hohepa measures the same shelf with his hand and gets $10$. Both counted carefully. The shelf did not change. Their hands are different sizes, so their units are different sizes, and the two numbers cannot be put side by side. A unit only lets people share a length when everyone uses the same one. A **standard unit** is a unit whose size is fixed and agreed, so that an inch in one town is the same as an inch in any other town. In the United States the everyday standard units of length are the **inch**, the **foot**, the **yard** and the **mile**. They are called **customary units**. Most of the rest of the world uses centimeters, meters and kilometers instead. This lesson gives each customary unit a size you can picture and a tool to measure with: an inch ruler, a foot ruler and a yardstick. It shows, by laying the tools end to end, that $12$ inches make a foot and $3$ feet make a yard. It ends with the mile, which is too long for any ruler, and with choosing the unit that fits the job. Reading a ruler is already known from measure-length. Every length here is a whole number of units. The lesson runs over four days, one unit a day. Day 1 is the inch (Examples 1 and 2). Day 2 is the foot (Examples 3 and 4). Day 3 is the yard (Examples 5 and 6). Day 4 is the mile and choosing a unit (Examples 7 and 8). ### [Customary weight units](https://math.wiki/customary-weight-units/) **Summary:** Ounces, pounds and tons, the United States units for weight **Question:** How heavy is a pound, and how can you tell whether a tent of 3 pounds or a tent of 44 ounces is the heavier one? ## Why this exists Varun is packing for a two-night hiking trip. He will carry everything on his back for hours, so what matters is not how much room each thing takes but how heavy it is. A big, puffy sleeping bag can be light. A small bottle of water can be heavy. His eyes cannot tell him which is which; a scale can. To write down how heavy something is, he needs a unit of weight. In the United States the units of weight are the **ounce**, the **pound** and the **ton**. They belong to the family met in the-customary-system. This lesson looks closely at those three units: how heavy each one feels, how to read a scale, how to pick a sensible unit, and how to compare and add weights that are written in different units. This lesson takes two days. Day 1 (Examples 1 to 3) is the three units, a picture for each, and reading a scale. Day 2 (Examples 4 to 7) is comparing, ordering and adding weights. ## Definition ### Day 1: the three units of weight The **weight** of an object is how heavy it is: how hard it pulls down on your hand or on a scale. Weight is not size. Two things can take up the same room and have very different weights. The customary units of weight, from lightest to heaviest: | unit | short form | equals | |---|---|---| | ounce | $\text{oz}$ | the smallest unit here | | pound | $\text{lb}$ | $16$ ounces | | ton | (none) | $2{,}000$ pounds | These two facts are exact: $$1 \text{ lb} = 16 \text{ oz} \qquad 1 \text{ ton} = 2{,}000 \text{ lb}$$ ### [Degrees](https://math.wiki/degrees/) **Summary:** A full turn is 360 of them; a right angle is 90. **Question:** One angle is here and another is on a page across town. How can you say exactly how far each one opens, with a number that anyone could check? ## Why this exists You can often tell which of two angles is bigger by laying one on top of the other (comparing angles by fitting). That only works when the two can be brought together, and it only ever answers "bigger" or "smaller". To say how much bigger, or to write an angle down so that someone far away can draw the same one, you need a number, and a number needs a unit. Length already solved this. A centimeter is a small fixed length, and a length measures however many centimeters fit along it end to end. Angles need the same thing: a small fixed amount of turn that can be counted. The unit in everyday use is the **degree**, and it is built from the one amount of turn that nobody has to measure: the full turn, where an arm swings all the way round and lands back where it started. ## Definition The size of an angle is the amount of turn from one of its arms to the other (an angle is an amount of turn). Cut a full turn into $360$ equal turns. Each one is a **one-degree angle**, and its measure is **one degree**, written $1^\circ$. An angle that turns through $n$ one-degree angles has a measure of **$n$ degrees**, written $n^\circ$. So a full turn measures $360^\circ$, and one degree is $\frac{1}{360}$ of a full turn. The small raised circle is the degree sign: $90^\circ$ is read "ninety degrees". The measure of $\angle ABC$ is written $m\angle ABC$, so $m\angle ABC = 50^\circ$ says that the angle turns through fifty one-degree angles. Naming angles explains why the angle and its measure get different symbols. ### [Density](https://math.wiki/density-as-mass-per-volume/) **Summary:** How much mass is packed into each unit of volume **Question:** A big log floats and a small pebble sinks, yet the log has far more mass. What does the pebble have more of? ## Why this exists Ask which is heavier, iron or wood, and the honest answer is "how much of each?" A pine log has more mass than an iron nail. Yet everyone feels that iron is the heavier material. What they mean is that iron packs more mass into the same amount of space. **Density** turns that feeling into a number. It compares materials fairly by asking each one the same question: how much mass is in one unit of volume? It is a unit-rate, like a speed or the flow-rate of a tap, and the same per-one reasoning that runs every unit rate runs this one. ## Definition The **density** of a material is its mass per unit of volume: $$\text{density} = \frac{\text{mass}}{\text{volume}}$$ In letters, $D = \dfrac{m}{V}$. With mass in grams and volume in cubic centimeters, density is in **grams per cubic centimeter**, written $\text{g/cm}^3$. With kilograms and cubic meters it is in **kilograms per cubic meter**, $\text{kg/m}^3$. Grams and kilograms are metric-mass-units. This lesson talks about mass, not weight; mass-versus-weight explains the difference. A density of $2.5\ \text{g/cm}^3$ says: every cubic centimeter of this material has a mass of $2.5$ grams. The same per-unit idea works on a map. **Population density** is the number of people per unit of **area**, usually people per square kilometer: $$\text{population density} = \frac{\text{people}}{\text{area}}$$ It measures how crowded a place is, the way density measures how tightly mass is packed. ### [Elapsed time](https://math.wiki/elapsed-time/) **Summary:** How long from then to now. **Question:** Story hour starts at a quarter past ten and ends at twenty to twelve. Neither clock reading says how long it lasted, so how do you get from two times of day to a length of time? ## When you reach for it A clock answers one question: what time is it right now? Reading it to the minute is the work of tell-time. But many everyday questions are about a different kind of time. How long was story hour? If the bus leaves at ten to three and the ride takes $45$ minutes, when does it get there? The sleigh ride ended at ten past four after $55$ minutes, so when did it start? Each of these joins two kinds of quantity. A **time of day**, such as $10{:}15$ a.m., names one moment. A **length of time**, such as $45$ minutes, says how much time passes between two moments. The length of time between a start and an end is called the **elapsed time**. There are three questions, depending on which of the three things is missing: | You know | You want | The question | |---|---|---| | start and end | the length | How long did it take? | | start and length | the end | When does it end? | | end and length | the start | When did it start? | All three are solved the same way, on a number line for the day, by jumping to the next hour mark. You reach for this whenever a plan has a start, an end, or a length in it: a library schedule, a school bus schedule, a ride that has to be back before dark. ## The procedure The lesson runs over four days. Part 1 separates a time of day from a length of time and puts both on a number line. Part 2 finds how long. Part 3 finds when something ends. Part 4 finds when something started, and reads a schedule, where all three questions turn up together. ### [Finding a missing side from the area](https://math.wiki/missing-side-from-area/) **Summary:** Working backwards from a known area and one side to the other side **Question:** Ljubica's dog run has to cover 72 square feet, and the side along the fence is 8 feet. How long is the other side, and how much fence goes around it? ## When you reach for it Most of the time an area problem gives you the sides and asks for the area. You reach for this move when the problem runs the other way: you know how much space a rectangle must cover, and you know one side, and you need the other side. Ljubica is planning a dog run. Her dog needs $72$ square feet of ground, and one side has to be $8$ feet long to fit along the yard fence. She cannot buy fencing until she knows the other side. Warrick has a swimming pool cover with its area printed on the label and wants to know how long it is. Both questions are about a missing side. The second half of the lesson brings in perimeter, the distance around a shape (see perimeter). A fence goes around a dog run, so the fence is a perimeter, while the grass inside is an area. Some problems give one and ask for the other. ## The procedure This lesson takes two days. Part 1 finds a missing side from the area. Part 2 finds a missing side from the perimeter, and goes back and forth between area and perimeter. ### Part 1: the other side from the area The area of a rectangle is its length times its width (see area-of-rectangles). The reason is the grid: a rectangle $8$ feet by $9$ feet holds $9$ rows of $8$ square feet, so $9 \times 8 = 72$ squares, and its area is $72$ square feet. $$ \text{area} = \text{length} \times \text{width} $$ Now hide one side. Ljubica knows the area, $72$ square feet, and one side, $8$ feet. Call the missing side $w$. The area fact still holds, so ### [Finding a missing side from the perimeter](https://math.wiki/missing-side-from-perimeter/) **Summary:** Working backwards from a known perimeter and the other sides **Question:** The fence is bought and its length is known, and so are all but one side of the shape it has to go around. How long must the last side be, and when is there no such side at all? ## When you reach for it In perimeter every side was known, and the sides were added to get the distance all the way around. Often the question comes the other way. Jurgis has $36$ meters of fence for a rectangular horse corral and wants it $11$ meters long: how wide can it be? Kotryna has a fixed length of border for a square herb garden: how long is each side? The perimeter is known, and so are some of the sides. One side, or one kind of side, is missing. The idea that settles every such question is the same: the perimeter is the sum of the sides. If the total is known and some of the parts are known, the missing part is the total minus the known parts, exactly as with any sum where one addend is unknown. Subtraction undoes addition. The lesson takes two days. Part 1 works with any shape with straight sides, a polygon, where one side is missing: subtract the known sides from the perimeter. It also writes the question as an equation with a letter for the missing side. Part 2 does rectangles and squares, where equal sides give shortcuts: half the perimeter of a rectangle is one length plus one width, and a square's perimeter divided by $4$ is its side. Part 2 ends with the cases that have no answer: no side in whole units, or no shape at all. ## The procedure ### Part 1: one side missing Going all the way around a polygon walks each side once, so the perimeter is the sum of the side lengths. Suppose a shape has five sides, four of them known, and the perimeter is known too. Then ### [Flow rate](https://math.wiki/flow-rate/) **Summary:** How much liquid passes a point in each unit of time **Question:** Water is running from a tap into a bath. How do you say how fast it is coming, and use that to know when the bath will be full? ## Why this exists Turn on a tap over an empty bath and you can watch the water rise, but watching does not tell you when to turn it off. To plan, you need one number that says how fast the water is coming. "Fast" here is not how quickly the water moves through the pipe. It is how much water arrives: how many liters land in the bath in each minute. That number is the **flow rate**, and it answers a whole family of practical questions. How long will a hose take to fill a pool? How much water does a garden's drip line deliver overnight? How long must a pump run to clear a flooded basement? Each has two measured amounts, a volume of liquid and a length of time, and the flow rate is what ties them together. It also works in reverse. You rarely see a liter of water go past; you see a full bucket, a meter reading or a stopwatch. Dividing the volume you collected by the time it took turns those into the flow rate, and the flow rate then predicts every other volume and every other time for that same flow. ## Definition The **flow rate** of a liquid is the volume of liquid that passes a point in one unit of time. You find it by dividing a volume by the time it took to pass: $$\text{flow rate} = \frac{\text{volume}}{\text{time}}$$ With letters, using $r$ for the rate, $V$ for the volume and $t$ for the time, $$r = \frac{V}{t}$$ Its unit is a volume unit per time unit, where "per" means "in each". A tap that delivers $12$ liters in each minute has a flow rate of $12$ liters per minute, written $12\ \text{L/min}$. A drip that releases $2$ milliliters each second has a flow rate of $2\ \text{mL/s}$. Liters per hour ($\text{L/h}$) and liters per second ($\text{L/s}$) are common too. ### [Iterating a single unit](https://math.wiki/iterating-one-unit/) **Summary:** Measuring by moving one copy of the unit along end to end and keeping count **Question:** There is only one rain boot, and the rug is much longer than one boot. How can one boot say how long the whole rug is? ## When you reach for it What a unit is measures by laying many copies of one unit along an object, end to end, with no gaps and no overlaps, and counting them. That needs enough copies to reach from one end to the other. Often there are not enough. A box of cubes runs out before the hallway does. And some units come one at a time: there is one hand, one foot, one paper strip cut for the job, one rain boot by the door. This lesson measures with **one copy** of the unit. Put it down at one end. Mark where it ends. Pick it up and put it down again so that it starts exactly at the mark. Count each time it is put down. Moving one copy along like this is called **iterating** the unit, which just means using it again and again. Iterating is the same measurement as the row of many copies, done one copy at a time. Every rule from what-a-unit-is still holds. What is new is that the row of copies is never all there at once. It lives in the marks, and the child has to build it, one placement at a time, without letting the copy skip ahead or slide back. It is taught over four days, one part a day: - **Part 1, mark and move:** the move itself, and why the number of placements is the length. - **Part 2, gaps and overlaps:** what happens to the count when the copy skips ahead or slides back. - **Part 3, a bit more:** what to say when the last placement does not fit. - **Part 4, a bigger unit:** why a longer boot gives a smaller count. ### [Length and units](https://math.wiki/length-and-units/) **Summary:** Measuring how long, and moving between units. **Question:** How long is it, in which unit, and how do you say the same length in a different unit? ## What this area is about A length is the number of equal units that fit along a thing, end to end with no gaps. This area turns that into reading a ruler, names the standard units people agree on, and changes a length from one unit to another without changing the length itself. ## How these fit together **The procedure.** Measuring length reads a ruler: a row of units already laid out, with the counting printed on it. **The units.** Customary length units (inch, foot, yard, mile) and metric length units (millimeter, centimeter, meter, kilometer) give each unit a fixed size and a sense of what it fits. **Reading more finely.** Measuring to the half and quarter inch reads an end that falls between inch marks. Length on a number line finds the distance between two points when neither sits at $0$: subtract. **Conversions.** Customary conversions multiply or divide by $12$, $3$, $5{,}280$ and the like. Metric conversions multiply or divide by a power of ten. Both cover weight and liquid units too, not only length. ## A path through it Measuring length comes first; everything else here builds on it. Then the two unit families and the number line. Half and quarter inches follow customary units and need fractions. Customary conversions need multiplication facts. Metric conversions wait for metric prefixes. Next: perimeter, segments and rays and conversion factors. ### [Length on a number line](https://math.wiki/length-on-a-number-line/) **Summary:** The distance between two points on a number line is the difference of the two numbers **Question:** A ruler counts length from zero. If two things stand somewhere along a long line, how far apart are they, and how do lengths that are joined or cut show up on that line? ## Why this exists Lehua and Saltan are walking on a boardwalk by the beach. Every meter along it there is a small marker post with a number on it, starting from $0$ at the steps. Lehua stops by one marker and Saltan stops by another, farther along. Nobody has a tape measure. How far apart are they? A child who can measure with a ruler knows how to read a length when one end sits at $0$. Here neither end is at $0$. The markers are a ruler laid along the ground, but the two children stand in the middle of it. The answer is in the two numbers: the distance between two points on a line is the **difference** of their numbers, the larger take away the smaller. That one fact turns a number-line into a tool for length problems. Two pieces of string laid end to end become two stretches of the line, one after the other, and their total is a sum. A piece cut off the end leaves a stretch that ends at a difference. Two lengths compared side by side differ by the stretch between their ends. A story problem about length can be drawn on the line, and the drawing shows whether to add or to subtract. The lesson runs over three days. Day 1 is the number line as a ruler, and the distance between two points (Examples 1 and 2). Day 2 is putting lengths together, cutting a piece off, and comparing (Examples 3 to 5). Day 3 is length story problems with a box for the unknown (Examples 6 and 7). Every length here is a whole number of units, and every number is $100$ or less. ### [Liquid volume: liters and milliliters](https://math.wiki/metric-capacity-units/) **Summary:** Milliliters and liters, the metric units for how much liquid a container holds **Question:** How much does a jug, a glass or a spoon hold, and how do you read the amount off the marks on a measuring cup when most of the marks carry no number? ## Why this exists Seydou's family keeps its drinking water in a big jug by the kitchen door. Awa works the counter at a juice bar, where every smoothie starts with a pour into the blender. Sundiata's class measures water in a tall, thin measuring cylinder. All three need to say **how much liquid** there is, in a way someone else can check. "A jugful" or "a cupful" does not do it, because jugs and cups come in every size. A unit fixes one size that everybody shares. For liquids, the two everyday metric units are the **liter** and the **milliliter**. This lesson gives each one a size you can picture, then reads the marks on the side of a container, where the real work is that most of the marks carry no number. the-metric-system-as-a-system made the liter the base unit for liquid volume, and metric-prefixes showed that milli- cuts a base unit into $1{,}000$ equal parts. Working out what one space on a marked scale stands for is the idea of reading-a-scale-between-marks; here it is used on cylinders, jugs and a blender. The lesson runs over three days. Day 1 is the two units and their sizes (Examples 1 and 2). Day 2 reads measuring cylinders and jugs marked in milliliters (Examples 3 to 5). Day 3 puts liters and milliliters on the same container (Examples 6 and 7). ## Definition ### Day 1: the liter and the milliliter **Liquid volume** is how much space a liquid takes up. The liquid volume a container holds when it is full is its **capacity**. Liquid volume is about space filled, not about how heavy the liquid is or how tall it stands in its container. ### [Mass and weight](https://math.wiki/mass-and-weight/) **Summary:** How much matter is in an object, and how hard gravity pulls on it **Question:** When you ask how heavy something is, are you asking how much stuff it holds or how hard gravity pulls on it? ## What this area is about "How heavy" hides two questions. Mass is how much matter an object holds, and it stays the same wherever the object goes. Weight is how hard gravity pulls on that matter, and it changes from Earth to the Moon. On Earth the two move together, so everyday speech uses one word for both. This area gives each its units and then pulls them apart. ## How these fit together **Units.** Metric mass units (milligram, gram, kilogram, tonne) measure mass, with benchmarks to feel each one and a balance or scale to read it. Customary weight units (ounce, pound, ton) are the United States family, with $1$ pound $= 16$ ounces and $1$ ton $= 2{,}000$ pounds. **The distinction.** Mass is not the same as weight separates the two ideas, and uses unit rates and the double number line to work out how much an object weighs under a different pull of gravity. ## A path through it The two unit lessons first. Neither needs the other: the metric one needs metric prefixes and benchmarks, the customary one needs the customary system. Mass versus weight comes later (grades $6$ to $8$), once unit rates are in hand. Next: density, which divides a mass by a volume, and story problems with grams, kilograms and liters. ### [Mass is not the same as weight](https://math.wiki/mass-versus-weight/) **Summary:** Mass does not change when an object is moved, but its weight depends on the pull of gravity there **Question:** An astronaut carries a rock from Earth to the Moon, and there it is far easier to lift. Has the rock changed, and if not, what has? ## Why this exists At the post office where Kerstin works, sacks of letters are weighed all day, and everyone gives the answer in kilograms. Now send one of those sacks to the Moon. An astronaut there lifts it with a fraction of the effort, yet not one letter has been taken out. Whatever "how heavy" means, part of it stayed the same on the journey and part of it changed. So "how heavy" hides two questions. How much stuff is there? And how hard is that stuff being pulled down? On Earth the two answers move together so closely that everyday speech uses one word for both. Away from Earth they come apart, and so do the instruments that measure them. Keeping them apart is ratio work, and the unit-rate and the double-number-line are exactly the tools it needs. ## Definition The **mass** of an object is how much matter it contains. It is measured in grams and kilograms (see metric-mass-units). Mass does not depend on where the object is: a $6$ kilogram rock has a mass of $6$ kilograms on Earth, on the Moon and on Mars. It changes only when matter is added or taken away. The **weight** of an object is how hard gravity pulls it down. A pull is a force, and forces are measured in **newtons**, written $\text{N}$, a unit named after Isaac Newton. On Earth an apple of about $100$ grams weighs about $1$ newton. Weight depends on where the object is, because different worlds pull with different strength. ### [Measurement](https://math.wiki/measurement/) **Summary:** Putting numbers on the world: length, area, volume, angle, time. **Question:** How do you put a number on how long, how heavy, how hot or how much space something is, so that anyone anywhere can check it? ## What this area is about To measure is to pick a unit and count how many copies of it fit. That one idea runs through every measurement: length, area, volume, mass, angle, temperature and time. This area builds it from comparing two objects by eye up to converting units and judging how many digits to trust. ## How these fit together **The foundation.** What measurement is covers comparing, units, reading a scale and how exact a reading can be. Systems of units covers the customary and metric families and how to convert between any two units. **One attribute each.** Length, area and perimeter, volume and capacity, mass and weight, angles, temperature and time and money. **Built from measurements.** Rates from measurements divide one measurement by another. Scale and size tracks lengths, areas and volumes when a whole object is scaled. Reasoning with units uses units to set up and check a multi-step problem. ## A path through it Start with the early lessons of what measurement is; its later lessons, length, mass, volume and systems of units interleave, each needing parts of the others. Area follows length, and volume follows area. Angles and temperature follow what measurement is. Time and money can run alongside. Mass follows systems of units. Rates come after mass, volume and time; scale and size after angles and volume. Reasoning with units waits for both foundation areas (it is grade $9$ work). ### [Measurement error](https://math.wiki/measurement-error/) **Summary:** The gap between a reading and the true amount, which every measurement carries **Question:** No ruler, scale or stopwatch gives the exact amount. How far off can a reading be, and when does being off actually matter? ## Why this exists Count the eggs in a box and you can be exactly right: there are $6$, not $6.02$. Measure the box with a ruler and you cannot. The edge of the box falls somewhere between two marks, and you write down the nearest one. The real length is a little more or a little less. That is not a mistake by the person measuring. It is built into every measurement of a length, a mass, a time, a volume or a temperature, because these amounts can take any value and a tool can only show finitely many. So the honest questions are "how far off could it be?" and "is that much off a lot, for this job?" This lesson answers both, and separates two words that sound alike, accuracy and precision, which name two different ways a reading can be good. ## Definition The **true value** $t$ is the amount actually there. The **measured value** $m$ is the reading. The **measurement error** is the gap between them. **Absolute error** is the size of that gap, in the same unit as the measurement: $$\text{absolute error} = |m - t|$$ The bars are absolute value: the absolute error is never negative. Whether the reading was too high or too low is said separately, in words. **Greatest possible error.** When a reading is taken to the nearest mark of a scale, and the marks are $u$ apart, the true value is at most half a space from the reading. That largest possible gap, $\tfrac{1}{2}u$, is the **greatest possible error** of the reading. A digital display works the same way, with $u$ the place value of its last digit: a stopwatch showing hundredths has $u = 0.01\text{ s}$. How finely a tool is marked is its precision, and reading a scale tells you what $u$ is. ### [Measurement is approximate: precision of a tool](https://math.wiki/precision-of-a-measuring-tool/) **Summary:** How finely a tool is marked sets the smallest difference it can possibly show **Question:** Two honest people measure the same pencil and get two different answers. Which one is right, and what does a measurement really tell you? ## Why this exists Anahita measures a pencil from her pencil case and says it is about $6$ inches long. Babak measures the same pencil and says it is $5\frac{3}{4}$ inches. Neither of them made a slip. They used different rulers. That happens with every kind of measuring: rulers, scales, thermometers, clocks. A tool can only answer with the marks it has, and the thing being measured almost never stops exactly on a mark. So every measurement is a choice of the nearest mark, and the answer you get depends on how close together the marks are. This lesson is about what that means. It explains why a measurement is always "about", why two objects can read the same and still be different, and how to pick a tool with marks fine enough for the job in front of you. It is taught over two days: the first is about the nearest mark, the second about what a tool can and cannot tell apart. ## Definition Measuring a length means counting how many copies of a unit fit along it, end to end, which is iterating one unit. A ruler is those copies already laid out, with a numbered mark where each one ends. The end of an object can stop anywhere, but the tool has marks only at certain places. So: - The **reading** of a measurement is the mark on the tool nearest to where the object ends (or where the pointer or the liquid stops). - How **finely** a tool is marked is the space between two neighboring marks. A ruler marked in quarter inches is finer than one marked in whole inches. A scale marked every gram is finer than one marked every $10$ grams. ### [Measuring and drawing angles with a protractor](https://math.wiki/read-a-protractor/) **Summary:** Lining the center and base line up with the angle and reading where the other arm crosses **Question:** An angle is drawn on a page. How do you find how many degrees it opens, and how do you draw a new angle that opens exactly the number of degrees you want? ## When you reach for it An angle is an amount of turn (an angle is an amount of turn), and degrees counts that turn in small equal steps: a full turn is $360^\circ$, a half turn is $180^\circ$ and a right angle is $90^\circ$. Counting $70$ tiny steps by hand would be slow. A **protractor** has the counting already printed on it. Reach for a protractor when an angle is drawn and you need its size as a number: the angle between two blades of a fan in a sketch, or between two roads on a map. Reach for it also the other way around, when you know the number and need to draw the angle: a road on a map that leaves a town at $75^\circ$ from another road, say. This lesson takes three days. Part 1 is the parts of a protractor and how to measure with it. Part 2 is the two rows of numbers, how to choose the right one, and how to estimate first. Part 3 is drawing an angle of a given size, arms that are too short, and turns of more than a half turn. ## The procedure ### Part 1: the parts of a protractor, and measuring an angle A protractor is a half circle, usually clear plastic. Its straight edge carries a line called the **base line**. Halfway along the base line is the **center mark**, a small mark or hole. Around the curved edge run tick marks, and beside them two rows of numbers from $0$ to $180$. Why a half circle, and why $180$? Picture a ray lying along the base line, pointing right, with its endpoint on the center mark. Now swing it about the center mark, all the way around to the left end. It has made a half turn, $180^\circ$. Every tick mark on the curved edge is where the swinging ray would point after turning some number of degrees. So the curved edge is a half turn cut into degrees, and every one of those degrees meets at the center mark. ### [Measuring angles](https://math.wiki/angle-measure/) **Summary:** Degrees, and reading a protractor. **Question:** An angle's arms can be drawn as long or as short as you like. So what exactly is being measured, and how do you put a number on it? ## What this area is about The size of an angle is not the length of its arms. It is how far one arm has to turn to land on the other. This area turns that idea into a number, the degree, and gives you a tool for reading and drawing it. ## How these fit together **The idea and the unit.** An angle is an amount of turn says what the size of an angle is. Degrees count that turn: a full turn is $360^\circ$ and a right angle is $90^\circ$. **The tool.** Measuring and drawing angles with a protractor reads the degrees off a drawn angle, or draws an angle of a given size. **A result to use.** Angle measures add: when an angle is split into parts, its measure is the sum of the parts, so a missing part is the whole minus the rest. **An application.** Bearings and navigation uses angles as directions of travel. It also needs sine, cosine and the Pythagorean theorem, so it is high school work. ## A path through it Turn first, then degrees, which builds on it. Then the protractor, then angles adding. Leave bearings until you have met right triangle trigonometry. Finish here and you can name and compare angles by size, sort them into acute, right and obtuse, and move on to angle pairs and circle graphs. ### [Measuring area with unit squares](https://math.wiki/tiling-with-unit-squares/) **Summary:** Covering a shape with same size squares that touch without gaps, then counting them **Question:** If everyone covers shapes with whatever unit they like, nobody's counts agree. Which unit should everyone use, and how do you count it when the squares do not fit the shape exactly? ## When you reach for it In area-as-covering the unit was free to choose: sticky notes, cards, paper triangles. That freedom is also the trouble with it. Eino says his drawing is $14$ and Hilkka says hers is $30$, and neither number means anything to anyone who has not seen their sticky notes. For a count of area to mean the same thing everywhere, everyone has to cover with the same unit. The unit everyone agrees on is a **square whose sides are one unit of length long**: a square $1$ centimeter on each side, or $1$ inch, $1$ foot or $1$ meter. Its area is called one **square centimeter** (or square inch, square foot, square meter). This skill is measuring area with these squares. You reach for it whenever a shape is drawn on grid paper, laid out in square tiles, or small enough to trace onto a grid: count the squares that cover it, and when the squares do not fit the shape exactly, put the part squares together, or estimate. The lesson takes three days. Part 1 sets up the square units and counts whole squares on a grid. Part 2 counts shapes made of whole squares and half squares. Part 3 estimates the area of a shape with a curved edge, where the squares can never fit exactly. ## The procedure ### Part 1: square units, and counting them on a grid **Why a square.** Any unit that covers without gaps or overlaps would measure area, so why a square? Two reasons. First, squares fit together edge to edge in rows with no gaps and no overlaps, and they keep doing so across a whole flat surface, however large. Round units cannot do this at all. ### [Measuring length](https://math.wiki/measure-length/) **Summary:** Lay units end to end; read a ruler. **Question:** A ruler is covered in marks and numbers. Which number says how long a crayon is, and what if the crayon does not start at the end, or the thing to measure is not straight at all? ## When you reach for it A length is the number of whole units that fit along a thing, laid end to end with no gaps and no overlaps. What a unit is showed this with a row of many copies, and iterating one unit showed it with one copy moved along and marked. Both work. Both are slow, and every placement is a chance to leave a gap. A **ruler** does the laying out ahead of time. It is a row of equal units already stuck together on a stick, with no gaps and no overlaps, and its numbers count the units for us. This lesson is about reading that row: which number is the length, why, and what to do when the thing does not sit at the start of the ruler or does not lie straight. Here the units have names. A **centimeter** and an **inch** are just two units, each the same size on every ruler. Only whole units are used: a length comes out as a whole number of units, or as "about" a whole number. It is taught over four days, one part a day: - **Part 1, what the marks mean:** a ruler as a row of units; reading a length that starts at $0$. - **Part 2, starting somewhere else:** counting the spaces, not the marks. - **Part 3, about:** an end that falls between two marks. - **Part 4, a curvy path:** measuring with a string. Do every part with real things: a centimeter ruler, an inch ruler, crayons, a few pieces of dry spaghetti, paper squares to cut out, and a long piece of string. ## The procedure ### Part 1: What the marks mean ### [Measuring to the half and quarter inch](https://math.wiki/measure-to-half-and-quarter-inch/) **Summary:** Reading a length that falls between the whole inch marks **Question:** A crayon ends between two inch marks on a ruler. How do you say how long it is, and how do you show the lengths of a whole box of crayons at once? ## When you reach for it A length is easy to say when its end lands right on an inch mark: the crayon is $3$ inches, the earthworm is $2$ inches. Most things do not end on an inch mark. An earthworm stretched along a ruler ends somewhere past $2$ and before $3$, and "about $2$" or "almost $3$" throws away information the ruler is holding. The marks between the inch marks are there to say how far past the last whole inch the end reaches. This lesson reads those marks. Part 1 uses a ruler with each inch cut in half. Part 2 uses a ruler with each inch cut into fourths, and names lengths such as $3\frac{1}{4}$ inches. Part 3 takes a pile of such measurements, a dozen earthworms or a box of crayons, and shows them all at once on a line plot whose marks go up by quarter inches, then reads the plot. Three ideas are already in place. An inch is a fixed length, and $12$ of them make a foot (customary-length-units). A fraction such as $\frac{1}{4}$ names one of $4$ equal parts of a whole (fraction-as-part-of-whole). A line plot puts one dot above a number line for each measurement (dot-plots-and-line-plots). ## The procedure ### Part 1: halves of an inch A ruler is a number line made of inches, with $0$ at the end you measure from (locate-a-fraction-on-the-number-line treats any number line the same way). Lay the object straight along the ruler with one end at $0$. The length is the number at the mark where the other end lands. ### [Measuring tools and reading their scales](https://math.wiki/reading-a-scale-between-marks/) **Summary:** Working out what one space on a marked scale stands for and reading a value that falls inside it **Question:** A meter stick prints only $10$, $20$, $30$ and so on, with a crowd of bare marks in between. How do you read a length that ends on one of the bare marks, or between two of them, and which tool should you pick up in the first place? ## When you reach for it A child who has measured with a ruler knows that each number on it counts the units that end at that mark. That lesson used a ruler with a number on every mark. Real tools are not so kind. A meter stick has a hundred marks and prints only ten numbers. A measuring tape may print a number only every five marks. Some rulers print only the even numbers. Before any of that, there is a choice to make: a short ruler, a long stick, or a tape that bends. This lesson covers both. It is taught over three days, one part a day: - **Part 1, the tools:** the inch ruler, the yardstick, the meter stick and the measuring tape, and how to pick the one that suits the job (Examples 1 and 2). - **Part 2, marks without labels:** working out what one space stands for, then counting on or back from the nearest label (Examples 3 to 5). - **Part 3, ends between marks:** naming the two marks, saying which one is nearer, and giving the length as "about" a whole number of units (Examples 6 and 7). Reach for it whenever a measuring tool is in a child's hand, which is every time a length is measured from now on. The units here are inches and centimeters, and every length is a whole number of them or "about" one. How big each unit is, and the other units in each family, belong to customary-length-units and metric-length-units. Have the real tools on the table: a ruler with inches on one edge and centimeters on the other, a yardstick, a meter stick, a measuring tape, and ribbon, string and paper strips to measure. ### [Measuring what you cannot reach](https://math.wiki/indirect-measurement-with-shadows/) **Summary:** Finding the height of something too tall to reach by comparing its shadow with a known object **Question:** A tree is far too tall to reach with a tape measure. How can its shadow, a mirror, or one angle tell you how tall it is? ## When you reach for it Some heights cannot be reached with a tape measure: a tree, a flagpole, the roof of a building, a monument. What you can still measure is on the ground near them: the length of a shadow, a distance you pace out, an angle read off a protractor. Finding a length from other measurements, without ever touching the length itself, is called **indirect measurement**. Every method in this lesson rests on one fact about similar figures: two triangles with the same angles have the same shape, so their sides are in the same ratio. Each method finds a **small triangle** you can measure completely, and a **large triangle** with the same angles that contains the height you want. The small triangle gives the ratio, and the ratio carries over to the large one. Reach for a **shadow** on a sunny day, when the shadow falls on level ground and you can measure it from the point under the top of the object. Reach for a **mirror** when there is no sun. Reach for **an angle and a distance** when you can stand a measured distance from the foot of the object and see its top. ## The procedure ### Why the shadow triangles have the same shape Stand a post upright on level ground in sunshine, near a tree. Each casts a shadow. The object, its shadow, and the ray of sunlight that just passes its top make a triangle. The two triangles have the same three angles. **A right angle at each foot.** The post and the tree are upright and the ground is level, so each meets the ground at $90^\circ$. ### [Metric length units](https://math.wiki/metric-length-units/) **Summary:** Millimeters, centimeters, meters and kilometers, the length units used almost everywhere else **Question:** A stuffed turtle, a bunk bed, an ant and the walk to a friend's house are all lengths. Which metric unit fits each one, how big is each unit, and how do the centimeter, the meter, the millimeter and the kilometer fit together? ## Why this exists Dzifa and Tawhiri both measure the same bunk bed. Dzifa says it is $2$. Tawhiri says it is $200$. Both are right, and both are useless to anyone else until they say *what* they counted: Dzifa counted meters, Tawhiri counted centimeters. A length is a number of units, and a unit only helps people share a length when its size is fixed and agreed. In the United States the everyday units are the inch, the foot, the yard and the mile (customary-length-units). Almost everywhere else, and in science everywhere, people use the **metric units**: the **centimeter**, the **meter**, the **millimeter** and the **kilometer**. They are worth learning for a second reason too. They fit together by tens, hundreds and a thousand, the same way ones, tens and hundreds fit together in a written number. This lesson gives each metric unit a size you can picture and a tool to measure with. The main units are the centimeter and the meter, measured with a centimeter ruler and a meter stick. That $100$ centimeters make a meter is *seen*, by laying ten-rods from a set of base ten blocks along a meter stick, not learned as a rule. The millimeter and the kilometer are met by their size. The lesson ends with choosing the unit that fits the job. Reading a ruler from $0$ is already known from measure-length, and every length here is a whole number of units. The lesson runs over four days. Day 1 is the centimeter (Examples 1 and 2). Day 2 is the meter (Examples 3 and 4). Day 3 is the millimeter (Examples 5 and 6). Day 4 is the kilometer and choosing a unit (Examples 7 and 8). ### [Metric mass units](https://math.wiki/metric-mass-units/) **Summary:** Milligrams, grams, kilograms and tonnes, the metric units for how much matter there is **Question:** A paper clip, a hamster and a melon are each heavy by a different amount. Which units say how much matter each one holds, and how do you get a number for it, first by feel and then with a balance or a scale? ## Why this exists Coumba runs a produce stand. A customer picks up a melon and asks, "How heavy is it?" Coumba could say "pretty heavy" or "about as heavy as a big book", but the customer's big book is not her big book. To answer in a way that means the same thing to everyone, she needs a unit: an amount of heaviness that everyone has agreed on, so that "two of them" says exactly how much. The metric units for how heavy a thing is are the **gram** and the **kilogram**. They measure **mass**, which is how much matter (how much stuff) an object holds. On Earth, an object with more mass is heavier to lift, so in this lesson "mass" and "how heavy" go together, and both are said plainly. Scientists keep mass and weight apart for a good reason that shows up off the Earth; that story is in mass-versus-weight. Everything in this lesson happens on the ground, at a produce stand and a pet shop. Knowing the units is only half of it. A person also needs a feel for them, so that "a $2$ gram bag of apples" sounds as wrong as it is, and a way to get a number: first an estimate by hand, then a measurement with a pan balance or a digital scale. This lesson takes three days. Day 1 (Examples 1 and 2) is the gram and the kilogram, a picture of each, and choosing the unit that suits an object. Day 2 (Examples 3 and 4) is estimating a mass by comparing it with an object whose mass you already know. Day 3 (Examples 5 to 7) is reading a digital scale and a pan balance. ### [Metric prefixes](https://math.wiki/metric-prefixes/) **Summary:** Kilo, centi, milli and the rest are names for a size, and they mean the same beside any base unit **Question:** Kilo, centi and milli turn up in front of meter, gram and liter. What does each little word part say about the size of a unit, and why does it say the same thing every time? ## Why this exists Sanjar signs up for a bike race. The poster says the course is $4$ km. Inaya reads the label on her cat's eye drops: "$1$ mL". Nilufar looks at the marks up the side of her baby brother's bottle: they are in mL too. The letters m, L, k keep turning up in new mixes, and nobody memorizes every mix one at a time. They do not have to. In each unit name, a short word part in front says **how big**, and the base unit after it says **what is measured**. There are only a few of these front parts, and each one always means the same size. Learn kilo-, centi- and milli- and you can read units you have never seen before. the-metric-system-as-a-system set out the base units and the steps of ten between metric units. This lesson looks closely at the front parts themselves. It takes two days. Day 1 (Examples 1 to 3) takes kilo-, centi- and milli- one at a time and reads each on a place value chart. Day 2 (Examples 4 to 7) puts the same front part beside different base units, sorts out which units people really use, meets the rest of the family, and uses the prefixes inside a problem. ## Definition ### Day 1: three prefixes A **prefix** is a word part put at the front of a word to change what it means, like re- in redo. In a metric unit the prefix names a **size**, and the base unit (meter, gram, liter) names **what** is measured. - **kilo-** means $1{,}000$ of the base unit. A kilometer is $1{,}000$ meters. Short form: k, as in km. - **centi-** means one of $100$ equal parts of the base unit. Cut a meter into $100$ equal parts and each part is a centimeter. Short form: c, as in cm. - **milli-** means one of $1{,}000$ equal parts of the base unit. Cut a liter into $1{,}000$ equal parts and each part is a milliliter. Short form: m, as in mL and mm. ### [Money](https://math.wiki/money/) **Summary:** Coins, bills, and making change. **Question:** A dime is smaller than a nickel but worth more. How do you know what each coin and bill is worth, make the same amount in different ways, and hand back the right change? ## When you reach for it A child who can count to a hundred can count a handful of coins and say "seven". That answers how many coins there are. It does not answer how much money there is, because coins are not all worth the same. Seven pennies buy almost nothing; seven quarters buy a lot. Money is the first place a child meets things that are counted by their **worth**, not by how many there are. So three jobs come together in this lesson, one for each day. - **Day 1: what each coin and bill is worth**, how to write an amount with the cent sign or the dollar sign, and why $100$ cents make one dollar. - **Day 2: the same amount in different ways.** Two piles can look nothing alike and still be worth the same. Some ways use fewer coins than others. - **Day 3: making change.** When someone pays more than the price, the change is the part that goes back. It is found by counting up from the price to the amount paid. All three rest on two things the child already has. One is skip-counting by fives, tens and twenty-fives. The other is the missing part: the price and the change together make the amount paid, which is subtraction as a missing part. Amounts here stay within $100$ cents, or are whole dollars. Writing dollars and cents together, as in $\text{\textdollar}3.40$, comes later in money-as-decimals. Real coins help more than anything printed. A small dish of pennies, nickels, dimes and quarters, and a few play bills, is enough for every example below. ### [Money written as decimals](https://math.wiki/money-as-decimals/) **Summary:** A price is a decimal number where the two digits after the point count hundredths of a dollar **Question:** A price sticker at a garage sale says 4.75. How can one number say "four dollars" and "seventy-five cents" at the same time? ## Why this exists Advika, Kunle and Soledad are running a garage sale. Every table holds old toys, books and lamps, and every item needs a price sticker. Kunle has also emptied his piggy bank to see what he can buy: a few dollar bills and a pile of quarters, dimes and pennies. Money comes in pieces of very different sizes. A dollar bill is worth $100$ pennies. A price such as "four dollars and seventy-five cents" uses two units at once, and two units make adding hard: you would have to add the dollars, add the cents, and then trade $100$ cents for a dollar whenever the cents got too big. Decimal notation fixes this. It writes the whole amount as **one number of dollars**, with the cents as the part after the point. Then every money sum and difference is just adding and subtracting decimals, with the same lined-up columns and the same trades of ten for one. ## Definition One dollar is $100$ cents. So one cent is one hundredth of a dollar: $$1 \text{ cent} = \tfrac{1}{100} \text{ dollar} = 0.01 \text{ dollar}$$ An amount of money written **as a decimal** counts dollars. The whole number part is the number of dollars, and the two digits after the point are the cents. The **dollar sign** goes in front of the number, so $\text{\textdollar}4.75$ is read "four dollars and seventy-five cents". An amount can also be written in cents alone, with the **cent sign** $\text{¢}$ after the number: $75\text{¢}$ is seventy-five cents, the same amount as $\text{\textdollar}0.75$. ### [Morning and afternoon on a twelve hour clock](https://math.wiki/am-and-pm/) **Summary:** The clock face comes round twice a day, so a label says which half of the day you are in **Question:** A clock face only has the numbers $1$ to $12$, but a day is much longer than twelve hours. When a clock says $7{:}00$, how do you know whether it means the morning or the evening? ## Why this exists A child who can tell time can look at a clock and say "seven o'clock". But seven o'clock comes twice every day: once in the morning, around the time many children wake up, and once in the evening, around the time many families finish dinner. The clock face looks exactly the same both times. If a friend says "come over at seven", the clock alone cannot say which seven is meant. The reason is that a clock face has only twelve hours on it, and a day has twenty-four. The hour hand goes all the way around the face once, and the day is only half over, so it goes around a second time. Every time a clock can show turns up twice. People solve this with two short labels. **a.m.** marks a time in the first half of the day and **p.m.** marks a time in the second half. With the label, "$7{:}00$ a.m." and "$7{:}00$ p.m." are two different times, and nobody has to guess. This lesson runs over two days: - **Part 1:** why each clock time happens twice, what a.m. and p.m. mean, and the two special times, noon and midnight, where one half of the day ends and the other begins. - **Part 2:** choosing a.m. or p.m. from what is happening at that time, and putting the events of one day in order when some are a.m. and some are p.m. A real clock helps, and so does a long strip of paper that the adult turns into a day line (the first picture below), with the child's own day marked on it: getting up, school, lunch, home, dinner, bed. ### [Multi-step problems with unit conversions](https://math.wiki/multi-step-conversion-problems/) **Summary:** Real problems that need a unit changed, such as centimeters to meters, partway through several steps. **Question:** A board is measured in feet, but the shelves you cut from it are measured in inches. How do you solve a problem when its amounts come in different units? ## When you reach for it Ziva is at the hardware store. She buys a board $8$ feet long and wants to cut shelves that are each $18$ inches long. How much board is left over? You cannot subtract $18$ inches from $8$ feet as they stand. The numbers count different things: one counts feet, the other counts inches. Before you add, subtract or compare, both amounts must be in the same unit. So the problem has an extra step in the middle: change a unit, then carry on. You reach for this move whenever a problem mixes units from one family: feet and inches, pounds and ounces, liters and milliliters, hours and minutes. The single conversion, such as $1$ foot $= 12$ inches, is reviewed in unit-conversion-customary and unit-conversion-metric. Word problems that already sit in one unit are in measurement-word-problems. This lesson puts the two together, in problems with several steps. ## The procedure This lesson takes four days. Day 1 works with length in feet, inches, yards and miles. Day 2 works with weight and capacity: pounds and ounces, gallons, quarts, pints and cups. Day 3 is metric units, where every change is a power of ten. Day 4 is time, answers written in two units at once, and choosing which unit to work in. ### Day 1: length, and the step in the middle These are the customary length facts. They are exact: a mile is defined as $5{,}280$ feet. | bigger unit | equals | |---|---| | $1$ foot | $12$ inches | | $1$ yard | $3$ feet, or $36$ inches | | $1$ mile | $5{,}280$ feet, or $1{,}760$ yards | ### [Perimeter](https://math.wiki/perimeter/) **Summary:** Add up the sides. **Question:** A farmer is buying fence for a goat pen, a strip of wood for a rabbit hutch, baseboard for a room. How much does it take to go all the way around a flat shape, and how do you find it when some sides are not marked? ## When you reach for it Edurne is putting up a fence around a goat pen. The fence goes along every edge of the pen, all the way around, and ends where it started. How much fence she needs is a length: the distance all the way around the pen. That distance is the pen's **perimeter**. The **perimeter** of a flat shape is the total length of its boundary, the distance you would walk if you went once around its edge and came back to where you began. Because the perimeter is a length, it is measured in units of length: centimeters, meters, inches, feet. You measure it the way you measure any length (see measure-length: lay a unit end to end and count, or read a ruler), only the thing being measured bends at the corners. Fence, a wood trim, a ribbon around a card, baseboard along the walls of a room: each of these is a perimeter question. Perimeter is not the same as area. In a picture of a field with a fence around it, the perimeter is the fence and the area is the field. Area counts the squares that cover the inside (area-as-covering); perimeter measures only the edge, in plain length units. The lesson takes three days. Part 1 finds the perimeter of any shape with straight sides by measuring or adding its sides. Part 2 does rectangles and squares, where equal sides give shortcuts. Part 3 handles shapes with square corners where some sides are not marked and have to be worked out from the others first. ### [Perimeter versus area](https://math.wiki/perimeter-versus-area/) **Summary:** Perimeter measures the edge in length units and area measures the inside in square units **Question:** A sheep pen needs fence around it and grass inside it. Does the amount of fence decide the amount of grass, and if the fence is fixed, which pen gives the sheep the most room? ## Why this exists Zivile keeps sheep. Their pen needs two things: a fence that goes all the way around it, and grass that covers the ground inside it. Fence is bought by the meter. Grass is laid in squares. Both depend on how big the pen is, so it is easy to think that one number tells both. It does not. Two pens can use exactly the same fence and hold very different amounts of grass. Two pens can hold exactly the same grass and need very different amounts of fence. The edge and the inside are two separate measures, and a question about a shape has to say which one it wants. perimeter measured the edge, and missing-side-from-perimeter worked it backward. area-as-covering and area-of-rectangles counted the squares inside. This lesson sets the two side by side. It takes three days. Part 1 tells the two measures apart and decides which one a question needs. Part 2 lists every rectangle with a given perimeter, or a given area, in an organized table, and sees how much the other measure changes. Part 3 reads those tables for the best choice: the most room for a fixed fence, and the least edge for a fixed set of squares. ## Definition The **perimeter** of a flat shape is the length of its edge, all the way around. It is a length, so it is measured in units of length: meters, feet, inches, centimeters. The **area** of a flat shape is the amount of flat space inside it: the number of unit squares that cover it with no gaps and no overlaps. It is counted in **square units**. A square meter is a square $1$ meter on each side; a square foot is a square $1$ foot on each side. ### [Rates from measurements](https://math.wiki/measured-rates/) **Summary:** Numbers made by dividing one measurement by another, such as speed or density **Question:** Some useful numbers are not a single measurement but one measurement divided by another. What do they tell you that neither measurement can? ## What this area is about Divide one measurement by another and you get a rate: how much of the first comes with each one unit of the second. A rate compares things fairly when the raw amounts differ in size. Each rate here is a unit rate with measured quantities on both sides, so the same per-one reasoning runs all of them. ## How these fit together Both children are definitions with the problems they answer. Flow rate is volume per unit of time, such as liters per minute. Knowing it lets you say how long a bath or a pool takes to fill, or how much arrives in a given time. Density is mass per unit of volume, such as grams per cubic centimeter. It compares materials, not objects: a log has more mass than a pebble, but the pebble packs more mass into each cubic centimeter. ## A path through it The two lessons mention each other, but neither depends on the other. Flow rate needs liters and milliliters and units of time. Density also needs mass units and the volume of a box, so if you are taking both, flow rate is the shorter step in. Finish here and any "per" quantity built from measurements reads the same way: a unit rate, with units on top and bottom that tell you whether to multiply or divide. ### [Reading a thermometer](https://math.wiki/reading-a-thermometer/) **Summary:** Taking a temperature off a marked scale where the liquid or the bar stops **Question:** The liquid in a thermometer stops at a mark with no number beside it. What temperature is it, and how much warmer is it than this morning? ## When you reach for it A thermometer is a narrow glass tube with a little bulb of colored liquid at the bottom. When the air or water around it gets warmer, the liquid swells and climbs the tube. When it gets colder, the liquid shrinks and drops. Beside the tube is a scale of marks, like a ruler stood on end. The temperature is the number on the scale where the top of the liquid stops. The trouble is that most marks have no number. A tube has room for a mark every degree or every $2$ degrees, but not room to write a number beside each one, so only some marks are labeled. To read the thermometer you have to work out what the unlabeled marks stand for. That is the skill from reading-a-scale-between-marks (find what one space is worth, then count spaces from a labeled mark), used on an upright scale. Two more things make a thermometer its own job. First, there are two common temperature scales, **degrees Fahrenheit**, written $^\circ\text{F}$, and **degrees Celsius**, written $^\circ\text{C}$. Many thermometers carry both, one on each side of the tube. Second, the question is often not just "how hot?" but "how much warmer than before?" or "how much did it cool?" That is a difference between two readings, found by counting up or by subtracting. Reach for this skill whenever a temperature has to be read off a scale, written down with its unit, or compared with another temperature on the same scale. ### [Scale and size](https://math.wiki/scale-and-size/) **Summary:** What happens to lengths, areas and volumes when a whole object is made bigger or smaller **Question:** Make a whole object bigger by some factor. What happens to its lengths, its area and its volume, and how can that measure something you cannot reach? ## What this area is about A scale copy multiplies every length by one number, the scale factor $k$. Lengths are the easy part. Area and volume do not grow by $k$. And because matching lengths keep one ratio, a small triangle you can measure tells you a height you can never reach. ## How these fit together **The rule for lengths.** Scale factor multiplies every length: sides, diagonals, heights and perimeter all get multiplied by the same $k$. **The result for area and volume.** Scale factor, area and volume shows that area is multiplied by $k^2$ and volume by $k^3$, because an area needs two lengths and a volume three. **An application.** Measuring what you cannot reach finds the height of a tree or a flagpole from a small triangle you can measure and a large one with the same angles, using a shadow, a mirror or an angle. ## A path through it Lengths first, then area and volume, which also needs surface area. Measuring what you cannot reach does not rest on the other two; it needs similar triangles, a protractor and angles of elevation, so it is high school work (grades $9$ to $12$). Finish here and you can predict how much paint, carpet or material a bigger or smaller copy needs, not just how long its sides are. ### [Scale factor multiplies every length](https://math.wiki/scale-factor-and-length/) **Summary:** Doubling a shape doubles every side, every diagonal and the perimeter **Question:** A shape is copied twice as large. Its sides double, but what about its diagonals, its height, its perimeter, and the distance between any two points on it? ## Why this exists A scale copy is made by multiplying the sides by one number, the scale factor. But a shape carries far more lengths than its sides: a kite has diagonals, a triangle has a height, a frame has a perimeter, a track has curved ends, and any two marked points have a distance between them. If each of those needed its own rule, every copy would have to be measured again. It does not. One number settles every length in the copy, straight or curved, drawn or not, and you can find any of them without a ruler. ## Definition When a figure is copied at scale factor $k$, **every length in the copy is $k$ times the matching length in the original.** "Every length" means every distance you could measure on the figure: sides, diagonals, a height, the perimeter, a curved edge, the distance between two marked points. In symbols, if $L$ is any length in the original and $L'$ is the matching length in the copy, $$L' = kL$$ How to find $k$ from one pair of matching sides is in the scale factor lesson; here it is put to work. ## Intuition Figure 1 shows a pentagon $ABCDE$ on a grid and its copy at scale factor $2$. The sides along grid lines can be counted. $AB$ covers $5$ squares and $A'B'$ covers $10$. Every grid step has become two grid steps, so every length made of grid steps has doubled. The slanted side cannot be counted. $CD$ goes $3$ squares across and $4$ up; $C'D'$ goes $6$ across and $8$ up. The dot at its middle splits it into two pieces, each going $3$ across and $4$ up: each piece is a copy of $CD$ slid into place. So $C'D'$ is two lengths of $CD$ end to end, twice as long, whatever $CD$ measures. The dashed diagonal tells the same story: $AC$ goes $5$ across and $2$ up, and the dot splits $A'C'$ into two copies of it. ### [Scale factor, area and volume](https://math.wiki/scale-factor-and-area/) **Summary:** Doubling every length makes four times the area, because an area needs two lengths **Question:** Make a rug twice as long and twice as wide. Is it twice as much rug? ## Why this exists Plans, models and photographs are scale copies: one number, the scale factor $k$, multiplies every length, each side, each diagonal and the perimeter. But the questions people ask about a plan or a model are often about something else. How much carpet covers the room on the plan? How much batter goes into a cake twice the size? How heavy would a giant be? Carpet and paint are measured by area, batter and weight by volume, and neither follows the scale factor. A rug twice as long and twice as wide is four times as much rug, and a cake twice as wide and twice as tall takes eight times the batter. This lesson finds the rule for area, then for volume, and then uses it to show why a giant built exactly like a person could not stand up. ## Definition A **scale copy** at scale factor $k$, where $k > 0$, has every length equal to $k$ times the matching length of the original. When a shape is copied at scale factor $k$: - its **area** is multiplied by $k^2 = k \times k$, the **area factor**; - if it is a solid, its **volume** is multiplied by $k^3 = k \times k \times k$, the **volume factor**, and its surface area, which is an area, by $k^2$. Area here is the number of unit squares that cover a shape, and volume the number of unit cubes that fill a solid. So the copy takes $k^2$ times as many squares to cover it and $k^3$ times as many cubes to fill it. ## Intuition **Area.** In Figure 1 a rectangle $3$ squares long and $2$ high is copied at scale factor $2$. The copy is $6$ long and $4$ high. ### [Significant figures](https://math.wiki/significant-figures/) **Summary:** The digits in a recorded measurement that the tool actually supported **Question:** A ruler, a scale or a stopwatch shows a number. Which of its digits did the tool really vouch for, and how many digits does an answer worked out from it deserve? ## Why this exists Rukmini measures the screen of her phone with a ruler marked in millimeters: $14.6\text{ cm}$ high and $6.8\text{ cm}$ wide. Her calculator multiplies them and shows $99.28$. The arithmetic is right, but $99.28\text{ cm}^2$ claims the area is known to the nearest hundredth of a square centimeter, and the ruler never knew either side that well. Measurement error wrote the uncertainty of a reading with a $\pm$. Significant figures do the same job with no $\pm$: the digits a measurement is written with say how fine the tool was. This lesson covers which digits count, how to write a measurement so nobody has to guess, and two rules of thumb (not laws) for how many digits a calculated answer deserves. ## Definition The **significant figures** of a recorded measurement are the digits the measuring tool actually supported. They start at the first nonzero digit and end at the digit in the place of the tool's precision (the space between its marks, or the last digit of its display), and every digit between those two counts, zeros included. Usually only the written number is in front of you, so the definition turns into rules for reading it: - **Nonzero digits always count.** $47.3$ has three significant figures. - **Zeros between nonzero digits count.** A kitchen scale reading of $207\text{ g}$ has three: the scale showed whole grams, and the tens digit it showed was $0$. - **Leading zeros never count.** In $0.0480\text{ kg}$ the zeros before the $4$ only place the decimal point. In grams the same mass is $48.0\text{ g}$ and those zeros have vanished: a digit that disappears when the unit changes was never measured. - **Trailing zeros after a decimal point count.** A lab thermometer reading of $20.0^\circ\text{C}$ has three. The thermometer showed tenths, and the tenth it showed was zero. - **Trailing zeros in a number with no decimal point are ambiguous.** A hedge written down as $1200\text{ cm}$ might have been measured to the nearest centimeter, ten centimeters or meter. The $1$ and the $2$ certainly count; the zeros might be measured digits or only placeholders. ### [Smaller units give bigger counts](https://math.wiki/unit-size-and-count-are-inverse/) **Summary:** The same object measures as more small units or fewer big ones, so size and count move opposite ways **Question:** One bench, measured once in feet and once in inches, gets two different numbers. Why, and how can you tell before measuring which number will be bigger? ## Why this exists Chukwuemeka and Aruzhan both measure the bench at the bus shelter. Chukwuemeka says it is $4$. Aruzhan says it is $48$. Neither made a slip. Chukwuemeka laid a foot ruler along the bench and counted feet. Aruzhan read an inch tape and counted inches. The bench is one bench; the two numbers belong to two different units. Children already met this idea with home-made units. In iterating-one-unit a longer rain boot fitted fewer times along the same rug, and what-a-unit-is states the rule: a bigger unit gives a smaller count. This lesson carries that rule over to the **standard units** children now measure with every day, inches and feet and centimeters and meters. Those units sit on rulers and tapes, often two of them on one ruler, and a child who measures one object twice in two of them will get two different numbers. This lesson is about that moment: predicting which number will be bigger before measuring, explaining the difference by the size of the unit, and running the idea backward, so that two counts of one object tell which unit is the bigger one. Every length here is a whole number of units, or the nearest whole number. Nothing is changed from one unit to another by a rule; each count is read off a tool. The lesson runs over two days. Day 1 measures the same object in two standard units and compares the counts (Examples 1 to 3). Day 2 reasons from the counts alone: ordering units, sizing an unknown unit, and comparing lengths given in different units (Examples 4 to 6). ### [Splitting a figure into rectangles](https://math.wiki/decompose-a-figure-into-rectangles/) **Summary:** Breaking an L shaped or stepped figure into rectangles and adding their areas **Question:** A tree house floor is shaped like an L, so it is not a rectangle and length times width does not fit it. How can its area still be found with the multiplication that works for rectangles? ## When you reach for it In area-of-rectangles, the area of a rectangle came from one multiplication: the side lengths say how many unit squares sit in each row and how many rows there are. That works because every row of a rectangle is the same length. Many floors, yards and rooms are not rectangles, but they are close. Efia's tree house floor is shaped like an L. Its rows are not all the same length, so no single product of two sides gives its area. Yet every corner of the floor is a square corner, and every side runs either straight across or straight up and down. A shape like that can always be cut into rectangles. So this is the move to reach for when a flat shape has **only square corners** but is not one rectangle: an L shape, a T shape, a shape that steps down like stairs. Cut it into rectangles, find the area of each with length times width, and add. Counting the unit squares one by one, as in tiling-with-unit-squares, would also work, and on a grid it is the check. ## The procedure The lesson runs over three days. Part 1 cuts a shape into rectangles and adds their areas, and says why adding is allowed. Part 2 finds the side lengths a plan leaves unmarked, which a cut usually needs. Part 3 cuts the same shape two different ways, checks both by counting squares, and meets one other route: a big rectangle with a corner taken away. ### Part 1: cut into rectangles, then add Here is Efia's tree house floor, drawn on a grid of squares $1$ foot on a side. ### [Splitting a rectangle into rows and columns of squares](https://math.wiki/partition-a-rectangle-into-squares/) **Summary:** A rectangle is cut into rows and columns of same-size squares, and the squares are counted. **Question:** A rectangle is to be cut into rows and columns of squares that are all the same size. Where do the cuts go, and how many squares does that make? ## When you reach for it Squares in rows and columns are all around a child. Ihapera's kitchen window is split by wooden bars into square panes. Tomiwa's hallway floor is laid with square tiles. Esinam's cupcake box has thin cardboard walls inside it that make a square cell for each cupcake. A sheet of graph paper is the same thing drawn in ink. In each one a rectangle has been cut up, by bars, by the edges of tiles or by walls, into squares that are all the same size and that sit in straight rows and straight columns. In arrays-of-objects a child set out separate things, corn plants or blocks, in rows and columns and counted them by adding the same number again for each row. This lesson runs the other way. It starts with one whole rectangle and asks the child to cut it up. That brings two jobs: - **Drawing the cuts** so that every piece is a square and every square is the same size. - **Counting the squares**, by rows or by columns, with a sum of equal addends. Both jobs matter later. The squares a child draws here are the same squares that will measure how much flat space a shape covers, and counting them by rows is the start of the multiplying that comes after. Neither of those ideas is taught here. This lesson is about cutting carefully and counting well. Give the child real things to work with: paper rectangles to fold, a ruler, square tiles or sticky notes to lay on a rectangle, and graph paper to check against. ### [Story problems with dollars and cents](https://math.wiki/money-story-problems/) **Summary:** Story problems with dollar bills and coins, using the right symbol for dollars and for cents. **Question:** A story says someone has some money, gets some more, spends some, or wants something that costs more than they have. How do you turn it into a number sentence, give the answer in cents or in dollars with the right sign, and know the answer fits the story? ## When you reach for it A child who can count a handful of coins knows how much money is in the hand. Money stories ask the next question. How much is there after more comes in? How much is left after something is bought? How much more is needed before something can be bought? Which of two things costs more, and by how much? Each of these is an adding or subtracting story, the same kinds a child has met with counters and cubes, only now the things being counted are cents or dollars. Two things are new. - **The answer has a unit, and the unit has a sign.** An answer of $18$ is not finished. It is either $18\text{¢}$ or $\text{\textdollar}18$, and those are very different amounts. The story says which. - **The answer can be checked against the world.** Money is something a child already has a feel for. A pack of stickers does not cost $\text{\textdollar}80$, and nobody has more left after spending than before. That feel catches a wrong operation quickly. The lesson runs over three days. - **Day 1: one-step stories in all and left.** Adding amounts that come in and taking away amounts spent, in cents or in dollars, with the right sign on the answer. - **Day 2: how much more.** How much more is needed to buy something, how much more one amount is than another, and how much there was at the start. - **Day 3: two-step stories.** Buying two things and getting change, or earning twice and then checking against a goal. ### [Story problems with grams, kilograms and liters](https://math.wiki/mass-and-liquid-volume-problems/) **Summary:** One-step story problems with masses and liquid volumes in the same unit, solved with any of the four operations. **Question:** A sack of seed, a bucket of water, a bottle of shampoo. When a story gives you grams or liters, how do you know whether to add, subtract, multiply or divide? ## When you reach for it The two lessons before this one gave names to two amounts. metric-mass-units measures how much matter something has, in grams (g) and kilograms (kg). metric-capacity-units measures how much liquid a container holds or how much liquid there is, in milliliters (mL) and liters (L). Once an amount has a number and a unit, it can go into a story: a feed store sells it by the bag, a horse barn pours it into buckets, somebody buys some and somebody uses some up. This lesson is about those stories, the kind that take one step. Each one gives two amounts and asks for a third, and the step is one of the four operations: add, subtract, multiply or divide. The amounts here always come in the **same unit**: grams with grams, liters with liters. Nothing has to be changed from one unit to another, and every answer is a whole number of that unit. You reach for this skill whenever a measured amount is joined, taken away, compared, made of equal copies, or shared out. The arithmetic is the same arithmetic used with counted things. What is new is that the numbers carry units, and the unit has to come along into the answer and make sense there. The lesson runs over three days. Day 1 adds and subtracts masses and liquid volumes. Day 2 multiplies and divides them. Day 3 reads amounts off a scale picture, compares two amounts, and checks every answer for its unit and its size. ## The procedure ### [Surface area, and how it compares with volume](https://math.wiki/surface-area-from-a-net/) **Summary:** Unfolding a solid flat and adding the areas of every face **Question:** How much paper covers a present exactly once, and why can two boxes that hold the same amount need different amounts of paper? ## When you reach for it Wrapping a present, painting a jewelry box, cutting card for a box: each asks how much flat material covers the outside of a solid exactly once. That amount is the solid's surface area. Filling the same box with sand asks a different question, about its volume. The lesson takes two days. Part 1 (Examples 1 to 4) finds surface area by laying a solid flat as a net and adding the areas of the pieces. Part 2 (Examples 5 to 7) sets surface area beside volume: what each measures, why one can change while the other stays the same, and how much surface a solid has for each unit of space inside it. ## The procedure ### Part 1: adding up the faces **What surface area is.** The surface area of a solid with flat faces is the total area of all its faces. It counts the unit squares needed to cover the outside once, with no gap and no overlap, so it is measured in square units: $\text{cm}^2$, $\text{m}^2$, $\text{in}^2$. **Why a net helps.** Unfolding a solid into a net bends the sheet at the folds but does not stretch or shrink any face, so each piece has the same area as the face it came from. The area of the net is therefore the surface area, and every net of the same solid gives the same answer. The net also shows every face at once, so none is missed and none is counted twice. **The area of each piece.** A rectangle is length times width, a triangle is half of base times height (area-of-triangles), and a trapezoid or parallelogram is found as in area-of-parallelograms-and-trapezoids. A length that is not given can be read from the edge it meets on folding, as in net-of-a-prism. ### [Systems of units](https://math.wiki/unit-systems/) **Summary:** A whole family of units built to fit together, so amounts can be traded between them **Question:** Why are there 12 inches in a foot but 1,000 meters in a kilometer, and how do you change any amount from one unit to another without changing the amount? ## What this area is about Units come in families built to fit together. The customary family grew out of old trade measures, so its steps are $12$, $3$, $16$ and so on. The metric family uses one base unit per attribute and steps of ten. This area describes both families, then gives one method that converts any unit to any other. ## How these fit together **The systems.** The customary system is the inch, pound and gallon family. The metric system has one base unit for each attribute. Metric prefixes such as kilo, centi and milli name a size, and mean the same beside any base unit. **The method.** Conversion factors are fractions such as $\frac{12 \text{ in}}{1 \text{ ft}}$ that equal $1$, so multiplying by one changes the unit and not the amount. **A result.** Converting square and cubic units uses the length factor twice for area and three times for volume, so $1 \text{ m}^2 = 10{,}000 \text{ cm}^2$. **Problems.** Multi-step conversion problems change a unit partway through a longer problem. ## A path through it The customary system follows customary length units; the metric system follows metric length units, and metric prefixes follow it. Conversion factors need customary conversions and unit rates. Square and cubic units come after conversion factors. Multi-step problems follow measurement word problems. Next: reasoning with units and metric conversions. ### [Telling time](https://math.wiki/tell-time/) **Summary:** Hours, minutes, and the two hands. **Question:** A clock face has two hands and twelve numbers. How do you read the time from it, and how do you write that time the way a digital clock shows it? ## When you reach for it Children hear times all day long. School starts at $8$ o'clock, lunch is at half past $11$, the bus comes at $3$. A time is only useful if everyone reads it the same way from a clock, and there are two kinds of clock to read. An **analog clock** has a round face with the numbers $1$ to $12$ and two hands that turn. A **digital clock**, like the one on a microwave or many alarm clocks, shows the time in digits, such as $7{:}30$. This lesson teaches reading and writing a time to the **hour** and to the **half hour** on both kinds of clock. It runs over four days: - **Part 1:** the clock face, the two hands and what each one counts, and times on the hour ("o'clock"). - **Part 2:** half past the hour, and why the hour hand sits halfway between two numbers. - **Part 3:** digital clocks, and writing a time as $3{:}00$ or $7{:}30$. - **Part 4:** matching an analog clock to a digital one, and drawing the hands for a time. The whole lesson rests on one fact: the two hands turn together, at different speeds, so where one of them points tells you where the other one must be. A real clock is the best thing to have on hand, ideally a geared teaching clock where turning the minute hand moves the hour hand along with it. A paper plate with two paper hands works, but its hands move on their own, so the adult has to keep the hour hand honest. ## The procedure ### Part 1: The clock face and the hour ### [Telling time to the nearest five minutes](https://math.wiki/tell-time-to-five-minutes/) **Summary:** Reading and writing the time from a clock with hands to the nearest five minutes, counting the minute hand by fives. **Question:** The long hand of a clock points at the $4$. Why does that mean twenty minutes and not four, and how do you read, write and draw a time like that? ## When you reach for it Most times in a day are not on the hour or the half hour. A piano lesson starts at twenty past four. Story time at the library starts at five past ten. In tell-time a child read a clock only when the long minute hand pointed at $12$ (o'clock) or at $6$ (half past). This lesson lets the minute hand point at **any** of the twelve numbers, and reads the time to five minutes. The key is one fact about the face. The long hand passes a number every five minutes, so the minutes are found by counting the numbers by fives: "five, ten, fifteen, twenty". That count is counting by fives, which the child already knows, put to a new use. The lesson runs over three days: - **Part 1:** reading the minute hand by counting fives, and writing the time as $4{:}20$. - **Part 2:** where the hour hand sits between two numbers, and which hour to read from it. - **Part 3:** drawing the hands for a time, and the names quarter past and quarter to. A geared teaching clock, where turning the minute hand moves the hour hand along with it, is the best thing to have on hand. Turn it slowly and let the child count as the long hand passes each number. ## The procedure ### Part 1: Counting the minute hand by fives **The marks around the face.** Look closely at a clock face. Besides the twelve numbers, there are small marks all the way around the rim, $60$ of them. The minute hand moves from one small mark to the next in one minute. One whole turn passes all $60$ marks, and one whole turn is one hour, which is why an hour has $60$ minutes (units-of-time). ### [Temperature](https://math.wiki/temperature/) **Summary:** How hot or cold something is, placed on a numbered scale **Question:** Hot and cold can be felt but not seen. How do you put a number on temperature, and what happens below zero or on a different scale? ## What this area is about A thermometer only shows a liquid or a bar rising and falling. A temperature scale marks it the same way everywhere, so a reading means the same thing to everyone. This area reads that scale, names the scales in use, goes below zero, and trades one scale for another. ## How these fit together **The procedure.** Reading a thermometer takes a temperature off a marked scale, working out what the unnumbered marks stand for. **The definitions.** Temperature scales sets up Celsius (at sea-level air pressure, water freezes at $0^\circ$ and boils at $100^\circ$), Fahrenheit ($32^\circ$ and $212^\circ$) and Kelvin, which starts at absolute zero. **Below zero.** Temperatures below zero and temperature change names readings under zero with negative numbers, compares them, and finds how far the temperature moved. **Changing scales.** Converting between Celsius and Fahrenheit uses a rule that both stretches, by $180$ Fahrenheit steps for every $100$ Celsius steps, and shifts, by $32$. ## A path through it Reading a thermometer first. Then the scales, and, separately, temperatures below zero, which also needs adding and subtracting integers. Converting comes last: it needs the scales and substituting into a formula. Finish here and you can read, compare and convert any everyday temperature. ### [Temperature scales: Celsius, Fahrenheit and Kelvin](https://math.wiki/celsius-scale/) **Summary:** A scale with zero at the freezing point of water and one hundred at its boiling point **Question:** Hot and cold can be felt but not seen. How do you put numbers on temperature so that every thermometer, anywhere, gives the same reading? ## Why this exists A thermometer is a thin tube of liquid that climbs when it warms and sinks when it cools. On its own it only says "higher" or "lower". Two tubes made in two places climb by different amounts, so a scratch on one glass means nothing to the owner of the other. To say how hot an oven should be, or how cold tonight will get, every tube has to be marked the same way. That is what a temperature scale does. It is a unit for warmth: an agreed step, the **degree**, counted from an agreed mark. This lesson builds the three scales in use today, Celsius, Fahrenheit and Kelvin, and shows why each is pinned to two temperatures that nature repeats exactly. ## Definition A **temperature scale** is made by choosing two **fixed points**, temperatures that come out the same every time and everywhere, giving each a number, and splitting the gap between them into equal steps called degrees. **The Celsius scale** puts $0$ at the temperature at which pure water freezes and $100$ at the temperature at which it boils, both at the air pressure found at sea level. The gap is split into $100$ equal steps, and one step is one degree Celsius, written $1^\circ\text{C}$. The steps carry on past both fixed points: above $100$ for anything hotter than boiling water, and below $0$ for anything colder than ice, where the readings are negative numbers. Eight degrees below freezing is $-8^\circ\text{C}$, read "negative eight degrees Celsius" or "eight degrees below zero". ### [Temperatures below zero and temperature change](https://math.wiki/negative-temperatures/) **Summary:** Reading and comparing temperatures under the zero of a scale **Question:** Zero on a thermometer is not the coldest it can get. How do you name a temperature below it, compare two of them, and say how far the temperature moved? ## When you reach for it A thermometer has a mark called zero, but zero is not the coldest anything can be. On the Celsius scale, zero is the temperature at which water freezes, and a winter night can be far colder. On the Fahrenheit scale, zero is a colder mark again, and a night can still be colder than that. The scale simply carries on downwards. Temperatures under the zero are named with negative numbers. Seven degrees below zero on the Celsius scale is written $-7^\circ\text{C}$ and read "negative seven degrees Celsius" or "seven degrees below zero". Temperatures above zero are positive, and $0^\circ\text{C}$ itself is neither. Reach for this skill whenever a reading lies below zero: to write it down, to decide which of two temperatures is colder, and to work out how much the temperature rose or fell, especially when it crossed zero on the way. A thermometer is a number line stood on end, so each of these questions is a question about integers, and adding and subtracting integers answers it. ## The procedure ### Reading a temperature below zero A temperature carries two pieces of information. Its **sign** says which side of zero it is on: positive above, negative below. Its **size**, the number without the sign, says how many degrees it is from zero. That size is the absolute value: $|-12| = 12$ because $-12^\circ\text{C}$ is $12$ degrees from zero, and $|5| = 5$. So "twelve degrees below zero" means start at zero and go down $12$ degrees, which is $0 - 12$, that is $-12$. Reading a thermometer below zero works the same way: count the marks down from zero to the top of the liquid. Below zero the numbers grow larger in size as you go down, which is the opposite of what happens above zero, so it pays to find the zero first and count from there. ### [The customary system](https://math.wiki/the-customary-system/) **Summary:** The inch, pound and gallon family used in the United States, grown out of old trade measures **Question:** Why are there 12 inches in a foot but 3 feet in a yard, and how do you know which of these units to use? ## Why this exists Gulsen works at a pet store. On one shelf she finds a leash marked $6$ feet, a bag of dog food marked $30$ pounds, and a fish tank marked $20$ gallons. Three different words, because the three labels measure three different things: how long, how heavy, and how much a container holds. A number alone tells you nothing. "The leash is $6$" could mean $6$ inches, far too short to walk a dog, or $6$ miles, far too long to hold. A measurement is a number together with a unit, and the unit says what one of the counted things is. The units used every day in the United States form a family called the **customary system**. This lesson meets the whole family, gets a feel for how big each unit is, and asks why the numbers that tie the units together are so uneven. This lesson takes two days. Day 1 (Examples 1 to 3) is the units, how big they are, and which one suits which job. Day 2 (Examples 4 to 7) is how the units compare in size, and why the steps between them are uneven. ## Definition ### Day 1: three families of units The **customary system** is the set of units for length, weight and capacity used in everyday life in the United States. Inside each family, every unit is an exact number of the next smaller one. The first column is the short way to write each unit. **Length** (how long, tall or far). You met these in customary-length-units. | unit | short form | equals | |---|---|---| | inch | $\text{in}$ | the smallest unit here | | foot | $\text{ft}$ | $12$ inches | | yard | $\text{yd}$ | $3$ feet | | mile | $\text{mi}$ | $1{,}760$ yards, which is $5{,}280$ feet | ### [The metric system and SI units](https://math.wiki/the-metric-system-as-a-system/) **Summary:** One base unit for each attribute, with every other unit ten times bigger or smaller **Question:** Why does the same word, kilo, show up on a scale at the veterinarian's and on a road sign, and what makes the metric units fit together so neatly? ## Why this exists Lenka works at a veterinarian's office. A beagle steps onto the floor scale and the screen reads $11$ kg. On her drive home, a road sign says the next town is $11$ km away. The two signs share a number and a letter k, and the k means the same thing on both: a thousand. Eleven kilograms is eleven thousands of grams. Eleven kilometers is eleven thousands of meters. That shared k is the whole idea of the metric system. The customary units grew up one at a time, each with its own odd number to the next: $12$ inches make a foot, $3$ feet make a yard, $16$ ounces make a pound. Each one has to be remembered on its own. The metric system was planned instead, as one system: one main unit for each thing you measure, every other unit made from it by tens, and the same short word naming the same size every time. Learn the pattern once and it works for length, mass and liquid volume alike. This lesson takes two days. Day 1 (Examples 1 to 3) is the base units, which attribute each one measures, and how to choose a unit of a sensible size. Day 2 (Examples 4 to 7) is the pattern of tens that ties the units together, and how it compares with the customary units. ## Definition ### Day 1: one base unit for each attribute An **attribute** is the feature of an object or event that you measure: how long it is, how much matter it has, how much liquid it holds, how long it takes. A unit is the amount you count in. The **metric system** is a system of units in which ### [Time and money](https://math.wiki/time-and-money/) **Summary:** Reading clocks and counting coins. **Question:** A clock face repeats twice a day and a dime is smaller than a nickel. How do you read the time, count money, and work with both? ## What this area is about Time and money are the two measurements children handle every day. Time is read from clocks and grouped into units that go by sixties and twenty-fours, not tens. Money is counted by worth, not by how many coins, and is written as a decimal. These are two separate strands that share an area. ## How these fit together **Reading clocks.** Telling time, then to the nearest five minutes. Morning and afternoon and twenty four hour time say which half of the day a time is in. **Units of time.** Units of time nest seconds up to years; calendars and dates group days into weeks, months and years. **Working with times.** Elapsed time finds how long from one time to another; time zones shift a time from one place to another. **Money.** Money gives each coin and bill its worth. Counting mixed coins and money story problems add and subtract amounts. Money as decimals writes four dollars and seventy-five cents as one decimal, $\text{\textdollar}4.75$, and currency exchange trades one country's money for another at a rate. ## A path through it Time: telling time first. Then five minutes, morning and afternoon, units of time and elapsed time, each building straight on telling time. Calendars follow units of time; twenty four hour time follows elapsed time; time zones come last. Money: money first. Mixed coins lead to story problems; decimals lead to currency exchange, which also needs proportions. ### [Time zones](https://math.wiki/time-zones/) **Summary:** The world is cut into zones whose clocks differ by set amounts, mostly whole hours but in some places half or three quarters of an hour **Question:** When it is lunchtime for you, it is the middle of the night for a friend on the other side of the world. How do you work out what time it is for them? ## Why this exists Baraka lives in Nairobi, in Kenya. Hoku lives in Honolulu, in Hawaii. Mitra lives in Kolkata, in India. They are pen pals, and they want to talk on a video call. When Baraka says "let's call at $19{:}00$", he means $19{:}00$ on his clock. On Hoku's clock it is a very different time. So how do they find a time that works for everyone? The reason is the sun. The Earth turns once a day, from west to east. So the sun rises in the east and seems to travel across the sky toward the west. **Noon** at a place is meant to be the time when the sun is highest in the sky there. When the sun is highest over Nairobi, it has not yet climbed high over places to the west, and it has already passed its highest point for places to the east. So noon cannot happen everywhere at once. Each place needs its own clock time, and the world needs a plan so that those clocks fit together. That plan is the system of time zones. These times are written in the twenty-four hour way, which gives each hour of the day its own number from $0$ to $23$ (twenty-four-hour-time). ## Definition A **time zone** is a region where everyone sets their clocks to the same time. Every zone is measured from one reference clock called **Coordinated Universal Time**, or **UTC** for short. Think of UTC as the clock the whole world agrees to count from. In winter, the clocks in London show UTC exactly. A zone's **offset** is how many hours its clocks are ahead of UTC or behind it. "UTC $+3$" means $3$ hours ahead of UTC. "UTC $-5$" means $5$ hours behind UTC. ### [Twenty four hour time and crossing midnight](https://math.wiki/twenty-four-hour-time/) **Summary:** Giving every hour of the day its own number so no morning or afternoon label is needed **Question:** The fire station's log says a call came in at "7:20". Was that in the morning or the evening, and how long is a night shift that starts on one day and ends on the next? ## When you reach for it Bogdana, Talara and Dakota work the night shift at a fire station. The shift starts in the evening and ends the next morning. Every call, every truck check and every meal goes into the station log with the time it happened. A clock with hands goes around twice a day, so "$7{:}20$" could be twenty past seven in the morning or twenty past seven in the evening. On a twelve-hour clock the label a.m. or p.m. says which half of the day you mean (am-and-pm). A label is easy to forget, and in a fire station log a forgotten label could put a call twelve hours off. So the station writes times on the **24-hour clock**, which gives every hour of the day its own number. Then no label is needed. The shift raises a second question. Bogdana starts at $22{:}15$ and goes home at $06{:}40$. The end time looks smaller than the start time, yet she worked most of the night. Finding how long a stretch of time lasts is elapsed-time: count from the start to the end. This lesson shows how to do that when the count runs through midnight into the next day. Reach for this lesson when a time is written on the 24-hour clock, when you need to change a time between the two clocks, or when a stretch of time crosses midnight. ## The procedure ### Part 1: every hour its own number On the 24-hour clock, the hour number is **the number of whole hours since midnight**. The day starts at midnight, written $00{:}00$. One hour later it is $01{:}00$, then $02{:}00$, and so on. Noon, twelve hours after midnight, is $12{:}00$. The clock does not start again at $1$ after noon; it keeps counting. One hour after noon is $13{:}00$, two hours after noon is $14{:}00$, and the last minute of the day is $23{:}59$. One minute after that it is midnight again, $00{:}00$ of the next day. ### [Units of time](https://math.wiki/units-of-time/) **Summary:** Seconds, minutes, hours, days, weeks, months and years, and how they nest inside one another **Question:** The movie runs $3$ hours, but the streaming app counts in minutes. How do you say the same amount of time in another unit, when the steps between the units are sixties and twenty-fours instead of tens? ## Why this exists Lowanna is planning a movie night, and Jorge and Mert are planning the sleepover that follows it. The popcorn needs a few minutes in the microwave, which counts in seconds. The movie runs a few hours, but the streaming app lists it in minutes. The sleepover lasts more than a day, and it is two weeks away. Time comes in sizes from a blink to a lifetime, and no single unit fits them all. Counting a movie in seconds gives a number too big to picture; counting popcorn time in hours gives a tiny part of one. So time has a family of units, each one made of a set number of the next smaller one. To compare two amounts of time, or to give one in the unit someone asked for, you change it from one unit to another. The amount of time stays the same; only the unit you count it in changes. ## Definition The **units of time** are the second, the minute, the hour, the day, the week, the month and the year. Each one is made of whole groups of a smaller one: | Fact | Read it as | |:-:|:-:| | $1$ minute $= 60$ seconds | sixty seconds make a minute | | $1$ hour $= 60$ minutes | sixty minutes make an hour | | $1$ day $= 24$ hours | twenty-four hours make a day | | $1$ week $= 7$ days | seven days make a week | | $1$ year $= 12$ months | twelve months make a year | | $1$ year $= 365$ days | except a leap year, which has $366$ | The units **nest**: a week holds days, a day holds hours, an hour holds minutes, and a minute holds seconds, like boxes inside boxes. A month is the one unit with no single size: it has $28$, $29$, $30$ or $31$ days, which is the business of calendars-and-dates. ### [Units, quantities and a sensible level of accuracy](https://math.wiki/reasoning-with-units/) **Summary:** Using units to set up and check a multi-step problem, choosing quantities to model a situation, and reporting a result to a sensible accuracy. **Question:** A problem hands you numbers with units attached. How do the units tell you what to multiply, what to divide, whether the answer can be right, and how many of its digits to believe? ## When you reach for it Most problems about the real world arrive as a handful of quantities, each a number with a unit: $14$ turns per minute, $6.4$ liters per hundred kilometers, $86{,}000$ people on $412$ square kilometers. The question asks for something in a different unit: turns in a year, kilometers on one liter, people per square kilometer of dry land. Three decisions stand between the givens and the answer: **which quantities matter**, **how to combine them** (mostly whether to multiply or divide, and which way up to write each fraction), and **how many digits the answer deserves**. Units help with all three: they show which way up a fraction must go, they show at once when a set-up does not produce the quantity asked for, and they separate exact numbers from measured ones. It builds on conversion factors and on significant figures, and it is the working method behind every rate in rates from measurements. ## The procedure ### Units multiply, divide and cancel like letters A quantity such as $80\text{ km/h}$ is a number times a unit, and the unit "kilometers per hour" is a quotient, $\frac{\text{km}}{\text{h}}$. Units obey the same rules as letters in algebra: they multiply, divide, and cancel when the same unit appears above and below a fraction bar. Driving $3$ hours at $80\text{ km/h}$: $$3\text{ h} \times \frac{80\text{ km}}{1\text{ h}} = 240\text{ km}$$ The hours cancel, leaving kilometers, which is what a distance is measured in. Dividing instead, $3\text{ h} \div \frac{80\text{ km}}{1\text{ h}}$, would give the unit $\frac{\text{h}^2}{\text{km}}$, which measures nothing anybody wants. The arithmetic gives no warning; the unit does. ### [Volume and capacity](https://math.wiki/volume-and-capacity/) **Summary:** How much space something takes up, or holds. **Question:** How much space does a solid take up, how much liquid does a container hold, and how do you find either without filling it? ## What this area is about Volume is how much space a solid takes up; capacity is how much a container holds. Liquid is measured by filling a unit container, such as a liter, and a solid by packing it with unit cubes. For a box, packing becomes a formula: three edge lengths give the volume. ## How these fit together **Liquid amounts.** Volume does not change when it is poured shows that water in a taller glass is not more water. Liters and milliliters and cups, pints, quarts and gallons are the units. **Volume of solids.** Volume as filling defines volume by counting unit cubes. Volume with unit cubes counts them in layers, and the volume of a box turns that into length times width times height, $V = lwh$. **Extending the formula.** Fraction edge lengths shows $V = lwh$ still holds when the edges are fractions. Solids made of boxes split a shape into boxes and add, or subtract a hollow part. **A different measure.** Surface area covers the outside of a solid, and two boxes with the same volume can need different amounts of it. ## A path through it Pouring needs only comparing two objects, so it can come first. Then liters and milliliters, then customary units and volume as filling. Unit cubes follow, also needing the area of a rectangle, then the box formula. Fraction edges, composite solids and surface area all come after it. Next: density, flow rate and converting cubic units. ### [Volume as filling](https://math.wiki/volume-as-filling/) **Summary:** How much space a solid takes up, found by filling it with copies of a unit cube **Question:** A building block, a jug of water and an oddly shaped stone all take up space. How do you say how much, and how do you compare them? ## Why this exists Length says how long something is. Area says how much flat space a shape covers, and you find it by covering the shape with unit squares and counting them. Neither one tells you how much room a solid thing takes up. A tall thin tower of blocks and a low wide wall of blocks can use exactly the same blocks, so their heights and their floor areas say nothing about which is "bigger". Volume answers that question. It measures space in three directions at once, and it is found the same way area is: choose a small piece of space as the unit, then count how many copies of it fill the solid. The piece is a cube. The same idea covers two other everyday questions. How much does a jug or a toy chest **hold**? That is the volume of the empty space inside it. And how much space does a stone take up, when its bumpy shape could never be filled with cubes? Sinking it in water turns that question into a reading on a measuring jug. This lesson takes the three in turn, one part each. ## Definition A **unit cube** is a cube whose edges are each $1$ unit long. Its volume is $1$ **cubic unit**. A cube $1$ centimeter on each edge is a cubic centimeter; a cube $1$ inch on each edge is a cubic inch. The **volume** of a solid is the number of unit cubes that fill it exactly, with **no gaps** and **no overlaps**. A solid that $10$ unit cubes fill has a volume of $10$ cubic units. The **capacity** of a container is the volume of the space inside it: how much it can hold. Liquids are measured in liters and milliliters: ### [Volume does not change when it is poured](https://math.wiki/volume-stays-the-same-when-poured/) **Summary:** The same water looks taller in a thin glass and shorter in a wide one, but there is just as much **Question:** When water is poured into a glass of a different shape, does the amount of water change? ## Why this exists Pour water from a short cup into a tall thin glass and it rises higher; pour it into a wide dish and it spreads out low. A young child usually judges "how much" by how high the water comes, so the thin glass looks like more. It is not: no water was added and none was taken away, so there is just as much as before. These three days let the child see that for themselves, and the tool that settles every argument is the same one each time: pour it back into the cup it came from. The child should already be able to compare two things side by side. Use plain words all week, "how much water", "more", "less", "just as much", and no measuring units. ## Day 1: The thin glass **You need:** two clear plastic cups of the same size (A and B), one tall thin clear glass, a pitcher of water, a towel. A drop of food coloring makes the water easy to see. Fill cup A and cup B to the same height while the child watches. Put them side by side and ask, "Is there more water in A, more in B, or just as much?" Let the child check the tops of the water against each other until they agree it is just as much. Everything that follows depends on this starting agreement. Now let the child pour cup A into the thin glass. Set the thin glass next to cup B and ask the same question: "Which has more water now, or is it just as much?" Many children point to the thin glass because the water is higher. Do not tell them they are wrong. Ask, "Did we add any water? Did we spill any?" Then say, "Let's find out. Pour it back into cup A." Put A beside B again. The water is at the same height as before (a drop or two left in the glass makes no difference you can see), so it was just as much the whole time. Ask, "So when it was in the thin glass, was there more water, or did it only look taller?" A child who still is not sure has not failed; pour it back and forth a few more times, or come back to it tomorrow. ### [Volume of a box](https://math.wiki/volume-of-a-rectangular-prism/) **Summary:** Finding how much space a rectangular box holds from its three edge lengths **Question:** You can measure the three edges of a box with a ruler. Is that enough to know how much it holds, without filling it? ## When you reach for it Masumi is renting a storage unit. Sanjay is buying an ice chest. Zacarias loads a shipping container at a dock. Each of them wants to know the same thing: how much space is inside? Nobody is going to fill a storage unit with foot cubes to find out. In volume-with-unit-cubes the volume of a box was found by counting cubes in layers: the cubes in one layer, times the number of layers. This lesson notices that those two numbers can be read straight off the box's edges. Then a ruler is enough. You reach for this move whenever you know the edges of a box and want its volume, or you know the volume and want an edge. The lesson runs over four days. Day 1 turns layers into the rule $V = l \times w \times h$. Day 2 writes the same rule as base area times height, and shows that any face of the box can be the base. Day 3 runs the rule backward to find a missing edge. Day 4 uses it in real problems, and compares boxes that hold the same amount but have different shapes. ## The procedure ### Day 1: from layers to length times width times height A **box** here means a solid with six flat rectangular faces that meet at square corners, like a shoebox or a brick. Mathematicians call it a **rectangular prism**. Every box has three kinds of edge: its **length** $l$, its **width** $w$ and its **height** $h$. Say the edges are whole numbers of some unit, such as feet. Pack the box with foot cubes. ### [Volume of a box with fraction edge lengths](https://math.wiki/volume-with-fractional-edge-lengths/) **Summary:** Packing a box with unit-fraction cubes shows that length times width times height still gives the volume when the edges are fractions. **Question:** A box is $2\tfrac{1}{2}$ inches long, so whole unit cubes no longer fit inside it. How do you find how much it holds? ## When you reach for it A box with whole-number edges can be filled with whole layers of unit cubes, and counting them gives the rule for the volume of a box: length times width times height. A box $4$ by $3$ by $2$ has $4 \times 3 = 12$ unit cubes in each layer and $2$ layers, so it holds $12 \times 2 = 24$ unit cubes and its volume is $24$ cubic units. Real boxes rarely measure whole units. A fudge box may be $1\tfrac{1}{2}$ inches deep, and a terrarium $1\tfrac{1}{4}$ feet wide. Unit cubes no longer fit exactly, so the counting that made $V = lwh$ true seems to break down. Reach for this lesson whenever a box has an edge that is a fraction or a mixed number. It shows two things. First, such a box can still be packed exactly, with smaller cubes whose edge is a unit fraction such as $\tfrac{1}{2}$ or $\tfrac{1}{4}$, and the count gives the volume. Second, that count always agrees with length times width times height worked out by multiplying fractions, so the rule you already know keeps working, and so does its shorter form $V = Bh$. Once that is settled, a missing edge can be found from the volume by division. ## The procedure ### Part 1: packing with smaller cubes **What a small cube is worth.** Cut a unit cube, a cube with edge $1$, in half along its length, its width and its height. You get $2 \times 2 \times 2 = 8$ small cubes, each with edge $\tfrac{1}{2}$. Eight equal pieces make one cubic unit, so each piece is $\tfrac{1}{8}$ of a cubic unit. ### [Volume of a solid made of boxes](https://math.wiki/volume-of-a-composite-solid/) **Summary:** Splitting a stepped solid into boxes and adding, or taking a hollow part away **Question:** A step stool, a medal podium and a planter box are not single boxes. How do you find how much space each one takes up, or holds? ## When you reach for it Seojun is building a wooden step stool. Pooja is measuring the podium where the medal winners will stand. Wapi is filling a planter box with soil. Each of them wants a volume, and none of the three things is a single box. volume-of-a-rectangular-prism gives the volume of one box: length times width times height. This lesson handles solids that are made of several boxes. There are two moves. You can **split** the solid into boxes, find each volume, and add. Or you can see the solid as **one big box with a box-shaped part taken away**, and subtract. You reach for this whenever a solid has steps, notches or hollows, but every face is still a flat rectangle meeting the others at square corners. It is the same thinking as area-of-composite-figures, one dimension up. The lesson runs over three days. Day 1 shows why the volumes of pieces add, and that two different ways of cutting give the same total. Day 2 finds edges the drawing does not label, and handles solids of three boxes, such as a medal podium. Day 3 takes a part away, for a planter box with walls and a hollow step stool, and uses both moves in word problems. ## The procedure ### Day 1: volume adds up Here is Seojun's step stool. It is not a box, because it has a step. But it is made of boxes. Imagine sawing it straight down where the step begins. The cut leaves two boxes: - a tall box A, $6$ in long, $9$ in deep and $12$ in tall; - a low box B, $8$ in long, $9$ in deep and $6$ in tall. ### [Volume with unit cubes](https://math.wiki/volume-with-unit-cubes/) **Summary:** A box holds whole layers of cubes, and each layer holds as many cubes as the base holds squares **Question:** A box is packed full of small cubes, but you can only see the outside. How can you know how many cubes are inside without taking them all out? ## Why this exists Lomasi, Nikhil and Yuna have a tub of wooden cubes, each one inch along every edge, and a stack of small shipping boxes. They want to know how many cubes each box holds. Counting one cube at a time is slow. Worse, once a box is full you can only see the cubes on the outside. The ones in the middle are hidden. In volume-as-filling the **volume** of a solid is the number of unit cubes that fill it. For a box, the cubes are not scattered. They sit in neat rows, and the rows sit in flat layers. That order turns counting into multiplying, and it lets you know how many cubes are inside without seeing them. The lesson runs over three days. Day 1 packs a box with whole layers of cubes. Day 2 names the units volume is measured in, and shows why they are cubed. Day 3 links the cubic centimeter to the milliliter, and shows what happens when the cube changes size. ## Definition A **unit cube** is a cube whose edges are each $1$ unit long. Its volume is $1$ **cubic unit**. The **volume** of a box is the number of unit cubes that pack it exactly, with no gaps and no overlaps. A **layer** is one flat level of cubes that covers the whole bottom of the box and is one cube thick. The unit cube can be any size, and its name comes from its edge: | Unit cube | Edge | Written | Read | |---|:-:|:-:|---| | cubic centimeter | $1$ cm | $1\ \text{cm}^3$ | one cubic centimeter | | cubic inch | $1$ in | $1\ \text{in}^3$ | one cubic inch | | cubic foot | $1$ ft | $1\ \text{ft}^3$ | one cubic foot | | cubic meter | $1$ m | $1\ \text{m}^3$ | one cubic meter | ### [What a unit is, and why we agree on standard ones](https://math.wiki/what-a-unit-is/) **Summary:** A unit is one chosen amount that you count copies of to get a measurement **Question:** Anyone can see that a pencil is longer than a crayon. How can we say how long the pencil is, with a number that someone else, even someone far away, could check? ## Why this exists A child can already tell which of two things is longer by standing them side by side (comparing two objects directly). But that only answers "which one?". It cannot answer "how long?", and it cannot help when one of the things is a table that will not move. A number can travel where the table cannot. To get one, we pick one small thing, a cube, a paper clip, a hand, and count how many copies of it fit along the table. That small thing is a **unit**, and counting copies of it is measuring. A count of hand-spans works well for the child who made it. It stops working as soon as the number has to reach someone else: a grandparent's hand is a different size, and a shop in another town has never seen the child's hand at all. So after saying what a unit is, this lesson has three more parts. It looks at the everyday things children measure with first (**non standard units**), at the one rule that makes any of them fair (**why units must all be the same size**), and at why people agreed long ago on units that are the same for everyone (**why standard units exist**). ## Definition ### What a unit is A **unit** is one fixed amount that we choose to measure with. For length it might be the length of one cube, one paper clip or one hand-span. "Fixed" means every copy of the unit is the same size: every cube in the box, every paper clip from the same packet. To **measure the length** of an object with a unit, lay copies of the unit along the object, end to end: ### [What measurement is](https://math.wiki/measurement-ideas/) **Summary:** Putting a number on something by picking a unit and counting how many copies of it fit **Question:** What does it actually mean to measure something, and how much can you trust the number you get? ## What this area is about To measure is to pick a unit and count how many copies of it fit. Before that comes deciding what you are measuring and comparing without numbers. After it comes a harder truth: no tool gives the exact amount, so every measurement is approximate. ## How these fit together **Comparing without numbers.** Attributes you can measure names length, mass and capacity as separate things. Comparing two objects lines them up; comparing with a middle object and balancing to compare mass extend it. **Units.** What a unit is and iterating one unit turn comparing into counting. Smaller units give bigger counts. Reading a scale between marks and benchmarks help read a tool and choose a unit. **How exact.** Precision of a tool, measurement error and significant figures say how far off a reading can be and how many digits to keep. **Using measurements.** Story problems with grams and liters and word problems with mixed units put measured amounts into arithmetic. ## A path through it Attributes, direct comparison, then the middle object and the balance. Then what a unit is, then iterating. After iterating: unit size and count, and precision, then error, then significant figures. Reading a scale follows what a unit is; benchmarks follow measuring length. The story problems wait for metric mass and liquid units; the word problems wait for unit conversions. Next: area, the protractor and the thermometer reuse the same unit idea. ### [Why the circle area formula works](https://math.wiki/why-the-circle-area-formula-works/) **Summary:** Thin wedges cut from a circle and interlocked make a near rectangle of radius by half the circumference **Question:** A circle has no straight sides to measure. Why should the space inside it be exactly pi times the radius times the radius? ## Why this exists A rectangle's area comes from counting rows of unit squares, and a parallelogram or a triangle is cut and rearranged until it is a rectangle, or half of one. All of those shapes have straight sides. A circle has none, so it seems no amount of cutting could turn it into a rectangle. Yet the area formula is exact: $$A = \pi r^2$$ The $\pi$ in it is the same number as in the circumference formula $C = 2\pi r$, which counts how many diameters go once around the circle. Nothing obvious links the distance **around** a circle to the space **inside** it. This lesson shows the link: a circle can be cut into pieces that come closer and closer to a rectangle whose sides are the radius and half the circumference. ## Definition The statement this lesson explains is: *The area inside a circle equals the area of a rectangle whose height is the radius and whose length is half the circumference.* In symbols, with $r$ the radius and $C$ the circumference, $$A = \tfrac12 C \times r$$ Because $C = 2\pi r$, half the circumference is $\pi r$, and the rectangle's area is $\pi r \times r = \pi r^2$. A **wedge** is a slice cut from the center to the edge, like a slice of a round cake. Its two straight sides are radii, each exactly $r$ long. Its curved edge is its **crust**, and the crusts of all the wedges together make the whole circumference. ## Intuition Cut a circle into $8$ equal wedges and shade the $4$ in the top half. Lay the plain wedges in a row, points up and crusts on the table, and drop each shaded wedge, point down, into the gaps between them. ### [Word problems with distance, time, volume, mass and money](https://math.wiki/measurement-word-problems/) **Summary:** Story problems with measurements, including fractions and decimals, solved with the four operations and converted to one unit. **Question:** At field day the relay legs are in meters and kilometers, the water coolers in liters and milliliters, and the snacks in dollars and cents. How do you answer a question that mixes them, and how do you know the answer is the right size? ## When you reach for it Radovan, Yindi and Narelle are helping at their school's field day. The relay track is marked in meters, but the long run is posted in kilometers. The water coolers hold liters, and the paper cups hold milliliters. The obstacle course is timed in minutes and seconds. At the snack table, prices are in dollars and cents. Every question they ask that day is a story with measurements in it. How far does the relay team run in all? How many cups does a cooler fill? Who finished the course faster? How much change comes back from a five-dollar bill? You reach for this skill whenever a story gives you measurements, asks for a new measurement, and the two do not come in the same unit. The conversions themselves are not new. Changing one unit to another is in unit-conversion-customary and unit-conversion-metric, and the units are described in the-customary-system, customary-weight-units, customary-capacity-units and units-of-time. Story problems that keep every amount in one unit are in mass-and-liquid-volume-problems. What is new here is putting the pieces together: reading what a story is asking, getting every amount into one unit, choosing the operation from what happens in the story, and checking that the answer makes sense. These are the facts used in this lesson. Each one is exact. | Measure | Facts | |:-:|:-:| | Length | $1$ km $= 1{,}000$ m, $1$ m $= 100$ cm, $1$ mile $= 5{,}280$ feet, $1$ yard $= 3$ feet, $1$ foot $= 12$ inches | | Mass and weight | $1$ kg $= 1{,}000$ g, $1$ pound $= 16$ ounces | | Liquid volume | $1$ L $= 1{,}000$ mL, $1$ gallon $= 4$ quarts, $1$ quart $= 2$ pints, $1$ pint $= 2$ cups | | Time | $1$ hour $= 60$ minutes, $1$ minute $= 60$ seconds | | Money | $1$ dollar $= 100$ cents | ## Number ### [Absolute value](https://math.wiki/absolute-value-as-distance/) **Summary:** How far a number sits from zero, with no regard for which side. **Question:** A valley floor at $-72$ meters and an account balance of $-45$ dollars both carry a minus sign, yet each has a size you can say out loud, 72 meters down and 45 dollars owed. How do you measure how far a number is from zero, whichever side it is on? ## Why this exists A mountain hut stands at $1{,}850$ meters above sea level, and the lowest point of a desert valley lies at $-72$ meters. An account that is overdrawn shows a balance of $-45$ dollars. Each number says two things at once: a side of zero (above or below sea level, money held or money owed) and a size (how far up, how far down, how much). Very often only the size matters. Which place is closest to sea level? Who owes the most? Whose score is furthest from zero? "Greater" cannot answer these, because "greater" means further right on the number line, and $-72$ is less than $-38$ even though the valley floor is further from sea level than a spring at $-38$ meters. A separate measure is needed: the size of a number with its side set aside. That measure is the **absolute value**. ## Definition The **absolute value** of a number is its distance from $0$ on the number line. It is written by putting the number between two upright bars, so $\lvert -6 \rvert$ is read "the absolute value of negative six". $$\lvert -6 \rvert = 6 \qquad \lvert 6 \rvert = 6 \qquad \lvert 0 \rvert = 0$$ A distance counts units and has no direction, so an absolute value is never negative. The definition works for every number on the line, fractions and decimals included: $\left\lvert -\tfrac{2}{7}\right\rvert = \tfrac{2}{7}$ and $\lvert -1.9 \rvert = 1.9$. A minus sign written **outside** the bars is not part of the absolute value. In $-\lvert 6 \rvert$ the bars are worked first, giving $6$, and then the minus sign takes the opposite, so $-\lvert 6 \rvert = -6$. The bars group what is inside them, just as parentheses do. ### [Acting out add and take-away stories within 10](https://math.wiki/story-problems-within-10/) **Summary:** A short story about joining or taking away is acted out with objects or fingers, drawn, and told as an equation. **Question:** How does a child turn a little story about things coming or going into something to act out, a drawing, and an equation? ## When you reach for it A story problem is a tiny story with a number question at the end: some things are there, some more come or some go away, and we ask how many there are now. A child solves it by acting it out with objects or fingers, then by drawing it, and finally by telling it as an equation such as $4 + 3 = 7$. This week moves through those three steps, and ends with children making up their own stories to match an equation. It rests on adding as putting together and subtracting as taking away. Keep every number at $10$ or below. ## Day 1: Acting out stories where more come **You need:** a strip of tape on the floor for a line at the drinking fountain, a sheet of paper with a pond drawn on it (the story mat), about $10$ counters. Ask $2$ children to stand in line at the tape. Tell the story slowly: "Two children are waiting for a drink. Five more children come to wait." Five more children walk over. Ask, "How many children are in line now?" Let the class count: $7$. Then move the story to the mat. Say, "Four fish swim in the pond," and have the child put $4$ counters on the pond. "One more fish swims in." The child slides $1$ more counter on. Ask, "How many fish now? Did the number of fish get bigger or smaller? Why?" The answer you want: bigger, because more fish came. **Practice.** Act out each story with counters, then answer. 1. $1$ bug sits on a leaf. $3$ more bugs land. How many bugs now? *Answer:* $4$. 2. $8$ apples are in a basket. Dad puts in $1$ more. How many apples now? *Answer:* $9$. 3. $6$ children sit on the rug. $1$ more sits down. How many now? *Answer:* $7$. 4. Use fingers: $3$ cats sit on a porch (hold up $3$), and $3$ more come (hold up $3$ on the other hand). How many cats? *Answer:* $6$ fingers are up, so $6$ cats. ### [Adding a one-digit number to a two-digit number by making a ten](https://math.wiki/add-within-100-making-a-ten/) **Summary:** Adding a one-digit number to a two-digit number such as 38 + 5 by filling up to the next ten, then adding what is left. **Question:** There are $47$ bees in a hive and $6$ more fly in. The $7$ ones and the $6$ ones make more than nine, so the tens change too. How do you find the total without counting on one bee at a time? ## When you reach for it The child already adds a one-digit number to a two-digit number when the ones stay small. In Adding two-digit numbers without regrouping, $43 + 5$ is $4$ tens and $3 + 5 = 8$ ones, so $48$. The ones join the ones and the tens are not touched. Now the ones get too big. There are $47$ bees in a hive and $6$ more fly in. The ones are $7$ and $6$, and $7 + 6 = 13$. Thirteen ones cannot sit in the ones place, because a ones digit goes only up to $9$. Ten of those ones are really a new ten. So the tens change too, and the answer is not "$4$ tens and $13$ ones" but $5$ tens and $3$ ones, which is $53$. This lesson is about that kind of sum: a two-digit number plus a one-digit number, where the ones make a new ten. The move that handles it is one the child already knows from Making ten and bridging through ten. There, $8 + 5$ was found by filling the $8$ up to $10$ and adding what was left. Here, $47 + 6$ is found by filling the $47$ up to the next ten, $50$, and adding what is left: $$47 + 6 = 47 + 3 + 3 = 50 + 3 = 53$$ The child needs three things first. The bonds to ten, so that $7$ is seen at once as needing $3$ (the bonds to ten). Splitting a small number into two parts, so that $6$ is $3$ and $3$. And Tens and ones, so that $50$ and $3$ more is fifty-three with no counting. The pictures are the same as before: ten frames, bundles of ten sticks or base-ten drawings of rods and loose squares, an open number line, and a hundred chart. ### [Adding and subtracting decimals](https://math.wiki/add-decimals-to-hundredths/) **Summary:** Totals where the places must line up before they are combined. **Question:** The rain gauge caught 3.4 cm of rain on Monday and 1.25 cm on Tuesday. Before you add them, which digit goes with which? ## When you reach for it The school weather station has a rain gauge: a clear tube marked in centimeters. Each morning one of the class reads how deep the water stands, to the nearest hundredth of a centimeter, writes it in the log and empties the tube. A week of the log might read $3.4$, $1.25$, $0$, $0.6$ and $2.08$. Two questions come up every week. **How much rain fell in all?** That is a sum. **How much more fell on one day than another**, or how much more is needed to reach some amount? That is a difference. Both are the whole-number addition and subtraction you already know, with one new demand: the digits have to be matched by **place**, not by where they happen to end. You reach for this skill whenever decimal amounts of the same unit are combined or compared: rainfall, lengths, masses, times on a stopwatch, and the prices in money written as decimals. It rests on decimal place value, on comparing decimals and on multi-digit subtraction. ## The procedure The lesson is taught in four parts, one a day. ### Part 1: line up the places In $3.4$ the $4$ means four tenths. In $1.25$ the $2$ means two tenths and the $5$ means five hundredths. Adding means putting together amounts of the **same** kind: ones with ones, tenths with tenths, hundredths with hundredths, just as $3$ tens and $5$ tens make $8$ tens. That is why tenths and hundredths written as fractions are first rewritten in one unit before they are added. ### [Adding and subtracting integers](https://math.wiki/add-subtract-integers/) **Summary:** Combining signed numbers; subtracting is adding the opposite. **Question:** Temperatures drop below zero and scores fall behind. How do you add and subtract numbers that can be negative? ## What this area is about An integer carries a size and a direction. Adding two of them combines two moves that may point opposite ways, so one can partly cancel the other. Subtracting asks for a difference even when the number taken away is negative. This area makes both work for every pair of integers, not only for whole numbers. ## How these fit together **A procedure.** Adding integers reads each integer as a move along the number line, then as counters that cancel in pairs, and turns both pictures into a rule. When the signs differ, the sum is the larger absolute value minus the smaller, with the sign of the number farther from zero. **A concept.** Subtracting integers shows that $a - b = a + (-b)$ for all integers $a$ and $b$, so every subtraction becomes an addition. It then uses differences to find distances on the number line and changes in height and money. ## A path through it Adding first, then subtracting, since the second turns every subtraction into a sum of the first kind. Finish here and you are ready for multiplying and dividing integers, order of operations with negative numbers, combining like terms with negative coefficients, and temperature changes below zero. ### [Adding and subtracting within 20](https://math.wiki/add-within-20/) **Summary:** Fluent facts, using ten as a landmark. **Question:** A child can find any sum up to twenty by counting from one. What ways of thinking let them know it instead, and why is ten at the center of most of them? ## What this area is about Every sum and take-away up to $20$, such as $8 + 5$ or $13 - 5$, can be found by counting. This area is about knowing them instead: from a picture, from a known fact, and above all from ten as a landmark. Each way is a reason the answer must be right. ## How these fit together **The first facts.** Adding within five and ten starts from seeing small groups at a glance, then uses five, bonds and doubles to reach ten. Subtracting within ten reads those facts backwards. **Counting, but less of it.** Counting on and counting back start from one part or the whole. **Facts from facts.** Doubles and near doubles: $6 + 6 = 12$ turns into $6 + 7 = 13$. **Using ten.** Ten more and ten less: only the tens change. Making ten finds $8 + 5 = 8 + 2 + 3 = 13$ and $13 - 5 = 13 - 3 - 2 = 8$. **All of them at once.** Facts within 20 from memory matches each fact with its quickest route, so every sum of two one-digit numbers, and every subtraction that undoes one, comes by recall or a short route in the head. Adding three numbers lives with the properties of addition. ## A path through it Adding within ten, then subtracting, counting on and back, and doubles. Then ten more and ten less, and making ten. Last, once fact families are known too, all the facts from memory. This opens adding two-digit numbers, the first step of multi-digit addition. ### [Adding and subtracting within 20 from memory](https://math.wiki/fluency-within-20/) **Summary:** Every sum and difference within 20 is known quickly, by recall or by a fast mental strategy. **Question:** There are a hundred sums of two one-digit numbers. How does a child come to know every one of them, and every subtraction that undoes one, without counting? ## When you reach for it Every adding and subtracting job from here on is built out of small facts. $46 + 38$ needs $6 + 8$ and $4 + 3$; $52 - 27$ needs $12 - 7$. A child who has to count out $6 + 8$ on fingers in the middle of a bigger problem often loses track of the bigger problem. So the aim now is that every sum of two one-digit numbers, and every subtraction that undoes one, is **known from memory**. "From memory" has two honest forms. Some facts are simply recalled: asked $5 + 5$, the child says $10$, and nothing happens in between. Others arrive by a route so short it runs in the head in a moment: asked $9 + 6$, the child thinks "nine needs one to make ten, five more, fifteen". Both count. What does not count yet is counting by ones. It is correct and always allowed, but it is slow, and $8 + 7$ by ones has seven steps to slip on. Routes that are used often turn into recall on their own. The routes were each taught in a lesson of their own: counting on, doubles and near doubles, making ten, the bonds of ten, fact-families, and subtraction as a missing part. This lesson does not teach them again. It puts all the facts in one picture, matches each fact with a route that suits it, shows how one known fact reaches its neighbors, and turns every subtraction into an addition the child already knows. ## The procedure The lesson has four parts, one for each day: the whole set of facts, choosing a route, reaching a fact from one you know, and subtracting by thinking of adding. A short note at the end says how recall is built. ### [Adding integers](https://math.wiki/add-integers-with-different-signs/) **Summary:** Totals where the two numbers partly cancel each other out. **Question:** An elevator three floors below the street goes up seven floors. Where does it stop, and how do you add two numbers that pull in opposite directions? ## When you reach for it Many quantities come with a direction. A temperature can be above zero or below it, a game score can be points ahead or points behind, a floor can be above the street or below it. The integers record both the size and the direction in one number: $5$ for five above, $-5$ for five below. As soon as two of these quantities are combined, you need to add integers. When both numbers point the same way, adding them is the addition you already know, with a sign attached. The new case is when they point opposite ways. An elevator going up $7$ floors from level $-3$, a player on $-27$ points who wins a hand worth $50$, a cave explorer at $-90$ meters who climbs $35$ meters: in each one, the second number partly undoes the first. The sum is what is left after the two cancel as far as they can. This lesson takes three days. **Part 1** reads a sum as a move along the number line. **Part 2** shows the same sums with counters, where the canceling can be seen as pairs that are worth nothing. **Part 3** turns both pictures into a way of adding any integers without drawing, using absolute value, and uses it on sums of several numbers, expressions and problems. ## The procedure ### Part 1: adding as moves on the number line Read an integer as a move along the number line. A positive number is a move to the right, a negative number a move to the left, and the size of the number, its absolute value, is how far the move goes. So $+4$ means "four to the right" and $-4$ means "four to the left". ### [Adding on an open number line](https://math.wiki/open-number-line-addition/) **Summary:** Jumping by tens and then by ones from one number up to the total. **Question:** The model railroad club has $38$ pieces of track laid and $45$ more in the box. Counting on forty-five times is slow, and no ruler or marked line is at hand. How can a quick drawing find the total? ## When you reach for it A child who adds two two-digit numbers needs a way to keep track. Counting on by ones from $38$ for forty-five counts is slow and a slip is easy. A ruler or a printed line would help, but it is rarely at hand, and a printed line from $0$ to $100$ is crowded. An **open number line** answers this. It is a plain line with no marks drawn ahead of time. The child marks only the number they start from, then draws a jump for each piece they add, and marks only where each jump lands. Nothing is to scale: a long jump means "a lot" and a short jump means "a little". The labels on the jumps carry the numbers, not the lengths. The lesson Number lines with different scales read lines with evenly spaced marks. An open line keeps the idea that bigger numbers sit to the right, and drops the spacing. The child has also met it twice already: hops of ten in Adding two-digit numbers without regrouping, and a stop at the next ten in Adding a one-digit number to a two-digit number by making a ten. This lesson puts those moves together for any two numbers whose total is at most $100$, including sums where the ones cross a ten, such as $38 + 45$. Reach for it when: - the child adds two two-digit numbers in their head and wants a record of each step; - the ones run past nine and a stop at a ten will help; - a number to add is close to a ten, such as $29$ or $48$, and a whole ten is easier to jump; - the question is "how many more?", which is a jump up from one number to another. ### [Adding two-digit numbers without regrouping](https://math.wiki/add-two-digit-numbers-no-regrouping/) **Summary:** Tens add to tens and ones add to ones with no bundling needed. **Question:** A bus has $43$ children on it and $5$ more get on. A parade float has $27$ paper flowers and $40$ more are added. Neither total needs counting one by one. What do you add to what, and why? ## When you reach for it A child who knows the facts within twenty can find $3 + 5$ quickly. Now the numbers get bigger: $43$ children are on a school bus and $5$ more get on. Counting on five from $43$ works, but the same child also meets $27 + 40$, and counting on forty times is slow and easy to get wrong. Both sums have a short way, and it is the same way for both. The short way rests on Tens and ones: a two-digit number is some tens and some ones. $43$ is $4$ tens and $3$ ones. $27$ is $2$ tens and $7$ ones. When we add, we put the tens with the tens and the ones with the ones. - In $43 + 5$, the $5$ is $5$ ones. It joins the $3$ ones, making $8$ ones. The $4$ tens stay as they were. So $43 + 5 = 48$. - In $27 + 40$, the $40$ is $4$ tens. It joins the $2$ tens, making $6$ tens. The $7$ ones stay as they were. So $27 + 40 = 67$. That is the whole idea, and this lesson spends four days making it solid. It covers two kinds of sum: - a two-digit number and a one-digit number, such as $43 + 5$, where the ones add to the ones; - a two-digit number and a whole number of tens ($10$, $20$, $30$ and so on), such as $27 + 40$, where the tens add to the tens. In every sum here the ones stay at $9$ or fewer, so no new ten is made. A sum such as $47 + 6$, where the ones run past nine, is a different move, named under Boundaries. The pictures are the ones the child already knows: bundles of ten sticks (or linking cubes snapped into trains of ten) with some loose ones, base-ten drawings of the same thing, a hundred chart, and an open number line. ### [Adding within five and ten](https://math.wiki/addition-within-10/) **Summary:** Totals up to ten. **Question:** Two groups make ten or fewer altogether. How can a child know the total by looking and thinking, instead of counting every one from the start? ## When you reach for it A child who knows what adding means can already find $3 + 2$ or $6 + 3$: put out one group, put out the other, push them together and count everything from one. That always works, and it is where every child starts. But it is slow, and the longer the count the easier it is to skip a counter or count one twice. This lesson is about the step after that, where the sums with a total of ten or less stop being counted and start being **known**. It happens in two stretches. First come the sums **within five**, a total of $5$ or less. These are small enough to see: two counters and three counters are a picture of five that the eye takes in at once. Past five that stops working. Few children can see eight loose counters at a glance, so the sums **within ten** are reached a different way, by thinking from facts already known: five as a landmark (one hand, one row of a ten frame), the number bond of a whole and its two parts, two equal rows for a double, and a full ten frame for ten. Knowing a fact is not the same as reciting it. A child who knows $3 + 2 = 5$ can see it in a picture, show it on a hand, tell a story for it, and say why it is right, and nothing in this lesson is timed. These sums come up constantly, in games with dice and dominoes, on fingers, in sharing a few snacks, which is exactly where they are best learned. ## The procedure There is no single method for these facts, and there should not be. A child who knows several ways to see a total picks whichever one the numbers make easy. None of the ways below is a list of steps; each is a reason the answer has to be what it is, resting on something already true about joining groups. ### [Adding, subtracting and multiplying complex numbers](https://math.wiki/multiply-complex-numbers/) **Summary:** Expanding as with binomials, then replacing i squared with negative one. **Question:** The square root of negative one is now a number. Can you still add and multiply the way you always have? ## When you reach for it Once $i$ is a number with $i^2 = -1$, the complex numbers $a + bi$ need arithmetic. You reach for it whenever two of them meet: a sum, a difference or a product to put back into the form $a + bi$, or a complex solution from the quadratic formula that you want to check by substituting it into the equation. Almost nothing new has to be learned. A complex number $a + bi$ behaves like a binomial whose "variable" is $i$, and you already know how to add and subtract polynomials and how to multiply two binomials. The one new fact is what happens when $i$ meets $i$: $i^2$ is not a new kind of term but the real number $-1$. ## The procedure **Why the old rules still apply.** The complex numbers are built so that the commutative, associative and distributive properties hold for them exactly as they do for real numbers. So every regrouping and every expansion you would make with $x$ in place of $i$ is allowed here too. The only difference comes at the end: where a polynomial keeps its $x^2$, here $i^2$ is replaced by $-1$. **Adding and subtracting.** The real parts are like terms of each other, and so are the imaginary parts. Reorder (commutative property), regroup (associative property) and collect: $$(a + bi) + (c + di) = (a + c) + (b + d)i$$ To subtract, take away both parts. The minus sign distributes over everything in the parentheses, just as $-(c + dx) = -c - dx$: $$(a + bi) - (c + di) = (a - c) + (b - d)i$$ ### [Addition and subtraction](https://math.wiki/addition-and-subtraction/) **Summary:** Putting together and taking apart. **Question:** Putting groups together and taking them apart look like two jobs. How are they one idea, and what does a child build on it? ## What this area is about Adding and subtracting are a child's first operations, from what the plus and minus signs mean to many-digit sums and differences. One picture threads through it: a whole made of two parts. Adding finds the whole; subtracting finds a part. ## How these fit together **What the signs mean.** Addition as putting together and subtraction as taking away come first. Subtraction gains two meanings: subtraction as the difference ("how many more?") and subtraction as an unknown addend, where $9 - 4$ asks "$4$ and what make $9$?" **The picture that links them.** Number bonds split a whole in two ways. Fact families read one bond as four sentences, or two for a double. **Facts and rules.** Adding and subtracting within 20 builds fluent facts with ten as a landmark. Properties of addition: order and grouping do not change a total. The missing number in an equation is the one that makes both sides the same amount, wherever the box sits: $11 = \square + 4$. **Bigger numbers and stories.** Multi-digit addition and multi-digit subtraction work place by place. Word problems ask which operation a story needs. ## A path through it Joining, taking away, bonds, facts within $20$, then the properties. Comparison comes between taking away and story problems. Missing parts follow the facts within $20$; then fact families, then story problems, and missing numbers in equations. Multi-digit addition follows the facts within $20$; multi-digit subtraction follows it and fact families. ### [Addition and subtraction problems in words](https://math.wiki/addition-and-subtraction-word-problems/) **Summary:** Stories whose structure decides which operation finds the missing amount. **Question:** A story gives some numbers and hides one. How do you tell from the story alone whether to add or subtract to find it? ## What this area is about A story problem never says "add" or "subtract". It says that things joined, left, sat side by side or were compared, and it hides one number. The work is reading the shape of the story: which amounts are parts, which is the whole, and which one is missing. Then the operation follows. ## How these fit together **First stories.** Acting out stories within 10 turns a little joining or taking-away story into objects, a drawing and an equation such as $4 + 3 = 7$. **Where the missing number sits.** Problems with the start, change or result missing gives the end of the story and hides the change or the start, within $20$. Total or result missing gives both numbers and asks how many in all or how many are left. Starting amount missing runs a story backward to its start, with two-digit numbers. **Comparing.** How many more and how many fewer covers stories where nothing moves: two amounts are compared, and the story may hide the gap or either amount. **Tools for harder stories.** Bar models are a drawing that shows parts and whole for any of these stories. Two-step problems need one answer found before the next. ## A path through it Start by acting stories out within $10$, then stories with the change or the start missing. Take total-or-result-missing and start-missing stories next, then comparing stories, then bar models, and finish with two-step problems. From here, stories that mix all four operations open up in problems using several operations. ### [Addition as putting together](https://math.wiki/addition-as-joining/) **Summary:** Two amounts combined make a new total. **Question:** There are some here and some more over there. How many are there when we put them all together? ## Why this exists A child who can count a set of objects can already find out how many there are when two groups are pushed together: push them together and count everything. Addition begins as a name for that act. It turns "three here and four there, how many now?" into something you can say, write down and later work out without touching anything. Once the act has a name and a sign, it can be talked about. $3 + 4 = 7$ is a sentence about any three things joined with any four things: ducks, blocks, children on a bus. That is the first time a child writes a fact that is true about more than the objects in front of them. ## Definition **Adding** is putting groups together and finding how many there are in all. The two amounts being joined are the **parts** and the amount we get is the **total**, sometimes called the **sum**. We write it with two signs: - the **plus sign** $+$ joins the parts. Read $3 + 4$ as "three plus four", or simply "three and four". - the **equals sign** $=$ says that the two sides are the same amount. Read $3 + 4 = 7$ as "three and four is the same amount as seven". Joining happens in two kinds of situation, and children meet both. In one, something **arrives**: there were three ducks and two more swim over. In the other, nothing moves at all: there are four red blocks and three blue blocks, and we ask how many blocks there are **altogether**. Both are the same addition, because both ask for the size of one group made from two. ### [Approximating irrational numbers](https://math.wiki/estimate-a-square-root-between-integers/) **Summary:** Using the perfect squares on either side to bracket the value. **Question:** The square root of 29 is not a whole number, and its decimal never ends. How can you tell how big it is, and where it sits among other numbers? ## When you reach for it A square garden has an area of $90$ square meters. Its side is the positive number that squares to $90$, which is $\sqrt{90}$, and $90$ is not one of the perfect squares. It is not a fraction either: the square root of a whole number that is not a perfect square is irrational, and its decimal never ends or repeats. You can still say how long the side is to the nearest tenth of a meter, and whether it is longer than some other length. Reach for this lesson whenever an irrational number has to become a size you can use: to measure with, to mark on a number line, to compare, or to estimate an expression such as $2 + \sqrt{3}$ or $\pi^2$. The method needs nothing but multiplication, and unlike a calculator it shows why the digits are right. This lesson is taught over two days. Part 1 traps a square root between two integers and then narrows the trap to tenths and hundredths (Examples 1 to 4). Part 2 places irrational numbers on the number line, puts them in order, and estimates expressions built from them (Examples 5 to 7). ## The procedure ### Part 1: trapping a square root **One fact does all the work.** For two positive numbers $a$ and $b$, $$a < b \quad\text{exactly when}\quad a^2 < b^2$$ Squaring keeps positive numbers in the same order. Here is why. If $0 < a < b$, multiply both sides of $a < b$ by the positive number $a$ to get $a^2 < ab$, and by the positive number $b$ to get $ab < b^2$. Together, $a^2 < ab < b^2$. In the other direction, if $a$ were not less than $b$, then $b \le a$, and the same argument with the letters swapped gives $b^2 \le a^2$. So $a^2 < b^2$ can only happen when $a < b$. ### [Arithmetic](https://math.wiki/arithmetic/) **Summary:** The four operations on whole numbers and decimals, and the rules they obey. **Question:** What are the four operations on whole numbers and decimals, and what rules do they obey? ## What this area is about Arithmetic is adding, subtracting, multiplying and dividing. Here the numbers are whole numbers and decimals: what each operation means, the facts and written methods that carry it out, the rules that let a calculation be rearranged, and the agreed order for reading a line with several operations in it. ## How these fit together **The operations.** Addition and subtraction put together and take apart: adding finds a whole, subtracting finds a part. Multiplication and division do the same for equal groups. Decimal arithmetic runs all four with a decimal point, where the new question is where the point goes. **A convention.** The order of operations fixes how to read a line such as $3 + 4 \times 5$: grouping first, then powers, then multiplying and dividing, then adding and subtracting, each pair from left to right. **A habit.** Estimation and mental math gets close quickly, checks whether an exact answer is the right size, and reshapes a problem into an easier one with the same answer. ## A path through it Addition and subtraction first, then multiplication and division. Decimal arithmetic and the order of operations both come after multiplication and division. Estimation and mental math run alongside the operations rather than after them: parts of it use adding or multiplying, and parts of those areas use estimates. Next come fractions and algebra, which writes the rules of arithmetic with letters. ### [Arrays](https://math.wiki/arrays-and-multiplication/) **Summary:** Rows and columns of objects show a product as a rectangle of items. **Question:** Things set out in neat rows and columns, like panes in a window or trees in an orchard, can be counted without counting one by one. What does the shape tell you? ## Why this exists Taavi looks at the big window in the hall and wants to know how many panes of glass it has. He could touch each pane and count, $1, 2, 3, \dots$ That works, but it is slow, and it is easy to skip a pane or count one twice. The window has a shape that helps. Its panes sit in straight rows, one row above another, and every row has the same number of panes. So Taavi can count one row, see that it has $7$ panes, count the rows, see that there are $3$, and know the whole window is $3$ groups of $7$. That is a multiplication, $3 \times 7$, and the window is a picture of it. Objects set out like this are called an **array**. Arrays are everywhere: eggs in a carton, tiles on a floor, trees planted in an orchard, stamps on a page. Each one turns "how many in all?" into "how many rows, and how many in each row?" Two small counts replace one big one. The array matters for more than counting. It is the picture that the rest of multiplication is built on: why the order of the factors does not matter, how a hard product splits into easy ones, and why the area of a rectangle is found by multiplying all come from looking at an array. ## Definition An **array** is a set of objects arranged in **rows** and **columns** so that every row has the same number of objects and every column has the same number of objects. A **row** runs across, from left to right. A **column** runs up and down. ### [Arrays: rows and columns of objects](https://math.wiki/arrays-of-objects/) **Summary:** Objects set out in equal rows and columns are counted by adding the same number again for each row. **Question:** Corn plants or toy blocks set out in equal rows and equal columns can be counted a row at a time. Why do the rows and the columns always give the same total? ## Why this exists Thabo's family grows corn. In spring they plant the seeds in straight rows, the same number of seeds in every row, and each seed comes up as one corn plant. Thabo wants to know how many plants are in the small patch by the gate. He could walk from plant to plant and count one by one, but it is easy to skip a plant or count one twice. The rows help. Thabo can count one row and find $5$ plants. Every row has the same number, so he can add $5$ again for each row instead of counting every plant. Three rows give $5 + 5 + 5 = 15$. One long count has become a short sum in which the same number is added again and again. Objects set out like this, in equal rows that also line up into equal columns, form an **array**. Arrays are all around a child: eggs in a carton, chairs set out for a show, stickers on a sheet, corn in a field, toy blocks pushed into neat lines on a mat. Each one turns "how many in all?" into "how many in a row, and how many rows?" The array matters beyond counting. A sum of the same number again and again is the first step toward multiplying, and the array is the picture that later work on multiplying is built on. Two rows of an array are also the plainest picture of an even number. Read everything here aloud with the child, and let them build it with blocks, counters or linking cubes on a mat or a table. ## Definition An **array** is a set of objects arranged in rows and columns so that every row has the same number of objects and every column has the same number of objects. ### [Bar models for story problems](https://math.wiki/bar-models-for-story-problems/) **Summary:** A bar split into parts shows how the numbers in a story fit together, which shows whether to add or subtract. **Question:** A story gives two numbers and asks for a third. How can one simple drawing show whether to add them or to take one from the other? ## When you reach for it A child who reads a story problem has to decide two things: which numbers belong together, and whether to add or to subtract. With small numbers they can act the story out with counters and watch the answer appear. With numbers like $38$ and $81$ that stops being practical. Nobody lays out $81$ counters. A **bar model** is a drawing that keeps the shape of the story and drops the counting. A long rectangle stands for an amount. Cut it into pieces and each piece is a part. A bracket over the whole bar holds the whole. A question mark goes wherever the story has a number missing. Once the bar is drawn, one question settles the calculation: **is the missing number the whole, or a part?** If the whole is missing, put the parts together and add. If a part is missing, take the known part away from the whole, or count up from the known part to the whole. Children have already met bars inside single kinds of story: a whole and two parts in parts and the whole, the missing change in change-unknown-problems, and two bars side by side in comparison-word-problems. This lesson puts the bar to work as **one tool for every kind** of adding and subtracting story within $100$: putting together, taking apart, adding on, taking away, and comparing. It is taught over four days, one part a day. The arithmetic itself (adding and subtracting two-digit numbers) is done the ways the child already knows; the bar's job is to decide which arithmetic to do. ### [Base ten blocks and trading ten for one](https://math.wiki/trade-ten-for-one/) **Summary:** Swapping ten rods for one flat, or one flat back into ten rods. **Question:** Ten ones make a ten. What do ten tens make, and can the same number be built out of blocks in more than one way? ## When you reach for it In tens-and-ones ten loose sticks were tied into one bundle, a ten, and a two-digit number was read as some tens and some ones. Past $99$ that is not enough. A heap of two hundred sticks would be twenty bundles, and twenty bundles are slow to count. The fix is the same move one place further left: ten tens are packed together into one bigger thing, a hundred. Base ten blocks let a child hold both moves in their hands. A small cube is one. A rod is ten cubes joined in a line. A flat is ten rods joined side by side. A child who swaps ten small pieces for one bigger piece, or one bigger piece for ten smaller ones, is doing the move that every later step of adding and subtracting bigger numbers depends on. Reach for this when a number up to $1{,}000$ has to be built, read off a pile of blocks, or broken into smaller pieces so that some can be taken away. The lesson runs over three days, one part a day. - **Part 1, the blocks and the trade.** What a unit, a rod and a flat are worth, and trading ten of one for one of the next, in both directions. - **Part 2, building a number.** Reading a pile of blocks as a number, and building a number as a pile, including numbers with a zero in them. - **Part 3, one number, many ways.** The same number built with different blocks: $163$ is $1$ hundred, $6$ tens and $3$ ones, and it is also $16$ tens and $3$ ones. A set of base ten blocks is best. Paper cut-outs work too: a small square, a strip of ten squares, a ten-by-ten square. A number cube (a die showing $1$ to $6$) is needed for the trading game, and beads on strings make a second picture of the same idea. Saying and writing three-digit numbers in words is its own lesson, numbers-to-1000, and the table of digit columns is The place value chart. This lesson is about the pieces themselves. ### [Base, exponent and the value of a power](https://math.wiki/base-and-exponent/) **Summary:** The lower number is the repeated factor and the raised one counts how many times. **Question:** When the same number is multiplied by itself again and again, how do you write it short, and how do you work out what it comes to? ## Why this exists Tali hears some news and phones three friends. Each of them phones three new people, and each of those phones three more, and so on. Count the calls round by round. Round one has $3$. Round two has $3 \times 3$, because each of the first three makes $3$ calls. Round three has $3 \times 3 \times 3$. By round twenty the product has twenty threes in it, a line long enough to lose count in. Folding a sheet of paper in half does the same thing with twos. Each fold doubles the number of layers, so after twenty folds the count of layers is twenty twos multiplied together. Multiplying the same number by itself again and again turns up so often that it has its own short way of writing: a small raised number that counts the factors. This lesson is about that notation and about the second question that comes straight after it, which is how large the result is. Multiplication facts are all it needs. ## Definition A **power** is a product in which every factor is the same number. It is written with two numbers: - the **base**, the number being multiplied, written at the usual size; - the **exponent**, a smaller number raised up beside the base, which counts how many factors of the base are multiplied together. $$3^4 = 3 \times 3 \times 3 \times 3$$ Here $3$ is the base and $4$ is the exponent: four factors, each of them $3$. The expression $3^4$ is called a power of $3$, and its value, $81$, is also called a power of $3$: $81$ is the fourth power of $3$. Writing $3^4$ is **exponential form**; writing $3 \times 3 \times 3 \times 3$ is **expanded form**. ### [Calculating in scientific notation](https://math.wiki/multiply-and-divide-in-scientific-notation/) **Summary:** Leading parts combine and the powers of ten add or subtract. **Question:** Two numbers are each written as a small number times a power of ten. How do you multiply, divide, add or subtract them without writing out all the zeros? ## When you reach for it A number in scientific-notation is a **leading part** between $1$ and $10$ times a power of ten, such as $2.5 \times 10^{13}$ or $7.5 \times 10^{-6}$. Writing a number this way makes it short; this lesson is about calculating with such numbers. Reach for it whenever a calculation mixes very large or very small numbers: how far light travels in a year, how many red blood cells are in a body, how many photos fit on a hard drive. Written out, $300\,000 \times 31\,500\,000$ is a long multiplication with many zeros to count. In scientific notation it is $3 \times 3.15$ and $5 + 7$, two small calculations. The work splits into two halves resting on two different facts. - **Multiplying and dividing** use the rules for powers with the same base: powers of ten multiply by adding exponents and divide by subtracting them. Leading parts and powers of ten are handled separately. - **Adding and subtracting** use the distributive property, which needs both numbers to have the **same** power of ten. So the first job is to make the powers match. Deciding which of two numbers is bigger is its own lesson. ## The procedure The lesson is taught over two days. Part 1 is multiplying and dividing. Part 2 is adding and subtracting, choosing a sensible unit, and reading a calculator. ### Part 1: multiplying and dividing **Multiplying.** For any leading parts $a$ and $b$ and any integer exponents $m$ and $n$, ### [Closure](https://math.wiki/closure-property/) **Summary:** A set is closed under an operation when the answer always lands back inside the set. **Question:** Adding two whole numbers always gives a whole number, but $4 - 9$ does not. Which operations can be carried out inside a set of numbers without the answer ever leaving it? ## Why this exists Start with the whole numbers $0, 1, 2, 3, \ldots$ and do arithmetic with them. Add two of them and you get another whole number. Multiply two of them and you get another whole number. Subtract, and sometimes it works, $9 - 4 = 5$, and sometimes it does not: $4 - 9$ has no answer among the whole numbers at all. Figure 1 shows why. Starting at $4$ and moving $9$ to the left runs off the end of the dots. Negative numbers fill that gap. Put each whole number's opposite beside it and you have the integers, and among the integers every subtraction has an answer: $4 - 9 = -5$. Division then breaks in the same way, since $9 \div 4$ is no integer, and fractions fill that gap, giving the rational numbers. Square roots break it once more, because no fraction squares to $2$ (square-roots-as-irrational), and the irrational numbers fill that one. Each new set was made so that an operation which used to throw its answer outside no longer does. That property is called **closure**. ## Definition A set of numbers is **closed under an operation** when carrying out that operation on any two members of the set, including a member with itself, always gives a member of the set. For addition and a set $S$, that reads $$ \begin{gathered} \text{if } a \in S \text{ and } b \in S, \\ \text{then } a + b \in S \end{gathered} $$ for every choice of $a$ and $b$, and the same with $-$, $\times$ or $\div$ in place of $+$. ### [Commutative, associative and distributive properties](https://math.wiki/commutative-property-of-real-numbers/) **Summary:** Addition and multiplication of any real numbers give the same result in either order. **Question:** You swap two terms, move parentheses, or multiply parentheses out. How can you be sure the expression is still worth the same, whatever numbers the letters stand for? ## Why this exists Algebra is mostly rewriting: a sum put in a handier order, parentheses multiplied out, a common factor pulled to the front. Each rewrite promises that the new expression is worth exactly what the old one was, **for every number the letters could stand for**. Trying numbers cannot keep that promise, since a thousand trials leave infinitely many real numbers untested. A short list of laws true of all real numbers at once does keep it, and three of them do most of the work: the **commutative**, **associative** and **distributive** properties. With whole numbers they are counting facts, met in the properties of addition. Here they are stated with letters for every real number, so that every step of a simplification can name the law that allows it. ## Definition The letters $a$, $b$ and $c$ stand for **any** real numbers: whole numbers, negatives, fractions, decimals, and irrational numbers such as $\sqrt{2}$. Each law says two expressions are equal for every choice of them. **The commutative properties: order does not matter.** Two numbers may be added, or multiplied, in either order. $$a + b = b + a \qquad\qquad ab = ba$$ **The associative properties: grouping does not matter.** Adding or multiplying three numbers, it does not matter which two are combined first. The numbers keep their order; only the parentheses move. $$\begin{aligned} (a + b) + c &= a + (b + c) \\ (ab)c &= a(bc) \end{aligned}$$ ### [Comparing and ordering decimals](https://math.wiki/compare-decimals/) **Summary:** Deciding which decimal is larger by comparing the places from the left. **Question:** One gymnast's cartwheel earns 0.7 of a point and another's earns 0.68. The second score has more digits. Which score is higher, and how can you tell without guessing? ## When you reach for it Bhumika, Kahu and Inti train at a gymnastics club, and on weekends they skate at the ice rink. Both sports are scored by judges, and the judges write their scores as decimals. At practice the coach scores each cartwheel from $0$ to $1$ point. Bhumika's cartwheel earns $0.7$ of a point and Kahu's earns $0.68$. Whose cartwheel scored higher? You reach for this skill whenever two or more decimals have to be put side by side: which score wins, which time is shorter, which amount of money is more, or how to list a set of numbers from least to greatest. You already know how to do this for whole numbers: line the numbers up by place and look at the biggest place first (compare-and-order-whole-numbers). Decimals work the same way, once you remember from tenths-and-hundredths what each place after the point is worth. Three symbols record the answer. $a > b$ says $a$ is greater than $b$. $a < b$ says $a$ is less than $b$. $a = b$ says they are the same number. The open side of $>$ and $<$ always faces the greater number (the symbols for comparing). ## The procedure The lesson is taught in three parts, one a day. ### Part 1: compare place by place from the left Put $0.7$ and $0.68$ on hundred grids, where the whole is a square cut into $100$ small squares. Seven full columns is $7$ tenths, which is $70$ small squares. Six full columns and eight more squares is $68$ small squares. ### [Comparing and ordering rational numbers](https://math.wiki/order-rational-numbers/) **Summary:** Place fractions, decimals and negatives on one line and compare them. **Question:** A golf average of one and a half under par, a lake one and a quarter meters below its usual mark, a fraction beside a decimal beside a percent. When numbers arrive in different forms and on both sides of zero, how do you tell which is greater? ## When you reach for it Numbers that describe one situation often arrive in different forms. A golf average might be $-1\frac{1}{2}$ for one player and $-1.3$ for another. A lake gauge might read $\frac{9}{10}$ of a meter above the usual mark one month and $1.15$ meters below it another. A survey might report $\frac{3}{8}$ of one group, $0.42$ of another and $38\%$ of a third. To say which is greater, or to put a whole list in order, the numbers have to be brought onto one line or into one form. Each piece of that is already in hand. Integers are compared by their places on the line (integers-on-a-number-line), negative fractions and decimals have their own places on it (negative-fractions-and-decimals), fractions are compared with each other (compare-fractions) and decimals with each other (compare-decimals), and a fraction, a decimal and a percent can be turned into one another (fraction-to-percent). This lesson joins them, so that any two rational numbers can be compared whatever their form and whatever their sign. It runs over two days. Part 1 compares pairs and orders lists. Part 2 writes and reads inequality statements about real situations, where "greater" has to be turned back into words such as "higher" or "better". ## The procedure ### Part 1: comparing and ordering **The one rule.** On a number line that runs left to right, of two different numbers the one further right is the greater. The statement $-2.3 < 0.55$ says that $-2.3$ lies to the left of $0.55$. This is the rule already used for integers, unchanged. It holds for every rational number because every rational number has exactly one point on the line. Everything below is a way of finding out which point is further right without drawing the line to perfect scale. ### [Comparing and ordering whole numbers](https://math.wiki/compare-and-order-whole-numbers/) **Summary:** Use place value to decide which number is bigger, and to put numbers in order. **Question:** Worku has $736$ reading points and Tuya has $763$. The two numbers use the same three digits. Which one is more, and how can you tell without counting every point? ## When you reach for it Worku, Tuya and Kawika are in a summer reading club. Every book read earns points, and the club keeps a running total for each reader. At recess the same three friends count their jumps with a jump rope and add them up over a week. Both games keep asking the same two questions. Who has more? And how do the totals line up, from least to greatest? Counting on from the smaller number answers the first question, but it is slow when the numbers are $489$ and $512$. Place value is faster. A three-digit number is some hundreds, some tens and some ones (numbers-to-1000), and each column of the place-value-chart counts a block ten times bigger than the column on its right. That makes the hundreds digit the one to look at first. Most of the time it settles the question alone; when it does not, the tens do, and only rarely the ones. Two-digit numbers were compared tens first in comparison-symbols, and the signs $>$, $<$ and $=$ were introduced there. This lesson uses those signs for numbers up to $1{,}000$ and does not reteach them. The lesson runs over three days. - **Day 1, column by column.** Comparing two three-digit numbers: hundreds first, then tens, then ones, and why that order is right. - **Day 2, numbers that look different.** A two-digit number against a three-digit one, and a pile of blocks that has not been traded yet. - **Day 3, putting numbers in order.** Ordering three or four numbers, and building the largest and smallest number from digit cards. ### [Comparing numbers and the symbols >, < and =](https://math.wiki/comparison-symbols/) **Summary:** Written signs that record which of two amounts is larger, or that they match. **Question:** Two towers of cubes are too tall to match up one cube at a time. How can the two written numbers alone say which tower has more, and how do we write that answer down? ## Why this exists Children already tell which of two small groups has more by matching them up or by counting (compare-quantities, compare-numbers-to-10). That works with seven cubes against five. It works badly with $45$ cubes against $54$. Matching that many cubes one to one is slow, and one slip spoils the answer. The written numbers already hold the answer. A two-digit number says how many tens and how many ones it has (tens-and-ones), and a ten is a lot more than a one. So we look at the tens first, and only look at the ones when the tens are the same. No cube has to be moved. Once we know which is more, we want to write it down in a way anyone can read. Three signs do that: $>$, $<$ and $=$. With them a whole comparison fits in a short sentence, $45 < 54$, read aloud as "forty-five is less than fifty-four". The lesson runs over four days. Day 1 compares two-digit numbers by their tens. Day 2 handles numbers with the same tens, and numbers that are equal. Day 3 brings in the three signs. Day 4 reads each comparison both ways and carries the idea past $100$, up to $120$. ## Definition Take two whole numbers. The **greater** one names more things. The **less** one names fewer things. If both name the same amount, they are **equal**. Three signs record which of these is true. | Write | Say | |---|---| | $54 > 45$ | fifty-four is greater than forty-five | | $45 < 54$ | forty-five is less than fifty-four | | $54 = 54$ | fifty-four is equal to fifty-four | ### [Comparing numbers in scientific notation](https://math.wiki/compare-numbers-in-scientific-notation/) **Summary:** The power of ten is checked first and the leading part only breaks a tie. **Question:** Two numbers are each written as a few digits times a power of ten. Which is bigger, and by how much? ## When you reach for it Scientific notation is built for numbers too large or too small to read at a glance: the mass of a planet, the width of a cell. Writing a number that way is half the point. The other half is that two such numbers can then be compared almost without arithmetic. Written out in full, the masses of the Earth and the Moon in kilograms have $25$ and $23$ digits, and deciding which is larger means counting them. In scientific notation they are about $$6 \times 10^{24} \text{ kg} \qquad\text{and}\qquad 7.3 \times 10^{22} \text{ kg}$$ and the exponents settle it at once: the Earth is heavier, even though $7.3$ is more than $6$. Reach for this whenever you need to say which of two measurements is larger, put several in order, or estimate roughly how many times larger one is than another. ## The procedure Every number in this section is positive and written in scientific notation, $a \times 10^{n}$ with $1 \le a < 10$ and $n$ an integer. Call $a$ the **leading part** and $n$ the **exponent**. ### Why the exponent decides The condition $1 \le a < 10$ traps the number between two neighboring powers of ten. Multiplying all three parts of $1 \le a < 10$ by the positive number $10^{n}$ keeps both inequalities pointing the same way: $$10^{n} \le a \times 10^{n} < 10^{n+1}$$ So every number with exponent $3$ lies from $10^{3}$ up to, but not including, $10^{4}$; every number with exponent $4$ lies from $10^{4}$ up to $10^{5}$; and so on. These stretches do not overlap, and each one sits wholly above the one before it. ### [Comparing quantities](https://math.wiki/compare-quantities/) **Summary:** Tell which group has more, fewer, or the same. **Question:** Two groups of things sit on a table. How can a child tell which group has more, which has fewer, or that they are the same? ## When you reach for it To compare two groups, a child asks "which has **more**, which has **fewer**, or are they **the same**?" There are two honest ways to answer: match the things up one to one and see which group has some left without a partner, or count each group and see which number comes later in the counting words. Reach for it once a child can match and count one to one; it is the step before comparing two written numbers. Keep every group at ten or fewer. Each day below takes about ten to fifteen minutes. ## Day 1: Line them up **You need:** plastic bottle caps in two colors (about ten dark and ten light), and the edge of a table or a strip of paper to line them along. Put $7$ dark caps in a row along the edge. Hand the child $5$ light caps and say, "Put one light cap right under each dark cap." When the light caps run out, ask, "Does every dark cap have a partner? Which color has more caps?" Then ask the other half: "Which color has fewer?" Say both sentences together: "There are more dark caps. There are fewer light caps." The two dark caps at the end with no partner are what make the dark group bigger. If every cap had a partner, you would say, "They are the same," or "There are the same number." No counting is needed today; the leftovers show the answer. **Practice.** Line up the caps, one light cap under each dark cap. 1. $6$ dark caps and $6$ light caps. *Answer:* every cap has a partner, so there are the same number. No cap is left over. 2. $4$ dark caps and $7$ light caps. *Answer:* $3$ light caps have no partner, so there are more light caps and fewer dark caps. 3. $8$ dark caps and $2$ light caps. *Answer:* $6$ dark caps have no partner, so there are more dark caps and fewer light caps. 4. Line up $9$ pasta shells and match each with a paper clip from a pile of $10$ clips. *Answer:* $1$ clip has no partner, so there are more clips and fewer shells. ### [Comparing two numbers to 10](https://math.wiki/compare-numbers-to-10/) **Summary:** Two written numbers from 1 to 10 are compared by counting or matching to say which is greater and which is less. **Question:** Two numeral cards lie on the table with nothing to count beside them. How can a child tell which number is greater and which is less? ## When you reach for it Once a child can compare two groups of things and can read the numerals $1$ to $10$, the next step is to compare two written numbers with no objects in sight: $9$ is **greater than** $3$, and $3$ is **less than** $9$. The child can always check by making each number with fingers and matching them, and slowly learns the faster way: the number said later when counting from one is the greater number. Each day below takes about ten to fifteen minutes. Use the words "greater than", "less than" and "equal" from the first day, and always say the sentence both ways. ## Day 1: Finger match **You need:** numeral cards $1$ to $10$, written large, with a short line under the $6$ and the $9$ so they cannot be read upside down. Sit facing the child. Each of you takes a card and shows that many fingers. Then press your fingertips together, one of yours against one of theirs, until someone runs out. Ask, "Who has fingers left over? Whose number is greater?" Then say both sentences together: "$5$ is greater than $3$. $3$ is less than $5$." The fingers are only a way to check. Keep pointing back to the cards: "So the card that says $5$ is greater than the card that says $3$." The goal is to know it from the numerals alone. **Practice.** Show the fingers, match them, then say both sentences. 1. The cards in the picture, $3$ and $5$. **Answer:** $2$ fingers are left over on the $5$ side, so $5$ is greater than $3$ and $3$ is less than $5$. 2. Cards $9$ and $6$. **Answer:** $3$ fingers are left over on the $9$ side, so $9$ is greater than $6$ and $6$ is less than $9$. 3. Cards $2$ and $4$. **Answer:** $2$ fingers are left over on the $4$ side, so $4$ is greater than $2$ and $2$ is less than $4$. 4. Cards $7$ and $7$. **Answer:** no finger is left over, so the numbers are equal: $7$ is equal to $7$. ### [Compensation when adding and subtracting](https://math.wiki/compensation-in-addition/) **Summary:** Adding a round number and then taking the extra back off. **Question:** A box of $299$ pencils arrives at a store that already has $457$. Is there a way to find the total in your head, by adding something rounder and then fixing it? ## When you reach for it Aulani runs the school store with her class. The store has $457$ pencils on the shelf, and a box of $299$ more arrives. How many pencils does the store have now? The answer is $457 + 299$. You could line the numbers up and add by place value, trading ones for a ten and tens for a hundred as in regrouping in addition. But $299$ is very close to $300$, and adding $300$ is easy: only the hundreds digit changes. So add $300$ instead, then fix the answer: $$457 + 299 = 457 + 300 - 1 = 757 - 1 = 756.$$ Adding $300$ added one pencil too many, so one is taken back. The store has $756$ pencils. This move is called **compensation**: change a number into a round one that is easy to work with, then make up for the change. You reach for it when one of the numbers sits just below or just above a multiple of ten or a hundred, such as $299$, $198$, $401$ or $99$, and you want the answer in your head. Prices that end in $99$ cents are made for it. The lesson has three parts, one for each day. Part 1 is compensation in addition. Part 2 is compensation in subtraction, along with a second way to keep a difference the same. Part 3 uses both on money. ## The procedure ### Part 1: adding a round number, then taking the extra back **The picture.** Think of the sum on an open number line: a line with only the numbers you need marked on it, not drawn to scale (as in adding on an open number line). Start at $457$. Adding $299$ means moving $299$ to the right. Make that move in two hops: a big hop of $300$ to land on $757$, and then a small hop of $1$ back to $756$. ### [Complex numbers](https://math.wiki/complex-numbers/) **Summary:** Numbers built from a square root of negative one. **Question:** No real number squares to a negative, so $x^2 + 9 = 0$ has no real solution. What happens if we allow a number whose square is $-1$? ## What this area is about The complex numbers add one new number, $i$, with $i^2 = -1$. Every complex number can be written $a + bi$ with $a$ and $b$ real, and the real numbers sit inside as the case $b = 0$. With them, every quadratic equation $ax^2 + bx + c = 0$ with real coefficients and $a \ne 0$ has complex solutions, and the old arithmetic still works. ## How these fit together **The definition.** Imaginary and complex numbers introduces $i$, writes $\sqrt{-a} = i\sqrt{a}$ for $a > 0$, and names the real and imaginary parts. **Arithmetic.** Adding, subtracting and multiplying complex numbers treats $a + bi$ like a binomial and replaces $i^2$ with $-1$. Conjugates and dividing complex numbers multiplies above and below by the conjugate $a - bi$, which makes the denominator real. **A picture.** The complex plane, modulus and argument plots $a + bi$ at the point $(a, b)$, with size $|z| = \sqrt{a^2 + b^2}$ and a direction angle. **A result.** Complex roots come in pairs: if a polynomial has real coefficients and $z$ is a root, so is its conjugate $\overline{z}$. ## A path through it Standard form first, then adding, subtracting and multiplying; dividing and the complex plane follow, in either order. The pairing of roots comes last, after the discriminant, which itself uses this area. Finish here and you are ready for the fundamental theorem of algebra and for polar form. ### [Complex roots come in pairs](https://math.wiki/conjugate-pairs-as-roots/) **Summary:** A polynomial with real coefficients has complex roots that are conjugates of each other. **Question:** A quadratic with a negative discriminant hands back two complex roots that differ only in the sign of i. Is that an accident of the formula, or does every polynomial do it? ## Why this exists When the discriminant of a quadratic is negative, the quadratic formula gives two complex roots, such as $1 + 2i$ and $1 - 2i$. They are conjugates: the same real part, opposite imaginary parts. That could be a quirk of the formula and its $\pm$. Cubics and quartics have roots too, and their formulas are far too long to read a pattern off. Yet the pairing survives, as long as every coefficient of the polynomial is a real number, and that has a practical payoff: one complex root hands you a second one free. For a cubic, two known roots out of three are enough to finish the job by hand. ## Definition A **polynomial with real coefficients** is one such as $2x^3 - x + 7$, in which every number multiplying a power of $x$ is real. A **root** of a polynomial $p(x)$ is a number $z$, real or complex, with $p(z) = 0$. The conjugate of $z = a + bi$, with $a$ and $b$ real, is $\overline{z} = a - bi$. When $b \ne 0$, $z$ and $\overline{z}$ are two different numbers, a non-real conjugate pair. **Conjugate pairs as roots.** If a polynomial has real coefficients and $z$ is a root, then $\overline{z}$ is also a root. For a non-real root this is a second, different root. For a real root, $\overline{z} = z$, and it says nothing new. ## Intuition ### For a quadratic, the $\pm$ makes the pair The discriminant lesson showed where the pair comes from. When $D = b^2 - 4ac$ is negative, the formula gives $x = p \pm qi$ with $p = -\frac{b}{2a}$ and $q = \frac{\sqrt{-D}}{2a}$, both real because $a$, $b$ and $c$ are. The $\pm$ adds the pure imaginary part $qi$ once and subtracts it once, so the two roots share the real part $p$ and have opposite imaginary parts. On the complex plane they are mirror images in the real axis. ### [Conjugates and dividing complex numbers](https://math.wiki/divide-complex-numbers/) **Summary:** Multiplying above and below by the conjugate turns the denominator real. **Question:** Adding, subtracting and multiplying complex numbers always lands back on a + bi. What does it take to divide one by another? ## When you reach for it Adding, subtracting and multiplying complex numbers never leaves standard form: the answer is always $a + bi$ with $a$ and $b$ real. A quotient such as $$\frac{3 + 4i}{1 - 2i}$$ is not in that form. It is a perfectly good number, but while $i$ sits below the line you cannot read off its real part or its imaginary part. Nor can you split it as $\frac{3}{1} + \frac{4i}{-2i}$: a fraction splits over a sum on top, $\frac{p + q}{r} = \frac{p}{r} + \frac{q}{r}$, never over a sum underneath. Reach for the conjugate whenever $i$ is in a denominator: dividing two complex numbers, finding a reciprocal, or solving an equation in which the unknown is multiplied by a complex number. The move is the one rationalizing a denominator makes with $\frac{1}{3 + \sqrt{2}}$: multiply above and below by a partner that turns the bottom into a plain real number. That is no coincidence, since $i$ is itself a square root, $\sqrt{-1}$. ## The procedure ### The conjugate The **conjugate** of the complex number $z = a + bi$, with $a$ and $b$ real, is $$\overline{z} = a - bi$$ The real part stays and the imaginary part changes sign. So the conjugate of $-6 + 2i$ is $-6 - 2i$. Flipping a sign twice undoes it, so $\overline{\overline{z}} = z$: the two numbers form a **conjugate pair**, each the conjugate of the other. A real number ($b = 0$) is its own conjugate, and a pure imaginary number $bi$ has conjugate $-bi$. ### [Converting between fractions and decimals](https://math.wiki/fraction-to-decimal-by-division/) **Summary:** Dividing the top by the bottom. **Question:** A quilt has 9 blue squares out of 25. A trail sign says 0.625 miles, and the map says five eighths of a mile. How do you move between a fraction and a decimal, and how do you know they name the same number? ## When you reach for it Eunji is sewing a quilt out of $25$ fabric squares, $5$ rows of $5$. Nine of the squares are blue, so $\frac{9}{25}$ of the quilt is blue. Her friend Benicio is sewing a quilt with $100$ smaller squares, and $36$ of his are blue. Which quilt is bluer? It is hard to tell from $\frac{9}{25}$ and $\frac{36}{100}$ as they stand. It is easy once both are written the same way. Ngozi has a different puzzle on a hiking trail. Her trail map says the first lookout is $\frac{5}{8}$ of a mile from the start. Her watch counts distance in decimals: $0.1$, $0.2$, $0.3$ miles. What number will the watch show when she reaches the lookout? Fractions and decimals are two ways of writing the same numbers. A fraction is a point on the number line (see fractions-on-number-line), and so is a decimal, whose digits after the point count tenths, hundredths and thousandths (see decimal-place-value and tenths-and-hundredths). You reach for this lesson whenever a number arrives in one form and you need the other: to compare two amounts, to put a number on a decimal scale or watch, to add a fraction to a decimal, or to write a measurement in its simplest form. This lesson takes three days. Part 1 renames a fraction so its bottom number is $10$, $100$ or $1000$, and then reads off the decimal. Part 2 divides the top number by the bottom number, which works for every fraction. Part 3 goes the other way, from a decimal back to a fraction in lowest terms, and lists the pairs worth knowing by heart. ### [Counting a large pile by grouping](https://math.wiki/counting-collections-by-grouping/) **Summary:** Making equal groups, usually tens, so a big collection can be counted quickly. **Question:** There is a big pile of popsicle sticks on the table, far too many to count one at a time without losing your place. How can you count it quickly, be sure the count is right, and know roughly how many there are before you start? ## When you reach for it A child who can count a set of objects touches each thing once while saying the next number word, and the last word said is how many. That works well for $8$ crayons or $15$ pom-poms. It starts to fail on a pile of $47$ popsicle sticks or a bag of $63$ pom-poms. The count is long, the things are all alike, and a single slip (a stick touched twice, a stick skipped, a lost place when someone talks) spoils the whole count with no way to tell where it went wrong. The fix is to **make equal groups as you count**. Put the sticks in groups of ten, then count the groups, "ten, twenty, thirty, forty", and count on through the few that are left over. Each small count only goes up to ten, so a slip costs one group, not the whole pile. The groups also stay on the table as a record of the counting already done. Reach for it whenever a collection is too big to count comfortably in one go: roughly anything past twenty, and every collection up to $120$ and beyond. Reach for its partner, **estimating**, before the counting starts: a sensible guess at how many, made by looking, which the count then confirms or corrects. The lesson has four parts, one for each day. The first is why a long count goes wrong, and how groups of ten fix it. The second is groups of two, five and ten, and why ten is the handiest. The third is recording a count and checking it by grouping a different way. The fourth is estimating before counting, then comparing the estimate with the count. ### [Counting a set of objects](https://math.wiki/counting-objects-to-20/) **Summary:** Touching each object once while saying the next counting word. **Question:** How does a child count a set of up to twenty things without skipping one or counting one twice? ## When you reach for it Counting a set means giving each object exactly one counting word, in order, so that no object is skipped and none gets two words. Up to ten a child can often do this by eye and by habit; between eleven and twenty the sets get big enough that keeping track becomes the real work, and the words themselves change shape. Reach for it once a child can say the counting words to twenty and has met one word for each object with small sets. Each day below takes about fifteen minutes. Every count this week is between eleven and twenty. ## Day 1: The words past ten **You need:** about $15$ toy cars, or any small things that can stand in a line along the edge of a table. Line the cars up along the table edge while the child watches. Count them together, touching each one: "one, two, ... ten." Then slow down and keep going: "eleven, twelve, thirteen." Point out that the words after ten sound different. "Eleven" and "twelve" are new words, and from "thirteen" on, each word ends in "teen". The touching does not change: one car, one word. Ask, "Which car got the word *eleven*? Which one got *twelve*?" Then, "What was the last word?" **Practice.** 1. Touch and count the counters in the picture above. *Answer:* $13$; the last counter gets "thirteen". 2. Line up $11$ cars and count them. *Answer:* $11$; the word right after "ten" is "eleven". 3. Line up $12$ crackers and count them. *Answer:* $12$, with "eleven" and "twelve" on the last two. 4. Line up $15$ cars and count them. *Answer:* $15$; the words after ten are "eleven, twelve, thirteen, fourteen, fifteen". ### [Counting and cardinality](https://math.wiki/counting-and-cardinality/) **Summary:** Numbers name how many. Counting is matching things to number words, one each. **Question:** How do counting words, said in order and matched to things, come to tell how many there are? ## What this area is about A number names how many. Counting gets there by matching each thing to one counting word, in order, with none skipped or said twice; the last word said is the size of the whole group. Then the words stretch to $1{,}000$, take written form, and let two amounts be compared. ## How these fit together **The words.** Counting words to 20, to 100, to 120, past 100, skip counting. **Counting things.** One-to-one correspondence, counting a set, how counting tells how many, counting out a number, one more and one less, counting on and back, counting a large pile by grouping, subitizing and ten frames. **Writing and naming.** Names, numerals and zero, writing numerals, ordinal numbers, the number line, even and odd. **Comparing.** Comparing groups, comparing numbers to 10, $>$, $<$ and $=$, the equal sign. ## A path through it Say the words to $20$, match one to one, count a set, then learn that the last word tells how many. Counting out, one more, counting on, and names and numerals follow; writing numerals and ordinals come after names. Subitizing precedes ten frames, and comparing groups precedes comparing numbers. Counting to $100$ opens skip counting, $120$ (once numerals are written), $1{,}000$, even and odd, and, after skip counting, grouping and the number line. Next come addition and subtraction and place value. ### [Counting by hundreds](https://math.wiki/skip-count-by-hundreds/) **Summary:** Counting in jumps of one hundred. **Question:** Craft sticks come in boxes of one hundred. How do you count a shelf of boxes without opening one, and why does only one digit change each time the count goes up by a box? ## When you reach for it The art room keeps its craft sticks in boxes of one hundred. Counting the sticks in seven boxes one at a time takes seven hundred counts. Counting the boxes takes seven: "one hundred, two hundred, three hundred". Each number said is one hundred more than the one before, because each box holds one hundred more sticks. That is skip-counting with the biggest jump a child meets before one thousand. Jumps of two, five, three and four are in skip-count-by-twos, and jumps of ten are in skip-count-by-tens. A count by hundreds has two pictures. On a number-line marked from $0$ to $1{,}000$, every jump is the same long stride. With base ten blocks, every jump is one more flat. A child reaches for a count by hundreds when: - counting things that come in hundreds, such as boxes of one hundred sticks or hundred flats; - counting on by hundreds from a number that is not a whole hundred, such as $62, 162, 262$; - counting back by hundreds, such as $900, 800, 700$, or from a number such as $845$; - counting a pile of hundreds, tens and ones quickly, the hundreds first. Counting by ones up to $1{,}000$, with the hundred chart pages where a whole page is one hundred more, is count-to-1000. Reading and writing three-digit numbers is numbers-to-1000, and the trade of ten rods for one flat is trade-ten-for-one. This lesson uses all three and reteaches none of them. ## The procedure ### [Counting by tens to one hundred](https://math.wiki/skip-count-by-tens/) **Summary:** Saying ten, twenty, thirty and on, one group of ten at a time. **Question:** How can a child count to one hundred by saying ten, twenty, thirty, one group of ten at a time? ## When you reach for it Counting by tens means saying one word for each whole group of ten: ten, twenty, thirty, and on up to one hundred. It works only when every group really holds ten, so the child first builds the groups with sticks, fingers or full ten frames, and then counts the groups, not the single things. Ten groups of ten make one hundred, which is the end of this week's count. The child should already count to one hundred by ones or be learning to, and know that a ten and some more makes a teen number. Each day takes about ten to fifteen minutes. ## Day 1: Bundles of ten sticks **You need:** about $40$ craft sticks (or straws) and some rubber bands. Count out $10$ sticks with the child, one by one, and put a rubber band around them. Say, "This is a bundle. How many sticks are in it?" The answer is ten, and the band means it stays ten, so nobody needs to count it again. Make three more bundles the same way. Now lay the bundles down one at a time. As each one goes down, say the new count: "ten, twenty, thirty, forty." Let the child put the bundles down while you both say the words, then let the child say them alone. Ask, "When I put down one more bundle, how many more sticks are there?" The answer is ten more, every time. Ask, "Is it faster to count the sticks or the bundles?" **Practice.** 1. Lay down $2$ bundles. *Answer:* "ten, twenty", so $20$ sticks. 2. Lay down $5$ bundles. *Answer:* the last word is "fifty", so $50$ sticks. 3. Lay down $7$ bundles. *Answer:* "seventy", so $70$ sticks. 4. Cover the numbers in the picture above and count the bundles by tens. *Answer:* "ten, twenty, thirty, forty", so $40$ sticks. ### [Counting by twos, fives, threes and fours](https://math.wiki/skip-count-by-twos/) **Summary:** Counting in jumps of two. **Question:** Rain boots stand in pairs by the door and tulips grow in rows of five. Why say "two, four, six" instead of counting every boot, and how can you tell in advance which numbers such a count will land on? ## When you reach for it Rain boots by a classroom door stand in pairs. A child can count them one boot at a time, "one, two, three, four...", and get the right answer. But the boots come in twos, and the eye already sees the pairs. Counting by twos says only the number at the end of each pair: "two, four, six, eight". The count is half as long, and there is half as much room for a slip. The same move works for any collection made of equal groups: tulips planted in rows of five, stools with three legs each, chairs with four. Say the number at the end of each group, and each number said is the size of one group more than the one before. That is skip counting, the subject of skip-counting, and this lesson covers jumps of two, five, three and four. Jumps of ten and of a hundred are skip-count-by-tens and skip-count-by-hundreds. A child reaches for skip counting in four situations: - counting a collection that comes in equal groups, such as pairs of boots or rows of tulips; - counting on in equal jumps from a number other than zero, such as $13, 18, 23$; - counting back in equal jumps, such as $40, 38, 36$; - deciding whether a count will ever land on a given number, by looking at the ones digits. The pictures are the number-line, where each count is a jump of the same length, and the hundred chart, the numbers $1$ to $100$ in rows of ten, where the squares a count lands on make a pattern. Counting itself, by ones and by tens, is count-to-100 and count-to-1000; counting on and back by ones is counting-on-from-a-number. ### [Counting on and counting back](https://math.wiki/counting-on-from-a-number/) **Summary:** Starting from a known amount and continuing the count instead of going back to one. **Question:** When a child already knows how many are in a group, how can they find the new amount without counting the whole group again from one? ## When you reach for it Once a group has been counted, its number stands for the whole group, so there is no need to count it again from one: say that number once and keep counting the new things. That is **counting on**. Taking things away one at a time while saying the words backward is **counting back**. These four days build both with real objects, first with a group hidden under a cup, then past ten, then backward, and finally with a written number standing in for the group, all within twenty. The child should already know that the last word said tells how many and be able to say the counting words starting from any number. Adding or taking away a single object is one more and one less; this week the child goes on for two, three or four steps. ## Day 1: Five under the cup **You need:** a paper cup, about $10$ counters (buttons or beans work). Count $5$ counters with the child and put them under the upside-down cup. Ask, "How many are under the cup?" The child should say "five" without lifting it. Then set $3$ more counters in a row beside the cup. Ask, "How many altogether?" Many children lift the cup and count from one. Show the shortcut: tap the cup and say "five" in a long, stretched voice, then touch each counter outside, "six, seven, eight." Lift the cup and count all of them to check that the answer is $8$. Ask, "Why did we say *five* for the cup?" You want to hear "because there are five under it." The cup's word stands for all five counters at once. ### [Counting on to add and counting back to subtract](https://math.wiki/counting-on-to-add/) **Summary:** Starting at the larger part and counting forward by the smaller one. **Question:** Eight counters are hiding under a cup and three more sit beside it. Do we have to tip them all out and count from one, or can we start at eight? And when two of seven are taken away, can we find what is left just by saying the numbers backwards? ## When you reach for it This lesson teaches two moves that mirror each other. **Counting on** adds by starting at one part and counting forwards through the other. **Counting back** takes away by starting at the whole and counting backwards, one number for each thing taken away. Both rest on the same fact about counting, and both are quick for the same reason: only a few numbers have to be said. **Counting on to add.** A child who adds by joining starts by counting all: make both groups, push them together, and count every object from one. That always works, but most of the counting goes on a group whose size is already known. If the cup holds eight, counting "one, two, three, up to eight" only tells you eight again. Counting on skips that part. Say the number of the first group, then keep counting through the second group only. For $8 + 3$ that is "eight ... nine, ten, eleven". It grows straight out of counting on from a number, where a child starts a count at five instead of at one, and it is the first way of adding that does not need every object in front of you. Reach for it when one part is small, one, two or three, sometimes four, and the other part is already known as a number: it is written down, it is hidden under a cup, or it is a group the child sees at a glance, like a full row of a ten frame or a whole hand of five fingers. The small part is the one you count, so it goes second. When the small part comes first, as in $2 + 9$, the child turns the sum around and counts on from $9$. That turning around is allowed, and the reason is the most important idea on the adding side of this lesson; it is explained under "Starting from the bigger number" in the next section. ### [Counting out a given number of objects](https://math.wiki/count-out-a-given-number/) **Summary:** Asked for a number up to 20, a child counts out exactly that many objects from a larger pile and stops. **Question:** Asked for seven, how does a child take exactly seven from a big pile and know when to stop? ## When you reach for it Counting a pile answers "how many are there?" Counting *out* turns that around: you start with a number, "bring me $7$", and have to make a group of exactly that size from a pile that holds more. The child has to keep the number in mind, move each object away from the pile as they count it, and stop when they say the number, even though there are plenty left. It rests on the last word said tells how many, and it is the skill behind every "get me" in daily life, from handing out napkins to building a number in addition. Each day below takes about ten to fifteen minutes. The pile is always bigger than the number asked for, so stopping is a real choice. ## Day 1: Spoons in the cup **You need:** a basket of $9$ plastic spoons and a cup. Say a number, "Put $3$ spoons in the cup", and let the child do it. Watch for two things. First, each spoon should leave the basket and go into the cup as its word is said: moving it is what keeps counted ones apart from the rest. Second, the child should stop at "three". Then ask, "How many are in your cup?" A child who answers "three" at once, without counting again, knows that the last word told them. If the child keeps going past the number, do not correct. Ask, "How many did I ask for?" and "How many are in the cup?" and let them fix it by putting spoons back. Keep numbers to $5$ today. **Practice.** Start each one with all $9$ spoons in the basket. ### [Counting past one hundred](https://math.wiki/count-to-1000/) **Summary:** Continuing the counting words through the hundreds up to one thousand. **Question:** After $120$ the count keeps going. What comes next, where are the hard places, and how does the count reach $1{,}000$? ## When you reach for it A child who can count to 120 can start at any number up to there and keep going. Plenty of counts go much further. A jigsaw puzzle with $1{,}000$ pieces, a step counter that adds one for every step on a walk, the pages of a thick book: the count runs on through the hundreds, all the way to one thousand. This lesson carries counting by ones from $120$ up to $1{,}000$. The child starts at any number, counts on or counts back, says each number, and writes the numbers as they go. The move is one they already have. Counting on means saying the next number, the one that is one more. Counting back means saying the number before, the one that is one less. Nothing about the move changes past $120$. What needs care are the places where the count **rolls over**: where a new ten starts, as at $139$ then $140$, and where a new hundred starts, as at $299$ then $300$. At those places more than one digit changes at once, and the child has to say a new word. Counting by ones is the slow, sure way along the count. Jumping a hundred at a time is skip-count-by-hundreds, and naming any three-digit number in words, digits and expanded form is numbers-to-1000. This lesson stays with one step at a time. The lesson runs over two days. - Day 1: counting on through the hundreds, the chart page of $101$ to $200$, and where a ten or a hundred rolls over. - Day 2: counting back, the last hundred up to $1{,}000$, and writing the numbers counted. ### [Counting to 100 and beyond](https://math.wiki/count-to-100/) **Summary:** Say the number words in order to one hundred, and start counting from any number. **Question:** Once a child can count to twenty, how do the counting words keep going all the way to one hundred? ## When you reach for it Past twenty, the counting words follow one pattern: a tens word (*twenty*, *thirty*, *forty*) followed by the ones words *one* to *nine*, and after *nine* the tens word changes. A child who already says the words to twenty needs two new things: the eight tens words up to *ninety*, and the moment each one takes over (twenty-nine, thirty). These four days build both, then lay the words out on the hundred chart, then let the child start counting from any number instead of always from one. Keep each session short and spoken. The goal is the words in order; counting a pile of a hundred things is not needed. ## Day 1: Past twenty, the ones start again **You need:** a strip of paper with boxes numbered $20$ to $40$, taped to the floor or the table. Walk a toy figure along the strip, one box per word, while the child says the words. At *twenty-one*, stop and ask, "What do you hear after *twenty*?" The child hears *one*, then *two*, *three*: the same words as before ten, now riding on *twenty*. Keep going. At *twenty-nine*, stop again and ask, "The ones are all used up. What comes next?" The answer is a new word, *thirty*, and then the ones start over: *thirty-one*, *thirty-two*. Ask, "Where does the new word start?" Point to the shaded $30$. **Practice.** 1. What comes after *twenty-three*? *Answer:* twenty-four. 2. What comes after *twenty-nine*? *Answer:* thirty. 3. Count from $31$ to $36$. *Answer:* thirty-one, thirty-two, thirty-three, thirty-four, thirty-five, thirty-six. 4. What comes after *thirty-four*? *Answer:* thirty-five. ### [Counting up to find a difference](https://math.wiki/count-up-to-subtract/) **Summary:** Starting at the smaller number and adding on until the larger one is reached. **Question:** To find $702 - 386$, do you have to take $386$ away, or can you start at $386$ and count up to $702$? ## When you reach for it Hlengiwe's family is holding a yard sale. They put $702$ items out on the tables in the morning, and by the afternoon $386$ have been sold. How many are still on the tables? The answer is $702 - 386$. You could line the numbers up and subtract by place value, but that needs a trade across the zero in $702$ (see subtracting three-digit numbers by place value). There is another way that needs no trade at all. Ask a different question: **how far is it from $386$ up to $702$?** Start at $386$ and count up, in a few big, easy hops, until you reach $702$. The hops together are the answer. This works because a subtraction and a missing addend are the same question: $702 - 386 = \square$ asks exactly what $386 + \square = 702$ asks. That idea was first met with small numbers in subtraction as finding an unknown addend; here it is put to work within $1{,}000$, on an open number line. You reach for counting up when the two numbers are close together, when the larger number is a round one or sits just past one (such as $702$, $800$ or $1{,}000$), and every time you hand back change. A cashier who counts change into your hand, "$2.70$, $3$, $4$, $5$", is counting up to subtract. The lesson has three parts, one for each day. Part 1 is counting up on an open number line. Part 2 is choosing when counting up is the quicker way. Part 3 is making change. ## The procedure ### Part 1: hopping from the smaller number to the larger one ### [Counting, reading and writing numbers to 120](https://math.wiki/numbers-to-120/) **Summary:** Counting on past 100 to 120 from any number, and reading and writing each numeral in that range. **Question:** After $99$ comes $100$. What comes after that, how is each number said, and how is it written, all the way to $120$? ## When you reach for it A child who can count to one hundred knows the counting words up to $100$ and the pattern inside them: one, two, three up to nine, then a new decade word, and the ones again. Many counts do not stop at $100$, though. The mailboxes along a long apartment hallway, the bowling pins stored at the back of a bowling alley, a big pile of anything: the count runs on, and the child needs the words that come next and the way to write them. This lesson takes the count past $100$ up to $120$. The child should be able to start at any number up to $120$ and count on from there, say each number aloud, and read and write each numeral from $100$ to $120$. Counting on is the move they already have: start from a number instead of from one and keep saying the next number. Two things are new. The words: after one hundred the count goes "one hundred one, one hundred two", and so on. The numerals: from $100$ on, every number is written with three digits. The words come easily, because they repeat a pattern the child already knows. The numerals take more care, because the spoken name has two parts ("one hundred" and "seven") but the written number is one numeral, $107$, not $100$ and $7$ set side by side. The lesson runs over four days. - Day 1: counting on across $100$. - Day 2: counting on across each new ten past $100$, at $110$ and at $120$. - Day 3: reading and writing the numerals from $100$ to $120$. - Day 4: the hundred chart grown to $120$, and counts that use all of it. ### [Cube roots and higher roots](https://math.wiki/cube-roots/) **Summary:** The number that gives the original when used three times as a factor, which works for negatives too. **Question:** A cube-shaped box holds a known amount. How long is each edge, and which number, used three times as a factor, gives that amount back? ## Why this exists A square root runs squaring backwards. Told that a square has area $49$, you ask which number times itself gives $49$, and the side is $7$. A cube asks the same question one dimension up. A cube-shaped water tank or a box packed with dice has a volume, and the volume is the edge used three times as a factor, $V = s \cdot s \cdot s = s^3$ (the volume of a prism with every edge equal). Going from edge to volume is multiplication. Going from volume back to edge needs a new operation, one that undoes cubing. That operation is the cube root. It also settles equations. To solve $x^3 = -512$ you need a number whose cube is $-512$, and a square root is no help: no number squared is negative. A cube root is, because a negative number cubed stays negative. Cubing and squaring differ in a second way: two different numbers can have the same square, but never the same cube. That is why $x^2 = 64$ has two solutions and $x^3 = 64$ has exactly one. ## Definition A number $b$ is a **cube root** of a number $a$ when $b$ cubed is $a$: $$b^3 = a$$ Every real number has **exactly one** real cube root, and it is written $\sqrt[3]{a}$. The small $3$ is the **index**; it says how many equal factors are being undone. So $$\sqrt[3]{a} = b \quad \text{means} \quad b^3 = a$$ For example $\sqrt[3]{64} = 4$ because $4^3 = 64$, and $\sqrt[3]{-64} = -4$ because $(-4)^3 = (-4)(-4)(-4) = -64$. A **perfect cube** is the cube of an integer: $0$, $1$, $8$, $27$, $64$ and so on, and also $-1$, $-8$, $-27$ and so on. The cube root of a perfect cube is an integer. Knowing the first ten cubes by sight makes the cube roots of perfect cubes quick to read off, and each negative cube is the opposite of the one shown: ### [Decimal arithmetic](https://math.wiki/decimal-arithmetic/) **Summary:** The four operations when numbers have a decimal point. **Question:** Prices, weights and measures rarely come out whole. How do the four operations work when the numbers have a decimal point? ## What this area is about Decimals are place value carried to the right of the point, so the four operations work just as they do for whole numbers. The new question each time is where the point goes. For adding and subtracting, line up the places. For multiplying, count decimal places. For dividing, carry the point straight up, and make a decimal divisor whole first. ## How these fit together **Adding and subtracting.** Adding and subtracting decimals lines up tenths with tenths and hundredths with hundredths, as in $3.4 + 1.25$. **Multiplying.** Multiplying a decimal by a whole number is equal groups, such as $2.65 \times 4$. Where the point goes in a product explains why the decimal places in the factors set the places in the answer. The standard algorithm multiplies as if the numbers were whole, then places the point. **Dividing.** Dividing a decimal by a whole number runs long division with the point carried straight up. Dividing by a decimal scales both numbers by a power of ten so the divisor is whole, and the standard algorithm does that with longer numbers, as in $3.15 \div 0.45$. **Using them.** Decimal problems in words decides which operation a story about money or measures needs. ## A path through it Add and subtract first. Then multiply by a whole number, see where the point lands, and learn the algorithm. Divide by a whole number next, then by a decimal. Word problems come last. ### [Decimal place value](https://math.wiki/decimal-place-value/) **Summary:** Digits to the right of the decimal point are tenths, hundredths, thousandths. **Question:** Place value counts ones, tens and hundreds. What do the digits to the right of a decimal point count? ## What this area is about Cut one whole into ten equal parts and each part is a tenth; cut a tenth into ten and each part is a hundredth, and so on. The digits after the decimal point count these pieces. Each place is worth one tenth of the place to its left, the same rule whole numbers already follow, carried past the point. ## How these fit together **The places.** Tenths and hundredths defines the first two places, $0.1 = \frac{1}{10}$ and $0.01 = \frac{1}{100}$. Thousandths adds the third, $0.001 = \frac{1}{1000}$, and writes decimals in standard, expanded and word form. **Using the places.** Decimals on a number line places a decimal by cutting the gap between two whole numbers into ten equal parts. Comparing and ordering decimals lines up the decimal points and compares place by place from the left. ## A path through it Tenths and hundredths come first. Thousandths, the number line and comparing follow, in any order. Finish here and you are ready for decimal arithmetic, rounding decimals, money written as decimals and, later, scientific notation. ### [Decimal problems in words](https://math.wiki/decimal-word-problems/) **Summary:** Stories about money, measures and rates that need decimal arithmetic. **Question:** A story hands you some decimals and a question. How do you tell whether to add, subtract, multiply or divide, and how do you know the answer makes sense? ## When you reach for it Wren keeps a fruit and vegetable stall at a Saturday farmers' market. Everything on her table is weighed in kilograms and priced in dollars and cents, so nearly every number she meets has a decimal point. One customer asks what $2.6$ kilograms of apples cost; another asks how much heavier one bag is than another. The arithmetic is already in hand: adding and subtracting decimals, multiplying them, and dividing them, by long division when the divisor is whole and by scaling first when it is not. A story adds a part before the arithmetic and a part after it. Before: the story never says "multiply", it says "what do $2.6$ kilograms cost", and you have to see the multiplication in it. After: the algorithm returns a number such as $5.357\ldots$, and the story decides whether that means $5$ costumes, $6$ meters of fabric to buy, or $5.36$ dollars. The lesson takes two days. Day 1 is one-step stories: choosing the operation, estimating first and checking that the answer fits. Day 2 is stories with several steps, and quotients that do not come out even. ## The procedure ### Day 1: the action in the story picks the operation **Read for what happens, not for a key word.** "Wren sold $2.5$ kg more plums than pears" contains "more", but if the plums are known and the pears are asked for, the answer comes from subtracting. What decides the operation is what the story does to the quantities, and there are only a few things it can do: ### [Decimals on a number line](https://math.wiki/decimals-on-a-number-line/) **Summary:** Placing a decimal between two whole numbers by splitting the gap into ten parts. **Question:** A frog's jump measures more than 3 meters but less than 4. On a tape marked only at 3 and 4, where exactly does a jump of 3.7 meters land? ## When you reach for it At the school frog-jumping contest, each frog makes three hops in a row, and a judge measures how far it got from the start line. A long tape lies on the grass with a big mark at every whole meter: $0$, $1$, $2$, $3$, and so on. Aditya's frog lands past the $3$-meter mark but before the $4$-meter mark. The judge writes down $3.7$ meters. Where on the tape is that? You reach for a number line whenever a decimal needs a place: to see how big it is, to see what it sits between, or to see which mark it is closer to. In tenths and hundredths you met the line from $0$ to $1$ cut into ten steps of $0.1$. This lesson moves the same idea to any gap between two whole numbers, and then cuts a tenth into ten again to place hundredths. One tool from earlier comes along. On any number line, the ticks step by a fixed amount, and you find it by dividing the gap between two labels by the number of equal parts between them; that is number lines with different scales. ## The procedure The lesson is taught in two parts, one a day. ### Part 1: tenths between two whole numbers Take $3.7$. The digit before the point says $3$ ones. The digit after it says $7$ tenths. So $$ 3.7 = 3 + \frac{7}{10} $$ That tells you two things at once. First, $3.7$ is more than $3$, because it is $3$ and a bit more. Second, it is less than $4$, because the bit more is only $7$ tenths, and it takes $10$ tenths to make another whole. So $3.7$ sits in the gap between $3$ and $4$. ### [Dividing a decimal by ten, one hundred or one thousand](https://math.wiki/divide-decimals-by-powers-of-ten/) **Summary:** Each factor of ten moves every digit one place to the right. **Question:** Dividing by ten makes a number ten times smaller. Where do its digits go, and what fills the empty places? ## When you reach for it Ignacio, Natsuki and Tanvi are on a class trip to the planetarium. Out front stands a rocket $56.3$ meters tall. In the lobby there is a model of it, made $100$ times smaller. How tall is the model? The question is $$56.3 \div 100$$ You reach for this move whenever a number is divided by $10$, $100$ or $1{,}000$. That happens when a group cost is shared among $10$ or $100$ people, when a length is cut into $10$ or $100$ equal parts, when a model is built $1{,}000$ times smaller, and when you need to undo a multiplication by $10$, $100$ or $1{,}000$. You never need long division for it. Place value does all the work. ## The procedure ### Day 1: every digit moves to the right In a place value chart, each place is worth ten times the place to its right. That is what multiplying and dividing by ten rests on. Read it the other way round: each place is worth one tenth of the place to its left. One ten split into ten equal parts is one one. One one split into ten equal parts is one tenth. One tenth split into ten equal parts is one hundredth. So dividing by $10$ takes one tenth of every digit's value. A $5$ in the tens place is worth $50$, and $50 \div 10 = 5$, which is a $5$ in the ones place. A $3$ in the tenths place is worth $0.3$, and one tenth of that is $0.03$, a $3$ in the hundredths place. **Every digit moves one place to the right.** The digits do not change and do not change their order. Only their places change. ### [Dividing by a decimal](https://math.wiki/divide-by-a-decimal-divisor/) **Summary:** Scaling both numbers by a power of ten so the divisor becomes whole. **Question:** Gunnhild has 4.2 liters of maple syrup and bottles that each hold 0.7 liter. How many bottles can she fill, and how do you divide when the number you divide by has a decimal point? ## When you reach for it Every spring, Gunnhild, Ludvik and Jokull help their grandmother make maple syrup. They collect sap from the trees and boil it down for hours until it turns thick and sweet. At the end of one day, Gunnhild has $4.2$ liters of syrup. Each bottle holds $0.7$ liter. How many bottles can she fill? That is a "how many groups" question. How many groups of $0.7$ fit in $4.2$? Questions like that are division (see what division means), so the answer is $4.2 \div 0.7$. You have already divided a decimal by a whole number, like $8.37 \div 3$, in divide-a-decimal-by-a-whole-number. There the size of the group, or the number of groups, was always a whole number. Now the number you divide by has a point in it. Three words help. In $4.2 \div 0.7$: - the **dividend** is $4.2$, the amount being split up; - the **divisor** is $0.7$, the size of one group; - the **quotient** is the answer, the number of groups. You reach for this lesson whenever the size of one group is a decimal with tenths. Liters of syrup per bottle. Meters of ribbon per bow. Kilograms of flour per bag. In every one, you ask how many of the small amounts fit in the big one. This lesson takes three days. Part 1 counts tenths on a strip. Part 2 uses an area model and breaks the dividend into friendly parts. Part 3 shows that $4.2 \div 0.7$ has the same answer as $42 \div 7$, and why, and it deals with half a group left over. ### [Dividing by a single digit](https://math.wiki/divide-by-one-digit/) **Summary:** Quotients where the divisor is a basic fact away. **Question:** Ariki has 5,632 pairs of shoes to set out on 8 shoe racks, the same number on each rack. No times-table fact is that big. How do you share a number this large, one place at a time? ## When you reach for it Meghna runs a card game club. She has $972$ game cards to deal out evenly to $4$ tables. How many cards does each table get? That is $972 \div 4$, a "how many in each group" question (see division-as-sharing). It is too big for a times-table fact. In partial-quotients you took away easy chunks and added them up at the end. This lesson does the same job in a tidier order. It shares the biggest pieces first, then the next biggest, one place at a time, and it writes the answer one digit at a time. The words, as before: in $972 \div 4 = 243$, the number being shared, $972$, is the **dividend**. The number of groups, $4$, is the **divisor**. The answer, $243$, is the **quotient**. What is left over, too small to give every group one more, is the **remainder**. Reach for this whenever a whole number of up to four digits is divided by a number from $1$ to $9$. Each step is a division fact, since the amount being shared at any one place is always small. ## The procedure This lesson takes four days, one part a day. ### Part 1: sharing one place at a time, from the left Think of $972$ as base-ten blocks: $9$ hundreds squares, $7$ ten rods and $2$ ones. Meghna shares them among $4$ tables, biggest blocks first. **Hundreds.** Share $9$ hundreds among $4$ tables. Each table gets $2$ hundreds, since $2 \times 4 = 8$, and $3$ hundreds each would need $12$. That uses $8$ hundreds and leaves $1$. ### [Dividing by a two-digit number](https://math.wiki/divide-by-two-digit-divisors/) **Summary:** Quotients where each step needs a trial multiple of a larger divisor. **Question:** When the divisor has two digits, no times table tells you each digit of the quotient. How do you find it? ## When you reach for it Akira works at a bookstore. A delivery of $1{,}768$ books has come in, and each shipping box holds $34$ books. How many boxes will she fill? That is $1{,}768 \div 34$. The layout is the one from the standard long division algorithm: divide, multiply, subtract, bring down, one place at a time. What changes with a divisor of two digits is the divide step. "How many $3$s are in $17$?" is a basic fact. "How many $34$s are in $176$?" is not. You have to make a good guess and then test it. Reach for this skill whenever the divisor is a two-digit number: books per box, seats per row, dollars per ticket. Partial quotients would also work, but it lets you take away any chunk you like, so it can take many rounds. Long division takes the biggest chunk at each place, so it finishes in one row per digit. This lesson takes four days. Day 1 is a quotient with one digit. Day 2 is quotients with two or three digits. Day 3 is what to do when a guess is wrong. Day 4 is longer dividends, zeros in the quotient, and what a remainder means in a story. ## The procedure ### Part 1: guessing a digit by rounding the divisor Start with a small one. Akira has $238$ books and boxes that hold $34$. How many boxes? **Where the digit goes.** Look at the dividend from the left. $2$ hundreds cannot fill a box of $34$ as hundreds. $23$ tens is still less than $34$. So the first place that can be divided is the ones: all $238$ ones. The quotient has just one digit, and it goes above the $8$. ### [Dividing decimals](https://math.wiki/divide-a-decimal-by-a-whole-number/) **Summary:** Quotients where the dividend carries a decimal point. **Question:** One morning's gas for the carpool costs 8.37 dollars, and three families share it equally. How do you split a number with a decimal point into equal parts, where does the point go in the answer, and what do you do when the division will not come out even? ## When you reach for it Haeun, Rajan and Pelumi live on the same street, and their families run a carpool to school. One morning the gas costs $8.37$ dollars. The three families split it equally. How much does each family pay? That is a sharing question, so it is division: $8.37 \div 3$. You already know how to share whole numbers with long division (see standard-long-division-algorithm). The new part is the decimal point. Where does it go in the answer? And what if the sharing does not come out even? You reach for this lesson whenever an amount with a decimal point is split into a whole number of equal parts. A bill shared by friends. Miles driven in a week, turned into miles per day. Gallons of gas used on several trips, turned into gallons per trip. In every one, the amount being shared has a point, and the number of parts is a whole number. This lesson takes three days. Part 1 shares a decimal with base-ten pieces and trades a one for ten tenths when the sharing gets stuck. Part 2 writes the same work as long division, with the point placed straight above. Part 3 rounds an answer that will not end, and shares among two-digit numbers of parts. ## The procedure ### Part 1: sharing a decimal, one place at a time Think of $8.37$ as base-ten pieces: $8$ ones, $3$ tenths and $7$ hundredths (see tenths-and-hundredths). A one is a large square. A tenth is a thin strip, one tenth of the square. A hundredth is a tiny square, one tenth of a strip. Now share the pieces among the three families, starting with the biggest pieces. ### [Dividing multi-digit decimals with the standard algorithm](https://math.wiki/divide-decimals-standard-algorithm/) **Summary:** Scale the divisor and dividend by the same power of ten so the divisor is whole, then divide with the long division algorithm. **Question:** A dowel is $3.15$ meters long and every peg cut from it must be $0.45$ meters. How do you divide when the divisor itself has a decimal point? ## When you reach for it Odile is a carpenter. She has a length of dowel $3.15$ meters long and wants pegs $0.45$ meters long. How many pegs does the dowel make? The question is $3.15 \div 0.45$, and this time the divisor has a decimal point. Dividing by a whole number is already in hand from divide-a-decimal-by-a-whole-number: the long division runs exactly as it does for whole numbers, and the point in the quotient goes straight above the point in the dividend. For $8.4 \div 4$, the $8$ ones give $2$ ones each and the $4$ tenths give $1$ tenth each, so $8.4 \div 4 = 2.1$. A decimal divisor is different. Long division finds each digit of the quotient by asking how many times the divisor fits into a whole number of some place, and "how many $0.45$s fit into $3$" is not a question one digit at a time can answer. The move in this lesson makes any decimal divisor whole without changing the quotient, then runs the long division you already know. Reach for it whenever the divisor has a decimal point. It is taught over three days. Day 1 makes the divisor whole. Day 2 writes zeros in the dividend, to make room for that move and to carry the division past the last digit. Day 3 deals with quotients that run long or never end. The words are the ones from whole-number division: in $3.15 \div 0.45 = 7$, the **dividend** is $3.15$, the **divisor** is $0.45$ and the **quotient** is $7$. ## The procedure ### [Divisibility and primes](https://math.wiki/divisibility-and-primes/) **Summary:** How whole numbers break into factors. **Question:** Which whole numbers divide a given number exactly, and what is every whole number made of? ## What this area is about One whole number divides another when the division leaves remainder $0$. Asking which numbers do that sorts the whole numbers greater than $1$ into primes, with exactly two factors, and composites, with more. Every composite number then breaks into primes in exactly one way, apart from order. ## How these fit together **Definitions.** Finding all factors lists the numbers that divide a number, and listing multiples goes the other way. Prime and composite numbers sorts numbers by how many factors they have, and perfect, abundant, deficient and amicable numbers by the sum of their factors other than themselves. **Shortcuts.** Divisibility rules read the last digit for $2$, $5$ and $10$; tests for three and nine read the digit sum. The sieve of Eratosthenes finds all primes up to a limit at once. **Procedures.** Prime factorization splits a number until only primes remain, and greatest common factor and least common multiple find what two numbers share. **The result.** Every whole number greater than $1$ has exactly one prime factorization, apart from order. ## A path through it Factors first. Then multiples, the divisibility rules (two, five and ten before three and nine), primes and composites, and the perfect and amicable numbers. Primes come before the sieve and prime factorization; factorization and multiples come before the greatest common factor and least common multiple, and factorization before uniqueness. ### [Divisibility rules](https://math.wiki/divisible-by-2-5-and-10/) **Summary:** The last digit alone settles these three. **Question:** A sock drive packs socks in pairs, in bundles of five and in boxes of ten. Can you tell from the count alone whether they will pack with none left over, without dividing? ## When you reach for it Rashmi runs a sock drive. The socks go out to shelters packed three ways: in pairs, in bundles of $5$, and in boxes of $10$. One week the drive collects $2{,}346$ socks. Will they pack into pairs with none left over? Into bundles of $5$? Into boxes of $10$? Each question asks whether $2{,}346$ is **divisible** by a number, which means it divides with remainder $0$. In find-all-factors-of-a-number every such question was a division. This lesson shows that for $2$, $5$ and $10$ one glance at the last digit settles it, and that for $4$, $8$ and $6$ a short look at a few digits does. The answer for $2{,}346$ comes from its last digit, $6$: the socks pack into pairs, but bundles of $5$ leave $1$ sock over and boxes of $10$ leave $6$. Reach for these tests when a number is too big to divide comfortably in your head and you only need to know whether it packs evenly, or how many are left over. They also speed up finding the factors of a number, since several of the small divisions become a glance. ## The procedure ### Part 1: the last digit decides 2, 5 and 10 Here are the three tests. Each one looks only at the **ones digit**, the last digit of the number. - **A whole number is divisible by $2$ exactly when its last digit is $0$, $2$, $4$, $6$ or $8$.** Those numbers are the **even** numbers, as in pair-up-to-test-even-odd. A number ending in $1$, $3$, $5$, $7$ or $9$ is **odd**, and pairs leave exactly $1$ over. - **A whole number is divisible by $5$ exactly when its last digit is $0$ or $5$.** - **A whole number is divisible by $10$ exactly when its last digit is $0$.** ### [Division facts](https://math.wiki/division-facts/) **Summary:** The quotients that match the known multiplication facts. **Question:** Fifty-four books go onto 9 shelves, the same number on each. Is there a way to know how many go on a shelf without dealing them out one at a time? ## When you reach for it division-as-sharing gave division its two meanings. Sharing asks how many are in each group; grouping asks how many groups there are. Either way the answer could be found by dealing things out, or by taking the same number away again and again. That always works, and it is slow. A **division fact** is a division you can know rather than work out, the same way $9 \times 6 = 54$ is a fact you know rather than count. The division facts are the quotients that match the multiplication-facts: a total up to $100$, divided by a number from $1$ to $10$, with a whole number answer from $0$ to $10$. Examples are $54 \div 9 = 6$, $48 \div 8 = 6$ and $7 \div 7 = 1$. Reach for them whenever a total has to be split evenly. Books shelved the same number to a shelf, chairs set out in equal rows for a music recital, cards dealt to players, a class split into teams. Reach for them also when checking a multiplication, and later inside every longer division, where each step is one of these facts. The key idea, in one sentence: **every division fact is a multiplication fact with one factor missing.** To find $54 \div 9$, ask "nine times what makes fifty-four?" The child already knows the multiplication facts, so the division facts come with them. This lesson takes four days, one part each. Part 1 turns a division into a missing factor. Part 2 gives ways to find the missing factor when it does not come to mind at once. Part 3 covers the special divisions with $1$ and $0$. Part 4 puts the box in other places and uses the facts to test which equal splits are possible. ### [Doubles and near doubles](https://math.wiki/doubles-facts/) **Summary:** The sums in which both parts are the same number. **Question:** Two groups have the same number of things in them. How many are there altogether, and why are those sums often the first ones a child knows without counting? And once a child knows that six and six make twelve, does she still have to count to find six and seven? ## When you reach for it A **double** is a sum in which both parts are the same number: ${4 + 4}$, ${7 + 7}$, ${10 + 10}$. The world is full of them, because so many things come in two matching halves. Two hands hold ${5 + 5}$ fingers. A spider has $4$ legs on each side. A box of twelve eggs has two rows of $6$. Two dice that land on the same face, the left and right wings of a butterfly, a child's own reflection: each one shows the same amount twice. This is why doubles are usually the first sums a child knows without counting. The picture is simple and it is always the same shape: two equal rows, each thing in one row with a partner in the other. Figure 1 shows ${4 + 4}$ both ways: as two rows of counters, and as a **number bond** with the whole in the top circle and the two parts below, the same picture as in Parts, wholes and number bonds. The only thing special about a double's bond is that both part circles hold the same number. A child reaches for a double in three kinds of situation. When the two parts are the same, the double is the answer. When two amounts are the same size and need putting together, such as two teams of $9$, the double finds the total. And when something is to be shared fairly between two, or a double is hiding inside a take-away, the double read backwards gives the answer. The doubles then do a second job, for the sums that sit right beside them. In ${6 + 7}$ the two parts are **next-door numbers**: seven comes straight after six when we count. A sum like that is called a **near double**. It is a double with one more on top, or, looked at from the other side, a double with one taken away. A child who knows ${6 + 6 = 12}$ can find ${6 + 7}$ in one short step instead of counting on six more from seven. ### [Doubling, halving and quick products](https://math.wiki/doubling-and-halving/) **Summary:** Reshaping a product into an easier one with the same value. **Question:** A train has $18$ cars, and each car carries $25$ crates of eggs. Is there a way to find $18 \times 25$ in your head, without writing out any steps? ## When you reach for it Harsha watches a freight train pull into the egg farm. It has $18$ cars, and each car carries $25$ crates of eggs. How many crates are on the train? The answer is $18 \times 25$. You could break it into partial products and add them up. But there is a quicker way. Change the problem into an easier one that has **the same answer**. Here, $$18 \times 25 = 9 \times 50,$$ and $9 \times 50$ is $9$ times $5$ tens, which is $45$ tens, or $450$. The train carries $450$ crates. The move is called **doubling and halving**: halve one factor, double the other, and the product does not change. This lesson is about that move, and about a few other quick products that work the same way. You reach for them when a problem has a factor that is close to a friendly number such as $10$, $50$ or $100$, and when you want an answer in your head. Part 1 is doubling and halving itself, and why it works. Part 2 uses the same ideas for quick products: times $5$, times $9$, and halving to divide by $2$ or $4$. ## The procedure ### Part 1: double one factor, halve the other **The picture.** Think of $18 \times 25$ as a rectangle that is $18$ squares tall and $25$ squares wide. Its area is the product: the number of unit squares inside. Now cut the rectangle in half across the middle. Each half is $9$ squares tall and still $25$ wide. Slide the bottom half up and set it beside the top half. The new rectangle is $9$ squares tall and $25 + 25 = 50$ squares wide. ### [Equal group problems](https://math.wiki/equal-groups-problems/) **Summary:** Stories about a number of groups holding the same amount each. **Question:** Six bags hold four favors each, or fifty-six slices are shared over eight tables. How do you know whether a story asks you to multiply or to divide? ## When you reach for it An **equal groups** story is a story about groups that all hold the same amount. Every story of this kind has three numbers in it: - the **number of groups** (for example, $6$ goody bags), - the **size of each group** ($4$ party favors in each bag), - the **total** ($24$ party favors in all). The story tells two of the three and asks for the third. That is the whole skill: picture the groups, see which of the three numbers is missing, and write one equation with a box where the missing number goes. Then find the box with a fact. The pieces are already in place. multiplication-as-groups gave multiplication its meaning as equal groups. division-as-sharing gave division its two questions, how many in each group and how many groups. division-facts made every division a multiplication with one factor missing, and inverse-relationship-of-multiplication-and-division showed that multiplying and dividing by the same number undo each other. This lesson puts all of that to work on stories. Reach for it whenever a story has equal groups in it: party favors packed into bags, marshmallows threaded onto sticks, pizza slices shared over tables, tents or chairs set out in equal rows, a roll of streamer cut into equal pieces, water carried in buckets of the same size. The groups can be bags, rows of an array, or equal measured amounts such as meters or liters. The lesson takes four days, one part each. Part 1 finds a missing total. Part 2 finds a missing group size or a missing number of groups. Part 3 reads the same story in an array. Part 4 uses measured amounts, and puts the box in all three places with the same three numbers. ### [Estimating a huge quantity from small facts](https://math.wiki/fermi-estimation/) **Summary:** Building a rough answer to an enormous question out of a chain of small guesses. **Question:** How many piano tuners work in a city of four million people? Nobody has counted, so how can anyone give an answer worth trusting? ## Why this exists Some questions ask for a number nobody has written down. How many times does a heart beat in a lifetime? How many grains of rice would fill a classroom? No table holds the answer, and counting is out of the question. What you do have is small knowledge, each piece guessable to within a factor of two or so: roughly how fast a heart beats, how big a grain of rice is. A chain of such facts builds the huge number, and the result is good enough to say whether a plan is possible, or whether a claim is off by a factor of a thousand. The method is named after the physicist Enrico Fermi, who was known for quick estimates of this kind. It rests on conversion factors, which cancel units along a chain, friendly numbers, which keep the arithmetic in your head, and scientific-notation, which keeps track of the zeros. ## Definition A **Fermi estimate** is an approximate value for a quantity, built by writing it as a product (sometimes with a quotient) of smaller quantities, estimating each from everyday knowledge, and multiplying them with their units. Four habits go with it. - **Every assumption is stated**, so anyone who disagrees can change that one number and redo the chain. - **Numbers keep one or two significant-figures**, because the inputs are known no better than that. - **The answer is judged by its order of magnitude.** - **Where possible it is bounded and checked**: a low and a high estimate fence it in, and a known figure, if one exists, tests it. ### [Estimating by rounding](https://math.wiki/rounding-to-estimate/) **Summary:** Round first, then compute, to check a result makes sense. **Question:** The packing house fills $487$ crates of apples a day. About how many crates is that in $6$ days, and how can you tell in a few seconds whether an exact answer is the right size? ## When you reach for it Jatin and Mere work at an apple packing house. Apples come in from the orchards, get packed into crates, and leave on trucks. Their friend Necalli runs the ticket window at an ice skating rink. All three need answers like "about how many crates this week?" and "about how much money for six tickets?" An exact answer takes paper and time. Often "about" is enough. There is a second reason to estimate. When you work out an exact answer, with paper or with a calculator, a quick estimate says what size the answer should be. If the two are close, the exact answer has the right size. If they are far apart, it is worth a second look. The tool is one you already have: rounding. To estimate, **round each number first, then compute** with the rounded numbers. How to round to a chosen place is in round-to-any-place; this lesson is about using it. ## The procedure ### Part 1: round first, then compute The packing house fills $487$ crates of apples a day. About how many crates is that in $6$ days? The exact answer is $487 \times 6$. That needs multi-digit-multiplication. But $487$ is close to $500$, and $500 \times 6$ is easy: $5 \times 6 = 30$, so $5$ hundreds times $6$ is $30$ hundreds. $$487 \times 6 \approx 500 \times 6 = 3{,}000$$ The sign $\approx$ is read "is about." It says the two sides are close, not equal. The packing house fills about $3{,}000$ crates in $6$ days. The exact number is $2{,}922$, and $3{,}000$ is close to it. ### [Estimating sums and differences](https://math.wiki/estimate-sums-and-differences/) **Summary:** Rounding first so a total or a gap can be judged quickly. **Question:** The canned food drive took in $287$ cans on Monday and $438$ on Tuesday. About how many is that, and how can a quick estimate tell you whether a worked-out total is the right size? ## When you reach for it Iara, Caue and Lucero run their school's canned food drive. On Monday $287$ cans come in, and on Tuesday $438$. The principal asks, "About how many so far?" Nobody needs the exact count to answer. "About $700$" or "about $730$" says it, and either can be worked out in your head in a few seconds. That is **estimating a sum**: round each number first, then add the easy rounded numbers. The same move estimates a **difference**: how many more books the book drive still needs, or how many more cans one class brought than another. The result is an **estimate**, a number close to the exact answer, written with the symbol $\approx$, read "is about equal to": $$287 + 438 \approx 730$$ You reach for an estimate in two situations. Sometimes "about" is all the question needs. And sometimes you have an exact answer, worked out on paper or read off a tally sheet, and you want to know whether it is the right size. An estimate cannot prove an answer right, but it can show quickly that an answer is far too big or far too small. This lesson uses whole numbers up to $1{,}000$ and rounds to the nearest ten or the nearest hundred, as taught in rounding-to-the-nearest-ten: find the two tens (or hundreds) a number sits between, and take the nearer one, with an exact halfway number rounding up. The exact sums and differences come from multi-digit-addition and multi-digit-subtraction. ### [Estimation and mental math](https://math.wiki/estimation-and-mental-math/) **Summary:** Getting close fast, and knowing when close is enough. **Question:** When does an answer only need to be close, and how do you get close in a few seconds? ## What this area is about Many questions need "about how many", not an exact count. An estimate answers them quickly, and it also checks an exact answer: if a worked-out result is far from the estimate, one of the two is wrong. Mental math is the companion habit of turning a problem into an easier one with the same answer, so it can be done without paper. ## How these fit together **Estimating by rounding.** Estimating by rounding rounds the numbers first and then computes. Estimating sums and differences applies that move to totals and gaps, written with $\approx$. **Choosing easier numbers.** Friendly numbers picks nearby numbers that work well together, such as a multiple of the divisor, and says whether the estimate is over or under. **Exact, in your head.** Mental strategies reshape a sum or product into an easier one with the same value, such as $457 + 299 = 457 + 300 - 1$. **Huge quantities.** Fermi estimation chains small guesses into a rough answer to a question nobody has counted. ## A path through it Estimating by rounding comes before friendly numbers, and friendly numbers before Fermi estimation. Sums and differences and mental strategies can come at any point. Afterward, mental strategies feed partial sums and counting mixed coins, and friendly numbers feed long division, where a two-digit divisor needs an estimate at each step. ### [Even and odd numbers](https://math.wiki/pair-up-to-test-even-odd/) **Summary:** Splitting a set into twos to see whether one object is left over. **Question:** Some piles of things can be put into pairs with nothing left over, and some always leave one thing on its own. How can we tell which kind a number is, even a big number we would rather not count out, and what happens when two such numbers are added together? ## When you reach for it On a field trip, children often walk in a **buddy line**: two by two, each child holding a partner's hand. Before the bus leaves, the teacher has one question. Will every child have a partner, or will one child be left on their own? The answer does not depend on how big the group is. Twelve children pair up exactly; thirteen leave one child alone. Forty pair up exactly; three do not. Every whole number falls into one of these two kinds, and the kinds have names. > A number is **even** when that many things can be put into pairs with nothing left over. A > number is **odd** when putting that many things into pairs leaves one thing over. Pairing is the test, and the definition says nothing else. Every other way of telling even from odd in this lesson, the doubles, the ones digit and the rules for sums, comes back to this picture and is true because of it. Two facts follow at once. First, the leftover is never more than one. If two things were still without partners, they could be partners for each other, so the pairing was not finished. When pairing is done there is either nothing left, and the number is even, or exactly one thing left, and the number is odd. Second, zero is even. With no things at all there is nothing to pair and nothing left over, and "nothing left over" is exactly what even means. A child reaches for this test whenever things go two by two: partners for a game, socks into pairs, two teams of the same size, a bag of marbles shared fairly between two friends. "Will everyone have a partner?" and "can this be shared equally between two?" are the same question, since sharing between two is handing out one pair at a time, one thing to each. ### [Every number has exactly one prime factorization](https://math.wiki/unique-factorization/) **Summary:** Whatever order the splitting happens in, the same primes appear the same number of times. **Question:** Two people break the same number into primes, starting in different places. Must they end with the same primes? ## Why this exists Chiara is a locksmith with an unusual system. Every key she cuts has a code number, and the code says how to cut it: write the code as a product of primes, then cut one notch for each prime, as deep as that prime, shallowest first. Code $140$ is $2 \cdot 2 \cdot 5 \cdot 7$, so its key has four notches, of depths $2$, $2$, $5$ and $7$. Her apprentices, Kofi and Anika, are both asked to cut code $140$. Kofi starts with $140 = 10 \cdot 14$; Anika starts with $140 = 4 \cdot 35$. Each keeps splitting until only primes are left, as a factor tree does, and Figure 1 shows what they find: the same four primes. The system only works if that always happens. If one code could break into two different lists of primes, two apprentices could cut two different keys from one code. ## Definition A **prime factorization** of a whole number $n$ greater than $1$ writes $n$ as a product of prime numbers. A prime is its own factorization: the factorization of $7$ is just $7$. Repeats are allowed and they count, so $2 \cdot 2 \cdot 5 \cdot 7$ and $2 \cdot 5 \cdot 7$ are different factorizations, of $140$ and of $70$. Two factorizations are **the same** when they use the same primes, each the same number of times, in any order: $2 \cdot 5 \cdot 2 \cdot 7$ and $7 \cdot 2 \cdot 2 \cdot 5$ are one factorization written two ways. The tidy form lists the primes from smallest to largest and gathers repeats into powers: ### [Expanded form](https://math.wiki/expanded-form/) **Summary:** Writing a number as the sum of what each of its digits is worth. **Question:** The counter at the museum door says $438$. How much of that number does each digit carry, and can the number be taken apart into those amounts and put back together again? ## Why this exists A museum has a counter at its front door. Every time a visitor walks in, the number on the counter goes up by one. At noon it shows $438$. Alemayehu, Oyuun and Kiele are on a class trip, and they want to know what that number is made of. The place-value-chart lesson answered one part of that: each digit is worth an amount that depends on its column. The $4$ is worth $400$, the $3$ is worth $30$ and the $8$ is worth $8$. This lesson puts those amounts back together. The number is the three amounts added up: $$438 = 400 + 30 + 8$$ Writing a number this way is called **expanded form**. It shows in one line what is hidden in the short way of writing. The short form $438$ is quick to write and read. The expanded form $400 + 30 + 8$ shows how much each digit carries. Both name the same number. Going the other way matters just as much. A sum of hundreds, tens and ones can be put back into a single number, even when the parts come in a jumble or a part is too big for one column. That is how blocks, bags and crates get counted, and it is the first step of adding and subtracting three-digit numbers by place value. The lesson runs over three days. - **Day 1, from a number to its parts.** Writing a three-digit number as hundreds plus tens plus ones, from the digits, from blocks and from words. - **Day 2, zero digits.** A digit $0$ is worth nothing, so its part is left out: $706 = 700 + 6$. - **Day 3, from the parts back to the number.** Parts given out of order, like $40 + 5 + 600$, and parts that are not one per place, like $400 + 17$. ### [Exponent notation](https://math.wiki/exponent-notation/) **Summary:** A small raised number counts how many times to multiply the base. **Question:** When the same number is multiplied by itself over and over, how do you write it short, and which of these powers are worth knowing by heart? ## What this area is about An exponent is shorthand for repeated multiplication. In $3^4$ the base $3$ is the factor and the exponent $4$ counts how many times it appears, so $3^4 = 3 \times 3 \times 3 \times 3 = 81$. The notation keeps long products short, and a few families of powers turn up so often that they are worth recognizing on sight. ## How these fit together **The notation.** Base, exponent and the value of a power names the parts of a power and works out what it equals. **Powers worth knowing.** Perfect squares and perfect cubes are numbers such as $49 = 7^2$ and $27 = 3^3$, which can be laid out as a full square or stacked as a full cube. Powers of ten such as $10^3 = 1{,}000$ are what each place value column is worth. ## A path through it Start with base and exponent. Perfect squares and cubes come after it; powers of ten can come before or after them. Next, the order of operations with powers says when a power is worked out, the laws of exponents calculate with powers without unpacking them, and square roots undo squaring. ### [Exponents and roots](https://math.wiki/exponents-and-roots/) **Summary:** Repeated multiplication, and undoing it. **Question:** How do you write a number multiplied by itself many times, and how do you get back to the number you started from? ## What this area is about An exponent counts repeated factors: $3^4 = 3 \times 3 \times 3 \times 3$. Combining powers by counting factors gives the laws of exponents, and running a power backward gives a root. A fractional exponent then joins the two, so that, for $a \ge 0$, $a^{1/2} = \sqrt{a}$. ## How these fit together **Notation.** Exponent notation names the base and the exponent and the powers worth knowing on sight. **Results.** The laws of exponents add exponents when powers with the same base multiply, subtract them when such powers divide (for a nonzero base), multiply them for a power of a power, and decide what zero and negative exponents must mean. **Undoing.** Square and cube roots undo squaring and cubing, and settle how many roots a number has, if any. **Applications.** Scientific notation writes a very large or very small number as $a \times 10^{n}$ with $1 \le a < 10$ and $n$ an integer. Rational exponents define $a^{m/n}$ through roots. ## A path through it Exponent notation comes first. The laws come before scientific notation; roots can come before or after the laws, but both come before rational exponents. Next come the Pythagorean theorem, exponential functions, multiplying monomials and simplifying square roots. ### [Fact families](https://math.wiki/fact-families/) **Summary:** One trio of numbers gives two addition sentences and two subtraction sentences. **Question:** A child knows that $3 + 4 = 7$. How many other adding and taking-away facts do they already know, without working anything out? ## Why this exists Children learning to add and take away meet a great many small facts, and at first each one seems to need its own work: $3 + 4$, then $4 + 3$, then $7 - 3$, then $7 - 4$. Counted out separately, that is four jobs. They are really one job. All four sentences are about the same three numbers, a whole of $7$ and its two parts $3$ and $4$, so a child who knows one of them already knows the other three. That set of sentences is a **fact family**. It cuts the number of facts a child has to hold in mind to about a quarter, and, more importantly, it gives every taking-away fact a partner addition fact to lean on. A child stuck on $7 - 3$ can ask "$3$ and what make $7$?" and use what they know about adding. ## Definition Take three numbers where two of them, the **parts**, make the third, the **whole**. The **fact family** for those numbers is the set of true adding and taking-away sentences that use those three numbers, each one once. For the parts $3$ and $4$ and the whole $7$ there are four: $$\begin{gathered} 3 + 4 = 7 \qquad 4 + 3 = 7 \\ 7 - 3 = 4 \qquad 7 - 4 = 3 \end{gathered}$$ The two adding sentences put the parts together to make the whole. The two taking-away sentences start from the whole and take one part away, leaving the other part. A fact family is the part-part-whole picture written as sentences, and the number bond is the quickest way to hold it: the whole in the top circle, the parts below, and all four sentences read off the same three circles. ### [Factors and multiples](https://math.wiki/find-all-factors-of-a-number/) **Summary:** Listing the whole numbers that divide it with nothing left over. **Question:** One hundred chairs are set out for the school photo, with every row the same length. In how many different ways can the rows go? ## When you reach for it Oddny is setting out chairs for the school photo. There are $100$ chairs, and she wants every row to have the same number of chairs, with no chair left over. Can she make $4$ rows? Can she make $3$ rows? How many different ways are there? Four rows works: $4$ rows of $25$ chairs use all $100$, because $4 \times 25 = 100$. Three rows does not: $3 \times 33 = 99$, so $3$ rows of $33$ leave $1$ chair over, and $3$ rows of $34$ would need $102$. The numbers of rows that work are exactly the numbers that divide $100$ with nothing left over. Those numbers are the **factors** of $100$. Reach for factors whenever a number of things must be split into equal groups, rows or teams with nothing left over, and you want to know every way it can be done. Chairs in rows, singers on choir risers, a class split into equal teams: each asks for the factors of a number. ## The procedure ### Part 1: a factor leaves remainder 0 In interpret-remainders-in-context a division gave a quotient and a remainder. Most of the time something is left over. Sometimes the remainder is $0$, and that is the case this lesson is about. **A factor of a whole number is a whole number that divides it with remainder $0$.** Since $100 \div 4 = 25 \text{ R } 0$, the number $4$ is a factor of $100$. Since $100 \div 3 = 33 \text{ R } 1$, the number $3$ is not. A remainder of $0$ means the groups fit exactly, so the division can be written as a multiplication with no leftover: ### [Facts for seven, eight and nine](https://math.wiki/facts-for-7-8-and-9/) **Summary:** The last and hardest block of the multiplication table. **Question:** After the zeros, ones, twos, threes, fours, fives, sixes and tens, only six facts are left. How can $9 \times 7$, $8 \times 9$ and $7 \times 7$ be built from facts you already know? ## When you reach for it By now a child knows the twos, fives and tens from facts-for-2-5-and-10, and the threes, fours and sixes from facts-for-3-4-and-6. Zero groups give $0$ and one group gives the other factor, so the zeros and ones are known too. The commutative property lets each fact count for both orders, so $7 \times 4$ is already known as $4 \times 7$. Once all of that is in place, only six facts of the table are still new: $$7 \times 7, \quad 7 \times 8, \quad 7 \times 9, \quad 8 \times 8, \quad 8 \times 9, \quad 9 \times 9$$ These are the largest products in the table and the ones that take longest to settle. This lesson builds each of them from a fact that is already known, with a move that fits the factor: - $9$ is one less than $10$, so a nine fact is a ten fact take away one group; - $8$ is $2 \times 2 \times 2$, so an eight fact is the other factor doubled three times; - $7$ is $5 + 2$, so a seven fact is a five fact and a two fact added. Each move rests on a property. The first and third are the distributive-property, which lets a factor be split into parts. The second is the associative property: in a product of three or more numbers, which pair is multiplied first does not change the answer. Reach for these moves whenever one of the six facts is half remembered. The setting is a scout troop and the relay race it runs at its summer field day. The lesson takes four days, one part each. The aim of the fourth day is the whole table: every product from $1 \times 1$ to $10 \times 10$ comes without rebuilding it, with the moves kept as the way to check. ### [Facts for three, four and six](https://math.wiki/facts-for-3-4-and-6/) **Summary:** The middle block of the multiplication table. **Question:** You already know your twos and your fives. How do they hand you $4 \times 7$, $6 \times 7$ and every other fact with a three, a four or a six in it? ## When you reach for it By now a child knows the twos, fives and tens from facts-for-2-5-and-10. Two properties turn those few facts into many more. The commutative property says the order of the factors does not change the product, so $8 \times 3$ is the same as $3 \times 8$. The distributive-property says a factor can be split into parts, each part multiplied, and the products added, so $3 \times 8$ is $2 \times 8$ and $1 \times 8$ added. This lesson uses both to fill in the middle of the multiplication table: the facts with a factor of $3$, $4$ or $6$. Each of these three numbers sits right next to a number whose facts are already known: - $3$ is $2 + 1$, so a three fact is a two fact and one more group; - $4$ is $2 + 2$, so a four fact is a two fact doubled; - $6$ is $5 + 1$, so a six fact is a five fact and one more group. It is also $3 + 3$, so a six fact is a three fact doubled. Reach for these moves whenever a fact is not known yet but a fact next to it is. That is the larger skill here, building an unknown fact from a known one, and it keeps working past this lesson. The setting throughout is a flower shop, where flowers are tied in bunches of $3$, $4$ or $6$, and buckets and shelves hold them in rows. The lesson takes four days, one part each. The aim of the fourth day is fluency: the product comes without rebuilding it, and the building move stays as the way to check it. ### [Facts for two, five and ten](https://math.wiki/facts-for-2-5-and-10/) **Summary:** The products that come straight out of the easiest skip counts. **Question:** Counting socks by twos, dimes by tens and nickels by fives is quick. How does each count turn into a multiplication fact you can just know? ## When you reach for it A child who knows what $8 \times 2$ means, $8$ equal groups of $2$ as in multiplication-as-groups, can always find it by counting. The aim now is to stop counting and know the product. The facts with a factor of $2$, $5$ or $10$ are the place to start, because each one sits at the end of a skip count the child already says out loud: - twos, from counting by twos: $2, 4, 6, 8, \dots$ - tens, from counting by tens: $10, 20, 30, 40, \dots$ - fives, from the same counting by fives: $5, 10, 15, 20, \dots$ Reach for these facts whenever things come in twos, fives or tens: pairs of socks, shoes or mittens; nickels, which are worth $5$ cents each; dimes, which are worth $10$ cents each; fingers on a hand; tens in a number. Reach for them also as stepping stones. Later facts are built from these ones: $4$ is two $2$s, and $6$ groups is $5$ groups and one more, which is the work of facts-for-3-4-and-6. This lesson takes four days, one part each. ## The procedure Each part below gives one family of facts, says why the shortcut gives the right product, and shows the picture to draw. The facts run from a factor of $0$ up to a factor of $10$. ### Part 1: twos are pairs and doubles A pair of socks is $2$ socks. So $8$ pairs of socks is $8$ groups of $2$, and the number of socks is $8 \times 2$. Count the socks by twos, one number for each pair, and stop when every pair has been counted. ### [Finding the missing number in an equation](https://math.wiki/unknown-number-in-an-equation/) **Summary:** The missing number in an addition or subtraction equation, wherever it sits, is found by asking what makes both sides the same. **Question:** In $\square + 6 = 13$ the box comes first, and in $11 = \square + 4$ it sits after the equal sign. How does a child find the missing number wherever the box is? ## When you reach for it A child who has learned that the equal sign means the same amount can now meet equations with a hole in them. The hole is a box, $\square$, and it stands for one number that nobody has said yet. The box can sit anywhere: $$\begin{gathered} \square + 6 = 13 \qquad 15 - \square = 9 \\ 11 = \square + 4 \qquad 8 + 5 = \square + 6 \end{gathered}$$ There is no story here, only the equation. (Stories that lead to equations like these have their own lesson, problems with the change or the start missing.) So the child cannot lean on "what happened first". What they can lean on is the meaning of the sign. Every one of these equations asks the same question: **which number in the box makes the two sides the same amount?** A child who reads $\square + 6 = 13$ as "something and six is the same as thirteen" has the whole problem in hand, wherever the box sits. Three ways find the number, and the child picks whichever is shortest for the numbers in front of them: - **a related fact**: an addition fact and a subtraction fact made of the same three numbers, as in fact families. If $13 - 6 = 7$ is known, so is the box in $\square + 6 = 13$. - **counting up** (or back) on a number line, stopping at $10$ on the way, as in counting on and making ten. - **a known bond**: a fact the child already owns, such as a double or a near double. Then one more move finishes every problem: **put the number back in the box and check that both sides are the same amount.** The check is not an extra; it is what "the missing number" means. ### [Finding them from prime factorizations](https://math.wiki/gcf-and-lcm-from-prime-factors/) **Summary:** Comparing the powers of each prime in the two factorizations. **Question:** Two clock gears with 72 and 84 teeth turn together. When will a marked tooth sit in its marked gap again, and can you tell without listing a long run of multiples? ## When you reach for it Mirela is restoring an old clock. Two of its gears mesh: one has $12$ teeth and the other has $18$. She paints a dot on one tooth of the small gear and on the gap of the large gear that the tooth sits in. As the gears turn, teeth pass the meeting point one at a time, the same number on each gear. The two dots meet again only when both gears have made whole turns at the same moment, that is, when the number of teeth that have passed is a multiple of $12$ and of $18$. The first time is their least common multiple. Listing multiples settles this pair quickly: $18$, $36$, and $36$ is a multiple of $12$. After $36$ teeth the small gear has turned $3$ times and the large one $2$ times. The next pair in the clock has $72$ and $84$ teeth. Listing still works, but only at the sixth multiple of $84$, and finding their greatest common factor by listing factors means writing out twelve factors of each. With bigger numbers listing becomes slow and easy to get wrong. This lesson finds both numbers straight from the prime factorizations by comparing the powers of each prime. Reach for it when the numbers are large, when they have many factors, or when the factorizations are already in front of you. ## The procedure ### Part 1: comparing the powers of each prime **Factorize both numbers and line up the primes.** A factor tree or a ladder gives $$72 = 2^3 \cdot 3^2 \qquad 84 = 2^2 \cdot 3 \cdot 7$$ ### [Friendly numbers and deliberate over and under estimates](https://math.wiki/compatible-numbers/) **Summary:** Swap in numbers that are easy to work with together. **Question:** The food drive has $4{,}186$ cans to share among $7$ trucks. Nobody can divide that in their head, so which nearby number can you divide? ## When you reach for it Nayeli, Satoshi and Wenjing run their school's food drive. Cans arrive by the hundred, get packed into boxes and go out on trucks. Wenjing also places the school's order of supplies: boxes, tape, labels and markers. All day they need answers like "about how many cans go on each truck?" and "is $60$ dollars enough?" An exact answer takes paper and time. An estimate takes a few seconds, as long as the numbers are easy to work with. You already know one way to estimate: round each number to a place, then compute (rounding-to-estimate). That works well for adding. For dividing and multiplying it often leaves numbers that are still awkward. This lesson adds three ideas, one for each part. - **Compatible numbers**, also called friendly numbers: numbers close to the real ones, chosen so that the computation becomes a basic fact you know. - **Benchmarks**: easy landmark numbers, such as $0$, $\tfrac{1}{2}$ and $1$, or $25$, $50$ and $100$, that other numbers are compared with. - **Overestimates and underestimates**: knowing whether an estimate is above or below the exact answer, how far off it is, and when that matters. ## The procedure ### Part 1: numbers chosen to fit a fact Seven trucks will share $4{,}186$ cans equally. About how many cans go on each truck? Try plain rounding first. To the nearest thousand, $4{,}186$ becomes $4{,}000$. But $4{,}000 \div 7$ is not a fact you know, because $7$ does not go evenly into $40$. To the nearest hundred, $4{,}186$ becomes $4{,}200$, and that one does work. The rounding was lucky. ### [Greatest common factor and least common multiple](https://math.wiki/gcf-and-lcm/) **Summary:** The biggest shared factor and the smallest shared multiple. **Question:** Two numbers share some factors and some multiples. How do you find the biggest factor and the smallest multiple they share? ## What this area is about The **greatest common factor** of two positive whole numbers is the largest number that divides both. The **least common multiple** is the smallest positive number that both divide. The first answers questions about splitting things into identical groups; the second answers questions about when two repeating things line up again. For $4$ and $6$ they are $2$ and $12$. ## How these fit together **By listing.** Greatest common factor by listing writes out the factors of each number and takes the largest they share. Least common multiple by listing writes out multiples of each and takes the first they share. **By prime factors.** Finding them from prime factorizations compares the powers of each prime in the two factorizations, which is faster when listing would run long, as with $72$ and $84$. ## A path through it Learn both listing methods first, in either order, then the prime factorization method. These two ideas come back in common denominators, in factoring out the greatest common factor, and in the Euclidean algorithm. ### [Greatest common factor by listing](https://math.wiki/gcf-by-listing-factors/) **Summary:** Writing out the factors of each number and taking the largest they share. **Question:** A food bank has 18 tins and 24 packets to share out into identical parcels, with nothing left over. How many parcels can it make, at most? ## When you reach for it On Saturday morning the food bank has $18$ tins of beans and $24$ packets of pasta to give out. Tamsin, who runs the packing table, wants every parcel to be exactly like every other, and she wants nothing left on the table at the end. How many parcels can she make? Start with the tins. If she makes some number of parcels, the $18$ tins are shared equally among them, so that number must go into $18$ with nothing left over: it is a factor of $18$. The packets ask the same of $24$. So the number of parcels has to be a factor of $18$ **and** a factor of $24$. Two parcels would work, and so would three, but Tamsin wants as many parcels as she can make, so that more families get one. She needs the largest number that is a factor of both. That number is $6$: six parcels, each with $3$ tins and $4$ packets. Reach for this whenever two or more amounts must be split into identical shares with nothing left over, and you want the most shares or the biggest share. The same number turns up when you simplify a fraction such as $\frac{18}{24}$ and when you pull a shared factor out of a sum such as $18 + 24$; both come up at the end of this lesson. ## The procedure ### A factor list, in pairs A **factor** of a whole number is a whole number that divides it with nothing left over, as in factors and multiples. To be sure of finding every factor, find them in pairs. Try $1$, $2$, $3$ and so on in turn. Each time a number divides $n$, the quotient divides $n$ too, because the two multiply back to $n$. So every success gives two factors at once. For $36$: ### [Grouping symbols](https://math.wiki/grouping-symbols/) **Summary:** Brackets that mark a part of an expression to be treated as one number. **Question:** The agreed order says multiply before you add. What do you write when the story needs the adding done first? ## Why this exists Dayo is buying paint at Jiaqi's paint store. He picks $3$ cans of blue and $4$ cans of white, and every can costs $5$ dollars. To find what he pays, you add the cans and then multiply by the price. Try writing that as one line with no extra marks: $3 + 4 \times 5$. The agreed order from numerical-expressions says multiply first, so that line names $3 + 20 = 23$. But Dayo does not pay $23$ dollars. He has $7$ cans at $5$ dollars each, and he pays $35$ dollars. The agreed order is a good one, and nobody wants to change it. What is missing is a way to say "do this part first" when a story needs it. That is what grouping symbols do: $$(3 + 4) \times 5 = 7 \times 5 = 35$$ The parentheses say: add $3$ and $4$ first, and treat the result as one number. The lesson takes three days. **Part 1** is parentheses: what they mean, how to work an expression that has them, and where to put them to reach a target value. **Part 2** is brackets and braces, for a group inside a group, worked from the inside out. **Part 3** is the fraction bar, which is a grouping symbol too, and parentheses that change nothing. ## Definition A **grouping symbol** marks a part of an expression that is to be worked out first and then treated as one number. There are four of them. - **Parentheses**, $(\;\;)$, are the most common: $(3 + 4) \times 5$. - **Brackets**, $[\;\;]$, and **braces**, $\{\;\;\}$, mean exactly what parentheses mean. They are used when one group sits inside another, so the eye can tell which closing mark goes with which opening mark: $2 \times [12 - (3 + 5)]$. - The **fraction bar** is the line in $\frac{12 + 8}{9 - 4}$. Everything above the bar is one group, everything below it is another, and the bar means divide the top by the bottom. ### [Higher roots](https://math.wiki/nth-roots/) **Summary:** Roots of any index, and how the index decides whether negatives are allowed. **Question:** A number multiplied by itself four, five or six times gives a known result. Which number was it, and when is there one answer, two, or none? ## Why this exists A square root undoes squaring and a cube root undoes cubing. Nothing stops at three. A quantity that is multiplied by the same factor every year for four years has grown by that factor used four times, and to recover the yearly factor from the four-year total you need a number whose fourth power is known. Equations such as $x^5 = -243$ and $x^6 = 729$ ask the same kind of question with five and six equal factors. The cube roots lesson met these higher roots in passing. This lesson treats them in full, and one fact runs through all of it: whether the index is even or odd decides almost everything. It decides whether a negative number has a root, whether an equation has one solution or two, whether undoing a power gives back $x$ or only $|x|$, and when the rules for products and quotients may be used at all. ## Definition Let $n$ be a whole number, $2$ or more. A number $b$ is an **$n$-th root** of a number $a$ when $b$ used $n$ times as a factor gives $a$: $$b^n = a$$ The radical sign names one particular root, the **principal $n$-th root**, written $\sqrt[n]{a}$. The number $n$ is the **index** and $a$ is the **radicand**. For the square root the index $2$ is left unwritten, so $\sqrt{a}$ means $\sqrt[2]{a}$. Which root the sign names depends on whether $n$ is odd or even. **Odd index** ($n = 3, 5, 7, \ldots$). Every real number $a$ has exactly one real $n$-th root, and $\sqrt[n]{a}$ is that root. It has the same sign as $a$. For example $\sqrt[5]{-243} = -3$ because $(-3)^5 = -243$. ### [How counting tells how many](https://math.wiki/cardinality-principle/) **Summary:** The final word said while counting a set names the size of the whole set. **Question:** After a child counts a group, what does the last counting word tell, and why does it stay the same when the group is moved or counted in a new order? ## Why this exists When a child counts a group, the last word said is not the name of the last thing touched: it is the size of the whole group. That number belongs to the group, so it stays the same when the things are moved, bunched up or spread out, and it comes out the same whichever thing is counted first. These four days take a child who can already count a set of objects and make that last word mean "how many", for groups up to $20$ in lines, rings and rows. Each day takes about ten to fifteen minutes. Ask "how many?" often, and ask "how do you know?" even more. ## Day 1: The last word is all of them **You need:** about $15$ pom-poms, a paper bag, a hand puppet (a sock works). Have the child drop pom-poms into the bag one at a time, saying a counting word with each drop. Stop at $5$ and fold the bag shut. Ask, "How many pom-poms are in the bag?" The child cannot see them or count them again, so the only way to answer is to remember the last word. That is the point: "five" is now a fact about what is in the bag. Next, pour out a new group and let the child count it. Then say, "Show me the five." A child who points at one pom-pom is treating "five" as that pom-pom's name. Sweep your hand around the whole group and say, "These are five. All of them." Have the child copy the sweep. Last, let the puppet count and get it wrong on purpose. It counts a group correctly and then says a different number, or it starts over from one when asked "how many". The child explains what the puppet should have said. ### [How many more and how many fewer](https://math.wiki/comparison-word-problems/) **Summary:** Stories that ask for the gap between two amounts. **Question:** A story says one child has some more than another. How do you tell whether to add or to take away? ## When you reach for it Some stories put two amounts side by side and talk about the gap between them. "Arlo has $3$ more apples than Wren." "Finn has $4$ fewer crayons than Bea." "The sunflower is taller than the tulip." Nothing joins, nothing leaves; two amounts are simply compared. These are **comparing stories**. A child has already met the gap itself in subtraction as the difference: when both amounts are known, the gap is the bigger take away the smaller. A story can hide any of the three numbers, though, not only the gap. It can give the gap and one amount and ask for the other amount. Then the question is whether to add or to subtract, and the words of the story do not settle it. "Tariq has $5$ more marbles than Suki" can call for $+5$ or for $-5$, depending on whose marbles we already know. What settles it is a picture of who has more. This lesson is about drawing that picture and reading the answer off it. Read every story aloud with the child, and let them build it with counters or cubes before any sum is written. ## The procedure ### Three numbers in every comparing story Rosa has $8$ stickers and Dev has $5$. Line the stickers up in two rows, Rosa's on top, and give each of Dev's stickers a partner from Rosa's row. Rosa's row now falls into two pieces. The first piece is the stickers that found a partner, and there are exactly as many of them as Dev has, $5$. The second piece is the stickers with no partner, $3$ of them: that is how many more Rosa has. So Rosa's row is a whole made of two parts, just like the pictures in parts and the whole: ### [Identity and inverse elements](https://math.wiki/identity-elements/) **Summary:** Zero leaves a sum unchanged and one leaves a product unchanged. **Question:** Is there a number that changes nothing when you add it or multiply by it, and can the effect of every number be undone? ## Why this exists Every equation you solve ends the same way: the letter stands alone on one side. Getting there means removing whatever has been done to the letter. In $n + 15 = 6$ something has been added to $n$; in $\frac{2}{5}n = 6$ it has been multiplied by something. Which number removes each, and what exactly is left when it has? The answers are two pairs of ideas. An **identity** is a number that does nothing: adding $0$ leaves a number as it was, and so does multiplying by $1$. An **inverse** is a number that undoes another: adding $-15$ undoes adding $15$, and multiplying by $\frac{5}{2}$ undoes multiplying by $\frac{2}{5}$. With these named, every step in solving an equation has a reason, and you can see which steps are always allowed and which one never is. ## Definition **Identities.** For every real number $a$, $$a + 0 = a \qquad\text{and}\qquad 0 + a = a.$$ Adding zero changes nothing, so $0$ is called the **additive identity**. In the same way, for every real number $a$, $$a \cdot 1 = a \qquad\text{and}\qquad 1 \cdot a = a,$$ and $1$ is called the **multiplicative identity**. **Inverses.** An inverse is defined by aiming at an identity. The **additive inverse** of $a$ is the number that, added to $a$, gives the additive identity $0$. It is the opposite of $a$, written $-a$: $$a + (-a) = 0.$$ The **multiplicative inverse** of a nonzero number $a$ is the number that, multiplied by $a$, gives the multiplicative identity $1$. It is the **reciprocal** of $a$, written $\frac{1}{a}$: ### [Imaginary and complex numbers](https://math.wiki/standard-form-of-a-complex-number/) **Summary:** Every complex number can be written as a real part plus an imaginary part. **Question:** No real number squares to a negative, so $x^2 + 9 = 0$ has no real solution. What if we allow a number whose square is $-1$? ## Why this exists Try to solve $x^2 + 9 = 0$. Subtracting $9$ gives $x^2 = -9$, and there the search stops. A positive number times itself is positive, a negative number times itself is positive too, and $0^2 = 0$. No real number has a negative square. Figure 1 shows it: the curve $y = x^2$ meets the level $y = 4$ twice, above $x = -2$ and $x = 2$, and never comes down to the level $y = -1$. For a long time that settled it: such equations had no solution. The trouble was that square roots of negative numbers kept turning up in the middle of problems with perfectly ordinary answers. In 1545 Gerolamo Cardano asked for two numbers that add to $10$ and multiply to $40$, and his method gave $5 + \sqrt{-15}$ and $5 - \sqrt{-15}$. He did not know what to make of them, yet if $\sqrt{-15}$ is treated as a number whose square is $-15$, they do add to $10$, and they multiply to $25 - (-15) = 40$. Later in the same century Rafael Bombelli took the cubic $x^3 = 15x + 4$, whose solution $x = 4$ anyone can check, and found that the known formula for cubics passes through $\sqrt{-121}$ on the way to it. Refusing those roots meant refusing a route to real answers. The way forward was the one taken before. Negative numbers were added so that $5 - 8$ has an answer, and fractions so that $3 \div 4$ has one. Here one new number is added, a number whose square is $-1$, the rules of arithmetic stay as they were, and we see what that forces. The result is the **complex numbers**, and in them every quadratic equation has solutions. ### [Integers](https://math.wiki/integers/) **Summary:** Whole numbers and their negatives: the counting numbers extended both ways. **Question:** When amounts run both ways from a starting point, how do numbers below zero behave? ## What this area is about The integers are the whole numbers together with their negatives: $\ldots, -2, -1, 0, 1, 2, \ldots$ They describe amounts measured from a starting point with room on both sides, such as temperatures below freezing or heights below sea level. Each integer has a side of zero and a size, and the four operations have to be made to work with both. ## How these fit together **Definitions.** Negative numbers name the amounts on the far side of zero. Opposites and zero pairs pair each number with the one the same distance away on the other side, so that $a + (-a) = 0$. Absolute value keeps only the size: the distance from $0$. **A skill.** Placing and comparing integers puts them on a line that runs both ways and reads which of two is greater. **Operations.** Adding and subtracting integers combines signed numbers, with subtracting as adding the opposite. Multiplying and dividing integers explains why the sign of a product of nonzero integers depends on how many factors are negative. ## A path through it Negative numbers first. Opposites and the number line follow, then absolute value (after the line), then adding and subtracting, and multiplying and dividing last. Next come rational numbers, where the same rules extend to signed fractions and decimals, and distance on a number line. ### [Irrational and real numbers](https://math.wiki/irrational-and-real-numbers/) **Summary:** Numbers that are not fractions, and the complete number line they fill in. **Question:** Fractions crowd the number line, but do they fill it, and what are the numbers that are left? ## What this area is about Between any two fractions there is another, so it is tempting to think fractions reach every point on the number line. They do not: the side of a square of area $2$ is $\sqrt{2}$, which is not a ratio of integers. Numbers like it are irrational, and the rational and irrational numbers together are the real numbers, which fill the whole line. ## How these fit together **Definition.** Irrational numbers shows why $\sqrt{2}$ cannot be a fraction, and that its decimal never ends or repeats. **Skills.** Sorting a number into its sets decides whether a number is whole, an integer, rational or irrational, often several at once. Approximating irrational numbers brackets a square root such as $\sqrt{29}$ between the perfect squares on either side. **Results.** The real number line matches every point of the line with a real number. Properties of real numbers names the rules for adding and multiplying that hold for every real number, which algebra relies on. ## A path through it Irrational numbers first. Sorting numbers, approximating square roots and the real number line can follow in any order, but sorting and the real line both come before the properties. Next comes proof by contradiction, where the proof that $\sqrt{2}$ is irrational is the leading example. ### [Irrational numbers](https://math.wiki/square-roots-as-irrational/) **Summary:** Why the square root of 2 cannot be a fraction. **Question:** A square of area $2$ is easy to draw. Is the length of its side a fraction, if only we look hard enough for the right one? ## Why this exists Between any two fractions there is another fraction, so fractions crowd the number line with no gap you could point to. It is natural to think they reach every point, so that every length would come out as an integer over an integer if only the ruler were fine enough. Figure 1 tests that belief. Join the midpoints of the sides of a $2$ by $2$ square. The tilted square inside has area $2$, because the four white corners together make two of the four small squares. So its side is a length $s$ with $s^2 = 2$, the number written $\sqrt{2}$. You can draw it with a ruler, and it sits at a definite point of the number line. Hunting for a fraction equal to it comes very close and never arrives. $\frac{140}{99}$ squared is $\frac{19600}{9801}$, just under $2$. $\frac{99}{70}$ squared is $\frac{9801}{4900}$, just over. This lesson shows that the miss can never be zero: no fraction has square $2$. Greek mathematicians knew this more than two thousand years ago, and it forced them to accept that lengths are not all ratios of whole numbers. ## Definition A number is **rational** when it can be written as $\frac{a}{b}$ with $a$ and $b$ integers and $b \neq 0$, as in what-makes-a-number-rational. An **irrational number** is a number on the number line that is not rational: no way of writing it as an integer over a nonzero integer exists. The same split can be read from decimals. Every rational number has a decimal that either stops, like $\frac{3}{8} = 0.375$, or repeats one block of digits forever, like $\frac{4}{37} = 0.\overline{108}$ (see repeating-decimal-notation), and every decimal of either kind is rational. So: ### [Laws of exponents](https://math.wiki/laws-of-exponents/) **Summary:** Multiply powers by adding exponents, divide by subtracting, power of a power by multiplying, and why. **Question:** Powers are shorthand for long products. How do you multiply, divide and raise them without writing the products out, and why do the rules work? ## What this area is about A power such as $2^{10}$ counts factors. The laws of exponents are what that counting gives when powers are combined: multiplying lines up two rows of factors, so exponents add; dividing cancels factors, so exponents subtract; a power of a power repeats a row, so exponents multiply. None of them needs memorizing apart from that reason, and the same reason decides what a zero or negative exponent has to mean. ## How these fit together **The first rule.** Multiplying and dividing powers with the same base gives $a^m \cdot a^n = a^{m+n}$ and, for $a \ne 0$, $a^m \div a^n = a^{m-n}$. **Extending it.** Powers of powers, products and quotients gives $(a^m)^n = a^{mn}$, and shows how a power spreads over a product or a fraction, as in $(5x)^3$. **New exponents.** Zero and negative exponents defines $a^0 = 1$ and $a^{-n} = \frac{1}{a^n}$, for $a \ne 0$, as the only meanings that keep the dividing rule true. ## A path through it Start from exponent notation. Learn the product and quotient rule first; the power-of-a-power rule and zero and negative exponents both follow from it, in either order. These laws make scientific notation, multiplying monomials and rational exponents possible. ### [Least common multiple by listing](https://math.wiki/lcm-by-listing-multiples/) **Summary:** Writing out multiples of each number and taking the first they share. **Question:** Two lighthouses flash at the same moment. One flashes every four seconds and the other every six. When do they next flash together? ## When you reach for it Rosa stands on a cliff between two lighthouses. The North light flashes every $4$ seconds and the South light every $6$ seconds, and both flash at the moment she starts her stopwatch. Counting from then, the North light flashes at these times, in seconds: $$4,\ 8,\ 12,\ 16,\ 20,\ 24,\ 28,\ \dots$$ These are the **multiples** of $4$. The South light flashes at the multiples of $6$: $$6,\ 12,\ 18,\ 24,\ 30,\ \dots$$ The two flash together whenever a time is on both lists. Those times, $12$, $24$, $36$ and so on, are the common multiples of $4$ and $6$, and the first of them, $12$ seconds, answers Rosa's question. The **least common multiple** of two whole numbers, neither of them $0$, is the smallest positive number that is a multiple of both. It is written $\text{LCM}(a, b)$, so $\text{LCM}(4, 6) = 12$. Three or more numbers have one too: the smallest positive number that is a multiple of all of them. The word "positive" matters, because $0 = 4 \times 0$ is a multiple of every number, and without it every least common multiple would be $0$. Reach for it whenever things repeat on different cycles and you want the first time they line up again: lights, bells, rotas, or a pile of $6$ cm tiles beside a pile of $8$ cm tiles. One quick sign that a question wants it: the answer is built by repeating the given numbers, so it is at least as big as the larger of them. ## The procedure ### [Listing multiples](https://math.wiki/multiples-of-a-number/) **Summary:** Writing out the numbers a given number divides into. **Question:** Yaretzi jumps rope in rounds of exactly $7$ jumps. Which totals can her count ever land on, and could it ever land on $30$? ## When you reach for it Yaretzi jumps rope in rounds. Every round is exactly $7$ jumps, and she counts as she goes. After $1$ round her count is $7$. After $2$ rounds it is $14$, after $3$ rounds $21$, and so on. Each round adds $7$ more. So which totals can her count be at the end of a round? Only the numbers $7 \times 1$, $7 \times 2$, $7 \times 3$, and so on. These are the **multiples** of $7$. Could her count be $30$ at the end of a round? Look at the number line: $7 \times 4 = 28$ and $7 \times 5 = 35$, and $30$ sits between them. No hop lands on $30$, so $30$ is not a multiple of $7$. Reach for multiples whenever something comes in equal amounts again and again: jumps in a round, beads in a repeat of a pattern, cookies in a box. The total after some number of whole repeats is always a multiple of the amount in one repeat. ## The procedure ### Part 1: listing the multiples of a number **A multiple of a whole number is that number times a whole number.** The multiples of $7$ are $$7 \times 1,\quad 7 \times 2,\quad 7 \times 3,\quad 7 \times 4,\quad \ldots$$ which are $7, 14, 21, 28, \ldots$ Writing them out is skip-counting by $7$: counting in equal jumps of $7$ instead of jumps of $1$. **Why each one is the last one plus the number.** Going from $7 \times 4$ to $7 \times 5$ adds one more group of $7$: $$7 \times 5 = 7 \times 4 + 7 \times 1 = 28 + 7 = 35$$ Splitting $5$ groups into $4$ groups and $1$ group like this is the distributive property. It is why the list can be made by adding $7$ each time, and it is why the hops on the number line are all the same length. ### [Long division](https://math.wiki/long-division/) **Summary:** Divide place by place, from the biggest, bringing down what is left. **Question:** A number is far too big for any times-table fact. How do you divide it, by one digit or by two, a piece at a time? ## What this area is about Long division splits one big division of whole numbers into a few small ones. It works from the biggest place down, trades whatever is left into the next place, and records the quotient one digit at a time, with a remainder when the division does not come out even. ## How these fit together **First methods.** Partial quotients takes away easy multiples of the divisor, such as $100$ or $10$ groups at once, and adds up how many groups were taken. Dividing by a single digit shares hundreds, tens and ones in turn, trading what is left into the next smaller place. **The algorithm.** The standard long division algorithm is partial quotients with no choosing: divide, multiply, subtract and bring down, one quotient digit in each place. **A harder divisor.** Dividing by a two-digit number keeps the same layout, but each quotient digit is now a guess, made by rounding the divisor, and then tested. ## A path through it Partial quotients first, since dividing by a single digit refers back to it; then dividing by a single digit; then the standard algorithm; then two-digit divisors. All of it rests on multi-digit multiplication. Finish here and you can divide a decimal by a whole number, turn a fraction into a decimal, and later carry the same steps over to polynomial long division. ### [Making ten and bridging through ten](https://math.wiki/making-ten-to-add/) **Summary:** Moving part of one addend across to fill a ten before adding the rest. **Question:** Eight and five make more than ten, and thirteen take away five drops below it. How can the ten hiding inside sums and differences like these be used to find the answer without counting one by one? ## When you reach for it This lesson has two halves that are one idea. The first half, making ten, adds two numbers whose total goes past ten. The second half, bridging back through ten, takes away across ten. Both stop at $10$ on the way, because from ten the rest can be said at once: $10 + 3$ is thirteen, and $10 - 2 = 8$ is a bond of ten. ### Adding past ten A child adding $3 + 4$ can count on from $4$ in three small steps, or simply know the answer. A sum like $8 + 5$ is harder. Its total is past ten, so counting on takes five counts, "nine, ten, eleven, twelve, thirteen", and each count is a chance to slip. Making ten is the move for these sums: the two numbers are each less than ten, but together they make more. The move rests on three things the child already has. The first is the pairs that make ten, from the bonds to ten: $8$ needs $2$ more to be $10$. The second is that a small number can be split into two parts, from the bonds to five and the bonds to ten: $5$ is $2$ and $3$. The third is that ten and some more is a teen number: $10 + 3$ is thirteen, with no counting at all (add-ten-to-a-number, Teen numbers as ten and some more). Put together, they give this. To find $8 + 5$, take $2$ of the $5$ to fill the $8$ up to $10$. That leaves $3$. And $10$ and $3$ make $13$. Two ten frames show it best (Figure 1). The $8$ dark counters sit in the first frame, with two boxes empty. The $5$ light counters go in next, filling the boxes in order, so $2$ of them finish the first frame and the other $3$ spill into the second. Now the first frame is a full ten and the second holds $3$. Nobody needs to count: a full frame and $3$ more is $13$. ### [Mental strategies](https://math.wiki/mental-strategies/) **Summary:** Make a ten, double and halve, break apart: computing without paper. **Question:** Some sums and products look hard on paper but are easy in your head. What makes them easy, and how do you turn a hard one into an easy one? ## What this area is about Mental math is not a faster way to run the paper method. It is a habit of changing a problem into an easier one with the same answer before working it out. A number close to a round one can be rounded and then fixed. A product can be reshaped, one factor halved and the other doubled, until it is one you already know. The skill is spotting when such a change pays off. ## How these fit together The area has two strategies, one for each pair of operations. **Adding and subtracting.** Compensation when adding and subtracting adds or subtracts a round number and then takes back the extra, so $457 + 299$ becomes $457 + 300 - 1$. **Multiplying.** Doubling, halving and quick products reshapes a product into an easier one with the same value, so $18 \times 25$ becomes $9 \times 50$, and halves to divide by $2$ or $4$. ## A path through it The two strategies stand apart; compensation usually comes first, since it needs only addition and subtraction. Both feed the written method of partial sums and counting a handful of mixed coins. ### [Multi-digit addition](https://math.wiki/multi-digit-addition/) **Summary:** Add by place value, regrouping when a column passes nine. **Question:** Once numbers have tens, hundreds and more, how do you add them without counting on one at a time? ## What this area is about Adding big numbers rests on one idea: add like places to like places, tens to tens and ones to ones. The only snag is a place that adds up to more than nine. Ten ones become one ten, ten tens become one hundred, and the rest of the sum goes on as before. Every method in this area is that idea written down a different way. ## How these fit together **Place by place, no trades.** Adding two-digit numbers without regrouping adds tens to tens and ones to ones, as in $43 + 5$ or $27 + 40$. **Making a new ten.** Adding by making a ten fills up to the next ten, as in $38 + 5$. Regrouping in addition is the rule behind it: no place ever holds more than nine. **Written methods.** The open number line jumps by tens, then ones. Partial sums adds each place on its own and then combines the parts. The standard algorithm works column by column from the right, carrying into the next place, for numbers of any length. ## A path through it Begin with two-digit sums that need no trade. Then make a ten, and learn regrouping. The open number line fits anywhere after that. Partial sums comes before the standard algorithm, which is where the area ends. With addition in hand, multi-digit subtraction runs the same ideas in reverse, and estimating sums checks any answer. ### [Multi-digit multiplication](https://math.wiki/multi-digit-multiplication/) **Summary:** Break numbers by place value, multiply the parts, add them up. **Question:** The facts you know stop at ten times ten. How do you multiply numbers that have several digits each? ## What this area is about Every large product of whole numbers is built from small ones. Split the factors by place value, multiply the parts using basic facts and shifts of place, and add the pieces. The distributive property is what guarantees the pieces add up to the whole product. ## How these fit together **The building block.** Multiplying by tens and hundreds gets products such as $40 \times 600$ from a basic fact, here $4 \times 6 = 24$, and a shift of places. **The method that shows why.** Partial products splits the factors by place value, multiplies every pair of parts, as the pieces of an area model, and adds the results. **The short record.** The standard multiplication algorithm does the same work with one row for each digit of the bottom factor, carrying into the next column as it goes. ## A path through it Multiplying by tens and hundreds and partial products come first, in either order; the standard algorithm comes last, since each of its rows is a shortened partial product. All of it starts from the multiplication facts. Finish here and you are ready for long division and for estimating by rounding to check that a product makes sense. ### [Multi-digit subtraction](https://math.wiki/multi-digit-subtraction/) **Summary:** Subtract by place value, regrouping when a column runs short. **Question:** When one number has several digits, how do you take another away when a place on top does not have enough? ## What this area is about Subtracting big numbers takes like places from like places: hundreds from hundreds, tens from tens, ones from ones. Trouble starts when a place on top is smaller than the place below it. Then one ten is opened up into ten ones (or one hundred into ten tens), and the subtraction goes on. Every lesson here leads up to that trade, uses it, or goes around it. ## How these fit together **No trades needed.** Subtracting multiples of ten treats $80 - 50$ as $8$ tens take away $5$ tens. Subtracting two-digit numbers without regrouping works when each place has enough. **The trade.** Regrouping in subtraction breaks a ten into ten ones when a place runs short. Subtracting three-digit numbers uses it with hundreds too. **Written methods.** The standard algorithm works column by column from the right for numbers of any length. Subtracting across zeros handles a trade when the next places hold nothing, as in $7{,}006 - 2{,}849$. **Another way.** Counting up starts at the smaller number and adds on to the larger one, which often skips the trade altogether. ## A path through it Start with multiples of ten and two-digit problems with no trade. Then learn regrouping, and take three-digit subtraction and the standard algorithm, then subtracting across zeros. Counting up fits in at any point after that. Next come subtracting decimals and, later, elapsed time, both built on this. ### [Multiplication and division](https://math.wiki/multiplication-and-division/) **Summary:** Repeated groups, and splitting into equal groups. **Question:** When every group holds the same amount, how do two numbers decide the total, and how do you work backward from a total? ## What this area is about Multiplication counts equal groups: when every group is the same size, the number of groups and the size of each decide the total. Division runs the other way, from a total back to the size of a group or the number of groups. Here the numbers are whole numbers. ## How these fit together **Meanings.** What multiplication means, rows and columns of objects, arrays, the area model, times as many and what division means. **Results.** Properties of multiplication, the distributive property (splitting a factor into parts), and how multiplying and dividing undo each other. **Facts and methods.** Multiplication facts through $10 \times 10$, division facts, multi-digit multiplication and long division. **Stories.** Equal group problems, what a remainder means and problems using several operations. ## A path through it Start with what multiplication means, then arrays and the properties, and the distributive property before the facts. The area model and the facts lead to multi-digit multiplication. On the division side: what division means, then division facts (after the multiplication facts), then undoing, equal group stories and multi-step problems. Remainders and multi-digit multiplication come before long division, which comes last. Times as many fits anywhere after the first star. Next come fractions, divisibility and primes, exponents and the order of operations. ### [Multiplication and division undo each other](https://math.wiki/inverse-relationship-of-multiplication-and-division/) **Summary:** Dividing by a number reverses multiplying by it. **Question:** A child knows that $7 \times 9 = 63$. Which division facts come free with it, and is there any division that no multiplication fact can ever answer? ## Why this exists In division-facts every division was turned into a multiplication with a box: $54 \div 9$ asks $9 \times \square = 54$. That habit works, but it leaves two questions open. The first is how much one fact is worth. A child who knows $7 \times 9 = 63$ knows more than one fact. The same three numbers give a second multiplication and two divisions, the way fact-families gave four adding and taking-away sentences from one train of cubes. Seeing all four at once means the multiplication table and the division facts are one thing to learn, not two. The second is why the box trick is allowed at all. It works because multiplying and dividing by the same number **undo each other**: multiply $8$ by $6$ and you get $48$, divide $48$ by $6$ and you are back at $8$. Undoing is how a division gets checked, and how a hidden starting number gets found. And the undoing has one place where it breaks. Multiplying by $0$ turns every number into $0$, so nothing can undo it. That is the whole reason a number cannot be divided by zero, and it is the last part of this lesson. The lesson takes three days. Part 1 builds multiplication and division fact families. Part 2 shows that multiplying and dividing by the same number undo each other. Part 3 explains what zero does to division. ## Definition Multiplication and division by the same number are **inverse operations**: each one undoes the other. Multiply a number by $6$, then divide the result by $6$, and you get the number you started with. Divide by $6$, then multiply by $6$, and again you are back where you began. ### [Multiplication facts](https://math.wiki/multiplication-facts/) **Summary:** Products through ten times ten, from understanding to fluency. **Question:** There are a hundred products up to ten times ten. How do you get from counting equal groups to simply knowing them? ## What this area is about Multiplication starts as counting equal groups. This area turns that counting into fluency with every product of two whole numbers from $0$ to $10$. No fact is learned alone: each new one is built from facts already known, and the commutative property lets every fact count in both orders. ## How these fit together **The starting facts.** Facts for two, five and ten come straight out of skip counts the child already says. **Facts built from known ones.** Facts for three, four and six use the distributive property: a three fact is a two fact and one more group. Facts for seven, eight and nine build the last six facts from known ones, for example a nine fact is a ten fact take away one group. **The table as a whole.** Patterns in the multiplication table explains why rows, columns and the diagonal repeat: mirror cells across the diagonal, square numbers, and where the odd products sit. ## A path through it Twos, fives and tens first, then threes, fours and sixes, then sevens, eights and nines. The patterns lesson comes last: only the first block is needed to start it, but it reads the whole table at once. Finish here and you are ready for division facts, multi-digit multiplication, powers such as $3^4$ and the order of operations. ### [Multiplying a decimal by ten, one hundred or one thousand](https://math.wiki/multiply-decimals-by-powers-of-ten/) **Summary:** Each factor of ten moves every digit one place to the left. **Question:** A pack of tortillas weighs $0.529$ kilograms. How much do $100$ packs weigh, and why can you find it without multiplying digit by digit? ## When you reach for it Graciela works at a tortilla factory. Tortillas go out in packs of $10$, cases of $100$ and pallets of $1{,}000$. One pack weighs $0.529$ kilograms. What does a stack of $100$ packs weigh? The question is $0.529 \times 100$. You could multiply it out digit by digit, but there is a much faster way, and it comes straight from how place value works. Multiplying by $10$, $100$ or $1{,}000$ never changes which digits you write. It only changes the place each digit sits in. You reach for this move any time one factor is $10$, $100$, $1{,}000$ or another power of ten: counting up packs, cases and pallets, changing kilograms to grams, or turning dollars into cents. It rests on one fact from times-ten-shifts-place-value: each place is worth ten times the place to its right. The places to the right of the point are tenths, hundredths and thousandths, as in tenths-and-hundredths and thousandths. ## The procedure This lesson takes two days. Day 1 multiplies by $10$, $100$ and $1{,}000$ on a place value chart and explains why it is the digits that move. Day 2 writes the factors as powers of ten, fills empty places with zeros, and uses the move in word problems. ### Day 1: every digit moves to the left Start with $0.529 \times 10$. The number $0.529$ is $5$ tenths, $2$ hundredths and $9$ thousandths, so it is a sum of three parts: $$0.529 = 0.5 + 0.02 + 0.009$$ To multiply a sum by $10$, multiply each part by $10$ and add. That is the distributive property. Each part is easy on its own. Ten tenths make one whole, so $5$ tenths times $10$ is $5$ ones. Ten hundredths make one tenth, so $2$ hundredths times $10$ is $2$ tenths. Ten thousandths make one hundredth, so $9$ thousandths times $10$ is $9$ hundredths. ### [Multiplying and dividing by ten, one hundred and one thousand](https://math.wiki/times-ten-shifts-place-value/) **Summary:** Each place is worth ten times the place to its right. **Question:** A button factory packs buttons in bags of ten, boxes of one hundred and cartons of one thousand. How can you know at once how many buttons are in $46$ boxes, and why do the digits simply slide over? ## Why this exists At a button factory, Divya, Nikau and Amaru pack buttons. A bag holds $10$ buttons. A box holds $100$ buttons. A carton holds $1{,}000$ buttons. All day long they ask questions like these: how many buttons are in $46$ boxes? How many bags can $4{,}700$ buttons fill? You could answer by adding $100$ again and again, $46$ times. That is slow, and it is easy to lose count. But our way of writing numbers has a pattern built into it that makes these questions quick. Every place is worth ten times the place to its right. Because of that, multiplying by $10$, $100$ or $1{,}000$ does not change the digits at all. It only moves them to new places. Dividing by $10$, $100$ or $1{,}000$ moves them back. This lesson takes two days. Part 1 looks at the places themselves: why a digit moved one place to the left is worth ten times as much, so that $700$ is ten times $70$. Part 2 uses that to multiply and divide by $10$, $100$ and $1{,}000$ in your head. ## Definition Each place in a whole number is worth **ten times** the place just to its right. Ten ones make one ten. Ten tens make one hundred. Ten hundreds make one thousand. Ten thousands make one ten thousand. You met this in numbers-to-one-million, where the places climb all the way to one million, and in trade-ten-for-one, where ten blocks are traded for one block of the next size. So the **same digit** is worth ten times as much when it sits one place further to the left. A $7$ in the tens place is worth $70$. A $7$ in the hundreds place is worth $700$, and ### [Multiplying and dividing integers](https://math.wiki/sign-rules-for-products/) **Summary:** Whether a product is positive or negative depends on how many factors are negative. **Question:** Three debts of four dollars make a debt of twelve. But what could it mean to multiply by a negative number, and why should a negative times a negative come out positive? ## Why this exists For counting numbers, $3 \cdot 4$ means three groups of four, and the answer is $12$. That meaning stretches without trouble to a negative second factor: three debts of $4$ dollars are $3 \cdot (-4) = (-4) + (-4) + (-4) = -12$, a debt of $12$ dollars. It does not stretch to a negative first factor. "Negative three groups of four" describes nothing you can lay out on a table. So once negative-numbers exist, the products $(-3)(4)$ and $(-3)(-4)$ have no meaning until somebody gives them one. The meaning is not chosen freely. Multiplication already obeys laws that every later piece of algebra leans on: the order of the factors does not matter (the commutative property), and multiplying a sum means multiplying each part (the distributive property). If those laws are to keep working once negative numbers join in, there is exactly one possible answer for every product of integers. This lesson finds those answers, shows why they are forced, and then carries them over to powers and to division. This lesson is taught in two parts. Part 1 is products: two factors, then many. Part 2 is powers of a negative number, quotients, and expressions that use all four operations. ## Definition **Multiplying two integers.** Multiply their sizes, ignoring the signs (the sizes are their absolute values). Then the sign of the product is: - **positive** when the two factors have the **same** sign, - **negative** when the two factors have **different** signs. ### [Multiplying and dividing powers with the same base](https://math.wiki/product-of-powers/) **Summary:** The exponents add because the repeated factors are simply lined up together. **Question:** When two powers of the same number are multiplied or divided, what happens to the exponents, and why? ## Why this exists An exponent is shorthand for a long product: $2^{10}$ stands for ten $2$'s multiplied together. The shorthand only earns its keep if you can calculate with it without unpacking it every time. Suppose a culture of bacteria doubles every hour. After $10$ hours it has grown by a factor of $2^{10}$, and over the next $6$ hours it grows by a further factor of $2^6$. Its total growth is the product $2^{10} \cdot 2^6$. You could write out all sixteen $2$'s and count them, but once the exponents run into the hundreds, or a letter stands where a number was, writing the factors out stops being an option. This lesson gives the two shortcuts. When powers of the same base are multiplied, the exponents add; when they are divided, the exponents subtract. Both come straight from what an exponent counts, and both need the bases to match. They are the first two of the laws of exponents, and the later laws are built from them. ## Definition For a number $a$ and a positive whole number $n$, the **power** $a^n$ is the product of $n$ factors, each equal to $a$ (this is exponent notation): $$a^n = \underbrace{a \cdot a \cdot \cdots \cdot a}_{n \text{ factors}}$$ The number $a$ is the **base** and $n$ is the **exponent**. A single factor is a power too: $a^1 = a$. So a base written with no exponent, such as $7$ or $x$, has exponent $1$. Two powers have the **same base** when their bases are the same number or the same letter. $5^3$ and $5^8$ have the same base, and so do $x^4$ and $x^9$. $2^5$ and $3^5$ do not, and neither do $x^4$ and $y^4$, even though their exponents match. ### [Multiplying and dividing rational numbers](https://math.wiki/multiply-and-divide-rational-numbers/) **Summary:** Products and quotients of signed fractions and decimals. **Question:** You can multiply $-3$ by $4$, and you can multiply $\frac{2}{3}$ by $\frac{3}{4}$. What happens when a number is negative and a fraction at the same time, and do the sign rules for integers still hold? ## When you reach for it You already know two separate things. From multiplying and dividing integers, $(-3)(4) = -12$ and $(-12) \div (-4) = 3$: the signs decide whether the answer is positive or negative. From multiplying fractions and dividing fractions, $\frac{2}{3} \cdot \frac{3}{4} = \frac{1}{2}$ and $\frac{1}{2} \div \frac{1}{4} = 2$: the numerators and denominators decide the size. This lesson puts the two together, for products and quotients such as $$\begin{gathered} \left(-\frac{3}{5}\right) \cdot \frac{10}{9} \\ (-1.35) \div (-0.9) \end{gathered}$$ in which a number is a rational number that is negative and not a whole number at once. Reach for it whenever a quantity that can go down is scaled or shared out: a snow depth that shrinks by $2\frac{2}{5}$ centimeters in three quarters of an hour, a balance that drops by $12.75$ dollars every week, a rate found by dividing a change by a time. You need it too whenever a formula is evaluated at a negative fraction, which is most of algebra from here on. The whole skill rests on one fact: **the sign and the size of a product or a quotient can be found separately.** The size is ordinary fraction or decimal arithmetic. The sign follows the same rules as for integers. The procedure below shows why the integer rules carry over unchanged, which is not obvious: $(-1)(-1) = 1$ was a fact about integers, and nothing so far says that $\left(-\frac{1}{2}\right)\left(-\frac{1}{2}\right)$ must be positive too. ### [Multiplying by tens and hundreds](https://math.wiki/multiply-by-multiples-of-ten/) **Summary:** Products such as 40 times 600, built from a basic fact and a shift of places. **Question:** A bus company has $40$ buses, and each one drives $600$ miles a month. How can the basic fact $4 \times 6 = 24$ give the total miles, and why does the answer end in exactly three zeros? ## When you reach for it Ishita runs a fleet of school buses. Aroha orders pencils for the schools the buses serve, and Xochi plans the trips. Their questions sound like these. Six buses each drive $400$ miles in a week: how many miles is that in all? Forty buses each drive $600$ miles in a month: how many miles? Thirty buses have $50$ seats each: how many seats? Each question is a product in which at least one factor is a multiple of ten, a hundred or a thousand, such as $400$, $600$ or $50$. Products like these come up whenever things are counted in tens and hundreds. They are also the pieces of every bigger product: partial-products splits a product like $47 \times 36$ into parts such as $40 \times 30$, and an estimate such as $500 \times 6$ in rounding-to-estimate is one of them. You already have the two tools this needs. One is a basic fact such as $6 \times 4 = 24$, a product of two one-digit numbers (multiplication-facts). The other is the shift from times-ten-shifts-place-value: multiplying by $10$, $100$ or $1{,}000$ moves every digit one, two or three places to the left. This lesson puts the two together over two days. Part 1 multiplies a one-digit number by a multiple of $10$, $100$ or $1{,}000$. Part 2 multiplies two multiples of ten. ## The procedure ### Part 1: a one-digit number times tens, hundreds or thousands Take $6 \times 400$, the miles for six buses that each drive $400$ miles. The number $400$ is $4$ hundreds, so $6 \times 400$ asks for $6$ groups of $4$ hundreds. Count the hundreds and there are $6 \times 4 = 24$ of them. ### [Multiplying decimals](https://math.wiki/multiply-a-decimal-by-a-whole-number/) **Summary:** Products such as a price repeated several times. **Question:** A taco costs 2.65 dollars and you want four of them. Each order uses 0.7 pound of filling and six orders are waiting. How do you multiply a number with a decimal point by a whole number, and how do you know where the point goes in the answer? ## When you reach for it Shuyi and Maximo run a taco truck. Every order of tacos gets $0.7$ pound of filling, and six orders are waiting at the window. How much filling should Shuyi scoop? A taco costs $2.65$ dollars, and Darshan wants four. How much does he pay? Both questions ask for the same thing: a number of equal groups, where each group is a decimal. That is multiplication, just as $6$ groups of $7$ is $6 \times 7$. The only new part is the decimal point. You reach for this lesson whenever one amount with a decimal point is repeated a whole number of times: a price times how many you buy, a weight per order times how many orders, a length per step times how many steps. You already have two tools that do the job. Adding $0.7$ six times works (see add-decimals-to-hundredths), but it is slow. And you can multiply whole numbers in columns (see standard-algorithm-for-multiplication). This lesson joins them: it counts tenths and hundredths the way you count anything else, and it shows why the point lands where it does. This lesson takes three days. Part 1 multiplies tenths by a whole number with base-ten pictures. Part 2 multiplies hundredths, first with an area model and then in columns. Part 3 puts the factors in either order, uses an estimate to check the point, and handles products that end in zero. ## The procedure ### Part 1: tenths times a whole number Start with Shuyi's filling: $6$ orders of $0.7$ pound each. One way is to add: ### [Multiplying multi-digit decimals with the standard algorithm](https://math.wiki/multiply-decimals-standard-algorithm/) **Summary:** Multiply as if the numbers were whole, then place the decimal point by counting decimal places and check with an estimate. **Question:** Molding costs $4.85$ dollars a meter and a customer wants $3.6$ meters. How do you multiply when both numbers have a decimal point? ## When you reach for it Ximena works the counter at a woodworking shop that sells molding and skirting board by the meter. When a customer wants a whole number of meters, the bill is a decimal times a whole number, which multiply-a-decimal-by-a-whole-number already handles: $3$ meters at $4.85$ dollars a meter is $4.85 \times 3$. But walls are rarely a whole number of meters long. A wall that needs $3.6$ meters of molding at $4.85$ dollars a meter asks for $4.85 \times 3.6$, and now both factors have a decimal point. The move in this lesson works for any two positive decimals, however many digits they have: multiply as if the points were not there, using the standard multiplication algorithm for whole numbers, then put the point back in the right place. It is taught over two days. Day 1 is the algorithm and placing the point by counting decimal places. Day 2 checks the answer with an estimate, deals with zeros in the product, and shows why counting places gives the right answer. ## The procedure ### Day 1: multiply the whole numbers, then count places Take $4.85 \times 3.6$. Multiplying by $100$ moves every digit of $4.85$ two places to the left, so $4.85 \times 100 = 485$; in the same way $3.6 \times 10 = 36$ (times-ten-shifts-place-value). These are whole numbers, so the ordinary algorithm multiplies them, one partial product for each digit of the lower factor: $485 \times 6 = 2910$ and $485 \times 30 = 14550$, which add to $485 \times 36 = 17460$. ### [Negative fractions and decimals on the number line](https://math.wiki/negative-fractions-and-decimals/) **Summary:** Parts of a whole can sit on the negative side of zero too. **Question:** Some amounts on the far side of a zero are not whole, such as a find part of the way down a meter of soil, or a month with two and a half centimeters less snow than usual. Where do they sit on the number line? ## Why this exists Negative numbers gave a name to each whole step left of zero: $-1$, $-2$, $-3$ and on. But the amounts that call for negative numbers are often not whole. A town's snowfall in a month can be $2.4$ centimeters below its average. A find in an archaeological dig can lie a quarter of a meter past the one-meter mark below the ground. Each is on the far side of a zero, and each falls between two integers. On the right of zero this was settled earlier: fractions and decimals fill the gaps between the whole numbers. This lesson carries them across zero, so that every fraction and every decimal has a twin on the left. Together with the integers they make up the **rational numbers**, the numbers you can place on a line by cutting the unit into equal parts and counting. ## Definition **Negative fractions.** For a fraction such as $\frac{3}{4}$, the number $-\frac{3}{4}$, read "negative three quarters", is the point $\frac{3}{4}$ of a unit to the left of $0$. To find it, split the stretch from $0$ to $-1$ into $4$ equal parts, as many as the denominator says, and count $3$ of those parts, as many as the numerator says, leftward from $0$. This is the same cutting and counting as in locating a fraction, done on the other side of zero. **Negative decimals.** A decimal that stops is a fraction with denominator $10$, $100$, $1000$ and so on, so it is placed the same way. $-0.7$ is seven tenths of a unit left of $0$, and $-0.07$ is seven hundredths of a unit left of $0$. ### [Negative numbers](https://math.wiki/negative-numbers/) **Summary:** Numbers less than zero, read as opposites on the number line. **Question:** Some amounts run both ways from a starting point, below freezing and above it, below sea level and above it. How do you write the ones on the far side of zero? ## Why this exists The whole numbers $0, 1, 2, 3, \ldots$ answer "how many" and "how far along". They start at $0$ and only go up, which is fine for apples and pages, but many amounts are measured from a starting point that has something on both sides of it. On a Celsius thermometer, $0$ is the freezing point of water, and the temperature can be $6$ degrees above it or $6$ degrees below it. Heights are measured from sea level, and a hill top is above it while the sea bed is below it. A bank account can hold money, or it can owe money. In each case "$6$" on its own is not enough: you have to say which side of the starting point you mean. Negative numbers are the numbers for the other side. They extend the number line past zero to the left, so that "$6$ below" becomes a single number, $-6$, that can be placed, compared and later added and subtracted like any other. They also give an answer to a question the whole numbers cannot answer at all, such as $3 - 5$, which is the story of why negative numbers were needed. ## Definition A number is **positive** when it is greater than $0$, and **negative** when it is less than $0$. The number $0$ is **neither** positive nor negative. On a number line, numbers grow to the right. To make room for numbers less than $0$, carry the line on to the left of zero, keeping the marks the same distance apart as on the right. The first mark to the left of $0$ is $-1$, the next is $-2$, and so on. In general, for a positive number $n$, the point $n$ units to the left of $0$ is the number $-n$, which is read as negative $n$. So $-3$ is three units left of zero, and $3$ is three units right of it. ### [Number](https://math.wiki/number/) **Summary:** What numbers are, and what you do with them: counting, place value, the four operations, negatives, fractions as numbers, and the real line. **Question:** What are numbers, how does each new kind extend the one before, and what can you do with them? ## What this area is about Numbers start as answers to "how many?" and grow each time a question has no answer yet. Whole numbers cannot go below zero, so the integers add negatives; one integer does not always divide another evenly, so the rational numbers add fractions; $\sqrt{2}$ is not a fraction, so the real numbers fill the line; no real number squares to $-1$, so the complex numbers add $i$. Along the way come writing numbers in place value and computing with them. ## How these fit together **Foundations.** Counting and cardinality makes counting words tell how many. Place value writes any whole number or decimal with ten digits. Arithmetic gives the four operations on whole numbers and decimals. **Structure.** Divisibility and primes breaks whole numbers into prime factors. Exponents and roots write repeated multiplication and undo it. **Number systems.** Integers, rational numbers, irrational and real numbers and complex numbers, each containing the one before. ## A path through it Counting first. Place value and arithmetic grow together; exponents come after place value. Divisibility needs arithmetic, place value and exponents. Integers (after exponents) and rational numbers (after divisibility) feed each other, and both come before the real numbers. Complex numbers need exponents and roots. Next come algebra, where letters stand for numbers and the rules of arithmetic still hold, measurement and statistics. ### [Number lines with different scales](https://math.wiki/number-line-intervals/) **Summary:** Reading a line whose tick marks step by two, five, ten or one hundred. **Question:** A number line from 0 to 1,000 with a mark for every number would never fit on a page. How can a line skip most of its marks and still say exactly where every number is? ## When you reach for it In locate-numbers-on-a-number-line every mark on the line is one more than the mark to its left. That works well for a line from $80$ to $90$. It does not work for the minutes a class reads in a month, or for the posts along a trail numbered up to $1{,}000$: a mark for every number would make the line far too long, and the marks too crowded to count. So most number lines in books, on posters and on maps leave marks out. Their marks step by $2$, by $5$, by $10$ or by $100$. A child who can count by twos and fives, by tens and by hundreds already has the counting. This lesson turns that counting into reading a line. It covers four things: - reading a line when you know its step, and naming a mark that has no label; - finding the step when it is not given, from two labeled marks; - placing a number that has no mark of its own, between two marks; - the rule that makes any of this possible: equal spaces, each standing for the same amount. Reach for it whenever a line shows only some of its numbers, which is nearly every number line a child will meet from now on. The next lesson, adding on an open number line, draws lines like these by hand to add and subtract within $100$. ## The procedure The lesson takes three days, one part a day. Two settings run through it. The first is a reading poster on the classroom wall: a long number line where the class puts a sticker to show how many minutes each child read in a week. The second is a hiking trail with numbered posts set at equal spaces along it, and a trail map that shows them as marks on a line. ### [Number names, numerals and zero](https://math.wiki/number-names-and-numerals/) **Summary:** The spoken word, the written symbol and a pile of objects can all name the same amount. **Question:** How do the spoken word "five", the written 5 and a pile of five things all mean the same number, and what number means none at all? ## Why this exists A number can show up in three forms: a word you say ("five"), a symbol you write ($5$, called a **numeral**), and an amount you can count (five shells). They are three names for one thing, and a child who moves easily between them can read a number on a page, say it, and know how many it means. **Zero**, written $0$, is the number for none: it answers "how many?" when there is nothing there. A child should already know that the last word said while counting tells how many. Forming the digits with a pencil is its own lesson, writing the numerals; here the child reads numerals and matches them, and needs only to point to a card. Each day takes about ten to fifteen minutes. ## Day 1: Three ways to say one number **You need:** index cards with one numeral each, $1$ to $5$, written large in marker; a small handful of dry pasta shells; three small paper bags. Before the lesson, put $2$ shells in one bag, $4$ in another and $5$ in the third. Lay the numeral cards face up. Hand the child a bag and say, "Pour them out and count them." When the count is done, ask, "How many shells?" and then, "Which card says that number?" The child sets the card beside the shells. Now point to three things in turn: the shells, the card, and your mouth as you say the word. "These shells are four. This card says four. And we say *four*. All three mean the same number." Ask the child to do the pointing and saying for the next bag. ### [Numbers in other bases](https://math.wiki/other-number-bases/) **Summary:** Place value still works when the bundling size is something other than ten. **Question:** We bundle in tens because we have ten fingers. What happens to place value if you bundle in fives, or in twos, instead? ## Why this exists Our numbers are built on bundles of ten. Ten ones make a ten, ten tens make a hundred, and a digit says how many of each bundle there are, as tens-and-ones and expanded-form show. Ten is a natural choice because we have ten fingers. But nothing in the rule needs the number ten. Chayton has a box of drinking straws and a bag of rubber bands. He puts a band around every five straws. When he has five bundles, he puts a bigger band around those, making one big bundle. He can then write down how many big bundles, bundles and loose straws he has, and that list of digits names the number of straws exactly. This is place value with a bundle size of five. Other bundle sizes are in use all around us. A computer stores every number with only two signals, on and off, so it bundles in twos. A clock bundles sixty seconds into a minute and sixty minutes into an hour. Seeing place value with a different bundle size also shows what was special about ten all along: nothing, except that it is the size we chose. ## Definition The bundle size is called the **base**. The base can be any whole number $2$ or more. In base five, the places from right to left are worth: - $1$ (a loose straw, the ones), - $5$ (a bundle of five), - $25 = 5 \times 5$ (a big bundle: five bundles of five), - $125 = 5 \times 5 \times 5$ (five big bundles), and so on. Each place is worth five of the place to its right. A digit tells how many of that place there are, and in base five the only digits are $0$, $1$, $2$, $3$ and $4$. ### [Numbers to one million](https://math.wiki/numbers-to-one-million/) **Summary:** Reading, writing and building numbers in the thousands and beyond. **Question:** A park welcomed three hundred four thousand, six hundred twenty-seven visitors. How do six digits hold a number that big, and how do you read it, write it and tell it apart from another big number? ## When you reach for it You already know how to read and write numbers up to $1{,}000$: hundreds, tens and ones (numbers-to-1000). But many real counts are much bigger. A national park can have hundreds of thousands of visitors in one summer. A state fair can sell tickets to more than half a million people over its run. To read a sign that says $304{,}627$ visitors, to write a number someone says aloud, or to tell which of two big counts is larger, you need the places past the thousands. The good news is that nothing new has to be invented. The same three places, hundreds, tens and ones, simply repeat. Once you see the pattern, a six-digit number is two three-digit numbers sitting side by side, with a name in between. This lesson takes three days. Part 1 meets the thousands period and the rule that every place is worth ten times the place to its right. Part 2 names the periods, uses commas and zeros, and reaches one million. Part 3 compares, orders and builds big numbers. ## The procedure ### Part 1: the thousands period, and ten times as much **Ten makes the next place.** Ten ones make one ten. Ten tens make one hundred. Ten hundreds make one thousand. The pattern does not stop there. Ten thousands make one **ten thousand**, which is $10{,}000$. Ten ten thousands make one **hundred thousand**, which is $100{,}000$. And ten hundred thousands make **one million**, $1{,}000{,}000$. So each place is worth ten times the place just to its right. Two places to the left is ten times ten, which is $100$ times as much. This one rule is what lets ten digits, $0$ to $9$, write any whole number at all: when a place fills up with ten, those ten are traded for one in the next place to the left. ### [Numerical expressions](https://math.wiki/numerical-expressions/) **Summary:** A written string of numbers and operations that names a single value. **Question:** Someone writes 3 + 4 × 5 on the board. Is that 35 or 23, and how does everyone who reads it get the same number? ## Why this exists Caio has $3$ tokens in his pocket at the arcade. He buys $4$ cups of tokens, and each cup holds $5$ tokens. To say how many tokens he has, you can write one short line: $$3 + 4 \times 5$$ That line is a **numerical expression**. It uses numbers and operation signs to name a number, and here it names Caio's total. But a line on paper is only useful if everyone who reads it gets the **same** number. Work from left to right, adding first, and you get $7 \times 5 = 35$. Multiply first and you get $3 + 20 = 23$. Caio has $23$ tokens, not $35$. So people who write math have agreed on an order for doing the operations. This lesson is about what an expression is, what that agreed order is, and why it is a good choice. The lesson takes three days. **Part 1** is what a numerical expression is, how it differs from an equation, and how two different expressions can name the same number. **Part 2** is why an order has to be agreed, and the order itself. **Part 3** uses the order on longer expressions, and compares expressions without working them out. ## Definition ### Part 1: expressions, values and equations A **numerical expression** is a group of numbers joined by operation signs: $+$, $-$, $\times$ and $\div$. It has **no equal sign**. It names one number, and that number is its **value**. Finding the value is called **evaluating** the expression. These are all numerical expressions: ### [One more and one less](https://math.wiki/one-more-one-less/) **Summary:** Adding or removing a single object moves the count one step along the number words. **Question:** When one thing is added to a group, or one is taken away, how can a child know the new count without counting everything again? ## Why this exists The counting words are not just a chant: each word names a group exactly one bigger than the word before it. So when one thing is added, the count is the next word, and when one thing is taken away, the count is the word before, with no need to count the whole group again. These four days build that link with cube towers and stones, line the towers up into a staircase, and end with groups the child cannot see, up to twenty. The child should already know that the last word said tells how many. ## Day 1: One more cube **You need:** about $12$ snap-together cubes (linking cubes or plastic bricks). Build a tower of $4$ cubes with the child and count it together: "one, two, three, four. Four cubes." Hold up one more cube and say, "I am going to put one more on top. How many will there be?" Let the child guess, snap it on, and then count to check. Ask, "What number comes after four when we count? How many cubes are in the tower now?" Both answers are "five". That is the idea of the day: one more cube makes the count the next counting word. **Practice.** 1. A tower of $2$ cubes, and one more goes on. *Answer:* $3$ cubes, the word after "two". 2. A tower of $6$ cubes, and one more goes on. *Answer:* $7$ cubes. 3. A tower of $9$ cubes, and one more goes on. *Answer:* $10$ cubes. 4. Build a tower of $5$, then put one more on while the child's eyes are closed. "How many now?" *Answer:* $6$. A child who says it without counting from one has the idea. ### [One-to-one correspondence](https://math.wiki/one-to-one-correspondence/) **Summary:** Each object gets exactly one count word. The last word said is how many. **Question:** When a child counts a pile of things, how do the counting words and the things fit together, and what does the last word tell you? ## Why this exists Counting is matching: each object gets exactly one counting word, in order, with none skipped and none counted twice. When the matching is done, the last word said tells how many there are, and it does not matter which object you started with. These four days build that match with the hands first (one counter for each cup), then with the words, then with the last word, and finally with things scattered on a table. A child only needs to be able to say the counting words in order to start. Keep every count at ten or below this week. ## Day 1: One for each **You need:** $5$ small paper cups, a handful of counters (buttons or beans work). Put $4$ cups in a row. Hand the child some counters and say, "Put one counter in each cup. Just one." Do not count anything yet. When every cup has one, ask, "Does every cup have a counter? Is any counter left over?" Then add a fifth cup and give no more counters. Ask, "Are there more cups or more counters? How do you know?" The child can say "that cup has no counter" without saying any number. That is the point of the day: two groups can be matched one to one, and the leftover shows which group is bigger. **Practice.** 1. $5$ cups and $5$ counters, one in each cup. *Answer:* every cup gets one and no counter is left, so there are the same number of cups and counters. 2. $5$ cups and $3$ counters. *Answer:* $2$ cups stay empty, so there are more cups. 3. $3$ children and $4$ crayons; give each child one crayon. *Answer:* $1$ crayon is left over, so there are more crayons than children. 4. Make a row of $4$ blocks. Under it, build a second row with one block under each block. *Answer:* the new row has $4$ blocks too, because every top block has exactly one partner. ### [Operations with rational numbers](https://math.wiki/add-and-subtract-rational-numbers/) **Summary:** Combining signed fractions and decimals. **Question:** A temperature of $-4.5$ degrees rises by $7\frac{3}{4}$ degrees. How do you add numbers that carry a sign and are not whole? ## When you reach for it You already add and subtract signed whole amounts, the integers, from adding and subtracting integers. You already add fractions and decimals that are positive, from adding fractions and decimal arithmetic. Real quantities often need both at once: a temperature of $-4.5$ degrees, a bank balance of $-23.40$ dollars, a depth of $-12\frac{1}{2}$ meters, a price that changed by $-0.48$. Reach for this skill whenever the numbers being added or subtracted can be negative and can also be fractions, mixed numbers or decimals. Nothing new has to be invented. The rational numbers sit on the same number line as the integers (negative-fractions-and-decimals), and addition and subtraction mean the same thing there. The work is in combining two things you know: the sign rules from integers and the common denominator from fractions. Multiplying and dividing signed fractions and decimals is its own lesson. This lesson is taught over two days. Part 1 is signed fractions and mixed numbers. Part 2 is signed decimals, fractions and decimals together, rearranging a long sum, and distance and change. ## The procedure ### Part 1: signed fractions and mixed numbers **The number line rule carries over.** Adding means what it meant for integers: $p + q$ is the point you reach by starting at $p$ and moving a distance $|q|$, to the right if $q$ is positive and to the left if $q$ is negative. Nothing in that rule needed the move to be a whole number of steps. Moving $\frac{5}{4}$ to the left is moving one and a quarter units to the left. ### [Opposites and zero pairs](https://math.wiki/opposites-and-zero-pairs/) **Summary:** Every number has a partner the same distance the other side of zero, and the pair adds to nothing. **Question:** A deposit and a withdrawal of the same size, a rise and a fall of the same size. Why do some pairs of amounts cancel each other out completely, and what does that let you do? ## Why this exists Put $20$ dollars into an account and then take $20$ dollars out, and the balance is back where it started. Walk $8$ steps forward and then $8$ steps back, and you are standing on the same spot. A temperature that rises $5$ degrees and then falls $5$ degrees ends at the temperature it began with. In each case two amounts have **undone** each other, and together they come to nothing. Negative numbers gave each of these amounts a number: the deposit is $20$ and the withdrawal is $-20$, the rise is $5$ and the fall is $-5$. Each pair is a number and its opposite. This lesson is about what happens when the two are put together: they always add to $0$. That one fact does a great deal of work. It is how sums like $8 + (-3)$ are worked out, by letting each negative cancel a positive and counting what is left. It lets a long sum be shortened by striking out pairs that cancel. It lets you find a missing number, since the only number that makes $0$ with $14$ is $-14$. And it is the first example of an *inverse*, an idea that runs right through algebra. ## Definition The **opposite** of a number $a$, written $-a$, is the number the same distance from $0$ on the other side of $0$. The opposite of $7$ is $-7$, the opposite of $-7$ is $7$, and the opposite of $0$ is $0$. A **zero pair** is a number together with its opposite, such as $7$ and $-7$. The sum of a zero pair is $0$: $$7 + (-7) = 0$$ ### [Order of operations](https://math.wiki/order-of-operations/) **Summary:** Grouping first, then powers, then multiply and divide, then add and subtract, and why there has to be a rule. **Question:** A line such as $3 + 4 \times 5$ could mean $35$ or $23$. What rule makes everyone read it the same way? ## What this area is about A written expression is only useful if every reader gets the same number from it. So there is an agreed order: grouping first, then powers, then multiplying and dividing from left to right, then adding and subtracting from left to right. The order is a convention, not a fact of nature, and grouping symbols let a writer override it. ## How these fit together **The rule.** Numerical expressions explains what an expression is and why the order is needed, settling $3 + 4 \times 5 = 23$. **Overriding it.** Grouping symbols mark a part to be worked first, as in $(3 + 4) \times 5$. **Translating.** Writing expressions from words turns a phrase like "add $8$ and $7$, then multiply by $2$" into $(8 + 7) \times 2$, and reads an expression back as words. **Extending it.** Order of operations with powers settles powers before multiplying, so $2 + 3^2 = 11$. Order of operations with negative numbers adds signs, and tells a minus that means subtract from one that means the opposite. ## A path through it Start with numerical expressions, then grouping symbols. Writing expressions from words and the order with powers both follow; negatives come last. With the order settled, evaluating expressions and formulas and all of algebraic expressions rely on it. ### [Order of operations with negative numbers](https://math.wiki/order-of-operations-with-negatives/) **Summary:** Expressions mixing signs, powers and brackets. **Question:** One minus sign can mean "subtract" and another can mean "the opposite of". When an expression has both, along with powers and parentheses, which happens first? ## When you reach for it Reach for this whenever an expression mixes negative numbers with more than one operation: a calculation like $-3 - 4(-5)$, a formula with a negative number put in for a letter, or a fraction whose top and bottom both have signs in them. Nothing about the order of operations changes when negatives appear. Grouping still comes first, then powers, then multiplication and division from left to right, then addition and subtraction from left to right, exactly as in the lesson on powers. What changes is the **reading**. Two questions decide almost every expression with negatives in it: - Is this minus sign a subtraction, or does it mean "the opposite of"? - What exactly does this power apply to? Once those are answered, each step is a calculation you already know: adding and subtracting signed numbers from add-subtract-integers, and the signs of products and quotients from sign-rules-for-products. ## The procedure ### One sign, two jobs A minus sign written **between** two quantities is subtraction: in $7 - 3$ it takes $3$ away from $7$. A minus sign written **in front of** a single quantity, with nothing before it, or just after another operation sign or an opening parenthesis, means the opposite of that quantity: in $-3$, $4 \cdot (-3)$ and $5 + (-2)$ it is attached to the $3$ or the $2$. The two jobs are tied together. Subtracting a number is the same as adding its opposite, ### [Order of operations with powers](https://math.wiki/order-of-operations-with-exponents/) **Summary:** Expressions where powers are settled before multiplying and dividing. **Question:** In 2 + 3 squared, is the 2 squared as well? When an expression mixes powers with other operations, which part is worked out first, and why that one? ## When you reach for it Reach for this whenever an expression holds a power together with other operations, such as $2 + 3^2$, $4 \times 5^3 - 1$ or a fraction with powers on its top and bottom. Written down, $2 + 3^2$ could be read two ways. Work the power first and it is $2 + 9 = 11$. Add first and it is $5^2 = 25$. A written expression is only useful if everyone who reads it gets the same number, so there has to be an agreed order. The agreed order is a convention, a choice people made, but it was not chosen at random: each step of it has a reason, and the reasons are what this lesson is about. What a power is and how to find its value is in base-and-exponent. ## The procedure The order, from first to last: - **grouping symbols**: parentheses and brackets, fraction bars, and anything raised as an exponent; - **powers**; - **multiplying and dividing**, as one level, from left to right; - **adding and subtracting**, as one level, from left to right. Inside a grouping symbol the same order applies again. Some people remember the levels by a word built from their first letters; the reasons below are what make the order hold together. ### Why powers come before multiplying A power is short for a product. $2 \times 3^2$ means $2 \times 3 \times 3$, which is $18$ whatever order its factors are multiplied in. Working the power first agrees: $2 \times 9 = 18$. Multiplying first would give $6^2 = 36$, which is $2 \times 3 \times 2 \times 3$: two factors of $2$, when only one was written. The exponent sits on the $3$ alone, so only the $3$ is repeated. Doing powers first keeps $3^2$ an honest short form of $3 \times 3$: swap one for the other anywhere in an expression and the value never changes. ### [Ordinal numbers](https://math.wiki/ordinal-numbers/) **Summary:** First, second and third describe position in a line rather than how many there are. **Question:** When we say "first", "second" and "third", what are we naming, and how is that different from saying how many? ## Why this exists The words **first, second, third, fourth** and so on name a *place* in a line, not an amount. "Four" tells how many things are in a group; "fourth" points at one single thing, the one that gets the word "four" when you count along the line from its start. So a line needs a start before its places mean anything, and these three days build that: naming places, telling a place from an amount, and seeing that the start of the line decides who is first. A child should already be able to count a set of objects and know that the last word said tells how many. Keep every line at ten or fewer. ## Day 1: First, second, third **You need:** $5$ small toys (stuffed animals or toy cars), a block or a book to be the "door". Put the door on the table and line the toys up from it, one after another. Say, "The line starts at the door. This one is next to the door, so it is **first**." Touch the next toy: "This one is **second**." Then "third", "fourth", "fifth". Point out that the place words sound like the counting words, except "first", "second" and "third", which have their own names. Ask, "Who is first? Who is third? Point to the fifth one." Then let the toys take turns going through the door, and ask before each one goes, "Which one goes now? First, second, or third?" **Practice.** Use the picture. The line starts at the flag. 1. Which shape is first? *Answer:* the dark circle, the one next to the flag. 2. Which shape is third? *Answer:* the triangle. 3. What place is the dark square in? *Answer:* fourth. 4. Which shape is last, and what place is it? *Answer:* the open circle, in fifth place. 5. Line up $4$ toys and have the child name each place out loud. *Answer:* "first, second, third, fourth", one place word for each toy. ### [Partial products](https://math.wiki/partial-products/) **Summary:** Breaking both factors by place, multiplying every pair, then adding the pieces. **Question:** Six rows of 34 seats is not a fact you know by heart. How do you break a big multiplication into small ones you do know? ## When you reach for it The front section of a school auditorium has $6$ rows, with $34$ seats in each row. How many seats is that? The answer is $6 \times 34$. That is not a fact you know by heart, since the facts you know stop at $10 \times 10$. So break the big multiplication into small ones you do know. Split a factor by place value, the way expanded-form writes it: $34 = 30 + 4$. Multiply each part. Then add the answers. Each small answer is called a **partial product**, because it is one part of the whole product. The words: in $6 \times 34 = 204$, the numbers being multiplied, $6$ and $34$, are the **factors**, and the answer, $204$, is the **product**. Two ideas you have met before make this work. - The area model: a rectangle whose sides are the two factors has the product as its area. Cutting the rectangle into pieces does not change its total area. - The distributive-property: multiplying a sum is the same as multiplying each part and adding, $a \times (b + c) = a \times b + a \times c$. The area model is the picture. The distributive property is the rule it pictures. Every time this lesson splits a factor, it is using the distributive property, and it will say so. Reach for partial products whenever a factor has more than one digit. ## The procedure This lesson takes four days, one part a day. ### Part 1: two digits times one digit, with an area model Draw the auditorium section as a rectangle, $6$ tall and $34$ wide. Now cut it with one line, where $34$ splits into $30$ and $4$. ### [Partial quotients](https://math.wiki/partial-quotients/) **Summary:** Taking away easy multiples of the divisor and adding up how many were taken. **Question:** Lalit has 1,674 beads and each bracelet takes 6. Counting out one bracelet at a time would take all day. How do you find how many bracelets he can make in a few big steps? ## When you reach for it Lalit makes friendship bracelets. Each bracelet takes $6$ glass beads, and he has $1{,}674$ beads. How many bracelets can he make? This is a "how many groups" question: how many groups of $6$ fit in $1{,}674$? That is the division $1{,}674 \div 6$ (see division-as-sharing, where division asks either how many in each group or how many groups). You could count out $6$ beads, then $6$ more, then $6$ more. But that takes hundreds of steps, and no times-table fact gives the answer. So take away many groups at once. Ten groups of $6$ is $60$ beads. A hundred groups of $6$ is $600$ beads. Those are easy to find, because each one is a basic fact and a shift of places (see multiply-by-multiples-of-ten). Take away a big easy chunk, then another, until no group of $6$ is left. Then add up how many groups you took. Each count is called a **partial quotient**, since it is one part of the whole answer. The words: in $1{,}674 \div 6 = 279$, the number being divided, $1{,}674$, is the **dividend**. The number it is divided by, $6$, is the **divisor**. The answer, $279$, is the **quotient**. If some is left over that is too small to make one more group, that amount is the **remainder**. Reach for partial quotients whenever the dividend is too big for a basic fact. ## The procedure This lesson takes four days, one part a day. ### Part 1: taking away easy chunks Start with all $1{,}674$ beads. A hundred bracelets use $100 \times 6 = 600$ beads, and $600$ is less than $1{,}674$, so Lalit can surely make $100$ bracelets. Take those beads away: ### [Parts, wholes and number bonds](https://math.wiki/number-bonds-to-10/) **Summary:** Knowing every pair of whole numbers that adds to ten. **Question:** Some things go into two groups. What can the two groups be, and if you can see one of them, how do you know the other without counting? ## When you reach for it A child who can count can find how many counters are on the table. They can also count just the dark ones, and just the light ones. Those look like three separate jobs with three separate answers. The idea of **parts and the whole** is that they are one fact: the dark counters and the light counters together *are* all the counters, so the three numbers are tied to each other, and knowing any two of them decides the third. That one picture is where adding and subtracting meet. Adding asks for the whole when both parts are known. Subtracting asks for a part when the whole and the other part are known. A child who sees both as questions about the same picture never has to learn subtraction as a new and harder thing, and fact-families and story problems all rest on it. **Number bonds** are the pairs of parts that make one particular whole, known so well that a child who sees three counters knows, without counting, that two more make five. This lesson builds them at three sizes, each on the one before. The pairs that make $5$ are $5$ and $0$, $4$ and $1$, and $3$ and $2$. The pairs that make $10$ are $10$ and $0$, $9$ and $1$, $8$ and $2$, $7$ and $3$, $6$ and $4$, and $5$ and $5$. Each pair can also be turned around. The pairs that make $20$ are hardly new at all: each one is a pair that makes ten with a full ten added to one part, so $3$ and $7$ becomes $13$ and $7$. ### [Patterns in the multiplication table](https://math.wiki/multiplication-table-patterns/) **Summary:** Rows and diagonals in the table repeat in ways that make facts predictable. **Question:** The multiplication table holds a hundred products. Why do its rows, columns and diagonal repeat in ways that let you predict a number before you work it out? ## Why this exists By now every product up to $10 \times 10$ has been found, one family at a time, in facts-for-2-5-and-10, facts-for-3-4-and-6 and facts-for-7-8-and-9. Written as a hundred separate facts they are a lot to carry. Laid out in one grid they stop being separate. The same numbers turn up in two places, one row is twice another, a column counts up in its tens and down in its ones, and the odd numbers sit in a neat checkerboard. Each of those patterns has a reason, and the reason is always the meaning of multiplication: equal groups, or an array of rows and columns, as in arrays-and-multiplication. That is the point of this lesson. A pattern that is only noticed is a guess about the next number. A pattern that is explained is a fact about every number it covers, and it can be used to predict a product, to check one, or to rebuild one that has slipped from memory. The lesson takes three days, one part each. Part 1 reads the table along its rows and across its diagonal. Part 2 compares whole rows and looks down the nines column. Part 3 asks which products are even and what the rows add up to. ## Definition The **multiplication table** is a grid with the factors $1$ to $10$ along the top and down the left side. The cell in **row** $a$ and **column** $b$ holds the product $a \times b$. So row $3$, column $7$ holds $3 \times 7 = 21$. A **row** of the table is the ten products $a \times 1, a \times 2, \dots, a \times 10$: the skip count by $a$. A **column** is the same thing read downward. ### [Perfect squares and perfect cubes](https://math.wiki/perfect-squares-and-perfect-cubes/) **Summary:** Numbers that are a whole number to the second or third power. **Question:** Alma has a pile of square paving slabs. When can every one of them be laid as one full square, and how can she tell without trying? ## Why this exists Alma is paving a courtyard with square slabs, and she wants them to form one full square, with no gaps and no slab left over. With $49$ slabs she can: $7$ rows of $7$. With $50$ she cannot, however she shuffles them. Ciro has the same question in a warehouse, where he stacks storage boxes, each a small cube, into one big solid cube. $27$ boxes make a block $3$ by $3$ by $3$; $28$ boxes do not. Some numbers come out as a full square or a full cube, and most do not. They turn up all through mathematics, in areas, volumes and square roots, so it pays to know the small ones on sight and to have a sure test for big ones. ## Definition A **perfect square** is a number you get by multiplying a whole number by itself. $49$ is a perfect square because $49 = 7 \times 7$. With an exponent this is written $7^2$ and read "seven squared". Perfect squares are also called **square numbers**. A **perfect cube** is a number you get by using a whole number as a factor three times. $27$ is a perfect cube because $27 = 3 \times 3 \times 3 = 3^3$, read "three cubed". They are also called **cube numbers**. Zero counts as both, since $0 = 0 \times 0$, but a courtyard of no slabs is not much of a square, so lists usually start at $1$. ## Intuition The names come from shapes. Lay $7$ rows of $7$ slabs and you have an array whose rows and columns are the same length: a square with $7$ slabs along each side. Counting the slabs is the area of a square with side $7$, which is $7 \times 7 = 7^2$. That is why the second power is called "squared". ### [Perfect, abundant, deficient and amicable numbers](https://math.wiki/amicable-numbers/) **Summary:** A pair of numbers where each one equals the sum of the other one's smaller factors. **Question:** Add up the factors of a number, leaving out the number itself. Does the total fall short, overshoot, or land exactly on the number, and can two numbers each produce the other? ## Why this exists The factors of $6$ other than $6$ itself are $1$, $2$ and $3$, and $1 + 2 + 3 = 6$. The factors give back the number they came from. Try the same with $8$ and you get $1 + 2 + 4 = 7$, one short. Try $18$ and you get $1 + 2 + 3 + 6 + 9 = 21$, three too many. Greek mathematicians studied numbers like $6$ more than two thousand years ago, and Greek writers gave the three kinds the names still used: **perfect**, **abundant** and **deficient**. Euclid's *Elements* (Book IX, Proposition 36) gives a rule that builds perfect numbers, and some questions it raised are still open today. There is a stranger pattern too. The factors of $220$, leaving out $220$, add to $284$, and the factors of $284$, leaving out $284$, add straight back to $220$. Each number is made from the other. That pair has been known since ancient times; later Greek writers credited it to the followers of Pythagoras. Pairs like it are called **amicable**, from the Latin for friendly. None of this is needed to do arithmetic. It is here because it is a good reason to get fast and careful at finding all the factors of a number, and because it shows how a simple question about whole numbers can stay unanswered for centuries. ## Definition Everything in this lesson is about whole numbers from $1$ up. The **proper factors** of a number are its factors other than the number itself. The proper factors of $12$ are $1$, $2$, $3$, $4$ and $6$; the number $12$ is a factor of $12$ but not a proper one. ### [Place value](https://math.wiki/place-value/) **Summary:** Where a digit sits decides what it is worth: ones, tens, hundreds. **Question:** The same digit can be worth $3$, $30$ or $0.3$. What decides which, and what does that let a written number do? ## What this area is about Our numerals bundle in tens: ten ones make a ten, ten tens a hundred, and the column a digit sits in decides what it is worth. Each place is worth ten times the place to its right, and the same rule, carried past the decimal point, gives tenths and hundredths. ## How these fit together **Building the places.** Teen numbers as ten and some more, tens and ones and trading ten for one make the bundles. **Reading and writing.** Numbers to 1,000, numbers to one million, the place value chart, expanded form and decimal place value. **Using the places.** Comparing and ordering, rounding, multiplying and dividing by ten, one hundred and one thousand, and the same moves for decimals: multiplying and dividing. **Other systems.** Other bases bundle in a size other than ten; tally marks and Roman numerals have no columns: each mark or letter has a fixed value, added or sometimes subtracted. ## A path through it Teen numbers, tens and ones, then trading and numbers to $1{,}000$. The chart and comparing follow, then expanded form and numbers to one million. Decimal place value needs expanded form and comparing; rounding needs decimals and numbers to a million. Shifting by ten comes after the chart and decimals, and before the decimal versions. Other bases follow expanded form; Roman numerals fit anywhere. Next come multi-digit addition, partial products and scientific notation. ### [Placing a number on a number line](https://math.wiki/locate-numbers-on-a-number-line/) **Summary:** Finding where a whole number sits on a marked line. **Question:** Counting tells you how many. How do you see where a number sits among all the others, and which of two numbers comes later? ## When you reach for it A child who can count and read and write numbers to 120 knows the numbers as a list said out loud: one comes after another. A **number line** turns that list into a place. Each number gets its own spot on a straight line, and the order of the list becomes left to right. Once the numbers are laid out like that, a child can *see* things that counting only says: that $14$ is a little past $10$, that $85$ sits halfway between $80$ and $90$, and that $17$ is further along than $12$. A good way to think of it is a ruler for numbers. A ruler has marks the same distance apart, and each mark has a number. A number line is the same: the marks are evenly spaced, and each mark is one more than the mark to its left. This lesson is about the three things that follow from that: - a number is a **point**, one mark on the line, not a space between marks; - a line does not need every mark labeled, and it does not need to start at $0$, because the missing labels can be found by counting marks from a label that is there; - a number further to the right is the greater number. The number line comes back again and again: for adding, for measuring, for lines that count by twos, fives and tens, and much later for fractions. Every one of those uses rests on placing a single number correctly, which is what this lesson does. Read it with the child, and build the lines on the floor, on paper strips and on a toy car track wherever you can. A line a child has walked along is understood in a way a printed one is not. ### [Placing and comparing integers on a number line](https://math.wiki/integers-on-a-number-line/) **Summary:** Placing positive and negative whole numbers on a line that runs both ways from zero. **Question:** Numbers now run both ways from zero. How do you find the exact place of a negative number on a line, read one off it, and tell which of two numbers is the greater? ## When you reach for it Any time numbers that run both ways from zero have to be pictured, read or put in order. A drawing of a volcanic island shows its summit above sea level and its foot on the sea floor below. The marks on a ship's hull climb up from the waterline and run down beneath it. Each is a number line, laid flat or stood on its end, and the questions are always the same two: where does this number go, and which of these two numbers is the greater? Negative numbers brought in the numbers below zero and the idea that a number to the left is less. This lesson turns both into a dependable skill for the integers, the whole numbers $0, 1, 2, 3, \ldots$ together with their negatives $-1, -2, -3, \ldots$: placing them on lines whose ticks count in ones, twos or hundreds, flat or upright, and comparing any two of them, including two negatives, with $<$ and $>$. ## The procedure The skill is taught over two days, one part each. ### Part 1: placing and reading integers **Every gap is worth the same.** The ticks on a number line are evenly spaced, and the amount one gap stands for, the **scale**, is the same everywhere on the line, on both sides of zero. That is what lets you count. With a scale of $1$, the point three ticks left of $0$ is $-3$, because "three units to the left of zero" is what $-3$ means. With a scale of $5$, three ticks left of $0$ is three steps of $5$, so $15$ units to the left, which is $-15$. ### [Powers of powers, products and quotients](https://math.wiki/power-of-a-power/) **Summary:** The exponents multiply because a group of factors is repeated. **Question:** A power is itself raised to a power, or a whole product or fraction is. How many factors does that make, and of what? ## Why this exists Multiplying powers with the same base answers one question: $a^m \cdot a^n$ is $a^{m+n}$, because two rows of factors line up into one longer row. Powers turn up in other shapes too. A square field whose side is $10^3$ meters has area $(10^3)^2$ square meters: a power, squared. A cube whose edge is $5x$ has volume $(5x)^3$: a product, cubed. And the fraction $\frac{2}{3}$ used as a factor four times is $\left(\frac{2}{3}\right)^4$: a quotient raised to a power. Multiplying everything out is slow with numbers and impossible with letters, since $x$ has no value. What is wanted is the answer written straight down as a single power, or a product of single powers, and it comes from the same place as the product of powers: an exponent counts factors, so count them. The count gives three rules. The exponents multiply. An exponent outside a product goes to every factor of it. An exponent outside a fraction goes to the top and to the bottom. The same count shows where the rules stop: at stacked exponents, and at an exponent outside a sum. ## Definition For a number $a$ and a positive whole number $n$, the power $a^n$ is the product of $n$ factors, each equal to $a$ (exponent notation): $$a^n = \underbrace{a \cdot a \cdots a}_{n \text{ factors}}$$ Here $a$ is the **base** and $n$ is the **exponent**. A number written with no exponent has exponent $1$: $a = a^1$. An exponent applies to the one thing written directly before it. When that thing is a set of parentheses, the exponent applies to the whole of what is inside: ### [Powers of ten](https://math.wiki/powers-of-ten/) **Summary:** Ten multiplied by itself, which is what each place value column is worth. **Question:** Ten ones make a ten, ten tens make a hundred, ten hundreds make a thousand. How can we write ten multiplied by itself many times, without counting a long row of zeros? ## Why this exists Fernanda is a bank teller. Loose dollar bills are slow to count, so the bank packs them. A **strap** holds $100$ bills. Ten straps are packed into a **brick**, so a brick holds $1{,}000$ bills. Each package is ten of the package before it. Ten keeps showing up because the way we write numbers is built on it. Ten ones make a ten, ten tens make a hundred, ten hundreds make a thousand, and it never stops. Each place is worth ten times the place to its right, as in times-ten-shifts-place-value. The trouble is that these numbers get long. A billion bills is written $1{,}000{,}000{,}000$. To read it you have to count the zeros one at a time, and one missed zero makes the number ten times too small. This lesson gives each of these numbers a short name that says, right in the name, how many times ten was multiplied: a billion is $10^9$. ## Definition A **power of ten** is the number we get by starting at $1$ and multiplying by $10$ some whole number of times. We write it as $10^n$. The small raised number $n$ counts the multiplications. $$10^n = 1 \times \underbrace{10 \times 10 \times \cdots \times 10}_{n \text{ tens}}$$ The $10$ is the **base** and $n$ is the **exponent**, the same words used in base-and-exponent. In this lesson $n$ is always a whole number: $0$, $1$, $2$, $3$ and so on. So $10^1 = 1 \times 10 = 10$, and $10^2 = 1 \times 10 \times 10 = 100$, and $10^3 = 1 \times 10 \times 10 \times 10 = 1{,}000$. ### [Prime and composite numbers](https://math.wiki/composite-numbers/) **Summary:** Whole numbers with more than two factors. **Question:** Some trays of seedlings can be planted in a rectangle of equal rows in several ways, and some only in one long row. Which numbers are which, and how can you tell? ## Why this exists Pranav is planting seedlings. He likes a neat bed: equal rows, every row the same length, no seedling left over. With $12$ seedlings he has choices. He can plant $2$ rows of $6$, or $3$ rows of $4$, or one long row of $12$. With $13$ seedlings he tries $2$ rows and has one left over. He tries $3$ rows and has one left over again. Every try fails except one long row of $13$ (or the same row turned, $13$ rows of one seedling each). So whole numbers come in two kinds. Some split into equal rows in more than one way. Others refuse to split at all, except into one row. The second kind turn out to be the building blocks that every other whole number is made from, so they get a name of their own. ## Definition In find-all-factors-of-a-number you listed every factor of a number by finding its factor pairs. Now count how many factors there are. **A prime number** is a whole number greater than $1$ that has **exactly two factors**: $1$ and the number itself. **A composite number** is a whole number greater than $1$ that has **more than two factors**. **The number $1$ is neither.** Its only factor is $1$, so it has just one factor. That is not two, so $1$ is not prime, and it is not more than two, so $1$ is not composite. | Number | Factors | Kind | |:-:|---|---| | $1$ | $1$ | neither | | $2$ | $1, 2$ | prime | | $7$ | $1, 7$ | prime | | $8$ | $1, 2, 4, 8$ | composite | | $13$ | $1, 13$ | prime | | $12$ | $1, 2, 3, 4, 6, 12$ | composite | ### [Prime factorization](https://math.wiki/factor-trees/) **Summary:** Splitting a number into factors again and again until only primes remain. **Question:** A mosaic artist can lay 132 square tiles as a rectangle in several ways. If you keep splitting 132 into smaller factors, what is it made of in the end? ## When you reach for it Folake makes mosaics from square tiles, and she lays each batch as a full rectangle, with no gaps and no tiles left over. A rectangle of tiles holds its number of rows times its number of columns, so every rectangle she can make from a batch is a factor pair of the number of tiles. Twenty-four tiles can be $1$ by $24$, $2$ by $12$, $3$ by $8$ or $4$ by $6$. Seven tiles can only be a single strip, $1$ by $7$, because $7$ is a prime number: its only factors are $1$ and itself. A factor pair is a way to split a number, and the pieces can be split again. Since $4 = 2 \cdot 2$ and $6 = 2 \cdot 3$, $$24 = 4 \cdot 6 = 2 \cdot 2 \cdot 2 \cdot 3$$ Now no piece splits any further, because $2$ and $3$ are prime. Writing a whole number greater than $1$ as a product of primes is finding its **prime factorization**. A prime that repeats is written with an exponent, so the prime factorization of $24$ is $2^3 \cdot 3$. A prime is its own prime factorization: the factorization of $7$ is just $7$. Reach for it whenever you need to know what a number is built from. This lesson gives two ways to find the primes: the factor tree (Part 1) and the ladder (Part 2). Later lessons use the result to list every factor of a number, to compare two numbers, and to simplify square roots. ## The procedure ### Part 1: factor trees **Start with any factor pair.** Write the number at the top. Below it, write two factors whose product is the number, neither of them $1$, and join each to the number with a branch. For $132$, Yuki starts with $6$ and $22$. This is allowed because $6 \cdot 22$ *is* $132$: a split changes how the number is written, not its value. The pair $1 \cdot 132$ is ruled out because it splits nothing; the $132$ comes back unchanged. ### [Problems using several operations](https://math.wiki/multi-step-problems-with-four-operations/) **Summary:** Stories that need two or more of the four operations in a chosen order. **Question:** Four packs of six movie tickets, and five of the tickets go unused. When a story needs one answer before it can give the next, how do you choose the steps and know the result makes sense? ## When you reach for it Some stories cannot be answered in one step. The number the story asks for depends on a number the story never states, and that hidden number has to be found first. Koldo's class goes to the movie theater. The teacher buys $4$ packs of movie tickets with $6$ tickets in each pack, and $5$ of the tickets go unused. How many tickets are used? No single fact answers this. The story does not say how many tickets there were in all. That number is hidden inside "$4$ packs of $6$", and only once it is known can the $5$ unused tickets be taken away. So the story has two steps, and the first step's answer is the second step's starting number. Reach for this whenever a story's question cannot be answered from the numbers it gives, but can be answered from a number that one of its own steps would produce. Two-step stories with adding and subtracting only are two-step-word-problems-add-subtract. Here any of the four operations can appear in either step: multiplying and dividing within $100$, and adding and subtracting within $1{,}000$. The pieces are already in place. equal-groups-problems showed when a story multiplies or divides, and letter-for-an-unknown wrote the number a story asks for as a letter. The lesson takes four days, one part each. Part 1 writes each step as its own equation with a letter. Part 2 chooses the steps and their order from what happens in the story. Part 3 checks that an answer is reasonable by rounding or by mental math. Part 4 writes both steps as one expression and settles which operation goes first. ### [Problems with the start, change or result missing](https://math.wiki/change-unknown-problems/) **Summary:** Stories that give a start and an end and ask what happened in between. **Question:** A story tells how many there were at the end, but not how many joined, left, or were there to begin with. How can a child find the missing number without guessing? ## When you reach for it Most early stories give the start and the change and ask for the end: "$5$ crabs, $3$ more come, how many now?" This lesson is about the stories that turn that around. The story tells the end, and the missing number is somewhere earlier. - **The change is missing.** "There were $5$ hermit crabs in the tank. Folau put some more in. Now there are $13$. How many did he put in?" - **The start is missing.** "Some fire trucks were at the station. $4$ drove out to a fire. $9$ are left. How many were there at first?" A child reading these stories cannot just act them out from the beginning, because the beginning, or the middle, is the very thing they do not know. Two things help. A picture of the story (a **part-part-whole bar**, or counters in a ten frame) shows which number is the whole and which numbers are parts. And an equation with a box, $\square$, written in the order the story happens, holds the place of the missing number until it is found. Then the box is filled by counting up or by thinking of the related fact. All the numbers here stay within $20$. The lesson is taught over four days: - **Part 1:** adding-to stories with the change missing, $5 + \square = 13$. - **Part 2:** taking-from stories with the change missing, $12 - \square = 7$. - **Part 3:** stories with the start missing, both $\square + 6 = 14$ and $\square - 4 = 9$. These are the hardest kind, because the number the child would act out first is the one that is gone. - **Part 4:** mixed stories. The child decides which number is the whole before doing anything else, and meets stories told out of time order. ### [Problems with the starting amount missing](https://math.wiki/start-unknown-problems/) **Summary:** Stories that give the change and the end and ask where things began. **Question:** A story tells what happened and how things ended, but not how they began. With numbers in the tens, how does a child go back to the start without guessing? ## When you reach for it Some stories hide their beginning. "Some toys were in the toy drive bin. $26$ more were given. Now there are $63$. How many toys were in the bin before?" The story tells the change and the end, and asks for the start. The child has met these stories already, within $20$, in problems with the start, change or result missing. The one-line recap: in an adding-to story the start is a **part** and the end is the whole, and in a taking-from story the start is the **whole**, so there the child adds even though the story took things away. What changes now is the size of the numbers. With $9$ trucks left and $4$ driven out, a child can try $12$, then $13$, and land on the answer by trying. With $63$ toys and $26$ given, trying starts does not work: there are too many to try, and each try is a two-digit sum. The child needs a reason for which way to go, and a sure way to get there. The reason is a simple one. **A story runs forward in time. To find its start, run it backward.** Whatever the story did, undo it. If toys were given, take them back off. If fish were let go, put them back in the bucket. Then run the story forward again from the number found, to check that it really ends where the story says. This lesson takes three days. Day 1 is adding-to stories, Day 2 is taking-from stories, and Day 3 is choosing between the two moves when the numbers are big and the story is told in a mixed-up order. ### [Problems with the total missing](https://math.wiki/result-unknown-problems/) **Summary:** Stories that give both parts and ask for the whole. **Question:** A story gives two numbers and asks how many there are in all, or how many are left at the end. How does a child decide whether to add or subtract, and know the answer fits the story? ## When you reach for it Many first-grade story problems ask one of two plain questions: **how many in all?** or **how many are left?** The two numbers in the story are both given, and the missing number is the last one, the total or the end. - **Putting together, total missing.** Nothing moves. Two groups are simply there, and the story asks about both at once: "Asta's bowl has $7$ red cherry tomatoes and $9$ yellow cherry tomatoes. How many cherry tomatoes are in the bowl?" - **Adding to, result missing.** Something happens: more arrive. "Trine has $8$ carrot sticks in her lunch. Arnfinn gives her $6$ more. How many does she have now?" - **Taking from, result missing.** Something happens: some go away. "Asta picked $16$ cherry tomatoes. She ate $7$ of them. How many does she have left?" These look like the easiest stories, and in a way they are: a child can act them out from the beginning, because the beginning is given. The work of the lesson is in three habits that make the harder stories later possible. First, decide **which number is the whole** (all of them together) and whether the story gives it or asks for it. Second, write an **equation with a box** for the missing number, in the order the story goes. Third, **check** the answer against the story: put it back in and see that the story still makes sense. All the numbers stay within $20$. The stories where the change or the start is missing came in the lesson before, problems with the start, change or result missing; here the box always sits at the end of the equation. The lesson is taught over four days: ### [Properties of addition](https://math.wiki/commutative-property-of-addition/) **Summary:** Two numbers added in either order give the same total. **Question:** A child who knows that 8 and 3 make 11, does she also know 3 and 8? And with three numbers to add, may she choose which two to put together first? ## Why this exists A child learning addition facts meets a great many of them. Every number from $0$ to $9$ added to every number from $0$ to $9$ makes a table of a hundred facts, and that is only the start. Learning each one separately would be slow and dull, and it would miss something a child can notice for herself: many of the facts are the same fact twice. Take $8 + 3$ and $3 + 8$. Eight blocks and three blocks, or three blocks and eight blocks: either way the same eleven blocks end up in the pile. A child who knows one of these facts knows the other, and a child who can count on can always start from the bigger number, since $3 + 8$ may be worked out as "eight ... nine, ten, eleven". The same freedom shows up when a story has three amounts in it. A tower of $7$ red, $5$ blue and $3$ yellow blocks asks for $7 + 5 + 3$. Going left to right means $7 + 5$ first, which crosses ten. But the $7$ and the $3$ make ten, and ten and five is fifteen, said at once. A child who may choose which two numbers to put together first can choose the easy pair. Zero needs a word of its own: a group with nothing in it adds nothing. And subtraction needs a warning, because none of this freedom carries over to taking away. This lesson gathers these facts together. They are called the properties of addition, and they are the reasons behind most of the shortcuts a child uses when adding. ## Definition There are three properties of addition, and one property that subtraction does not have. ### [Properties of multiplication](https://math.wiki/commutative-property-of-multiplication/) **Summary:** Turning an array on its side gives the same number of items. **Question:** Four rows of nine seats and nine rows of four seats look like two different rooms. Why do they hold the same number of people, and what other changes leave a product the same? ## Why this exists Isko helps set up the school theater. The seating plan on the wall shows $4$ rows with $9$ seats in each row. Nkiru is looking at the same plan from the side of the room, and to her it shows $9$ rows with $4$ seats in each. Isko works out $4 \times 9$; Nkiru works out $9 \times 4$. Do they have to count again to be sure they agree? They do not. Nobody moved a seat, so the number of seats cannot have changed. That simple fact, that a product does not care which factor comes first, is the first of three rules in this lesson. Each rule says that some change to a multiplication leaves its value alone: - swapping the order of two factors, - changing which pair of three factors is multiplied first, - multiplying by $1$, which changes nothing, and by $0$, which always gives $0$. These rules matter because they cut the work of learning the facts. Once $9 \times 4$ is known, $4 \times 9$ is known too, so most facts come in pairs and only about half need learning. And when three numbers are multiplied, the rules let you pick the easiest pair to start with, such as a $5$ and a $2$ that make $10$. ## Definition The numbers being multiplied are the **factors** and the answer is the **product**, as in arrays-and-multiplication. **The commutative property of multiplication.** Two factors can be multiplied in either order and the product is the same. For example, $$4 \times 9 = 9 \times 4$$ ### [Properties of real numbers](https://math.wiki/properties-of-real-numbers/) **Summary:** Commutative, associative, distributive, identities and inverses, named and used. **Question:** Which rules about adding and multiplying hold for every real number at once, and why can algebra rely on them? ## What this area is about Algebra is mostly rewriting: an expression or an equation turned into a handier one that is worth the same. Trying numbers can never show a rewrite is safe for every value of the letters. A short list of properties true of all real numbers can, and this area names them so that each step of a simplification or a solution has a reason. ## How these fit together **Laws for rewriting.** Commutative, associative and distributive properties: order and grouping do not change a sum or a product, and $a(b + c) = ab + ac$ for all real $a$, $b$ and $c$. **Numbers that do nothing, and numbers that undo.** Identity and inverse elements: adding $0$ and multiplying by $1$ change nothing; every real $a$ has an opposite with $a + (-a) = 0$, and every $a \ne 0$ has a reciprocal with $a \cdot \frac{1}{a} = 1$. Zero has no reciprocal. **Staying inside a set.** Closure asks whether an operation on two members of a set always gives a member: the whole numbers are closed under addition but not under subtraction. ## A path through it There is no forced order. The rewriting laws come first naturally, then identities and inverses, which explain each step of solving an equation. Closure comes last, since it uses sorting numbers into sets and subtracting integers. Finish here and every rewrite in later algebra can name the property that allows it. ### [Rational exponents](https://math.wiki/unit-fraction-exponents/) **Summary:** A power of one half means a square root, and one third means a cube root. **Question:** An exponent counts how many times a number is used as a factor. What could it mean to use 1331 as a factor one third of a time? ## Why this exists A whole-number exponent counts factors: $11^3 = 11 \cdot 11 \cdot 11 = 1331$. A count of factors can be three or four, but it cannot be one third. Yet fractional exponents turn up as soon as questions are asked in reverse, or as soon as a formula from science is written down. The time a planet takes to go around the Sun, for instance, measured in suitable units, is its average distance from the Sun raised to the power $\tfrac32$ (Example 8). The way forward is the one already taken for the zero exponent: instead of asking what the symbol counts, ask what it has to equal if the laws of exponents are to keep working. The power of a power law says $(a^m)^n = a^{mn}$. If that law is to hold when $m = \tfrac13$, then $$\left(1331^{1/3}\right)^3 = 1331^{\frac13 \cdot 3} = 1331^1 = 1331.$$ So $1331^{1/3}$ has to be a number whose cube is $1331$. There is exactly one real number like that, the cube root $\sqrt[3]{1331} = 11$. Nobody chose this meaning. The law forced it, and with it every fraction gets a meaning as an exponent. ## Definition **Unit fractions.** For a whole number $n \ge 2$, the power $a^{1/n}$ is the $n$th root of $a$: $$a^{1/n} = \sqrt[n]{a}.$$ It is defined exactly when $\sqrt[n]{a}$ is a real number: for every real $a$ when $n$ is odd, and for $a \ge 0$ when $n$ is even. So $a^{1/2} = \sqrt{a}$ and $a^{1/3} = \sqrt[3]{a}$: the denominator of the exponent is the index of the root. ### [Rational numbers](https://math.wiki/rational-numbers/) **Summary:** Any number that can be written as one integer over another. **Question:** Integers, fractions, negatives and decimals each came with their own rules. Is there one kind of number that takes them all in, and how do you work with it? ## What this area is about A rational number is one that can be written $\frac{a}{b}$ with $a$ and $b$ integers and $b \ne 0$. Integers, fractions, mixed numbers and decimals that stop or repeat all qualify, on both sides of zero. This area joins them into one family on one number line, and does arithmetic with signed fractions and decimals. ## How these fit together **What they are.** What makes a number rational gives the test. Negative fractions and decimals places the negative ones left of zero. **As decimals.** Converting between fractions and decimals divides top by bottom. Terminating and repeating decimals adds the bar for digits that repeat forever, and turning a repeating decimal into a fraction runs it back. A result: a fraction in lowest terms terminates exactly when its denominator has no prime factors but $2$ and $5$. **Procedures.** Comparing and ordering, adding and subtracting, multiplying and dividing, and rational numbers in real situations. ## A path through it The definition and negatives first, then comparing. Converting to decimals comes before repeating decimals, which come before turning them back into fractions; which fractions terminate can follow the converting. Adding and subtracting comes before multiplying and dividing and before the word problems. Finish here and you can sort a number into its sets and see why $\sqrt{2}$ is not rational. ### [Rational numbers in real situations](https://math.wiki/rational-numbers-in-context/) **Summary:** Stories about balances, temperatures and changes that use signed fractions and decimals. **Question:** A river is $0.6$ meters below flood stage, a share price fell $1.40$ dollars, a runner finished a kilometer $2.4$ seconds ahead of her target. How do words like these become signed numbers, and how does the answer get back into words? ## When you reach for it Many measurements are made from a reference point and can land on either side of it. A river gauge gives the water's height above or below flood stage, the level at which the river starts to spill over its banks. A golf score is counted over or under par, the number of strokes the course is expected to take. A share price moves up or down from where it closed the day before. A runner's time for each kilometer is over or under her target. Clocks around the world are set ahead of or behind one reference time. In each case a single signed number says two things at once: how far from the reference point, and on which side. The arithmetic is already in hand: adding and subtracting and multiplying and dividing signed fractions and decimals. What this lesson adds is the move between the story and the numbers: deciding what zero stands for, turning words such as "below", "under", "fell" and "slow" into signs, choosing the operation the question asks for, and reading the sign of the answer back as words. ## The procedure The lesson is taught in two parts. The first sets up signed quantities and works with sums, differences and order. The second works with products, quotients and averages, and with problems of several steps. ### Part 1: zero, direction and change **Choose a zero and a positive direction, and say what they are.** A signed number is a position measured from a reference point, the way negative numbers mark positions on either side of $0$ on a number line. So the first decision is what $0$ stands for (flood stage, par, the target pace, last night's closing price) and which side counts as positive. Usually the choice is natural: up, above, ahead and gain are positive; down, below, behind and loss are negative. A reading $0.6$ meters below flood stage is then $-0.6$, and $\frac{3}{4}$ of a meter above it is $\frac{3}{4}$. The choice is not forced. Counting downward as positive would flip every sign in the problem, every answer would flip with it, and the answers read back into words would be the same. What matters is to make the choice once and keep to it. ### [Reading and writing numbers to 1,000](https://math.wiki/numbers-to-1000/) **Summary:** Hundreds, tens, ones, in words, digits and expanded form. **Question:** Page $805$ and page $85$ share the digits $8$ and $5$, and the words "eight hundred five" never say "zero". How do three digits, and the words we say for them, tell exactly how many hundreds, tens and ones a number has? ## When you reach for it Radu is reading a thick book. The number at the bottom of his page is $247$. At a fun run, Pendo pins on a race bib that says $315$, and as she crosses the finish line the announcer reads her number out loud. Each time, three digits stand for one number. Someone has to turn the digits into words, or turn spoken words back into digits, and get the same number both ways. This lesson reads and writes whole numbers from $100$ up to $1{,}000$. A two-digit number is some tens and some ones, as in tens-and-ones. A third digit, on the left, counts a bigger group: hundreds. The lesson runs over three days. Day 1 builds a hundred out of ten tens and reads a three-digit number from base-ten blocks. Day 2 writes the same numbers in words and turns words back into digits. Day 3 handles a zero in the tens or ones place, as in $805$, $508$ and $580$, and ends at one thousand, which is ten hundreds. You reach for it whenever a count goes past $99$: page numbers, bib numbers, seats in a hall, the steps on a long stair. ## The procedure ### Day 1: hundreds, tens and ones **A hundred is ten tens.** Base-ten blocks come in three sizes. A small square is one. A rod is a column of ten small squares: one ten. A flat is a big square made of ten rows of ten small squares: one hundred. Lay ten rods side by side and they cover exactly one flat (Figure 1). So $$10 \text{ tens} = 1 \text{ hundred} = 100$$ ### [Regrouping in addition](https://math.wiki/regrouping-in-addition/) **Summary:** Ten ones become one ten so that no place ever holds more than nine. **Question:** Anka has $47$ pretzel rods and Naranbaatar brings $36$ more. The ones make $13$, but a place holds only one digit. Where do the extra ones go? ## Why this exists The child can already add two two-digit numbers when nothing spills over. In Adding two-digit numbers without regrouping, $32 + 45$ is $3 + 4 = 7$ tens and $2 + 5 = 7$ ones, so $77$. The child can also add a one-digit number that makes a new ten, as in $38 + 5$ from Adding a one-digit number to a two-digit number by making a ten. Now both numbers have tens and ones, and the ones are big. At Anka's farm stand, pretzel rods are sold in bags of ten. Anka has $4$ bags and $7$ loose rods, which is $47$. Naranbaatar brings $3$ bags and $6$ loose rods, which is $36$. Bags go with bags: $4 + 3 = 7$ bags. Loose rods go with loose rods: $7 + 6 = 13$ loose rods. "$7$ bags and $13$ loose" is a true count, but it is not how the stand keeps its rods. Ten of the loose rods fill a new bag. Then there are $8$ bags and $3$ loose rods: $83$ pretzel rods. The digits need the same fix. The ones place holds one digit, from $0$ to $9$. Thirteen ones cannot be written there. Writing the $7$ tens next to the $13$ ones gives $713$, which is seven hundred thirteen, nowhere near the real count. **Regrouping** is the fix: ten ones become one ten, so no place ever holds more than nine. ## Definition **Regrouping** in addition means trading ten of one place for one of the next place to the left, so that every place holds at most nine: - ten ones are traded for one ten; - ten tens are traded for one hundred. ### [Regrouping in subtraction](https://math.wiki/regrouping-in-subtraction/) **Summary:** One ten is broken into ten ones when a place does not have enough. **Question:** Iluka has $54$ buttons, $5$ cards of ten and $4$ loose, and a customer wants $27$. There are only $4$ loose buttons and she needs to hand over $7$. Where do the other loose buttons come from? ## Why this exists The child can already subtract two-digit numbers when each place has enough. In Subtracting two-digit numbers without regrouping, tens are taken from tens and ones from ones, and nothing has to be opened up. The child also knows the trade itself from Base ten blocks and trading ten for one: one rod can be swapped for ten units. Now the ones run short. At the craft store, Iluka keeps buttons sewn on cards of ten. She has $5$ cards and $4$ loose buttons, which is $54$. A customer wants $27$ buttons: $2$ cards and $7$ loose ones. The $2$ cards are easy. The $7$ loose buttons are not, because Iluka has only $4$. She does not turn the customer away. She has plenty of buttons, just not loose ones. So she snips the buttons off one card. Now she has $4$ cards and $14$ loose buttons. That is still $54$ buttons; nothing was added or lost. But now there are enough loose ones. She hands over $2$ cards and $7$ loose buttons and keeps $2$ cards and $7$ loose: $27$ buttons. **Regrouping in subtraction** is that move: when a place does not have enough to take from, one of the next place up is broken into ten. ## Definition **Regrouping** in subtraction means breaking one of a place into ten of the place to its right, so that a place has enough to take from: - one ten is broken into ten ones; - one hundred is broken into ten tens. The number stays the same. Only the way it is split into tens and ones changes. Some books call this "borrowing", "unbundling" or "breaking a ten". It is the same idea, and it is regrouping in addition run backward: there ten ones become one ten, here one ten becomes ten ones. ### [Remainders and what they mean](https://math.wiki/interpret-remainders-in-context/) **Summary:** Deciding whether a leftover is dropped, rounded up or reported as it stands. **Question:** Thirty-four campers ride four to a canoe. Division says 8 with 2 left over, so why does the camp need 9 canoes? ## When you reach for it The camp is going out on the lake. There are $34$ campers, and each canoe holds $4$. How many canoes does the camp need? Division asks how many groups of $4$ fit in $34$. Eight groups of $4$ is $32$, and nine groups of $4$ is $36$, which is more than $34$. So $8$ canoes can be filled, and $34 - 32 = 2$ campers are still on the dock. Those $2$ campers are the **remainder**: what is left over when no more full groups can be made. Nobody is left behind, so the camp needs a ninth canoe for the last $2$. The answer to the question is $9$, even though the division "said" $8$. The same division can answer other questions about the same trip: - How many canoes go out completely full? $8$. - How many canoes does the camp need? $9$. - How many campers ride in the canoe that is not full? $2$. One division, three different answers. Reach for this lesson whenever a division in a story does not come out even. The division always gives the same two numbers, the quotient and the remainder. The story decides what to do with them. ## The procedure ### Part 1: what a remainder is Dividing $34$ by $4$ can ask how many groups of $4$ fit in $34$, or how many each person gets when $34$ is shared among $4$; division-as-sharing has both meanings. Either way, make equal groups until you cannot make another one. Here are the canoes, one dot for each camper. Eight full groups of $4$ use up $32$ campers, and $2$ are left. Two campers cannot fill a group of $4$, so we stop. We write ### [Rounding](https://math.wiki/rounding/) **Summary:** Replace a number with a nearby one that is easier to work with. **Question:** An exact number often carries more detail than the moment needs. Which simpler, nearby number should stand in for it? ## What this area is about Rounding replaces a number with the closest number that has only zeros from a chosen place onward. The number sits between two such neighbors, and a halfway mark between them decides which is closer; a number exactly on the halfway mark rounds up. The idea is the same for whole numbers and for decimals. ## How these fit together All three are procedures, each widening the one before. Rounding to the nearest ten or hundred finds the two neighbors on a number line and compares distances. Rounding to any place carries that to thousands and beyond, including roundings that carry into the next place, such as $396{,}500 \approx 400{,}000$ to the nearest ten thousand. Rounding decimals rounds to the nearest whole number, tenth or hundredth, as when $7.648 \approx 7.6$ to the nearest tenth. ## A path through it Tens and hundreds first, then any place, then decimals, which also need decimal place value. Finish here and you can estimate sums and differences, estimate by rounding before multiplying, and see why a measurement is only as exact as the tool that made it. ### [Rounding decimals](https://math.wiki/rounding-decimals/) **Summary:** Naming the nearest tenth, hundredth or whole number. **Question:** The scale at the cider stand reads $7.648$ pounds of apples, but the price sign only has room for one digit after the point. Which shorter number is closest to the real weight? ## When you reach for it Bukola, Esteban and Minseo run a cider stand. Their scale shows weights to the thousandth of a pound, like $7.648$ pounds of apples. Their jugs are filled from a tank whose meter reads to the hundredth of a gallon, like $13.95$ gallons. Most of the time nobody needs all those digits. The price sign has room for one digit after the point, and a customer only wants to know "about how much". Rounding a decimal means trading it for a shorter number that is as close as possible. You say which place to keep: the nearest whole number, the nearest tenth, or the nearest hundredth. You already round whole numbers to a ten or a hundred (round-to-any-place). Decimals work the same way, one place at a time, because each place is ten times smaller than the one to its left (decimal-place-value, thousandths). ## The procedure ### Two neighbors and a halfway mark Take $7.648$ and round it to the nearest whole number. Look only at whole numbers. The number sits between $7$ and $8$, so those are its two neighbors. One of them is the answer, whichever is closer. The point exactly between the neighbors is the halfway mark, $7.5$. Anything to the left of $7.5$ is closer to $7$. Anything to the right is closer to $8$. The dot for $7.648$ is to the right, so $$7.648 \approx 8$$ to the nearest whole number. The sign $\approx$ is read "is about". Now round the same number to the nearest tenth. The neighbors are the two numbers that stop at the tenths place on either side of it: $7.6$ and $7.7$. The halfway mark between them is $7.65$. ### [Rounding to any place, including decimals](https://math.wiki/round-to-any-place/) **Summary:** Choosing the nearest number that is zero from a named place onward. **Question:** The car's odometer says $386{,}451$ miles. Nobody needs every digit to say how far the car has gone. Which simpler number is closest, and how do you pick it for a ten, a thousand or a hundred thousand? ## When you reach for it Reza, Losana and Palesa are on a long road trip. The car's odometer counts every mile the car has ever driven, and a trip meter counts the miles since they left home. Along the way, signs give the population of each town they pass: how many people live there. These numbers are exact, but most of the time nobody needs every digit. "We have driven about $600$ miles" or "that town has about $8{,}000$ people" says enough. Rounding trades a number for a nearby number that ends in zeros from some place onward. You choose the place: the nearest ten, hundred, thousand, ten thousand or hundred thousand. The answer is the number of that kind that is **closest** to the one you started with. You have already met this for tens and hundreds (rounding-to-the-nearest-ten: a number sits between two multiples of ten and goes to the closer one). This lesson does the same thing at every place, using the places of numbers-to-one-million. The lesson takes three days. Part 1 rounds to the nearest ten, hundred and thousand on a number line. Part 2 rounds large numbers to the nearest ten thousand and hundred thousand, including the rounding that carries into the next place. Part 3 chooses a sensible place, says "about", and takes a short look at a decimal. ## The procedure ### Part 1: two neighbors and a halfway mark **The two neighbors.** To round $638$ to the nearest hundred, look only at numbers that end in two zeros: $500$, $600$, $700$, $800$. The number $638$ sits between $600$ and $700$. Those two are its **neighbors** for this place, and the answer is one of them, whichever is closer. ### [Rounding to the nearest ten or hundred](https://math.wiki/rounding-to-the-nearest-ten/) **Summary:** Naming the closest multiple of ten. **Question:** At the county fair gate, $347$ people came through in the first hour. Is that nearer $340$ or $350$, nearer $300$ or $400$, and how do you decide when a number sits exactly in the middle? ## When you reach for it Aiyetoro counts people at the gate of the county fair. In the first hour, $347$ people come through. When the fair manager asks how the morning is going, nobody needs to hear "three hundred forty-seven." "About $350$" says the same thing in a way that is easy to hear, remember and work with. "About $300$" is rougher, but still true enough for a sign on the barn wall. Each of those is **rounding**: replacing a number with a nearby number that is easier to use. The nearby numbers we round to here are the tens ($10$, $20$, $30$, and so on) and the hundreds ($100$, $200$, $300$, and so on). Rounding to the nearest ten asks which ten is closest. Rounding to the nearest hundred asks which hundred is closest. You reach for it whenever an exact number is more detail than the moment needs: telling someone a count, writing a quick note, or, in the next lesson, estimating a sum or a difference before working it out. The whole move rests on one picture, the number line from number-line-intervals, with its tick marks stepping by ones, tens or hundreds. This lesson uses whole numbers up to $1{,}000$. The lesson runs over three days. Part 1 rounds to the nearest ten. Part 2 rounds to the nearest hundred. Part 3 handles the edges: rounding that carries into the next hundred, small numbers, and working backward from a rounded number to the numbers it could have come from. ## The procedure ### [Saying the counting words to 20](https://math.wiki/rote-counting-to-20/) **Summary:** Reciting the number words in order, before they are attached to any objects. **Question:** How does a child learn to say the counting words from one to twenty in the right order, and to start from any of them? ## When you reach for it Before a child can count things, the counting words have to come out in the same order every time: one, two, three, all the way to twenty. This lesson is only the words, said aloud with claps, jumps and steps, with nothing to count yet. The hard part is past ten: *eleven* and *twelve* sound like no number before them, and *thirteen* and *fifteen* are not "threeteen" and "fiveteen". Each day below takes about ten minutes. Say the words with the child, never as a quiz, and let a wrong word be fixed by saying the right one together. ## Day 1: The words to ten **You need:** nothing but your hands and voice. Clap and count together from one to ten, one clap for each word, slowly. Then try it loud, then in a whisper, then as a rocket launch (start at one and shout "ten!" at the end). The beat helps: one clap, one word, so no word gets rushed or dropped. Now stop partway and let the child go on. Say "one, two, three, four" and pause. Ask, "What comes next? Keep going to ten." Do this from different stopping places. **Practice.** 1. Count to $10$ with a clap on each word. *Answer:* one, two, three, four, five, six, seven, eight, nine, ten, with $10$ claps. 2. "I say one, two, three. You go on to ten." *Answer:* four, five, six, seven, eight, nine, ten. 3. What word comes right after *six*? *Answer:* seven. 4. What word comes right before *nine*? *Answer:* eight. Saying the words up to it, "seven, eight, nine", lets the child find it. ### [Scientific notation](https://math.wiki/scientific-notation/) **Summary:** Very big and very small numbers as a number between 1 and 10 times a power of ten. **Question:** Some numbers are far too big or too small to read at a glance. How do you write them, compare them and calculate with them without counting zeros? ## What this area is about The distance to the Sun and the width of a hair both come with long runs of zeros, and the size of such a number hides in how many there are. Scientific notation writes a number as $a \times 10^{n}$, with $1 \le a < 10$ and $n$ an integer. The exponent $n$ carries the size; the leading part $a$ carries the digits that matter. ## How these fit together **The form.** Writing and reading scientific notation defines the form and moves between it and ordinary notation, for large numbers and for small ones, where the exponent is negative, and reads a calculator display such as $\texttt{4.2E-6}$. **Using it.** Comparing numbers in scientific notation works with positive numbers in this form: the larger exponent gives the larger number, and the leading part only breaks a tie; it also estimates how many times larger one is than the other. Calculating in scientific notation multiplies and divides by combining the leading parts and adding or subtracting the exponents, then adds and subtracts once the powers of ten match. ## A path through it Writing and reading the form comes first, and it leans on powers of ten and zero and negative exponents. Comparing and calculating follow, in either order. Finish here and very large and very small measurements, the kind found in science texts and on a calculator screen, can be read, ordered and combined without writing out a single long string of zeros. ### [Skip counting](https://math.wiki/skip-counting/) **Summary:** Counting in equal jumps rather than one at a time. **Question:** Why count things one at a time when they come in pairs, fives or tens? ## What this area is about Skip counting says only the number at the end of each equal group: "two, four, six" for pairs, "ten, twenty, thirty" for tens. The count is shorter, so there is less room for a slip. It works only when every group really holds the same amount, and the numbers it lands on follow a pattern a child can learn to predict. ## How these fit together **By tens.** Counting by tens to one hundred counts groups of ten up to $100$, one word for each full group. **Smaller jumps.** Counting by twos, fives, threes and fours counts pairs, rows of five and other small groups, and asks which numbers such a count will land on. **By hundreds.** Counting by hundreds counts boxes of $100$, and shows why only one digit changes at each step. ## A path through it Count by tens first, then by hundreds. Counting by twos and fives can come before hundreds or after. Skip counting is the first step toward multiplication, and it leads on to facts for two, five and ten and listing multiples. ### [Sorting a number into its sets](https://math.wiki/classify-real-numbers/) **Summary:** Deciding whether a number is whole, an integer, rational or irrational. **Question:** Is $-\frac{8}{4}$ a fraction or an integer, and is $\sqrt{49}$ irrational because it has a root sign? Given any number, which of the named families does it belong to? ## When you reach for it The numbers met so far came in families, each built by adding to the one before: counting numbers, then zero, then the negative integers, then fractions and decimals, and most recently numbers such as $\sqrt{2}$ that are not fractions at all. So one number usually belongs to several families at once. $5$ is a counting number, and also an integer, a rational number and a real number. Sorting a number means listing every family it belongs to. You need it whenever a statement depends on the kind of number in play: a count of people must be a whole number, and "solve in the integers" throws out an answer such as $\frac{7}{2}$. It takes two things: knowing how the families sit inside one another, and seeing past the way a number is written to the number itself. ## The procedure ### The six sets - The **natural numbers**, also called the **counting numbers**: $\mathbb{N} = \{1, 2, 3, \ldots\}$. The braces list the members, as in roster notation, and the dots mean the list goes on forever. In this lesson they start at $1$. - The **whole numbers**: $0, 1, 2, 3, \ldots$, the natural numbers with $0$ added. This set has no standard letter. - The **integers**: $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$, the whole numbers together with their opposites. - The **rational numbers**, $\mathbb{Q}$: every number that can be written as $\frac{a}{b}$ with $a$ and $b$ integers and $b \neq 0$. As what-makes-a-number-rational shows, these are exactly the numbers whose decimals either stop or repeat. - The **irrational numbers**: the real numbers that are not rational. Their decimals never stop and never repeat. $\sqrt{2}$ is one, and so is $\pi$. - The **real numbers**, $\mathbb{R}$: the rational and irrational numbers together, which are exactly the numbers that label the points of the number line. ### [Square and cube roots](https://math.wiki/square-and-cube-roots/) **Summary:** The number that, squared or cubed, gives you back the one you started with. **Question:** You know the area of a square or the volume of a cube. How do you get back to the length of its side? ## What this area is about Squaring and cubing take a length to an area or a volume. A root runs that backwards: $b$ is a square root of $a$ when $b^2 = a$, and a cube root of $a$ when $b^3 = a$. Much of the area is about how many answers there are, and when there are none. ## How these fit together **Square roots.** Square roots reads roots straight off the perfect squares, and fractions and decimals built from them. A positive number has two square roots, one positive and one negative; $\sqrt{a}$ names the one that is not negative, and a negative number has no real square root. **Cube roots.** Cube roots and higher roots undoes cubing. Every real number has exactly one real cube root, so a negative radicand is allowed: $\sqrt[3]{-64} = -4$. **Any index.** Higher roots treats $\sqrt[n]{a}$ for whole numbers $n \ge 2$. Whether $n$ is odd or even decides almost everything: an odd index gives exactly one real root for every real $a$, while an even index gives a real root only when $a \ge 0$. ## A path through it Square roots first, then cube roots, then higher roots: each builds on the one before. Finish here and you can simplify a square root, see why $\sqrt{2}$ is irrational, use the Pythagorean theorem, solve quadratics with square roots and read rational exponents such as $a^{1/2}$. ### [Square roots](https://math.wiki/roots-of-perfect-squares/) **Summary:** Reading a root straight off the squares that are already known. **Question:** You know the area of a square. How long is its side? ## When you reach for it Squaring takes a number and multiplies it by itself: $6^2 = 6 \times 6 = 36$. Many questions run the other way. You are told the result, $36$, and asked what number was squared to give it. The plainest case is a square whose area you know. Thirty-six square tiles can be laid out as a square, and the question "how many tiles along each side?" asks for the number whose square is $36$. The answer is $6$ (Figure 1). Undoing a square is called taking a **square root**. This lesson takes square roots of the numbers whose roots can be read straight off a square you already know: the perfect squares $1, 4, 9, 16, 25, \dots$, and fractions and decimals built from them, such as $\frac{25}{36}$ and $0.64$. For a number like $50$, which is not the square of a whole number, see estimate-a-square-root-between-integers. ## The procedure ### Two numbers square to the same thing A **square root** of a number $a$ is a number whose square is $a$. Both $5^2 = 25$ and $(-5)^2 = (-5)(-5) = 25$, because the product of two negative numbers is positive. So $25$ has two square roots, $5$ and $-5$. The same happens for every positive number: if $r$ is a square root of $a$, then so is $-r$, since $(-r)^2 = r^2$. A positive number has exactly two square roots, one positive and one negative, and they are opposites. Zero has only one square root, $0$, because $0^2 = 0$ and every other number squares to something positive. ### [Subitizing and ten frames](https://math.wiki/subitizing-small-groups/) **Summary:** Naming the size of a very small group instantly. **Question:** How can a child know there are three without counting one, two, three? ## When you reach for it **Subitizing** means naming how many are in a very small group at a glance, without counting them one by one: you see three dots and simply know "three". It only works for small groups, so a larger group is seen as small groups put together ($5$ and $2$ make $7$), which is what dice, dominoes and ten frames are built for. Reach for it once a child can count a small set; it makes counting faster, and it is the first step toward seeing numbers as parts. Each day below takes about ten to fifteen minutes. Keep it a game: a quick look, a guess out loud, then a second look to check. ## Day 1: Quick looks at one, two and three **You need:** a few index cards with $1$, $2$ or $3$ dots drawn in marker, and a sheet of paper to cover them. Hold up a card for about two seconds, then cover it. Ask, "How many dots did you see?" Then ask the more important question: "How did you know?" A child who says "I just saw it" is subitizing. A child who tried to touch and count needs a shorter look, not a longer one; the quick flash is what nudges the eyes to take in the whole group at once. Mix the arrangements. Three in a row and three in a triangle are both three, and saying so is the point. Flash fingers too: hold up $2$ fingers, hide your hand, and ask for the number. Then let the child flash fingers at you. **Practice.** Flash each card for two seconds. 1. Card A. How many? **Answer:** $2$. 2. Card B. How many? **Answer:** $3$, in a triangle. 3. Card C. How many? **Answer:** $1$. 4. Card D. How many? **Answer:** $3$, in a row. Ask whether B and D show the same number. They do: both are $3$, arranged differently. 5. You flash $3$ fingers, then hide them. How many? **Answer:** $3$. ### [Subtracting across zeros](https://math.wiki/subtract-across-zeros/) **Summary:** Differences where the places being borrowed from hold nothing. **Question:** A fossil museum has $7{,}006$ fossils and lends out $2{,}849$ of them. The ones run short, but the tens and hundreds hold nothing to unbundle. Where does the trade come from? ## When you reach for it Bahar works at a fossil museum. The museum owns $7{,}006$ fossils, and it lends $2{,}849$ of them to a school for a year. How many fossils stay at the museum? The answer is $7{,}006 - 2{,}849$. You already know the standard subtraction algorithm: line up the places, start at the ones, and when a place on top runs short, **unbundle** one unit from the place to its left. That one trade is regrouping in subtraction. Here the ones run short: $6$ ones, and you need to take away $9$. The usual move is to unbundle a ten. But the tens place holds $0$. So does the hundreds place. There is no ten to unbundle, and no hundred either. This lesson is about that case: **subtracting across zeros**. It comes up all the time, because round numbers are full of zeros. A goal of $1{,}000$ visitors, a fundraising goal of $10{,}000$ dollars, a box of $500$ labels: every one of them, minus some amount, asks you to unbundle through zeros. There are two good ways to think about it. Part 1 walks left to the first place that has something, and passes the trade back down one place at a time. Part 2 changes both numbers first, so that no trade is needed at all. ## The procedure ### Part 1: walking left to the first nonzero place Take $7{,}006 - 2{,}849$, lined up on the right. $$ \begin{array}{r} 7{,}006 \\ -\;2{,}849 \\ \hline \end{array} $$ **Ones.** The top has $6$ ones and you need to take away $9$. Not enough. Look left for something to unbundle. The tens hold $0$. The hundreds hold $0$. The thousands hold $7$. That is the first place with anything in it, so the trade starts there. ### [Subtracting integers](https://math.wiki/subtract-integers-as-adding-the-opposite/) **Summary:** Taking a number away gives the same result as adding its opposite. **Question:** How do you take away a number you do not have, such as subtracting $-3$ from $4$? ## Why this exists Among the whole numbers ($0$, $1$, $2$, $3$ and so on), $3 - 5$ has no answer: no whole number added to $5$ gives $3$. The integers were made so that it does. But they bring a new puzzle. Read as "take away", $4 - 2$ is easy: four things, take two. Now try $4 - (-3)$. It asks you to take three negatives away from a collection that has no negatives in it. And $-8 - 3$ asks you to take three positives from a collection of negatives only. This lesson shows that every subtraction of integers can be turned into an addition, and the addition is one you already know how to do from adding integers. Then it puts subtraction to two uses: finding how far apart two numbers sit on the number line, and finding changes in height and money when some of the values are below zero. Changes in temperature have a lesson of their own, negative-temperatures. ## Definition In plain words: **subtracting a number gives the same result as adding its opposite.** $$a - b = a + (-b)$$ This holds for all integers $a$ and $b$. The opposite of $b$, written $-b$, is the number the same distance from zero on the other side (opposites-and-zero-pairs). The opposite of a negative number is positive, $-(-b) = b$, so subtracting a negative number is the same as adding a positive one: $$a - (-b) = a + b$$ The rule rests on what subtraction means. $a - b$ is the number that, added to $b$, gives $a$ (subtraction-as-unknown-addend). That meaning was true for whole numbers and stays true, word for word, for integers. "Take away" is one picture of it, and it still works with integers once zero pairs are allowed, as the next section shows. ### [Subtracting multiples of ten](https://math.wiki/subtract-multiples-of-ten/) **Summary:** Taking away whole tens from a multiple of ten, such as 70 minus 30, by counting tens. **Question:** Eighty take away fifty sounds like a big subtraction. Why can a child who knows that eight take away five is three answer it at once? ## When you reach for it A child who has met tens-and-ones knows that $80$ is $8$ tens and no ones, and that $50$ is $5$ tens. A question such as $80 - 50$ looks far bigger than anything within twenty. It is not. Eight tens take away five tens leaves three tens, for the same reason that eight things take away five things leaves three things. The thing being counted is a ten instead of a single bead, and nothing else changes. So the fact $8 - 5 = 3$, which the child already knows, does all the work, and the answer is $3$ tens, which is $30$. That is the whole idea of this lesson: **a ten is a unit**, a thing that can be counted, added and taken away like any other thing. This lesson keeps to numbers that are whole tens, from $10$ up to $90$, on both sides of the minus sign: $70 - 40$, $90 - 60$, $60 - 60$. There are no loose ones anywhere, so nothing ever has to be broken apart. The lesson takes four days. - **Day 1** builds the idea with things that come in tens: strings of ten beads, bags of ten animal crackers, boxes of ten paint brushes. - **Day 2** shows the same subtraction on the tens column of a hundred chart and as jumps of ten back along a number line. - **Day 3** turns a subtraction into an addition with a missing number, $50 + \square = 80$, and puts the four related facts together. - **Day 4** uses all of it to solve story problems. The child has already met ten more and ten less, which takes one ten away from any number. Here a whole group of tens goes at once, and the pictures from that lesson, the chart and the jumps, come back as ways to see it. ### [Subtracting three-digit numbers by place value, including across zero](https://math.wiki/subtract-within-1000/) **Summary:** Subtracting within 1,000 hundreds from hundreds, tens from tens and ones from ones, trading a hundred or a ten when needed, even across a zero. **Question:** The apple barn has $642$ apples, packed in crates of a hundred, bags of ten and a few loose, and a store orders $278$. With only $2$ loose apples and $4$ bags on hand, how can the order be filled, and how many apples are left? ## When you reach for it Any time a three-digit amount is taken away, compared with another, or split into a known part and an unknown part, the question is a subtraction within $1{,}000$. A number such as $642$ is $6$ hundreds, $4$ tens and $2$ ones (numbers-to-1000), which expanded-form writes as $600 + 40 + 2$. To take $278$ away from it, take hundreds from hundreds, tens from tens and ones from ones. When a place does not hold enough, trade: one ten for ten ones, exactly as in regrouping-in-subtraction, or, new here, one hundred for ten tens. When the place you need to trade from is empty, trade twice in a row. One picture carries the whole lesson. At a pumpkin farm, Wanjiku runs the apple barn. Apples are packed in crates of $100$, bags of $10$, and a few loose apples. Opening a crate gives ten bags. Opening a bag gives ten loose apples. Neither move changes how many apples there are, and every trade in this lesson is one of those two moves. Base ten blocks say the same thing: a flat is a hundred, a rod is a ten, a unit is a one, and a flat can be swapped for ten rods or a rod for ten units. The lesson runs over four days. Day 1 takes away with no trade. Day 2 trades a ten, a hundred, or both. Day 3 meets a zero in the tens place. Day 4 meets two zeros and checks every answer by adding back. Each day moves from blocks to quick drawings (a square for a hundred, a stick for a ten, a dot for a one) to writing by place. The column method that crosses out digits comes later, in the standard subtraction algorithm; here every trade is shown with pieces or written as hundreds, tens and ones, so the child can see what each trade means. ### [Subtracting two-digit numbers without regrouping](https://math.wiki/subtract-two-digit-no-regrouping/) **Summary:** Each place has enough to take from, so nothing is unbundled. **Question:** The bookmobile has $87$ books on board and children borrow $34$ of them. Counting back thirty-four times would take a long time. Is there a shorter way to see what is left, and why does it give the right answer? ## When you reach for it A child who knows Subtracting within ten can find $7 - 4$ at once. Now the numbers are bigger. The bookmobile has $87$ books and children borrow $34$. Counting back thirty-four steps from $87$ works, but it is slow and one slip spoils it. There is a shorter way, and it rests on Tens and ones: a two-digit number is some tens and some ones. $87$ is $8$ tens and $7$ ones. $34$ is $3$ tens and $4$ ones. Taking away $34$ means taking away $3$ tens and $4$ ones. Tens come from the tens and ones come from the ones: - $8$ tens take away $3$ tens leaves $5$ tens; - $7$ ones take away $4$ ones leaves $3$ ones. So $5$ tens and $3$ ones are left, and $87 - 34 = 53$. The two small facts, $8 - 3$ and $7 - 4$, are ones the child already knows. This lesson covers the case where **each place has enough**: the first number has at least as many tens as the second, and at least as many ones. In $87 - 34$ there are $8$ tens to take $3$ from and $7$ ones to take $4$ from, so nothing has to be broken apart. When a place does not have enough, as in $82 - 34$ (only $2$ ones, and $4$ to take), a ten has to be opened first. That is a different move, named under Boundaries. It is the subtracting twin of Adding two-digit numbers without regrouping: the same rule, tens with tens and ones with ones, run the other way. The pictures are ones the child knows: bundles of ten sticks with loose sticks, base-ten rods and units, a quick drawing of the same thing, and an open number line. ### [Subtracting within ten](https://math.wiki/subtraction-within-10/) **Summary:** Taking away from ten or less. **Question:** Ten take away seven. Can a child simply see that three are left, without counting back seven times? ## When you reach for it A child who understands taking away can find $7 - 3$: put out seven counters, move three away, count what is left. That always works, and it should stay the thing the child falls back on. But counting every time is slow, it is easy to lose count, and it keeps the child's attention on the counting instead of on the numbers. This lesson is the next step: knowing each take-away with numbers up to $10$ because the child can *see* it, in a ten frame, on their fingers, or in a number bond, without counting one at a time. It is the partner of adding within ten, and it is what the child reaches for every time a subtraction comes up later, inside bigger problems as much as on its own. Quick here means quick because it is seen, not quick because it is rushed. It grows from many short, relaxed turns with objects and pictures, each one talked through, never from racing a clock. After enough of those turns, most children simply know that $10 - 7$ is $3$, and can still show why if asked. ## The procedure Underneath, every subtraction here is the same: the first number is the whole, the number taken away is one part, and the answer is the other part. That is Parts and the whole. What changes from fact to fact is how the child finds the other part quickly. For the adult, it helps to know how much there is. Every whole from $0$ to $10$, with anything from $0$ up to that whole taken away, gives $66$ take-aways, from $0 - 0$ to $10 - 10$. That sounds like a lot, but none of them is new. Each is a split of the whole into two parts, one of the bonds to five or to ten read from the whole, and each is an adding fact read backwards: $7 - 3 = 4$ because $4 + 3 = 7$. A handful of pictures does nearly all the work, and each one works for a reason. ### [Subtraction as an unknown addend](https://math.wiki/subtraction-as-unknown-addend/) **Summary:** Taking away and finding the missing part are the same question. **Question:** To work out 13 take away 9, do we have to take 9 away at all, or can we ask what goes with 9 to make 13? ## Why this exists A child who has learned taking away can find $13 - 9$ by making $13$, removing $9$ one at a time and counting what is left. It works, but it is slow, and with that many to move or count backwards it is easy to lose track. There is a second way to see the same question. Instead of taking $9$ away, ask: "$9$ and how many more make $13$?" Now the child starts at $9$ and counts up: $10$, then $11, 12, 13$. That is four steps, and four is the answer. A subtraction has turned into an addition with a missing part, and children usually know their addition facts before their subtraction. Some stories ask for a missing part from the start. "I have $8$ stickers and I want $11$. How many more do I need?" Nothing is being taken away, yet the answer is $11 - 8$. Seeing subtraction as a missing part is what lets a child recognize that this, too, is a subtraction. ## Definition **Subtraction finds a missing part.** When we know a whole and one of its two parts, subtracting the known part from the whole gives the other part. So $$9 - 4 = \square$$ asks the same thing as $$4 + \square = 9$$ Read the first as "nine take away four" and the second as "four and what make nine". Both have the same answer, $5$. The box stands for a number we have not found yet, and it can sit anywhere in the sentence. A number bond shows why. The whole, $9$, sits in the top circle. One part, $4$, sits below it, and the other part is empty. Taking $4$ away from $9$ leaves exactly that other part. Joining $4$ with that other part makes $9$. Either way, the empty circle is the number we want. ### [Subtraction as taking away](https://math.wiki/subtraction-as-taking-away/) **Summary:** Removing part of an amount leaves the rest. **Question:** Some of a group goes away. How many are left, and how do we write that down? ## Why this exists A child who can count and who has met putting groups together already knows what happens when a group gets bigger. Things also go the other way: apples get eaten, birds fly off, children get off the bus. Taking away is the name for that, and subtraction is the way to write it down. It is usually the first meaning of subtraction a child meets, and the easiest to act out, because the child can do it with their own hands: make a pile, move some off, count what stays. ## Definition **Taking away** means starting with a group, removing some of it, and finding how many are left. We write it with the **minus sign**, $-$, read "minus" or "take away". The number sentence $$7 - 3 = 4$$ is read "seven take away three is four" or "seven minus three equals four". Each number has a job, and they come in the order the story happens: - $7$ is how many there were at the start, - $3$ is how many were taken away, - $4$, after the equals sign, is how many are left. With counters, as in Figure 1, the child makes the starting group, crosses out or pushes away the ones that leave, and counts the rest. Crossing out is worth teaching early: the taken ones stay visible, so the child can check both numbers afterwards. ## Intuition Taking away is a before-and-after story. Before, there is one group. Something happens. After, there is a smaller group, and the ones that left are somewhere else. A tower of cubes shows this well (Figure 2). The child builds a tower of $6$, lifts $2$ off the top and sets them aside, then counts the tower again: $4$. Nothing vanished. The $6$ cubes are still on the table, split into the $4$ that stayed and the $2$ that left. That split is the idea behind parts and the whole, and it is why subtraction and addition are so closely tied: put the $2$ back and the tower is $6$ again. ### [Subtraction as the difference](https://math.wiki/subtraction-as-comparison/) **Summary:** How much more one amount is than another. **Question:** Two groups sit side by side and nothing is taken away. How do you say how many more one has than the other? ## Why this exists A child's first subtraction is taking away: seven birds sit on a fence, three fly off, four are left. Something leaves, and we count what stays. Then a different question turns up. Mia has seven shells and Sam has four. How many more does Mia have? Nobody gives anything away. Both piles stay on the table the whole time. And yet the answer is still $7 - 4 = 3$. This lesson is about that second meaning of the minus sign. It answers "how far apart are these two amounts?", and the answer has a name, the **difference**. Children use it every day (who has more crayons, which tower is taller, how many more days until a birthday), and it is the kind of subtraction that later measures lengths, reads bar graphs and finds distances on a number line. ## Definition The **difference** between two amounts is how many more the bigger one has. It is also how many fewer the smaller one has: that is the same gap, seen from the other side. When the two amounts are the same, the difference is $0$. We find it by subtracting the smaller number from the bigger one: $$7 - 4 = 3$$ Read aloud, "seven minus four equals three" now says two things at once: Mia has $3$ more shells than Sam, and Sam has $3$ fewer shells than Mia. The words that call for it are "how many more", "how many fewer", "how many less", "how much taller" or "longer", and "how far apart". ## Intuition The picture behind the difference is **matching**. Line the two groups up, one row above the other, and give every counter in the short row a partner in the long row. This is the same pairing a child uses to decide which group is bigger at all (one-to-one-correspondence and Comparing quantities); now we also count what is left over. ### [Tally marks and Roman numerals](https://math.wiki/roman-numerals/) **Summary:** Letters that stand for fixed values and are added or subtracted according to position. **Question:** Before there were digits like ours, how could people write down how many, or which year, so that anyone could read it later? ## Why this exists The oldest way to keep count is to make one mark for each thing. One sheep, one scratch. One vote, one line. This is one-to-one-correspondence: each thing gets exactly one mark, so the number of marks is the number of things. These marks are called **tally marks**, and people still use them to count votes, laps or birds. One mark per thing works, but a long row of marks is hard to read. Is it $17$ marks or $18$? So tally marks are drawn in bundles of five. Then you can count the bundles by fives and only the last few marks one at a time. The Romans took the next step. Instead of drawing ten marks, they wrote one symbol that means ten. Instead of a hundred marks, one symbol that means a hundred. Their symbols, called **Roman numerals**, are still all around us. They show the year on monuments and cornerstones and at the end of movie credits. They number the chapters of some books and the hours on many clock faces and sundials. Reading them is a useful skill, and seeing how they work shows, by contrast, what is so clever about our own digits. ## Definition **Tally marks.** One upright mark stands for one thing. The fifth mark in each group is drawn slanting across the four before it, which closes a **bundle of five**. A tally is read by counting the bundles by fives and then counting the single marks by ones. **Roman numerals.** Seven symbols, each with a fixed value: ### [Teen numbers as ten and some more](https://math.wiki/teen-numbers-as-ten-and-ones/) **Summary:** Thirteen is one ten and three ones, which the English number word hides. **Question:** How is a number from eleven to nineteen a full ten and some more? ## Why this exists Every number from $11$ to $19$ is ten ones and some more ones: $13$ is a full ten and $3$ more, so $13 = 10 + 3$. The English words hide this ("thirteen" does not say "ten and three", and "eleven" and "twelve" give no clue at all), so the child needs to see and build it with objects. This week the child makes a full ten and some extra ones, takes a teen number apart into ten and some more, and records it as $10 + 3 = 13$. The child can already say the counting words to twenty, count a set of objects and read a ten frame. Each day takes about ten to fifteen minutes. You need two ten frames drawn on paper, about $20$ counters, and about $20$ straws or craft sticks with a few rubber bands. Keep the talk to "ten ones and some more ones"; calling the ten a single unit and reading the digits as tens and ones comes later. ## Day 1: A full frame and some more **You need:** two ten frames side by side and $20$ counters. Ask the child to fill the first frame, one counter in each box, top row first. "How many? Is it full?" It holds $10$, and the child knows that from the empty boxes being gone, not from counting. Now put $2$ counters in the second frame. "How many in all?" Let the child count on from the full frame if needed: "ten, eleven, twelve." Then say it the new way: "Twelve is a full ten and $2$ more." Clear the second frame and build other numbers the same way. The left frame never changes; only the right frame does. Ask each time, "How many in the full frame? How many more?" The first answer is always $10$, and that sameness is the point. ### [Ten frames](https://math.wiki/ten-frames/) **Summary:** A two by five grid showing a number as how full it is and how far from ten. **Question:** How does a frame of ten boxes show how big a number is and how far it is from ten? ## Why this exists A **ten frame** is a card with two rows of five boxes, $10$ boxes in all, filled one counter per box, top row first. Because every row is $5$ and the whole frame is $10$, a filled frame tells two things at once: how many counters there are, and how many empty boxes are left before ten. The child met the frame while naming small groups at a glance; this week the child builds numbers on it, reads $6$ to $9$ as "five and some more", finds how far a number is from ten, and compares two frames. Each day takes about ten to fifteen minutes. Use a ten frame drawn on paper (two rows of five boxes, each box big enough for a counter) and about $20$ counters, pennies or beans. Stay with numbers from $0$ to $10$; numbers past ten come later. ## Day 1: Building a number on the frame **You need:** one ten frame and $10$ counters. Say a number and let the child build it: one counter in each box, top row from left to right first, then the bottom row the same way. Watch the order more than the total. A child who drops counters anywhere has a pile in a box, not a ten frame, so gently move them into the top row and say, "We always start here." Then switch jobs: you build a number, and the child reads it. Ask after each build, "How many counters? How did you know without counting them all?" and "Is the top row full yet?" Show the empty frame and ask how many counters it holds. The answer is $0$, and an empty frame is a fine way to show zero. ### [Ten more and ten less](https://math.wiki/add-ten-to-a-number/) **Summary:** Adding ten makes one more ten: the ones stay the same and the tens go up by one, so 36 becomes 46. **Question:** A child can count on from $36$ ten times to get $46$, or count back ten times to get $26$. Why does a whole ten never need all that counting, and why does only one digit change? ## When you reach for it Ask a child for $4 + 10$ and many will hold up four fingers and count on, "five, six, seven...", all the way to fourteen. That works, but it takes ten careful counts, and a single slip gives $13$ or $15$. There is a much shorter way, and it is worth teaching on purpose because it is one of the first places a child puts to work what our way of writing numbers is built on: groups of ten. The shortcut rests on one fact from Tens and ones, where teen numbers are first seen as ten and some more. A number such as $14$ is one ten and $4$ ones; $36$ is $3$ tens and $6$ ones. Adding ten adds one whole ten and no loose ones. So the loose ones stay exactly as they were, and the count of tens goes up by one. Written in digits, the ones digit does not change and the tens digit goes up by one: $36 + 10 = 46$. Taking ten away is the same thing backwards: $46 - 10 = 36$. (The one surprise, a tens digit of $9$, is Example 10.) The child does not need to count ten times. They only need to ask "how many tens now?" This lesson covers four situations, in the order a child meets them: - ten added to a number from $0$ to $9$, as $10 + 4$ or $4 + 10$, which makes a teen number; - ten added to a teen number or any larger two-digit number, as $13 + 10$ or $36 + 10$; - ten more and ten less than any number, the two moves side by side, and how they sit next to one more and one less; - counting on or back by tens from a number that does not end in $0$, as $7, 17, 27, 37$. ### [Tens and ones](https://math.wiki/tens-and-ones/) **Summary:** A two-digit number is some tens and some ones. **Question:** A heap of thirty-four sticks is slow to count one at a time. What changes if we tie them in tens, and why is the number written with a $3$ and a $4$? ## When you reach for it A child who can count-to-100 can count a heap of $34$ sticks one at a time. It is slow, and one lost place means starting again from one. Now tie the sticks into bundles of ten as they are counted, the way counting-collections-by-grouping does. The heap becomes $3$ bundles and $4$ sticks left over, and nobody has to keep track of a long count: "ten, twenty, thirty" for the bundles, then "thirty-one, thirty-two, thirty-three, thirty-four" for the loose sticks. That is also exactly what the written number says. In $34$ the $3$ counts the bundles and the $4$ counts the loose sticks. This lesson is about reading a two-digit number that way, as some tens and some ones, and about going back the other way, from a number to the bundles that make it. It is the first real piece of place-value, and most of what comes later about adding, subtracting and comparing bigger numbers is built on it. Reach for it whenever a number bigger than ten has to be counted, built, read or written down. The lesson has three parts, and they come in the order a child meets them. The first is **bundling ten ones into one ten**: ten single things can be tied or snapped together and treated as one thing, a ten. The second is **teen numbers as ten and some more**: $14$ is one ten and $4$ more, which the word "fourteen" hides. The third carries the idea past twenty, to every number up to $99$. ### [Tenths and hundredths](https://math.wiki/tenths-and-hundredths/) **Summary:** The first two places right of the point, worth one tenth and one hundredth of a whole. **Question:** A race is 100 meters long and you stop at the 70-meter mark. What part of the race have you run, and how can that part be written with a point instead of a fraction bar? ## Why this exists The school track has a straight that is exactly $100$ meters long, with a painted mark every $10$ meters. Tevita runs from the start and stops at the $70$-meter mark. How much of the straight has he run? You can already say it with a fraction. The marks cut the straight into $10$ equal parts and he has run $7$ of them, so he has run $\frac{7}{10}$ of it. But a fraction needs a top, a bar and a bottom, and when the bottom is $10$ or $100$ there is a shorter way. We write $$ \frac{7}{10} = 0.7 $$ and read it "seven tenths". The dot is the **decimal point**. Digits to its left count wholes, as they always have. Digits to its right count the small pieces you get by cutting a whole into ten, and then into ten again. This lesson builds the first two of those places, **tenths** and **hundredths**. It uses two pictures you will see again and again: a strip or number line cut into ten parts, and a square cut into a hundred small squares. A number line with marks that step by some fixed amount is in number lines with different scales; here the step is one tenth. ## Definition Cut one whole into $10$ equal parts. One part is **one tenth**: $$ 0.1 = \frac{1}{10} $$ Cut one whole into $100$ equal parts. One part is **one hundredth**: $$ 0.01 = \frac{1}{100} $$ A **decimal** writes a number of tenths and hundredths with a point. The first place to the right of the point is the **tenths place**. The second place is the **hundredths place**. ### [Terminating and repeating decimals](https://math.wiki/repeating-decimal-notation/) **Summary:** A line drawn above the digits that repeat forever. **Question:** Dividing $7$ by $12$ never comes out even, and the digits run on forever. How can such a decimal be written down exactly, compared with others, and read back as a fraction? ## Why this exists Dividing one whole number by another sometimes stops: $11 \div 16 = 0.6875$. Sometimes it does not. Divide $7$ by $12$ and the quotient begins $0.58333$, and however long you carry on, another $3$ comes. A decimal that never ends cannot be written out in full, and writing $0.58333$ alone names a different number, a terminating one that is only close to $\frac{7}{12}$. Two things are needed: a way to write a never-ending decimal exactly in a few symbols, and a way to work with it once it is written down: compare it, round it, add it, and turn it back into the fraction it came from. This lesson does both, in two parts. - **Part 1** turns fractions into decimals by division, sorts the results into decimals that stop and decimals that repeat, and introduces the bar that marks the repeating digits. - **Part 2** puts the bar to use: reading it back out to compare and round, and reading it back into a fraction. ## Definition To write a fraction $\frac{a}{b}$ as a decimal, divide $a$ by $b$ (fraction-to-decimal-by-division), carrying the long division past the decimal point by writing zeros after it, as in divide-a-decimal-by-a-whole-number. A decimal is **terminating** if from some place on every digit is $0$, so it can be written out in full with a last digit. Both $0.6875$ and $-3.04$ terminate, and so does every integer. A decimal that does not terminate is **repeating** if, from some place after the point, one block of digits repeats forever with nothing else after it. That block is the **repeating block**, and its **length** is the number of digits in it. ### [Tests for three and nine](https://math.wiki/divisible-by-3-and-9/) **Summary:** The sum of the digits settles these two. **Question:** Can you tell whether a big number splits evenly into threes or nines without doing the division? ## When you reach for it Felipe threads his bracelets in a repeating pattern of $3$ beads, and every one of Amara's bracelets takes $9$ beads. Each of them has a jar of beads and the same question: will the beads come out even, with none left over? That is the question "is this number divisible by $3$?" or "is it divisible by $9$?". A number is **divisible by $9$** when it is $9$ times a whole number, so it splits into groups of $9$ with nothing left; divisible by $3$ means the same with groups of $3$. For $2$, $5$ and $10$ the last digit settles it, as in the tests for 2, 5 and 10. For $3$ and $9$ the last digit is no help. $13$ and $33$ both end in $3$, but only $33$ is a multiple of $3$. $19$ and $99$ both end in $9$, but only $99$ is a multiple of $9$. Every digit matters, so the test for these two uses all of them at once. Reach for it when a number is too big to divide comfortably in your head and you only need to know whether $3$ or $9$ goes in exactly, or how many are left over. Long division would answer the same question, with many more steps. ## The procedure Add the digits of the number. The result is the number's **digit sum**. Then: **A whole number is divisible by $9$ exactly when its digit sum is divisible by $9$. A whole number is divisible by $3$ exactly when its digit sum is divisible by $3$.** The digit sum of $5382$ is $5 + 3 + 8 + 2 = 18$, and $18$ is divisible by $9$, so $5382$ is divisible by $9$ (in fact $5382 = 9 \times 598$). If the digit sum is still too big to judge, add its digits again. That is allowed because the digit sum is a whole number like any other, and the test applies to it as well. ### [The area model](https://math.wiki/area-model-of-multiplication/) **Summary:** A rectangle whose side lengths are the factors and whose area is the product. **Question:** Can a multiplication you do not know yet, like $4 \times 16$, be drawn as a floor, cut into pieces you do know, and measured? ## Why this exists In area-of-rectangles, a rectangle's area was found by multiplying its side lengths. This lesson turns that around. Any multiplication of whole numbers can be drawn as a rectangle: the two factors are the side lengths and the product is the area. That picture is called the **area model**. Why turn it around? Because a rectangle can be cut. Kwaku wants to know $4 \times 16$, which is not a fact anyone memorizes. Drawn as a rectangle $4$ by $16$, it can be cut into a $4$ by $10$ piece and a $4$ by $6$ piece. Both of those are easy, and the two pieces together are the whole rectangle. The cut is the distributive-property, which was first seen in arrays of dots. Here it is seen as area, in square units, with only the side lengths drawn. The area model also lets a multiplication be checked (does the picture hold that many squares?) and solved when one factor is missing (how long must the other side be?). And it gives one picture for two kinds of story: things set out in rows, like chairs, and surfaces covered by squares, like a floor. ## Definition The **area model** of a multiplication $a \times b$ is a rectangle with one side $a$ units long and the other side $b$ units long. Its area is $a \times b$ square units, because it holds $a$ rows of $b$ unit squares (see area-of-rectangles). In this lesson $a$ and $b$ are whole numbers. The squares do not have to be drawn. An area model is usually a plain rectangle with its two side lengths written beside it. The side lengths are the factors; the area inside is the product. The drawing does not need to be to scale, since the labels, not a ruler, say how long each side is. ### [The complex plane, modulus and argument](https://math.wiki/plot-complex-numbers/) **Summary:** The real part reads across and the imaginary part reads up. **Question:** A real number is a point on a line. A complex number has two parts. Where does it live, how big is it, and which way does it point? ## When you reach for it A real number is a point on a number line. A complex number $a + bi$ in standard form carries two real numbers, its real part $a$ and its imaginary part $b$, and one line cannot hold two independent numbers. A plane can. Once each complex number is a point, questions that sounded strange get plain answers. How big is $-6 + 2.5i$? It is a distance from $0$. In which direction does it lie? It is an angle. How far apart are two complex numbers? It is the length of a segment. You reach for the complex plane whenever size, direction or distance matter: to compare the sizes of complex numbers, to see what adding and subtracting does, to describe a set of complex numbers by a condition such as $|z| = 3$, and as the first step toward writing a complex number by its length and angle. ## The procedure **Plotting.** Draw a horizontal axis, the **real axis**, and a vertical axis, the **imaginary axis**. The number $a + bi$ is plotted at the point $(a, b)$: real part across, imaginary part up. Two complex numbers are equal exactly when both parts match, which is exactly when they are the same point, so every complex number gets one point and every point names one complex number. A real number $a = a + 0i$ lands on the real axis, and the real axis is the old number line sitting inside the plane. A pure imaginary number $bi$ lands on the imaginary axis. **Modulus.** The **modulus** of $z = a + bi$ is its distance from $0$. The distance formula from $(0, 0)$ to $(a, b)$ gives ### [The distributive property](https://math.wiki/distributive-property/) **Summary:** Multiply a sum by multiplying each part: the idea behind every multiplication method. **Question:** You know $6 \times 5$ and you know $6 \times 3$. Is that enough to find $6 \times 8$? ## Why this exists Ronja is planting a garden bed. It has $6$ rows, with $8$ lettuce plants in each row. How many plants is that? It is $6 \times 8$, but Ronja does not know that fact yet. She knows her fives, her twos and her tens, and she can work out $6 \times 3$ by skip-counting: $3, 6, 9, 12, 15, 18$. She could count all the plants one by one, or add $8 + 8 + 8 + 8 + 8 + 8$. Both work, and both are slow. Here is a faster way. Lay a string across the bed after the fifth plant in every row. On the left of the string are $6$ rows of $5$ plants. On the right are $6$ rows of $3$. Those are two facts Ronja knows: $6 \times 5 = 30$ and $6 \times 3 = 18$. Together that is $30 + 18 = 48$ plants. That move, cutting a product you do not know into products you do, is the **distributive property**. It is how most of the multiplication table gets built from a few easy facts, and later it is the idea under every way of multiplying bigger numbers. ## Definition Multiplying a number by a sum gives the same result as multiplying that number by each part of the sum and then adding the products. For Ronja's bed: $$6 \times (5 + 3) = (6 \times 5) + (6 \times 3)$$ The **parentheses** say "work this out first, and treat it as one number". So $6 \times (5 + 3)$ is $6 \times 8$. On the right side, each product in parentheses is found first, and then the two are added: $30 + 18$. One factor stays whole and the other is split into parts. The factor that stays whole, here the $6$, multiplies **every** part. The word *distribute* means to hand out: the $6$ is handed out to the $5$ and to the $3$. ### [The equal sign means the same amount](https://math.wiki/equal-sign-means-same-amount/) **Summary:** Both sides of an equation name one amount, so the sign is not an order to write an answer. **Question:** A child sees $3 + 5 = \square$ and writes $8$. Shown $3 + 5 = 8 + 2$, the same child may say it is right. What does the equal sign really say? ## Why this exists Children meet the equal sign in sentences like $4 + 3 = 7$, again and again, always with the sum on the left and one number on the right. From that steady diet a child can build a rule the teacher never meant: "the equal sign means *now write the answer*." The rule works on every sentence they have seen so far, which is why it sticks. It breaks the moment a sentence is written any other way. A child holding that rule reads $7 = 4 + 3$ as backwards, reads $3 + 5 = 8 + 2$ as "three and five is eight, then add two", and later fills $8 + 4 = \square + 5$ with $12$. None of these is a hard sum. The trouble is only the sign. This lesson gives the sign its real meaning early, while the numbers are small: **the two sides name the same amount.** That one idea is what lets a child find a missing number anywhere in a sentence, and years later it is the whole of algebra, which works by keeping two sides equal. The lesson takes four days. - **Part 1:** the balance. The same amount in each pan keeps the beam level, and the equal sign says that in writing. One number on each side, or one number and a sum. - **Part 2:** true and false. An equation can be checked, and it can be false. The single number can sit on either side. - **Part 3:** a sum on each side, such as $6 + 3 = 5 + 4$. - **Part 4:** deciding without finding both totals, by looking at how the two sides are built. ## Definition ### [The number line](https://math.wiki/number-line/) **Summary:** Numbers as positions on a line, evenly spaced, growing to the right. **Question:** Counting gives numbers in order. What changes when each number gets its own place on a line? ## What this area is about A number line turns the counting list into a picture. Each number gets a spot, the spots are evenly spaced, and the numbers grow to the right. Once numbers are laid out like that, a child can see which of two numbers is bigger, how far apart they are, and what lies between them. ## How these fit together **Placing numbers.** Placing a number on a number line puts whole numbers on a line where each mark is one more than the last, and reads which number comes later. **Changing the scale.** Number lines with different scales reads lines whose marks step by $2$, $5$, $10$ or $100$, so a line from $0$ to $1{,}000$ still fits on a page. ## A path through it Place numbers on a line marked in ones first, then read lines with bigger steps. The number line comes back again and again: adding on an open number line, decimals on a number line, integers, and later the axes of every graph. ### [The partial sums method](https://math.wiki/partial-sums-method/) **Summary:** Adding each place on its own and then combining the parts. **Question:** One class brings $246$ cans and $138$ bottles to a school bottle drive, and the numbers are too big to count on. Can each number be broken into hundreds, tens and ones, each place added on its own, and the pieces put back together? ## When you reach for it A school is holding a bottle drive. Every class brings in cans and bottles, and each Friday the counts go up on the hall board. Tumaini's class brought $246$ cans and $138$ bottles. How many is that in all? Counting on $138$ from $246$, one at a time, would take all afternoon. The child already has two tools that make it quick. From expanded-form they know a number is the sum of what its digits are worth: $246 = 200 + 40 + 6$. From Adding two-digit numbers without regrouping they know that tens join tens and ones join ones. The **partial sums method** puts the two together: - break each number into hundreds, tens and ones; - add the hundreds to the hundreds, the tens to the tens and the ones to the ones; - add those answers, the **partial sums**, to get the total. For $246 + 138$ the partial sums are $200 + 100 = 300$, $40 + 30 = 70$ and $6 + 8 = 14$. Then $300 + 70 + 14 = 384$. Reach for it whenever two or more whole numbers are put together and no basic fact gives the answer. It is also the method the child will lean on most while bigger sums are new: every step is a fact they know, done in one place at a time, and every partial sum can be seen in the blocks. The earlier lesson kept the ones at nine or fewer. Here that limit goes away. When the ones make $14$, the partial sum is simply $14$, and it is added in at the end like any other part. No special step is needed. ### [The place value chart](https://math.wiki/place-value-chart/) **Summary:** A table of columns in which a digit's column decides what it is worth. **Question:** The cards $5$, $3$ and $7$ can be laid down as $537$ or as $357$. The cards are the same both times, so what decides what each one is worth? ## Why this exists Selamawit, Ivar and Haruka play a card game. Each card shows one digit, $0$ to $9$. A player lays three cards in a row and reads the number they make. With the cards $5$, $3$ and $7$, a player can lay down $537$, or $357$, or $375$. Those are three different numbers, made from the same three cards. Only the positions changed. So a written number carries more than its digits. It also carries, without saying so, what each position counts. In numbers-to-1000 the positions were read left to right: hundreds, then tens, then ones. A place value chart writes those names down. It is a table with a heading over each column, and each digit sits under the heading that says what it counts. Room numbers in a school building, page numbers, the cards in the game: every whole number up to $1{,}000$ fits in it. The lesson runs over three days. - **Day 1, a digit and its value.** The chart, and the difference between a digit (the symbol on the card) and its value (what it is worth in its column). - **Day 2, zeros and filling the chart.** Why $609$ needs its $0$ while $69$ needs none in front, and moving between a number, the chart and a block picture. - **Day 3, nine at most, then a fourth column.** Why a column never holds more than nine, and why ten hundreds need a thousands column. Digit cards (two or three of each digit), a three-column chart drawn on paper with cells wide enough for a card, and base ten blocks are all the lesson needs. ### [The real number line](https://math.wiki/real-number-line/) **Summary:** Every point on the line is a real number; rationals and irrationals fill it together. **Question:** Fractions crowd the number line so tightly that there is always another one between any two. Do they fill the line, and if not, what fills the rest? ## Why this exists A number-line starts from two choices: a point called $0$ and a point to its right called $1$. Copies of the stretch from $0$ to $1$, laid end to end in both directions, place the integers, and cutting that stretch into equal parts places every fraction, which is how rational numbers get their points. Fractions are crowded. Between $\frac{1}{3}$ and $\frac{1}{2}$ sits $\frac{5}{12}$, between $\frac{1}{3}$ and $\frac{5}{12}$ sits $\frac{3}{8}$, and the squeezing never stops. It is natural to guess that a line so packed with fractions has no room left for anything else. The guess is wrong. Draw a square on the stretch from $0$ to $1$, take its diagonal, and swing the diagonal down onto the line with a compass. Its far end lands on a definite point, and no fraction names that point. Its distance from $0$ is $\sqrt{2}$, and square-roots-as-irrational proves that $\sqrt{2}$ is not a fraction. So the fractions, crowded as they are, leave holes. The real numbers are what you get by refusing to leave any holes: one number for every point on the line. This lesson is about that line, what fills it, and the order along it. ## Definition Fix a line, a point on it named $0$, and a point to its right named $1$. The distance from $0$ to $1$ is the unit of length. The **real number line** matches points with numbers. A point at distance $d$ to the right of $0$ is the number $d$, a point at distance $d$ to the left of $0$ is the number $-d$, and the point $0$ is the number $0$. Every point gets exactly one number this way, and every number gets exactly one point. ### [The sieve of Eratosthenes](https://math.wiki/sieve-of-eratosthenes/) **Summary:** Crossing out multiples in order so only the primes are left standing. **Question:** You want every prime up to $120$. Is there a way to find them all at once, without testing each number on its own? ## When you reach for it At the lantern festival, $120$ lanterns hang in a long row, numbered $1$ to $120$. The plan is to leave lit only the lanterns whose numbers are prime and to put the rest out, in rounds. Testing each number on its own would work, but slowly: to decide whether $119$ is prime by itself you try divisor after divisor. Recall from composite-numbers that a **prime** is a whole number greater than $1$ with exactly two factors, $1$ and itself, and a **composite** number is a whole number greater than $1$ with more than two. The **sieve of Eratosthenes**, a method that carries his name, finds every prime up to a chosen number in one sweep. Instead of asking "what divides this number?" over and over, it asks "which numbers does this prime divide?", and those it can list by counting up in steps. A hard question asked many times becomes an easy one asked a few times. Reach for it when you want **all** the primes in a range: to make a table, to see how the primes are spaced, or to have the small primes ready for prime factorization. To test a single number, the same idea works in a smaller form (Example 3). ## The procedure ### Running the rounds Write $1$ to $120$ on a board, ten to a row. Every lantern starts lit. **Put out $1$.** It is not prime, for the reasons at the end of this section. **Round $2$.** The first lit number is $2$. Circle it: no whole number other than $1$ and $2$ can divide $2$, so it is prime. Now cross out every multiple of $2$ after $2$ itself: $4, 6, 8, \ldots, 120$. Each has $2$ as a factor besides $1$ and itself, so each is composite. That is $59$ lanterns. ### [The standard addition algorithm](https://math.wiki/standard-algorithm-for-addition/) **Summary:** Adding column by column from the right and carrying into the next place. **Question:** Zahra's school collected $48{,}675$ cans in its fall recycling drive and $27{,}548$ cans in the spring. How do you add numbers this long, place by place, when a column adds up to more than nine? ## When you reach for it Zahra's school runs a recycling drive twice a year. In the fall the students collected $48{,}675$ cans. In the spring they collected $27{,}548$ cans. How many cans is that in all? The answer is the sum $48{,}675 + 27{,}548$. No basic fact gives it, and the numbers are too long to hold in your head. The **standard addition algorithm** adds whole numbers of any length. You write one number under the other, lined up by place value. Then you add ones to ones, tens to tens, hundreds to hundreds, and so on, **starting at the right**. When a column adds up to ten or more, you **carry**: ten of that place become one of the next place, and that one is added into the next column. Reach for it whenever amounts are put together. "In all", "altogether" and "total" describe joining. So does "how many more" turned around: if one school collected $6{,}948$ more cans than another, you add to find its count. The word "more" alone does not say whether to add or subtract; the story does. The words: in $48{,}675 + 27{,}548 = 76{,}223$, the numbers being added are **addends** and the answer $76{,}223$ is the **sum**. You already have the two facts the method rests on. The first is **place value**: each place is worth ten of the place on its right, the columns of a place-value-chart, and you can read numbers up to a million. The second is the **addition facts** up to $9 + 9 = 18$: every column ends as one of them, plus a carry. Two earlier ways of adding lead straight here. The partial-sums-method adds each place on its own and then combines the parts. Regrouping in addition trades ten ones for one ten, so that no place holds more than nine. ### [The standard long division algorithm](https://math.wiki/standard-long-division-algorithm/) **Summary:** Working one place at a time, recording a digit of the quotient at each step. **Question:** A number is far too big for any times table. How do you divide it, one digit at a time? ## When you reach for it A shipping depot has $852$ boxes to load onto $3$ trucks, the same number on each truck. No times-table fact gives $852 \div 3$, and handing the boxes out three at a time would take hundreds of rounds. Partial quotients would get there, by taking away $3 \times 200$, then $3 \times 80$, then $3 \times 4$, but you have to choose the chunks yourself and add them up at the end. The **standard long division algorithm** is partial quotients with no choosing. It works through the dividend one place at a time, from the biggest place down, and each step writes exactly one digit of the quotient, in its own place. Reach for it whenever a whole number is too big to divide in your head, by a divisor of one digit or two, and whenever you want the remainder as well as the quotient. With a one-digit divisor each step is a basic fact, as in divide-by-one-digit; with a two-digit divisor each step needs an estimate first. The words: in $852 \div 3 = 284$, the number being shared, $852$, is the **dividend**; the number it is divided by, $3$, is the **divisor**; the answer, $284$, is the **quotient**. What is left when no more equal groups can be made is the **remainder**. On paper the divisor goes outside a bracket, the dividend inside it, and the quotient on top. Each digit of the quotient is written straight above the last digit of the part of the dividend it came from, which is how the quotient keeps its place value. ### [The standard multiplication algorithm](https://math.wiki/standard-algorithm-for-multiplication/) **Summary:** Stacking the factors and recording one shortened partial product per digit. **Question:** Partial products work, but they fill many lines. How do you multiply two big numbers with just one line for each digit of the bottom factor? ## When you reach for it The home bleachers at a stadium have $36$ sections, and each section has $214$ seats. How many seats is that in all? The answer is $214 \times 36$, and no times-table fact gives it. Partial products can do it: split both factors by place value, multiply every pair of parts, and add the pieces. Here that is six products to find and add. It works, but it fills a page. The **standard multiplication algorithm** does the same work in a shorter record. You stack the factors in columns by place value. Then you write **one row for each digit of the bottom factor**, and you add the rows. Reach for it whenever both factors are too big to multiply in your head. The words: in $214 \times 36 = 7{,}704$, the numbers being multiplied, $214$ and $36$, are the **factors**, and the answer, $7{,}704$, is the **product**. Each row of the algorithm is itself a partial product. Two facts carry the whole method. The **distributive property** lets you split a factor by place value and multiply the parts one at a time: $214 \times 36 = 214 \times 30 + 214 \times 6$. And **regrouping** is a place-value trade: ten ones make one ten, ten tens make one hundred, and so on. The algorithm makes that trade every time a column holds ten or more. ## The procedure ### Part 1: from partial products to the shortened record Start with one digit on the bottom: $268 \times 7$. Split $268$ by place value as $200 + 60 + 8$. The distributive property says you may multiply each part by $7$ and add: ### [The standard subtraction algorithm](https://math.wiki/standard-algorithm-for-subtraction/) **Summary:** Subtracting column by column from the right, unbundling where needed. **Question:** A trail starts $2{,}738$ feet above sea level and ends on a peak $6{,}415$ feet above sea level. How do you find the climb, place by place, when the top number keeps running short? ## When you reach for it A hiking trail starts at a trailhead $2{,}738$ feet above sea level. It climbs to the top of a peak that is $6{,}415$ feet above sea level. How many feet does the trail climb? The climb is the difference between the two heights, $6{,}415 - 2{,}738$. No basic fact gives that, and the numbers are too big to hold in your head comfortably. The **standard subtraction algorithm** handles numbers of any size. You write one number above the other, lined up by place value. Then you subtract ones from ones, tens from tens, hundreds from hundreds, and so on, **starting at the right**. When a place on top has fewer than the place below it, you **unbundle** one unit from the place to its left: one ten becomes ten ones, one hundred becomes ten tens. That trade is regrouping in subtraction, where one ten is broken into ten ones because a place does not have enough. Reach for it in two kinds of problem. **Taking away** asks how many are left: a hiker at $7{,}429$ feet walks down $3{,}615$ feet. **Comparing** asks how much more: how much taller is one peak than another. Both are answered by the same subtraction, as in taking away and comparing. The words: in $6{,}415 - 2{,}738 = 3{,}677$, the answer $3{,}677$ is the **difference**. Two facts carry the whole method. First, **place value**: each place is worth ten of the place on its right, the same columns you see in a place-value-chart. So one unit of a place can always be traded for ten units of the next place to the right, and the number stays the same. Second, the **subtraction facts** up to $18 - 9$: every column ends as one of them. ### [Thousandths: reading, writing and expanded form](https://math.wiki/thousandths/) **Summary:** Places further to the right, each worth one tenth of the place before it. **Question:** A bobsled run is timed to the thousandth of a second. In a time like 52.374 seconds, what is each digit worth, and how do you say the number out loud? ## Why this exists In a bobsled race, a sled of two or four riders slides down an icy track of curves, and a clock times each run. The runs are very close. Two sleds can finish within a few hundredths of a second of each other, and sometimes in the same hundredth. Suppose Aarav's sled and Camila's sled both show $52.37$ seconds on a clock that stops at hundredths. Who won? That clock cannot say. A finer clock reads $52.374$ seconds for one and $52.378$ seconds for the other, and now the race has a winner. The new digit lives in a new place, one step further right than the hundredths: the **thousandths** place. You already know tenths and hundredths, the first two places to the right of the point, and expanded form for whole numbers, where a number is written as the sum of what its digits are worth. This lesson adds the third place, shows the rule that makes every place fit with its neighbors, and then writes decimals three ways: in digits, in expanded form, and in words. ## Definition Cut one whole into $1000$ equal parts. One of those parts is **one thousandth**: $$ 0.001 = \frac{1}{1000} $$ The **thousandths place** is the third place to the right of the decimal point. A digit there counts thousandths, so the $4$ in $52.374$ means $4$ thousandths, $\frac{4}{1000}$, or $0.004$. Here are the places this lesson uses, with what one of each is worth. | Place | As a decimal | As a fraction | |---|:-:|:-:| | tens | $10$ | $10$ | | ones | $1$ | $1$ | | tenths | $0.1$ | $\frac{1}{10}$ | | hundredths | $0.01$ | $\frac{1}{100}$ | | thousandths | $0.001$ | $\frac{1}{1000}$ | ### [Times as many](https://math.wiki/multiplicative-comparison/) **Summary:** Describing one amount as a number of copies of another. **Question:** One pile has 7 stickers and another has 28. Saying "21 more" is true, but how do you say that the big pile is four of the small one? ## Why this exists Killa has $7$ stickers. Chirag has $28$. The quick way to compare them is to find the gap: $28 - 7 = 21$, so Chirag has $21$ more. That is true, and it is the comparison taught in subtraction-as-comparison. But it leaves something out. Look at a second pair. One sheet has $100$ stickers and another has $121$. The gap is $21$ again. Yet the two pairs are not alike at all. A sheet of $121$ is only a little bigger than a sheet of $100$. A pile of $28$ is four whole piles of $7$. The gap cannot tell these two pairs apart. So there is a second way to compare: count how many copies of the smaller amount it takes to make the bigger one. Chirag's $28$ is $4$ copies of Killa's $7$, so Chirag has **$4$ times as many** stickers as Killa. This kind of comparison says how big one amount is *next to* the other, and it is the comparison that a building "three times as tall" or a recipe "twice as much flour" uses. ## Definition "One amount is $4$ times as many as another" means the first amount is made of $4$ copies of the second, put together. In symbols, $28$ is $4$ times as many as $7$ because $$28 = 4 \times 7$$ In general, for whole numbers $a$, $b$ and $n$: **$a$ is $n$ times as many as $b$** when $a = n \times b$. The number $n$ says how many copies of $b$ make $a$. The same words work for any kind of amount that can be copied: "times as tall", "times as long", "times as heavy", "times as far". Both amounts must count or measure the same kind of thing in the same unit. This is a **multiplicative comparison**, because it compares by multiplying. A comparison by a gap ("$21$ more", "$21$ fewer") is an **additive comparison**, because it compares by adding on. ### [Turning a repeating decimal into a fraction](https://math.wiki/repeating-decimal-to-fraction/) **Summary:** Multiply a repeating decimal by a power of ten and subtract to cancel the repeating tail, leaving a fraction. **Question:** A decimal like $0.575757\ldots$ never ends, so it cannot be read off as so many tenths or hundredths. Which fraction is it, exactly, and how can you be sure? ## When you reach for it Dividing turns any fraction into a decimal that either stops or repeats, as repeating-decimal-notation showed. This lesson runs the other way: you are handed a repeating decimal, say $0.\overline{57} = 0.575757\ldots$, and you want the fraction it equals. A terminating decimal is easy to turn back, because place value already writes it as a fraction: $0.57$ is $\frac{57}{100}$. A repeating decimal has no last place, so that reading fails. The earlier lesson gave a shortcut, the **block over nines**, which says $0.\overline{57} = \frac{57}{99}$, and checked it by dividing. This lesson gives the reason the shortcut works, and a method that also handles the decimals the shortcut does not fit directly: blocks that start late, whole-number parts, negatives, and the puzzling $0.\overline{9}$. The whole method is one idea. Multiply the decimal by a power of ten so that its digits slide along until the repeating tail lines up with the tail of the original. Two numbers with the same tail differ by an ordinary number, so subtracting removes the part that never ends, and what is left is a short equation to solve. It is taught over two days: - **Day 1:** blocks that start straight after the decimal point, one digit long and longer. - **Day 2:** blocks that start later, which need two multiplications; answers as mixed numbers; and why $0.\overline{9} = 1$. ## The procedure ### [Two-step addition and subtraction problems](https://math.wiki/two-step-word-problems-add-subtract/) **Summary:** Stories that need one answer worked out before the next. **Question:** A story changes an amount twice, or compares two amounts and then asks about both. The question cannot be answered from the numbers in the story directly. What has to be found first, and how does a child keep track of it? ## When you reach for it A one-step story gives two numbers and asks about a third that can be found from them at once. A **two-step story** asks a question that the numbers in it cannot answer directly. Something else has to be found first, and the story never says what. Read this one aloud: "The food pantry had $45$ cans. $28$ more cans were brought in. Then $37$ cans went out to families. How many cans does the pantry have now?" The question is about the end of the day. The $37$ cans went out of a pile, but the story never says how big that pile was. Before anyone can take $37$ away, they need the answer to a question the story does not ask out loud: **how many cans were there after the $28$ came in?** That is the **hidden question**. Answer it, and the story becomes an ordinary one-step story with a known start. The whole lesson turns on that idea. A child who has met stories with the start, change or result missing and stories that compare two amounts already has every single step. What is new is seeing that a story holds two of them, one after the other, and that the answer to the first is a number in the second. The arithmetic stays within $100$ and uses the adding and subtracting the child already knows: an open number line, partial sums, and regrouping in addition and subtraction. The lesson is taught over four days: - **Part 1:** two changes, one after the other. Something is added and something is taken away. - **Part 2:** two changes of the same kind, and three parts of one whole. Here the order of the steps can be chosen. - **Part 3:** a comparison, then a total or a change. The first step finds an amount the story only describes ("$17$ more than Seble"). - **Part 4:** the missing number sits inside the second step, and the child checks whether an answer fits the story. ### [What division means](https://math.wiki/division-as-sharing/) **Summary:** How many in each group, or how many groups: two questions, one operation. **Question:** Twenty-eight beads are shared equally among 4 friends, and thirty buttons are sewn onto cards of 5. One story asks how many in each group, the other how many groups. Why is the same operation the answer to both? ## Why this exists Multiplication puts equal groups together. Four friends each holding $7$ beads have $4 \times 7 = 28$ beads in all, as multiplication-as-groups shows. There we knew the groups and asked for the total. Real problems often run the other way. The total is in your hands, and something about the groups is missing. At the craft club there are $28$ beads to share fairly among $4$ friends: how many does each friend get? Or there are $30$ buttons to sew onto cards, $5$ to a card: how many cards can be filled? Neither question is answered by putting groups together. Both are answered by taking a total *apart* into equal groups. That is what division does. It starts from the total and finds the missing part of an equal-groups picture. The surprise of this lesson is that the two questions above, "how many in each group?" and "how many groups?", are answered by the same operation, and the reason is a picture you can draw. ## Definition **Division** splits a total into equal groups. The division $28 \div 4$ (read "twenty-eight divided by four") has two meanings, and they always give the same number: - **Sharing.** Split $28$ into $4$ equal groups. $28 \div 4$ is how many are in each group. - **Grouping.** Split $28$ into groups of $4$. $28 \div 4$ is how many groups there are. The numbers have names. In $28 \div 4 = 7$, the total $28$ is the **dividend**, the $4$ we divide by is the **divisor**, and the answer $7$ is the **quotient**. ### [What makes a number rational](https://math.wiki/what-makes-a-number-rational/) **Summary:** A number that can be written as one integer divided by another. **Question:** Whole numbers, negatives, fractions, mixed numbers and decimals look like different kinds of number. Is there one test that takes them all in, and is there anything it leaves out? ## Why this exists By now you have met several families of numbers, each learned on its own: the counting numbers, zero, the negatives that make up the integers, fractions such as $\frac{3}{4}$, mixed numbers such as $2\tfrac{1}{3}$, and decimals such as $0.75$ and $-4.07$. Each came with its own rules, and it is fair to ask whether $-6$ is "a fraction", whether $0.75$ is, or whether these are really different things. They are not different things. Every one of them can be written as one integer divided by another, and that single description is what this lesson is about. It does three jobs. It joins the families into one, so one arithmetic serves them all. It gives a test you can run on any number you meet. And it marks an edge: some numbers, such as the length of the diagonal of a square with sides of length $1$, pass no such test, and they lead on to the rest of the number line. ## Definition A number is **rational** if it **can be written** as a fraction whose numerator is an integer and whose denominator is a nonzero integer: $$\frac{a}{b}, \qquad b \neq 0$$ with $a$ and $b$ both integers. The set of all rational numbers is written $\mathbb{Q}$, for *quotient*. So $\frac{3}{4} \in \mathbb{Q}$ is read "three quarters is in the set of rational numbers". The word *rational* comes from *ratio*, a comparison of two whole amounts; it has nothing to do with being sensible. Three parts of the definition do the work. ### [What multiplication means](https://math.wiki/multiplication-as-groups/) **Summary:** Three groups of four, as an array, as repeated addition, as area. **Question:** Some things come in groups that are all the same size, like muffins in pans or cans on shelves. How can you find how many there are in all without counting every one? ## Why this exists A child who can add can find how many cans are on three shelves by counting every can. That works, but it is slow, and it throws away something the child can see: every shelf holds the same number of cans. When the groups are all the same size, the total depends on only two numbers, how many groups there are and how many are in each group. Know those two and the total is decided. This lesson is about that one idea, **equal groups**, and three ways to find the total that they make: adding the same number again and again, skip counting, and making equal hops along a number line. All three are the same thing seen from different sides. Together they are what multiplication means, long before the child meets a multiplication sign or learns a single multiplication fact. A child who knows that "three groups of four" means $4 + 4 + 4$ will later see each fact as a picture, not a string of words. The lesson runs over four days. Day 1 is what equal groups are and how adding finds the total. Day 2 finds the total faster by skip counting, including groups of ten. Day 3 shows the same groups as equal hops on a number line. Day 4 asks why the groups must be equal, and shows that three groups of four and four groups of three make the same total. Read everything aloud with the child, and let them build it with counters, cubes, buttons or small cups. ## Definition A **group** is a set of things kept together: the muffins in one pan, the cans on one shelf, the counters inside one loop. Groups are **equal groups** when every group holds the same number of things. ### [Where the point goes in a product](https://math.wiki/decimal-places-in-a-product/) **Summary:** The number of decimal places in a product comes from the places in the factors. **Question:** Wash-and-fold costs $1.3$ dollars a pound and the bag weighs $2.5$ pounds. When both numbers have a decimal point, where does the point go in the product, and why there? ## Why this exists Wale runs a laundromat. Customers drop off a bag of laundry, and Wale washes, dries and folds it for $1.3$ dollars a pound. Gauri's bag weighs $2.5$ pounds. What does she pay? The cost is the price of one pound times the number of pounds, $1.3 \times 2.5$. If the bag weighed a whole number of pounds, multiply-a-decimal-by-a-whole-number would finish the job: $3$ pounds would cost $3 \times 1.3 = 3.9$ dollars. But a scale does not stop at whole pounds. Now both factors have a decimal point. Multiplying the digits is not the hard part: $13 \times 25 = 325$. The hard part is where the point goes. Is Gauri's bill $32.5$ dollars, $3.25$ dollars, or $0.325$ dollars? Only one of those is right, and this lesson shows how to know which, using pictures of tenths and hundredths first and then a pattern that works every time. The lesson takes three days. Day 1 multiplies tenths by tenths on a hundredths grid. Day 2 uses area models for bigger factors and finds the pattern in the number of decimal places. Day 3 places the point by an estimate and by the count, handles zeros in front, and solves laundromat problems. ## Definition The digits after a decimal point are its **decimal places**. $2.5$ has one decimal place (tenths), $0.45$ has two (hundredths), and $0.056$ has three (thousandths). A whole number such as $6$ has zero decimal places. **Where the point goes.** Multiply two decimals as if the points were not there. Then the product has as many decimal places as the two factors have together, counted before any zero at the end of the product is dropped. ### [Which fractions give terminating decimals](https://math.wiki/which-fractions-terminate/) **Summary:** A fraction in lowest terms stops exactly when its denominator has only twos and fives as prime factors. **Question:** Dividing $13$ by $16$ stops at $0.8125$, but dividing $9$ by $28$ never stops. Can you tell which will happen before doing any division at all? ## Why this exists Turning a fraction into a decimal by dividing (fraction-to-decimal-by-division) ends in one of two ways. Either the division comes out even and the decimal stops, as $\frac{13}{16} = 0.8125$ does, or the remainders come round again and the digits repeat forever; what-makes-a-number-rational proves there is no third ending. It does not say which ending a given fraction has, and dividing to find out is slow: after twenty digits you still cannot be sure the next one is not the last. Once the fraction is in lowest terms, the answer is decided by the denominator alone, and by one fact about it: which primes it is built from. With the denominator broken into primes (factor-trees), you can say at a glance that $\frac{1}{1024}$ stops after exactly ten digits and that $\frac{9}{28}$ never stops, without one step of long division. ## Definition A decimal **terminates** if it stops: from some place on, every digit after the point is $0$. Both $0.8125$ and $-2.5$ terminate, and so does every integer (repeating-decimal-notation names the other kind). A fraction is in **lowest terms** when its numerator and denominator have no common factor bigger than $1$ (equivalent-fractions). So $\frac{1}{2}$ is in lowest terms and $\frac{3}{6}$ is not, since $3$ divides both $3$ and $6$. ## Intuition Decimal places are fractions with special denominators: tenths, hundredths, thousandths. A decimal that stops after $k$ digits is a whole number of $10^k$ths (powers-of-ten); for instance $0.8125 = \frac{8125}{10000}$. So the question "does $\frac{a}{b}$ terminate?" is really the question "is $\frac{a}{b}$ equal to some fraction whose denominator is a power of ten?" ### [Writing and reading numerical expressions from words](https://math.wiki/write-expressions-from-words/) **Summary:** Turning a phrase like add 8 and 7, then multiply by 2 into an expression, and reading an expression without working it out. **Question:** Someone says "add 8 and 7, then multiply by 2". How do you write that as one line that everyone reads the same way, and how do you read a line back as words without working it out? ## When you reach for it At the school scavenger hunt, every clue a team finds is worth points. Minh's team finds $8$ clues in the gym and $7$ clues in the library. Then the leader says: "Your clues count double today." So the team adds $8$ and $7$, then multiplies by $2$. You could write the work as two lines, $8 + 7 = 15$ and then $15 \times 2 = 30$. But one line is shorter, and it shows the whole plan at once: $$2 \times (8 + 7)$$ The parentheses matter. Without them, $2 \times 8 + 7$ multiplies first, as numerical-expressions agreed, and names $16 + 7 = 23$. Only the $8$ gym clues would count double. With the parentheses from grouping-symbols, the $8 + 7$ is one number, and all $15$ clues count double. You reach for this skill in two directions: - **Words into an expression.** A sentence or a story says what to do. You write one expression that does exactly that, in that order. - **An expression into words.** You read an expression and say what it means, often without working it out. "$2 \times (8 + 7)$ is twice as large as $8 + 7$" needs no adding at all. The lesson takes three days. **Part 1** turns a phrase into an expression, and shows when parentheses are needed. **Part 2** reads an expression in words and compares two expressions without evaluating them. **Part 3** writes expressions for stories with several steps, and finds two different expressions for the same story. ## The procedure ### [Writing and reading scientific notation](https://math.wiki/write-in-scientific-notation/) **Summary:** Recording a number as one digit, a decimal part, and a power of ten. **Question:** How can a number like 150,000,000 or 0.00007 be written so that its size shows at a glance, with no zeros to count? ## When you reach for it Some quantities are too large or too small for ordinary place value to be comfortable. The distance from Earth to the Sun is about $150{,}000{,}000$ km, and a human hair is about $0.00007$ m wide. Written out like that, the size of each number lives in its count of zeros, and counting zeros is slow: at a glance, is the first one fifteen million or a hundred and fifty million? Scientific notation moves that count into one small number, an exponent, and keeps the digits that matter at the front: $$150{,}000{,}000 = 1.5 \times 10^{8}$$ $$0.00007 = 7 \times 10^{-5}$$ You reach for it whenever a number has a long run of zeros on either side of the decimal point: writing down a measurement from astronomy or from a microscope, reading a science text, or reading a calculator that has run out of room on its screen. It is also the first step before comparing such numbers or calculating with them. It rests on powers-of-ten, and for small numbers on zero and negative exponents. The lesson runs over two days. Day 1 sets out the form, writes large numbers in it and back again, and explains why the leading number is kept between $1$ and $10$. Day 2 does the same for small numbers, using negative powers of ten, and reads calculator displays such as $\texttt{4.2E-6}$. ## The procedure ### Day 1: the form, and large numbers **The form.** A number is written in **scientific notation** when it is written as ### [Writing the numerals 0 to 10](https://math.wiki/write-numerals-0-to-10/) **Summary:** Forming the written digits that stand for the first ten amounts. **Question:** How does a child learn to write the numerals from 0 to 10 so anyone can read them? ## When you reach for it A **numeral** is the written mark for a number, like $7$, and a child who can write $0$ through $10$ can record a count so that someone else can read it later. Reach for this once the child knows the names and can match each numeral to an amount; here the job is the hand's: forming each numeral the same way every time, starting at the top, facing the right way, and writing $10$ as a $1$ followed by a $0$. Each day below takes about ten to fifteen minutes. Build or trace each numeral big with the whole arm first, then small with a pencil. Keep a written strip of $0$ to $10$ in view as the model the whole time. ## Day 1: Numerals made of straight lines **You need:** a handful of craft sticks, and a shallow tray with a thin layer of salt. Start with $1$, $4$ and $7$, because they use only straight lines. Lay sticks on the table to build each one, and count the sticks aloud as you place them. Then the child builds it beside yours. Next, trace the numeral in the salt tray with one finger. Say the path while moving: for $7$, "across the top, then slant down"; for $4$, "down and across, lift, then straight down." Point out that every numeral starts at the top. Shake the tray to erase and go again. If a numeral comes out facing the wrong way, place the model strip next to it and trace it together. **Practice.** 1. Build $1$ with sticks. How many sticks? **Answer:** $1$. 2. Build $7$ with sticks. How many sticks? **Answer:** $2$, one across the top and one slanting down. 3. Build $4$ with sticks. How many sticks? **Answer:** $3$. 4. Trace $7$ in the salt. Where does your finger start? **Answer:** at the top left, moving across. 5. Which of $1$, $4$ and $7$ makes you lift your finger partway? **Answer:** $4$. It takes two strokes; $7$ uses two sticks but only one stroke, since the finger turns the corner without lifting. ### [Zero and negative exponents](https://math.wiki/zero-exponent/) **Summary:** The rule for dividing powers forces the zero exponent to stand for one. **Question:** Dividing powers subtracts their exponents. What do $10^0$ and $10^{-3}$ have to mean so that rule never breaks? ## Why this exists An exponent counts factors. $10^3$ means $10 \cdot 10 \cdot 10$, three factors of $10$, and that is all exponent notation says. Under that meaning $10^0$ and $10^{-3}$ are nonsense: you cannot write down no factors, and you certainly cannot write down minus three of them. Yet the rule for dividing powers with the same base produces exactly these symbols. That rule subtracts the exponents, $\frac{a^m}{a^n} = a^{m-n}$, and it was proved for $m$ larger than $n$. Use it anyway on two quotients whose values nobody doubts: $$\begin{aligned} \frac{10^4}{10^4} &= 10^{4-4} = 10^0 \\ \frac{10^2}{10^5} &= 10^{2-5} = 10^{-3} \end{aligned}$$ The first quotient is $1$, because a nonzero number divided by itself is $1$. The second is $\frac{100}{100\,000} = \frac{1}{1000}$. So there are two ways forward. Either the subtraction rule carries a list of cases where it may not be used, or the symbols $10^0$ and $10^{-3}$ are given meanings, and the only meanings that keep the rule true are the values it has just produced. This lesson takes the second way. The meanings are not found by counting factors, since there are none to count. They are **chosen**, and the choice is forced: any other choice would break a rule that already works. The pay-off is that powers of ten, and of every other nonzero number, now run in both directions. A millimeter is $10^{-3}$ meters and a micrometer is $10^{-6}$ meters, and the same rules that multiply $10^3$ by $10^2$ handle them. ## Probability ### [Arrangements and factorials](https://math.wiki/permutations-of-all-objects/) **Summary:** The number of orders for a whole set of different objects is the factorial of how many there are. **Question:** Eight different flags are hung one above another on a pole. How many orders are there, and why does the count grow so fast when one more flag is added? ## Why this exists People line things up all the time: books on a shelf, runners in a relay, songs in a playlist, flags on a pole. Each time the same question comes up. If every one of the objects is used exactly once, in how many different orders can they go? The fundamental counting principle already answers it: fill the places one at a time and multiply. With every object used, the product runs all the way down to $1$, and the same shape of product turns up so often that it has its own name and symbol, the **factorial**. This lesson is about that one product: what it counts, why $0!$ has to be $1$, how quickly it grows, and what changes when the objects sit around a circle instead of in a row. ## Definition An **arrangement** of a set of different objects is a way of putting **all** of them in a row, so that each object is used exactly once and the order matters. Another name for an arrangement is a **permutation** of the set. The objects must be different from one another, the way the elements of a set are. For a positive whole number $n$, **$n$ factorial** is the product of the whole numbers from $n$ down to $1$: $$\begin{aligned} n! = n &\times (n - 1) \times (n - 2) \\ &\times \cdots \times 2 \times 1 \end{aligned}$$ So $1! = 1$, $2! = 2 \times 1 = 2$, $3! = 3 \times 2 \times 1 = 6$ and $4! = 4 \times 3 \times 2 \times 1 = 24$. **Zero factorial** is defined to be $$0! = 1$$ and the reasons for that choice are in "Why it is true" below. The factorial is defined only for the whole numbers $0, 1, 2, 3$ and so on. ### [At least one problems](https://math.wiki/probability-that-none-occur/) **Summary:** Multiplying the chances of each separate failure to get the chance that every attempt misses. **Question:** Six fireworks are set out for a show. What is the chance that at least one of them goes off, when counting every way that could happen would take all afternoon? ## When you reach for it Ffion sets out $6$ fireworks. Each one goes off with probability $0.85$, and the fireworks do not affect one another. What is the chance that at least one of them goes off? "At least one" is a crowded event. It holds every way of getting one firework to go off, two, three, four, five or six, and each of those can happen in many orders: exactly two going off alone can be any $2$ of the $6$ fireworks. Adding all of that up is a long job with many chances to slip. Its opposite is short. The only way to fail to get at least one is for **none** of the fireworks to go off: firework one fails, and firework two fails, and so on to firework six. That is a single run of failures, and a run of independent events is one product. So the move is: $$P(\text{at least one}) = 1 - P(\text{none})$$ Reach for it whenever a question says "at least one", "one or more", "any of them", or "not all miss": an alarm that must go off at least once, a message sent several ways that must arrive at least once, a test that must catch at least one fault. Reach for it again, the other way round, when you need to know **how many tries** it takes to make "at least one" likely enough, because the same formula with $n$ tries puts the unknown in an exponent. ## The procedure ### Why "none" is the complement Every outcome of the experiment has some number of successes: $0$, or $1$, or $2$, and so on. An outcome with $0$ successes is in "none"; an outcome with any other number is in "at least one". No outcome is in both and none is left out, so the two events are complements, and complements have probabilities that add to $1$: ### [Basic probability](https://math.wiki/basic-probability/) **Summary:** Counting what can happen, and the fraction of it you want. **Question:** Nobody can say how a single toss will land. What can be said, and how does it become a number? ## What this area is about A probability is a number from $0$ to $1$ that says how likely an event is. There are two ways to get one: reason it out from a model of the situation, or run the process many times and count. This area builds both. ## How these fit together **Words and notation.** What randomness means, the scale of chance, outcomes and events, the complement ($P(\text{not } A) = 1 - P(A)$) and odds. **Probability from a model.** Listing a sample space and counting outcomes, which works only when every outcome is equally likely; probability models for when they are not; and area, when a point lands anywhere in a region with no part favored. **Probability from trials.** Experimental probability counts results. The law of large numbers says that over many independent trials with the same chance, the running share settles near that chance. Designing a simulation and estimating from its trials put that to work. ## A path through it Randomness first, then the chance scale and outcomes and events together (each refers to the other). Next the sample space and counting outcomes, then the complement, which probability models need. Area and odds follow, then experimental probability. The law of large numbers also assumes compound events, which build on the lessons above, so take that area next. Then the law of large numbers, designing a simulation and, last, estimating from it. ### [Bayes' rule](https://math.wiki/bayes-rule-with-a-tree-diagram/) **Summary:** Finding the wanted branch probability and dividing it by the total probability of all matching branches. **Question:** A check that is almost always right raises an alarm. How likely is it that the alarm is real, and why can the answer be far smaller than the check's own success rate? ## When you reach for it Information often arrives the wrong way round. A factory knows how its inspection camera behaves on a faulty circuit board and on a good one: it knows $P(\text{flagged} \mid \text{faulty})$ and $P(\text{flagged} \mid \text{good})$. The person holding a flagged board wants the other direction, $P(\text{faulty} \mid \text{flagged})$: given what the camera said, what is the board really like? These two conditional probabilities are different numbers, and they can be wildly different. Bayes' rule is the move that turns one into the other. Reach for it whenever - the situation splits into cases (faulty or good, which supplier, which coin), and you know the chance of each case, - you know how likely some piece of evidence is **within** each case, and - the evidence has now been seen, and you want the chance of a case **given** the evidence. The question "given the result, what was the cause?" is the signal. It covers rare faults and the false alarms they bring with them, reading back from an effect to the supplier that produced it, and updating a judgment when a second piece of evidence arrives. ## The procedure The method rests on three things already known: the formula for a conditional probability, the general multiplication rule and the law of total probability. Nothing new is assumed; Bayes' rule is these three used in the right order. **Draw the tree in the direction the information comes.** The first fan of the tree is the partition into cases, labeled with their chances $P(B_i)$. The second fan, from each case, is the evidence, labeled with the chances given that case, $P(A \mid B_i)$. This is the order the numbers are known in, so every branch gets a number that was actually given. The branch you want, a case given the evidence, appears nowhere on this tree, which is exactly why a calculation is needed. ### [Chance in words and the 0 to 1 scale](https://math.wiki/chance-in-words/) **Summary:** Saying how likely something is using impossible, unlikely, even chance, likely and certain, before any number is used. **Question:** One person says rain is "likely", another says "probably not", a third gives a percent. How can chance be put on one scale, so that everyone means the same thing and two chances can be compared? ## Why this exists People talk about chance mostly in words: "no way", "probably not", "fifty-fifty", "probably", "definitely". Words are loose. When one person says rain is "likely", do they mean more likely than not, or nearly sure? Two people can use the same word and mean different things. What randomness means showed that a random event cannot be called in advance, yet follows a steady pattern over many tries. This lesson gives that pattern a ruler: a line from $0$ to $1$, with the everyday words given exact places on it. Once a chance sits on the line, two chances can be compared without arguing about words. ## Definition An **event** is something that might or might not happen: it rains tomorrow, the school bus is late, a raffle ticket drawn without looking is yours. Its exact meaning, built from a list of outcomes, is in outcomes and events. The **probability** of an event is a number from $0$ to $1$ that says how likely the event is. A larger number means a more likely event. | Word | Probability | What it says | |---|:-:|---| | impossible | $0$ | cannot happen | | unlikely | between $0$ and $\tfrac{1}{2}$ | less likely to happen than not | | even chance | $\tfrac{1}{2}$ | as likely to happen as not | | likely | between $\tfrac{1}{2}$ and $1$ | more likely to happen than not | | certain | $1$ | must happen | Even chance is also called "fifty-fifty". "Very unlikely" and "very likely" are used near $0$ and near $1$; they have no exact border. ### [Combinations](https://math.wiki/combination-notation/) **Summary:** The symbol for combinations records how many items there are and how many are being chosen. **Question:** A pizza can have any $4$ of $9$ toppings. Picking the toppings one after another gives thousands of orders, so why are there only $126$ different pizzas? ## Why this exists Many counting questions ask for a choice, not a line-up. A pizza gets $4$ toppings, a committee gets $5$ members, a hand of cards gets $5$ cards. Nobody cares which topping went on first: a pizza with mushrooms, olives, peppers and onions is the same pizza however the cook picked them. The fundamental counting principle counts choices made one after another, so it counts ordered lists. For a choice where order does not matter that overcounts, and by a known amount: each choice of $r$ objects shows up once for every one of its arrangements, which is $r!$ times. Dividing that out gives the number of **combinations**, one of the most used numbers in mathematics. This lesson is about that number: its symbol, its formula and why the formula divides, the fact that choosing what to take is the same as choosing what to leave, choosing from several groups at once, and how to tell whether a question wants order to matter. ## Definition A **combination** of $r$ objects from a set of $n$ different objects is a choice of $r$ of them in which order does not matter. In the language of sets, it is a **subset** with exactly $r$ elements: $\{A, B\}$ and $\{B, A\}$ are the same subset, because a set is decided only by what is in it. The number of combinations of $r$ objects from $n$ different objects is written $$\binom{n}{r}$$ and read "$n$ choose $r$". Here $n$ and $r$ are whole numbers with $0 \le r \le n$. The same number is also written $C(n, r)$ or ${}_{n}C_{r}$; all three mean the same count. ### [Combinations with repetition: stars and bars](https://math.wiki/combinations-with-repetition/) **Summary:** Choosing k items from n kinds with repeats allowed is the same as placing k stars among n minus 1 bars. **Question:** A box holds twelve donuts and the shop makes five kinds, as many of each as you like. How many different boxes are there, without listing them? ## When you reach for it A combination chooses $k$ different objects and ignores their order. Many choices allow repeats instead: a box of donuts may hold three jelly donuts, a basket may hold five apples, a roll of dice may show two sixes. What is chosen is a number of items of each **kind**, the items of one kind are alike, and only how many of each kind matters, not the order they were picked in. Such a choice is a **selection of $k$ items from $n$ kinds with repeats allowed** (also called a multiset of size $k$). The same count answers questions about equations in natural numbers, nondecreasing sequences, terms of a polynomial and identical dice: each is matched one to one with a row of stars and bars, counted by one binomial coefficient. Throughout, $\mathbb{N} = \{0, 1, 2, \ldots\}$ is the set of natural numbers, and "positive integers" means $1, 2, 3, \ldots$. ## The procedure This lesson takes two days. Day 1 is the stars-and-bars count and why it is true (Examples 1 to 3); Day 2 is lower bounds, other things the same count counts, and choosing among the four basic counts (Examples 4 to 7). ### Day 1: from a selection to an equation Name the kinds $1, 2, \ldots, n$. A selection of $k$ items is fixed by how many of each kind it holds: $x_1$ of kind $1$, $x_2$ of kind $2$, and so on. Each $x_i$ is in $\mathbb{N}$ (a kind may be left out) and together they make $k$. So the selections correspond one to one with the solutions of ### [Combining random variables](https://math.wiki/variances-add-for-independent-variables/) **Summary:** If two random variables do not affect each other, the variance of their sum is the sum of the variances. **Question:** A bottle is filled and then capped, and each step takes a random amount of time. If you know the mean and the spread of each step, what are the mean and the spread of the total time, and of the gap between the two? ## Why this exists A random variable $X$ has a mean $\mu_X$ and a variance $\operatorname{Var}(X) = \sigma_X^2$, with standard deviation $\sigma_X$. adding-a-constant-shifts-the-mean showed that a changed variable $a + bX$ has mean $a + b\mu_X$ and standard deviation $|b|\sigma_X$. That handles one variable. Many quantities are built from two or more random variables. At Teuila's small guava juice shop, each bottle is filled by Teuila's filler in $X$ seconds and then capped by Losa's capper in $Y$ seconds, so a bottle takes $X + Y$ seconds. Whether the filler runs ahead of the capper depends on the gap $X - Y$. When Sefo loads four crates into the van, the load is a sum of four weights. Two steps in sequence are a compound event, and a small table of pairs can be listed, but the list grows fast and cannot be written at all for a continuous variable. This lesson gives the rules that skip the list. The lesson runs over three days: Day 1 is sums, Day 2 is differences and what happens without independence, Day 3 is normal variables and totals of several. The shop, its people and every probability are made up. ## Definition Let $X$ and $Y$ be random variables attached to the same chance process. Their **sum** $X + Y$ is the random variable whose value, for each outcome, is the value of $X$ plus the value of $Y$. Their **difference** $X - Y$ is defined the same way with subtraction. $X$ and $Y$ are **independent** when knowing the value of one does not change the probabilities for the other. For discrete variables this means that for every pair of values, ### [Complements and probabilities that add to one](https://math.wiki/complement-notation/) **Summary:** Writing the event that something does not happen, and reading the rule that the two probabilities add to one. **Question:** A forecast gives the chance of rain tomorrow. What is the chance that it stays dry, and why is that number already settled by the first one? ## Why this exists A forecast gives a $36\%$ chance of rain tomorrow. Nobody prints the chance of no rain beside it, and nobody needs to. Tomorrow will either have rain or not, never both, so the two chances share the whole of the certainty between them. Once one number is known, the other is fixed. The same idea saves work. "At least one head" in a toss of several coins covers many outcomes, while its opposite, "no heads", is a single one. Count whichever side is smaller. This lesson names the event "$A$ does not happen", gives it notation, and shows why its probability and the probability of $A$ always add to $1$. Underneath sits a second fact: the probabilities of all the outcomes in a sample space add to $1$. ## Definition Start with an experiment, its sample space $S$, and an event $A$, which is a collection of outcomes from $S$. The **complement** of $A$ is the event made of every outcome in $S$ that is not in $A$. It happens exactly when $A$ does not happen. This lesson writes the complement as $\text{not } A$. Other books write $A'$ (read "$A$ prime"), $A^c$ (read "$A$ complement") or $\overline{A}$ (read "$A$ bar"). All four name the same event. For one spin of the prize wheel in Figure 1, the eight equal sections are the outcomes. Let $M$ be the event "lands on miss", the three shaded sections. Then $\text{not } M$ is "lands on pen, cap or mug", the five unshaded sections. The second idea of the lesson is about the probabilities themselves. Each outcome of a sample space has a probability from $0$ to $1$, and **the probabilities of all the outcomes add to $1$**. A list of numbers that does not total $1$ cannot be the probabilities of a complete sample space. ### [Compound events](https://math.wiki/compound-events/) **Summary:** Two or more things happening, together or in sequence. **Question:** When two things might happen, together or one after the other, how do their separate chances combine? ## What this area is about Most real questions join events: this "or" that, this "and then" that, "at least one". Counting every outcome again works for small cases and fails quickly. This area gives the rules that build the chance of a joined event from the chances of its parts, and says exactly when each rule holds. ## How these fit together **Language.** Union and intersection give "or" and "and" their exact meanings, and joint and marginal probability reads them from a two-way table. **"Or" rules.** Mutually exclusive events add, because they cannot happen together. The general addition rule subtracts the overlap for any two events. **"And" rules.** Independent events multiply. Tree diagrams lay out stages that need not be equally likely. Dependent events, such as draws without replacement, update the chance after each stage, and the general multiplication rule $P(A \text{ and } B) = P(A) \times P(B \mid A)$, for $P(A) > 0$, covers every case. **A procedure.** "At least one" is $1$ minus the chance that none occur. **A sub-area.** Conditional probability finds $P(B \mid A)$ for $P(A) > 0$, and reverses it with Bayes' rule. ## A path through it Union and intersection and joint probability first, then the two addition rules. Then independent events, tree diagrams, dependent events and the general multiplication rule, then "at least one". Conditional probability comes last, since it uses nearly all of these. ### [Conditional probability](https://math.wiki/conditional-probability/) **Summary:** The probability of A, given that B happened. **Question:** Once you learn that one event has happened, how does the chance of another one change? ## What this area is about New information changes a chance. If you know an event $A$ has happened, the outcomes outside $A$ are ruled out, and the chance of $B$ has to be measured inside $A$ alone. That new number is the conditional probability $P(B \mid A)$, defined only when $P(A) > 0$. This area computes it, uses it to build an overall chance from separate cases, and turns it around. ## How these fit together **The definition.** Finding a conditional probability divides the chance that both happen by the chance of the event conditioned on: $P(B \mid A) = P(A \text{ and } B) / P(A)$, for $P(A) > 0$. It also tests whether knowing $A$ changes the chance of $B$ at all. **A result.** The law of total probability splits the outcomes into cases that do not overlap and together cover everything, then adds each case's share of the event. **A procedure.** Bayes' rule reverses the condition: from the chance of the evidence within each case to the chance of a case given the evidence. ## A path through it Take them in this order: the formula first, then the law of total probability, then Bayes' rule, which divides one path of a tree by the total that the law of total probability supplies. Finish here and you can later see what a p-value does and does not say: it is a chance computed on the condition that the null hypothesis is true. ### [Counting](https://math.wiki/counting-principles/) **Summary:** How many ways, without listing them all. **Question:** How do you count the ways something can happen when there are far too many to list? ## What this area is about Listing every outcome works for a handful, not for the millions of codes a lock can take. This area counts without listing, by multiplying the choices made at each step and then correcting for what that multiplication counts twice or wrongly. The counts feed probability: when outcomes are equally likely, a chance is a count over a count. ## How these fit together **The basic move.** The fundamental counting principle multiplies the number of choices at each stage. **Ordered arrangements.** Permutations of a whole set ($n!$), permutations of some objects ($P(n, r)$), and arrangements with a rule attached. **Choices where order does not matter.** Combinations $\binom{n}{r}$; Pascal's triangle, which builds them by addition; the binomial theorem, where they reappear in $(a + b)^n$; and combinations with repeats allowed. **Applications.** Hypergeometric counting finds chances for draws without replacement, and the inclusion-exclusion principle counts the union of many overlapping sets. ## A path through it Counting in stages first, then permutations of a whole set, of some objects, and with restrictions. Combinations come next, then Pascal's triangle and the binomial theorem, then combinations with repetition. Hypergeometric counting needs combinations and drawing without replacement. Inclusion-exclusion goes last: its proof uses the binomial theorem. Finish here and the binomial distribution can count successes in repeated trials. ### [Counting with restrictions](https://math.wiki/counting-with-a-position-restriction/) **Summary:** Filling the position that carries the rule first, then counting the free positions that remain. **Question:** Seven cars park in a row of seven spaces, and the two red cars must be side by side. How many ways are there, when a rule ties some objects to places or to each other? ## When you reach for it The factorial counts every order of a set, and $P(n, r)$ counts the rows of $r$ objects chosen from $n$. Both assume that any object may go in any place. Real arrangements usually come with a rule attached: a given speaker must go last, the leader must hold a first-aid certificate, two cars must park side by side, a word must not start with a given letter, a number must be even. A rule like that is a **restriction**. Reach for this move whenever the question still asks "how many orders" or "how many rows", but some of the rows are not allowed. The tools are the ones already known, the fundamental counting principle and the complement; what is new is the order in which the places are filled, and the habit of choosing which side of the rule to count. ## The procedure **Fill the restricted place first.** The counting principle multiplies the number of choices at each stage, and that is only allowed when the number at a stage is the same whatever happened before. A free place does not care what went earlier, but a restricted place does. Suppose the leader of a hiking group of $10$ must be one of the $3$ members with a first-aid certificate. If the leader is chosen first there are exactly $3$ choices, and then the free places take what is left. If the leader were chosen last, the number of certificate holders still available would depend on whether some had already been used in the earlier places, and the product would have no single factor to write down. So the place carrying the rule goes first, and the free places, which always have "however many are left", go after. ### [Cumulative probability](https://math.wiki/probability-of-an-interval-of-values/) **Summary:** Finding the chance the variable lands between two values by subtracting cumulative probabilities. **Question:** A distribution table gives the chance of each single value. How do you get the chance that the variable lands anywhere in a range, without adding a fresh string of entries every time? ## When you reach for it A probability distribution table lists each value of a discrete random variable with its probability, and a question about a range of values, such as "at most $4$" or "between $2$ and $5$", is answered by adding the probabilities of the values in that range. That always works, but each new range means a new string of additions. The **cumulative distribution** does the adding once. Its entry at $x$ is the probability that the variable is at most $x$. With that table in hand, the probability of any interval is one subtraction of two entries, and "at least" questions are one subtraction from $1$. Reach for it when a question asks about a range of values, when you are handed a cumulative table (software often gives one), and when you want a middle value of the variable: the cumulative table shows where the probability passes one half. The examples use one made-up setting. Shota looks after the jukebox in a diner. Let $X$ be the number of songs a customer plays on one visit, counting only customers who use the jukebox, so $X$ is at least $1$: | $x$ | $P(X = x)$ | |:-:|:-:| | $1$ | $0.10$ | | $2$ | $0.20$ | | $3$ | $0.25$ | | $4$ | $0.20$ | | $5$ | $0.15$ | | $6$ | $0.10$ | ## The procedure ### Running totals The **cumulative distribution function** of $X$ is $$F(x) = P(X \le x)$$ the probability that $X$ comes out at most $x$. For a discrete variable, $F$ at a possible value is the sum of the probabilities of that value and every smaller one, so a cumulative table is a column of running totals: each entry is the previous entry plus the probability of the new value. That adding is allowed because the events $X = 1$, $X = 2$ and so on are disjoint, and the probability of a union of disjoint events is the sum of their probabilities. ### [Dependent events and drawing without replacement](https://math.wiki/dependent-events/) **Summary:** The first outcome changes the odds of the second; multiply with the updated probability. **Question:** Two prizes are drawn from a hat of names, and nobody may win twice. Once the first name is out of the hat, what has happened to everybody else's chances? ## Why this exists The multiplication rule for independent events multiplies two chances because the second stage does not care how the first one went. A coin does not remember the last toss. Plenty of everyday draws are not like that. Two raffle prizes are drawn from a hat of names, and a name that has won is kept out so that nobody wins twice. A coach picks a captain, then a vice-captain from the players who are left. Someone eats a chocolate from a box, then takes another. Each time the first draw takes something away, so the second is made from a smaller, different collection, and its chances are no longer those of the first. This lesson finds the new chances by recounting what is left, carries them on a tree, compares drawing with and without putting back, and checks the tree against a plain count of outcomes. ## Definition **Drawing with replacement** means each item drawn is put back and mixed in before the next draw. Every draw is made from the same full collection. **Drawing without replacement** means each item drawn is kept out. Each draw is made from what the earlier draws left behind, so the collection has one item fewer every time. Two events are **dependent** when knowing how one turned out changes the probability of the other. They are **independent** when it does not, which is the case the multiplication rule handles. Drawing without replacement makes the draws dependent. A box holds $10$ chocolates, $4$ caramel and $6$ mint, all wrapped alike, and you take one without looking. The chance it is caramel is $\frac{4}{10}$. If it was caramel, the box now holds $9$ chocolates and $3$ of them are caramel, so the chance the next one is caramel is $\frac{3}{9}$. If it was mint, $4$ caramels remain among $9$, and the chance is $\frac{4}{9}$. The chance for the second draw depends on the result of the first, which is exactly what dependent means. ### [Designing a simulation](https://math.wiki/design-a-simulation/) **Summary:** Setting up a chance process whose outcomes match the real situation being studied. **Question:** You cannot wait through a thousand real mornings to see how often a bus is late. What can stand in for them? ## When you reach for it Kenelm's bus is late on $30\%$ of mornings. How often will it be late on two or more mornings in the same school week? Watching real weeks takes a year, and counting is awkward: a week has $2^5 = 32$ patterns of late and on time, and they are not equally likely. A **simulation** is a third way. You pick a chance process you can run in minutes (a coin, a spinner, a number cube, cards or random digits) and set it up so that each of its outcomes has the same probability as the real outcome it stands for. One complete run of it is a **trial**, and you record what each trial gave. Reach for one when the real thing is slow or impossible to repeat, or when the question is about a **compound event**, several chance events at once, such as "late on at least two of five mornings". This lesson designs simulations and runs a few trials by hand. Turning many trials into a number is running a simulation and estimating a probability, and why more trials help is the law of large numbers. ## The procedure ### Day 1: a device, an assignment and a trial **What a simulation copies.** The real situation comes with a probability model: outcomes, each with a probability. The simulation copies the model and nothing else. A coin is nothing like a baby, but if the model gives each child a chance $\frac{1}{2}$ of being a girl, heads has exactly that chance too. So every design is judged by one requirement: **each simulated outcome must have the same probability as the real outcome it stands for.** ### [Expected value](https://math.wiki/expected-value-as-a-weighted-mean/) **Summary:** Each value is weighted by how likely it is, so likelier values pull the average toward themselves. **Question:** A claw machine usually gives nothing, sometimes a small prize, and now and then a big one. What is one play worth on average, and what does that average actually promise? ## Why this exists Vardan stands at an arcade claw machine (made-up numbers throughout this lesson). A play costs $2$ dollars. The sign says the claw grabs a big plush worth $15$ dollars with probability $0.08$, a small one worth $5$ dollars with probability $0.12$, and otherwise nothing. Is a play worth $2$ dollars? Averaging the three prize values, $\frac{15 + 5 + 0}{3} \approx 6.67$ dollars, says yes, and is badly wrong: it treats the big plush as if it came up a third of the time, when nothing comes up $80\%$ of the time. A fair average must give each value a say in proportion to how often it happens. That average is the **expected value**. It turns a whole distribution into one number, says what a game pays per play in the long run, and lets a business compare choices whose outcomes are uncertain. The lesson takes two days: the first builds the expected value and its long-run meaning, the second uses it on games and decisions, and marks where it stops being a good guide. ## Definition A random variable $X$ attaches a number to each outcome of a chance process, and its probability distribution lists the possible values $x_1, x_2, \ldots, x_n$ with their probabilities $p_1, p_2, \ldots, p_n$, which are not negative and add to $1$. The **expected value** (or **mean**) of $X$ multiplies each value by its probability and adds the products: $$ \begin{aligned} E(X) = \mu_X &= x_1 p_1 + x_2 p_2 + \cdots + x_n p_n \\ &= \sum x_i p_i . \end{aligned} $$ ### [Experimental probability and relative frequency](https://math.wiki/experimental-probability-from-data/) **Summary:** Working out a probability by counting how often something actually happened in recorded trials. **Question:** A thumbtack can land point up or point down, and nothing about its shape says how often each happens. How do you put a number on that chance? ## When you reach for it Some chances can be counted. A fair cube has six faces that nobody can tell apart once it is rolling, so each face gets $\frac{1}{6}$, and the probability of an event is the number of outcomes in it over the number of outcomes in all. That count needs a model whose outcomes are equally likely, or at least have known chances (see probability-models). Many chance processes have no such model. A tossed thumbtack lands either point up (resting on its flat head) or point down (tipped over, with the point touching the table). Two outcomes, no reason to think them equally likely, and nothing about the tack says how often each happens. The same goes for a dropped paper cup, the next bird at a feeder, or a player's next free throw. There is nothing to count, so the chance has to be learned from what actually happens. That is when you reach for this skill: repeat the process many times, count, and use the share of times something happened as an estimate of its probability. It also runs the other way: a probability says about how many times to expect the event in a given number of further tries. ## The procedure ### Day 1: relative frequency Each repetition of the process is a **trial**: one toss of the tack, one drop of the cup, one bird landing, one cap checked, one free throw. The number of trials in which an event happened is its **frequency**, and its share of all the trials is its **relative frequency**: ### [Finding a conditional probability](https://math.wiki/conditional-probability-formula/) **Summary:** Dividing the chance that both happen by the chance of the event being conditioned on. **Question:** Once you know that one thing has happened, how do you work out the new chance of another, and how can you tell whether knowing it changed anything at all? ## Why this exists The general multiplication rule builds the chance of two things together out of the chance of the first and the chance of the second given the first: $$P(A \text{ and } B) = P(A) \times P(B \mid A)$$ Often the information comes the other way around. A bakery knows what share of its orders include a cake, and what share include a cake and are delivered; it wants to know how likely an order with a cake is to be delivered. A garden center has a two-way table of the plants it has sold and wants the chance that a fern it sold was bought as a gift. A tree diagram shows the chance of each whole path, and the chance on one branch is missing. Each time the wanted number is a probability given an event, and the known numbers are a joint probability and the probability of that event. Dividing one by the other is the whole of this lesson. It also answers a second question: whether knowing that $A$ happened changes the chance of $B$ at all, which is what it means for two events to be dependent. ## Definition The **conditional probability** of $B$ given $A$, written $P(B \mid A)$, is the probability that $B$ happens when it is known that $A$ has happened. The bar is read "given", and the event after the bar is the one known to have happened. Knowing $A$ shrinks the possibilities to the $A$ cases only, and $P(B \mid A)$ is the share of those cases in which $B$ also happens. As a formula, for any events $A$ and $B$ with $P(A) > 0$, ### [Geometric probability](https://math.wiki/probability-from-an-area-model/) **Summary:** Comparing the area of a target region with the area of the whole region to get a probability. **Question:** A raindrop can land anywhere on a patio, and there are far too many landing spots to count. How likely is it to land in the drain? ## When you reach for it Counting outcomes needs a list of equally likely results, like the six faces of a die. A raindrop falling on a patio has no such list: it can come down at any point, and there are more points than any count could reach. So can a hailstone striking a flat roof, a parachutist coming down in a field, or a ball rolled across a floor until it stops. Reach for area when the outcome is **a position inside a region** and nothing favors one part of the region over another. Then the chance of landing in a part is that part's share of the area. A spinner is the same idea on a disc, where each slice's share is set by its angle. ## The procedure ### What "at random in a region" means A point **lands at random in a region** when it is certain to land somewhere in the region, and any two parts with the same area are equally likely to receive it, whatever their shape and wherever they sit. That is a claim about the situation, not something the mathematics promises. Rain falling evenly on a small patio fits it; a parachutist steering for the middle of a field does not. ### Why probability becomes area Cover the whole region with a grid of small equal squares. In the figure the region is a $6$ by $4$ rectangle, cut into $24$ unit squares, and the shaded target covers $6$ of them. Equal squares are equally likely, and exactly one of them receives the point, so the counting rule applies: ### [Inclusion-exclusion for three or more sets, and derangements](https://math.wiki/inclusion-exclusion-principle/) **Summary:** Add the single sets, subtract the overlaps of pairs, add back the overlaps of triples, and so on; the same count gives the arrangements that leave nothing in its place. **Question:** Several groups overlap in every possible way. How do you count what is in at least one of them, or in none? ## When you reach for it The two-set count handles "in $A$ or in $B$". Many questions involve three properties, or ten, or $n$: integers divisible by at least one of several numbers, arrangements in which at least one object stays where it was, functions that miss at least one value, solutions of an equation in which some variable is too large. In each case the single sets and their intersections are easy to count, while the union is not. Often the target is the opposite count, the elements with **none** of the properties, where each property names a kind of element to throw out. Inclusion-exclusion turns the easy counts into the hard one: add the single sets, subtract the intersections of two, add back the intersections of three, and keep alternating. ## The procedure ### Day 1: three sets, then any number Throughout, the sets are finite subsets of a universal set $U$. **Three sets.** For finite sets $A$, $B$, $C$, $$\begin{aligned} &\lvert A \cup B \cup C\rvert \\ &\quad = \lvert A\rvert + \lvert B\rvert + \lvert C\rvert \\ &\quad\quad - \lvert A \cap B\rvert - \lvert A \cap C\rvert - \lvert B \cap C\rvert \\ &\quad\quad + \lvert A \cap B \cap C\rvert \end{aligned}$$ *Proof.* Each term on the right counts certain elements once, with a plus or a minus sign. Take any element $x$ of $U$ and add up what $x$ contributes. If $x$ lies in none of the sets, it is in no term and contributes $0$, which is right, since it is not in the union. If $x$ lies in exactly one set, it appears in one single-set term and in no intersection: it contributes $1$. If $x$ lies in exactly two, say $A$ and $B$ but not $C$, it appears in $\lvert A\rvert$, $\lvert B\rvert$ and $\lvert A \cap B\rvert$ and nowhere else: $1 + 1 - 1 = 1$. If $x$ lies in all three, it appears in all three single terms, all three pair terms and the triple term: $3 - 3 + 1 = 1$. So every element of the union is counted exactly once and every other element zero times. $\square$ ### [Independent events and the multiplication rule](https://math.wiki/multiplication-rule-for-independent-events/) **Summary:** When one event does not change the chance of the other, multiply the two probabilities together. **Question:** A coin is tossed and a number cube is rolled. Knowing how the coin landed tells you nothing about the cube, so how likely is it that both come out the way you want? ## Why this exists Many chance experiments happen in stages: toss a coin and then roll a number cube, spin a spinner twice, draw a card and then draw again. A table of every outcome, as in list-a-sample-space, answers any question about such an experiment, but the tables grow fast. Two spins of a ten-section spinner give $100$ outcomes; three rolls of a number cube give $216$. Often the stages have nothing to do with each other: the coin does not know what the cube will do. Then the chance that both stages go a certain way is the chance for the first times the chance for the second. This lesson says when that short cut is allowed, shows why it agrees with counting, and stretches it to three or more stages. ## Definition Two events are **independent** when knowing whether one of them happened does not change the probability of the other. If they are not independent, they are **dependent**. Independence usually comes from stages that cannot affect each other. A coin toss and a cube roll are independent. So are two spins of the same spinner, because the spinner has the same sections both times. **Sampling with replacement** means drawing one item from a collection, noting it, putting it back, mixing, and drawing again. The second draw is made from exactly the same collection as the first, so its chances are the same whatever came first: draws made with replacement are independent. If the item is kept out instead, the collection has changed and the draws are dependent; that case belongs to dependent-events. ### [Joint and marginal probability](https://math.wiki/joint-and-marginal-probability/) **Summary:** A joint probability sits inside a table cell and a marginal probability sits in a row or column total. **Question:** One passenger is picked at random from the 400 on a ferry. How likely is it that they hold a first-class return ticket, and how likely that they hold a return ticket of either class? ## Why this exists A two-way table counts a group of people, or things, by two variables at once. You already know how to read its counts and its totals. Now put the group in a hat and draw one member at random, every member equally likely. "How likely is the member drawn to be such and such?" is a probability question, and the table answers it with one division. The questions come in two kinds. Some are about **both** variables at once: is the passenger in first class **and** holding a return ticket? Others are about **one** variable only: does the passenger hold a return ticket, whatever the class? The first kind is answered by a single cell of the table, the second by a row or column total. These two kinds of probability have names, and the relationship between them (the total is the sum of its cells) is what every later table calculation leans on. ## Definition Take a two-way table with grand total $N$, and pick one member of the group at random, each of the $N$ equally likely. Let $A$ be a category of the row variable and $B$ a category of the column variable. A **joint probability** is the probability that the member drawn falls in category $A$ **and** in category $B$. It is the cell where row $A$ meets column $B$, divided by the grand total: $$P(A \text{ and } B) = \frac{\text{cell for } A \text{ and } B}{N}.$$ A **marginal probability** is the probability of one variable's category on its own, with no attention paid to the other variable. It is a row total or a column total, divided by the grand total: ### [Linear transformations of a random variable](https://math.wiki/adding-a-constant-shifts-the-mean/) **Summary:** Adding the same amount to every value moves the mean by that amount and leaves the spread alone. **Question:** Every value of a random variable is changed by the same rule, as when a day's count of rentals becomes a day's profit. Do the mean and standard deviation have to be worked out again from scratch, and does anything else about the distribution change? ## Why this exists A random variable $X$ comes with a probability-distribution-table, a mean $\mu_X = \sum x\,P(x)$, and a variance $\operatorname{Var}(X) = \sum (x - \mu_X)^2 P(x)$ with standard deviation $\sigma_X = \sqrt{\operatorname{Var}(X)}$. Often the number that matters is not $X$ itself but something computed from it. Leire rents kayaks at a lake dock for $12$ dollars each and pays $30$ dollars a day for her spot, so her daily profit is $Y = 12X - 30$, where $X$ is the number of rentals. A time in minutes becomes a time in hours; a score becomes a z-score. A new table for $Y$ and every weighted sum recomputed would work, but for a rule of the form $a + bX$ the mean and standard deviation of $Y$ follow from those of $X$ in one line each. The same fact for a list of data was the subject of add-a-constant-to-every-value; here it is proved for a random variable from the weighted sums, extended to negative multipliers, and used to standardize a random variable. The lesson runs over two days. Day 1 is the rules for $a + bX$; Day 2 is standardizing and money models. All settings are made up: Leire's kayak dock, where Haizea leads guided trips and Ekaitz runs a snack cart. ## Definition A **linear transformation** of a random variable $X$ is a new random variable $$Y = a + bX$$ where $a$ and $b$ are constants, the same for every outcome. Whenever $X$ takes the value $x$, $Y$ takes the value $a + bx$. The constant $a$ is **added** to every value; $b$ **multiplies** every value. ### [Listing a sample space: lists and tables](https://math.wiki/list-a-sample-space/) **Summary:** Writing out every possible outcome of an experiment once each, in an order that shows none was missed. **Question:** Two number cubes are rolled. How can you write down every way they can land, and be sure you missed none and wrote none twice? ## When you reach for it The sample space of an experiment is the list of all its possible outcomes. For one toss of a coin it is short enough to write without thinking: heads (H) or tails (T). For one roll of a number cube it is $1$, $2$, $3$, $4$, $5$, $6$. Many experiments happen in two stages: toss a coin and then spin a spinner, roll two number cubes, choose a main course and then a drink. Now each outcome has two parts, and the outcomes pile up quickly: two number cubes can land in $36$ ways. Written down as they come to mind, a list that long easily has a gap or a repeat, and nothing on the page shows where. So write the sample space in an order that shows its own completeness: every outcome once, and a count that can be checked. That list comes before any probability found by counting the outcomes in an event; here the job stops at the list and its count. ## The procedure ### What one outcome of two stages looks like An outcome of a two-stage experiment records what happened at **both** stages. Tossing a coin and then spinning the spinner below can give heads and $2$, written H2. "Heads" on its own is not an outcome of this experiment, because it does not say what the spinner did. For two number cubes, write an outcome as an **ordered pair**: the first cube's number, then the second cube's. The order matters. $(2, 5)$ means the first cube shows $2$ and the second shows $5$, and $(5, 2)$ is the other way around. They are different outcomes even when the cubes look alike, just as red $2$ beside blue $5$ is a different sight from red $5$ beside blue $2$. Decide which stage is written first, and keep to it for the whole list. ### [Mean and standard deviation of a binomial](https://math.wiki/binomial-mean/) **Summary:** The expected number of successes is the number of trials times the chance of success on one trial. **Question:** A count of successes over many independent tries could be described by listing every probability. Is there a quick way to say how many successes to expect, and how far the count typically strays from that? ## Why this exists Ruta runs the evening classes at a cooking school. Each class takes $40$ reservations, and each booked guest fails to show with chance $0.1$, independently of the others (a made-up chance). The number of no-shows $X$ is a binomial count (recognize-a-binomial-setting): $n = 40$ trials, a "success" being a no-show, with $p = 0.1$. Ruta's real questions are about center and spread. How many empty places should she expect, and how much does that number move from night to night? The mean $\mu_X = \sum x \, P(x)$ and the standard deviation $\sigma_X$ answer both, but computed from the table they need all $41$ probabilities from binomial-probability-of-exactly-k, then $41$ products, then $41$ squared deviations. For a binomial count that work collapses to two short formulas in $n$ and $p$. ## Definition Let $X$ be the number of successes in $n$ independent trials, each a success with the same probability $p$. The **mean of the binomial** is its long-run average count, $\mu_X = np$. The **standard deviation of the binomial** is the typical distance of the count from that mean, $\sigma_X = \sqrt{np(1-p)}$. These are the same $\mu_X$ and $\sigma_X$ defined for every discrete random variable; the binomial setting simply lets them be computed from $n$ and $p$ alone. ## Intuition If one guest in ten misses, then over many nights $40$ reservations lose about one tenth of $40$, that is $4$ guests, on average. The mean is the share that fails times the number of tries. ### [Mean and standard deviation of a geometric random variable](https://math.wiki/geometric-mean-number-of-trials/) **Summary:** On average the first success arrives after one over p tries, with a standard deviation of the square root of one minus p, over p. **Question:** If each try succeeds with the same small chance, how many tries does the first success take on average, and how far from that average does a single wait usually land? ## Why this exists A yogurt brand prints a prize code under $1$ in $25$ lids, each lid independently of the others (made-up data, as is every number in this lesson). Thida opens lids one at a time until she finds a code. Her count of lids $X$ is a geometric random variable with $p = 0.04$, and geometric-distribution-formula already gives every probability: $P(X = k) = (1 - p)^{k - 1}p$ and $P(X > k) = (1 - p)^k$. That list has no end, so it does not answer the plain questions directly. How many lids does a search take on average? How much do searches differ from one another? The mean and the standard deviation would need an infinite table of products. For a geometric wait the whole sum collapses to two short formulas in $p$. ## Definition Let $X$ be a geometric random variable with probability of success $p$ on each trial, $0 < p \le 1$, counting trials up to and including the first success. Its **mean** is the long-run average of $X$ over many repetitions of the whole search, $$\begin{aligned} \mu_X = {} & 1 \cdot P(X = 1) + 2 \cdot P(X = 2) \\ & + 3 \cdot P(X = 3) + \cdots \end{aligned}$$ and its **variance** $\sigma_X^2$ is the long-run average of $(X - \mu_X)^2$, weighted the same way. Its **standard deviation** $\sigma_X$ is the square root of the variance, measured in trials. ## Intuition Picture a long row of lids, $10{,}000$ of them. About $p \times 10{,}000 = 400$ carry a code. The searches split that row into about $400$ stretches, each ending at a code, and together they use up all $10{,}000$ lids. So a stretch averages $\tfrac{10{,}000}{400} = 25$ lids. Codes per lid is $p$; lids per code is $\tfrac{1}{p}$. This is the same long-run reading that gives a binomial count the mean $np$: in $n$ lids expect $np$ codes, so the lids per code is $\tfrac{n}{np} = \tfrac{1}{p}$. ### [Mutually exclusive events and the addition rule](https://math.wiki/mutually-exclusive-events/) **Summary:** Cannot both happen; their probabilities add. **Question:** One marble is pulled from a bag. It cannot be red and blue at the same time, so how does the chance of "red or blue" come from the chance of each color on its own? ## Why this exists Basic probability finds the chance of an event by counting its outcomes. Often, though, an event is described in pieces: "red or blue", "a sum of $5$ or two even numbers", "the train is late or canceled". Each piece may already have a known chance, and recounting from scratch every time wastes that work. Worse, sometimes there is nothing to count: a chance like $0.13$ for "the train is early" comes from records, not from a list of equally likely outcomes. So the question is when the chance of "$A$ or $B$" can be built straight from the chance of $A$ and the chance of $B$. The answer is simple when the two events cannot happen together: add them. This lesson says exactly when that is allowed, why it works, and why "cannot happen together" is a different idea from "has no effect on each other". ## Definition Two events $A$ and $B$ on the same trial are **mutually exclusive** when they cannot both happen on that trial. In the language of sample spaces and events: no outcome belongs to both of them, so their intersection is the empty set, $$A \cap B = \varnothing$$ Mutually exclusive events are also called **disjoint**, the word used for sets with no member in common in union-and-intersection. Several events are mutually exclusive when **every pair** of them is: no outcome lies in two of them at once. The event "$A$ or $B$" happens when at least one of $A$ and $B$ happens. In probability, "or" always includes the case where both happen, unless the problem says otherwise. Its probability is written $P(A \text{ or } B)$, or $P(A \cup B)$ with the union sign. For mutually exclusive events the "both" case never arises, so the two readings of "or" give the same event. ### [Odds](https://math.wiki/odds-in-favor-and-against/) **Summary:** Odds in favor put the successes first and odds against put the failures first, so the two are reversed. **Question:** A chance can be told as a share of everything that might happen, or as the ways it happens set against the ways it does not. How do the two descriptions fit together, and how do you move from one to the other? ## Why this exists A probability answers the question "what share of all the outcomes make the event happen?" It is a part compared with the whole. That is not the only natural way to talk about chance. People often compare the two sides directly: the ways the event can happen against the ways it cannot. "Three ways it happens for every five ways it does not" is a complete description of a chance, and it never mentions the total of eight. That second description is called **odds**. It is a part-to-part ratio, where a probability is a part-to-whole fraction, and the two carry exactly the same information. The point of this lesson is to see why, and to move between them without losing anything on the way. Sports commentary and games of chance often state chances as odds, and some later statistics (the odds ratio in medicine, logistic models) is built on odds rather than on probabilities, so the conversion is worth knowing well. ## Definition Take an experiment and an event $E$. Every outcome either makes $E$ happen or makes its complement $\text{not } E$ happen, never both. When the outcomes are equally likely, count them: say $s$ outcomes make $E$ happen and $f$ outcomes do not. - The **odds in favor of** $E$ are $s : f$, read "$s$ to $f$". The ways it happens come first. - The **odds against** $E$ are $f : s$. The ways it does not happen come first. So the odds against are the odds in favor written in the other order. ### [Pascal's triangle](https://math.wiki/pascals-triangle-and-combinations/) **Summary:** Every entry of the triangle is a combination count, and the building rule is a fact about choosing. **Question:** Working out $\binom{n}{r}$ from factorials one number at a time is slow. Is there a way to get every combination count at once, using nothing harder than addition? ## Why this exists Combinations gave a formula for the number of ways to choose $r$ objects from $n$ different objects: $$\binom{n}{r} = \frac{n!}{r!\,(n - r)!}$$ The formula answers one question at a time, and each answer costs a run of multiplications and a division. But combination counts are not isolated numbers. Laid out in rows, one row for each $n$, they fit together so tightly that each row can be written down from the row above it by adding neighbors. No factorial is needed at all. The arrangement is called **Pascal's triangle**, after Blaise Pascal, who wrote a treatise on it in the 1650s, though mathematicians in India, Persia and China had written down versions of the table long before him. The adding rule that builds it is not a coincidence of arithmetic. It is a fact about choosing, and seeing why it holds is the main idea of this lesson. The same view of choosing also answers a question the formula alone does not: how many subsets does a set have altogether, of every size at once? ## Definition **Pascal's triangle** is the table whose row $n$, for $n = 0, 1, 2, \dots$, lists the combination counts $$\binom{n}{0},\ \binom{n}{1},\ \binom{n}{2},\ \dots,\ \binom{n}{n}$$ from left to right, each row centered under the one before. Row $n$ has $n + 1$ entries. Rows are numbered from $0$, so the top row, a single $1$, is row $0$; and the places in a row are numbered from $0$ as well, so the entry in place $r$ of row $n$ is $\binom{n}{r}$, the number of $r$-element subsets of an $n$-element set. ### [Permutations](https://math.wiki/permutations-of-some-objects/) **Summary:** Counting the ordered ways to pick and place only part of a set, leaving the rest unused. **Question:** A club of fifteen members elects a president, a vice-president and a treasurer. How many outcomes are there, when only three of the fifteen are placed and the other twelve are left out? ## Why this exists The factorial counts the orders of a whole set, every object used once. Very often only some of the objects are placed. A club of $15$ fills three offices, a contest gives out a first and a second prize, a lock code uses five of the ten digits. The rest of the set is left out, and yet the order of the ones chosen still matters: being president is not being treasurer. The fundamental counting principle counts these too, by filling the places one at a time, but the product now stops part-way down. That stopped product turns up so often that it gets its own symbol, $P(n, r)$, and a formula as a quotient of factorials. Working with that formula means simplifying factorial expressions, the way rational expressions are simplified by canceling common factors. The same dividing idea then answers a second question the factorial alone cannot: how many different rows can be made when some of the objects are exactly alike, such as the letters of a word with a repeated letter. ## Definition A **permutation of $r$ objects chosen from $n$** is a row of $r$ of the $n$ different objects, each used at most once, where the order matters. The other $n - r$ objects are not used. Here $n$ and $r$ are whole numbers with $0 \le r \le n$. The number of such permutations is written $$P(n, r)$$ and read "the number of permutations of $n$ objects taken $r$ at a time". Other books write ${}_{n}P_{r}$ or $P^{n}_{r}$ for the same number. When $r = n$ every object is used, so $P(n, n)$ is the number of arrangements of the whole set. ### [Probability](https://math.wiki/probability/) **Summary:** Measuring how likely something is. **Question:** How do you put a number on how likely something is, and what can you do with that number once you have it? ## What this area is about Probability measures how likely an event is, on a scale from $0$ (impossible) to $1$ (certain). It starts with single events, combines them, counts outcomes too many to list, and then attaches numbers to outcomes so that a whole distribution has a center and a spread. ## How these fit together **Foundations.** Basic probability finds a chance from a model (counting equally likely outcomes, or a region) or from repeated trials. Compound events give the rules for "or", "and" and "at least one", and hold conditional probability, defined when the event conditioned on has probability above $0$. **A tool.** Counting finds how many ways something can happen without listing them: permutations, combinations, the binomial theorem. **Distributions.** Random variables attach a number to each outcome and describe its mean and spread, including the binomial and geometric distributions. **College level, not yet written.** Foundations: probability from axioms, counting models and conditioning. Random variables: discrete and continuous distributions, expectation and variance. Several random variables: joint distributions, conditional expectation and moment generating functions. Limit theorems: why averages settle down, and why they look normal. ## A path through it Basic probability and compound events go together: start with outcomes, events and counting outcomes, take compound events, then return for the law of large numbers and simulation. Counting comes next, then random variables. At college level the order is foundations, random variables, several random variables, then limit theorems. ### [Probability models: equally likely or not](https://math.wiki/probability-models/) **Summary:** A probability model is a sample space together with a probability attached to every outcome in it. **Question:** A spinner has three colors, but its sections are different sizes. How do you say how likely each color is, and how can you tell whether a list of chances could possibly be right? ## Why this exists Counting the outcomes in an event and dividing by the total is a quick way to find a probability, but it rests on a claim that is easy to miss: every outcome must be exactly as likely as every other. A fair number cube earns that claim. Plenty of chance devices do not. The spinner on the right of the figure below lands on red, blue or yellow, yet red has the smallest section and yellow the largest, so nobody should expect each color a third of the time. A coin bent in a door hinge still lands heads or tails, but there is no longer any reason to think the two are equally likely. What is needed is a way to write down the chances for any experiment, equal or not, and a test that any such list must pass before it can be trusted. That is a probability model. ## Definition A **probability model** for an experiment has two parts: the sample space, which lists every possible outcome, and a probability for each of those outcomes. The probabilities must obey two rules: - every probability is between $0$ and $1$, with $0$ and $1$ themselves allowed; - the probabilities of all the outcomes add to exactly $1$. A model is usually written as a table, with the outcomes in one row and their probabilities in the row below. For a fair coin: | Outcome | heads | tails | |---|:-:|:-:| | Probability | $\frac{1}{2}$ | $\frac{1}{2}$ | $P(\text{heads})$ is read "the probability of heads". With outcomes $o_1, o_2, \dots, o_n$, the two rules say ### [Probability of an event](https://math.wiki/probability-of-an-event-with-several-outcomes/) **Summary:** Counting how many outcomes make the event happen and comparing that to the total number of outcomes. **Question:** A prize is taped under one of fifty seats, and your family sits in three of them. How do you turn that into one number for your chance of winning, and how is that number written? ## When you reach for it The words of the chance scale say roughly how likely something is. Often you can give the exact number instead. A prize is taped under one of fifty seats, and your family sits in three of them. A bag holds twelve counters, five of them red. A shuffled playlist picks its first song from forty. In each case you know every result that can happen, you know which ones you are hoping for, and nothing makes one result come up more often than another. That last condition is the one this lesson needs. When every outcome in the sample space is **equally likely**, the probability of an event is found by counting: how many outcomes make the event happen, out of how many there are. The words **outcome**, **sample space** and **event** are the ones from sample spaces, outcomes and events: an outcome is one single result, the sample space $S$ is the set of all of them, and an event is any collection of outcomes. Over two days: Part 1 counts and writes the answer three ways; Part 2 counts in two stages, compares events, and works backwards from a probability to a count. ## The procedure ### Part 1: counting the outcomes in an event **The rule.** If every outcome in the sample space $S$ is equally likely, then for any event $A$ $$P(A) = \frac{\text{number of outcomes in } A}{\text{number of outcomes in } S}$$ $P(A)$ is read "the probability of $A$". The top counts the outcomes that make $A$ happen, and the bottom counts every outcome there is. ### [Probability with counting formulas](https://math.wiki/hypergeometric-counting-problems/) **Summary:** Counting how many ways to draw a set number of each type when items are taken without replacement. **Question:** A committee of $5$ is picked at random from $8$ seniors and $7$ juniors. There are thousands of possible committees, so how do you find the chance that exactly $3$ of the $5$ are seniors without listing them? ## Why this exists Drawing without replacement already has a method: follow the draws one at a time, and multiply each chance by the next one worked out from what is left. That works well for two or three draws. A committee of $5$ chosen from $15$ people is five draws, and the question "exactly $3$ seniors" does not say *which* three of the five places the seniors fill. A tree for it has $32$ paths, and ten of them end in the outcome you want. Very often the order of the draws does not matter at all. A committee is a group, a hand of cards is a group, a sample of apples pulled from a basket for inspection is a group. When the outcome is a group, there is a shorter road: count groups. Combinations count how many groups of a given size there are, and the fundamental counting principle counts the groups that have the right make-up. The probability is one count divided by the other. This lesson builds that count, uses it for "exactly", "at least" and "at most" questions, checks it against drawing in order, and carries it to collections with three or more kinds of item. ## Definition A collection of $N$ different items is split into two kinds: $K$ items of the kind you are interested in, and the other $N - K$. A group of $n$ items is taken **at random without replacement**, which means every group of $n$ items has the same chance of being the one taken. This is the same as drawing the items one at a time without putting any back and then forgetting the order they came in. ### [Random variables](https://math.wiki/random-variables/) **Summary:** A number attached to each outcome, and the distribution of those numbers. **Question:** When a chance process produces a number, how do you describe every value it might take, where it centers and how far it spreads? ## What this area is about Often what matters about an outcome is a number attached to it: shots of espresso in a drink, calls until a pledge, dollars won. That number is a random variable $X$. This area lists its values with their probabilities, then summarizes them with a mean and a standard deviation, and names two patterns that turn up again and again. ## How these fit together **Describing one variable.** A probability distribution table gives each value of $X$ and its probability, and cumulative probability answers questions about a range. **Center and spread.** Expected value $\mu_X = \sum x \, P(x)$ is the long-run average. The variance and standard deviation measure spread. **Results for changed and combined variables.** Changing every value by a rule $a + bX$ gives the new mean $a + b\mu_X$ and standard deviation $|b|\sigma_X$: adding moves only the center, multiplying rescales both. Variances add for a sum or difference, when the variables are independent. **Two named distributions.** The binomial distribution counts successes in a fixed number $n$ of independent trials with the same chance $p$. The geometric distribution counts independent trials, each with the same chance $p$, until the first success, and its mean is $1/p$. ## A path through it The distribution table first, then cumulative probability, expected value and variance. Next, adding a constant, then variances adding. The binomial distribution needs the table and the variance; the geometric formula follows it, then the geometric mean. ### [Random variables and probability distributions](https://math.wiki/probability-distribution-table/) **Summary:** Laying out the values in one row and their probabilities in the next so the whole distribution is visible. **Question:** A coffee shop pours drinks with zero, one, two, three or four shots of espresso. How do you write down, all at once, how likely each number of shots is, and what can you read off once you have? ## When you reach for it A probability model lists the outcomes of a chance process with a probability for each, every probability between $0$ and $1$ and all of them adding to $1$. Many questions, though, are not about the outcome itself but about a number that comes with it. Sopio runs a coffee shop. When a customer walks in, she does not care which drink they order so much as how many shots of espresso it takes, because that number decides how fast her beans run out. Reach for a random variable and its distribution whenever the thing you care about is a number that depends on chance: the shots in the next drink, the larger of two numbers drawn from a jar, the sum of two draws. The distribution puts every possible value and its probability side by side, so that any question about the variable ("at least two shots?") becomes a matter of adding entries. This lesson runs over two days. Part 1 defines a random variable, writes its distribution as a table, checks that the table is valid, and answers questions by adding. Part 2 builds a distribution from a sample space, writes it as a function, and draws it as a probability histogram. ## The procedure ### Part 1: random variables and the distribution table **A random variable** is a rule that gives a number to each outcome of a chance process. It is written with a capital letter, such as $X$, and a particular value it can take is written in lowercase, such as $x$. "Let $X$ be the number of espresso shots in the drink of a randomly chosen customer" names the variable; $x = 2$ is one value it can take. The variable is a function in the sense of a function: each outcome (a customer's order) gets exactly one number. ### [Running a simulation and estimating a probability](https://math.wiki/estimate-a-probability-from-trials/) **Summary:** Dividing the number of trials that gave the outcome by the number of trials run. **Question:** A simulation has been designed. How do its trials become a number, and how far can that number be trusted? ## When you reach for it Verena shoots rounds of three arrows at an archery club. How likely is she to put at least two arrows of a round in the gold? A simulation for this can be designed as in designing a simulation: a random digit for each arrow, three digits for a round. What is left is to run it, keep a record, and turn the record into a number. That number is an **estimate** of a probability. Reach for one when counting the outcomes of a compound event is slow or out of reach: a round of arrows, a trip of several fish, a wait for a first success, a set of stickers to collect. It also checks an exact answer from a tree diagram, since the two should agree. ## The procedure ### Day 1: running trials and recording them **Run the design exactly as it was fixed.** The assignment, the length of a trial and the starting place are settled before any digit is read. Read in order, cut the digits into trials, and judge a trial only once all of it is in. A trial cut off by the end of the digits is not counted, because nobody knows how it would have ended. **Record every trial.** A results table has one line per trial: what the device gave and what that means. Record a count (the number of golds) rather than only yes or no where you can, since a count answers several questions later. **Gather the record in a tally table.** List every possible result once and make one stroke $\mathsf{I}$ for each trial that gave it, drawing every fifth stroke across the other four, $\cancel{\mathsf{IIII}}$. A result's **frequency** is its number of strokes. Every trial lands in exactly one row, so the frequencies add up to the number of trials, which checks the tallying. ### [Sample spaces, outcomes and events](https://math.wiki/outcome-versus-event/) **Summary:** An outcome is one single result, while an event is any collection of outcomes you care about. **Question:** A number cube is about to be rolled. "It shows a 4" and "it shows an even number" might both come true, but they are not the same kind of statement. What is the difference, and how is each one written down? ## Why this exists Before a number cube is rolled, you can already ask questions about the roll. Will it show a $4$? Will it show an even number? After the roll, one look at the top face answers both. The questions are not alike, though. "A $4$" names exactly one face. "An even number" names three faces, $2$, $4$ and $6$, and comes true whichever of them turns up. A roll of $6$ is one thing that happened, yet it makes "even" and "more than $4$" true at the same time. Probability will soon put a number on questions like these. First two things must be settled exactly: what are all the things that can happen, and which of them does a question count? The words for these are **outcome**, **sample space** and **event**. The lesson on what randomness means described random processes; this one names their pieces. ## Definition An **experiment** is an action whose result is not known until it is carried out, such as rolling a number cube, spinning a spinner or drawing one card from a pile. Each time it is carried out is a **trial**. An **outcome** is one single result of one trial. The cube shows $4$; the spinner lands on blue; the card drawn is the $7$. The **sample space** of an experiment is the set of all its possible outcomes. It is usually called $S$. A set is written by listing its members between curly brackets, called **braces**, with commas between them; this is roster notation. For one roll of a number cube, ### [Shape of a binomial and the normal approximation](https://math.wiki/normal-approximation-to-a-binomial/) **Summary:** When the counts are large enough, the binomial distribution is close enough to a normal curve to use it. **Question:** A binomial count over hundreds of trials has hundreds of possible values, each with its own small probability. When does its bar chart take on a recognizable shape, and when can a single normal curve stand in for all those bars? ## Why this exists Ilze runs a citrus packing line. Each grapefruit that passes her station is blemished with chance $0.12$, independently of the others (a made-up chance), and a crate holds $400$ of them. The number of blemished grapefruit $X$ in a crate is a binomial count (recognize-a-binomial-setting) with $n = 400$ and $p = 0.12$. Janis buys crates for juice and pays less for any crate with $40$ or fewer blemished fruit, since those go to the fresh market instead. He wants $P(X \le 40)$. The binomial formula answers it, but only as a sum of $41$ terms, from $P(X = 0)$ to $P(X = 40)$, with binomial coefficients as long as $56$ digits. Technology does that sum without complaint. Still, a statistician needs to know what the distribution looks like, and for large counts there is a much simpler description: a normal curve, fixed by just the mean and standard deviation. This lesson takes two days. Day 1 is the shape of a binomial distribution: which way it leans, and why it straightens out as $n$ grows. Day 2 is the normal approximation and the condition that says when it can be trusted. ## Definition A binomial distribution's **shape** is the pattern of its probability bar chart. It is **skewed right** when the bars pile up at low counts and trail off in a long tail toward high counts, **symmetric** when the bars on either side of the center match, and **skewed left** when the long tail runs toward low counts. ### [The binomial distribution](https://math.wiki/binomial-distribution/) **Summary:** Count successes in a fixed number of independent tries. **Question:** When you repeat the same chance experiment a fixed number of times, how likely is each possible count of successes? ## What this area is about Many questions count successes: how many of $12$ seeds sprout, how many of $40$ guests fail to show. When there is a fixed number $n$ of trials, each trial has two outcomes, the trials are independent, and every trial succeeds with the same probability $p$, the count $X$ follows one pattern, the binomial distribution. One formula then gives every probability, with no list of the $2^n$ ways the trials can come out. ## How these fit together **The setting.** Recognizing a binomial setting checks those four conditions. Everything else here depends on them holding. **The formula.** The probability of exactly $k$ successes is $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$, and sums of it give "at most" and "at least". **Center and spread.** The binomial mean and standard deviation are $\mu = np$ and $\sigma = \sqrt{np(1 - p)}$, with no table needed. **An approximation.** The normal approximation lets a normal curve with that mean and standard deviation stand in for the bars when $np \ge 10$ and $n(1 - p) \ge 10$. ## A path through it In order: recognize the setting, then the formula for exactly $k$, then the mean and standard deviation, then the normal approximation, which uses both. Finish here and the geometric distribution counts the trials until the first success, and the sampling distribution of a sample proportion builds on the same counts. ### [The binomial probability formula](https://math.wiki/binomial-probability-of-exactly-k/) **Summary:** Putting the number of trials, the success chance and the wanted count into the binomial formula. **Question:** Eight independent tries, each a success with the same chance. What is the chance of exactly two successes, or of at most one, or at least three, without listing all 256 ways the tries can come out? ## When you reach for it A bakery puts a slip of paper in every fortune cookie, and each slip is "lucky" with probability $0.2$, independently of every other cookie (a made-up setting). Kavitha buys a box of $8$. Let $X$ be the number of lucky slips in her box. The count can be any of $0, 1, 2, \dots, 8$, and the question is how likely each value is. The box of cookies is a binomial setting: a fixed number of trials ($n = 8$ cookies), two outcomes on each (lucky or not), trials that are independent, and the same chance of success on every trial ($p = 0.2$). Checking those four conditions is the work of recognize-a-binomial-setting, and it comes first. Once they hold, $X$ has a binomial-distribution with $n = 8$ and $p = 0.2$, and one formula gives the probability of any count. This lesson builds that formula and then uses it, together with the complement and technology, for questions about a range of counts. You reach for it whenever a question asks how many successes occur in a fixed number of independent, identical tries: "exactly $k$", "at most $k$", "at least $k$", "between $a$ and $b$". ## The procedure The lesson runs over two days. Day 1 builds the formula for exactly $k$ successes. Day 2 adds questions about several counts at once. ### Day 1: exactly $k$ successes Take the lead question: what is $P(X = 2)$, the chance that exactly $2$ of the $8$ slips are lucky? **One arrangement.** Write L for a lucky slip and N for a plain one, and record the cookies in the order Kavitha opens them. One way to get exactly two lucky slips is LLNNNNNN. Because the cookies are independent, the multiplication rule for independent events gives its probability as a product of eight factors, $0.2$ for each L and $1 - 0.2 = 0.8$ for each N: ### [The binomial setting](https://math.wiki/recognize-a-binomial-setting/) **Summary:** Testing a situation against each of the four conditions before any binomial formula is used. **Question:** Twelve seeds go into a tray and you count how many sprout. When does a count like that follow one standard pattern, so that a single formula can give every one of its probabilities? ## When you reach for it Bolormaa, Ganbaatar and Oyuna run the seed trays at a community garden. Bolormaa fills a tray with $12$ seeds of one kind of bean, one seed to a cell. From the seed company's germination tests she takes each seed to sprout with probability $0.85$, each seed on its own (made-up data, as is every number in this lesson). The question she cares about is a count: how many of the $12$ will sprout? Call that count $X$. It can be any whole number from $0$ to $12$, and it is a random variable: its value is not known until the tray has had its week. Finding its probabilities from scratch means listing outcomes of a compound event with $12$ stages, which is hopeless by hand: each seed has two outcomes, so a tray has $2^{12} = 4{,}096$ possible patterns of sprouts and failures. Many counts like this share one structure, and when they do, a single formula gives every probability (the next lesson). This lesson is the test that comes first: checking whether a count has that structure, called the **binomial setting**. Reach for it whenever a question asks "how many of these will..." and before any binomial formula is used. Day 1 checks the four conditions on settings that pass; Day 2 studies settings where one fails. ## The procedure A chance process is a **binomial setting** when it is a run of repeated **trials** and all four of these hold. - **Binary outcomes.** Each trial ends in one of two outcomes, called **success** and **failure**. A seed sprouts or it does not. - **Independent trials.** Knowing how some trials came out does not change the probabilities for the others. Bolormaa's seeds sit in separate cells and do not affect one another. - **Number of trials fixed.** The number of trials, $n$, is set before the process starts. The tray holds $n = 12$ seeds whatever happens. - **Same chance of success.** Every trial has the same probability of success, $p$. Each seed sprouts with $p = 0.85$. ### [The binomial theorem](https://math.wiki/binomial-coefficients/) **Summary:** The numbers multiplying each term of an expanded binomial are combination counts. **Question:** Multiplying out $(a + b)^{10}$ by hand takes nine long multiplications. Why are the numbers in front of its terms exactly the combination counts $\binom{10}{r}$, and how does that let you write down any one term at once? ## Why this exists Multiplying two binomials gives $(a + b)^2 = a^2 + 2ab + b^2$. Multiplying by $a + b$ once more gives $$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$$ and each further power costs another, longer multiplication. Reaching $(a + b)^{10}$ this way means nine rounds of multiplying and collecting like terms, with plenty of room for a slip, and a question such as "what is the coefficient of $x^6$ in $(x - 2)^{10}$?" would need the whole expansion to answer one number. The coefficients $1, 2, 1$ and $1, 3, 3, 1$ are rows of Pascal's triangle, and this is no accident. Every coefficient in the expansion of $(a + b)^n$ is a combination count $\binom{n}{r}$, for a reason that is about choosing. The **binomial theorem** states this, and with it any power of a binomial can be written down directly, and any single term found without the others. ## Definition A **binomial** is a sum of two terms, such as $a + b$, $x - 2$ (which is $x + (-2)$) or $2x^2 + 5y$. The **binomial theorem** says that for any numbers $a$ and $b$ and any whole number $n$ (that is, $n = 0, 1, 2, \dots$), $$\begin{aligned} (a + b)^n &= \binom{n}{0}a^n + \binom{n}{1}a^{n - 1}b \\ &\quad + \binom{n}{2}a^{n - 2}b^2 + \cdots \\ &\quad + \binom{n}{n - 1}ab^{n - 1} + \binom{n}{n}b^n \end{aligned}$$ In sigma notation, which writes a sum by giving its typical term and the range of the counter $r$, $$(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n - r} b^r$$ ### [The fundamental counting principle](https://math.wiki/counting-choices-in-stages/) **Summary:** Multiplying the number of options at each step to count all the ways a whole sequence can turn out. **Question:** A suitcase lock has four dials. How can you count every possible code without writing a single one down? ## When you reach for it Many things are built by making choices one after another. A sandwich is a bread, then a filling, then a sauce. A lock code is a first digit, then a second, then a third. An organized list writes every result down, which suits six sandwiches but not the millions of license plates a country needs. The counting principle finds the count without the list. Reach for it when the things being counted are built in **stages**, each stage can go a known number of ways, and that number stays the same however the earlier stages went. It answers "how many", never "which ones" and never "how likely": putting chances on the stages is the work of probabilities on a tree diagram. ## The procedure ### Two stages: multiply **The fundamental counting principle.** If a first choice can be made in $m$ ways, and then, whatever the first choice was, a second choice can be made in $n$ ways, then the two choices together can be made in $m \times n$ ways. A small sandwich bar uses white or brown bread and one filling: cheese, egg or ham. Each path from the starting dot to an end of the tree below is one sandwich. Here is why the count is a product. Sort the sandwiches by their bread. Every sandwich has exactly one bread, so it falls into exactly one of $m = 2$ groups. Inside a group the bread is fixed and the filling runs through its $n = 3$ choices, one sandwich each, so every group holds $3$ sandwiches. Two groups of three make $2 \times 3 = 6$, and in general $m$ groups of $n$ make $m \times n$. It is the argument that showed an organized list is complete, one block for each first choice, with the writing dropped. ### [The general addition rule](https://math.wiki/why-subtract-the-overlap/) **Summary:** Outcomes in both events get counted twice when the probabilities are added, so one copy is removed. **Question:** Two events can happen together. If the chance of each is known, why does adding them give too much for the chance that at least one happens, and by exactly how much? ## Why this exists When two events cannot happen together, the chance that one or the other happens is the sum of their chances: that is the rule for mutually exclusive events. Most pairs of events are not like that. A zoo visitor can see the penguins and the reptile house on the same visit; a roll of two number cubes can show a $6$ and have a large sum at the same time. Adding the two chances then gives the wrong answer, and sometimes an impossible one: two events with chances $0.6$ and $0.7$ would add to $1.3$. Recounting from scratch is not always possible either. A chance such as "$55\%$ of visitors see the penguin pool" comes from records, not from a list of equally likely outcomes. What is needed is a rule that turns $P(A)$, $P(B)$ and one more number into $P(A \text{ or } B)$, for any two events at all. The rule repairs the sum by removing what it counted twice, exactly as counting a union does for the sizes of sets. ## Definition Let $A$ and $B$ be two events on the same trial. In the language of unions and intersections of events: - "$A$ or $B$" is the event that **at least one** of $A$ and $B$ happens, possibly both. Its probability is written $P(A \text{ or } B)$ or $P(A \cup B)$. - "$A$ and $B$" is the event that **both** happen. Its probability is written $P(A \text{ and } B)$ or $P(A \cap B)$. This is the **overlap**. The **general addition rule** says that for any two events $A$ and $B$, ### [The general multiplication rule](https://math.wiki/general-multiplication-rule/) **Summary:** The chance of both events is the chance of the first times the chance of the second given the first. **Question:** Two things both have to happen, and the first one changes the chances of the second. Is there one rule for the chance of both that works every time? ## Why this exists When two events are independent, the multiplication rule for independent events gives the chance of both: multiply the two chances. When they are not, the lesson on dependent events showed what to do for draws from a jar or a hat: recount what is left after the first draw and multiply by the new chance. Many pairs of events are not draws at all. A town's commuters are counted, and $40\%$ of them take the train; among the train takers, $15\%$ cycle to the station. Nothing is taken out of anything, and yet the chance of cycling clearly depends on whether the person takes the train. Or a cable car shuts more often on windy days than on calm ones. Here there is nothing to recount, but there is still a "chance of the second, once the first is known". This lesson gives that chance a name and a symbol, and states one rule for the chance of both that covers every case: independent events, draws without replacement, tables of counts, and chains of three or more events. ## Definition The probability of $B$ **given** $A$, written $P(B \mid A)$, is the probability that $B$ happens when it is known that $A$ has happened. It is the share of the $A$ cases in which $B$ also happens. The bar is read "given", and the event after the bar is the one known to have happened. It only makes sense when $P(A) > 0$, since there must be some $A$ cases to take a share of. In the commuter town, write $T$ for "takes the train" and $C$ for "cycles to the station". Then $P(T) = 0.4$ and $P(C \mid T) = 0.15$. The $0.15$ is a share of the train takers, not of the whole town. ### [The geometric distribution](https://math.wiki/geometric-distribution-formula/) **Summary:** The chance the first success lands on a given trial is a run of failures followed by one success. **Question:** A caller keeps dialing until someone pledges. How likely is it that the first pledge comes on the third call, or not until after the tenth, when the number of calls is not fixed in advance? ## Why this exists Rojin, Baran and Zhilwan volunteer at the phone bank of a community radio station. Each call they make ends in a pledge or it does not, and on their main list each call ends in a pledge with probability $0.2$, independently of every other call (made-up data, as is every number in this lesson). A caller works down the list until the first pledge comes in. The question here is not "how many pledges in $10$ calls". That count has a fixed number of trials and belongs to the binomial setting, where $n$ is set before the process starts. The question is "on which call does the first pledge come?" Now the number of calls is decided by the outcomes themselves: it could be $1$, or $3$, or $25$, with no largest value. A simulation, as in estimate-a-probability-from-trials, can estimate the chance of a long wait by running many pretend calling runs. This lesson gives the exact chance, from $p$ alone. Day 1 builds the setting and the probability that the first success lands on a given trial. Day 2 answers "more than", "at most" and "at least" questions with a shortcut, and separates a geometric count from a binomial one. ## Definition A chance process is a **geometric setting** when it is a run of trials in which - **binary outcomes:** each trial ends in success or failure; - **independent trials:** the outcome of one trial does not change the chances for any other; - **same chance of success:** every trial has the same probability of success, $p$, with $0 < p \le 1$; - **count until the first success:** the trials go on until the first success, and the variable is the number of trials it takes. ### [The law of large numbers and the gambler's fallacy](https://math.wiki/law-of-large-numbers/) **Summary:** As the number of trials grows, the relative frequency of an outcome settles near its true probability. **Question:** A fair coin has just landed heads five times running. Is tails now more likely? And if it is not, how do heads and tails still come out about even in the long run? ## Why this exists Experimental probability and what randomness means both showed a running share jumping about at first and then settling. Two questions were left open. Settling near what? And what does the settling say about the next trial? The second question has a famous wrong answer. After a long run of black on a wheel, someone bets on red because red is "due". After four daughters, a relative says the next child is surely a son. A lottery number not drawn for two years gets called "overdue". Each time the idea is that chance must balance its books. The settling itself has been tested directly. During the Second World War a mathematician held prisoner passed the time by tossing a coin $10{,}000$ times, recording every result, and published an account afterwards; the share of heads wandered early on and ended close to one half. This lesson says exactly what that settling is, and why it gives no support at all to "due". ## Definition Trials of a chance process are **independent** when the result of one has no effect on the chances in any other, as in independent events: spins of a wheel, tosses of a coin, draws from a bag with the counter put back each time. **The law of large numbers.** Repeat a chance process in independent trials under the same conditions, and let $p$ be the probability of one outcome. As the number of trials grows, the relative frequency of that outcome tends to get closer to $p$: with many trials, landing far from $p$ becomes very unlikely. The law is about the share, $\frac{\text{count}}{n}$, never the count itself. ### [The law of total probability](https://math.wiki/law-of-total-probability/) **Summary:** Splitting a population into non-overlapping groups and adding each group's share of the event. **Question:** An event can happen in several different situations, and its chance is different in each one. What is its chance overall, before you know which situation you are in? ## Why this exists The general multiplication rule gives the chance that two things both happen: the chance of the first, times the chance of the second given the first. Often, though, the question is about one event on its own. A canteen buys its bread from three bakeries, and each bakery sends a stale loaf now and then, some more often than others. What is the chance that a loaf in the canteen is stale? No single stale rate answers that, because there are three of them. The answer has to depend on the rates and also on how much of the bread comes from each bakery: a bakery with a high stale rate matters little if it supplies only a few loaves. The law of total probability is the rule that puts those two kinds of information together. It splits the situation into cases that do not overlap, finds the event's share inside each case, and adds. ## Definition A **partition** of the possible outcomes is a list of events $B_1, B_2, \ldots, B_n$ such that - no two of them can happen together (they are mutually exclusive), and - one of them always happens (together they cover every outcome). So every outcome lies in exactly one of the cases. The bakeries are a partition of the loaves: each loaf came from exactly one bakery. Any event $B$ and its complement "not $B$" form a partition with two cases. The **law of total probability** says that if $B_1, B_2, \ldots, B_n$ is a partition and every $P(B_i) > 0$, then for any event $A$, ### [Tree diagrams and organized lists](https://math.wiki/tree-diagram-probabilities-on-branches/) **Summary:** Writing a probability on every branch and multiplying along a path to get the chance of that whole path. **Question:** An experiment happens in stages, and the results need not be equally likely. How do you see every way it can turn out, and the chance of each way? ## When you reach for it Many experiments happen in stages. A customer picks an ice cream flavor and then a cone or a tub. A cyclist meets one traffic light and then another. A forecast covers Saturday and then Sunday. An outcome of the whole experiment is one result from each stage, in order, and before you can find a probability you need to see all of those outcomes, each written once. For two stages, an organized list or a table does that. A **tree diagram** does the same job as a picture that keeps working for three or more stages, where a table runs out of directions. It also does a second job that a list cannot: each branch of the tree has room for the probability of its own result, so the tree still gives the right answer when the outcomes are **not** equally likely, which is exactly where counting outcomes stops working. ## The procedure This lesson takes two days. Part 1 draws the tree and reads it as a list of outcomes. Part 2 writes a probability on every branch and uses those numbers to find chances. ### Part 1: drawing the tree and reading its paths An ice cream stand sells three flavors, mango (M), lemon (L) and pistachio (P), each served in a cone (C) or a tub (T). An order is one flavor followed by one way of serving it. **Start at a point and draw one branch for each result of the first stage.** Here that is three branches, with Mango, Lemon and Pistachio written at their ends. Between them they must cover every flavor on offer, each once, because exactly one of them is what happens at the first stage. ### [Union and intersection of events](https://math.wiki/union-and-intersection-of-events/) **Summary:** The union is everything in either event and the intersection is only what is in both. **Question:** A spinner is about to be spun. You win if it lands on an even number or on a multiple of 5. Which landings win, which win for both reasons at once, and how likely is each? ## Why this exists An event is a set of outcomes, and its probability, when the outcomes are equally likely, is a count of that set over the count of the whole sample space. That handles one description at a time: "the spinner lands on an even number". Real questions join descriptions. You win if the number is even *or* a multiple of $5$. A ticket is valid if it is for today *and* for this seat. A part passes if it is *not* cracked. Because events are sets, the joining words already have exact meanings waiting for them. "Or" is the union of the two sets, "and" is their intersection, and "not" is the complement inside the sample space. Once a compound description is turned into a set, finding its probability is the same counting as before. The work this lesson does is the turning: reading a sentence with "and", "or" and "not" in it and finding exactly which outcomes it names. ## Definition Let $A$ and $B$ be events in the same sample space $S$. **The event "$A$ or $B$"** is the set of outcomes that are in $A$, in $B$, or in both. It is the union $A \cup B$. It happens when at least one of $A$ and $B$ happens. **The event "$A$ and $B$"** is the set of outcomes that are in $A$ and also in $B$. It is the intersection $A \cap B$. It happens when both happen on the same trial. **The event "not $A$"** is the set of outcomes of $S$ that are not in $A$. It is the complement $A^{c}$. It happens exactly when $A$ does not. ### [Variance and standard deviation of a random variable](https://math.wiki/variance-of-a-discrete-random-variable/) **Summary:** Weighting each squared distance from the mean by its probability and adding the results. **Question:** Two random variables can have the same mean and still behave very differently. How do you measure how far a random variable's values typically land from its mean? ## When you reach for it A probability-distribution-table lists the values of a random variable $X$ and their probabilities, and the mean $\mu_X = \sum x \, P(x)$ says where $X$ balances in the long run. The mean says nothing about spread. A player who always takes $3$ strokes on a hole and a player who swings between a hole in one and $5$ strokes can share the mean $3$. To tell them apart you need a measure of how far the values typically fall from $\mu_X$: the **variance** $\sigma_X^2$ and its square root, the **standard deviation** $\sigma_X$. You reach for them whenever the question is about consistency, risk, or how much one result may differ from what is expected. All the settings below are made up: a mini golf course run by Josu, where Oihane and Garazi play. ## The procedure ### Day 1: the variance and standard deviation from a table Start from the distribution and its mean. For each value $x$, the **deviation** $x - \mu_X$ says how far that value is from the mean and on which side. The deviations cannot simply be averaged: weighted by probability they always add to zero, because $\sum (x - \mu_X) P(x) = \sum x P(x) - \mu_X \sum P(x)$, which is $\mu_X - \mu_X = 0$. Squaring each deviation makes every distance count as positive, exactly as for data in compute-standard-deviation-by-hand. Then each squared deviation is weighted by how likely its value is, and the results are added: ### [What randomness means](https://math.wiki/what-randomness-means/) **Summary:** An outcome is random when nobody can say which one will happen, even though the list of possible ones is known. **Question:** A coin is about to be tossed. Nobody can say how it will land, yet there is still a lot you can say. What exactly, and what not? ## Why this exists Some questions have answers you can work out before anything happens. What is $3 + 4$? What day comes after a Tuesday? When you press the light switch in a room with a working bulb, does the light come on? Think carefully and you know. Other questions cannot be settled in advance, however carefully you think. Which face of a coin lands up? Which number a die shows? Which name is pulled out of a hat? You can list what might happen, but not which of those things will. That second kind would be hopeless to study if it had no rules at all. It does have rules, of a strange sort. One toss of a coin tells you nothing ahead of time, but a thousand tosses behave in a way you can describe quite exactly. **Randomness** is the name for that mix: unpredictable one at a time, regular in bulk. Everything in probability is built on it, so it is worth being exact about what it means before any probability is ever written as a number. ## Definition One performance of an action, such as one toss of a coin, is a **trial**. What happens on a trial, such as heads, is its **outcome**. A **random process** (also called a **chance process**) is an action that can be repeated, under the same conditions each time, with these three properties. 1. **The possible outcomes are known in advance.** Before the trial you can list everything that might happen. 2. **A single outcome cannot be predicted.** No one can say for certain which outcome the next trial will give. 3. **Many trials show a steady pattern.** Over a long run of trials, the share of trials that give each outcome settles down. ## Statistics ### [Association in a two way table and segmented bar graphs](https://math.wiki/association-in-a-two-way-table/) **Summary:** Two variables are associated when the conditional shares differ from row to row. **Question:** People answer two questions, each with a few possible answers. How can you tell whether the answer to one goes with the answer to the other? ## Why this exists A two-way table records two categorical variables for the same people: whether each student has a library card, say, and whether each one reads for fun. The natural question is whether the two go together. Do card holders read for fun more often than everybody else? Counting cannot answer that: if $150$ students have no card and only $50$ have one, the no-card group can hold more readers simply because it holds more students. And "has a card" has no place on a number line, so there is no scatter plot to draw. What works is comparing the share of each group giving a particular answer, and a segmented bar graph puts those shares side by side. ## Definition Two categorical variables recorded on the same people (or things) are **associated** in a set of data when the conditional relative frequencies differ noticeably from one row of the table to the next, or from one column to the next. They are **not associated** in that data when the conditional relative frequencies are about the same in every row. A row's share for an answer is its count for that answer divided by the row total. How big a gap counts as "noticeable" is a judgment at this level (a point or two usually reads as none, tens of points as a clear one), which the test for independence later turns into a calculation. A **segmented bar graph**, also called a stacked $100\%$ bar graph, draws one bar for each group. Every bar is the same height, standing for $100\%$ of its group, and is cut into pieces whose heights are that group's conditional shares. The difference between two percentages is measured in **percentage points**: $70\%$ and $30\%$ are $40$ percentage points apart. ### [Bar graphs and pictographs](https://math.wiki/bar-graphs-and-pictographs/) **Summary:** One bar or picture per category. **Question:** You have counted how many of each kind there are. How can a picture show those counts so anyone sees the biggest at a glance and can still read the exact numbers? ## What this area is about A list of votes or a tally holds every count, but nobody can see the winner by looking at it. A picture graph and a bar graph both give each category its own row or bar, so a longer one means more. A picture graph shows the count with small pictures. A bar graph shows it with a solid bar and a number scale beside it. ## How these fit together **Picture graphs.** Picture graphs and their key draws one picture per thing and uses a key to say what one picture stands for. Scaled picture graphs lets one picture stand for several things, say $10$, so half a picture stands for half of that, $5$. **Bar graphs.** Reading a bar graph gets a count out of a bar by following its end across to the scale. Making bar graphs goes the other way, from a table of counts to bars of the right height with a labeled scale. ## A path through it Picture graphs and their key come before scaled picture graphs. Read a bar graph before you make one. The two pairs can be taken in either order, though most readers meet pictures first. Bars return in histograms, where they stand for number ranges, and in the segmented bar graphs of association in a two-way table. ### [Bias](https://math.wiki/bias-in-sampling/) **Summary:** Ways a sample systematically misses the population. **Question:** A survey's answer can be wrong in the same direction every time it is run. Where does that lean come from? ## What this area is about A sample never matches its population exactly, and chance error goes up one time and down the next. **Bias** is different: a way of collecting data that keeps favoring some answers, so its results lean the same way every time it is used. A bigger sample does not cure it. This area names where the lean comes from, so that from a study's description you can say who was missed or how answers were bent, and which way that pushes the result. ## How these fit together The three children are three places a lean can enter. **Who is chosen.** Convenience and voluntary response samples let the collector, or the respondents themselves, decide who is in the sample. **Who could be chosen, and who replies.** Undercoverage and nonresponse are the gaps a chance draw cannot close: people missing from the list it draws from, and people drawn who never answer, with the response rate $\frac{r}{n}$ to measure the second. **How they answer.** Response bias comes from the answers themselves: sensitive questions, wanting to please, poor memory, and leading or loaded wording. ## A path through it The three can be taken in any order; undercoverage reads best after random sampling, since it is about the gaps a random draw leaves. Finish here and you can judge how data were gathered before trusting them, which observational studies and experiments builds on. ### [Bias and variability of an estimator](https://math.wiki/unbiased-estimator/) **Summary:** A statistic whose sampling distribution is centered at the parameter it is meant to estimate. **Question:** Several formulas turn the same sample into an estimate. Which one aims at the truth, and which one lands close to it? ## Why this exists A city's rental scooters carry serial numbers $1, 2, 3, \dots, N$, and the company will not say how large $N$ is. Sophea writes down the numbers on five scooters picked at random: $$61, \quad 187, \quad 233, \quad 302, \quad 415$$ (made-up data). Each of three people turns the same five numbers into a different estimate of $N$. Vannak says $415$: the largest number seen. Bopha doubles the sample mean. Sophea adds to the largest number an allowance for the scooters above it. All three are reasonable, and they disagree. One sample cannot settle it, because nobody knows $N$. What can be judged is the *method*. Sampling variability showed that a statistic, recomputed over many samples, makes a pile with a center and a spread. Lay that pile next to the parameter: does it sit on the parameter or to one side, and how widely does it scatter? Those two questions, bias and variability, are how estimators are compared. ## Definition An **estimator** is a rule that turns a sample into a guess at a parameter, such as "take the largest number". The value it gives for one sample is an **estimate**, written with a hat: $\hat{N} = 415$. The estimator's **sampling distribution** is the pile of estimates from every possible random sample of the same size from the same population, each sample counted once (all samples equally likely). - The **bias** of an estimator is the mean of its sampling distribution minus the parameter. An estimator is **unbiased** when its bias is $0$: its sampling distribution is centered at the parameter. A negative bias means it runs low on average, a positive bias high. - The **variability** of an estimator is the spread of its sampling distribution, measured by that distribution's standard deviation. ### [Bivariate data](https://math.wiki/bivariate-data/) **Summary:** Two variables at once: is there a relationship, and how strong? **Question:** Two quantities are measured on the same individuals. How do you tell whether they move together, how tightly, and what a fitted line can be trusted to say? ## What this area is about Bivariate data pairs two numbers, $x$ and $y$, for each individual. This area turns a scatter plot into a number for strength, a line for prediction, and a check on whether that line was the right model. Each tool here describes association only: none, by itself, shows that $x$ causes $y$. ## How these fit together **Measuring the pattern.** Correlation gives one number, $r$, between $-1$ and $1$, for the direction and strength of a straight-line pattern. Lurking and confounding variables says why a strong $r$ still does not prove cause. **Fitting a line.** A line of best fit is a trend line drawn to follow the dots. Least squares regression gives exactly one line by rule, provided the $x$-values are not all equal. **Judging the line.** Residuals, $y - \hat{y}$, show how far off predictions are and whether a line was the right shape. Outliers and influential points asks whether one point is steering the slope. Reading regression output finds the line and its fit in a software table. **When the plot curves.** Transforming to linearity fits exponential models through $\log y$; log-log transforms fit power models through $\log x$ and $\log y$. ## A path through it Start with a line of best fit and correlation, in either order. Confounding variables can come any time after correlation. Least squares regression needs both, then outliers and residuals follow it. After residuals come reading output and transforming to linearity, and log-log transforms last. ### [Blocking: randomized block and matched pairs designs](https://math.wiki/blocking-in-an-experiment/) **Summary:** Grouping similar units first so a known source of variation does not hide the treatment effect. **Question:** Some units are known to differ before an experiment starts, like vans that crawl downtown and vans that cruise the highway. How can the treatments be compared so that this known difference stays out of the comparison? ## Why this exists At the end of completely-randomized-design, one shuffle sent $6$ of the $12$ guitar players to one group and $2$ to another. Nothing was unfair about that, but a variable known to matter was left to chance, and its variation swelled the noise. When such a variable is known before the experiment starts, it can be handled on purpose. Ilinca runs a made-up delivery fleet of $24$ vans and wants to compare three tire treads by fuel economy, in miles per gallon (mpg): the standard tread now on the vans (the control), a low-resistance "eco" tread, and a deep "grip" tread. Before she starts, she knows that route matters far more than tread: downtown vans stop at every corner and get about $14$ mpg, highway vans about $23$. If a shuffle put most highway vans on one tread, that tread would look good for a reason that has nothing to do with tires. Even with even luck, a $9$ mpg spread between routes can bury a tread effect of $1$ mpg. The fix is to group the vans by route first and randomize inside each group. Day 1 builds this **randomized block design**; Day 2 takes it to blocks of two, the **matched pairs design**. All data in this lesson are made up. The pieces it rests on: units and treatments say what gets what, a control-group gives the baseline, and random-assignment lets chance form the groups so that they differ only by chance. ## Definition A **block** is a group of experimental units that are alike in some variable expected to affect the response, formed **before** treatments are assigned. The blocking variable is not a treatment: the experimenter does not impose it and is not studying it, only keeping its effect out of the comparison. ### [Box plots](https://math.wiki/box-plots/) **Summary:** Five numbers, one picture: minimum, quartiles, maximum. **Question:** A list of numbers is hard to take in. How can five numbers summarizing it become one picture of where the data sit and how far they spread? ## What this area is about A box plot draws a numerical data set's five-number summary over a number line: a box from the lower quartile $Q_1$ to the upper quartile $Q_3$, a line at the median, and whiskers out toward the extremes. Each of the four pieces holds about a quarter of the values, so the picture shows where the middle half sits and how spread out each part is, without drawing every value. ## How these fit together **The basic picture.** Making and reading box plots is the procedure: find the quartiles, draw the box and whiskers on a scale, then read center and spread back off the plot, including what a box plot cannot show. **A variation.** Box plots that show outliers keep the same box but stop each whisker at the last value inside the fences $Q_1 - 1.5\,\text{IQR}$ and $Q_3 + 1.5\,\text{IQR}$, and plot each value beyond them as its own dot. **A use.** Comparing box plots puts two boxes on one shared scale to compare the center, spread and skew of two groups measured the same way. ## A path through it Quartiles and the interquartile range $\text{IQR} = Q_3 - Q_1$ come first. Then make and read a single box plot, and after that either the outlier version or the side-by-side comparison, in either order. Finish here and you can compare two distributions by their centers and spreads. ### [Box plots that show outliers](https://math.wiki/modified-box-plot/) **Summary:** A box plot whose whiskers stop at the last value inside the fence, with far values plotted alone. **Question:** One house on the street sold for far more than the others. How can a box plot show that house without stretching the picture of all the rest? ## When you reach for it A box plot runs its whiskers out to the smallest and largest values. When one value sits far from the rest, its whisker stretches all the way out to it, and the picture suggests a quarter of the data spread thinly along that whole stretch, when in fact one value sits out there alone. A **modified box plot**, a box plot that shows outliers, ends each whisker before such values and draws each of them as a dot of its own. Reach for it whenever a data set might hold values far from the rest: house prices, where one large house can sell for far more than its neighbors, or museum visitors, where one school trip can fill a quiet day. It draws with the 1.5 IQR rule. That rule sets two fences, $Q_1 - 1.5\,\text{IQR}$ below the box and $Q_3 + 1.5\,\text{IQR}$ above it, where $\text{IQR} = Q_3 - Q_1$ is the interquartile range; a value below the lower fence or above the upper fence is an outlier. Quartiles here, as in comparing box plots, are the medians of the lower and upper halves of the ordered data, with the median left out of both halves when the count is odd. All the data in this lesson are made up. ## The procedure **The box is unchanged.** Order the data and find its five-number summary as for any box plot. The box still runs from $Q_1$ to $Q_3$ with a line at the median; only the whiskers change. **The fences come from the box.** Find $\text{IQR} = Q_3 - Q_1$, multiply it by $1.5$, subtract the result from $Q_1$ and add it to $Q_3$. The fences are measured from the quartiles, not the median, because the box already holds the middle half of the data: the question is how far beyond the box a value has strayed, measured in box widths. The fences are worked out but not drawn on the finished plot. ### [Cautions about significance](https://math.wiki/statistical-vs-practical-significance/) **Summary:** A difference can be too small to matter and still be detected by a large enough sample. **Question:** A test on 400,000 website visitors says a new checkout button "significantly" raises sales. Does that mean the gain is big, or worth paying for, and what changes when twenty such tests are run at once? ## Why this exists Leticia sells acai bowls through a made-up online shop in Brazil. Her web developer Vinicius tries changes on the site, and her analyst Murilo tests them: each visitor is shown one version, chosen by random-assignment, so any real difference in buying can be put down to the version. All data in this lesson are made up. The reject-or-fail-to-reject lesson ended every test the same way: compare the p-value with $\alpha$, then say whether there is convincing evidence for the alternative. That decision answers exactly one question: could chance alone plausibly explain the data? It does not say how big the effect is, whether it is worth acting on, or how the answer changes when many tests are run. Those three gaps are the subject here. ## Definition A result is **statistically significant at level $\alpha$** when its p-value is at or below $\alpha$: if the null hypothesis were true, data at least this far from it would turn up with probability at most $\alpha$. It is a statement about chance, not about size or importance. A difference is **practically significant** when it is large enough to matter for the decision at hand: more sales than the change costs, a health benefit worth a side effect. It is judged from the size of the effect, estimated with a confidence interval, against a standard that comes from the context, never from the p-value. **Multiple comparisons** means running many tests on one study, for example comparing several versions each against the original, or one comparison on many outcomes. Each test at level $\alpha$ has chance $\alpha$ of rejecting a true null hypothesis, so across many tests some false rejections become likely. ### [Center and spread](https://math.wiki/center-and-spread/) **Summary:** One number for the middle, one number for how spread out. **Question:** A data set is a list of different numbers. Which single number says where it sits, which says how far it wanders, and how do you choose between them? ## What this area is about A list of numbers is summarized by a center (a typical value) and a spread (how far values wander from it). The right pair depends on the data's shape and on whether a few extreme values should count. ## How these fit together **Centers.** The mean is a fair share and a balance point; the median and mode need no adding. **Spreads.** Range and IQR cut sorted data at fixed places. Mean absolute deviation averages distances from the mean, and standard deviation squares them first. **Extreme values.** The 1.5 IQR rule is the procedure that flags outliers, and resistant measures is the result that says which summaries one extreme value drags along. **Describing and comparing.** Shape names the outline, describing a distribution puts shape, center and spread into sentences, then comparing two distributions and the gap between means in MADs set two groups side by side. **Changing scale.** Shifting and rescaling says how $\bar{x}$ and $s$ move, and standardizing puts any data set whose values are not all equal on a scale with mean $0$ and standard deviation $1$. ## A path through it Mean and median and mode come first, in either order. Range and IQR follow the median, then the 1.5 IQR rule, resistant measures, and shape. MAD can come any time after the mean; standard deviation needs MAD and resistant measures, then shifting, then standardizing. Shape leads to describing, then comparing two distributions, then comparing in MADs, which also needs MAD. ### [Checking whether data look normal](https://math.wiki/assess-normality-with-a-plot/) **Summary:** Comparing the data against what a normal model would give, by graph rather than by assumption. **Question:** A normal model makes everything easy to calculate, but only if the data really are close to normal. How can you tell, from the data alone, whether they are? ## When you reach for it The normal-distribution turns two numbers, a mean and a standard deviation, into a whole curve, and from the curve you can read the share of values in any range. That is a great deal of power from very little, and it is only honest when the data are close to normal. Many of the methods that come later, such as the conditions for inference about a mean, ask you to check exactly this before you lean on a normal model. The lesson on the normal distribution already rules some variables out by thinking about them: a mean far from the median, or a hard floor at $0$ only a standard deviation or so below the mean, means the model cannot fit. This lesson goes further and puts the data themselves next to the model. You reach for it whenever you hold a list of values and want to know whether a normal curve describes them, and you want the answer from the data, not from a hope. There are three checks, from rough to sharp: - the **shape** of a histogram (or dot plot), - the **share** of values within one and two standard deviations of the mean, compared with the $68\%$ and $95\%$ a normal curve gives, and - a **normal probability plot**, which puts every value next to the value a normal model would place at the same percentile. ## The procedure ### Looking at the shape A normal curve is one symmetric hump that falls away smoothly on both sides. So the first thing to look for in a histogram is one peak, roughly the same tail on each side, and no value standing apart from the rest. A histogram that is clearly skewed, has two peaks, or has an isolated bar far from the others already says no. ### [Chi-square tests](https://math.wiki/chi-square-tests/) **Summary:** Do observed counts match what a claim predicts? **Question:** Counts in several categories come back uneven. Are they too far from what a claim predicts to blame on chance? ## What this area is about A chi-square test checks counts of a categorical variable, or of two at once, against the counts a claim predicts. It gathers all the gaps between observed counts $O$ and expected counts $E$ into one number, $\chi^2 = \sum \frac{(O - E)^2}{E}$, and asks how often a value that large would arise by chance if the null hypothesis were true. The P-value is trustworthy only when the data come from random sampling, random assignment or a random process, every expected count is at least $5$, and a sample drawn without replacement is at most $10\%$ of its population. ## How these fit together **The tool.** Expected counts and the chi-square statistic builds $E$ from the claim, computes $\chi^2$, and introduces the chi-square curves, one for each number of degrees of freedom. **One variable.** The goodness of fit test asks whether one categorical variable follows a claimed set of proportions. **Two variables.** Tests for independence and homogeneity carry the same statistic to a two-way table. Independence asks whether two variables recorded on one random sample are associated in the population; homogeneity asks whether one variable has the same distribution in several populations or treatment groups. The design, not the table, says which. ## A path through it The statistic first, then goodness of fit, then independence and homogeneity. Finish here and you can add chi-square tests to the list when you choose an inference procedure. ### [Choosing a display for a question](https://math.wiki/choose-a-display/) **Summary:** Picking the graph that answers the question asked, given whether the data are categories or numbers. **Question:** Bar graph, circle graph, dot plot, stem-and-leaf plot, histogram or box plot. Which one answers the question you were asked about your data? ## When you reach for it You have data for a statistical-question and want to draw it. Six displays are on offer. None is wrong, but each shows some features of the data and hides others, so the choice rests on two things: what kind of data you have, and what you want to find out. Each display has its own lesson, linked below; this one teaches how to choose. All the data are made up. ## The procedure ### First, what kind of data Ask whether each value names a group or is a number (categorical-vs-numerical-data). The answer splits the displays in two, because a display for numbers is built on a number line, and categories have no place on one. Favorite colors cannot go on a dot plot, and $120$ potato masses would make a bar graph with a bar for nearly every potato. ### Displays for categories A bar graph gives each category a bar whose height is its count, all read against one scale. Bars side by side on one baseline are easy to compare, so a bar graph answers "which is most common?" and "how many more?". It shows counts, not shares. A circle graph gives each category a sector sized by its share of the whole, so it answers "what share?" at a glance, but hides the counts unless the total is given. It fits only when the parts make one whole: every case lands in exactly one category. ### Displays for numbers A dot plot puts one dot over each value on a number line. For a small set it shows everything: every value, the most common one, gaps, clusters and shape. It works best when values repeat on a short scale, so that dots stack, and it crowds as the set grows. ### [Choosing the right procedure](https://math.wiki/choose-an-inference-procedure/) **Summary:** Matching the number of samples, the kind of variable and the design to the procedure that fits. **Question:** A city department sends a question and a description of how its data were gathered. Before any arithmetic, which of the inference procedures fits, and how does the question itself say so? ## When you reach for it The research office of the city of Barcelona (a made-up office) receives study requests from city departments: parks, transit, libraries, markets. Laia, Arnau and Mireia read each request and the plan for gathering the data, and before anyone computes anything they must decide which procedure answers it. Each procedure has its own lesson; this lesson is about the choice that comes first. Each statistic estimates one parameter under one kind of design, so a correct calculation with the wrong procedure answers a question nobody asked. ## The procedure ### Day 1: the variable, the groups, estimate or claim **Name the parameter first.** Read the request for its population, its variable and the summary it wants. A categorical variable (satisfied or not, which metro line) is summarized by a proportion $p$, or by a whole distribution when there are more than two categories. A quantitative variable (a measured number with units) is summarized by a mean $\mu$. The question decides, not the raw data: a visit length in minutes is quantitative, but "the share of visits over an hour" is a proportion. **Count the groups.** One population or process: a one-sample procedure. Two populations, or two treatments formed by random assignment: a two-sample procedure, whose parameter is a difference, $p_1 - p_2$ or $\mu_1 - \mu_2$. A categorical variable with three or more categories, or compared across three or more groups, calls for chi-square: goodness of fit for one group against a claimed distribution, homogeneity for several groups. ### [Circle graphs](https://math.wiki/read-a-circle-graph/) **Summary:** Turning a wedge back into a percent and then into a count of the group. **Question:** A whole group splits into a few parts. How can one picture show at a glance what share of the whole each part is? ## When you reach for it Some questions about data ask "what share?" rather than "how many?" What part of all the tickets went to adults? Does sleep take more or less than a third of a day? A **circle graph**, also called a pie chart, answers that kind of question with a picture: the whole circle stands for the whole group, and each category gets a slice whose size shows its share. It suits categorical data that split one whole into a few parts, with every item in exactly one category. Reading a circle graph means turning a slice back into a fraction or a percent, and then into a count. Making one goes the other way: from counts to shares to angles at the center. ## The procedure ### What a slice means A **sector** is the part of a circle between two radii, shaped like a slice of pie. In a circle graph the sectors fill the circle without overlapping, one per category, and the rule that makes the picture honest is this: **each sector's share of the circle equals its category's share of the data.** The share of the circle is measured by the angle at the center. A full turn is $360^\circ$ (degrees), so a sector whose angle is $a$ degrees is $\frac{a}{360}$ of the circle. Its area is that same fraction of the circle's area, which is why the eye can judge it; sectors and their areas makes that exact later on. So one share can be named three ways, and they are always equal: $$\text{share} = \frac{\text{count}}{\text{total}} = \frac{\text{percent}}{100} = \frac{\text{angle}}{360^\circ}$$ ### [Cluster samples and choosing a design](https://math.wiki/cluster-sample/) **Summary:** Splitting the population into naturally occurring groups and taking every member of a few of them. **Question:** The people you want to ask live in a hundred and fifty buildings across a city, and nobody has a list of them. How can you still give every one of them a known, fair chance without knocking on doors all over town? ## When you reach for it A stratified random sample splits the population into groups and draws some members at random from every group. That needs a list of every member, and it sends interviewers into every group. Sometimes neither is practical. A city's housing office may list its $150$ apartment buildings, about $40$ households each, without listing the households, and an interviewer who must walk to each sampled door spends more time traveling than asking. Reach for a **cluster sample** when the population already comes in ready-made groups (buildings, city blocks, classrooms, shipping crates), you can list the groups even if you cannot list the members, and reaching a group is the expensive part. You choose a few whole groups at random and take everyone in them. ## The procedure **The definition.** Split the population into groups called **clusters**, so that every member is in exactly one cluster. Take a simple random sample of the clusters, then measure every member of each chosen cluster. The clusters are numbered and drawn exactly as individuals are drawn for a simple random sample: the chance device picks clusters, not people. **Each member's chance.** A member is in the sample exactly when its cluster is chosen. If $k$ of $K$ clusters are drawn, each cluster's chance is $\frac{k}{K}$, so each member's chance is $\frac{k}{K}$ too, in a large cluster or a small one. When clusters differ in size, the sample size is not fixed in advance: it is the total size of whichever clusters come up. ### [Collecting data with a survey](https://math.wiki/collect-data-with-a-survey/) **Summary:** Asking the same question of many people and recording every answer. **Question:** You want to know what a whole class thinks, so you ask them. What makes the question you ask, and the way you write down the answers, good enough to trust? ## When you reach for it A statistical question such as "Which dish on the cafeteria menu do the students in our class like best?" expects a different answer from each student, and the answers are in the students' heads. To get them you **survey** the students: ask each one the same question, in the same words, and write down every answer. Each person is an individual and each answer is one value of the variable asked about. The lesson takes two days. Part 1 plans the survey: the question and its choices, who to ask, and how to record and count the answers. Part 2 looks at wording, because the same people asked about the same thing in different words can give different data. ## The procedure ### Part 1: the question each person is asked The statistical question is about the group. The **survey question** is put to each person, so it is about that person: "Which dish on the cafeteria menu do you like best?" All the answers together answer the statistical question. A survey question is **open** when people answer in their own words, and **fixed-choice** when they pick from a list. Open answers must be sorted into groups before counting, and the sorter has to decide whether "noodles" and "pasta" are the same answer. Fixed choices are already the groups, so counting is quick, but only if **every possible answer fits exactly one choice**: - **No gaps.** Every truthful answer has a choice. A last choice such as "something else" catches what the list cannot name. - **No overlaps.** No answer fits two choices, or two people with the same answer can tick different boxes and the counts stop meaning anything definite. - **One answer each** ("choose one"), unless the question is meant to allow several. ### [Comparing box plots](https://math.wiki/compare-box-plots/) **Summary:** Putting two boxes on the same scale to compare center, spread and skew. **Question:** Two groups were measured the same way. How can one picture show which group runs higher, which is more spread out, and whether the difference is real? ## When you reach for it You have measured the same thing in two groups, such as the reaction times of two classes, and want to say how the groups differ. Two box plots drawn one above the other on one number line show at a glance where each group sits, how spread out it is, and which way it leans. As a reminder, a box plot draws a data set's five-number summary: a box from the lower quartile $Q_1$ to the upper quartile $Q_3$ with a line at the median, and whiskers out to the minimum and maximum. Each of the four pieces holds about a quarter of the values, and the box's width is the interquartile range, $\text{IQR} = Q_3 - Q_1$. Quartiles here are the medians of the lower and upper halves of the ordered data, with the median left out of both halves when the count is odd. ## The procedure ### Day 1: two box plots on one scale In the ruler-drop test one person lets go of a ruler and the other catches it; how far it fell gives the reaction time. Two classes of $12$ took the test, in milliseconds (ms), so smaller is quicker. Morning class: $165$, $172$, $180$, $188$, $194$, $198$, $204$, $210$, $218$, $226$, $248$, $270$. Afternoon class: $172$, $196$, $208$, $216$, $222$, $226$, $230$, $234$, $238$, $244$, $250$, $256$. With $12$ ordered values, the median is the mean of the $6$th and $7$th, and each quartile the mean of the $3$rd and $4$th values of its half. | | Morning | Afternoon | |---|:-:|:-:| | Minimum | $165$ | $172$ | | $Q_1$ | $184$ | $212$ | | Median | $201$ | $228$ | | $Q_3$ | $222$ | $241$ | | Maximum | $270$ | $256$ | ### [Comparing two distributions](https://math.wiki/compare-two-distributions/) **Summary:** Saying how two groups differ in center, spread and shape, on the same scale, with units. **Question:** Two groups were measured in the same way. How do you say which runs higher, which is more spread out, and whether the difference is real or just the luck of the sample? ## When you reach for it You have measured the same thing in two groups: hours of sleep at two ages, lengths of fish in two lakes, prices of tickets at two theaters. The question is about the groups, not about any one value: which runs higher, which is more spread out, whether the shapes differ, and whether a difference between the samples says anything about the populations behind them. Day 1 describes and compares; Day 2 chooses the measures and judges the difference. The tools are all met before: dot plots and histograms; the mean and the median for center; the range and the interquartile range for spread; and shape. Describing one distribution this way is describe-a-distribution-in-context. Two box plots on one scale compare five-number summaries; here the plots show every value, or every bin. ## The procedure ### Day 1: describing two distributions side by side In a made-up town, random samples of $15$ people aged $11$ to $13$ and $15$ people aged $16$ to $18$ gave their hours of sleep on a school night, to the nearest half hour. Aged $11$ to $13$: $7.5$, $8$, $8$, $8.5$, $8.5$, $8.5$, $9$, $9$, $9$, $9.5$, $9.5$, $9.5$, $10$, $10$, $10.5$ Aged $16$ to $18$: $5.5$, $6$, $6.5$, $6.5$, $7$, $7$, $7.5$, $7.5$, $7.5$, $8$, $8$, $8.5$, $8.5$, $9$, $9.5$ **Put both on one scale.** Draw the two dot plots one above the other, over the same number line. The eye compares positions, so a position must mean the same number in both plots. Give each plot its own number line, stretched to fit its own data, and both clusters sit in the middle of their plots and look alike. On one scale the shift is the first thing you see. ### [Comparing two populations: the gap between means in MADs](https://math.wiki/compare-populations-with-mad/) **Summary:** Measure how far apart two distributions are by expressing the difference of their means as a multiple of their mean absolute deviation. **Question:** The means of two groups differ by 4 grams. Is that a big difference or a small one? ## Why this exists Two teams do a standing long jump, and one team's mean jump is $20$ cm longer. Is that a lot? The $20$ cm alone cannot say. If every jumper lands within a few centimeters of the team's mean, a $20$ cm gap splits the teams cleanly apart. If each team's jumps scatter across a meter, the same $20$ cm is lost in the scatter. And if two tomato varieties differ by $4$ g in mean mass, is that a bigger difference than the $20$ cm? Centimeters and grams cannot be compared at all. One fix answers both. Measure the gap between the means with a ruler made from the data itself: the spread. This lesson uses the mean absolute deviation as the ruler and turns "the means differ by $20$ cm" into "the means differ by $2.5$ MADs", a plain number that says how much the two groups overlap. It builds on comparing two distributions, where center and spread were described side by side. ## Definition Two measures from earlier lessons do the work, recalled here only. - The mean of a data set is the sum of its values divided by how many values there are. - The mean absolute deviation, or $\text{MAD}$, is the mean of the distances from each value to the mean: how far a typical value sits from the middle. Take two groups whose MADs are about the same. The **gap between the means in MADs** is the distance between the two means divided by a MAD: $$\text{gap} = \frac{\text{larger mean} - \text{smaller mean}}{\text{MAD}}$$ ### [Comparing two treatments by re-randomizing](https://math.wiki/compare-treatments-by-rerandomizing/) **Summary:** Deciding whether a difference between two treatment groups is bigger than chance assignment alone would produce, by shuffling the data many times. **Question:** The group that got the treatment did better. Could the luck of who landed in which group have done that on its own? ## When you reach for it A randomized experiment ends with two groups and two summaries, and the summaries differ. The group given the treatment recalled more words, or lasted longer, or left a tip more often. Experiments with random assignment can show cause, but only once one doubt is out of the way: the groups were formed by chance, and chance alone never splits a set of subjects perfectly evenly. Some difference would appear even if the treatment did nothing at all. So the question is not "do the groups differ?" They always do. It is: **is this difference bigger than the luck of the assignment would usually produce?** Re-randomizing answers it by doing the assignment again, many times, on the same results, and watching how big the differences come out when the treatment is given no part to play. It is the same move as simulating a sampling distribution, with one change: there the chance was in which members were sampled, here it is in which subjects were put in which group. Reach for it whenever an experiment used random assignment to two treatments and you want to judge a difference in means or in proportions between the groups. All the experiments and numbers in this lesson are made up. ## The procedure **Say what "no effect" would mean.** Suppose the treatment makes no difference at all. Then every subject would have produced exactly the same result whichever group chance had put them in. The $20$ numbers recorded would be the same $20$ numbers; only the labels on them, "treatment" and "control", came from the shuffle. This supposition, that the treatment does nothing, is the model being tested. The rest of the procedure asks what the data would look like if it were true. ### [Completely randomized design and replication](https://math.wiki/completely-randomized-design/) **Summary:** Assigning every unit to a treatment purely at random, with no grouping first. **Question:** You have units, treatments and a way to assign by chance. How do you build a whole experiment from them, and how many units does each treatment need before a real difference shows through the noise? ## When you reach for it You want to know whether some treatments cause a difference in a response, you can impose the treatments yourself, and you know of nothing about the units that should be handled first. The pieces are already in place: the units and treatments, a control group to compare against, and random-assignment, which uses chance so that the groups differ only by chance. A **completely randomized design** is the plainest way to put these together into one experiment, and **replication** is what lets its result be judged. Khoa teaches beginners at a made-up music school and wants to know how adults best learn to switch between ukulele chords. All data in this lesson are made up. ## The procedure **The design.** In a completely randomized design, all the units are assigned to the treatments by chance alone, with no grouping of the units beforehand. Usually the group sizes are planned and one chance process makes every way of splitting the units into groups of those sizes equally likely; letting chance decide each unit on its own, as a die roll does, also qualifies, though the group sizes then come out unequal (Example 4). Each unit then receives its treatment, the response is measured, and the groups are compared. Written out, a design names five things, and each has a reason. - **The units**, so the reader knows what was compared. Each unit must be able to receive any treatment, and no unit's treatment may spill onto another. - **The treatments**, including the control. Without a control there is nothing to measure a gain against; with it, every other group is judged by its difference from the control. - **The random assignment of all units at once.** One shuffle of the whole list forms every group, so no trait of a unit can steer it toward a treatment. How to carry out the shuffle, and why it balances the groups, is in random-assignment. - **The response**, said precisely: what is measured, when, and by whom. Everything else that could affect the response is kept the same for every unit, so it cannot differ between groups. - **The comparison**, usually of group means: each treatment's mean against the control's mean. ### [Computing a standard deviation, sample and population](https://math.wiki/compute-standard-deviation-by-hand/) **Summary:** Working through the distances from the mean, squaring, averaging and taking the root. **Question:** The variance measures spread in squared units. How do you get a spread in the data's own units, and does it matter whether the numbers are the whole group or only a sample of it? ## When you reach for it You have a list of numbers and want one number for how far they typically sit from their mean, in the data's own unit: pages, years, hours. The variance is the mean of the squared deviations, but it is in squared units. Its square root, the **standard deviation** (standard-deviation), is back in the data's unit, and it is the measure of spread used throughout statistics. There are two versions. When the numbers are the whole group you care about, a **population**, divide the sum of the squared deviations by the count, as the variance lesson did. When the numbers are a **sample** standing in for a bigger group, divide by one less than the count. This lesson runs over two days: Day 1 is the population standard deviation, Day 2 the sample one and how to choose between them. ## The procedure ### Day 1: the population standard deviation For a whole population of $N$ values, the mean is written $\mu$ (mu) and the standard deviation $\sigma$ (sigma). Greek letters mark numbers that describe a whole population, as parameter-vs-statistic explains. The standard deviation is the square root of the variance: $$\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}$$ Read it from the inside out. $x - \mu$ is one value's deviation from the mean. Squaring removes its sign. $\sum$, a capital sigma, means "add this up over every value". Dividing by $N$ gives the variance $\sigma^2$, and the square root brings the unit back. ### [Conditional relative frequencies](https://math.wiki/conditional-relative-frequency/) **Summary:** The share of one row or one column that lands in a given cell. **Question:** Of the museum visitors who took a guided tour, what share bought something at the shop? And is that the same number as the share of shoppers who took a tour? ## When you reach for it A two-way table counts one group of people or things by two questions at once. A made-up museum asked $80$ visitors on their way out whether they took a guided tour and whether they bought something at the shop. | Table 1 | bought something | bought nothing | Total | |---|:-:|:-:|:-:| | took a tour | $24$ | $8$ | $32$ | | no tour | $6$ | $42$ | $48$ | | Total | $30$ | $50$ | $80$ | The four inner counts are the joint frequencies and the totals around the edge are the marginal frequencies (marginal-frequencies). Now ask: "of the visitors who took a tour, what share bought something?" The answer is not $24$ out of $80$. The question's first words narrow the whole down to the $32$ tour visitors, and the answer is a share of them. You reach for a conditional relative frequency whenever a question names a group and asks what share of that group has some other property, and whenever groups of different sizes have to be compared. ## The procedure ### Part 1: a cell over its row total or its column total A **conditional relative frequency** is the count in one cell of a two-way table divided by the total of the row, or of the column, that the question picks out. That row or column is the condition: the group the question has narrowed down to. $$\text{share} = \frac{\text{count in the cell}}{\text{total of the group}}$$ This is the same part over whole as any share (find-the-percent-one-number-is-of-another); what is new is only that the whole is a row or column total rather than the grand total. For the tour visitors, the group is the "took a tour" row, total $32$, and those who also bought something are the cell $24$: ### [Conditions for one mean](https://math.wiki/conditions-for-one-mean-inference/) **Summary:** A random sample, small against the population, from a population that is normal or a sample large enough. **Question:** An interval or a test for a mean will run on any list of numbers. What must be true of the data before its answer means what it says? ## Why this exists Jiwoo runs the weighing shed at a ginseng farm in Korea (made-up data throughout this lesson). One field holds about $5{,}000$ four-year roots, and she wants $\mu$, the mean fresh weight of a root there. She digs up a simple random sample of $12$ roots and weighs each, in grams: $$38,\ 42,\ 45,\ 47,\ 49,\ 51,\ 52,\ 55,\ 58,\ 61,\ 66,\ 74$$ Their mean is $\bar{x} \approx 53.17$ grams and their standard deviation $s_x \approx 10.31$ grams. The next lessons turn numbers like these into a confidence interval and a significance test for $\mu$. Both rest on the t-distribution: the standardized mean $$t = \frac{\bar{x} - \mu}{s_x/\sqrt{n}}$$ follows a $t$ distribution with $n - 1$ degrees of freedom when the values are independent draws from a normal population. The arithmetic runs just as smoothly when that is false. The conditions for one mean say when it is close enough to true for the stated confidence level to be trusted. ## Definition Inference about a population mean $\mu$ with $t$ procedures needs three **conditions**. - **Random.** The data come from a random sample of the population, or from a randomized experiment, one that uses random-assignment. - **Ten percent.** When the sample is drawn without replacement, $n \le 0.10\,N$. - **Normal or large sample.** At least one holds: the population is known to be close to normal; or $n \ge 30$, the usual guide; or, for a smaller sample, a plot of the data shows no strong skew and no outliers. ### [Conditions for one proportion](https://math.wiki/conditions-for-one-proportion-inference/) **Summary:** A random sample, a sample small against the population, and enough expected successes and failures. **Question:** A sample gave a share. Before that share is stretched into a claim about the whole population, what must be true of how the sample was taken? ## Why this exists A school district has $4000$ high school students (made-up data throughout this lesson). Adeola, who works in the district office, wants $p$, the share of them with a part-time job. She takes a simple-random-sample of $200$ students and finds $54$ with a job, so $\hat{p} = \frac{54}{200} = 0.27$. The last few lessons turn one number like this into a range of plausible values for $p$: the standard-error, the critical-value $z^*$, the margin-of-error and the confidence-level it promises. All of them rest on the sampling-distribution-of-a-sample-proportion: $\hat{p}$ is centered at $p$, has standard deviation $\sqrt{\frac{p(1-p)}{n}}$, and is close to normal in shape. Those three facts are not automatic. Each holds only for data gathered in a certain way, and the arithmetic of an interval runs just as smoothly when they fail. The conditions for one proportion say whether the arithmetic means anything. ## Definition Inference about a population proportion $p$ from a sample proportion $\hat{p}$, by the normal methods of this course, needs three **conditions**. - **Random.** The data come from a random sample of the population, or from a randomized experiment, one that uses random-assignment of subjects to treatments. - **Ten percent.** When the sample is drawn without replacement, it is at most a tenth of the population: $n \le 0.10\,N$. - **Large counts.** The sample holds at least $10$ successes and at least $10$ failures. For a confidence interval this is checked with the sample's own counts: ### [Conditions for regression inference](https://math.wiki/conditions-for-regression-inference/) **Summary:** A true linear relationship, independent observations, equal spread of residuals, and residuals near normal. **Question:** A sample's slope comes with a standard error and a t distribution. What has to be true of the data before those numbers can be believed? ## Why this exists Pinar studies tea gardens near Rize, on the Black Sea coast of Turkey. For a random sample of plots she records the nitrogen fertilizer each plot received, $x$ kilograms per hectare, and its leaf yield, $y$ kilograms per plot, and fits the least-squares line $\hat{y} = a + bx$. Her slope $b$ is a statistic; what she wants is $\beta$, the slope for every plot the estate could grow. sampling-distribution-of-the-slope described how $b$ varies from sample to sample: centered on $\beta$, with standard deviation $\sigma/(s_x\sqrt{n - 1})$, normal in shape. Each of those facts was proved from a model of the population. If the model is wrong for Pinar's gardens, the standard error and the $t$ distribution built on it describe some other population, not hers. This lesson names the conditions that model requires and shows how each is checked on the data, mostly with a residual plot. It plays the part that conditions-for-one-mean-inference plays for a mean. ## Definition **The regression model.** For each value of $x$ the responses have mean $\mu_y = \alpha + \beta x$ and standard deviation $\sigma$ about that line; $\alpha$, $\beta$ and $\sigma$ are parameters. The sample estimates them with $a$, $b$ and $s$, the standard deviation of the residuals $y - \hat{y}$. **The conditions for inference about a slope** are five, and their first letters spell LINER. - **Linear.** The true mean of $y$ is a straight-line function of $x$. Checked by a scatter plot that shows no curve and a residual plot with no curved pattern. - **Independent.** The observations are independent of one another. When cases are sampled without replacement, the sample is at most $10\%$ of the population (the ten-percent-condition). - **Normal.** At each $x$ the responses vary normally about the line. Checked by a histogram or dot plot of the residuals that shows no strong skew and no outliers. - **Equal spread.** The standard deviation $\sigma$ is the same at every $x$. Checked by a residual plot whose vertical spread does not fan out or narrow. - **Random.** The data come from a random sample or a randomized experiment (random-sampling, random-assignment). ### [Confidence intervals](https://math.wiki/confidence-intervals/) **Summary:** A range of plausible values for the population parameter, and what the percent really means. **Question:** One random sample gives one estimate, almost surely a little off. How do you report a range that honestly says how far off it might be? ## What this area is about A confidence interval reports an estimate from a random sample together with a distance: $\text{point estimate} \pm \text{margin of error}$. The values inside are the plausible values of the parameter in light of that sample. Its percent, the confidence level, belongs to the method, not to one computed interval: it is the share of all possible random samples for which the method would capture the parameter. The margin of error covers only the chance variation of random sampling, not bias from a poorly drawn sample. ## How these fit together **Building blocks.** Standard error estimates the spread of a statistic's sampling distribution from the sample itself, such as $\text{SE}_{\hat{p}} = \sqrt{\hat{p}(1 - \hat{p})/n}$. Point estimates and the margin of error multiply it: $\text{ME} = (\text{multiplier}) \times \text{SE}$. **Meaning.** Confidence level and interpreting an interval says what the percent is a statement about, and what a report may honestly claim from one interval. **Precision.** Critical value gives the exact multiplier $z^*$ for any chosen level, when the statistic's sampling distribution is close to normal. ## A path through it Standard error, then margin of error, then confidence level, then critical value. Finish here and you can build a one-sample $z$ interval for a proportion and a one-sample $t$ interval for a mean. ### [Confidence level and interpreting an interval](https://math.wiki/confidence-level/) **Summary:** The share of all possible samples for which this method would capture the true parameter. **Question:** A computed interval either contains the true value or it does not. So what is the "95%" in a 95% confidence interval a statement about, and what can a report honestly say from one interval? ## Why this exists Kaisa works for a state wildlife agency that tracks invasive zebra mussels. The state has $12{,}000$ lakes, far too many to inspect, so the agency inspects a random sample of $250$ and finds mussels in $60$ of them (made-up data). As in margin-of-error, the estimate is the sample proportion give or take about $2$ standard errors: $$\hat{p} = \frac{60}{250} = 0.24, \qquad 0.24 \pm 0.054$$ roughly $0.186$ to $0.294$, and the agency's report calls this a $95\%$ confidence interval. That percentage hides a real puzzle. The true share $p$ of the state's lakes with mussels is one fixed number. The interval is two fixed numbers. Either $p$ lies between them or it does not, and nothing about that is left to chance. So "$95\%$" cannot be the chance that this particular interval is right, the way a forecast gives a chance of rain tomorrow. It is a statement about something else, and a report that puts it in the wrong place claims more than the data support. Part 1 of this lesson pins down what the confidence level measures. Part 2 turns to what one interval lets the agency say and decide. ## Definition A **confidence interval** for a parameter is a range of values computed from a random sample by a fixed method. In this lesson the method is $$\hat{p} \pm 2\,\text{SE}_{\hat{p}}, \qquad \text{SE}_{\hat{p}} = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$$ with the standard-error built from the sample. ### [Control groups, placebos and blinding](https://math.wiki/control-group/) **Summary:** A group receiving no treatment or the usual one, to compare the others against. **Question:** People in a trial get better whether or not the new treatment works. How can an experiment tell how much of the improvement the treatment caused? ## Why this exists A sore throat almost always clears on its own within a week or so. Suppose $90$ people suck a new throat lozenge and their throats clear in $4.4$ days on average. Did the lozenge do that? The number cannot say. Their throats would have cleared anyway, and people who believe they have been treated often feel better sooner for that reason alone. A result about one group has nothing to be measured against. An experiment imposes treatments on experimental units (the subjects, here) and compares their responses. This lesson is about what the treatment group is compared WITH, and about keeping that comparison honest: the **control group**, the **placebo** it often receives, and **blinding**, which keeps knowledge of who got what from leaking into the results. Almaz, Rahel and Kalkidan run a made-up trial. They recruit $180$ volunteers who have just come down with a sore throat. By chance, $90$ get the new lozenge and $90$ get a placebo lozenge that looks, tastes and is packed exactly like it but contains no active ingredient. Each volunteer sucks one lozenge four times a day, and the response is the number of days until the sore throat clears. All data in this lesson are made up. ## Definition **Part 1.** A **control group** is a group in an experiment that does not receive the treatment being tested, so that its response gives the baseline the treatment group is compared against. It can receive: ### [Convenience and voluntary response samples](https://math.wiki/convenience-sample/) **Summary:** Taking whoever is easiest to reach, which usually leaves out whole parts of the population. **Question:** The people easiest to ask are right in front of you, and some people are keen to answer without being asked. Why can their answers tell you the wrong thing about everyone? ## Why this exists To learn about a large group you usually look at part of it, a sample from the population. Choosing that part well takes work, and two shortcuts save it: ask whoever is near you, or put out an open invitation and take whoever answers. Both are cheap, and both can be badly wrong, because the people near you, or the people who bother to answer, are often unlike the rest in the very thing you are asking about. This lesson names the two shortcuts and shows how to tell, from a study's description, who was left out or counted too often, and which way that pushes the result. ## Definition A **convenience sample** is made of the individuals easiest for the person collecting the data to reach: those nearby, passing by, or already at hand. The collector decides who is in it, and decides by what is easy. A **voluntary response sample** is made of the individuals who choose to answer an open invitation: they phone in, click, text, or post a form back. Nobody is picked; each respondent picks himself or herself, so it is also called a **self-selected sample**. A sample **represents** its population when its mix matches the population's mix in whatever matters for the question. A way of choosing is **biased** when it keeps favoring some parts of the population over others, so its results lean the same way time after time (the general idea is bias). The lean that comes from letting people choose whether to answer is called **voluntary response bias**. ### [Correlation](https://math.wiki/correlation/) **Summary:** Direction and strength of a linear relationship, and the number r. **Question:** Two scatter plots both climb from left to right. How can one number say which cloud hugs a straight line more tightly? ## What this area is about Looking at a scatter plot, people agree quickly on direction and form, but not on strength: one calls a cloud "fairly tight", another "quite loose". Correlation settles that with a single number, $r$, which runs from $-1$ to $1$. Its sign gives the direction of a straight-line pattern and its size gives the strength. It measures only straight-line patterns, and it says nothing about cause. ## How these fit together **The definition.** The correlation coefficient says what $r$ is, what values near $1$, $0$ and $-1$ look like, and that it always lies between $-1$ and $1$. **Results.** Properties of the correlation coefficient says what does not move $r$ (changing units, swapping the two variables) and what does: a single unusual point can move it a lot. **The procedure.** Correlation from standardized values opens the box: $r$ is the sum of the products of the two variables' $z$-scores, divided by $n - 1$, and that formula shows why it can never pass $1$ or $-1$. ## A path through it Start with the correlation coefficient itself. Then take its properties and the $z$-score computation, in either order. With $r$ in hand you can build the least-squares line, whose slope is $r$ times a ratio of spreads, measure fit with $r^2$, and learn why correlation is not causation. ### [Correlation from standardized values](https://math.wiki/r-from-standardized-values/) **Summary:** Finding r as the average product of the two variables' z-scores. **Question:** Technology reports a correlation r for a cloud of points. Where does that number come from, and why can it never pass 1 or -1? ## When you reach for it The correlation coefficient $r$ measures how tightly the points of a scatter plot cluster about a straight line, and technology reports it in one keystroke. This lesson opens the box. You reach for the formula when you want to compute $r$ by hand for a small data set, when you want to see which points push $r$ up and which pull it down, and when you want to know *why* $r$ has no units, why it does not care which variable is called $x$, and why it never leaves the interval from $-1$ to $1$. Each of those facts is visible in the formula and nowhere else. The building block is the z-score: how many standard deviations a value sits from its mean. For paired data $(x, y)$ each variable gets its own, $$z_x = \frac{x - \bar{x}}{s_x}, \qquad z_y = \frac{y - \bar{y}}{s_y}.$$ ## The procedure For $n$ points, the correlation is $$r = \frac{1}{n-1} \sum z_x z_y .$$ In words: standardize each variable, multiply the two z-scores of each point, add the products, and divide by $n - 1$. Each part has a reason. **Why z-scores.** A z-score is a distance from the mean measured in standard deviations, so it has no units: centimeters divided by centimeters. Both variables now sit on one common scale with mean $0$ and standard deviation $1$, and a height and a weight can be multiplied without the answer depending on whether heights were in centimeters or inches. **Why products.** The point $(\bar{x}, \bar{y})$ is the center of the cloud. Lines through it split the plot into four boxes (Figure 1). A point in the top-right box has both values above their means, so $z_x > 0$ and $z_y > 0$ and the product is positive. A point in the bottom-left box has both below, two negatives, and the product is again positive. A point in the top-left or bottom-right box has one z-score of each sign, and its product is negative. A point on either center line has a z-score of $0$ and contributes nothing. ### [Correlation is not causation: lurking and confounding variables](https://math.wiki/confounding-variables/) **Summary:** When two possible causes change together so their effects cannot be told apart. **Question:** Two things rise and fall together. When does that mean one causes the other, and when is something else pulling both? ## Why this exists A scatter plot or an association in a two-way table shows that two variables go together, and the correlation coefficient $r$ says how tightly a straight-line pattern holds. The next question is nearly always "so does one cause the other?" A seaside shop sells more sunscreen in the weeks it sells more ice cream. Fires with bigger crews do more damage. Should the shop push ice cream to sell sunscreen, and should the fire chief send fewer firefighters? The patterns in those data are real. The conclusions are not. This lesson is about the gap between "these go together" and "this one makes that one happen", and about reading a claim and saying exactly what its data can and cannot support. ## Definition Two variables are **associated** when knowing the value of one tells you something about the value of the other; when the pattern is a straight line, they are **correlated**. One variable **causes** a change in another when changing the first, with everything else left as it was, changes the second. **Correlation is not causation.** An association, however strong, does not by itself show that either variable causes the other. Four kinds of explanation produce the same pattern between an $x$ and a $y$, and more than one can be at work at once: | Explanation | What happens | |---|---| | Cause | changing $x$ changes $y$ | | Reverse direction | changing $y$ changes $x$ | | A third variable | something else changes both $x$ and $y$ | | Coincidence | the pattern belongs only to this small or hand-picked set of data | ### [Counting each group and ordering groups by how many](https://math.wiki/count-and-order-categories/) **Summary:** After sorting, each group is counted and the groups are put in order from most to fewest. **Question:** Once a pile has been sorted into groups, how does a child find out which group has the most and which has the fewest? ## When you reach for it After a child has sorted a pile into groups, the next question is "how many in each group, and which group has the most?" Each group is counted, and then the groups are lined up side by side (cube towers or rows matched one to one) so that most, fewest and "the same" can be seen as well as counted. This lesson puts three or four groups in order; comparing just two groups or two numbers is compare-quantities and compare-numbers-to-10. Start each day with groups that are already sorted, and keep every group at ten or fewer. ## Day 1: Count each group **You need:** four small boxes or paper plates, and a mixed handful of small things already sorted into them, one kind per box (dark buttons, light buttons, square tiles, triangle blocks). Point to one box and ask, "How many are in this box?" Let the child count by moving each piece out of the box as it gets its word, the way they count one word per thing. Then put a number card (or a sticky note with the number) beside the box. Do this for every box. Ask, "Which box has the most? How do you know?" A child who points to the box with the biggest number is using the counts. Keep the number cards out; tomorrow you will check them by lining the groups up. **Practice.** Use the picture. 1. How many are in box A? *Answer:* $4$ dark counters. 2. How many are in box B? *Answer:* $6$ open counters. 3. How many are in box C? *Answer:* $3$ squares. 4. How many are in box D? *Answer:* $7$ triangles. 5. Which box has the most? *Answer:* D, with $7$ triangles, the biggest of the four numbers. ### [Critical value](https://math.wiki/critical-value/) **Summary:** The multiplier from the reference distribution that turns a standard error into a margin of error. **Question:** A confidence interval reaches out some number of standard errors from the estimate. For a chosen level, such as exactly 95% or 90% or 98%, how many standard errors is that, and where does the number come from? ## Why this exists Maricel runs a community center that is thinking of starting a weekend Tagalog class. She asks a random sample of $400$ residents of the neighborhood, and $252$ say they are interested (made-up data). The estimate and its standard-error are $$\begin{aligned} \hat{p} &= \frac{252}{400} = 0.63 \\ \text{SE}_{\hat{p}} &= \sqrt{\frac{0.63 \times 0.37}{400}} \approx 0.0241 \end{aligned}$$ In confidence-level the interval reached out "about $2$ standard errors" for "about $95\%$", and $1$ and $3$ standard errors gave about $68\%$ and $99.7\%$. Those are rounded readings of the normal curve, and they offer only three levels. Maricel's board wants exactly $95\%$, and a grant form asks for $90\%$. She needs the exact multiplier for any level she chooses. That multiplier is the critical value. ## Definition For a statistic whose sampling distribution is close to normal, the **critical value** $z^*$ for confidence level $C$ is the positive number with area $C$ under the standard normal curve between $-z^*$ and $z^*$. The area left outside, $1 - C$, splits equally into two tails of $\frac{1 - C}{2}$ each, because the curve is symmetric about $0$. So $z^*$ is the point with area $$\Phi(z^*) = 1 - \frac{1 - C}{2} = \frac{1 + C}{2}$$ to its left. It is found by reading a normal table backwards, as in area-under-a-normal-curve, or with a calculator's inverse normal command (often called invNorm) applied to $\frac{1 + C}{2}$ with mean $0$ and standard deviation $1$. ### [Cumulative frequency graphs and percentiles](https://math.wiki/cumulative-frequency-graph/) **Summary:** A rising graph showing how many values fall at or below each point. **Question:** Eighty waits between geyser eruptions are sorted into classes ten minutes wide. Below what wait do half of them fall, and what share of them last less than 95 minutes? ## When you reach for it A relative-frequency-histogram shows how many values fall in each class. Many questions are about everything below a point instead: what share of waits are shorter than $95$ minutes, or below what wait the shortest half fall. A histogram answers them only by adding bars, and only at class boundaries. A **cumulative frequency graph** (also called an **ogive**, said "OH-jive") does the adding once and draws the result as a rising line, so any such question becomes a reading off one curve. The graph plots, above each value $x$, the number of data values less than $x$. Plotted as a count it is a cumulative frequency graph; divided by the total $n$, so that its scale runs from $0$ to $100$ percent, it is a **cumulative relative frequency graph**. The two have the same shape, and one figure can carry both scales. You reach for it in two directions. From a value to a percent: what is the **percentile rank** of a $65$ minute wait, the percent of waits at or below it? From a percent to a value: what wait is the $90$th **percentile**, the value with $90$ percent of the data at or below it? The second direction gives the median and the quartiles, which are the $50$th, $25$th and $75$th percentiles. find-the-value-at-a-percentile works with both ideas on a list of values; this lesson works when only the grouped counts are known. All the data in this lesson are made up. Nino, Giorgi and Ketevan are timing a geyser. ### [Data and variables](https://math.wiki/data-basics/) **Summary:** Recorded facts about a group of individuals, where a variable is one thing recorded about each individual. **Question:** Someone hands you a pile of answers about a group. What has to happen before those answers can tell you anything? ## What this area is about Data are facts recorded about a group of individuals, and a variable is one thing recorded about each of them. Before any graph or average, data have to be gathered from a question worth asking, labeled so each value means something, and sorted and counted. This area covers those first steps, from sorting buttons to a table of shares. ## How these fit together **Sorting and counting.** Sorting objects into categories puts a mixed pile into groups by one rule; counting and ordering categories finds which group has the most and which the fewest. **Asking.** A statistical question expects its answers to vary; collecting data with a survey asks it of each person in fair words and records every answer. **Reading a data table.** Individuals and variables are the rows and columns of a data table. Categorical and numerical data tells values that name a group from values that count or measure an amount. **Summarizing counts.** Frequency tables count each category; relative frequency tables turn each count into a share of its total, such as $\frac{28}{80} = 0.35$, so groups of different sizes compare fairly. ## A path through it Within each pair, the first comes first: sorting before counting and ordering, the question before the survey, individuals and variables before data types, counts before shares. Finish here and you are ready to make a bar graph or a histogram, choose a display, find the mean from a frequency table, and spot response bias in a survey. ### [Data displays](https://math.wiki/data-displays/) **Summary:** Pictures of data: which one to draw, and how to read it. **Question:** Data can be drawn a dozen ways. Which picture answers the question you were asked, and how do you read one someone else drew? ## What this area is about A list or table of data rarely shows its pattern at a glance; a display does. Each display suits one kind of data and shows some features while hiding others, so this area is about matching the picture to the data and the question, and about reading a picture honestly. ## How these fit together **Categorical data.** Bar graphs and pictographs show counts per category, and circle graphs show each category's share of the whole. **One numerical variable.** Dot plots and stem and leaf plots keep every value. Histograms group many values into equal bins, and box plots draw the five-number summary. **Two variables.** Two-way tables count individuals by two categorical variables, scatter plots plot two numerical ones as points $(x, y)$, and line graphs follow one measurement over time. **Reading and choosing.** Reading a graph starts with the title, labels and units. Misleading graphs shows how a picture with correct numbers can still mislead, and choosing a display matches the picture to the question. ## A path through it Bar graphs come first. After them, in any order: dot plots, reading a graph, and two-way tables. Stem and leaf plots follow dot plots; histograms need both bar graphs and dot plots. Reading a graph leads to misleading graphs and choosing a display. Circle graphs, line graphs and scatter plots have no required place here, and box plots need quartiles from center and spread. ### [Data tables, tallies and frequency tables](https://math.wiki/frequency-tables/) **Summary:** A table listing each value or category beside the number of times it occurred. **Question:** A month of weather cards, a basket of lunch choices, a class full of answers about pets. How do you turn a pile of separate answers into a few numbers you can read at a glance? ## When you reach for it Every school morning Vieno's class looks out the window and puts one picture on the class weather chart: a sun for a sunny day, a cloud for a cloudy day, a cloud with rain for a rainy day. After a month the chart is full. Ask the class "what was the weather like this month?" and the chart, as it stands, does not quite answer. It is a list of separate days. To answer, the days have to be sorted by kind of weather and each kind counted. A collection of answers like this, one for each day or one for each child, is called **data**. When every answer is one of a few kinds (sunny, cloudy or rainy; dog, cat or fish), the kinds are called **categories**. This lesson turns data like that into a short table of categories and counts, and then reads the table to answer questions. Here the data always fall into at most three categories. The lesson takes four days. - **Day 1** sorts the data into categories and counts each one. - **Day 2** keeps the count with tally marks, bundled in fives. - **Day 3** writes the counts in a table, and reads a table someone else made. - **Day 4** answers questions from the table: how many in all, how many in one category, and how many more or fewer in one category than in another. The child already counts objects (count-to-100) and has met sorting objects into groups by one feature. The questions on Day 4 need only adding and subtracting within twenty and adding a one-digit number to a two-digit number. ### [Deciding and concluding from a p-value](https://math.wiki/reject-or-fail-to-reject/) **Summary:** Comparing the p-value with the chosen cutoff and saying which hypothesis the evidence supports. **Question:** A sample says more customers buy tea than the owner believed, with a p-value of 0.020. What does she decide, what may she then say about all her customers, and what may she not say? ## When you reach for it Ebru runs a made-up simit bakery in Istanbul. She has long believed that $25\%$ of her customers also buy a glass of tea with their simit. Her nephew Kaan wanted to check, so before collecting any data they wrote down the hypotheses (see null-and-alternative-hypotheses) and a significance level. With $p$ the proportion of all the bakery's customers who buy tea, $$H_0\colon p = 0.25 \qquad H_a\colon p \neq 0.25 \qquad \alpha = 0.05$$ A random sample of $320$ customers then showed $98$ buying tea, and the test statistic and the p-value came out at $z \approx 2.32$ and $0.020$. Recall from p-values that the p-value is the chance, if $H_0$ were true, of a sample at least as far from $0.25$ as this one, and that $\alpha$ is the cutoff for "small enough". The numbers are finished; the thinking is not. Ebru now needs three things: a **decision** (reject $H_0$ or not), a **conclusion** in words about her customers, and a sense of what the decision does not say. Part 1 of this lesson makes the decision and writes the conclusion. Part 2 matches a two-sided test at level $\alpha$ with a confidence interval at level $1 - \alpha$, a match that for proportions is close but not exact. ## The procedure **The cutoff comes first.** The significance level $\alpha$ is fixed before the data are seen, chosen from the cost of a false alarm: a smaller $\alpha$ when acting on a wrong rejection would be expensive. The reason it must come first is the guarantee behind it. If $H_0$ is true, a share $\alpha$ of all samples give a p-value at or below $\alpha$; a cutoff picked after seeing the p-value can be fitted to the result and guarantees nothing. ### [Density curves: area, mean and median](https://math.wiki/density-curves/) **Summary:** A smooth curve standing in for a histogram, with total area one under it. **Question:** With thousands of values, the bars of a histogram blur into one smooth outline. How do you read proportions, a median and a mean off a curve that has no bars at all? ## Why this exists A relative frequency histogram answers "what share of the values lies in this bar?". It answers it only for whole bars. Ask for the share of waits between $2.5$ and $3.5$ minutes when the bars run from $2$ to $4$, and the histogram has nothing to say, because a bar does not show how its values spread inside it. The picture also depends on a choice that has nothing to do with the data: move the bin edges or change their width and the bars come out different. Now imagine a great many values, sorted into narrower and narrower bins. The jagged staircase of bar tops settles down into one smooth outline. That outline no longer depends on any choice of bins, and it can be cut anywhere: between $2.5$ and $3.5$ as easily as between $2$ and $4$. A **density curve** is that outline, made exact. It is a model of a distribution, an idealized picture of how the values of a numerical variable are spread, in which every proportion is an area. It is also the shape every later lesson on the normal distribution is built on. A normal curve is one particular density curve, and every question asked of it is a question about area. ## Definition A **density curve** is the graph of a function that - never goes below the horizontal axis, and - has exactly $1$ unit of area between it and the horizontal axis. It models a numerical variable: the proportion of values between any two numbers $a$ and $b$ is the area under the curve from $a$ to $b$. Writing $X$ for the value of one item picked at random, that area is written $P(a \le X \le b)$, and the same number is the probability that the item's value lands between $a$ and $b$. ### [Describing a distribution](https://math.wiki/describe-a-distribution-in-context/) **Summary:** Reporting shape, center, spread and unusual features together, in the words of the setting. **Question:** You have a plot and a page of summaries. What do you write so that someone who never saw the data knows what it says, and can check it? ## When you reach for it A statistical-question such as "How long are the lizards at this site?" is answered by data, but a list of fifteen numbers or a dot plot does not answer it by itself. Someone has to say, in sentences, what the data shows, so that a reader who never saw the plot can picture it and check it. That account is a **description of the distribution**: the context, the shape, a center with a matching spread, and anything unusual, each in the units of the setting. It comes at the end of any question about one variable. Setting two groups side by side is compare-two-distributions. This lesson takes two days. **Part 1** builds a description from a display and its summaries. **Part 2** judges and improves short descriptions, and starts from raw data. ## The procedure ### Part 1: the pieces of a description | Piece | What it says | |---|---| | Context | what was measured, on which individuals, how, in what units, and how many | | Shape | symmetric, skewed left or right, uniform, one peak or two | | Center and spread | one matching pair, chosen for the shape | | Unusual features | outliers tested with the fences, gaps, clusters | | The answer | a last sentence back to the question | **Context.** Name the individuals and the variable (individuals-and-variables), with the unit: "$11$" means nothing, "$11$ cm, snout to tail tip" does. Say how the values were measured, since a lizard measured to the base of its tail and one measured to the tip give different numbers. Say how many there are: the count tells the reader how far to trust the shape. ### [Describing a scatter plot](https://math.wiki/describe-a-scatter-plot/) **Summary:** Saying the direction, the form, the strength and any unusual points, in context. **Question:** The dots are on the plot. What does the whole cloud say about how the two measurements go together, and what can it not tell you? ## When you reach for it plot-paired-data turns each individual into one dot. Once the dots are down, somebody wants to know what the plot shows: do thicker trees tend to be taller? Do colder months bring bigger heating bills? The answer is a few sentences about the two variables, in their own units, so that someone who never saw the plot knows what it showed. You reach for this whenever a scatter plot of bivariate-data is in front of you, before drawing a line through it or claiming anything from it. The description has a fixed set of parts: the overall pattern (its **direction**, **form** and **strength**), then anything that breaks the pattern (**clusters** and **outliers**), then a limit on what it means: two variables that go together need not affect each other. The lesson takes two days, the overall pattern first and the rest second. ## The procedure **Part 1: direction, form and strength.** Two variables are **associated** when knowing one tells you something about the other. On a scatter plot that shows as a pattern in the cloud, and three words sum it up. **Direction: which way does the cloud run?** Read from left to right. If the dots tend to climb, the association is **positive**: larger values of one variable go with larger values of the other. If they tend to fall, it is **negative**: larger values of one go with smaller values of the other. If they neither climb nor fall, there is **no association**: knowing the value across does not help you guess the value up. Direction belongs to the whole cloud, so a positive association allows a few pairs in which the dot further right is lower. That is what "tend" means. ### [Distributions](https://math.wiki/distributions/) **Summary:** Models for how values are spread, and the normal curve above all. **Question:** Values spread out in a pattern. When can one smooth curve stand in for that pattern, and what does it let you work out? ## What this area is about A distribution is the pattern of how values are spread. This area replaces a histogram with a smooth model, above all the normal curve $N(\mu, \sigma)$, and reads shares of values as areas under it. Those areas hold only when the data, or the statistic, really is close to normal, so checking that is part of the work. ## How these fit together **Definitions.** Density curves are smooth outlines with total area $1$, where area is proportion. The normal distribution is the symmetric bell pinned down by $\mu$ and $\sigma$, and the standard normal is the one with $\mu = 0$ and $\sigma = 1$. **Procedures.** z-scores and the empirical rule measure distance from the mean in standard deviations. Normal calculations go from a value to a share and back. Checking whether data look normal decides whether a normal model is fair. **Statistics as variables.** Sampling distributions describe how $\bar{x}$ or $\hat{p}$ varies over random samples, and the central limit theorem says when the shape for $\bar{x}$ is close to normal even though the population is not. ## A path through it The normal distribution and density curves come first, in either order. z-scores follow the normal distribution; the standard normal needs all three; normal calculations come last of these. Checking normality can come any time after the normal distribution, and sampling distributions need it. The central limit theorem and sampling distributions interleave: the theorem comes after sampling variability and before the sampling distribution of $\bar{x}$. ### [Dot plots](https://math.wiki/dot-plots-and-line-plots/) **Summary:** One dot per data point, stacked over its value. **Question:** A list of measurements is hard to read. What picture shows every value at once, already sorted and counted? ## What this area is about A dot plot, also called a line plot, is a number line with one dot (or X) above it for each data value, placed over that value. Equal values stack, so the height of a stack is how often that value came up. It suits numerical data with a modest number of values, few enough that every one can be drawn, and it shows clusters, gaps, peaks and how far apart the smallest and largest values are. ## How these fit together **Reading.** Reading dot plots counts the stack over each value and reads clusters, gaps and peaks off a finished plot. **Making.** Making a dot plot runs the same idea the other way: mark a number line in equal steps from the smallest value to the largest, then place one mark per value. **An extension.** Line plots with fractions are the same display on a number line marked in halves, quarters or eighths, for lengths measured to the nearest $\frac{1}{2}$, $\frac{1}{4}$ or $\frac{1}{8}$ inch. ## A path through it Read a finished plot first, then make one from a list, then move to fractional number lines once fractions on a number line are familiar. Finish here and you can group values into bins for histograms, find the median of a list with an even count, and compare two groups with back-to-back stem-and-leaf plots. ### [Expected counts and the chi-square statistic](https://math.wiki/chi-square-statistic/) **Summary:** Adding up, over all cells, the squared gap between observed and expected divided by expected. **Question:** A beekeeper expects bees to visit four flower colors equally often, and the counts come back uneven. How can one number say how far a whole set of counts sits from what a claim predicts, and how large that number tends to be when the claim is true? ## When you reach for it Nikos keeps bees on Crete. In a test garden he planted equal areas of yellow, white, purple and blue flowers, and he recorded which color each of $200$ honeybee visits went to (made-up data): | color | visits | |:-:|:-:| | yellow | $62$ | | white | $41$ | | purple | $48$ | | blue | $49$ | | total | $200$ | If the bees have no preference, each color should get about a quarter of the visits, $50$ each. The counts are not $50$, but chance alone never gives exactly $50$ either. Before anyone can ask whether these gaps are too large to be chance, the four gaps have to be gathered into one number. A test-statistic for one proportion did that for a single gap; here there are four gaps at once, and the same need arises for a whole two-way table of counts. Reach for this lesson whenever counts in categories are set against a claim: a claimed set of proportions for one categorical variable, or a claim that two categorical variables are not associated. Day 1 finds what the claim predicts, the **expected counts**, and combines the gaps into the **chi-square statistic**. Day 2 describes the **chi-square distributions**, which say how large that statistic tends to be when the claim is true. ## The procedure ### Day 1: expected counts and the statistic **The claim is a null hypothesis.** In null-and-alternative-hypotheses the surf school's claim, $p = 0.80$, was a null hypothesis about one proportion. Nikos's claim is a null hypothesis about four proportions at once: ### [Experimental design](https://math.wiki/experimental-design/) **Summary:** Control, randomization, replication, blocking. **Question:** You can impose a treatment on purpose. How do you set up the comparison so that a difference in the results can only have come from the treatment or from chance? ## What this area is about An experiment imposes treatments and measures what happens, which is what lets it show cause and effect. That only works if the plan keeps everything else out of the comparison. This area is that plan, built from four ideas: control, randomization, replication and blocking. ## How these fit together **The vocabulary.** Experimental units and treatments names the parts: units, factors and their levels, treatments, and the response variable. **The principles.** A control group, often given a placebo and kept blind, sets the baseline. Random assignment lets chance alone split the units into groups, so that nothing else steers who gets which treatment. **The designs.** A completely randomized design assigns all units by chance alone, with no grouping first, and adds replication, enough units per treatment for a real difference to show. Blocking groups units that are alike before randomizing within each group; matched pairs is its smallest case. **Reading the result.** Comparing treatments by re-randomizing asks whether the difference between groups is bigger than the luck of the assignment would usually produce. ## A path through it Units and treatments first, then the control group, then random assignment, then the completely randomized design, then blocking. Re-randomizing comes last, once hypotheses and simulated sampling distributions are familiar. Finish here and paired data analyzes the matched pairs design that blocking builds. ### [Finding the median](https://math.wiki/median-of-an-even-list/) **Summary:** When two values sit in the middle, the median is halfway between them. **Question:** Line the values up from smallest to largest. Which number sits in the middle, and what if two of them share the middle? ## When you reach for it Abril has nine podcast episodes saved, of lengths $42, 35, 51, 38, 47, 60, 33, 45$ and $40$ minutes. She wants one number that says how long a typical episode is. One good answer is the episode in the middle: put the lengths in order from shortest to longest and take the one with as many episodes before it as after it. That number is the **median**. The median is one of the measures of center in median-and-mode. The mean balances the distances on either side; the median simply stands in the middle of the sorted list. You reach for it whenever the data are numbers and the question is "what value splits the data into a lower half and an upper half?" It works on a plain list, and it works just as well on a dot plot or a frequency table, where the values are already in order and only need counting. The **median** of a numerical data set is the middle value once the values are sorted from smallest to largest. When the number of values is even, there are two middle values, and the median is the number halfway between them. Either way, as many values sit before the median in the sorted list as sit after it. ## The procedure **Sort first.** A list is usually written in the order the values were collected: first day, second day, and so on. That order says nothing about size, so the entry in the middle of the written list can be any value at all. The median is about position by size, so the values must be put in order before anything is counted. Every value is kept, repeats included: if $3$ appears twice, it is written twice, because each occurrence is one piece of data. ### [Fitting linear and exponential models with technology](https://math.wiki/fit-a-function-to-data-with-technology/) **Summary:** Using a calculator or spreadsheet to fit a least-squares line or an exponential curve to data and reading what its numbers say. **Question:** Two people draw two different lines through the same scatter plot. Is there one line, or one curve, that everybody can agree on, and how does a calculator find it? ## When you reach for it A line-of-best-fit drawn by eye is quick, but two careful people draw two slightly different lines, and neither can say why theirs is the right one. When you have paired data on a scatter plot (see plot-paired-data) with a clear trend (describe-a-scatter-plot), a calculator or spreadsheet can fit **one** model that anyone using the same numbers will get, in seconds. On Day 1 that model is a straight line, the least-squares line. On Day 2 it is an exponential curve $y = ab^x$, for data that grow or shrink by roughly constant ratios. The skill is not pressing the buttons. It is knowing what the machine chose, reading its numbers in context, and deciding which of the two models the data and the situation call for. Every data set in this lesson is made up. ## The procedure ### Day 1: the least-squares line **What the machine makes small.** Take any line through a scatter plot. Each data point has a **vertical gap** to it: the point's actual $y$ minus the line's $y$ at the same $x$. A point above the line has a positive gap, a point below a negative one. Adding the gaps does not judge a line, because positive and negative gaps cancel: every line through the center of the cloud, the point of means $(\bar{x}, \bar{y})$ below, has gaps that total zero, however badly it is tilted. So each gap is **squared**, which makes it positive and makes a big miss count far more than a small one, and the squares are added. The **least-squares line** is the line whose total of squared vertical gaps is as small as possible. For any data with at least two different $x$-values there is exactly one such line, and technology finds it with formulas you do not need here. Later this unit the gaps get a name, residuals, in interpret-a-residual. ### [Goodness of fit test](https://math.wiki/chi-square-goodness-of-fit-test/) **Summary:** Testing whether one categorical variable follows a claimed set of proportions. **Question:** A genetics class crosses pea plants and predicts that the offspring seeds come in a 9 to 3 to 3 to 1 ratio of four types. When the counts come back uneven, how do you decide whether they still fit the prediction, and if they do not, which seed types are off and in which direction? ## When you reach for it Solomiia, Yaryna and Ostap are in a high school genetics class in Ukraine. They cross pea plants that each carry one dominant and one recessive version of two genes: seed shape (round over wrinkled) and seed color (yellow over green). Mendel's model predicts that the offspring fall into four seed types in the ratio $9:3:3:1$. The class raises $320$ offspring and sorts them by seed type (made-up data): | seed type | predicted share | observed | |:-:|:-:|:-:| | round yellow | $\tfrac{9}{16}$ | $190$ | | round green | $\tfrac{3}{16}$ | $52$ | | wrinkled yellow | $\tfrac{3}{16}$ | $60$ | | wrinkled green | $\tfrac{1}{16}$ | $18$ | | total | $1$ | $320$ | chi-square-statistic already gives the expected counts $E = np$, the statistic $\chi^2 = \sum \frac{(O - E)^2}{E}$, its df $k - 1$, and the area to its right under a chi-square curve. This lesson adds the hypotheses, the conditions, and a conclusion about the seeds. Reach for the **chi-square goodness-of-fit test** when one categorical variable is recorded on one random sample, or on the outcomes of one random process, and a claim names the proportion in every category. The question is whether the claimed distribution fits. Two categorical variables recorded on the same individuals call instead for the chi-square-test-for-independence. ## The procedure ### Day 1: from claim to conclusion **Hypotheses.** The null hypothesis states the whole claimed distribution, one proportion for every category. For the cross, with $p$ standing for the long-run proportion of all offspring of this cross that have a given seed type, ### [Histograms](https://math.wiki/histograms/) **Summary:** Group values into bins, then bar the counts. **Question:** When numerical data are too many or too spread out to draw one by one, how can bars still show where they crowd together? ## What this area is about A histogram cuts the number line into bins of equal width that run end to end, then stands a bar on each bin as tall as the number of values in it. It is for numerical data, and it helps most when the values are many or rarely repeat, so a dot plot would show no clusters. The price is that a bar says how many values fall in a bin, never which ones. ## How these fit together **Reading and making.** Reading a histogram gets counts and totals from the bars, with a fixed rule for a value on an edge. Making a histogram goes the other way, from a list to a frequency table to bars, and asks how wide the bins should be. **Shares instead of counts.** Relative frequency histograms give each bar the height $\frac{f}{n}$, a bin's count over the total, so groups of different sizes can be compared; the heights add to $1$. **Running totals.** Cumulative frequency graphs and percentiles add the bars from the left and draw a rising line, so the share of values below any point is read off one curve. ## A path through it Read a histogram first (making one reuses its edge rule), then make one. Relative frequency builds on making, and cumulative frequency on relative frequency. Finish here and you can describe the shape of a distribution, and the same bar picture leads on to probability distribution tables and density curves. ### [Hypothesis testing](https://math.wiki/hypothesis-testing/) **Summary:** Assume nothing is going on, see how surprising the data would be. **Question:** A sample disagrees with a claim. Is the gap too large to blame on the ordinary variation between random samples? ## What this area is about A hypothesis test starts by assuming a claim about a parameter, then asks how surprising the sample would be if that claim were true. If data at least this extreme would be rare under the claim, that counts as evidence against it. The reasoning holds only when the data come from a random sample, random assignment or a random process, and it measures chance alone: it says nothing about how large or important a difference is, and a decision can still be wrong. ## How these fit together **Setting up.** Null and alternative hypotheses state the claim $H_0$ (an equality, such as $p = 0.80$) and the alternative $H_a$ the study looks for, with $<$, $>$ or $\neq$. **Measuring.** Test statistic counts how many standard deviations the sample statistic sits from the null value, computed as if $H_0$ were true. **Deciding.** Deciding and concluding from a p-value compares the p-value with a significance level $\alpha$ fixed in advance. Cautions about significance separates "detected" from "large enough to matter". **Errors.** Type I and Type II errors name the two wrong decisions, and power is the chance of rejecting $H_0$ when the true value is a given alternative. ## A path through it Hypotheses, then the test statistic, then p-values (outside this area). After that, the decision and its cautions on one branch, and the two errors then power on the other. Finish here and you can run a one-sample $z$ test for a proportion. ### [Individuals, variables and units](https://math.wiki/individuals-and-variables/) **Summary:** The rows of a data table are the individuals studied and the columns are the variables recorded about them. **Question:** Someone hands you a page of numbers. Before you can learn anything from it, what do you need to know about each number? ## Why this exists Here are three numbers somebody wrote on a scrap of paper: $$14.5 \qquad 410 \qquad 12$$ Nothing can be learned from them. Is $410$ a length, a mass, a count? Of what? Measured how, in which unit? A number gets its meaning from three things that travel with it: **which thing** it describes, **which feature** of that thing it records, and **the unit and method** of the recording. A data table keeps all three attached, and every later piece of statistics starts by reading one. ## Definition Data are facts recorded about a group of things. An **individual** is one of the things the data describe: a person, an animal, an object, a place, a day, an event, whatever was studied. Statistics books also say **case** or **observational unit**. A **variable** is one feature recorded about every individual, whose value can differ from one individual to the next. Its values may be words (a turtle's species) or numbers (its mass in grams); sorting variables by that difference is the job of categorical-vs-numerical-data. A **data table** gives each individual one row and each variable one column. The cell where a row meets a column holds one **value**: that variable for that individual. Many tables begin with a column that only names each row, such as a tag; in this lesson it is not counted as a variable, since it only says which individual the row is. Some books do count it. An **observation** is everything recorded about one individual, so one row. The **number of observations**, written $n$, is the number of individuals, which is the number of rows under the heading. ### [Inference for means](https://math.wiki/inference-for-means/) **Summary:** Estimating and testing claims about the average value in a population. **Question:** A sample has an average. What does it say about the average of the whole population, or about what a treatment did to it? ## What this area is about A sample mean $\bar{x}$ is known; the population mean $\mu$ is not, and another sample would give another $\bar{x}$. This area turns sample means into an interval for $\mu$ or a test of a claimed value. The population standard deviation $\sigma$ is almost never known, so the sample's $s_x$ stands in for it, and every procedure here uses $t$, not $z$. ## How these fit together **Foundations.** The t distribution is the curve that $t = \frac{\bar{x} - \mu}{s_x/\sqrt{n}}$ follows for samples from a normal population, with $n - 1$ degrees of freedom. The conditions for one mean say when that can be trusted: a random sample, or a randomized experiment using random assignment; $n \le 0.10N$ when sampling without replacement; and a population close to normal, or $n \ge 30$, or a smaller sample whose plot shows no strong skew and no outliers. **One mean.** The one-sample interval estimates $\mu$ and sizes a sample; the one-sample test checks a claimed value $\mu_0$. **Two means.** The interval and the test for a difference of means compare two independent groups, each checked against the conditions. **Paired data.** Paired data reduces linked measurements to one column of differences and runs one-sample inference on it. ## A path through it The t distribution and the conditions first, then the one-sample interval and test. Next the two-sample interval, then the two-sample test, and paired data last. ### [Inference for proportions](https://math.wiki/inference-for-proportions/) **Summary:** Estimating and testing claims about the share of a population with some feature. **Question:** A sample found that some share of people have a feature. What does that say about the share in the whole population, or about what a treatment changed? ## What this area is about A sample proportion $\hat{p}$ is known; the population proportion $p$ is not, and another sample would give another $\hat{p}$. This area turns one or two sample shares into a range of plausible values for $p$, or into a verdict on a claimed value. Every procedure here leans on the normal curve, so every one starts by checking when that curve fits. ## How these fit together **The groundwork.** The conditions for one proportion are three: a random sample, or a randomized experiment using random assignment; $n \le 0.10N$ when sampling without replacement; and large counts, at least $10$ successes and $10$ failures, checked with $\hat{p}$ for an interval and with the null value $p_0$ for a test. **One proportion.** The one-sample interval estimates $p$ and sizes a survey; the one-sample test checks a claimed $p_0$. **Two proportions.** The interval and the test for a difference of proportions compare two groups, with the conditions checked in each and the two samples independent. The test checks large counts with the pooled proportion, since its null hypothesis says the two shares are equal. ## A path through it The conditions first, then the one-sample interval and test. Next the two-sample interval, then the two-sample test. Finish here and choosing an inference procedure asks which of these, or of the tests for means, fits a study. ### [Inference for slopes](https://math.wiki/inference-for-slopes/) **Summary:** Deciding what a sample regression line says about the true relationship in the population. **Question:** A sample's fitted line rises. What does its slope say about the true rate of change across the whole population? ## What this area is about A least-squares line $\hat{y} = a + bx$ describes one sample. Behind it sits a population line $\mu_y = \alpha + \beta x$, and another sample would give another slope $b$. This area does for the slope $\beta$ what inference for means did for $\mu$: a range of plausible values, or a verdict on a claimed value, usually $\beta = 0$, no linear relationship. ## How these fit together **The model.** The sampling distribution of the slope shows how $b$ varies from sample to sample around $\beta$, which is what gives it a standard error. **The conditions.** The conditions for regression inference are five, spelling LINER: a linear trend; independent observations, with the sample at most $10\%$ of the population when drawn without replacement; normal responses about the line, judged from residuals with no strong skew and no outliers; equal spread at every $x$; and a random sample or random assignment. **The procedures.** The interval for a slope estimates $\beta$ as $b \pm t^* SE_b$, read from computer output, with $n - 2$ degrees of freedom. The test for a slope asks whether a sample slope this far from $0$ could arise by chance if $\beta = 0$. ## A path through it The sampling distribution of the slope first, then the conditions, then the interval, then the test. Finish here and a fitted slope becomes evidence about how $y$ changes with $x$ across the whole population, not just a description of the sample. ### [Interpreting a residual](https://math.wiki/interpret-a-residual/) **Summary:** Reading a residual as how far above or below the prediction an actual value fell. **Question:** A fitted line predicted one value and the real value came out different. How far off was the line, on which side, and what does that gap say about the thing measured? ## When you reach for it A walking group timed itself on eight hiking trails, from $2$ to $9$ km long (made-up data, listed in full in Example 3). Technology fits the regression line $$\hat{y} = 14x + 10$$ where $x$ is the trail length in kilometers and $\hat{y}$ is the predicted walking time in minutes. The $6$ km trail took $103$ minutes, but the line predicts $14(6) + 10 = 94$. The walk took $9$ minutes more than the line said. Is that a big miss? Which trail does the line get most wrong? The gap between what happened and what the line predicted is the **residual**. You reach for it whenever a fitted line meets real data: to say how far off one prediction was, on which side, and which points the line fits worst. Getting the line belongs to fit-a-function-to-data-with-technology, using it to predict to predict-with-a-regression-line, and reading its slope to interpret-the-slope-of-a-regression-line. A line-of-best-fit was already judged by its vertical gaps; this lesson names those gaps and measures them. ## The procedure **Actual minus predicted.** For a data point $(x, y)$, put $x$ into the line to get the predicted value $\hat{y}$, read "y-hat". The hat marks a prediction, not a measurement. Then $$\text{residual} = y - \hat{y} = \text{actual} - \text{predicted}$$ The residual is in the units of $y$: minutes for the trails, dollars for a price. **The sign says the side.** The order is fixed, actual first, so the sign carries meaning. A positive residual means the actual value was more than predicted: the point lies above the line and the line underestimated. A negative residual means less than predicted: the point lies below the line and the line overestimated. A residual of $0$ means the point is on the line. ### [Interpreting a standard deviation](https://math.wiki/interpret-standard-deviation-in-context/) **Summary:** Saying in the setting's own units how far a typical value sits from the mean. **Question:** A machine fills juice bottles to $500$ mL on average. How far from $500$ mL should any one bottle be expected to land? ## When you reach for it A standard deviation arrives as one number with a unit, such as $2$ mL. You reach for this skill when that number must become a statement about the data: what a typical value looks like, which group is more spread out, whether one value is far out, and whether the standard deviation is the right summary at all. ## The procedure **What the number measures.** The standard deviation is the square root of the variance, the mean of the squared distances from the mean $\mu$ over all $N$ values: $$\sigma = \sqrt{\frac{\sum (x - \mu)^2}{N}}$$ Squaring and then taking the root brings the answer back to the units of the data, so the standard deviation of volumes in milliliters is in milliliters. It is roughly the distance a typical value sits from the mean. The working itself, and the sample version $s$, which divides by $n - 1$ and comes out a little larger, belong to compute-standard-deviation-by-hand. Every data set here is a whole group (one batch, one week), so it divides by $N$; each interpretation reads the same for $s$. **Typical, not exact, not a maximum.** Being the root of a mean of squares, the standard deviation blends all the distances. Often no value sits exactly one standard deviation from the mean, and some values sit further out. It is never smaller than the mean absolute deviation (MAD), because squaring gives the larger distances more weight, and the two are equal only when every value sits the same distance from the mean. ### [Interpreting the slope and intercept](https://math.wiki/interpret-the-slope-of-a-regression-line/) **Summary:** Saying what a one unit change in the explanatory variable predicts about the response, in context. **Question:** A straight line has been fitted through a cloud of data points. What do its slope and its intercept actually say about the things that were measured? ## When you reach for it Ten adults on an online typing course logged their hours of practice and then took a speed test (the numbers are made up). On a scatter plot the points drift upwards from left to right, and a straight line has been fitted through them, a line-of-best-fit: $$y = 4x + 22$$ where $x$ is the hours of practice and $y$ is the predicted speed in words per minute. What do the $4$ and the $22$ say about typing? Reading $m$ and $b$ in $y = mx + b$ is familiar from read-slope-and-intercept, and linear-function-from-a-graph gives them meaning in context for an exact line. A fitted line differs in one way that changes every sentence: the points do not sit on it. So the slope becomes a *predicted* change, true *on average* rather than for each individual, and the intercept sometimes means nothing at all. Drawing the line belongs to line-of-best-fit, describing the cloud to describe-a-scatter-plot. ## The procedure **The line gives predictions.** In $y = 4x + 22$, the $y$ is the speed the line predicts, not a speed anyone measured. The learner with $4$ hours of practice typed $39$ words per minute, while the line gives $4(4) + 22 = 38$. Statistics books mark a predicted value with a hat, $\hat{y}$, read "y-hat", to keep it apart from a measured one. In this lesson the $y$ of a fitted line always means the prediction. **The slope is the predicted change for one more unit of $x$.** Slope is rise over run, so on a run of $1$ the rise is the slope itself. From $5$ hours to $6$ hours the line rises from $42$ to $46$, which is $4$ words per minute. In words: for each extra hour of practice, the predicted typing speed is about $4$ words per minute higher. ### [Interval for a difference of means](https://math.wiki/two-sample-t-interval-for-a-difference-of-means/) **Summary:** Estimating how far apart two population averages are. **Question:** Two groups have different averages in their samples. How far apart are the means of the two whole populations, or, in an experiment, how much did one treatment change the mean response compared with the other? ## When you reach for it Zawadi grows two macadamia varieties on her orchard in Kenya, and a buyer pays more for heavier nuts. She wants to know how much heavier, on average, a variety A nut is than a variety B nut across this season's whole harvest. She weighs a random sample of $20$ variety A nuts, mean $7.8$ grams, and an independent random sample of $18$ variety B nuts, mean $7.0$ grams (made-up data throughout this lesson). The gap in the samples is $0.8$ grams, but another pair of samples would give another gap. What range of gaps between the two whole harvests do these samples support? The two-sample interval for a difference of means answers that, for two populations sampled separately (Day 1) or two treatment groups in an experiment (Day 2). It is the one-sample-t-interval-for-a-mean, $\bar{x} \pm t^* s_x/\sqrt{n}$, with two samples in place of one. Its standard error comes from sampling-distribution-of-a-difference-of-means, whose Day 2 showed that for independent random samples $\bar{x}_1 - \bar{x}_2$ centers at $\mu_1 - \mu_2$ with standard deviation $\sqrt{\sigma_1^2/n_1 + \sigma_2^2/n_2}$. The parallel lesson for shares is the two-sample-z-interval-for-a-difference-of-proportions. ## The procedure ### Day 1: two independent random samples The population standard deviations are unknown, so the sample standard deviations stand in for them, and, as for one mean, that extra uncertainty calls for the t-distribution in place of the normal curve: ### [Interval for a difference of proportions](https://math.wiki/two-sample-z-interval-for-a-difference-of-proportions/) **Summary:** Estimating how far apart two population shares are. **Question:** Two groups have different shares of some feature in their samples. How far apart are the shares in the two whole populations, or, in an experiment, how much did one treatment change the share compared with the other? ## When you reach for it Maartje works for a regional energy office and wants to compare two Dutch cities: what share of households in Utrecht have solar panels on their roof, and what share in Groningen? She cannot visit every household, so she takes an independent random sample in each city: $156$ of $400$ Utrecht households have panels, and $117$ of $450$ Groningen households do (made-up data throughout this lesson). The sample shares are $0.39$ and $0.26$, a gap of $0.13$. A different pair of samples would give a different gap, so the question is what range of gaps between the two whole cities this pair supports. That is the job of the two-sample interval for a difference of proportions. It is the move whenever a yes-or-no feature is compared across two groups: two populations sampled separately, or two treatment groups in an experiment. It is the one-sample-z-interval-for-a-proportion with two samples in place of one: the same plan of naming the parameter, checking conditions, computing and concluding, with a standard error built for a difference. The lesson takes two days: Day 1 compares two populations from two random samples, and Day 2 compares two treatments in a randomized experiment. ## The procedure ### Day 1: two independent random samples The sampling-distribution-of-a-difference-of-means lesson worked out how $\hat{p}_1 - \hat{p}_2$ behaves over repeated pairs of independent random samples: it centers at $p_1 - p_2$, its variance is the sum of the two variances, and with enough successes and failures in both samples its shape is close to normal. An interval follows the usual pattern, estimate plus or minus a critical-value times a standard-error. The true standard deviation contains the unknown $p_1$ and $p_2$, so the sample shares stand in for them: ### [Interval for a slope](https://math.wiki/t-interval-for-a-slope/) **Summary:** Estimating the true rate of change as the fitted slope give or take a margin of error. **Question:** A sample of trees gives a fitted slope. What range of values for the true rate of change across the whole orchard does that one slope support? ## When you reach for it Jordi grows almonds in Catalonia. He takes a random sample of $14$ trees from his orchard of $2{,}400$ and records each tree's trunk circumference $x$, in centimeters, and its kernel yield $y$ that season, in kilograms (made-up data). Bigger trunks go with bigger harvests, and a statistics program prints: $$\small\begin{array}{lrrrr} \text{Predictor} & \text{Coef} & \text{SE Coef} & \text{T} & \text{P} \\ \hline \text{Constant} & -0.505 & 1.127 & -0.45 & 0.662 \\ \text{Circumference} & 0.052 & 0.011 & 4.61 & 0.001 \end{array}$$ $$\small\text{S} = 0.851 \quad \text{R-Sq} = 63.9\% \quad \text{R-Sq(adj)} = 60.9\%$$ As regression-computer-output showed, the slope sits in the Coef column of the Circumference row: in these $14$ trees, each extra centimeter of circumference goes with $0.052$ kg more predicted yield. But Jordi cares about all $2{,}400$ trees, and a different $14$ would have given a different slope. What he wants is the rate for the whole orchard, given as an estimate with a margin of error. The interval for a mean in one-sample-t-interval-for-a-mean had the shape estimate $\pm\ t^* \times$ standard error; the interval for a slope has the same shape, with the slope as the estimate. ## The procedure ### Day 1: the interval from computer output **The parameter.** The model behind regression inference, from sampling-distribution-of-the-slope, is a population line $\mu_y = \alpha + \beta x$: at each circumference, the mean yield of all trees of that circumference in the orchard lies on this line. Its slope $\beta$ is the parameter. In context, $\beta$ is the change in mean kernel yield for each additional centimeter of trunk circumference, among all trees in Jordi's orchard. The sample slope $b$ estimates it. ### [Joint and marginal frequencies](https://math.wiki/marginal-frequencies/) **Summary:** The row and column totals of a two way table, each describing one variable on its own. **Question:** When every person answers two questions, how do you count them without losing which answers went together? ## Why this exists A frequency table counts one variable. Ask the same people a second question and a second table counts that, but two separate tables lose which answers came from the same person. At a made-up camp, $12$ campers each chose the lake or the forest trail for an afternoon walk. $5$ chose the lake, and $6$ are under $12$. How many campers under $12$ chose the lake? Any number from $0$ to $5$ fits those two counts. To answer, you must count the campers by both answers at once, which is what a two-way table does. It is where every question about whether two variables are connected starts. ## Definition The data are two categorical variables recorded for each member of a group of **individuals** (people, orders, animals). Each individual has exactly one category of each variable. A **two-way table** has one row for each category of one variable and one column for each category of the other. The cell where a row meets a column counts the individuals with **both** that row's category and that column's category. A column on the right holds the row totals, and a row along the bottom holds the column totals. For the $12$ campers (made-up data): | | Lake | Forest trail | Total | |---|:-:|:-:|:-:| | Under 12 | $4$ | $2$ | $6$ | | 12 and over | $1$ | $5$ | $6$ | | Total | $5$ | $7$ | $12$ | - A **joint frequency** is the number in one cell of the body: how many individuals have one category of the first variable *and* one of the second. The $4$ counts campers under $12$ who chose the lake. - A **marginal frequency** is a row or column total: how many individuals have one category of one variable, whatever their other answer. The $5$ counts all campers who chose the lake. These totals sit in the margins, hence the name. - The **grand total**, in the bottom right corner, is the number of individuals. ### [Least squares regression](https://math.wiki/least-squares-regression/) **Summary:** The line that makes the squared vertical misses as small as possible. **Question:** Everyone draws a slightly different line through the same scatter plot. Which rule picks one line that everybody agrees on, and what can that line be trusted to say? ## What this area is about A line drawn by eye through a scatter plot is a guess, and no two guesses match. Least-squares regression replaces the guess with a rule: choose the line $\hat{y} = a + bx$ that makes the sum of the squared vertical misses as small as possible. As long as the $x$-values are not all equal, that rule gives exactly one line, which can then be read, used for predictions, and judged. ## How these fit together **The definition.** The least squares criterion states the rule and shows why it picks a single line. **A result.** Slope from $r$ and the spreads writes that line from five summary numbers, slope $b = r\,\dfrac{s_y}{s_x}$, and explains regression to the mean. **Procedures.** Fitting models with technology gets the line (or an exponential curve) from a calculator or spreadsheet. Interpreting the slope and intercept says what the numbers mean in context, and predicting with a regression line uses it for cases nobody measured, with a warning about extrapolation. ## A path through it Take the criterion first, then the slope from $r$ and the spreads. Fitting with technology, interpreting the slope and predicting can follow in that order. Next, residuals show what the line missed, influential points show what can drag it, and the sampling distribution of the slope leads to inference. ### [Line graphs](https://math.wiki/read-a-line-graph/) **Summary:** Getting a value at a given time, and saying where the line rises, falls or stays flat. **Question:** A bean plant is measured every Saturday. How can one picture show how its height changed, and still let you read the height at any week? ## When you reach for it Xiu planted a bean seed and measured the plant with a ruler every Saturday for eight weeks. Yoshiro weighed his puppy, Pepper, once a month as she grew. Each of them ended up with a list of pairs: a time, and the number measured at that time. That is a data table, like the tables you already know, except that each row pairs a time with a measurement instead of a category with a count. A **line graph** turns such a table into a picture. Time runs along the bottom, the measurement goes up the side, each reading becomes a point, and the points are joined in time order. You reach for a line graph when the question is about change over time. How tall was the plant at week $5$? When did it reach $14$ centimeters? When did it grow the most? Did the puppy ever lose weight? Which of two puppies was heavier, and when did that change? A table holds the same numbers, but the line shows the shape of the change at a glance. This lesson takes three days. - **Part 1:** how a line graph is built, reading its scales, reading the value at a given time, and finding the times a value was reached. - **Part 2:** where the line rises, falls or stays flat, how big each change is, and why a reading between two points is only an estimate. - **Part 3:** double line graphs, which put two quantities on the same axes, and making a line graph from a table. ## The procedure ### Part 1: reading a value, and finding a time ### [Line of best fit](https://math.wiki/line-of-best-fit/) **Summary:** Eyeball a line through the cloud and use it to predict. **Question:** The dots on a scatter plot climb together, but not along any one straight line. Which line sums them up, and what can it tell you about a case nobody measured? ## When you reach for it A scatter plot puts one dot on the grid for each case: one puppy-week, one evening, one test flight. Often the dots rise or fall together in a band that runs roughly straight, and then a single straight line can stand in for the whole cloud. That line is a **line of best fit**, also called a **trend line**: a straight line drawn on a scatter plot to follow the overall pattern of the dots, passing as close to them as it can overall. Reach for it when the scatter plot looks linear, which is a matter of reading its form, direction and strength, and you want a short summary of the pattern or a prediction for a case nobody measured: the puppy's mass in the week it was not weighed, a cricket's chirping rate at a temperature nobody recorded. Unlike writing a linear rule from a story, where every point sits on the line, real measurements scatter, and the task is choosing a line that misses them by as little as it can. The lesson takes two parts. Part 1 draws a line by eye and judges between two lines. Part 2 writes the line's equation, uses it to predict, and asks when a prediction can be trusted. ## The procedure ### Part 1: drawing a line by eye **First decide whether a line belongs at all.** A straight line can only sum up a pattern that is straight. If the dots bend, any straight line runs below them in one stretch and above them in another. If the dots show no pattern, there is nothing for a line to follow. ### [Line plots with fractions](https://math.wiki/line-plots-with-fractions/) **Summary:** A dot plot whose number line is marked in halves, quarters or eighths. **Question:** Ten seashells are measured to the nearest quarter inch. How can every length be shown in one picture, and what can that picture tell you? ## When you reach for it On a nature walk, Akua picks up seashells and Kagiso picks up pinecones. In science class, Laleh's class measures ladybugs. Every one of these things has a length, and almost none of the lengths is a whole number of inches. A shell might be $1\frac{1}{2}$ inches long, a ladybug only $\frac{3}{8}$ of an inch. A list of ten or twelve lengths is hard to take in. You cannot see at a glance which length came up most, or how far apart the shortest and the longest are. A **line plot** shows all of them at once. It is a dot plot: a number line with one dot stacked above a value each time that value was measured. The only new thing here is that the number line is marked in fractions: halves, quarters or eighths of an inch. Reach for a line plot with fractions when you have many measurements of the same kind, made to the nearest half, quarter or eighth of a unit, and you want to see them together, answer questions about them, or share their total out evenly. The lesson takes three days. Part 1 measures to a fraction of an inch and makes the plot. Part 2 reads the plot and works with the numbers on it. Part 3 uses the plot to share a total out equally. ## The procedure ### Part 1: measuring and plotting **Measuring to a fraction of an inch.** A ruler cuts each inch into equal parts. On most rulers the inch is cut into $8$ equal parts, so the smallest step is $\frac{1}{8}$ inch. The ticks have different heights to help you count: the inch tick is the longest, the half inch tick is next, then the quarter inch ticks, and the eighth inch ticks are the shortest (measure-to-half-and-quarter-inch). ### [Log-log transforms and power models](https://math.wiki/log-transform-of-both-variables/) **Summary:** Plotting log against log to straighten a power relationship. **Question:** Taller redwoods have thicker trunks, but height grows more slowly than diameter and the straight line leaves an arch in its residual plot. Taking the log of height alone does not fix it. What does? ## When you reach for it transforming-to-achieve-linearity fitted the least-squares line to $(x, \log y)$ and undid the log to get an exponential model $\hat{y} = A B^x$, the right shape when $y$ is multiplied by the same factor for each one-unit step in $x$. Many relationships work differently: $y$ is multiplied by the same factor each time $x$ is *multiplied* by a fixed amount. Doubling a tree's trunk diameter does not double its height; it multiplies it by roughly the same factor whether the trunk goes from $20$ to $40$ centimeters or from $100$ to $200$. That is a **power model**, $$\hat{y} = A\,x^{p}, \qquad x > 0,$$ with a constant $A > 0$ and an exponent $p$. Reach for it when the straight line leaves a curve in its residual plot, the curve survives taking $\log y$ alone, and the log of both variables gives a plot that is straight with a residual plot that has no pattern. semi-log-plots showed an exact power law becoming a straight line on log-log axes. Here the data scatter, so the line is the least-squares line and the residual plots do the deciding. ## The procedure **Why both logs.** If $y = A x^{p}$ exactly, the laws of logarithms give $$\log y = \log A + p \log x,$$ a straight line in $\log x$ with slope $p$ and intercept $\log A$. So when the data follow a power model with scatter, the pairs $(\log x, \log y)$ lie in a straight band, and the least-squares line $$\widehat{\log y} = a + b \log x$$ ### [Making a dot plot](https://math.wiki/make-a-dot-plot/) **Summary:** Drawing a number line and stacking one mark above each value every time it appears. **Question:** Ten pencils have been measured, and their lengths sit in a list. How do you turn that list into a picture where every pencil can be seen, sorted and counted? ## When you reach for it Mereana takes ten pencils from the class pencil cup and measures each one to the nearest inch (measure-length). She writes the lengths down as she goes: $5, 7, 4, 6, 5, 3, 7, 5, 6, 5$. That list is true, but it is hard to see anything in it. Which length came up most? Are the pencils mostly long or mostly short? To answer, you would have to sort and count the list in your head. A **line plot** answers those questions with a picture. It is a number line, like the edge of a ruler, with one X above it for each thing measured, placed above that thing's length. Equal lengths stack up. The same picture is also called a **dot plot**; this lesson says line plot, as read-a-dot-plot does. That lesson reads a finished plot. This one draws a plot from a list, which is the same idea run the other way: each number in the list becomes one X. Reach for a line plot when several things of one kind have been measured with the same unit, or when one thing has been measured several times, and you want to see all the measurements together. The lesson takes three days. Part 1 draws a plot from a list: the number line, the Xs, the title and the label. Part 2 starts earlier, with measuring or with a table, and looks closely at where the number line should start. Part 3 checks a plot against its list, adds new measurements, and reads the finished plot back. ## The procedure ### Part 1: from a list to a line plot ### [Making a histogram and choosing the bin width](https://math.wiki/make-a-histogram/) **Summary:** Grouping values into equal intervals and drawing a bar for the count in each. **Question:** Twenty numbers in a list say little at a glance. How do you turn them into bars, and how wide should each bar be? ## When you reach for it Fidel wrote down how many texts he sent each day for twenty days. Twenty numbers in a row are hard to take in. A **histogram** turns them into a picture: the number line is cut into **bins** of equal width that run end to end, and on each bin stands a bar as tall as the number of values in it. read-a-histogram reads a histogram already drawn. This lesson draws one, from a list of numbers to a frequency table of bins and from the table to bars, and then asks how wide the bins should be. You reach for it whenever you have numerical data and want to see where the values crowd together and where they thin out. It helps most when the values are spread over a long stretch, or measured in decimals so that few of them repeat: a dot plot of such data has one dot over almost every value and no clear picture. ## The procedure ### Day 1: from a list to bars **Bins.** A bin is a stretch of the number line, written "$20$ to $30$". Every bin in one histogram has the same **bin width**, here $30 - 20 = 10$, and the bins run end to end with no gaps and no overlaps, so the edges go up in equal steps: $0, 10, 20, 30$ and so on. **The edge rule.** The number $30$ ends one bin and starts the next. As in read-a-histogram, **a value on an edge goes in the bar to its right**: the bin "$20$ to $30$" holds every value $x$ with $20 \le x < 30$, and $30$ itself is counted in "$30$ to $40$". So every value lies in exactly one bin. For whole-number data, "$20$ to $30$" holds $20, 21, \dots, 29$, and some tables write that same bin as "$20$ to $29$". ### [Making and reading box plots](https://math.wiki/make-a-box-plot/) **Summary:** Drawing a box from the lower to the upper quartile with whiskers out to the extremes. **Question:** A five-number summary is five numbers in a row. How can they become one picture that shows at a glance where the data crowd together and where they spread out? ## When you reach for it The five-number summary, the minimum, $Q_1$, the median, $Q_3$ and the maximum, says a lot about a data set, but five numbers in a row are hard to take in at a glance. A **box plot** (also called a box-and-whisker plot) draws them over a number line: a box from $Q_1$ to $Q_3$, a line across the box at the median, and a line called a **whisker** out from each side of the box to the minimum and to the maximum. Reach for one when you want a quick picture of where the middle half of the data sits and how spread out each part is, above all for a data set too large to draw value by value. The lesson runs over two days: Day 1 draws box plots, Day 2 reads them and finds what they cannot show. The quartiles are found as in quartiles: sort the data, cut it at the median into a lower and an upper half (leaving the median out of both when the count is odd), and take the median of each half. All the data in this lesson are made up. ## The procedure ### Day 1: drawing a box plot **A scale comes first.** Draw a number line from at or below the minimum to at or above the maximum, marked in equal steps, with the units named. Every part of a box plot is read against this scale. The line need not start at $0$: a scale from $3$ to $9$ kilograms shows melons from $3.2$ to $9$ kilograms far more clearly than one from $0$ to $10$ that squeezes them into a corner. **The box runs from $Q_1$ to $Q_3$.** Draw a rectangle just above the number line, its left edge over $Q_1$ and its right edge over $Q_3$. Its height means nothing. The box marks where the middle half of the data lies, and its width is the interquartile range, $\text{IQR} = Q_3 - Q_1$. ### [Making and reading residual plots](https://math.wiki/residual-plots/) **Summary:** Plotting each residual against the explanatory value to see what the line missed. **Question:** A line has been fitted to some data and the dots sit close to it. How can you tell whether a straight line was the right shape to fit at all? ## When you reach for it Technology has given you the least-squares line for a scatter plot (the line of best fit it computes), and you want to know whether a line was a fair choice. The scatter plot alone often cannot tell you: when the dots climb across a wide range, a gentle bend hides inside them. A residual is actual minus predicted, $y - \hat{y}$, positive when the dot is above the line and negative when it is below. A **residual plot** puts one dot at $(x,\ y - \hat{y})$ for every case, with a horizontal line at $0$. It is the scatter plot with the fitted line laid flat and the gaps stretched to fill the picture, so what the line missed is the only thing left to see. Reach for it whenever you must decide whether a line describes the data or whether to choose another family. A correlation near $1$ or $-1$ does not settle that: the correlation coefficient measures how tightly the dots hug a line, and Example 2 has $r$ close to $1$ and a curve all the same. ## The procedure **Making one.** For each case, put its $x$ into the line to get the prediction $\hat{y}$, then subtract: the residual is $y - \hat{y}$. A table with columns $x$, $y$, $\hat{y}$ and $y - \hat{y}$ keeps each actual value next to its prediction. Then plot each pair $(x,\ y - \hat{y})$, with $x$ across as before and residuals up the side, and draw the horizontal line at residual $0$. That line stands for the fitted line itself, since a dot exactly on the fitted line has residual $0$. ### [Making bar graphs](https://math.wiki/make-a-bar-graph/) **Summary:** Turning a frequency table into bars of the right height with a labeled scale. **Question:** You have counted how many of each kind there are. How do you draw those counts so that anyone can see them at a glance, and read the exact numbers back? ## When you reach for it Feruza's class has a sidewalk chalk art day. Each child picks one color of chalk to start with, and Feruza keeps the count in a table (frequency-tables): Blue $7$, Pink $5$, Yellow $3$, Green $6$. The table holds every number, but a reader has to compare four numbers in their head to see which color won. A **bar graph** shows the same counts as bars, one for each color, and the longest bar is the winner before anyone reads a number. The child has already read bar graphs (read-a-bar-graph): find the scale, follow the end of a bar across to it, read the number. Making one runs that the other way. Start from a count, and draw a bar whose end sits level with that count on the scale. Every part a reader relies on (a title, labels, a scale from $0$ in equal steps) is now something the maker has to put there. You reach for this whenever counts sorted into a few kinds need to be shown: a class vote, a tally from the playground, things brought in for a project. Every graph here has at most four categories, and the counts come from a table, a tally, or a plain list. The lesson takes three days. Day 1 builds a bar graph from a table or a tally on a scale that counts by $1$, with bars up the page and across it. Day 2 checks a finished graph against its table, and starts from a list in which one category was chosen by nobody. Day 3 is short: a scale that counts by $2$ or $5$ when the counts are big, and one look at a double bar graph. ### [Mean](https://math.wiki/mean/) **Summary:** Add them up, share them out evenly. **Question:** A list of numbers is all different. Is there one number that can speak for the whole list, and what does it really mean? ## What this area is about The mean is the number you get by adding the values and dividing by how many there are. For $4$, $5$, $5$, $8$ and $13$ it is $35 \div 5 = 7$. That one calculation has two meanings, and each one explains something the other does not: it is a fair share, and it is a balance point. ## How these fit together **Two definitions, one number.** Mean as fair share pools the total and shares it out evenly, which is where the "add up, then divide" rule comes from. Mean as balance point places the values on a number line and finds where the distances above and below cancel, which shows how the mean moves when one value changes. **A procedure.** Mean from a frequency table finds the mean when each value comes with a count, and extends the idea to a weighted mean where some values count for more than others. ## A path through it Fair share first, then balance point, then the frequency table. The two meanings can be read in either order, but the computing rule is easiest to meet as a fair share. The mean is the center that mean absolute deviation measures spread around, and later the average value of a function is the same fair share taken over a continuous interval. ### [Mean absolute deviation](https://math.wiki/mean-absolute-deviation/) **Summary:** Average distance from the mean. **Question:** Two punnets of strawberries have the same mean mass. Does that make them alike, and what one number says how spread out each one is? ## Why this exists Didier and Dilys each buy a punnet of six strawberries and weigh them one by one. Didier's weigh $13$, $14$, $15$, $15$, $16$ and $17$ grams; Dilys's weigh $8$, $10$, $15$, $15$, $20$ and $22$ grams. Both punnets hold $90$ grams, so both have a mean of $\frac{90}{6} = 15$ grams. By the mean the two punnets are the same. Tip them out, though, and the difference is plain: Didier's strawberries are all much the same size, while Dilys's run from tiny to large. A center alone cannot tell these apart. What is missing is a second number that says how far the values tend to sit from the center. This lesson builds one, the **mean absolute deviation**, out of the distances from the mean that the mean as a balance point already measures. The first day defines it, computes it, reads it off a dot plot and says what it means. The second compares data sets with the same mean, shows why the MAD adds the distances on both sides of the mean rather than taking one side from the other, and watches the MAD respond to a new value. ## Definition A value's **distance** from the mean is how far apart the two sit on the number line: the larger minus the smaller. For a value $x$ and a mean $m$ it is written with absolute value bars, $\lvert x - m \rvert$, and read "the distance between $x$ and $m$"; it is the same whichever of the two is larger. With $x = 3$ and $m = 5$ the distance is $5 - 3 = 2$. As in the mean as a balance point, a value below the mean is on the **low side**, a value above it on the **high side**, and a value equal to the mean has distance $0$ and sits on neither side. ### [Mean as balance point](https://math.wiki/mean-as-balance-point/) **Summary:** The place on the number line where the distances above and the distances below cancel out. **Question:** A list of numbers is spread along a line. Is there one point where it balances, and how does that point move when one number changes? ## Why this exists Five friends compare their weekly pocket money: $4$, $5$, $5$, $8$ and $13$ dollars. What amount is typical? One answer is the fair share: put all the money in one pile and share it out equally. The pile holds $35$ dollars, and five equal shares are $7$ dollars each. That number is the **mean**. A calculation hides where its answer sits. Mark the five amounts on a number line and $7$ lands in an odd spot: three friends below it, only two above, and nobody actually gets $7$ dollars. Why there? This lesson gives the mean a place on the line: the point where the data balance, like a seesaw on its pivot. That picture explains why one large value drags the mean toward it, exactly how far the mean moves when one value changes, and how to recover a value that went missing when only the mean was kept. The first day recaps the fair share and builds the balance picture. The second follows a changed value and finds a missing one. ## Definition The **mean** of a list of numbers is the total of the values divided by how many values there are: $$\text{mean} = \frac{\text{total of the values}}{\text{number of values}}$$ It is the fair share: if the total were shared out so that every value became the same, each would be the mean. The full lesson on computing it is the mean as a fair share; the lines above are all this lesson needs. A value's **distance** from a point is how far apart the two sit on the number line: the larger number minus the smaller. Measured from the mean, a value below it is on the **low side**, a value above it on the **high side**, and a value equal to the mean has distance $0$ and sits on neither side. ### [Mean as fair share: computing the mean](https://math.wiki/mean-as-fair-share/) **Summary:** The amount each individual would get if the total were shared out evenly. **Question:** Three friends have stacks of cards of different heights. If they pooled their cards and shared them out evenly, how many would each one get, and what does that one number say about all of them? ## Why this exists Chiamaka, Ilker and Noelani each keep a stack of trading cards. Chiamaka has $7$, Ilker has $2$ and Noelani has $6$. Who has "about how many"? No single stack answers that. Chiamaka's is the biggest, Ilker's the smallest, and neither speaks for the group. Here is a question that does have one answer: if the three friends put all their cards in one pile and shared the pile out evenly, how many would each get? That number uses every stack, and it is the same for everyone. It is called the **mean** of the numbers. The mean lets you describe a whole list with one number. It lets you compare two lists of different lengths, like five days of cookies against seven. And because it is tied to the total, it lets you work backward: if you know the mean and how many values there are, you know the total, and from the total you can find a value that went missing. The lesson takes three days. Part 1 levels towers and finds the mean as the total divided by how many. Part 2 meets means that are not whole numbers, including means of fractions. Part 3 works backward from the mean to a missing value, and asks what happens to the mean when a value changes or a new one is added. ## Definition The **mean** of a list of numbers is the amount each one would be if the total were shared out evenly among them. Sharing a total evenly into a number of equal parts is division (division as sharing: the total is split into equal groups, one group for each). So the mean is found by adding all the values and dividing by how many values there are: ### [Mean from a frequency table and weighted mean](https://math.wiki/mean-from-a-frequency-table/) **Summary:** Finding the mean when each value carries a count instead of being listed one at a time. **Question:** A table says how many times each value happened instead of listing every value. How do you find the mean without writing the whole list out, and what if some values should count for more than others? ## When you reach for it Duncan keeps a small flock of hens and notes how many eggs he collects each day. After twenty days his notebook could hold twenty numbers, but most of them repeat, so he keeps a frequency table instead: each number of eggs, beside the number of days it happened. A dot plot holds the same information as stacks of dots. The mean is the total of the values divided by how many values there are. A table or a dot plot writes down neither number, but both can be read straight off it. Sometimes values are meant to count for different amounts. A spelling bee may make its final round worth half of the overall score, and a farm stand may sell many boxes of eggs at one price and few at another. The mean that respects those differences is the **weighted mean**, and it is the same calculation with the counts renamed **weights**. ## The procedure **Each column stands for a value repeated.** In Duncan's table, the column "$8$ eggs on $6$ days" stands for the list $8, 8, 8, 8, 8, 8$. Adding those six values means adding $8$ six times, which is $8 \times 6 = 48$, because multiplication is repeated addition. So each column puts value $\times$ frequency into the total, and the total of all the values is the sum of those products. **The count is the total frequency.** Every day is one value, so the number of values is the sum of the frequency row. It is not the number of columns: Duncan's table has five columns only because five different egg counts happened, but it holds twenty values. Then the definition of the mean, unchanged, reads ### [Mean, median and skew: choosing the center](https://math.wiki/mean-and-median-in-a-skewed-distribution/) **Summary:** The long tail pulls the mean toward it while the median stays near the bulk of the data. **Question:** When data has a long tail, the mean and the median stop agreeing. Which one is the center, and how does the shape tell you where each will land? ## Why this exists Marius looked up the populations of the $15$ towns in his region, to the nearest thousand people. In thousands, sorted, they are $$\begin{aligned} &2,\ 3,\ 3,\ 3,\ 4,\ 4,\ 4,\ 5,\\ &5,\ 6,\ 6,\ 8,\ 10,\ 15,\ 27 \end{aligned}$$ The median is the eighth value, $5$ thousand. The total is $105$ thousand, so the mean is $\frac{105}{15} = 7$ thousand. Two correct calculations, two different answers, and $11$ of the $15$ towns are smaller than the mean. So is a typical town home to $5$ thousand people or to $7$ thousand? That depends on the shape of the data and on what the number is for. ## Definition The **mean** is the total of the values divided by how many there are. The **median** is the middle value of the sorted list, or halfway between the two middle values. Both are measures of **center**: one number standing for the whole data set. **Choosing a center** means deciding which one describes the data for the job at hand: - the **median**, when you want a typical value and the data is skewed or has outliers, since it stays with the bulk of the values; - the **mean**, when the total matters (sharing a cost evenly, or finding a total from a count), and when the data is roughly symmetric, so that the two nearly agree and the mean uses every value. A center is reported with a matching **spread**: the median with the IQR, and the mean with the MAD, which measures distances from the mean. The shape words are those of skewed-distributions. ### [Median and mode](https://math.wiki/median-and-mode/) **Summary:** The middle value, and the most common one. **Question:** The mean is not the only way to say what is typical. What other numbers can stand for a whole data set, and when are they the better choice? ## What this area is about Two more answers to "what is typical?" need no adding at all. The median is the middle value once the data are in order. The mode is the value that occurs most often. The median ignores how far the extreme values sit, and the mode works even for categories, like colors, that cannot be added or ordered. ## How these fit together **A procedure.** Finding the median sorts the data and takes the middle value; with an even number of values, two share the middle and the median is halfway between them. For $n$ values the middle sits at position $\frac{n+1}{2}$. **A definition.** Mode and bimodal data defines the most common value, and says what it means when two separate values are both common: often, two groups mixed together. ## A path through it The two can be taken in either order. The median comes first here because it is the one the rest of the branch builds on. The median is the first cut of the quartiles, which lead to the interquartile range and box plots. Two-peaked data return in uniform and bimodal shapes, part of describing the shape of a distribution. ### [Misleading graphs](https://math.wiki/misleading-graphs/) **Summary:** Displays whose scale, cut axis or picture sizes make a difference look bigger or smaller than it is. **Question:** Every number on the graph is right. How can the picture still say something false, and how big is the false impression? ## Why this exists The eye reads a graph by comparing sizes before it reads any tick label. A bar twice as tall says "twice as much"; a line climbing evenly says "steady growth". That is what makes a graph quick to read, and it is also what lets a graph mislead while every number on it is correct. Reading a graph you did not draw starts with the title, the axes and the key. This lesson compares what the picture seems to say with what the numbers say, and measures the gap. "It looks $4$ times bigger, but it is really $1.25$ times bigger" is a claim anyone can check. ## Definition A graph is **misleading** when the comparison its picture makes (lengths of bars, areas of pictures, sizes of slices, steepness of a line) does not match the comparison the data make, even though each value shown is correct. For two amounts, the two comparisons are two ratios: - the **drawn ratio**: the size of one drawn shape divided by the size of the other; - the **true ratio**: one value divided by the other. On an honest bar graph or picture graph the two ratios are equal. When they differ, the drawn ratio is the size of the false impression. ## Intuition Figure 1 shows the same two sales figures twice. On the left the value axis starts at $1{,}100$, so each bar shows only the part of its value above $1{,}100$, and the eye compares those parts. On the right the axis starts at $0$ and the bars are close in height, as the numbers are. ### [Mode and bimodal data](https://math.wiki/mode-and-bimodal-data/) **Summary:** The most common value, and what it means when two separate values are both common. **Question:** Jelly bean colors cannot be added up or put in order, yet "which color is most common?" still has an answer. What is that answer, and what does it mean when two values stand out at once? ## Why this exists Asa tips a bag of jelly beans onto the table and sorts them by color. What color is a typical bean? The mean cannot answer: there is no adding red to green and sharing the total out. The median cannot answer either: colors have no order, so there is no middle color. Yet the question has a plain answer, the color that turns up most often. Numbers have a most common value too: an answer that really was given, more often than any other. And when two values stand out at once, far apart, that is news in itself, because it often means the data holds two groups mixed together. ## Definition The **frequency** of a value is the number of times it appears in the data. The **mode** of a data set is the value, or category, with the highest frequency. - If two or more values tie for the highest frequency, each of them is a mode. - If the data holds at least two different values and every value has the same frequency, no value stands out, and the data has **no mode**. - Data with exactly two modes is called **bimodal**, meaning "two modes". The word is also used more loosely, for data whose plot shows two separate peaks, each standing well above the values around it, even when one peak is a little taller than the other. Example 6 shows such a case. The mode is a value, never a count. If $7$ shells is the answer given most often, and it was given $3$ times, then the mode is $7$ shells and $3$ is its frequency. ### [Normal calculations: areas and percentiles](https://math.wiki/area-under-a-normal-curve/) **Summary:** Turning a range of values into z-scores and reading off the share of the distribution that falls there. **Question:** A normal model describes a real measurement in grams or seconds. How do we get from it the share of values in a given range, or the value that cuts off a given share? ## When you reach for it A normal model $N(\mu, \sigma)$ has been fitted to a real measurement: bag weights, brewing times, fish lengths. Two kinds of question then come up, and they run in opposite directions. - **From a value to a share.** What share of bags weigh under $1{,}000$ g? What share of brews take between $141$ and $168$ seconds? The range is given in the measurement's own units, and the answer is an area under the density curve. - **From a share back to a value.** What weight do only the heaviest $10\%$ of bags exceed? Where do the middle $50\%$ of brewing times lie? Now the share is given, and the answer is a percentile in the measurement's units. No normal curve has a simple formula for its areas, and no table could be printed for every choice of $\mu$ and $\sigma$. The standard normal distribution has one table that serves them all, because every normal model turns into it when its values are written as z-scores. So both kinds of question are answered by passing through $z$: into it on the way to a share, and out of it on the way back to a value. This lesson takes two days, one for each direction. ## The procedure The table below holds every standard normal area this lesson uses. Each entry is $\Phi(z) = P(Z < z)$, the area to the left of $z$ under $N(0, 1)$, rounded to four places; where it overlaps the table in standard-normal-distribution, it is the same table. ### [Null and alternative hypotheses](https://math.wiki/null-and-alternative-hypotheses/) **Summary:** A claim of no effect set against the claim the researcher is investigating, both about a parameter. **Question:** A surf school says that 80% of its beginners stand up on their first lesson. Before anyone looks at data, what exactly is the claim being tested, and what would count against it? ## Why this exists A made-up surf school on Oahu advertises that $80\%$ of beginners stand up on their first lesson. Makoa, thinking of signing up, doubts it. He watches $50$ beginners and sees $36$ of them stand up, a sample proportion of $\frac{36}{50} = 0.72$. Is that evidence that the advertisement is wrong? Not by itself. The $0.72$ is a statistic, worked out from the $50$ beginners he watched; the advertisement is about a parameter, the share of all the school's beginners (see parameters and statistics). Even if the true share were exactly $0.80$, a sample of $50$ would rarely land on exactly $0.80$, because statistics vary from sample to sample. So the question has to be asked more carefully: is $0.72$ further below $0.80$ than chance alone would usually carry a sample? To ask that, Makoa first has to say precisely which claim he is testing and what he suspects instead. Those two statements are the **null hypothesis** and the **alternative hypothesis**, and every hypothesis test starts by writing them down. ## Definition A **hypothesis** in statistics is a claim about a parameter of a population, such as a population proportion $p$ or a population mean $\mu$. The **null hypothesis**, written $H_0$, is the claim of no effect, no change, no difference, or the value someone has claimed. It is always written as an equality: the parameter equals one particular number, called the **null value**. ### [Observational studies and experiments](https://math.wiki/observational-vs-experimental/) **Summary:** Watching versus assigning, and what each can and cannot conclude. **Question:** Plants given fertilizer grew bigger. Did the fertilizer do that, or were those plants different to begin with? ## Why this exists Oktavia grows lettuce. She gives fertilizer to some of her plants, and those plants end up heavier. It is tempting to say the fertilizer did it. But suppose she gave the fertilizer to the plants that already looked strongest. Then the fertilized plants were ahead before any fertilizer touched them, and their extra mass proves nothing. Every question of the form "does this cause that?" runs into the same wall. Two groups differ in the thing you care about, and they may also differ in a dozen things nobody measured. How the data was collected decides whether you can get past that wall. This lesson sorts studies into three kinds and says what each kind is allowed to conclude. All the studies and numbers in it are made up. ## Definition A **sample survey** asks a sample questions in order to describe a population: how many lettuces shoppers buy in a month, how many books customers read in a year. Its aim is a number for the whole population, which is the business of populations-and-samples and simple random samples. An **observational study** records what its subjects already do or already are, and imposes nothing. The researcher watches, measures or looks up records, and usually compares groups that formed on their own: people who keep a vegetable garden against people who do not. An **experiment** imposes something on its subjects on purpose and then measures what happens. The thing imposed is a **treatment**, and whatever it is given to (people, plants, anything) are the **subjects** of the experiment. ### [One sample interval for a mean and sample size](https://math.wiki/one-sample-t-interval-for-a-mean/) **Summary:** Estimating a population average as the sample average give or take a margin of error. **Question:** A random sample of animals has been weighed. What range of values for the mean of the whole herd does the sample support, and how many animals must be weighed to pin that mean down to within a chosen margin? ## When you reach for it Illari runs a llama ranch in the highlands of Peru, and every year each animal is shorn and its fleece weighed (made-up data throughout this lesson). One herd has $600$ llamas. The wool buyer wants an estimate of $\mu$, the mean fleece weight for the whole herd this year, before shearing is finished, so Illari shears a simple random sample of $15$ llamas first. Their mean fleece is $4.2$ pounds. The buyer does not care about these $15$ animals; the price rests on all $600$, and another $15$ would have given another mean. The one-sample interval for a mean turns one sample mean into a range of plausible values for the population mean, with a stated confidence level. It is the move for any measured quantity in one random sample: weights, times, lengths, scores. Its second use comes before any data exist: run backwards, the same formula says how many individuals to measure so that the interval comes out no wider than wanted. The parts are already in place. conditions-for-one-mean-inference says when the sample mean behaves well enough for this: a random sample, no more than a tenth of the population, and a population close to normal or a sample large enough. t-distribution supplies the curve to use when the population's spread has to be estimated from the sample, with $n - 1$ degrees of freedom. critical-value turns a level into a multiplier. The same plan for a share instead of a mean is one-sample-z-interval-for-a-proportion. ### [One sample interval for a proportion and sample size](https://math.wiki/one-sample-z-interval-for-a-proportion/) **Summary:** Estimating a population share as the sample share give or take a margin of error. **Question:** A random sample found that some share of the people asked have a feature. What range of values for the share in the whole population does that sample support, and how many people must be asked to pin the share down to within a chosen margin? ## When you reach for it The public library system of Sapporo wants to know what share of its cardholders borrowed an e-book last year. There are about $180{,}000$ cardholders, far too many to ask, so Yuto, who studies the system's records, asks a simple random sample of $500$ and finds that $210$ did (made-up data throughout this lesson). The sample share is $0.42$. The system does not care about these $500$ people; it cares about all $180{,}000$, and a different sample would have given a different share. The one-sample interval for a proportion turns one sample share into a range of plausible values for the population share, with a stated confidence level. It is the move for any yes-or-no feature counted in one random sample: the share who borrowed, voted, own, attended. Its second use comes before any data exist. When a survey is being planned, the same formula, run backwards, says how many people to ask so that the interval comes out no wider than wanted. The parts are already in place. conditions-for-one-proportion-inference says when the sample share has a sampling distribution close to normal: a random sample, a sample no more than a tenth of the population, and at least $10$ successes and $10$ failures. critical-value supplies the multiplier $z^*$ for a chosen level. This lesson puts them together, in two days: the interval itself, then the sample size. ## The procedure ### Day 1: the interval ### [One sample test for a mean](https://math.wiki/one-sample-t-test-for-a-mean/) **Summary:** Testing a claimed population average against what the sample found. **Question:** For years the terraces have averaged 5.8 metric tons of rice per hectare. Under a new water-sharing schedule, twelve plots average 6.15. Is that real evidence the mean has risen, when the spread of all yields is not known and has to be guessed from the same twelve plots? ## When you reach for it Someone claims a value for the mean of a quantitative variable in a population, and a sample is meant to test it. A rice cooperative in Bali farms several hundred terrace plots, and its records give a long-run mean yield of $5.8$ metric tons per hectare (every number in this lesson is made-up data). This season it tries a new schedule for sharing irrigation water between the terraces. Wayan, who keeps the cooperative's records, picks $12$ plots at random and finds a mean yield of $6.15$ metric tons per hectare. That is above $5.8$, but a sample mean never lands exactly on the population mean. The question is whether $6.15$ is further from $5.8$ than chance alone would plausibly carry it. The one-sample $t$ test answers that. It uses the same reasoning as the test for a proportion, with one change: the population standard deviation $\sigma$ is unknown, so the sample's $s_x$ stands in, and the statistic follows a t-distribution instead of the normal curve. The lesson runs over two days. Part 1 carries out a full one-sided test. Part 2 adds two-sided tests, their exact match with the t interval, calculator and spreadsheet output, and a case where a condition fails. ## The procedure **The parameter and the hypotheses.** Name the parameter in context first: $\mu$ is the mean yield, in metric tons per hectare, of all the cooperative's plots under the new schedule. The null hypothesis gives the claimed value $\mu_0$; the alternative says what the investigator suspects, chosen before the data are seen (null-and-alternative-hypotheses): ### [One sample test for a proportion](https://math.wiki/one-sample-z-test-for-a-proportion/) **Summary:** Testing a claimed population share against what the sample found. **Question:** For years one kiwifruit in ten has been too small to export. After a dry summer, a sample of 400 fruit has 52 undersized. Is that real evidence the share has gone up, or ordinary sampling luck? ## When you reach for it Someone states a value for the share of a population with some feature, and a sample is meant to test it. Tamatea grows kiwifruit in New Zealand. In past seasons $10\%$ of the orchard's fruit has been too small for export (all numbers in this lesson are made-up data). After a dry summer he suspects the share is higher this year. He picks a random sample of $400$ fruit from across the orchard and finds $52$ undersized, a sample proportion of $0.13$. Thirteen percent is more than ten, but a sample never lands exactly on the population value, so the question is whether $0.13$ is further from $0.10$ than chance alone would plausibly carry it. The one-sample $z$ test for a proportion answers that. It brings together pieces from earlier lessons: hypotheses (null-and-alternative-hypotheses), the three conditions (random, independent, large counts checked with the null value), the test-statistic (how many standard deviations the sample sits from the claim, as the geothermal spa's $89$ of $250$ massage bookings showed), the p-value, and the decision of reject-or-fail-to-reject. The lesson runs over two days. Part 1 carries out a full one-sided test. Part 2 adds two-sided tests, a test that fails to reject, technology output, and what to do when a condition fails. ## The procedure **The parameter and the hypotheses.** The test is about a population, so it begins by naming the parameter in context: $p$ is the proportion of all fruit on Tamatea's orchard this season that is too small for export. The null hypothesis gives the claimed value $p_0$, and the alternative says what the investigator suspects, chosen before looking at the data: ### [Outliers and the 1.5 IQR rule](https://math.wiki/iqr-outlier-rule/) **Summary:** Marking as an outlier any value more than one and a half interquartile ranges beyond a quartile. **Question:** One value sits far from all the others. How far is far enough to call it an outlier, and what should be done with it once it is found? ## When you reach for it Heddwyn measures eleven young sharks that a research boat catches and releases. Ten are between $61$ and $84$ centimeters long; one is $131$ centimeters. A value that sits far from the rest of the data is an **outlier**. An outlier can come about in three ways: - **A mistake:** a length misread, a digit typed wrong, a value written in the wrong unit. - **A rare but real case:** a shark that really is that long. - **A member of a different group:** an adult among young sharks, or an animal of another kind counted in by accident. The numbers alone cannot say which happened; checking the record can. But first people must agree which values count as far, and by eye, on a dot plot, two people can disagree. The **1.5 IQR rule** settles it with arithmetic, so everyone flags the same values. Reach for it whenever a data set seems to hold a value far from the rest, before reporting a summary of it or drawing a box plot that shows outliers. The rule uses the quartiles $Q_1$ and $Q_3$ and the interquartile range $\text{IQR} = Q_3 - Q_1$, found as in that lesson: sort the data, split it at the median, and when the count is odd leave the median out of both halves. ## The procedure **The ruler is the IQR.** The IQR is the width of the middle half of the data, the ordinary values, so measuring "far" in IQRs fits the rule to each data set: $10$ millimeters beyond the middle half is far when that half spans $2$ millimeters, and not far when it spans $50$. The range would be a poor ruler, because the far value is itself one end of the range: the farther out it sits, the wider the range, so the ruler would grow with the very thing it measures. With six or more values the quartiles do not use the smallest or largest value, so the IQR stays put however far out an extreme value sits. ### [Outliers, leverage and influential points](https://math.wiki/outliers-vs-influential-points/) **Summary:** A point far from the pattern is an outlier, while a point that visibly moves the line is influential. **Question:** One point sits apart from the rest of a scatter plot. Does it matter to the fitted line, and how can you tell? ## Why this exists A least-squares line makes the sum of the squared residuals as small as possible (least-squares-criterion). Squaring makes a big miss very expensive, so the line will bend a long way toward a point it would otherwise miss badly. One unusual point can therefore decide the slope that is meant to describe all the others. With one variable, an outlier is a value far from the rest of its list, and the iqr-outlier-rule marks it. With two variables a point can be unusual in two different directions: its $y$ can sit far from the pattern of the other points, or its $x$ can sit far from the other $x$ values. These two kinds behave very differently, and only one of them tends to move the line. This lesson names the kinds and gives the test that tells them apart: remove the point and fit again. ## Definition Take a least-squares line $\hat{y} = a + bx$ fitted to a set of points, with residuals $y - \hat{y}$ (residuals) and residual standard deviation $s$ (regression-computer-output). - An **outlier** in regression is a point whose $y$ is far from the pattern of the other points: its residual is large. A common guideline flags a point whose residual is more than $2s$ away from $0$. - A point has **high leverage** when its $x$ is far from the mean $\bar{x}$ of the other points' $x$ values. A z-score for $x$, $z = \dfrac{x - \bar{x}}{s_x}$ computed from the other points, measures how far. - A point is **influential** when removing it and fitting the line again changes the slope, the intercept or the predictions noticeably. **Removing a point and refitting** is the test. ### [p-values and the significance level](https://math.wiki/p-values/) **Summary:** The probability of data at least this extreme if the null were true, and what that does not mean. **Question:** A new furnace makes fewer flawed vases in one sample. If the furnace were really no better, how often would a sample look this good by luck alone, and how small does that chance have to be before it counts? ## Why this exists A made-up glassworks in Bohemia has blown vases for years in an old furnace, and $8\%$ of its vases come out with bubbles trapped in the glass. Zdenka, who runs the works, tries a new furnace that is meant to hold a steadier heat. She takes a random sample of $300$ vases from the new furnace, and $15$ of them have bubbles: $$\hat{p} = \frac{15}{300} = 0.05$$ Her hypotheses were written before she looked (see null-and-alternative-hypotheses): $p$ is the proportion of all vases from the new furnace that have bubbles, and $$H_0\colon p = 0.08 \qquad H_a\colon p < 0.08$$ The sample came in at $0.05$, below $0.08$. But samples vary. Even if the new furnace were no better at all, some samples of $300$ would land below $0.08$ by chance, and a few would land well below it. The question Zdenka needs answered is a number: if the new furnace really makes bubbles in $8\%$ of its vases, how often would a sample of $300$ come out at $0.05$ or lower? That number is the **p-value**. The second question, how small it must be before she takes it seriously, is answered by the **significance level**. ## Definition The lesson runs over two days. Part 1 defines the p-value and finds it from a test statistic; Part 2 sets the significance level, reads the p-value as a conditional probability, and finds it by simulation. **The p-value.** Suppose $H_0$ is true. The **p-value** is the probability of getting a statistic at least as extreme as the one actually observed, where "extreme" means far from the null value in the direction (or directions) that $H_a$ names. For a test about one proportion $p$, with null value $p_0$, the extreme direction comes from the alternative: ### [Paired data: interval and test for a mean difference](https://math.wiki/paired-data/) **Summary:** Two measurements linked on the same individual or matched pair, reduced to one column of differences. **Question:** Ten swimmers each swim 100 meters once in each of two swimsuits. The swimmers differ from one another by many seconds, while a swimsuit might change a time by less than one. How can the twenty times show what the suit does? ## Why this exists A swim club in Athens wants to know whether a new tech swimsuit makes its swimmers faster over $100$ meters than the club's standard suit (every number in this lesson is made-up data). Katerina, the head coach, has ten swimmers each swim $100$ meters twice, once in each suit, with a long rest between. For each swimmer a coin decides which suit comes first, so tiredness or warm-up on the second swim cannot always favor the same suit. This is the matched pairs design of blocking-in-an-experiment, where each delivery van carried both kinds of tire: here each swimmer is a block of two swims. The fastest swimmer finishes in about $57$ seconds and the slowest in about $76$, while a suit might change a time by a fraction of a second. Compared as two groups, any suit effect is buried under the gap between swimmers. But each swimmer's two times belong together, and the change from one to the other is measured against that same swimmer. Analyzing those changes is what makes a small effect visible. Part 1 (day 1) defines paired data, reduces the pairs to one column of differences, and runs the one-sample $t$ procedures on it. Part 2 (day 2) analyzes the same times wrongly, as two independent samples, to show what pairing buys and why, and settles how to tell from a design which procedure fits. ## Definition Two sets of measurements are **paired** when each measurement in one set belongs with exactly one measurement in the other, through the way the data were collected: two measurements on the same individual (one swimmer in two suits, a score before and after), or one on each member of a matched pair (two swimmers matched by best time, one assigned to each treatment at random). ### [Parameters and statistics](https://math.wiki/parameter-vs-statistic/) **Summary:** A parameter describes the whole population and a statistic describes the sample in hand. **Question:** You wanted a number about a whole group, but you measured only part of it. Is the number you worked out the number you wanted? ## Why this exists A farmer wants the mean height of the sunflowers in her field. There are thousands of them, so she measures $25$ and works out the mean of those $25$ heights. She now holds a number, but not the number she wanted: hers describes the $25$ she measured, and the one she wanted describes every plant in the field. They are the same kind of number about the same plants, yet reporting one as the other claims more than she knows. Every study that measures part of a group (see populations and samples) holds these two numbers. Statistics names them so they cannot be confused: the number about the whole group is a **parameter**, and the number from the part is a **statistic**. ## Definition A **parameter** is a number that describes a whole population. It is worked out, or could be worked out, from the values of every individual in the population. Usually nobody knows it, because nobody has measured everyone. At any one moment a population has one value of each parameter. A **statistic** is a number worked out from the sample in hand. Once the sample is measured, the statistic is known. A different sample holds different individuals, so it usually gives a different value. This lesson uses two kinds of number, and each comes as a pair. | Kind | The parameter | The statistic | |---|---|---| | a mean | **population mean**: the mean of the values of every individual | **sample mean**: the mean of the values in the sample | | a proportion | **population proportion**: the share of all the individuals with some trait | **sample proportion**: the share of the sample with it | ### [Percentiles](https://math.wiki/find-the-value-at-a-percentile/) **Summary:** Going from a given percent back to the data value that sits there. **Question:** Where does one value sit among all the others, and which value sits at a given place in the line? ## When you reach for it A marathon finishing time of $252$ minutes means little on its own. Was it quick or slow for that race? The useful statement is where it sits: "$60$ percent of the runners finished in that time or less." That is a percentile. You reach for percentiles when one value has to be placed among many, and for the reverse question: which time marks the quickest tenth of the field, or the middle? The quartiles already do this at three places: the lower quartile $Q_1$ a quarter of the way up the ordered data, the median half way, the upper quartile $Q_3$ three quarters of the way, with the interquartile range $Q_3 - Q_1$ measuring the middle half. Percentiles allow any place, not just those three. They are how a large data set, such as a race with thousands of runners, is summarized, and they are the curves on a baby's growth chart. ## The procedure ### What a percentile is The $p$th percentile, written $P_p$, is a value with about $p$ percent of the data at or below it, and so about $100 - p$ percent above it. Three percentiles already have names: $$P_{25} = Q_1, \qquad P_{50} = \text{median}, \qquad P_{75} = Q_3.$$ The word "about" matters. With $20$ values, each value is $5$ percent of the data, so the share at or below a point moves in steps of $5$ percent and may never equal $42$ percent exactly. The rule below says which value to report. ### The percentile rank of a value ### [Picture graphs and their key](https://math.wiki/picture-graph-key/) **Summary:** The note saying how many things one picture stands for. **Question:** Twenty children each voted for a favorite fruit. How can one page show every vote at once, so that anyone can see which fruit won, and by how much? ## Why this exists Vladislava's class votes for a favorite fruit. Each child says one fruit, and the teacher writes it down. At the end there is a list of twenty words: banana, grapes, strawberry, banana, and so on. Every vote is on that list, but nobody can look at it and say which fruit won. To answer, you have to sort and count the whole list again. A **picture graph** (also called a pictograph) does the sorting and counting once, on paper, so that it never has to be done again. Each fruit gets its own row, and each vote becomes one small picture in its fruit's row. Now the answer can be seen. The longest row won, and the rows can be compared side by side. The graph also carries a small note called the **key**. It says what one picture stands for. In this lesson the key always says the same thing: one picture stands for one thing. That might seem too obvious to write. It is not. The key is the promise that lets a reader trust the count: it tells them that counting the pictures in a row gives the number of votes, and not some other number. A reader who meets a graph they did not make reads the key first, every time. This lesson takes three days. Day 1 reads a picture graph. Day 2 makes one from a tally or a list. Day 3 asks the graph questions: how many in all, how many more, how many fewer. Each graph here has at most four categories. ## Definition A **picture graph** shows how many things fall into each of a few groups, called **categories**, using one picture for each thing counted. It has: ### [Plotting paired data: explanatory and response variables](https://math.wiki/plot-paired-data/) **Summary:** Turning two measurements on the same individual into one point. **Question:** You measured two things about each person. How do you draw both at once, so that each person is still one mark you can point to? ## When you reach for it Some questions are about two measurements at once. Do runners who train longer run faster? Do hotter days sell more lemonade? Does a car lose value as it ages? A list of training times and a separate list of run times cannot answer these, because the question is about how the two go together for each runner. That is bivariate-data: two variables recorded on the same individuals. The picture for it is a scatter plot. Each individual becomes one point on a coordinate-plane, placed across by one measurement and up by the other, so the two numbers that belong together stay together. Saying what the whole cloud of points shows is the next lesson, describe-a-scatter-plot. ## The procedure **One individual, one point.** A table of bivariate data has one row per individual and one column per variable, as in individuals-and-variables. Six students had their arm span (fingertip to fingertip, arms stretched sideways) and their height measured. These numbers are made up. | Student | Arm span (cm) | Height (cm) | Point | |---|:-:|:-:|:-:| | Paloma | $162$ | $160$ | $(162, 160)$ | | Quentin | $175$ | $177$ | $(175, 177)$ | | Roswitha | $158$ | $161$ | $(158, 161)$ | | Sanjeev | $181$ | $178$ | $(181, 178)$ | | Tilde | $169$ | $166$ | $(169, 166)$ | | Uriah | $172$ | $174$ | $(172, 174)$ | Paloma's row becomes the ordered pair $(162, 160)$: arm span first, height second. Both coordinates come from the same person, so the point is Paloma herself, seen through two measurements. The data are called **paired** for this reason, and a scatter plot can only be drawn when each value is known to belong with its partner. ### [Point estimates and the margin of error](https://math.wiki/margin-of-error/) **Summary:** How far the estimate might reasonably sit from the true value, given the method's variability. **Question:** A sample gives one number, and that number is almost surely a little off. How far off might it reasonably be, and what makes that distance bigger or smaller? ## Why this exists A city transit agency wants to know what share of all its riders are satisfied with the service. Mehwish, who runs the survey, cannot ask every rider, so she asks a random sample of $600$ riders and $384$ say they are satisfied (made-up data). The sample proportion is $$\hat{p} = \frac{384}{600} = 0.64$$ Reporting "$64\%$ of riders are satisfied" claims too much. Another random sample of $600$ would give a slightly different share, and the true share $p$ among all riders is almost surely not exactly $0.64$. Reporting nothing but "somewhere between $0$ and $1$" claims too little. What the agency needs is the single best number together with an honest distance: how far that number might reasonably sit from the truth, given how much samples of this size vary. The idea first appeared in simulate-a-sampling-distribution, where a margin came from the spread of a pile of simulated samples. Here there is no pile and no simulation, only the one sample in hand, and the spread comes from its standard-error. ## Definition A **point estimate** is a single value of a statistic, computed from the sample, used as the estimate of a parameter. The sample proportion $\hat{p}$ is the point estimate of the population proportion $p$, and the sample mean $\bar{x}$ is the point estimate of the population mean $\mu$. Both are unbiased estimators when the sample is random. The **margin of error** is the distance, on each side of the point estimate, within which the estimate is likely to lie from the parameter, taking into account only the chance variation of random sampling. It is a multiple of the standard error: ### [Populations, samples and census](https://math.wiki/populations-and-samples/) **Summary:** The whole group you care about, and the part you actually measure. **Question:** You want a fact about a whole group, but you can only look at part of it. When does the part tell you the truth about the whole? ## Why this exists Some questions are about a group far too large to look at one member at a time. What percent of the fish in a lake carry a parasite? How long do the light bulbs in a shipment last? What do the adults of a city think of a plan for a new park? To answer exactly, you would have to check every fish, burn out every bulb and ask every adult. Measuring every member of a group like this is called a census, and often it cannot be done. It costs too much, it takes too long, the group changes while you are counting, or measuring destroys the thing measured: a bulb tested until it burns out can no longer be sold. So you measure part of the group and use what you find to say something about the whole. That move is the starting point of statistics, and it raises the question this lesson answers: when can the part stand in for the whole, and when does it mislead? The answer depends first on **how** the part was chosen, and only after that on how big it is. ## Definition A statistical question is always about some group. That group is the population. The **population** is the entire set of individuals that a statistical question is about. Each member of it is an **individual**. Individuals are often people, but they can just as well be fish, light bulbs, days or coin tosses (see individuals and variables). The number of individuals in the population is the **population size**, written $N$. ### [Power and what raises it](https://math.wiki/power-of-a-test/) **Summary:** The chance a test detects an effect of a given size when that effect is really there. **Question:** A new dough might tear less often. If it really does, how likely is a taste-kitchen trial to notice, and how many dumplings must the kitchen boil to be fairly sure it will? ## Why this exists A made-up pierogi kitchen in Krakow knows from long records that $15\%$ of its dumplings tear when boiled. Wojciech, the head cook, has a new dough. Malgorzata, who runs the kitchen, will switch only on evidence, so they plan a trial: boil a random sample of $n$ dumplings made with the new dough and test $$H_0\colon p = 0.15 \qquad H_a\colon p < 0.15$$ where $p$ is the proportion of new-dough dumplings that tear, at $\alpha = 0.05$ (a one-sample-z-test-for-a-proportion). Type I and Type II errors already gave the two ways this test can go wrong, and showed that $\beta$, the chance of missing a false $H_0$, is computed at one specific alternative value. Before boiling anything, Malgorzata wants the question turned around: if the new dough really is better, what is the chance the trial says so? A trial that would usually miss a real improvement is a waste of flour. That chance is the power, and it is the number used to choose $n$. ## Definition The **power** of a test at a specific alternative value $p_a$ is the probability that the test rejects $H_0$ when the true parameter value is $p_a$: $$\begin{aligned} \text{power} &= P(\text{reject } H_0 \mid p = p_a) \\ &= 1 - \beta \end{aligned}$$ Rejecting and failing to reject are the only two decisions, so at the same $p_a$ the power and the Type II error probability $\beta$ add to $1$. Like $\beta$, power has no single value: it is a number for each alternative value, and it also depends on $n$ and $\alpha$. ### [Predicting with a regression line, and extrapolation](https://math.wiki/predict-with-a-regression-line/) **Summary:** Putting a value of the explanatory variable into the equation to get a predicted response. **Question:** A line has been fitted to the data. What does it say about a case nobody measured, and how far can that answer be trusted? ## When you reach for it A nurse measured nine children, one of each age from $2$ to $10$ years (made-up data). As (age in years, height in cm) the pairs are $(2, 90)$, $(3, 93)$, $(4, 101)$, $(5, 109)$, $(6, 112)$, $(7, 118)$, $(8, 127)$, $(9, 130)$ and $(10, 137)$. Technology, as in fit-a-function-to-data-with-technology, gives the least-squares regression line $$\hat{y} = 77 + 6x$$ where $x$ is the age in years and $\hat{y}$, read "y-hat", is **the predicted value** of the height in centimeters. The hat separates a height the line predicts from a height somebody measured, which is written plain $y$. You reach for this line when you know the explanatory variable for a case and want its response: a child's age is known and the height is not. Sometimes the question runs the other way, from a target response back to the explanatory value. Either way, a second question comes with the first: is this value of $x$ where the data are? In line-of-best-fit predictions were read from a line drawn by eye; here the line is the least-squares line and the prediction is computed. ## The procedure ### Substituting the explanatory value The equation is the line, written as a rule: for each $x$ it gives the height of the line above that $x$. So a prediction is an evaluation. For a child aged $5.5$ years, $$\hat{y} = 77 + 6(5.5) = 77 + 33 = 110$$ The answer carries the response's units, centimeters, and is reported in a sentence that names both variables: "The line predicts a height of about $110$ cm for a child aged $5.5$ years." A number with no units and no context answers nothing. ### [Properties of the correlation coefficient](https://math.wiki/properties-of-r/) **Summary:** It has no units, does not change if the variables are swapped, and never leaves the interval from negative one to one. **Question:** If one scientist measures in centimeters and another in inches, or one puts a variable across and the other puts it up, do they get the same r? And can one unusual individual change it? ## Why this exists correlation-coefficient-r puts one number, $r$, on the direction and strength of a straight-line pattern in a scatter plot. Before leaning on a number like that, you want to know what moves it. If one scientist measures birds in centimeters and another in inches, can they compare their values of $r$? Does it matter which variable goes across? Can one unusual individual change the verdict? The answers decide when two values of $r$ can be compared at all, and when $r$ should not be trusted until somebody has looked at the plot. ## Definition This star collects the properties of the correlation coefficient $r$ of paired data $(x, y)$: what leaves it alone, what changes it, and which values it can take. Two words are used throughout. Changing the **units** of a variable multiplies every value by the same positive number (centimeters to inches divides by $2.54$), and sometimes adds a constant as well (degrees Celsius to degrees Fahrenheit multiplies by $1.8$ and adds $32$). A summary is **resistant** when a few extreme values barely move it (resistant-vs-nonresistant-measures): the median is resistant, the mean is not. ## Intuition $r$ measures how tightly the dots cluster about a straight line, judged in the cloud's own shape. Change the units and the scales get new numbers; redraw the plot to fit its frame and every dot sits exactly where it did. Add a constant to every value and the cloud slides along an axis without changing shape. Neither can move $r$. ### [Quartiles and the five number summary](https://math.wiki/quartiles/) **Summary:** The three values that cut an ordered data set into four parts of about equal size. **Question:** The median cuts sorted data into a lower half and an upper half. Where do the cuts go for four quarters, and how wide is the middle half of the data? ## Why this exists Carys keeps a reading log of how many pages she reads each week. Her median week says what a typical week looks like, but not how much her weeks vary. A reader who reads $44$ to $46$ pages every week and one who swings between $10$ and $90$ can share a median of $45$. Telling them apart needs a measure of **spread**: how far apart the values are. The quickest measure is the distance from the smallest value to the largest, but it rests on the two most extreme values only, so one unusual week can make it huge. A steadier picture comes from cutting the sorted data into four parts with the same number of values in each, and asking how wide the middle two parts are. The three cut points are the quartiles. ## Definition Sort the data from smallest to largest and find the median, as in median-of-an-even-list. The median splits the sorted list into a **lower half** and an **upper half**: - When the number of values $n$ is even, the lower half is the first $\frac{n}{2}$ values and the upper half the last $\frac{n}{2}$. - When $n$ is odd, the median is the single middle value. It is **left out of both halves**, so each half holds $\frac{n-1}{2}$ values. The **first quartile** $Q_1$ is the median of the lower half, and the **third quartile** $Q_3$ is the median of the upper half. The median itself is sometimes called the second quartile. Together they cut the sorted data into four parts holding about the same number of values, the **quarters** of the data. ### [r squared and the standard deviation of the residuals](https://math.wiki/coefficient-of-determination/) **Summary:** The share of the variation in the response that the regression line accounts for. **Question:** A least-squares line has been fitted. How much of the variation in the response does it actually account for, and how far off are its predictions, typically? ## Why this exists Hodan keeps eight beehives. At the end of the season she records, for each hive, the number of frames of drawn comb in its honey boxes and the honey she harvests, and fits a least-squares line. The line rises, and the correlation says the dots cluster fairly closely around it. But a beekeeper wants plainer answers. Of all the differences in harvest from hive to hive, how much does the number of frames account for? And when she uses the line to predict next year's harvest, how far off should she expect to be? Neither the slope nor $r$ answers those questions directly. This lesson builds two numbers that do. The first, **r squared**, compares the line's misses with the misses of the best guess made without $x$ at all. The second, $s$, puts the size of a typical miss in pounds. The lesson takes two days: r squared on the first, $s$ and how the two numbers differ on the second. ## Definition Without the explanatory variable, the best single prediction of every $y$ is the mean $\bar{y}$. Its misses are the deviations $y - \bar{y}$, and the **total sum of squares** collects them: $$\text{SST} = \sum (y - \bar{y})^2.$$ With the least-squares line, each miss is a residual $y - \hat{y}$, and the **sum of squared errors**, met in judge-a-models-fit, collects those: $$\text{SSE} = \sum (y - \hat{y})^2.$$ The **coefficient of determination**, written $r^2$ and read "r squared", is the share of $\text{SST}$ that the line removes: ### [Random assignment and confounding](https://math.wiki/random-assignment/) **Summary:** Using chance to decide which unit gets which treatment, so the groups start out alike. **Question:** An experimenter has to split the units into treatment groups somehow. Which ways of splitting can the comparison trust, and how do you check a split once it is made? ## Why this exists Mariam runs a made-up poultry farm with $24$ pens of laying birds, numbered $1$ to $24$ along the barn: pens $1$ to $12$ on the south side, $13$ to $24$ on the north side. She wants to know whether a new feed raises egg production compared with the standard feed, so $12$ pens will get each. The pens are the experimental units, and the standard-feed pens are the control group, the baseline the new feed is compared against. All data in this lesson are made up. Lasha, who hauls the feed, suggests the easy split: the truck parks at the south end, so give the new feed to pens $1$ to $12$. Tornike, who knows the birds, offers to pick the $12$ pens that look healthiest. Both plans let something besides the feed decide who gets it, and that something can also affect laying. observational-vs-experimental explains why chance is the cure; this lesson is about carrying it out, checking it with numbers, and what it cannot do. ## Definition **Random assignment** decides which unit gets which treatment by a chance process that gives every possible division of the units into groups of the planned sizes the same chance. With $24$ pens and two groups of $12$, every set of $12$ pens is equally likely to be the new-feed group. Three ways to do it: - **Slips in a hat.** Write each unit's label on an identical slip, mix well, draw $12$; those units get the new feed and the rest the standard feed. - **A table of random digits.** Label the units with the same number of digits ($01$ to $24$), read the table in blocks of that many digits, and use a block only if it is a label not already chosen: skip repeats and blocks out of range. This is the reading taught in simple-random-sample. - **Software.** Shuffle the list of labels and take the first $12$; or draw random integers from $1$ to $24$, skipping repeats, until $12$ are chosen. ### [Random sampling](https://math.wiki/random-sampling/) **Summary:** Every member equally likely, so the sample looks like the population. **Question:** You can only ask a few members of a population. How do you choose them so the few speak fairly for everyone? ## What this area is about When the person collecting data picks the sample, or the eager pick themselves, the result leans toward whoever was reached. Random sampling hands the choice to chance, so no one's preferences tilt it and anyone can check it was fair. The four methods here differ in what you need in hand (a full list, known groups, an ordered line) and in what they cost. ## How these fit together **The basic method.** A simple random sample of $n$ from $N$ gives every member, and every group of $n$ members, the same chance of being chosen. It needs a list of the whole population. **Variations that use structure.** A stratified random sample splits the population into strata that answer differently and draws a simple random sample inside each, so every group gets its share. A systematic sample takes a random start and then every $k$-th individual, with $k = \frac{N}{n}$ (rounded down when not whole), for individuals that arrive in order. A cluster sample draws whole ready-made groups at random and measures everyone in them, for when the groups can be listed but the members cannot. ## A path through it The simple random sample first. Then the systematic and stratified samples, and the cluster sample after the stratified one, since it reuses the idea of splitting into groups but draws whole groups. Finish here and bias shows what goes wrong without chance, and sampling distributions study how a random sample's result varies from draw to draw. ### [Range and interquartile range](https://math.wiki/range-and-iqr/) **Summary:** How far the data spans, and how far its middle half spans. **Question:** Two data sets share the same middle. How do you say which one is more spread out, without letting one strange value decide? ## What this area is about A center says where data sit; a spread says how far they wander. The simplest spread is the range, largest value minus smallest. One extreme value can stretch it a long way, so a sturdier measure looks only at the middle half of the data: the interquartile range, $\text{IQR} = Q_3 - Q_1$. Both come from cutting sorted data at fixed places. ## How these fit together **Definitions.** Quartiles and the five number summary cuts sorted data into four parts of about equal size at $Q_1$, the median and $Q_3$. With the minimum and maximum these make the five-number summary, and the range and the IQR are read straight off it. **A procedure.** Percentiles generalizes the cuts from quarters to any percent: where one value sits among the others, and which value sits at a given percent. ## A path through it Quartiles first, then percentiles, since $Q_1$ and $Q_3$ are the $25$th and $75$th percentiles (the two methods can differ slightly for some list sizes). The five-number summary is what a box plot draws, and the IQR powers the $1.5\,\text{IQR}$ outlier rule. Percentiles return in normal calculations. ### [Reading a bar graph](https://math.wiki/read-a-bar-graph/) **Summary:** Getting a count out of a bar by following it across to the scale. **Question:** A bar on a graph has no pictures in it to count. How can you tell what number it stands for? ## When you reach for it Ekow's class voted for a favorite season, and the teacher drew the votes as a **bar graph**. There is one bar for each season. A longer bar means more votes, so the winner can be seen at a glance. But a bar is a solid block. There are no pictures inside it to count, the way there are in a picture graph. So how does a reader find out that the Summer bar stands for $9$ students and not $8$ or $10$? The answer is the **scale**: a row of numbers along one edge of the graph, starting at $0$ and going up in equal steps. A bar graph is a picture graph whose pictures have been pushed together into one bar, with the counting moved out to the edge. The bar shows *how far*. The scale says *how many* that is. Reading a bar graph means carrying the end of each bar over to the scale and reading the number there. You reach for this skill whenever a bar graph is in front of you: a class vote, a weather chart, a count of anything sorted into a few kinds. Every bar graph in this lesson has at most four bars. Making one is a separate skill, make-a-bar-graph. The lesson takes three days. Day 1 names the parts of a bar graph and reads bars on a scale that counts by $1$, with bars drawn up the page and across it. Day 2 reads a scale that counts by $2$, including a bar that stops between two labeled lines. Day 3 asks the graph questions (how many in all, how many more, how many fewer) and looks at what a graph cannot tell you. ### [Reading a graph: titles, labels and units](https://math.wiki/read-a-graph-you-did-not-draw/) **Summary:** Working out what a display shows by reading its title, axes and key before reading any value. **Question:** Someone hands you a bar graph you have never seen. Before you read a single bar, what do you need to know to be sure what it says? ## When you reach for it Most graphs you meet were drawn by somebody else: a poster in the hallway, a page in a science book, a chart a classmate made after a survey. The person who drew it knew what it was about. You only have what is printed on the page. Making a graph is covered in make-a-bar-graph: a table of counts becomes one bar per category, each bar as tall as its count on a numbered scale. Reading a graph runs that the other way, and it starts with the words, not the bars. A bar on its own is just a shaded box. It becomes a number only once you know what the numbers on the side count by, and it becomes a fact only once you know what was counted. This lesson takes two class days. Part 1 reads the frame of a bar graph and then its values, including a bar that stops between two marks. Part 2 looks at the same data drawn on two different scales, at picture graphs and their keys, and at graphs with words missing. ## The procedure ### Part 1: the frame first, then the values Every bar graph has a frame of words and numbers around the bars. There are four parts to it. - **The title** says what was counted, and often where or when. "Students in Each Field Day Event" tells you the bars count students and that each bar is one event. - **The category labels** under the bars name what each bar stands for: one event, one person, one job. - **The label on the value axis** (the side with the numbers) says what the numbers count, and in what unit: students, laps, minutes, bottles. - **The scale** is the list of numbers up that side. What matters most is how much one space between two marks stands for. ### [Reading a histogram](https://math.wiki/read-a-histogram/) **Summary:** Getting the count in an interval from a bar, and seeing that individual values are no longer shown. **Question:** Thirty books, sorted by length into bars. What can the bars tell you about the books, and what have they hidden? ## When you reach for it Ayla counted the pages in every book on her class shelf. A list of thirty page counts is hard to take in; a **histogram** shows numerical data like this as a picture (categorical-vs-numerical-data): the number line is cut into **bins** of equal width that run end to end, and on each bin stands a bar as tall as the number of values that land in it, the bin's **count** or **frequency**. You reach for this skill whenever data come to you as a histogram: to say how many values lie in a stretch of the number line, how many there are in all, or where the median lies. One bar is read as on a bar graph (read-a-bar-graph): follow its top across to the scale. What is new is that the axis along the bottom is a number line, and that the separate values have gone. A bar says how many, never which. ## The procedure **What one bar says.** In Ayla's histogram (Example 1) the bar on the bin from $200$ to $300$ pages reaches $10$, so $10$ books have a page count in that bin. Whether they have $210$ pages or $290$, the bar does not say: inside a bin, the values could be anywhere. **A value on an edge.** The number $300$ ends one bin and starts the next, so a book of exactly $300$ pages needs a rule. In this course **a value on an edge goes in the bar to its right**. The bin "$200$ to $300$" holds every page count $x$ with $200 \le x < 300$, and a book of exactly $300$ pages is counted in the $300$ to $400$ bar. So every value belongs to exactly one bin. ### [Reading dot plots: clusters, gaps and peaks](https://math.wiki/read-a-dot-plot/) **Summary:** Counting the stack of dots above a value to see how often that value occurred. **Question:** Twelve worms from a garden dig have been measured, and their lengths sit in a list of twelve numbers. What picture lets you see at a glance how long the worms are, which length came up most, and how far apart the shortest and longest are? ## When you reach for it Tasi digs up a garden bed and finds $12$ worms. She lets each one stretch out along a ruler and writes down its length to the nearest inch: $5, 3, 6, 5, 4, 9, 6, 5, 7, 4, 5, 6$. Every worm is on that list, but the list does not answer much. Which length came up most? How long was the longest worm? Are most of the worms about the same size? To answer, you would have to sort the list and count it again, one number at a time. A **line plot** shows all twelve lengths at once. It is a number line, like the one on a ruler, with one dot above it for each worm, placed above that worm's length. When two worms have the same length, their dots stack up. The picture shows the whole list, already sorted and already counted. The same display is also called a **dot plot**. The two names mean the same thing, and this lesson says line plot. A line plot is a close cousin of the picture graph and the bar graph, but it is used for a different kind of list. A picture graph counts things in groups that have names, such as favorite fruits. A line plot counts **measurements**, which are numbers, so the groups sit in number order along a number line, and the space between them means something. That is what lets a line plot answer questions like "how many worms are longer than $5$ inches?" and "how far apart are the shortest and the longest?" Reach for a line plot whenever many things of one kind have been measured with the same unit and you want to see the measurements together. This lesson reads line plots; drawing one from a list is make-a-dot-plot. It takes three days. Part 1 reads the stacks and finds how many in all. Part 2 asks the plot questions: longer, shorter, most common, longest and shortest. Part 3 looks at the shape: where the dots bunch up, where there is an empty stretch, and where the tallest stack stands. ### [Reading regression output](https://math.wiki/regression-computer-output/) **Summary:** Finding the slope, intercept, residual spread and r squared in a printed regression table. **Question:** Software has fitted the line and printed a table of numbers. Which of them are the line, which say how well it fits, and which can wait? ## When you reach for it Zsofia measures twelve dragonflies (made-up data), recording body length $x$ and wingspan $y$, both in millimeters, as pairs $(x, y)$: $(42, 65)$, $(49, 77)$, $(50, 72)$, $(51, 70)$, $(52, 82)$, $(53, 80)$, $(55, 83)$, $(57, 75)$, $(62, 93)$, $(64, 94)$, $(66, 95)$, $(71, 103)$. The dots rise in a fairly tight straight band. She hands the data to a statistics program, which fits the least-squares line and prints this: $$\small\begin{array}{lrrrr} \text{Predictor} & \text{Coef} & \text{SE Coef} & \text{T} & \text{P} \\ \hline \text{Constant} & 8.686 & 8.494 & 1.02 & 0.331 \\ \text{Body length} & 1.317 & 0.150 & 8.77 & 0.000 \end{array}$$ $$\small\text{S} = 4.134 \quad \text{R-Sq} = 88.5\% \quad \text{R-Sq(adj)} = 87.3\%$$ Nobody computes a line through a real data set by hand; a program or calculator does it, and the person reading the result was often not the one who ran it. A printout like this is how a fitted line is reported in articles, lab notes and textbooks. This lesson reads one: where the intercept and slope sit, which entries describe the fit, how to recover $r$, and which entries belong to later work. Running the fit yourself is a different job, fit-a-function-to-data-with-technology; here the output is already in your hands. ## The procedure **The line is in the Coef column.** Each row of the table is one term of the line $\hat{y} = a + bx$. The row labeled Constant holds the intercept $a$, the term with no variable attached. The other row is named after the explanatory variable, and its Coef is the slope $b$, the number that multiplies that variable. So Zsofia's line is ### [Relative frequency histograms](https://math.wiki/relative-frequency-histogram/) **Summary:** A histogram whose bar heights are shares of the total instead of counts. **Question:** Forty earthworms from a back garden and 250 from an orchard. How can one picture show whose worms tend to be longer? ## When you reach for it A histogram groups numerical data into bins of equal width and draws one bar per bin, as tall as the number of values in it; make-a-histogram is where the bins are chosen. A count answers "how many?". That is the right question for one group and the wrong one for two groups of different sizes. Teodora measured the $40$ earthworms she dug up in her back garden, and Melchior measured $250$ from a large orchard. (All the data in this lesson is made up.) Drawn as counts, the orchard has more worms of every length simply because it has more worms. Its bars tower over the garden's, and setting them side by side says nothing about whose worms tend to be longer. relative-frequency-tables met the same problem with categories and solved it by replacing each count with its share of the total. A **relative frequency histogram** does the same for a histogram: the same bins, but each bar's height is the share of the data in that bin, written as a decimal or a percent. Two groups of any sizes then stand on one footing. ## The procedure **Equal bins, and a rule for the edges.** Exactly as for a count histogram. A value landing on the edge between two bars has to belong to one of them, so state a rule and keep it. In this lesson **a value on an edge goes in the bar to its right**: the bin "$8$ to $10$ cm" holds lengths $x$ with $8 \le x < 10$, and a worm exactly $10$ cm long is counted in the $10$ to $12$ bar. The last bar has no bar to its right, so it keeps a value sitting on its top edge: a worm of exactly $16$ cm would go in the $14$ to $16$ bar. ### [Relative frequency tables](https://math.wiki/relative-frequency-tables/) **Summary:** A table giving each category as a fraction or percent of the total instead of a raw count. **Question:** Two schools ran the same survey, and one school is about three times the size of the other. How can their answers be compared fairly? ## When you reach for it A frequency table answers "how many?" for each category of one categorical variable. That is enough while you look at one group. It stops being enough the moment you compare two groups of different sizes. Here is a made-up survey. Two schools asked every student, "Which kind of book do you like best?", and each student gave exactly one answer. Each school's column is its own frequency table. | Book | Ashby | Brookfield | |:-:|:-:|:-:| | adventure | $28$ | $70$ | | mystery | $20$ | $105$ | | comics | $24$ | $50$ | | poetry | $8$ | $25$ | | total | $80$ | $250$ | Brookfield has more readers of every kind of book, simply because it has more students. So the counts cannot answer the question people usually mean: at which school is adventure the more popular choice? For that you need the adventure readers as a **part of each school**: $28$ out of $80$ against $70$ out of $250$. A relative frequency table gives every category that way, as a share of its own total, so groups of any size sit on the same scale. You reach for it whenever the question is "what part of the group?" rather than "how many?". ## The procedure ### Frequency and relative frequency The **frequency** of a category is the number of responses that fall in it. Its **relative frequency** is that frequency divided by the total number of responses: $$\text{relative frequency} = \frac{\text{frequency}}{\text{total}}$$ ### [Residuals](https://math.wiki/residuals/) **Summary:** Actual minus predicted, and what their pattern reveals. **Question:** A fitted line got some points right and missed others. What do the misses say about the line, and about whether a line was the right choice at all? ## What this area is about Every point in a regression has a residual: actual minus predicted, $y - \hat{y}$. A positive residual means the point sat above the line, a negative one below. One residual describes one individual. All of them together describe the line: whether its shape was right, and how far off its predictions typically are. ## How these fit together **The procedure for one point.** Interpreting a residual reads a single miss in context: how far, on which side, and in what units. **The procedure for all of them.** Residual plots graph each residual against the explanatory variable. A shapeless band says a line was a fair choice; a curve says it was the wrong shape, and a fan says its predictions grow less sure as $x$ grows. **Two summaries.** $r^2$ and the standard deviation of the residuals turns the misses into two summaries: the share of the variation in the response the line accounts for, and the typical size of a prediction error. ## A path through it Interpret one residual first, then make residual plots, and finish with $r^2$ and the standard deviation of the residuals. This is where regression inference starts: the conditions for inference are checked on residuals, regression output reports their summaries, and a curved residual plot is the cue for transforming to linearity. ### [Resistant and non resistant measures](https://math.wiki/resistant-vs-nonresistant-measures/) **Summary:** A summary is resistant when a few extreme values barely move it. **Question:** One value in a data set is far bigger than the rest, maybe by mistake. Which summaries does it drag along with it, and which barely notice it? ## Why this exists Magda's household pays a water bill every month. For ten months the bills, in dollars, are $40, 43, 36, 45, 39, 41, 34, 44, 38, 40$. Now suppose a pipe under the garden had leaked in the month that cost $45$ dollars, so that bill came to $115$ dollars, and every other month stayed the same. Here are five summaries of the ten bills, both ways. | Summary | Before the leak | With the leak | |---|:-:|:-:| | median | $40$ | $40$ | | IQR | $5$ | $5$ | | mean | $40$ | $47$ | | range | $11$ | $81$ | | MAD | $2.6$ | $13.6$ | Two summaries did not move at all. Three moved a long way: the mean of $47$ dollars is more than any ordinary month cost, and a MAD of $13.6$ dollars says the bills typically sit $13.6$ dollars from their center, though nine of the ten lie between $34$ and $44$. The $115$ is an outlier by the 1.5 IQR rule: the quartiles are $38$ and $43$, so the fences are $30.5$ and $50.5$. An extreme value may be a mistake, a rare but real case, or a member of another group. Before a summary is trusted, it helps to know which summaries one such value can push around. This lesson sorts the five. ## Definition An **extreme value** is a value at one end of a data set, the largest or the smallest, especially one far from the rest. A summary is **resistant** when one or a few extreme values barely move it, however far out they sit. It is **not resistant** when a single extreme value, moved far enough, can push it as far as you like. ### [Response bias and question wording](https://math.wiki/response-bias/) **Summary:** When people answer untruthfully or inaccurately, whatever the sampling did. **Question:** A survey can ask exactly the right people and still get the wrong answer. How can the way a question is asked push the answers away from the truth? ## Why this exists A survey has two jobs: ask the right people, and get true answers from them. The first job, choosing who is asked, is the subject of convenience and voluntary response samples, and the fix there is to let chance choose. Chance cannot do the second job. Once a person is chosen, the answer still has to travel from what is true, through memory, feelings and a reading of the question, onto the form, and it can be bent on the way. A person may shade an embarrassing answer, give the answer the asker seems to hope for, or guess when they cannot remember. The question itself can lean on them too. So the convenience-sample lesson is about **who answers**; this lesson is about **how they answer**. ## Definition **Response bias** is a lean in a survey's results that comes from the answers themselves: the people asked give answers that differ from the truth, and the differences tend in one direction. That one direction is what makes it bias rather than plain error. | Source | What can happen | Which way it pushes | |---|---|---| | A sensitive question | The true answer is embarrassing | toward the answer that looks better | | Wanting to please the asker | People guess what the asker hopes for | toward the hoped-for answer | | Poor memory | People cannot recall, so they guess | leans only if something tilts the guess | | Leading or loaded wording | The question praises, blames or assumes | toward the side the words favor | | A double question | Two questions get one answer | the answers cannot be read | | The order of the options | Some places in a list may be chosen more | toward the option in that place | ### [Sampling and study design](https://math.wiki/sampling-and-studies/) **Summary:** How data was collected decides what it can tell you. **Question:** Two studies report the same number. How does the way each one collected its data decide what that number can tell you? ## What this area is about Before any calculation, the way data were collected sets the limits of what they can say. A sample speaks for a population only if it was chosen fairly, and a difference between groups shows cause and effect only if the treatments were imposed and assigned at random. This area is about planning a study, and reading someone else's, with those limits in view. ## How these fit together **Definitions.** Populations, samples and census separates the whole group from the part you measure. Parameters and statistics separates the number you want about the whole, such as $\mu$ or $p$, from the one you can compute, such as $\bar{x}$ or $\hat{p}$. **Collecting a sample.** Random sampling lets chance choose the sample, so no one's preferences tilt it. Bias names the ways a collection method leans the same way every time, which a bigger sample does not cure. **Studying cause.** Observational studies and experiments says why a study shows cause only when it imposes treatments and assigns them at random, and experimental design builds that comparison from control, randomization, replication and blocking. ## A path through it Populations and samples come first. Parameters and statistics, and random sampling, follow in either order. Bias comes after random sampling, then observational studies and experiments, then experimental design. Next: sampling distributions, confounding variables and null and alternative hypotheses. ### [Sampling distribution of a difference](https://math.wiki/sampling-distribution-of-a-difference-of-means/) **Summary:** The pattern made by the gap between two independent sample means. **Question:** At one farmers market more shoppers pay cash than at the other. If a random sample is taken at each, how far apart will the two sample shares land, and how often will they rank the markets the wrong way around? ## Why this exists Riverside and Hilltop, two neighboring towns, each hold a Saturday farmers market. The two markets, their shoppers and every number in this lesson are made up. The market association wants to compare them: at which market do more shoppers pay cash, and at which do shoppers spend more? Pema, Dolma and Lhakpa run its surveys, and each survey takes one random sample at each market. The answer to such a question is a gap between two sample statistics, such as the Riverside sample share minus the Hilltop sample share. Draw new samples and the gap changes. To know whether an observed gap reflects a real difference or only the luck of two draws, we need the pattern the gap makes over all the pairs of samples that could have been drawn. That pattern is the **sampling distribution of a difference**. Two earlier lessons supply the pieces. In sampling-distribution-of-a-sample-proportion, the share $\hat{p}$ of composting households in a sample centered at $p$ with standard deviation $\sqrt{p(1-p)/n}$, close to normal under the large-counts condition. In sampling-distribution-of-a-sample-mean, the mean weight $\bar{x}$ of a sample of coffee bags centered at $\mu$ with standard deviation $\sigma/\sqrt{n}$, normal when the population is normal or $n$ is large. This lesson puts two such statistics side by side. Day 1 is a difference of two proportions, Day 2 a difference of two means. ### [Sampling distribution of a sample mean](https://math.wiki/sampling-distribution-of-a-sample-mean/) **Summary:** The pattern made by the sample mean over many samples of the same size. **Question:** A coffee roaster's machine leaves about one bag in four under 12 ounces. Why, then, is a sample of 9 bags that averages under 12 ounces a sign that the machine has gone wrong? ## Why this exists A coffee roaster's machine fills bags labeled $12$ ounces. No machine is exact: from bag to bag the fill weight varies. The roaster cannot weigh every bag, so each day it weighs a random sample and looks at the sample mean $\bar{x}$. A low $\bar{x}$ might be bad luck or a machine that has drifted. Telling the two apart needs one thing: how $\bar{x}$ varies from sample to sample when the machine is working as it should. That pattern is the **sampling distribution of the sample mean**, one case of the general idea in sampling-distributions. The central-limit-theorem showed that for large random samples this pattern is close to normal, centered at $\mu$ with standard deviation $\sigma/\sqrt{n}$. This lesson turns that into a working tool. Day 1 is the center and spread of $\bar{x}$, when they hold, and normal calculations for a population that is itself normal. Day 2 is the condition for using a normal model when the population's shape is not known: a population known to be normal, or a sample large enough, judged from a plot of the sample as in assess-normality-with-a-plot. The roaster, its people and every number here are made up. Arben, Besnik and Drita work there. ## Definition Fix a population with mean $\mu$ and standard deviation $\sigma$, and a sample size $n$. The **sampling distribution of the sample mean** is the distribution of the values $\bar{x}$ takes over all possible random samples of size $n$ from that population. Its mean is written $\mu_{\bar{x}}$ and its standard deviation $\sigma_{\bar{x}}$. ### [Sampling distribution of a sample proportion](https://math.wiki/sampling-distribution-of-a-sample-proportion/) **Summary:** The pattern made by the sample share of successes over many samples of the same size. **Question:** A county says that 35% of its households compost. A random sample of 150 households finds 40% composting. Is that gap a surprise, or the ordinary wobble of a sample? ## Why this exists Kadri works in a county waste office. The county's records say $35\%$ of its households compost food scraps (a made-up figure, as is every number in this lesson). Kadri surveys a random sample of $150$ households and finds $60$ that compost, a share of $60/150 = 0.40$. Has composting grown, or did her sample happen to catch a few extra composters? One sample cannot answer that by itself. What answers it is the pattern the sample share makes over all the samples Kadri could have drawn: where it centers, how far it typically strays, and what shape it takes. That pattern is the **sampling distribution of a sample proportion**, the proportion case of sampling-distributions. The lesson takes two days. Day 1 finds the center and spread of the sample share from the binomial count, and the condition the spread needs. Day 2 adds the condition for a normal shape and uses the normal model to find chances. Tiit and Maarja, who also work in the office, run the other surveys. ## Definition Each household in a sample either composts (a **success**) or does not. If $X$ is the number of successes in a sample of size $n$, the **sample proportion** is $$\hat{p} = \frac{X}{n}$$ The **population proportion** $p$ is the share of successes in the whole population; here $p = 0.35$. The **sampling distribution of** $\hat{p}$ is the distribution of the values $\hat{p}$ takes over all possible random samples of size $n$ from the population. Its mean is $\mu_{\hat{p}}$ and its standard deviation $\sigma_{\hat{p}}$. ### [Sampling distribution of the slope](https://math.wiki/sampling-distribution-of-the-slope/) **Summary:** The pattern the fitted slope makes over many samples from the same population. **Question:** Ten test plots gave a fitted slope. Ten other plots would have given a different one. How far can a sample's slope be from the true slope of the whole population? ## Why this exists Nahuel runs ten quinoa test plots in the Andes. For each plot he records the rainfall over the growing season, $x$ millimeters, and the yield, $y$ kilograms, and fits the least-squares line $\hat{y} = a + bx$. His slope is $b = 0.0096$: about $0.0096$ more kilograms per plot for each extra millimeter of rain. That number describes ten plots. What Nahuel wants to know is how yield depends on rain for all the plots that could be grown this way. Ten other plots would have given another slope; that is sampling-variability again, now for a slope instead of a mean or a proportion. Before a sample slope can be trusted, we need to know how much slopes wander from sample to sample, and around what. That pattern is the sampling distribution of the slope, and it is what makes an interval or a test for a slope possible. ## Definition **The population regression model.** For each value of $x$ there is a whole population of responses. Their mean lies on a straight line, $$\mu_y = \alpha + \beta x,$$ and individual responses scatter above and below that line with the same standard deviation $\sigma$ at every $x$. Here $\alpha$ and $\beta$ are parameters: fixed, usually unknown numbers about the population. In this lesson the made-up population of quinoa plots has $$\mu_y = 2.0 + 0.010x, \qquad \sigma = 0.6 \text{ kilograms}.$$ A plot that gets $300$ millimeters of rain has mean yield $2.0 + 0.010(300) = 5.0$ kilograms, and single plots at $300$ millimeters typically miss that by about $0.6$ kilograms. ### [Sampling distributions](https://math.wiki/sampling-distributions/) **Summary:** Take many samples, compute the statistic each time, look at how it varies. **Question:** Two random samples from the same population give different answers. How far from the truth could either one be? ## What this area is about A statistic such as a sample mean $\bar{x}$ or a sample proportion $\hat{p}$ changes from one random sample to the next. Its sampling distribution is the pattern of its values over all random samples of one size from one population, drawn the same way: where they center, how far they spread, and what shape they make. That pattern is what tells you how much one sample's answer is worth. ## How these fit together **The idea.** Sampling variability and simulating it shows the change from sample to sample, and simulating a sampling distribution turns it into a margin of error. **Judging estimators and draws.** Bias and variability of an estimator separates aiming at the parameter from landing close to it. The ten percent condition lets draws without replacement be treated as independent when $n \le 0.10\,N$. **Results.** The sampling distributions of $\hat{p}$ and of $\bar{x}$ give the center and spread, and the conditions under which the shape is close to normal (the large-counts condition for $\hat{p}$; a normal population or a large sample for $\bar{x}$). Sampling distribution of a difference does the same for $\hat{p}_1 - \hat{p}_2$ or $\bar{x}_1 - \bar{x}_2$ from two independent samples. ## A path through it Sampling variability first. Then simulation, bias and the ten percent condition, in any order. Then the proportion and mean formulas (the mean after the central limit theorem), and the difference last. ### [Sampling variability and simulating it](https://math.wiki/sampling-variability/) **Summary:** Different samples from the same population give different statistics, and that spread is predictable. **Question:** Two fair random samples from the same bag give different answers. Which one is right, and how far from the truth could either of them be? ## Why this exists A sample is taken because checking the whole population is too slow, too costly or impossible. The number the sample gives, a statistic, then stands in for the population's number, the parameter. But a second random sample of the same size gives a different statistic, and a third gives a third. None of them was done wrongly: chance decides which members land in each sample. So every estimate has to answer a question: if the next sample would have said something else, how much is this one worth? **Sampling variability** is the name for that change from sample to sample. Drawing many samples from a population whose answer is known shows that it is not chaos: the statistics pile up around the true value, tighter when the samples are bigger. That pattern lets one sample say "the answer is probably within about this much of what I found". ## Definition **Sampling variability** is the variation of a statistic from one random sample to another, when every sample has the same size and is drawn from the same population by the same method. To see it, you **simulate**: draw many random samples of one size (with real objects, with random digits or a device that stands in for the real draw, or with a computer), work out the statistic for each, and plot all the values, usually as a dot plot with one dot per sample. Two features of that plot matter: - its **center**, where the pile sits, read as its middle or as the mean of all the plotted values; - its **spread**, how far the dots stretch on either side, described by the smallest and largest values or by how many dots land within some distance of the center. ### [Scaled picture graphs](https://math.wiki/scaled-picture-graphs/) **Summary:** A picture graph where one symbol stands for several items, so half a symbol is half that many. **Question:** Forty students voted for tacos. Do you really have to draw forty little pictures to show it? ## When you reach for it Tahnee's class ran a lunch vote, and $40$ students chose tacos. A picture graph that draws one picture for each student would need a row of $40$ plates for tacos alone. That row would be hard to draw, hard to count, and too long for the page. A picture graph always comes with a key, the note that says how many things one picture stands for (picture-graph-key). So far the key has said "one picture, one thing". A **scaled picture graph** changes the key: one plate stands for $10$ students. Now the tacos row needs only $4$ plates, because four groups of $10$ make $40$. Reach for a scaled picture graph when the counts are too big to draw one by one, but you still want rows that can be compared at a glance. Reading one is multiplication. Each symbol is a group of the same size, and a row of symbols is equal groups, which is exactly what $4 \times 10$ means (multiplication-as-groups: $4$ groups of $10$). ## The procedure This lesson takes three days, one part a day. Part 1 reads a row. Part 2 asks questions that compare or combine rows. Part 3 turns a table of counts into a graph, which means choosing a key. ### Part 1: reading a row **Read the key first.** The key says what one whole symbol stands for. In the lunch vote below, one plate stands for $10$ students. Every plate in the graph stands for the same $10$; that is what makes the graph fair. **A row of whole symbols is equal groups.** The tacos row has $4$ plates, and each plate is a group of $10$ students. Four groups of $10$ is ### [Scatter plots](https://math.wiki/scatter-plots/) **Summary:** Two variables, one point per case. **Question:** Two things were measured on each individual. How can one picture show how the two go together? ## What this area is about A scatter plot shows two numerical variables recorded on the same individuals. Each individual becomes one point $(x, y)$ on the coordinate plane, across by one measurement and up by the other, so the two numbers that belong together stay together. The whole cloud of points then shows whether, and how, the variables are associated. An association on a plot does not by itself show that one variable affects the other. ## How these fit together **A procedure.** Plotting paired data turns each row of a table into one point, with the explanatory variable across and the response variable up when one is meant to explain the other. **A description.** Describing a scatter plot reads the finished cloud: its direction (positive, negative or none), its form, its strength, and any clusters or outliers, all stated in the variables' own units and context. ## A path through it Plot the points first, then describe the cloud they make. Finish here and you can measure a linear association with correlation, draw a line of best fit through the cloud, and meet the least-squares criterion, one exact rule for which line fits best. ### [Shape of a distribution](https://math.wiki/shape-of-a-distribution/) **Summary:** Symmetric, skewed, uniform, bimodal, and which center to trust. **Question:** Two plots can share a center and a spread and still look nothing alike. How do you name the outline of a data set, and what does the outline tell you? ## What this area is about The first thing the eye takes from a dot plot or histogram is its outline. A few words cover most real data: symmetric, skewed to the left or right, uniform, bimodal. Naming the shape turns a picture into a sentence, often hints at where the data came from, and decides which center and spread are worth reporting. ## How these fit together **Definitions.** Symmetric and skewed distributions names a shape with one long tail after the side the tail runs to. Uniform and bimodal shapes names the flat shape and the two-humped one, which often means two groups were mixed. **A result.** Mean, median and skew shows that a long tail pulls the mean toward it while the median stays with the bulk, so in skewed data the median is usually the better typical value, while the mean still serves when the total matters. ## A path through it Symmetric and skewed shapes first, then mean and median under skew. Uniform and bimodal shapes can come at any point. Shape, center and spread together make up describing a distribution. Smoothing an outline into a curve leads to density curves and the most important symmetric shape of all, the normal distribution. ### [Shifting and rescaling a data set](https://math.wiki/add-a-constant-to-every-value/) **Summary:** Shifting all the data moves the center by that amount and leaves the spread alone. **Question:** Every value in a data set has the same number added to it, or is multiplied by the same number, as when the units change. Do the mean and the standard deviation have to be worked out again from scratch? ## When you reach for it A data set often has to be changed as a whole. A shop adds the same delivery fee to every order. A thermometer that reads $1.5^\circ\text{C}$ too high has $1.5$ taken off every reading. Times in minutes are reported in hours, lengths in centimeters in meters, a fridge log kept in Celsius is reported in Fahrenheit. In each case one rule is applied to every value: add a constant, multiply by a constant, or both. The summaries of the new data follow from the old ones without touching a single value. That matters when there are hundreds of values, or when only the summary was published. All the data in this lesson are made up. ## The procedure Write the old values as $x_1, x_2, \ldots, x_n$, with mean $\bar{x}$ and standard deviation $\sigma_x$. ### Adding a constant Add the same number $c$ to every value: $y_i = x_i + c$. Every one of the $n$ values gains $c$, so the total gains $nc$, and dividing by $n$, $$\begin{aligned} \bar{y} &= \frac{(x_1 + c) + \cdots + (x_n + c)}{n} \\ &= \frac{x_1 + \cdots + x_n}{n} + \frac{nc}{n} = \bar{x} + c. \end{aligned}$$ Adding $c$ does not change the order of the values, so the value in each position of the sorted list is the old one plus $c$. The minimum, the maximum, the median and both quartiles therefore all move by $c$, and so does any percentile, found the way a percentile is read from a sorted list. Spread does not move. The range is $(\text{max} + c) - (\text{min} + c)$, and the $c$ cancels; the interquartile range $Q_3 - Q_1$ loses its $c$ in the same way. For the standard deviation, look at the deviations from the mean: ### [Simple random samples and the sampling frame](https://math.wiki/simple-random-sample/) **Summary:** A sample drawn so that every group of the chosen size is equally likely. **Question:** You can only ask a few members of a group. How do you choose them so that nobody's preferences, not even your own, decide who is asked? ## When you reach for it You want to know something about a whole population and can only ask or measure a sample of it. Convenience and voluntary response samples let ease or eagerness decide who is in the sample, and the result leans toward whichever corner of the population was reached. A **simple random sample** is the fix: chance, and nothing else, decides who is in it, so nobody's preferences can tilt it. Reach for it when you have a list of the whole population, or can make one, and want a result that speaks for all of it by a method anyone can check was fair. ## The procedure ### Day 1: what makes a sample simple random, and how to draw one **The definition.** A sample of $n$ members from a population of $N$ members is a **simple random sample** when a chance process chooses it so that - every member has the same chance of being in the sample, and - every group of $n$ members has the same chance of being the sample. No member can be chosen twice (the sample is drawn *without replacement*), so it always holds $n$ different members. The second condition is the heart of it. Choose $2$ of four friends, Montague, Nell, Orlaith and Piers, by drawing two slips from a hat. These are all the groups that can come out: | Group | Group | Group | |---|---|---| | Montague, Nell | Montague, Orlaith | Montague, Piers | | Nell, Orlaith | Nell, Piers | Orlaith, Piers | The hat favors no slip, so it favors no pair: each of the $6$ groups has chance $\frac{1}{6}$. Each friend is in $3$ of them, so each friend's chance is $\frac{3}{6} = \frac{1}{2}$, which is $\frac{2}{4}$: $2$ places shared among $4$ friends. In general, every draw fills exactly $n$ places and no member is favored, so the places are shared evenly, and each member's chance is ### [Simulating a sampling distribution and a margin of error](https://math.wiki/simulate-a-sampling-distribution/) **Summary:** Drawing many samples by chance and collecting their statistics to see the pattern they make. **Question:** One poll of 50 people says 38% are in favor. How far from the truth could that be, and how would anyone find out? ## When you reach for it A study has one sample and one statistic: a poll's share in favor, a survey's mean. To report it honestly it needs a second number, how far the statistic is likely to sit from the parameter it estimates. Sampling variability showed that random samples scatter around the parameter, tighter for bigger samples. This lesson turns that picture into a number. The tool is a **simulated sampling distribution**: take a population whose parameter you know, or a model of one, draw a great many random samples of the size the study used, compute the statistic for each, and look at the pile they make. The pile's standard deviation measures how far one sample typically lands from the truth, and twice it gives a **margin of error**: the distance within which most samples, about $95$ in $100$, land. Reach for it when you need to say "about how close" a sample statistic is, when you want to judge whether a result is surprising under some claim, or when you want to know how large a sample must be for a given precision. ## The procedure **Model the population.** A simulation can only copy a population it can describe. For a yes-or-no question that means a share $p$; for a measurement it means the whole list of values, or their shares. If the real parameter is unknown, use a value close to the sample's own, or a cautious value (see Boundaries). The model has to be right for the simulation to mean anything. ### [Slope from r and the spreads, and regression to the mean](https://math.wiki/slope-from-r-and-standard-deviations/) **Summary:** The slope is r times the ratio of the response spread to the explanatory spread. **Question:** If all you know about two variables is their means, their standard deviations and the correlation r, can you write down the least-squares line? And why are its predictions always closer to the mean than the values they come from? ## When you reach for it Narek, Gayane and Ashot weigh and measure the tortoises at a sanctuary. Their report keeps only summaries (made-up data): | | mean | standard deviation | |---|:-:|:-:| | shell length $x$ (cm) | $\bar{x} = 30$ | $s_x = 4$ | | mass $y$ (kg) | $\bar{y} = 9$ | $s_y = 2.5$ | and the correlation $r = 0.85$. The raw pairs are gone. The least-squares line has slope $b = S_{xy}/S_{xx}$ and passes through the point of averages $(\bar{x}, \bar{y})$, but $S_{xy}$ and $S_{xx}$ are built from the raw pairs. This lesson rewrites the slope in terms of the five summaries, so a line can be written from a report, a table in a paper, or a calculator's summary statistics. The rewrite also answers a second question. Measured in standard deviations, every prediction the line makes is closer to the mean than the value it was made from. That is **regression to the mean**, and the formula shows exactly where it comes from. ## The procedure **The slope.** The correlation is the sum of the products of z-scores divided by $n - 1$. Writing the z-scores out, $$\begin{aligned} r &= \frac{1}{n-1}\sum \frac{x_i - \bar{x}}{s_x}\cdot\frac{y_i - \bar{y}}{s_y} \\ &= \frac{S_{xy}}{(n-1)\,s_x s_y}, \end{aligned}$$ so $S_{xy} = (n-1)\,r\,s_x s_y$. The standard deviation of $x$ satisfies $s_x^2 = S_{xx}/(n-1)$, so $S_{xx} = (n-1)\,s_x^2$. Divide the first by the second; the $n - 1$ and one $s_x$ cancel: ### [Sorting objects into groups](https://math.wiki/sort-objects-into-categories/) **Summary:** Putting objects into groups by one shared feature and counting how many are in each group. **Question:** How does a child put a mixed pile into groups by one rule, check the groups, and say how many are in each? ## When you reach for it **Sorting** means putting a mixed pile into groups by one rule, such as color, size, kind, or what a thing is used for. A good rule gives every object exactly one group, so anyone can check where each one goes. Once the pile is sorted, each group is counted, which turns a jumble into a few clear numbers. Reach for it once a child can count a small group and say how many. Each day below takes ten to fifteen minutes. Keep every group at ten or fewer, and always say the rule out loud before anything is sorted. ## Day 1: Sorting by a rule you are given **You need:** about ten buttons or counters in two colors, and two bowls. Mix the counters on the table. Point to one bowl and say, "Dark ones go here." Point to the other: "White ones go here." Let the child pick up one counter at a time, say its color, and drop it in its bowl. When the table is empty, ask, "Is anything left on the table? Is each counter in its own bowl?" Now count. Pour out one bowl, count it, and ask, "How many dark counters?" Then the other bowl. Each answer is the last counting word said for that group. **Practice.** 1. Sort the counters in the picture by color. How many in each group? **Answer:** $5$ dark and $4$ white. 2. Mix $6$ yellow cubes and $2$ green cubes and hand them over. Sort by color and count each bowl. **Answer:** $6$ yellow and $2$ green, and no cube is left on the table. 3. Mix $3$ spoons and $6$ forks. Sort them by what they are. **Answer:** $3$ spoons and $6$ forks. 4. Mix $3$ socks, $2$ mittens and $4$ hats, with three spots on the floor. Sort them by kind. **Answer:** $3$ socks, $2$ mittens and $4$ hats. A rule can make three groups as easily as two. ### [Standard deviation](https://math.wiki/standard-deviation/) **Summary:** The typical distance from the mean, squared and rooted, and why squared. **Question:** The values above the mean and below it cancel when you add their distances. How do you still get one honest number for how spread out the data are? ## What this area is about The standard deviation is the typical distance of a value from the mean, in the data's own units. Signed distances from the mean always sum to $0$, so they are squared before averaging, which gives the variance; its square root brings the units back. For a sample, $s = \sqrt{\dfrac{\sum (x - \bar{x})^2}{n - 1}}$. ## How these fit together **The definition.** Variance of a data set is the average squared distance from the mean, and explains why squaring is the fix for the cancelling. **Procedures.** Computing a standard deviation works through distances, squares, average and root, and says when to divide by $n$ (a whole population) and when by $n - 1$ (a sample). Interpreting a standard deviation turns the number into a sentence in the setting's units: what a typical value looks like, which group varies more, and when one value is far out. ## A path through it Variance first, then the computation, then interpretation. The standard deviation is the yardstick of most later statistics: it sets the scale of the normal distribution, shows how shifting and rescaling affect spread, and is built into correlation. ### [Standard error](https://math.wiki/standard-error/) **Summary:** The estimated spread of a statistic's sampling distribution, computed from the sample. **Question:** The spread of a statistic's sampling distribution depends on the very parameter we are trying to estimate. How can one sample say how far off it might be? ## Why this exists A university genetics lab keeps a large culture of fruit flies, some with curly wings. Ioane wants the share of curly-winged flies in the whole culture, the parameter $p$. Counting every fly is out of the question, so he takes a random sample of $200$ flies and finds $46$ with curly wings (made-up data): $$\hat{p} = \frac{46}{200} = 0.23$$ The estimate $0.23$ is almost surely not exactly $p$. How far off might it be? Sampling variability gave the answer in principle: the spread of the sampling distribution of $\hat{p}$, the pile of sample proportions from all possible samples of $200$. For random samples that spread has a formula, $$\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}$$ and there is the snag. The formula needs $p$, the number Ioane is trying to find. If he knew $p$ he would not need the sample. The way out is to put the estimate where the unknown parameter was. That plug-in number, computed from the one sample in hand, is the standard error. ## Definition The **standard error** of a statistic is an estimate, computed from the sample data, of the standard deviation of that statistic's sampling distribution. It is written $\text{SE}$ with the statistic as a subscript. Two numbers must be kept apart: | | Uses | Can it be computed from the sample alone? | |---|---|:-:| | standard deviation of the sampling distribution, $\sigma_{\hat{p}}$ or $\sigma_{\bar{x}}$ | the parameter, $p$ or $\sigma$ | no | | standard error, $\text{SE}_{\hat{p}}$ or $\text{SE}_{\bar{x}}$ | the statistic, $\hat{p}$ or $s_x$ | yes | ### [Standardizing a data set](https://math.wiki/standardizing-a-data-set/) **Summary:** Subtracting the mean and dividing by the standard deviation to get a set with mean zero and spread one. **Question:** Two data sets sit on different scales, in different units or with different spreads. How can every value in each be put on one common scale, where the middle is always 0 and a step of 1 always means the same thing? ## When you reach for it Raw numbers carry their scale with them. A disc golf hole $70$ meters long, dough that takes $36$ minutes to rise and a score of $9$ out of $10$ cannot be set side by side: the units differ, and even within one unit a "big" value depends on what is typical for that set. Two judges can both mark out of $10$ and still mean different things by a $9$ if one of them spreads marks widely and the other keeps them close together. **Standardizing** a data set rewrites every value as a distance from the set's mean, measured in standard deviations. Whatever the data were, the new set has mean $0$ and standard deviation $1$, and it has no unit at all. Reach for it when you want to: - put two or more data sets on one common scale, so their values can be compared or combined fairly; - read every value at a glance as "above or below typical, and by how much"; - prepare data for a method that expects that common scale, such as finding a correlation from standardized values or reading a normal curve as the standard normal curve. Describing one value's position by its standardized value is the subject of z-scores. This lesson does the whole set at once and asks what the new set looks like. All the data here are made up. ## The procedure Standardizing uses the two moves of shifting and rescaling a data set, one after the other, chosen so that the result has mean $0$ and standard deviation $1$. ### [Statistical inference](https://math.wiki/inference/) **Summary:** From a sample to a claim about the population, with honest uncertainty. **Question:** A random sample gives one answer, and another sample would give a different one. What can that one answer honestly say about the whole population? ## What this area is about Inference reasons from a statistic you know, such as $\bar{x}$ or $\hat{p}$, to a parameter you do not, such as $\mu$ or $p$. It does that in two ways: a range of plausible values, or a verdict on a claimed value. Both rest on the sampling distribution of the statistic, so both hold only when the data come from random sampling, random assignment or a random process, and the method's conditions are checked. ## How these fit together **The two ideas.** Confidence intervals report $\text{estimate} \pm \text{margin of error}$. Hypothesis testing assumes a claim and asks how surprising the sample would be under it, and p-values and the significance level measure that surprise and set the bar. **Procedures by parameter.** Inference for proportions uses $z$ for one or two shares. Inference for means uses $t$, because $s$ stands in for an unknown $\sigma$. Inference for slopes does the same for a regression slope $\beta$. Chi-square tests compare observed counts in categories with the counts a claim predicts. **Choosing.** Choosing the right procedure reads the question and the data design to pick one of these before any arithmetic. ## A path through it Confidence intervals come first. Hypothesis testing and p-values follow and interleave: p-values come after the hypotheses and the test statistic, before the decision. Then proportions and means, in either order. Slopes and chi-square tests follow means. Choosing a procedure comes last, after proportions, means and chi-square tests. ### [Statistical questions and variability](https://math.wiki/statistical-question/) **Summary:** A question worth asking of data because the answers vary from one individual to the next. **Question:** One student goes to bed at one time, but a whole class goes to bed at many different times. What kind of question asks about the class, and how is it answered? ## Why this exists Ask Bodil what time she went to bed last night and she gives one answer, say $9{:}30$ pm. That settles it. Now ask every student in Bodil's class. The answers are not all the same: some went to bed at $8{:}30$, some at $10{:}00$, one at $11{:}00$. No single time is "the answer". The answer is the whole collection of times, and the job becomes describing it: where the bedtimes bunch up, how far they run, which times are rare. Questions of the second kind are what statistics is built for. They come up whenever the thing asked about differs from one person, object or day to the next. This lesson is about recognizing such a question, turning a one-answer question into one, and taking a first look at the answers that come back. ## Definition A **statistical question** is a question that expects its answers to vary, and is answered by collecting data and looking at all the answers together. Two tests tell whether a question is statistical. - **Can the answer differ from one individual to the next?** If the question is about a group of individuals (people, pairs of shoes, nights, tries) and different individuals can give different answers, it is statistical. If there is one fixed answer, it is not. - **Is it settled by collecting many values?** A statistical question is answered by gathering one value from each individual and describing them all. A question settled by one look, one count or one measurement is not statistical. ### [Statistics](https://math.wiki/statistics/) **Summary:** Collecting, describing, and drawing careful conclusions from data. **Question:** Data about a group vary from one individual to the next. How do you collect them, describe them, and say what they show without claiming more than they can support? ## What this area is about Statistics is the study of data that vary: how to gather, summarize and picture them, and how to reason from a sample to a population with an honest measure of uncertainty. The conclusions hold only as far as the way the data were collected allows. ## How these fit together **Starting out.** Data and variables asks a statistical question, records individuals and variables, and counts categories in tables. **Describing data.** Data displays picture data, from bar graphs to box plots and scatter plots, and center and spread summarize them with numbers such as the mean, median, IQR and standard deviation. **Collecting well.** Sampling and study design says when a sample speaks for its population and when a study can show cause and effect. **Models.** Distributions replace a histogram with a smooth curve, above all the normal curve $N(\mu, \sigma)$, and describe how a statistic varies from sample to sample. **Two variables.** Bivariate data measures association with $r$ and fits a line $\hat{y} = a + bx$, without by itself showing cause. **Conclusions.** Statistical inference turns a sample into confidence intervals and hypothesis tests about $\mu$, $p$ or a slope $\beta$. ## A path through it Data and variables come first. Data displays and center and spread interleave next, then distributions, then bivariate data, then inference. Sampling and study design runs alongside: it can start right after data and variables, random sampling comes before sampling distributions, and its experiments lessons draw on confounding and on hypothesis tests. ### [Stem and leaf plots](https://math.wiki/back-to-back-stem-and-leaf-plots/) **Summary:** Two groups sharing one column of stems, with leaves running out to the left and the right. **Question:** A list of numbers shows nothing at a glance. How can you draw it so its shape shows, without losing a single value? ## When you reach for it Celyn writes down the best clearance of each of fifteen jumpers at a high jump practice. As a list it says little: where do most jumps sit, and is one far from the rest? A dot plot would answer that. A **stem and leaf plot** does the same job with the digits themselves. Each value is split into a stem and a leaf, the stems go down one column, and the leaves sit in rows beside them. The rows make a picture of the data's shape, and every value can still be read back exactly. It suits a small set of numbers, a few dozen at most. Two groups measured in the same way can share one column of stems, with leaves running out to each side: a **back-to-back** stem and leaf plot. ## The procedure ### Splitting each value By place-value, every digit of a number has a place. The **leaf** is the value's last digit, and the **stem** is all the digits in front of it. | Value | Stem | Leaf | |:-:|:-:|:-:| | $113$ | $11$ | $3$ | | $97$ | $9$ | $7$ | | $7$ | $0$ | $7$ | | $6.4$, leaf in tenths | $6$ | $4$ | A one-digit number has stem $0$. Values with the same stem lie in the same stretch of ten: $110$ to $119$ all have stem $11$. So one row of the plot holds values that are close together, and every row covers an equal stretch, ten possible leaves from $0$ to $9$, like the marks on a number line. ### Building the plot Celyn's fifteen heights, in centimeters, are $118, 104, 125, 131, 112, 97,$ $120, 116, 108, 127, 113, 122,$ $135, 119, 110$. The smallest is $97$ and the largest $135$, so the stems run from $9$ to $13$. They are written once each, down a column from smallest to largest, with a line to their right. Then each value's leaf goes in its stem's row: $118$ puts an $8$ in row $11$. Last, the leaves of each row are put in order from smallest to largest, evenly spaced. ### [Stratified random sample](https://math.wiki/stratified-random-sample/) **Summary:** Splitting the population into similar groups first, then sampling at random inside each. **Question:** You already know which group each member of a population belongs to, and the groups answer differently. How do you make sure every group gets its fair share of the sample instead of leaving it to luck? ## When you reach for it A simple random sample lets chance alone choose who is asked, so every group of $n$ members is equally likely. That fairness has a cost: by luck the sample may hold too many of one kind of member and too few of another, and if those kinds answer differently, the result swings with the mix. Reach for a **stratified random sample** when, before sampling, you can sort every member of the population into groups by something you already know (class year, building, region), and that something is likely tied to the answer you are after. Then you stop leaving the mix to chance: you fix how many come from each group and let chance choose only *which* members. ## The procedure **The definition.** Split the population into groups called **strata** (one group is a **stratum**) so that every member is in exactly one stratum. Then take a separate simple random sample inside each stratum. The combined members form a stratified random sample. Write $N_h$ for the size of stratum $h$, $N$ for the whole population, $n_h$ for the number sampled from stratum $h$ and $n$ for the whole sample. **Choosing the strata.** Good strata are *alike inside and different from each other* on the thing being measured. If first-years mostly answer one way and seniors another, class year is a good stratifying variable: inside a stratum the answers vary little, so a modest sample pins that stratum down, and the differences between strata, the part an unlucky mix would distort, are handled by design. A variable unrelated to the answer, such as birth month, makes strata that all look like the population, and splitting by it changes almost nothing. ### [Symmetric and skewed distributions](https://math.wiki/skewed-distributions/) **Summary:** A shape with one long tail, named for the side the tail runs to. **Question:** Two dot plots can have their tallest stack in the same place and still look nothing alike, one balanced and one leaning to a side. How do you say which way data leans, and why does so much real data lean? ## Why this exists After a large earthquake the ground goes on shaking for days, in smaller quakes called aftershocks. Enid counted the aftershocks felt in her town on each of $24$ days: on most days none, one or two, on a few days many. The median says where the middle of those counts is, but not how the values are arranged: piled up at one end and thinning out toward the other, or balanced on both sides of the middle. That arrangement is the **shape** of the data. It often points back at the thing measured (a day cannot have fewer than $0$ aftershocks, and that floor bends the whole picture), and it decides which center describes the data fairly. ## Definition The **distribution** of a numerical data set is the way its values are spread along the number line: which values occur, and how often. A dot plot or a histogram draws it. The **peak** is where the values crowd most, the tallest stack or bar. A **tail** is a stretch at one end of the distribution where the values thin out, fewer and fewer, moving away from the peak. - A distribution is **symmetric** when a vertical line can be drawn through the plot so that the two sides are mirror images: folded along the line, each stack lands on a stack of the same height. Real data is almost never exactly symmetric, so the working phrase is **roughly symmetric**: the two sides nearly match and the two tails are about the same length. - A distribution is **skewed right** when the values crowd toward the low end and a long tail runs out toward the high values, to the right on the number line. - A distribution is **skewed left** when the values crowd toward the high end and a long tail runs out toward the low values, to the left. ### [Systematic sample](https://math.wiki/systematic-sample/) **Summary:** Picking a random start and then taking every so many individuals down a list. **Question:** Boards come off a saw line one after another, and nobody has a list of them until the shift is over. How can chance choose a fair sample while they pass? ## When you reach for it A simple random sample lets chance alone choose, so that every group of $n$ individuals is equally likely to be the sample. To draw one you need the whole list in hand and $n$ separate random numbers. Often the individuals arrive in order instead: boards off a saw, cars through a gate. Nobody can number them in advance. A **systematic sample** needs only one random number: pick a random starting point, then take every $k$-th individual down the list or down the line. Every individual gets the same chance, and the sample behaves like a simple random sample whenever the order has nothing to do with what is measured. ## The procedure **The interval.** A population of $N$ individuals stands in some order, and the sample is to hold $n$ of them. The **sampling interval** is $$k = \frac{N}{n}.$$ At the sawmill where Aitor checks boards for moisture, a shift produces $N = 3{,}600$ boards and he wants $n = 120$, so $k = \frac{3600}{120} = 30$: one board in every $30$. **The random start.** Choose a whole number $s$ from $1$ to $k$ at random, each equally likely (a random digit table read in pairs, a calculator, or $30$ slips in a hat). The sample is $$s,\ s + k,\ s + 2k,\ \ldots,\ s + (n - 1)k.$$ The last one is at most $k + (n-1)k = nk = N$, so the sample never runs off the end of the list and always holds exactly $n$ individuals. The start must be random: if Aitor always began at board $1$, no chance would be involved, and boards in the other $29$ positions would never be checked. ### [Test for a difference of means](https://math.wiki/two-sample-t-test-for-a-difference-of-means/) **Summary:** Testing whether two groups really differ in average value. **Question:** Two groups have different averages in their samples. Is the gap big enough to show that the population means really differ, or that a treatment really changed the mean, or could chance alone have produced it? ## When you reach for it Malee buys longan fruit from orchards in northern Thailand and pays by weight. She believes orchard A's fruit run heavier than orchard B's. A random sample of $25$ fruit from A averages $11.8$ grams and a random sample of $22$ from B averages $10.9$ grams (made-up data throughout this lesson). Even two orchards with equal means would give samples that differ a little. Is $0.9$ grams more than chance would plausibly produce? The two-sample $t$ test answers that, for two populations sampled separately (Day 1) or two treatment groups in an experiment (Day 2). From the one-sample-t-test-for-a-mean it takes the shape: a null value, a $t$ statistic, a p-value from a $t$ curve. From the two-sample-t-interval-for-a-difference-of-means it takes the standard error $\sqrt{s_1^2/n_1 + s_2^2/n_2}$ and the two choices of degrees of freedom. The parallel test for shares is the two-sample-z-test-for-a-difference-of-proportions. ## The procedure ### Day 1: two independent random samples **Parameters and hypotheses.** $\mu_1$ and $\mu_2$ are the means of a stated variable, with units, over two stated populations. The null hypothesis says there is no difference, and the alternative says what the investigator suspected before seeing the data: $$H_0\colon \mu_1 - \mu_2 = 0 \qquad H_a\colon \mu_1 - \mu_2 > 0, \ < 0, \text{ or } \neq 0$$ The hypotheses are about the population means, never the sample means, which are already known. Naming which group is $1$ fixes the order of subtraction. ### [Test for a difference of proportions](https://math.wiki/two-sample-z-test-for-a-difference-of-proportions/) **Summary:** Testing whether two groups really differ in the share with some feature. **Question:** Two samples show different shares of some feature. Is the gap real, a sign that the two populations or the two treatments truly differ, or could chance alone have made it? ## When you reach for it Amani studies small businesses in Kenya. She wants to know whether the share of small shops that accept mobile money payments differs between two towns, Nakuru and Eldoret. She takes an independent random sample of shops in each: $132$ of $200$ Nakuru shops accept mobile money, and $117$ of $220$ Eldoret shops do (made-up data throughout this lesson). The sample shares are $0.66$ and about $0.532$. The gap is about $0.13$, but two other samples would give another gap. Is this one too large to be chance, if the two towns really had the same share? That is a yes-or-no question about $p_1 - p_2$, and the two-sample $z$ test for a difference of proportions answers it. It joins two earlier lessons. The two-sample-z-interval-for-a-difference-of-proportions estimated $p_1 - p_2$ as $\hat{p}_1 - \hat{p}_2$ plus or minus a margin, with a standard error built from both samples. The one-sample-z-test-for-a-proportion measured how far a sample share sat from a claimed value, in standard deviations computed as if the null hypothesis were true. Here the claim is that the two shares are equal. Day 1 runs the test on two random samples; Day 2 runs it on a randomized experiment, adds one-sided alternatives, and sets the test beside the interval. ## The procedure ### Day 1: two independent random samples **Parameters and hypotheses.** Name $p_1$ and $p_2$ in context: the proportions of two stated populations with a stated feature. The null hypothesis is no difference, and the alternative is what the investigator set out to find, chosen before the data are seen: ### [Test for a slope](https://math.wiki/t-test-for-a-slope/) **Summary:** Testing whether there is any real linear relationship, usually against a slope of zero. **Question:** A sample of plots shows a fitted line that rises. Is the rise real across the whole coast, or could a flat population line have produced a sample slope this steep by chance? ## When you reach for it Somchai restores mangroves on the coast of Thailand. The project has $320$ planted plots, and he takes a random sample of $16$ (made-up data). For each plot he records $x$, the planted seedling density in seedlings per square meter, and $y$, the number of crab burrows per square meter, a sign that the mud is coming back to life. His program prints: $$\small\begin{array}{lrrrr} \text{Predictor} & \text{Coef} & \text{SE Coef} & \text{T} & \text{P} \\ \hline \text{Constant} & 2.777 & 0.732 & 3.79 & 0.002 \\ \text{Density} & 0.844 & 0.311 & 2.71 & 0.017 \end{array}$$ $$\small\text{S} = 1.225 \quad \text{R-Sq} = 34.5\% \quad \text{R-Sq(adj)} = 29.8\%$$ In the sample, denser plots have more burrows: $0.844$ more per square meter for each extra seedling per square meter (Figure 1). But sixteen plots scatter widely, and even if burrows had nothing to do with density across all $320$ plots, a sample would rarely give a slope of exactly $0$. The question is the one one-sample-t-test-for-a-mean asked of a mean, now asked of a slope: is the sample result too far from the claimed value to blame on sampling alone? t-interval-for-a-slope gave the plausible slopes as $b \pm t^*\,SE_b$; the test weighs one claimed slope, almost always $0$. ## The procedure ### Day 1: the test from computer output **The parameter and the hypotheses.** As in sampling-distribution-of-the-slope, the model is a population line $\mu_y = \alpha + \beta x$ giving the mean of $y$ at each $x$. Here $\beta$ is the change in mean burrows per square meter for each additional seedling per square meter, among all $320$ plots. A slope of $0$ means the mean number of burrows is the same at every density: no linear relationship. So the usual test is ### [Test statistic](https://math.wiki/test-statistic/) **Summary:** How many standard errors the observed result sits from what the null hypothesis expected. **Question:** A sample of 250 spa visits found 35.6% with a massage booked, against a claimed 30%. Is a gap of 5.6 percentage points big or small? ## Why this exists A geothermal spa in Iceland has a booking system that reports $30\%$ of visitors book a massage (all numbers in this lesson are made-up data). Bjarki, who manages the treatment rooms, thinks the true share is higher. In the language of null-and-alternative-hypotheses, the claim and his suspicion are $$H_0\colon p = 0.30 \qquad H_a\colon p > 0.30$$ where $p$ is the proportion of all visits that include a massage. He takes a random sample of $250$ visits from the year's records and finds $89$ with a massage: $$\hat{p} = \frac{89}{250} = 0.356$$ That is $0.056$ above the claim. Is $0.056$ a lot? On its own the number cannot say. Sample proportions vary from sample to sample, so even if $p$ really were $0.30$ a sample would rarely land exactly on it. A gap means something only next to the typical size of that variation. The standard-error lesson measured such spread for an estimate; here the spread is measured in the world the null hypothesis describes, and the gap is counted in those units. That count is the test statistic. ## Definition A **test statistic** measures how far a sample's statistic sits from the null value, in units of the statistic's standard deviation computed as if the null hypothesis were true: $$\text{test statistic} = \frac{\text{statistic} - \text{null value}}{\text{SD under } H_0}$$ Here "SD under $H_0$" is the standard deviation of the statistic's sampling distribution, worked out as if $H_0$ were true. When that standard deviation depends on a parameter $H_0$ does not fix, as for a mean, a standard error stands in for it (see Results). ### [Tests for independence and homogeneity](https://math.wiki/chi-square-test-for-independence/) **Summary:** Testing whether two categorical variables recorded on one sample are associated. **Question:** A survey sorts adults by age group and by the tea they prefer, and the tea shares look different from one age group to the next. Is that a real association among all adults, or the kind of difference random sampling produces anyway? And when separate samples are taken from three cities instead, how does the same arithmetic answer a different question? ## When you reach for it Kourosh, Roya and Farhad want to know whether tea preference in Isfahan changes with age. In a two-way table of a sample, association shows as conditional shares that differ from row to row. But a sample's shares always differ a little, even when the variables are unrelated in the population. The question here is whether the differences are too large to blame on chance. The chi-square-goodness-of-fit-test answered that kind of question for one categorical variable against a claimed distribution. This lesson carries the same statistic, conditions and upper-tail P-value over to a two-way table, where there are two designs. - **Test for independence.** One random sample, each individual classified on two categorical variables. Question: are the two variables associated in the population? - **Test for homogeneity.** Separate random samples from two or more populations, or two or more groups formed by random assignment, each classified on one categorical variable. Question: is that variable's distribution the same in every population or treatment? The arithmetic of the two is identical. What differs is the design that produced the table, and so the hypotheses and the conclusion. Choosing among all the procedures is the subject of choose-an-inference-procedure. ## The procedure ### Day 1: the test for independence **Hypotheses.** For one random sample classified by two variables, ### [The central limit theorem](https://math.wiki/central-limit-theorem/) **Summary:** Averages of enough independent draws look normal, whatever the population looked like. **Question:** Order totals at a food co-op are lopsided, mostly small with a few big ones. Why would the mean of 36 of them follow a bell curve, and how far can that be trusted? ## Why this exists A sampling distribution is the pattern a statistic makes over many random samples. Simulation can draw one, but only for a population that is fully known, and a real study knows its population only through its one sample. To say how far a sample mean is likely to sit from the population mean, a study needs the shape of the sampling distribution without seeing the population. The central limit theorem supplies it. Whatever shape the population has, skewed, lumpy or flat, the mean of a large enough random sample is approximately normal, and the normal curve's areas from area-under-a-normal-curve then answer questions about it. That is why the normal curve runs through nearly every inference method for means. The lesson runs over two days. Day 1 is what the theorem says and how to use it. Day 2 is what it needs, and how large "large enough" has to be. The co-op, its people and every number in it are made up. ## Definition Take $n$ independent random draws $X_1, X_2, \ldots, X_n$ from one population with mean $\mu$ and a finite standard deviation $\sigma$, and form the sample mean $$\bar{x} = \frac{X_1 + X_2 + \cdots + X_n}{n}$$ **The central limit theorem.** As $n$ grows, the sampling distribution of $\bar{x}$ gets closer and closer to a normal distribution with mean $\mu$ and standard deviation $\sigma/\sqrt{n}$: $$\bar{x} \approx N\!\left(\mu, \frac{\sigma}{\sqrt{n}}\right) \;\text{ for large } n$$ ### [The correlation coefficient](https://math.wiki/correlation-coefficient-r/) **Summary:** A number between negative one and one measuring how closely points cluster about a straight line. **Question:** Two clouds of dots both climb. Which one keeps closer to a straight line, and can a single number say so? ## Why this exists When you describe a scatter plot by eye, you name its direction, its form and its strength. Direction and form are usually easy to agree on. Strength is not: one person calls a cloud "fairly tight", another "quite loose", and neither can say which of two climbing clouds is tighter when they look alike. A line-of-best-fit shows the trend, but not how closely the points keep to it. The correlation coefficient settles this with one number, $r$, that measures how closely the points cluster about a straight line. It is for two numerical variables measured on the same individuals. When both variables sort people into categories, the tool is association in a two-way table instead. ## Definition The **correlation coefficient** $r$ of a set of paired numerical data is a number from $-1$ to $1$ that measures how closely the points of its scatter plot cluster about a straight line. - Its **sign** gives the direction: $r > 0$ when the cloud climbs from left to right, $r < 0$ when it falls. The sign of $r$ always matches the sign of the slope of the line of best fit. - Its **size**, the distance of $r$ from $0$, gives the strength. $r = 1$ or $r = -1$ only when every point lies on one straight line. $r$ near $0$ means there is no straight-line pattern. In practice $r$ is found with technology: a graphing calculator, a spreadsheet or a statistics app takes the list of pairs and reports $r$, often alongside the line of best fit. A formula for $r$ exists, built from standardized values of the two variables, and is met in a statistics course (r-from-standardized-values). In this lesson $r$ is always rounded to two decimal places. ### [The least squares criterion and its line](https://math.wiki/least-squares-criterion/) **Summary:** The line is chosen to make the total of the squared vertical misses as small as possible. **Question:** Three people draw three different lines through the same scatter plot. What exact rule picks one line as the best, so that everyone who follows it gets the same line? ## Why this exists Itziar, Iker and Maialen counted how often fireflies flashed on six summer evenings and noted the air temperature each time (made-up data): | $x$ (°F) | $60$ | $65$ | $70$ | $75$ | $80$ | $85$ | |---|:-:|:-:|:-:|:-:|:-:|:-:| | $y$ | $8$ | $11$ | $13$ | $14$ | $17$ | $21$ | Here $x$ is the temperature in degrees Fahrenheit and $y$ the flashes per minute. Each of them drew a line by eye through the scatter plot, and the three lines differ. Looking harder will not settle it. What settles it is a number that scores any line, and a rule that the best line is the one with the best score. The **least-squares criterion** is that rule. With it, "the regression line" names one line that anyone computes the same way, and everything that follows in regression (slope from $r$, r squared, computer output) describes that one line. This lesson takes two days. **Part 1** sets up the score: square each vertical miss and add, and see why squares rather than the misses themselves or their sizes. **Part 2** finds the line with the lowest score: it passes through the point of averages $(\bar{x}, \bar{y})$, its slope sits at the vertex of a parabola, and because the misses it counts are vertical, predicting $x$ from $y$ needs a different line. Examples 1 to 3 go with Part 1, Examples 4 to 6 with Part 2. ## Definition Take data points $(x_1, y_1), \dots, (x_n, y_n)$ and any candidate line $\hat{y} = a + bx$. The **residual** of the $i$th point is actual minus predicted (interpret-a-residual), ### [The normal distribution](https://math.wiki/normal-distribution/) **Summary:** The bell curve: symmetric, described by mean and standard deviation. **Question:** Lengths of needles, heights of people, errors in a careful measurement all pile up in the same symmetric hump. Is there one curve for all of them, and what does it take to pin it down? ## Why this exists Measure the length of a few thousand needles from one pine tree and draw a histogram. Most needles are close to some typical length, fewer are a little longer or shorter, and very few are far off in either direction. The histogram is a single hump, roughly symmetric, falling away smoothly on both sides. The same hump turns up for the heights of adults of one sex, the masses of coins fresh from a mint, and the errors when one quantity is measured over and over with care. A density curve can model any of these, but there are endless curves to choose from. What makes this shape so useful is that one family of curves fits all of these humps, and a curve in that family is fixed completely by just two numbers: where its center is and how spread out it is. Those are the mean and the standard-deviation. Once you know them, you know the whole curve, and with it every proportion the model gives. That family is the **normal distribution**, and it is the model most of statistics is built on. ## Definition A **normal distribution** is a model for a numerical variable whose density curve is a symmetric, single-peaked bell. Each normal curve is fixed by two numbers: - its **mean** $\mu$, the center, where the peak stands, and - its **standard deviation** $\sigma$, a positive number giving how far the values typically spread from the mean. The normal distribution with mean $\mu$ and standard deviation $\sigma$ is written $N(\mu, \sigma)$. Because these two numbers settle everything about the curve, they are called its **parameters**. ### [The standard normal distribution](https://math.wiki/standard-normal-distribution/) **Summary:** The normal curve with mean zero and standard deviation one, which every normal curve becomes after standardizing. **Question:** Every normal model has its own mean and standard deviation, so it seems every one needs its own table of areas. Is there one curve whose areas answer the questions for all of them? ## Why this exists A normal model answers questions with areas: the share of values below some cut is the area under the curve to the left of it. But the curve has no simple area formula. Its areas have to be worked out numerically, and there is a different curve for every mean and every standard deviation. A printed table for each one would fill a library, and every new model would need a new table. Standardizing removes the problem. Measure every value by its z-score, the number of standard deviations it lies from the mean, and every normal curve turns into the same curve: the one with mean $0$ and standard deviation $1$. So a single table of areas, worked out once, serves every normal model there is. That one curve is the **standard normal distribution**, and this lesson is about reading it: areas to the left, to the right and between two cuts, and the cut that has a given area to its left. ## Definition The **standard normal distribution** is the normal distribution with mean $0$ and standard deviation $1$, written $N(0, 1)$. A variable with this distribution is usually called $Z$, and its values are z-scores. Its density curve is the graph of $$\varphi(z) = \frac{1}{\sqrt{2\pi}}\, e^{-\frac{z^2}{2}}.$$ The share of the distribution below a value $z$ is the area under this curve to the left of $z$. That area is written $\Phi(z)$, so $$\Phi(z) = P(Z < z).$$ Here $P(\dots)$ is read "the probability that", or equally "the share of the distribution for which". $\Phi$ is the capital Greek letter phi, and $\varphi$ is the small one. ### [The t distribution and degrees of freedom](https://math.wiki/t-distribution/) **Summary:** A bell shape with heavier tails than the normal, used when the population spread is estimated from the sample. **Question:** A sample mean is judged by how many standard deviations of the mean it sits from a claimed value, but that standard deviation needs the population's spread, which nobody knows. Put the sample's own standard deviation in its place, and what distribution does the result follow? ## Why this exists Morag runs a sugarhouse in Vermont, where maple sap is boiled down to syrup. Sweeter sap needs less boiling for each gallon of syrup, so she wants to know the mean sugar content of the sap across all her trees. For planning she has assumed $2.0$ percent. Catriona measures the sap of a random sample of $8$ trees (made-up data, in percent sugar): $$\begin{gathered} 2.4,\ 1.9,\ 2.2,\ 1.7, \\ 2.6,\ 2.0,\ 2.3,\ 1.9 \end{gathered}$$ The sample mean is $\bar{x} = 2.125$. In sampling-distribution-of-a-sample-mean, a sample mean from a normal population was judged by standardizing it, $z = \dfrac{\bar{x} - \mu}{\sigma/\sqrt{n}}$, and that $z$ follows the standard-normal-distribution; this is how a light sample of coffee bags was shown to be unusual. Here the population standard deviation $\sigma$ of the sugar content is not known. The best Morag has is the sample's own standard deviation, $s \approx 0.301$, which gives the standard-error $\dfrac{s}{\sqrt{n}} \approx 0.106$. So the ratio she can compute is $\dfrac{\bar{x} - \mu}{s/\sqrt{n}}$. Is it standard normal? It is not, and the reason is that its denominator is now a statistic: it changes from sample to sample along with the numerator. This lesson describes the distribution it does follow. ## Definition For a random sample of size $n$ from a population with mean $\mu$, the **t statistic** is $$t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$$ ### [The ten percent condition](https://math.wiki/ten-percent-condition/) **Summary:** Sampling without replacement behaves like independent draws as long as the sample is a small part of the population. **Question:** Picking people one at a time without putting anyone back changes the odds on every later pick. When is that change small enough to ignore? ## Why this exists A mountain trail race has $1500$ entrants, and $450$ of them are from out of state (made-up data). A race official, Bishnu, wants to learn about the entrants by asking a random sample of them. He samples **without replacement**: once a runner is picked, that runner is set aside, because asking the same person twice tells him nothing new. The formulas a statistics course builds next, the binomial distribution and the standard deviation of a sample proportion, assume the draws are **independent**: the chance of a success on each draw does not depend on what the earlier draws gave. Without replacement it does depend, since each pick removes a runner and changes the mix left for the next pick. There are two ways out. One is to redo every formula for draws without replacement, which can be done (hypergeometric-counting-problems does the exact counting) but is heavier. The other is a test that says when the independent formulas are close enough to the truth to use. The ten percent condition is that test. ## Definition In plain words: a random sample drawn without replacement may be treated as if its draws were independent when the sample is at most one tenth of the population. With $N$ the size of the population and $n$ the size of the sample, the **ten percent condition** is $$n \le 0.10\,N, \quad \text{or equivalently} \quad N \ge 10n.$$ The condition is checked only when sampling without replacement from a finite population. Draws with replacement, and the outcomes of a random process such as spinning a spinner, are independent already, and the condition has nothing to say about them. ### [Transforming to linearity: logs and exponential models](https://math.wiki/transforming-to-achieve-linearity/) **Summary:** Re-expressing one or both variables so a curved relationship becomes a straight one. **Question:** The counts climb faster every year and the least-squares line leaves a curve in its residual plot. Can the ordinary line-fitting machinery still describe data like that? ## Why this exists The least-squares line is the model regression hands you ready-made: formulas for $a$ and $b$, residuals that balance about $0$, and r squared and $s$ to summarize the fit. Much data does not lie along a line. A population that grows by a fixed percent each year adds more animals every year than the year before, so its scatter plot bends upward and the residual plot of the straight-line fit shows a U. One could invent a new fitting method for every curve. **Re-expression** is the cheaper idea: change the scale of a variable so the curve becomes straight, fit the ordinary least-squares line on the new scale, judge it with the usual residual plot, then translate the line back into a curve on the original scale. For growth by a constant percent, the change of scale is the logarithm of the response. This lesson follows one made-up data set, manatee counts at a warm-water spring, through all of it: on Day 1 the transformation and the fit, on Day 2 judging the fit and predicting with it. Power models, which need the logarithm of both variables, come next in log-transform-of-both-variables. ## Definition To **transform** (re-express) a variable is to replace every value $y$ by $f(y)$ for one fixed function $f$, keeping each case paired with its own $x$. The **log transformation of the response** replaces $y$ by $\log y$ (base $10$) or $\ln y$ (base $e$), and needs every $y$ to be positive. Recall from logarithms that $\log y$ is the power $10$ must be raised to in order to give $y$, so $\log 1000 = 3$. ### [Two-way tables](https://math.wiki/two-way-tables/) **Summary:** Counts for two categorical variables at once, and the relative frequencies inside. **Question:** Every person answered two questions with a few possible answers each. How do you count them so you can tell whether the answers go together? ## What this area is about A two-way table counts one group of individuals by two categorical variables at once: one row per category of one variable, one column per category of the other, and each cell counts the individuals with both. It keeps which answers came from the same person, which two separate frequency tables lose. Turning its counts into shares of the right total is what lets you say whether the two variables are associated in that data. ## How these fit together **Definitions.** Joint and marginal frequencies name the parts: a joint frequency is one cell, and the marginal frequencies are the row and column totals, each describing one variable on its own. **A procedure.** Conditional relative frequencies divide a cell by its row total or its column total, the group the question narrows to. The two choices answer different questions and usually give different numbers. **A definition built on them.** Association in a two-way table calls the variables associated in the data when the conditional shares differ noticeably from row to row, and draws those shares as a segmented bar graph. ## A path through it Joint and marginal frequencies, then conditional relative frequencies, then association. Finish here and you can read joint and marginal probabilities and conditional probability from a table, and later test an association with a chi-square test for independence. ### [Type I and Type II errors](https://math.wiki/type-ii-error/) **Summary:** Failing to reject a null hypothesis that was actually false. **Question:** A test on a sample of brake cables can send back a good shipment or let a bad one through. How often does each mistake happen, and which one should the factory fear more? ## Why this exists A made-up bicycle factory in the Netherlands buys brake cables by the shipment. Its rule is that a shipment is acceptable if at most $2\%$ of its cables are faulty. Joris, who runs incoming inspection, cannot test every cable, so from each shipment he takes a random sample of $500$ and tests $$H_0\colon p = 0.02 \qquad H_a\colon p > 0.02$$ where $p$ is the proportion of faulty cables in the whole shipment, at significance level $\alpha = 0.05$. A p-value is the chance, if $H_0$ is true, of a sample proportion at least as extreme as the one observed; Joris rejects $H_0$ when the p-value is at or below $\alpha$, and then the shipment goes back to the supplier. The decision rests on $500$ cables, not on the whole shipment, so it can be wrong. A good shipment can, by bad luck, give a sample with many faulty cables and be sent back. A bad shipment can give a sample that looks fine and be put on the assembly line. These are two different mistakes with two different costs. This lesson names them, says how often each one happens, and shows how the choice of $\alpha$ trades one against the other. ## Definition The lesson runs over two days. Part 1 names the two errors, finds the chance of the first, and weighs their consequences. Part 2 computes the chance of the second at a specific alternative. **The four outcomes.** The truth is either "$H_0$ true" or "$H_0$ false", and the test's decision is either "reject $H_0$" or "fail to reject $H_0$". Crossing them gives four outcomes, two right and two wrong: ### [Types of data: categorical, numerical, discrete, continuous](https://math.wiki/categorical-vs-numerical-data/) **Summary:** Data that name a group versus data that count or measure an amount. **Question:** A notebook of data comes back with some columns of words and some of numbers. Which columns can you add up, which can you only count, and how can you tell before doing any arithmetic? ## Why this exists Olavi runs a fruit stand and writes one line in his notebook for each customer. Some columns hold words (apples, card, cash), some hold numbers ($6$, $1.2$). Adding the pieces of fruit customers bought tells him how many he sold. Suppose he also wrote down each customer's loyalty card number, say $417$ and $856$. Adding those gives $1{,}273$, which is not a number of anything. Both columns hold numbers, yet only one takes arithmetic. The difference is what a value **is**: an amount, or a name for a group. That decides which summaries and graphs can be used at all. ## Definition Here a **variable** is one feature recorded about each individual, in the sense of individuals-and-variables. A column that only names each row, such as a customer number, is a tag; individuals-and-variables does not count it as a variable. Books that do count it call it categorical, a group of one per row. A tag is never numerical. A variable is **categorical** when each value names a group, or **category**, that the individual belongs to. The values are usually words (plum, cash, collie), but they may be letters or numbers used as labels. A variable is **numerical** when each value is a number that **counts or measures an amount**, in a unit. Adding, subtracting and averaging the values give answers that mean something. **The test.** Ask: does an average of these values make sense? The average here is the mean: add the values and share the total out equally. If the result is an amount of the same thing, the variable is numerical. If it is meaningless, the variable is categorical. ### [Undercoverage and nonresponse](https://math.wiki/undercoverage/) **Summary:** When part of the population had no chance of being chosen at all. **Question:** A poll draws its people by chance, so every voter seems to have a fair shot. What about the voters who were never on the list it drew from, or who were chosen and never answered? ## Why this exists A simple random sample takes the choice of who is asked away from the person collecting the data, and that removes one large source of bias. But the chance draw is made from a list, and it lands on people who must then answer. Two gaps are left that no amount of careful drawing can close: people who were never on the list, and people who were drawn and never replied. This lesson names both gaps, shows when they push a result one way, and shows how to put a number on how far the second gap could move it. ## Definition The **population** is the whole group a study wants to describe. The **sampling frame** is the list the sample is actually drawn from (recapped from simple-random-sample). **Undercoverage** happens when some members of the population are missing from the frame, or are less likely than others to be reached by the method, so they have little or no chance of being in the sample. **Nonresponse** happens when an individual chosen for the sample cannot be reached or will not take part. If $n$ individuals are chosen and $r$ of them respond, the **response rate** is $$\frac{r}{n}.$$ Each becomes **undercoverage bias** or **nonresponse bias** when the people left out differ from the people counted in the thing being measured, so the result leans the same way every time the method is used. ## Intuition Seini runs a made-up phone poll for a county. She wants the percent of the county's registered voters who support a ballot measure to pay for new bus routes. Her list is every landline number in the county's voter file. She dials $1{,}000$ numbers chosen at random from it, and $380$ voters answer the survey. ### [Uniform and bimodal shapes](https://math.wiki/uniform-and-bimodal-distributions/) **Summary:** A flat shape where every value is about as common, and a two peaked shape that often means two groups. **Question:** Some plots stand level like a wall and some rise in two humps with a dip between. What do those two shapes say about where the data came from? ## Why this exists A dot plot or a histogram shows a whole data set at once, and the first thing the eye takes in is its outline: where it is tall, where it is short, where there is nothing. Naming that outline turns a picture into a sentence someone else can use. Two outlines carry a story. A plot that stands level often comes from fair chance, such as a spinner or the month a person happens to be born in. A plot with two humps often comes from two groups mixed together, such as two breeds of cow in one herd. The name tells you the next question: "is anything favored here?" for the first, "what are the two groups?" for the second. ## Definition A **distribution** is the values a data set takes together with how often each occurs. Its **shape** is the outline of its dot plot or histogram. A **peak** is a stack or bar clearly taller than the stacks or bars beside it. A difference of a dot or two in a small data set does not make a peak. A distribution is **uniform** when its stacks or bars are roughly level across the whole range: every value, or every bin, is about equally common, and no peak stands out. A distribution is **bimodal** when it has two clear peaks with a dip or a gap between them. The peaks need not be equally tall. This is the looser of the two meanings of "bimodal" in mode-and-bimodal-data: it describes the plot, not an exact tie between two modes. A plot with a single peak is called **unimodal**; whether it is symmetric or has a long tail on one side is the subject of skewed-distributions. ### [Units, treatments, factors and levels](https://math.wiki/experimental-units-and-treatments/) **Summary:** What or who receives a condition, and the conditions being compared. **Question:** Before an experiment can compare anything, what exactly is being compared, on what, and how is the result measured? ## Why this exists An experiment imposes something on purpose and measures what happens, which is what lets it show cause and effect (observational-vs-experimental). But "we tried fertilizer on some soybeans" is not yet a plan. Tried it on what: single plants, rows, whole fields? How much fertilizer, and compared with what? Was anything else changed at the same time? What was measured at the end? Farrukh, Ismoil and Abebe are planning a soybean trial on a field split into small test plots. They want to know how fertilizer and watering affect the harvest. Before they can randomize anything or count anything, they need words for the parts of an experiment: the things that get treated, the conditions that are set, the possible settings of each, and the outcome. This lesson names those parts and shows one thing that testing two conditions together can reveal and testing them one at a time cannot. All data in it are made up. ## Definition The **experimental units** are the individuals an experiment assigns treatments to and measures: the smallest thing that can receive a treatment of its own. In the soybean trial each unit is a **plot**, a small rectangle of field, because fertilizer and water are given plot by plot. When the units are people, they are called **subjects** (or participants). A **factor** is an explanatory variable that the experimenter sets. Each value a factor is set to in the experiment is a **level** of that factor. The trial has two factors: ### [Variance of a data set](https://math.wiki/variance-of-a-data-set/) **Summary:** The average squared distance from the mean, whose square root is the standard deviation. **Question:** Every value in a data set sits some distance above or below the mean. How can all those distances be rolled into one number, when the ones above and the ones below cancel out? ## Why this exists The mean says where the middle of a data set is. It says nothing about how far the values stray from that middle. Two paper planes might both average $10$ m a throw, one landing between $9$ and $11$ m every time and the other anywhere from $3$ to $17$ m. One number for that spread lets the two be compared. The natural start is each value's distance from the mean, with a sign, and then the average of those. That fails at once: these signed distances always add to $0$, so their average is always $0$, whatever the data. The signs have to go. Taking absolute values gives the mean absolute deviation (MAD), the mean distance of the values from their mean. Squaring gives the **variance**. Squares are much easier to handle in algebra than absolute values, and the square root of the variance, the standard deviation, is the measure of spread used throughout statistics. ## Definition The **deviation** of a value $x$ is how far it sits from the mean $\bar{x}$, with a sign: $$\text{deviation} = x - \bar{x}$$ A value above the mean has a positive deviation, a value below it a negative one, and a value equal to the mean has deviation $0$. The **variance** of a data set of $n$ values is the mean of the squared deviations: square each value's deviation, add the squares, and divide by $n$. It is written $\sigma^2$, read "sigma squared": $$\sigma^2 = \frac{\text{sum of the squared deviations}}{n}$$ ### [z-scores and the empirical rule](https://math.wiki/z-scores/) **Summary:** How many standard deviations from the mean. **Question:** A moth with a wingspan of 81 mm and a cabbage weighing 9 kg are both big for their kind. Which is more out of the ordinary, when one is measured in millimeters and the other in kilograms? ## Why this exists A moth of one species has a wingspan of $81$ mm. Is that large? The number alone cannot say. If wingspans of this species are usually close to $80$ mm, then $81$ is ordinary; if they are usually close to $50$ mm, it is a giant. Even knowing the mean is not enough: $9$ mm above the mean is a lot when wingspans barely vary and nothing special when they vary widely. What answers the question is distance from the mean measured in the right unit, and the right unit is the standard-deviation, the typical distance of a value from the mean (interpreting a standard deviation). Counting in standard deviations does more than that. Millimeters and kilograms cannot be compared, but "one and a half standard deviations above the mean" means the same thing for a moth, a cabbage or a phone call. That count is the value's **z-score**. And for a variable with a normal model, the z-score alone tells you roughly what share of values lie beyond it, through three numbers that are worth knowing by heart: the **empirical rule**. ## Definition The **z-score** of a value $x$ is the number of standard deviations it lies above or below the mean. For a distribution with mean $\mu$ and standard deviation $\sigma$, $$z = \frac{x - \mu}{\sigma}.$$ For a sample described by its mean $\bar{x}$ and standard deviation $s$, the same idea reads $z = \frac{x - \bar{x}}{s}$. Either way, the top is a distance from the mean and the bottom is the size of one standard deviation. ## Trigonometry ### [Adding and subtracting vectors](https://math.wiki/vector-addition-triangle-rule/) **Summary:** Placing the second arrow's tail at the first arrow's head, with the sum running from first tail to last head. **Question:** One trip is followed by a second trip in a different direction. Which single trip has the same effect as both, and how is it found from the two arrows or from their components? ## Why this exists Kele, Tala and Zaid go geocaching: hunting for small containers hidden in a park, using a GPS that gives positions in meters east and north of the trailhead. Kele walks to one cache, then turns and walks to the next. Each walk is a vector, so much east and so much north. Two questions follow at once. Where is Kele now, as a single trip from the trailhead? And which trip takes Tala, who is standing at a different cache, straight to Kele? The first question asks for the sum of two vectors and the second for their difference. Both have a picture (arrows placed end to end) and a calculation (components added or subtracted), and the two always agree. ## Definition The **sum** $\mathbf{u} + \mathbf{v}$ is found by the **triangle rule**, also called the **tip-to-tail rule**: draw $\mathbf{u}$, place the tail of $\mathbf{v}$ at the head of $\mathbf{u}$, and draw the arrow from the tail of $\mathbf{u}$ to the head of $\mathbf{v}$. That arrow is the sum, also called the **resultant** of $\mathbf{u}$ and $\mathbf{v}$. More vectors are added the same way, each tail at the previous head, and the resultant runs from the first tail to the last head. In components, with $\mathbf{u} = \langle a, b \rangle$ and $\mathbf{v} = \langle c, d \rangle$, $$\mathbf{u} + \mathbf{v} = \langle a + c,\ b + d \rangle$$ and in $\mathbf{i}$, $\mathbf{j}$ form the coefficients of $\mathbf{i}$ add, and so do those of $\mathbf{j}$. ### [Amplitude and reflection](https://math.wiki/amplitude/) **Summary:** Half the distance between the highest and the lowest value of a wave. **Question:** A guitar string swings farther from rest when it is plucked harder, and half a vibration later it bulges the other way. How can one number in front of sine describe both? ## Why this exists The graphs of $y = \sin x$ and $y = \cos x$ (graph-sine-from-the-unit-circle) never rise above $1$ or fall below $-1$, because they record the coordinates of a point on a circle of radius $1$. Most things that repeat swing by other amounts. When a guitar string vibrates, its middle may move $2$ millimeters from rest or a tenth of that, depending on how hard it was plucked, and half a vibration after the string bulges up it bulges down by the same amount. To describe such motion with sine and cosine, a wave needs a size other than $1$, and it needs a way to be turned upside down. Both come from one number multiplied onto the output: $y = a\sin x$ and $y = a\cos x$. This lesson says what that number does to the graph and names the size it controls, the amplitude. ## Definition The graph of a periodic function such as $y = a\sin x$ swings between a highest value $M$ and a lowest value $m$. Its **amplitude** is half the distance between them: $$\text{amplitude} = \frac{M - m}{2}$$ It is the distance from the line halfway between the highest and lowest values, the **midline**, up to a highest point, and equally down to a lowest point. For the waves in this lesson the midline is the $x$-axis, $y = 0$; moving it is the subject of vertical-shift-of-a-sinusoid. Amplitude is a distance, so it is never negative. For any real number $a \ne 0$, the graphs of $$y = a\sin x \qquad\text{and}\qquad y = a\cos x$$ ### [Amplitude, period and phase shift](https://math.wiki/amplitude-period-phase/) **Summary:** How tall, how long, and how far shifted, read off the formula. **Question:** Real waves swing by different amounts, repeat at different rates, start at different times and balance at different levels. Where does each of those live in the formula? ## What this area is about The graphs of $y = \sin x$ and $y = \cos x$ swing between $-1$ and $1$, balance on the $x$-axis and repeat every $2\pi$ (with $x$ in radians). A general sinusoid $y = a\sin\big(b(x - h)\big) + k$, with $a \ne 0$ and $b \ne 0$, changes each of those, one number at a time, and every number can be read off the formula. ## How these fit together **Two definitions on the output.** Amplitude and reflection: the amplitude is $|a|$, and $a < 0$ flips the wave over its midline. Midline and vertical shift: adding $k$ moves the wave so it balances on $y = k$, without changing the amplitude. **Two procedures on the input.** Finding the period: multiplying the input by $b$ gives period $\frac{2\pi}{|b|}$. Phase shift: the sideways slide $h$ can only be read once the input is factored as $b(x - h)$, so in $\sin\left(3x - \frac{\pi}{2}\right)$ it is $\frac{\pi}{6}$, not $\frac{\pi}{2}$. ## A path through it Amplitude comes before the midline lesson, which uses it. The period comes before the phase shift, which first squeezes the wave by $b$. With all four, you can write an equation from a wave and combine a sine and a cosine into one wave. ### [Angles and their measure](https://math.wiki/angles-and-their-measure/) **Summary:** How an angle is named and measured once it is placed on the coordinate plane instead of inside a shape. **Question:** In a triangle an angle never passes 180 degrees. What changes when an angle becomes an amount of turning, which can go around more than once, or backward? ## What this area is about Geometry measures an angle as a figure, between $0^\circ$ and $180^\circ$. Trigonometry needs angles that turn: any amount, either direction, pinned to the coordinate plane. It also needs a unit that comes from the circle itself, the radian, which makes the formulas joining an angle to a length come out clean. A full turn is $360^\circ$, or $2\pi$ radians. ## How these fit together **Two definitions.** Angles in standard position put the vertex at the origin and the initial side on the positive $x$-axis; counterclockwise is positive, clockwise negative, and coterminal angles share a terminal side, differing by whole turns. Radian measure measures an angle as arc over radius, $\theta = \frac{s}{r}$, and converts between radians and degrees, minutes and seconds. **What radians buy.** Arc length and sector area in radians gives $s = r\theta$ and $A = \frac{1}{2}r^2\theta$, which hold only with $\theta$ in radians. Angular and linear speed separates how fast an object turns from how fast one point of it travels; with $\omega$ in radians per unit of time, $v = r\omega$. ## A path through it Radian measure comes before arc length and angular speed, since both are built on it. Standard position is not a prerequisite of either, but radian measure and angular speed refer back to it, so it is a natural first lesson. Next, these angles go onto the unit circle, where sine and cosine become coordinates, and into polar coordinates; the turning wheel model uses angular speed. ### [Angles in standard position and coterminal angles](https://math.wiki/standard-position-angle/) **Summary:** An angle whose vertex sits at the origin and whose initial side lies along the positive x axis. **Question:** A radar sweep has turned through more than a full circle, or turned the other way. How do you say exactly where it points, and how far it has turned to get there? ## Why this exists In geometry an angle is a figure: two rays from one point, measured with a protractor somewhere between $0^\circ$ and $180^\circ$ (see angle-measure). That is enough for a triangle. It is not enough for anything that turns. A radar sweep starts pointing east and turns. After a while it has gone once around and a little more. Another sweep turns the other way. A protractor reading of the final position says where each one points, but not how it got there, and two sweeps pointing the same way may have turned through very different amounts. The fix is to treat an angle as a **rotation** and to pin it to the coordinate plane. Every angle starts in the same place, the direction of the turn is recorded with a sign, and the amount can be any number of degrees at all. This is the picture the rest of trigonometry is built on. ## Definition An angle is in **standard position** when its vertex is at the origin and its **initial side** lies along the positive $x$-axis. The ray where the rotation ends is the **terminal side**. The measure of the angle is the amount of rotation from the initial side to the terminal side, with a sign for the direction: - a **positive angle** turns **counterclockwise**; - a **negative angle** turns **clockwise**; - the angle $0^\circ$ does not turn at all, so its terminal side is the positive $x$-axis. A full turn is $360^\circ$, and nothing stops the rotation at one turn. An angle of $410^\circ$ goes all the way around and then $50^\circ$ more. Angles are often named with Greek letters such as $\theta$. ### [Angles of elevation and depression](https://math.wiki/two-observer-elevation-problems/) **Summary:** Using sightings of the same object from two places to find a height or distance neither one gives alone. **Question:** You can see the top of a tower but cannot walk to its foot. How can angles, measured from where you are allowed to stand, tell you how tall it is? ## When you reach for it Some lengths cannot be measured with a tape: the height of a cliff, a tower on the far bank of a river, the distance from a lighthouse out to a boat. Angles can be measured from where you stand, with a clinometer or a surveyor's instrument, and solving a right triangle turns one angle and one length into all the rest. This lesson is about where those right triangles are in the world, and what to do when one triangle is not enough. Every problem here is built from the same three lines: a **horizontal** line through the eye of the person looking, a **vertical** line through the thing looked at, and the **line of sight** joining the two. A horizontal line and a vertical line meet at a right angle, so the three always make a right triangle, and the angle that gets measured is always taken from the horizontal. Reach for one sighting when one of the two legs is known: the distance along the ground to the foot of a building, or the height of the cliff you are standing on. Reach for the same triangle, read as rise over run, when a road or a roof is described by its grade. And reach for **two sightings** when the foot of the object cannot be reached, so that neither leg is known. One angle then gives only the shape of the triangle, not its size, and a second angle measured from a second place, a known distance away, is what fixes the height. ## The procedure ### Angles of elevation and depression ### [Angular and linear speed](https://math.wiki/angular-speed/) **Summary:** How fast something turns, measured as angle swept per unit of time. **Question:** Two points ride the same turning wheel, one near the center and one at the rim. They turn together, so why does the rim point move so much faster? ## Why this exists A pottery wheel spins at $150$ revolutions per minute. That one number says how fast the wheel turns, but not how fast the clay moves under the potter's hands: clay near the center barely creeps, while clay at the rim of a wide plate races past. A turning object needs two speeds. **Angular speed** is how fast it turns, the angle swept per unit of time, and it is the same for every point of the object. **Linear speed** is how fast one point travels along its circle, distance per unit of time, and it grows with the distance from the center. Radians join the two. Because a radian measures an angle by arc over radius (radian measure), turning speed times radius gives travel speed with no stray factor. The same link answers a practical question: when a belt or chain connects two wheels of different sizes, how fast does the second one turn? ## Definition An object turning about a fixed center sweeps an angle $\theta$ in a time $t$. Its **angular speed** is the angle per unit of time: $$\omega = \frac{\theta}{t}$$ The letter $\omega$ is the Greek omega. Angular speed may be given in radians per second, degrees per second, or **revolutions per minute** ($\text{rpm}$), the unit printed on most motors. One revolution is a full turn, $2\pi$ radians or $360^\circ$. A point of the object at distance $r$ from the center travels along an arc of length $s$ in the same time. Its **linear speed** is the distance per unit of time: ### [Applications of oblique triangles](https://math.wiki/triangulation/) **Summary:** Locating a point by measuring the angles to it from the two ends of a known baseline. **Question:** You cannot stretch a tape to the far bank, the hut across the ice or the top of an antenna. How can you still find where it is, how far, and how big? ## Why this exists A tape measure needs both ends in your hands. The far bank of a river, a hut across a frozen lake, the top of an antenna: you can stand at one end of the distance, never the other. Angles, though, can be measured from where you stand. Triangulation trades the distance you cannot walk for one you can, called a **baseline**, plus angles sighted from its ends. The triangle that results almost never has a right angle, so the tools are the law of sines, the law of cosines, the sine area formula and Heron's formula. This lesson is about using them on real ground: setting up the triangle and choosing which tool the measurements allow. ## Definition **Triangulation** locates a point $R$ from a known segment $PQ$, the **baseline**, by measuring the angles $\angle QPR$ and $\angle PQR$ at its two ends. When $\angle P + \angle Q < 180^\circ$, two angles and the side between them is ASA, so exactly one triangle fits on each side of the line $PQ$, and $R$ is fixed once you know which side it is on. An **application of oblique triangles** is any measurement whose triangle has no right angle: locating a point, correcting a course, measuring land, finding a height you cannot reach. Each is solved by asking which three parts of a triangle the measurements give, and choosing the tool from that. ## Intuition Stand at $P$ and sight the boulder: it lies somewhere on that ray, but you do not know how far along. Walk to $Q$ and sight it again: a second ray. Two rays from different points cross at most once, and the crossing is the boulder. The baseline turns two directions into a position. ### [Arc length and sector area in radians](https://math.wiki/arc-length-in-radians/) **Summary:** Finding the distance along a circle by multiplying the radius by the angle in radians. **Question:** Once an angle is measured in radians, how long is its arc and how much area does its sector hold? ## When you reach for it Reach for these formulas whenever a piece of a circle is cut off by a central angle measured in radians: the edge of a folding fan, the paper the fan carries, the strip a pendulum sweeps. The arc length and sector area lessons did the same jobs in degrees, where every answer carried a factor of $\frac{\pi}{180}$ or $\frac{1}{360}$. With radian measure that factor disappears, and the formulas become short enough to solve backward, combine, and compare. ## The procedure ### Arc length The radian measure of a central angle is defined as arc over radius, $\theta = \frac{s}{r}$. Multiplying both sides by $r$ gives the arc: $$s = r\theta$$ Nothing new has been assumed; this is the definition read the other way. It holds only with $\theta$ in radians, and $s$ comes out in the same unit as $r$. ### Sector area A sector with angle $\theta$ is the fraction $\frac{\theta}{2\pi}$ of a full turn, because a full turn is $2\pi$ radians. Equal central angles cut equal slices, so the sector holds that same fraction of the circle's area $\pi r^2$: $$A = \frac{\theta}{2\pi} \cdot \pi r^2 = \frac{1}{2}r^2\theta$$ Replacing $r\theta$ by $s$ gives a third form, which needs no angle at all: $$A = \frac{1}{2}r s$$ It reads like the area of a triangle with base $s$ and height $r$, and that is the picture: a thin sector is nearly a tall thin triangle, as in the area of a circle. ### Running the formulas backward ### [Area from two sides and the included angle](https://math.wiki/derive-area-from-two-sides-and-an-angle/) **Summary:** The sine of the included angle turns one side into the height used by the usual triangle area formula. **Question:** The area of a triangle is half its base times its height. What if you know two sides and the angle between them, but not the height? ## When you reach for it The area of a triangle is $\tfrac{1}{2}(\text{base})(\text{height})$. The base is usually a side you know; the height usually is not. When you know two sides and the angle between them (SAS), the height is one sine away, and the area follows without solving the rest of the triangle. Any triangle whose parts you can find leads here too. AAS and ASA triangles give their missing sides by the law of sines, and SSS triangles give an angle by the law of cosines. Once two sides and the angle between them are known, this move gives the area. As in those lessons, the triangle is $ABC$, and each side is named by the lowercase letter of the angle across from it. ## The procedure ### The height is a side times a sine Take side $CB$, of length $a$, as the base. Drop the height from $A$ to the line $CB$, meeting it at $D$, and call its length $h$. If $C$ is acute and $D$ falls on the base, then $CDA$ is a right triangle with hypotenuse $CA = b$ and the side $h$ opposite the angle $C$. The sine ratio gives $\sin C = \dfrac{h}{b}$, so $h = b\sin C$. Then $$\begin{aligned} \text{Area} &= \frac{1}{2}(\text{base})(\text{height}) \\ &= \frac{1}{2}a\,(b\sin C) = \frac{1}{2}ab\sin C. \end{aligned}$$ This is the same altitude that proved the law of sines; there it was compared with a second expression for itself, and here it is multiplied by the base. ### When the included angle is obtuse ### [Cofunction identities](https://math.wiki/cofunction-identities/) **Summary:** Each function of an angle equals its partner function of the complementary angle. **Question:** The cosine of an acute angle is the sine of its complement. Does that swap survive for every real number t, where no right triangle can be drawn? ## Why this exists In a right triangle the two acute angles add to $90^\circ$, and the side opposite one is the side adjacent to the other. sine-ratio read that as $\sin A = \cos(90^\circ - A)$, the "co" in cosine standing for complement, and secant-ratio did the same for secant with cosecant and tangent with cotangent. That argument needs a triangle, so it only covers angles between $0$ and $\frac{\pi}{2}$. On the unit circle every real number $t$ has a sine and a cosine, and $t = 2.5$ or $t = -1$ is no angle of any right triangle. Is $\sin\left(\frac{\pi}{2} - 2.5\right)$ still $\cos 2.5$? It is, and the reason is a fourth mirror of the unit circle, the line $y = x$, alongside the three in unit-circle-symmetry. Once the swap holds for every $t$, it combines with the even and odd identities and with periodicity to turn any quarter-turn shift, such as $\sin\left(t - \frac{\pi}{2}\right)$ or $\cos\left(\frac{\pi}{2} + t\right)$, into a single function of $t$. ## Definition In radians, two numbers are **complementary** when they add to $\frac{\pi}{2}$, so the complement of $t$ is $\frac{\pi}{2} - t$. Unlike the angles of a triangle, $t$ may be any real number, and its complement may be negative or larger than $\frac{\pi}{2}$. The six trig functions come in three **cofunction pairs**: sine and cosine, tangent and cotangent, secant and cosecant. The **cofunction identities** say that each function of $t$ equals its partner function of the complement of $t$, for every real $t$ at which the two sides are defined. They are identities in the sense of what-an-identity-is. In degrees the complement of $\theta$ is $90^\circ - \theta$. ### [Combining a sine and a cosine into one wave](https://math.wiki/linear-combination-as-a-single-sinusoid/) **Summary:** A sine term plus a cosine term of the same period is itself one wave with a new amplitude and shift. **Question:** A sine wave plus a cosine wave of the same period: is the sum another wave, and if so, how tall is it and where does it peak? ## Why this exists Expand $5\sin(x + 1)$ with the sine sum formula and you get $5\cos 1\,\sin x + 5\sin 1\,\cos x$. A shifted sine wave is a mix of $\sin x$ and $\cos x$. Mixes like that also turn up on their own. A weight on a spring that starts away from rest and already moving, two sound sources a little apart, two parts of an alternating current circuit: each produces a sine term plus a cosine term of the same period. The questions are the ones asked of any wave. How far does it swing? When is it largest? When does it reach a given value? None of them is easy to read from $3\sin x + 2\cos x$, because two terms change at once. Running the sum formula backward turns the mix into one wave, $R\sin(x + \alpha)$. A single wave answers all three questions at sight: its amplitude is $R$, and its peaks and its solutions come from one sine. ## Definition Let $a$ and $b$ be real numbers, not both $0$. The sum $a\sin x + b\cos x$ is a **linear combination** of $\sin x$ and $\cos x$: each is multiplied by a constant and the results are added. Writing it **as a single sinusoid** means finding a number $R > 0$ and an angle $\alpha$ with $$a\sin x + b\cos x = R\sin(x + \alpha)$$ for every real $x$. The two numbers are fixed by $$\begin{aligned} R &= \sqrt{a^2 + b^2} \\ \cos\alpha &= \frac{a}{R}, \quad \sin\alpha = \frac{b}{R} \end{aligned}$$ Both conditions on $\alpha$ are needed. Together they pick out one angle in each turn of $2\pi$; either one alone allows two. ### [Component form of a vector](https://math.wiki/component-form-of-a-vector/) **Summary:** Writing a vector as the horizontal and vertical change from its tail to its head. **Question:** Every arrow that makes the same trip is the same vector. What short name can every one of those arrows share? ## Why this exists A directed line segment is one drawing of a vector, and the same vector has endless drawings, one starting at every point of the plane. Talking about "the vector drawn from $(2, 7)$ to $(5, 9)$" picks one drawing out of all of them and makes you carry two points around when the vector itself is only a trip. On a grid, every drawing of the same vector goes the same distance across and the same distance up. Those two numbers are what all the drawings share, so they can serve as the vector's name. That name is the vector's component form, and it turns arrows into numbers you can compute with. ## Definition Let $P(x_1, y_1)$ be the initial point (tail) and $Q(x_2, y_2)$ the terminal point (head) of a directed segment. The **component form** of the vector it represents is $$\overrightarrow{PQ} = \langle x_2 - x_1,\ y_2 - y_1 \rangle$$ terminal minus initial, one coordinate at a time. A vector $\mathbf{v} = \langle a, b \rangle$ has **horizontal component** $a$ (the change in $x$) and **vertical component** $b$ (the change in $y$). A negative component means the arrow goes left or down. Angle brackets mark a vector; parentheses mark a point. The point $(5, 3)$ is a location. The vector $\langle 5, 3 \rangle$ is a trip, $5$ right and $3$ up, which can start anywhere. The drawing of $\langle a, b \rangle$ whose tail is the origin $O(0, 0)$ is its **position vector** drawing. Its head is $(0 + a, 0 + b) = (a, b)$, so the position vector of the point $(a, b)$ is $\langle a, b \rangle$. ### [Composing trig and inverse trig functions](https://math.wiki/simplify-a-composition-to-an-algebraic-expression/) **Summary:** Drawing a reference triangle from the inner inverse so the outer function becomes an expression in x. **Question:** If all you know about an angle is that its sine is $\\frac{5}{13}$, can you find its cosine without ever finding the angle? ## When you reach for it An inverse trig function hands back an angle. Very often that angle is not wanted for itself: it is fed straight into another trig function, as in $\cos\left(\arcsin\frac{5}{13}\right)$. This is a composition, the output of $\arcsin$ becoming the input of $\cos$. A calculator can grind out a decimal, but the exact value, $\frac{12}{13}$, comes from a picture, and when the inside holds a variable, as in $\tan\left(\arccos\frac{x}{3}\right)$, only the picture works. The result is then an algebraic expression in $x$, with no trig left in it. You reach for this move whenever a trig function is applied to an inverse trig function of a different kind, when a formula such as $\cos(\arcsin x)$ needs to be written without trig, and later, in calculus, wherever an angle was introduced only to be removed again. It rests on the slices chosen in the inverse trig functions: $\arcsin$ returns an angle in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, $\arccos$ one in $[0, \pi]$, and $\arctan$ one in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. ## The procedure **Name the inner angle.** Write $\theta$ for the inner expression. In $\cos\left(\arcsin\frac{5}{13}\right)$, let $\theta = \arcsin\frac{5}{13}$. By the definition of arcsine this says two things: $\sin\theta = \frac{5}{13}$, and $\theta$ lies in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$. The question has become "find $\cos\theta$", and both facts are needed to answer it. ### [Converting an equation between the two systems](https://math.wiki/convert-an-equation-between-polar-and-rectangular/) **Summary:** Substituting the conversion relations to rewrite a whole curve in the other coordinate system. **Question:** A curve is described by an equation in x and y, but the problem is about distances and directions from one center, or the other way round. How do you rewrite the whole curve in the other system, and how do you recognize it afterward? ## When you reach for it The relations from polar-to-rectangular-conversion, $$\begin{aligned} x &= r\cos\theta, & y &= r\sin\theta, \\ x^2 + y^2 &= r^2, & \tan\theta &= \frac{y}{x} \end{aligned}$$ move one point at a time between the two systems. Used on a whole equation, they move a whole curve. The circle $x^2 + y^2 = 49$ needs two variables in rectangular form but is simply $r = 7$ in polar form: every point at distance $7$. The other way, an unfamiliar polar equation such as $r = 6\cos\theta - 8\sin\theta$ turns out, once converted, to be a circle whose center and radius can be read off. Rewrite in polar form when the curve is about distance from one center or about direction; rewrite in rectangular form to recognize a curve you have only seen in polar form. Throughout, the pole sits at the origin and the polar axis runs along the positive $x$-axis. ## The procedure ### What a converted equation has to say A point lies on the graph of a polar equation when **at least one** of its polar names satisfies the equation. A point has many names, $(r, \theta)$, $(r, \theta + 2\pi)$, $(-r, \theta + \pi)$ and so on, and the pole is $(0, \theta)$ for every $\theta$; only one name needs to work. The relations $x = r\cos\theta$ and $y = r\sin\theta$ hold for **every** name of a point, negative $r$ included. So if a rectangular equation in $x$ and $y$ holds at a point, it holds after $x$ and $y$ are replaced by $r\cos\theta$ and $r\sin\theta$, for every name of that point; and if the substituted equation holds for one name, the point's $x$ and $y$ satisfy the original. The two equations describe the same set of points. Nothing else is needed to justify the substitution. ### [Converting between polar and rectangular](https://math.wiki/polar-to-rectangular-conversion/) **Summary:** Turning a distance and an angle into x and y with cosine and sine. **Question:** An instrument reports where something is as a distance and an angle, but the map is a grid of east and north. How do you turn one kind of name into the other, both ways? ## When you reach for it Genji maps an archaeological dig. A surveying instrument called a total station stands on a tripod over a fixed point, the datum, and for each find it reports a **distance** in meters and an **angle**. Surveyors usually count angles clockwise from north; in this lesson, to match the mathematician's convention, the team's angles are taken counterclockwise from grid east. The site plan, though, is a grid of meters east and north of the datum. Every find arrives in polar coordinates (plot-a-polar-point) and has to be drawn in rectangular ones. Going back the other way, to set out a planned test pit with the instrument, a grid position has to become a distance and an angle. This lesson moves single points between the two systems, in both directions. Throughout, the pole sits at the origin and the polar axis runs along the positive $x$-axis. ## The procedure ### From polar to rectangular Take a point with polar name $(r, \theta)$ and suppose first that $r > 0$. The ray at angle $\theta$ crosses the unit circle at $(\cos\theta, \sin\theta)$: that is the definition of cosine and sine (cosine-and-sine-as-coordinates), and it holds for every angle, not only acute ones. The point $r$ units out along the same ray is that unit-circle point stretched away from the origin by the factor $r$, which keeps it on the ray and multiplies its distance by $r$. Stretching by $r$ multiplies both coordinates by $r$, so ### [Cosine of a sum or difference](https://math.wiki/sum-formula-for-cosine/) **Summary:** The cosine of two angles added is not the sum of the cosines but a mix of both functions. **Question:** The exact cosines of $60^\\circ$ and $45^\\circ$ are known. Is that enough to find the exact cosine of $105^\\circ$, and if so, how do the pieces combine? ## Why this exists The unit circle gives exact sines and cosines at a handful of angles: the multiples of $30^\circ$ and $45^\circ$, or of $\frac{\pi}{6}$ and $\frac{\pi}{4}$. An angle such as $105^\circ$ is not on that list, but it is $60^\circ + 45^\circ$, and both of those are. What is missing is a rule for the cosine of a sum. The first guess, that the cosine of a sum is the sum of the cosines, is false. Take $a = b = 60^\circ$: $$\begin{aligned} \cos(60^\circ + 60^\circ) &= \cos 120^\circ = -\frac{1}{2} \\ \cos 60^\circ + \cos 60^\circ &= 1 \end{aligned}$$ Cosine is not multiplication by a number, so it does not distribute over a sum. The true rule needs the sines of both angles as well as the cosines, and it comes straight out of the distance formula on the unit circle. ## Definition For all real numbers $a$ and $b$ (or angles in degrees), the **difference formula for cosine** and the **sum formula for cosine** are the identities $$\begin{aligned} \cos(a - b) &= \cos a\cos b + \sin a\sin b \\ \cos(a + b) &= \cos a\cos b - \sin a\sin b \end{aligned}$$ They are identities in the sense of what-an-identity-is: true for every pair $a, b$, with nothing excluded. Read in words, the cosine of a difference is "cosine cosine plus sine sine", and the cosine of a sum has the same two products with a minus sign between them. Together they are often called the **sum and difference formulas for cosine**. ### [Directed line segments](https://math.wiki/directed-line-segment/) **Summary:** A segment with an arrow on it, carrying both a length and a direction. **Question:** A segment tells you how far apart two points are. How do you also say which way you went? ## Why this exists The segment $\overline{AB}$ and the segment $\overline{BA}$ are the same set of points. A segment has a length and nothing more: it does not know which end it starts from. A move does care. An air hockey puck sliding from one end of the table to the other has not made the same trip as a puck sliding back. To describe a move, a force or a velocity you need two facts at once: **how far** and **which way**. So pick one end as the start, the other as the finish, and put an arrow on the segment. That arrow is a directed line segment, the first way to see a vector. ## Definition A **directed line segment** is a segment with one endpoint chosen as its start. The directed segment from $P$ to $Q$ is written $\overrightarrow{PQ}$. Its **initial point** (or tail) is $P$ and its **terminal point** (or head) is $Q$. It is drawn as the segment $\overline{PQ}$ with an arrowhead at $Q$. A directed segment carries two things. - Its **length** is the length $PQ$ of the segment $\overline{PQ}$. - Its **direction** is the way the arrow points. Two directed segments of positive length have the **same direction** when they lie on parallel lines (or on one line) and point the same way. They have **opposite directions** when they lie on parallel lines (or on one line) and point opposite ways. Swapping the ends gives $\overrightarrow{QP}$, the **opposite** of $\overrightarrow{PQ}$: same length, opposite direction. ### [Double angle formulas](https://math.wiki/double-angle-formula-for-sine/) **Summary:** Setting both angles equal in the sine sum formula gives twice sine times cosine. **Question:** The sum formulas handle the sine and cosine of any two angles added together. What do they say when the two angles are the same, so that the angle is simply doubled? ## Why this exists The sum formulas (sum-formula-for-cosine and sum-and-difference-formulas-for-sine) take two angles $a$ and $b$. The case that comes up more than any other is $b = a$: an angle doubled. A point on a wheel that turns twice as fast, a wave of twice the frequency, a launch angle that appears in a range formula as $2\theta$: each needs $\sin 2x$ or $\cos 2x$ written in terms of $\sin x$ and $\cos x$. Doubling does not pass through the sine. At $x = 90^\circ$, $\sin 2x = \sin 180^\circ = 0$, while $2\sin x = 2$. Setting $b = a$ in the sum formulas gives the true rules, and the Pythagorean identity then gives cosine three useful shapes. ## Definition The **double angle formulas** are the identities $$\begin{aligned} \sin 2x &= 2\sin x\cos x \\ \cos 2x &= \cos^2 x - \sin^2 x \\ &= 1 - 2\sin^2 x \\ &= 2\cos^2 x - 1 \end{aligned}$$ which hold for every real $x$ (or angle in degrees), and $$\tan 2x = \frac{2\tan x}{1 - \tan^2 x}$$ which holds whenever $\tan x$ and $\tan 2x$ both exist: $x$ is not an odd multiple of $\frac{\pi}{2}$, and $x$ is not $\frac{\pi}{4} + k\cdot\frac{\pi}{2}$ for any integer $k$. They are identities in the sense of what-an-identity-is. The three lines for $\cos 2x$ are one identity in three forms: pick the form whose ingredients are known. ## Intuition Take an acute angle $\theta$ and build an isosceles triangle with two sides of length $1$ meeting at $O$ with angle $2\theta$. Its area can be found two ways. ### [Eliminating the parameter](https://math.wiki/eliminate-the-parameter/) **Summary:** Combining the two equations to get a single relation between x and y. **Question:** A pair of parametric equations traces a path one moment at a time. Which curve is that path, written as a single equation in $x$ and $y$? ## When you reach for it A pair $x = f(t)$, $y = g(t)$ (see parameter-and-orientation) gives the curve one point at a time. To say which curve it is, a line, a parabola, a circle, an ellipse or a hyperbola, you want an equation in $x$ and $y$ alone, the kind whose graph you already know. Getting rid of $t$ to reach that equation is **eliminating the parameter**. The trade has two sides. The single equation names the shape, but it forgets the starting point, the direction and the timing. And it can describe more than the path: the equation $y = 4 - x^2$ is a whole parabola, while a parametric pair may trace only half of it. So the finished answer is always an equation **together with** the part of its graph that is traced and the direction of travel. ## The procedure **Solve one equation for $t$ and substitute.** At every point of the curve, both equations hold with the same value of $t$. If one of them can be solved for $t$ with a single answer, as in solve-for-a-variable, that expression is the $t$ of the point, and putting it into the other equation leaves an equation in $x$ and $y$ that every point of the curve satisfies. Choose the equation that gives one answer. From $x = 3t - 2$ there is only $t = \frac{x + 2}{3}$, while $y = t^2 + 1$ gives $t = \pm\sqrt{y - 1}$, two candidates, and picking one silently drops half the curve. **Use an identity.** When the coordinates are trigonometric, solving for $t$ would bring in inverse functions with restricted outputs. Instead, isolate the trigonometric functions and feed them into an identity that holds for every $t$, such as the Pythagorean identities ### [Exact values on the unit circle](https://math.wiki/exact-values-at-quadrantal-angles/) **Summary:** Reading the values of the six functions where the terminal side lies on an axis. **Question:** A lighthouse beam turns until it points along $\\frac{4\\pi}{3}$, or straight down the page at $\\frac{3\\pi}{2}$. At which angles can all six trigonometric values be written exactly, with no calculator, and how are they read off the circle? ## When you reach for it On the unit circle every angle $\theta$ has a terminal point $(x, y)$, with $\cos\theta = x$ and $\sin\theta = y$ (cosine-and-sine-as-coordinates), and the other four functions are built from the same two numbers (undefined-values-of-tangent-and-secant). For most angles those coordinates are decimals a calculator has to supply. For two families of angles they are exact numbers you can write down by hand: - the **quadrantal angles**, the multiples of $\frac{\pi}{2}$ (or $90^\circ$), whose terminal side lies on an axis; - the multiples of $\frac{\pi}{6}$ and of $\frac{\pi}{4}$ (or $30^\circ$ and $45^\circ$), whose reference angle is $\frac{\pi}{6}$, $\frac{\pi}{4}$ or $\frac{\pi}{3}$. You reach for this whenever a trig value is wanted at one of these angles: to evaluate an expression exactly, to place a point a given distance away in a given direction, to check a calculator, or as the first step in solving an equation such as $\cos\theta = -\frac{1}{2}$. Exact values also carry structure a decimal hides: $\tan\frac{\pi}{3} = \sqrt{3}$ says the line at that angle has slope $\sqrt{3}$, which $1.732$ only approximates. The lesson runs over two days. Part 1 places the sixteen exact points on the circle. Part 2 reads all six functions at any of those angles. ## The procedure ### Part 1: the quadrantal angles A quadrantal angle ends on an axis, so its terminal point is where that axis meets the circle of radius $1$. Going counterclockwise from the positive $x$-axis the four points are ### [Finding the period from the equation](https://math.wiki/period-from-the-b-value/) **Summary:** Reading how much a wave is squeezed or stretched sideways from the coefficient on the input. **Question:** A hummingbird's wings sweep back and forth dozens of times a second. When a wave like that is written as y = sin(bx), where in the formula is its speed, and how long does one cycle last? ## When you reach for it Reach for this move whenever a sine or cosine has a number multiplying its input, as in $y = \cos\left(\frac{2}{3}x\right)$ or $y = \sin(4\pi x)$, and you need to know how long one cycle lasts: to sketch the graph, to place its key points, to say how often something repeats, or to choose the number that makes a model repeat at the right rate. The graphs of sine and cosine have period $2\pi$. Multiplying the input by a constant $b$ keeps the shape of the wave and changes only how wide a cycle is. ## The procedure **One cycle of the inside.** Take $b > 0$. The graph of $y = \sin u$ makes one full cycle while $u$ runs from $0$ to $2\pi$. In $y = \sin(bx)$ the sine is applied to $u = bx$, so one cycle runs while $bx$ goes from $0$ to $2\pi$, that is, while $x$ goes from $0$ to $\frac{2\pi}{b}$. The same is true of cosine. **Why that is exactly the period.** Let $b$ be any nonzero number. A shift $p$ is a repeat shift of $\sin(bx)$ when $\sin\big(b(x + p)\big) = \sin(bx)$ for every $x$. As $x$ runs over all real numbers so does $u = bx$, so this says $\sin(u + bp) = \sin u$ for every $u$: the number $bp$ is a repeat shift of sine. By period-and-cycle, the repeat shifts of sine are exactly $2\pi k$ for nonzero integers $k$. So $p = \frac{2\pi k}{b}$, and the smallest positive choice gives the period of both $\sin(bx)$ and $\cos(bx)$: $$P = \frac{2\pi}{|b|}$$ **What happens to the graph.** If $(x_0, y_0)$ is on $y = \sin x$, then $\left(\frac{x_0}{b}, y_0\right)$ is on $y = \sin(bx)$, because $\sin\left(b \cdot \frac{x_0}{b}\right) = \sin x_0 = y_0$. Every point keeps its height and has its $x$-coordinate divided by $b$. For $b > 1$ the graph is squeezed toward the $y$-axis, a **horizontal compression** by the factor $\frac{1}{b}$. For $0 < b < 1$ it is pulled away from the $y$-axis, a **horizontal stretch** by the factor $\frac{1}{b}$. The number inside acts on $x$ in reverse, unlike the vertical stretches made by multiplying the output: to feed the sine the same input $x_0$, the new $x$ only has to be $\frac{x_0}{b}$. ### [Forces and velocities as vectors](https://math.wiki/force-and-velocity-vectors/) **Summary:** Combining pushes or motions acting in different directions, such as a plane in a crosswind. **Question:** A plane flies through air that is itself moving, and several ropes pull on one load at once. What single motion, or single pull, do they add up to? ## When you reach for it Some quantities have a size and a direction. A **velocity** is a speed together with a direction of motion, and a **force** is a push or a pull of a certain strength in a certain direction. Both are vectors, and when two or more of them act together their combined effect is their sum, found exactly as in adding vectors. Three friends meet the idea in three ways. Aigul flies a seaplane between lakes; her instruments show which way the nose points and how fast the plane moves through the air, but the air itself is moving. Baurzhan takes a motorboat across a river whose water is flowing downstream. Dauren rigs ropes and cables: several pulls on one ring, a chandelier hung from two cables, a cart held on a ramp. In each case the question is the same: what single vector has the effect of all of them together, and, when something stays put, what must the forces be for that sum to be zero? ## The procedure ### Velocities add A plane's **air velocity** $\mathbf{a}$ is its motion through the air: its magnitude is the **airspeed** and its direction is the **heading**, the way the nose points. The **wind** $\mathbf{w}$ is the motion of the air over the ground. The plane's motion over the ground is $$\mathbf{g} = \mathbf{a} + \mathbf{w}$$ whose magnitude is the **ground speed** and whose direction is the **true course**. The reason is the triangle rule. In one hour the plane moves $\mathbf{a}$ through the air while the air carries everything in it by $\mathbf{w}$, so the plane ends up $\mathbf{a}$ followed by $\mathbf{w}$ from where it started, tip to tail. A boat works the same way: its velocity through the water plus the current's velocity gives its velocity over the riverbed. ### [Graphing sine and cosine](https://math.wiki/graph-sine-from-the-unit-circle/) **Summary:** Plotting the height of the terminal point against the angle to see where the wave shape comes from. **Question:** A reflector on a turning bicycle wheel rises, falls and rises again. If its height is plotted against the angle the wheel has turned, what shape comes out, and why that shape? ## When you reach for it On the unit circle a sine is found one angle at a time: choose the angle, find its terminal point, read the $y$-coordinate (cosine-and-sine-as-coordinates). Many questions are about all angles at once. Where is the sine positive? Where is it growing? Which angles between $0$ and $20$ put the terminal point at the very top of the circle? Which angles have $\sin x = \frac{1}{2}$? A single point answers none of these quickly. A graph answers them all by looking, the way the graph of any function shows its whole behavior at once. Measured in radians, an angle is a real number (circular-functions-of-a-real-number), so sine and cosine are functions from real numbers to real numbers and can be drawn on ordinary axes. This lesson draws $y = \sin x$ and $y = \cos x$ themselves, with no stretching or shifting. It runs over two days. Part 1 (Examples 1 to 4) builds the sine graph from the circle. Part 2 (Examples 5 to 7) builds the cosine graph and the five key points that pin down one cycle of either. One point about letters. On these graphs $x$ is the input, an angle in radians, exactly as for any function graphed in the plane, and $y$ is the output. The $x$ on the axis is not the $x$-coordinate of the terminal point. ## The procedure ### Part 1: unrolling the circle into the sine graph Let the angle $x$ grow from $0$ to $2\pi$. Its terminal point starts at $(1, 0)$ and travels once counterclockwise around the unit circle. At every moment the point has a height, its $y$-coordinate, which is $\sin x$. Plot each pair $(x, \sin x)$: the angle across, the height up. That is the whole idea. The graph is a record of the height of a point going around a circle, with the circle's turn laid out flat along the horizontal axis. ### [Graphing tangent and the reciprocal functions](https://math.wiki/asymptotes-of-the-tangent-graph/) **Summary:** The vertical lines the tangent graph races toward at the angles where cosine is zero. **Question:** A camera on a pivot follows a car along a straight road. As the camera turns toward pointing along the road, the car it aims at runs off toward the horizon. What does the graph of tangent look like, with its gaps, and how do the graphs of secant, cosecant and cotangent come out of sine and cosine? ## Why this exists The sine and cosine graphs are smooth waves with a value at every input. The other four trigonometric functions each divide by a coordinate, and at some angles that coordinate is $0$, so the function has no value there. A list of missing inputs says where the graph breaks, but not what the graph does just before and just after a break. That is what this lesson draws. Near each missing input the graph shoots up or down without bound, hugging a vertical line it never touches: a vertical asymptote, the same kind of line that asymptotes describes for rational functions. Knowing where the asymptotes stand and which way each branch runs is most of what there is to know about these four graphs. This lesson takes two days. **Part 1** draws tangent and cotangent. **Part 2** draws secant and cosecant from the cosine and sine curves. ## Definition A **vertical asymptote** of the graph of $f$ is a vertical line $x = a$ such that, as $x$ moves toward $a$ from one side or both, the values $f(x)$ grow without bound, positive or negative. For each trigonometric function, with $k$ standing for any integer: - $y = \tan x = \frac{\sin x}{\cos x}$ has a vertical asymptote at every zero of cosine, $x = \frac{\pi}{2} + k\pi$. - $y = \cot x = \frac{\cos x}{\sin x}$ has a vertical asymptote at every zero of sine, $x = k\pi$. - $y = \sec x = \frac{1}{\cos x}$ has the same asymptotes as tangent, $x = \frac{\pi}{2} + k\pi$. - $y = \csc x = \frac{1}{\sin x}$ has the same asymptotes as cotangent, $x = k\pi$. ### [Height on a turning wheel](https://math.wiki/ferris-wheel-model/) **Summary:** Describing a rider's height above the ground as a wheel turns at a steady rate. **Question:** A rider boards an observation wheel at the bottom and the wheel turns at a steady rate. What formula gives the rider's height at every moment of the ride, and when is the rider above a given height? ## When you reach for it Reach for this model whenever a point rides a wheel that turns at a steady rate and you need its height as a function of time: a rider on an observation wheel, a paddle on a millwheel, a bucket on a water lift. The height rises and falls on a repeating cycle, so it is a sinusoid, and every number in its formula is a measurement of the wheel: - the **amplitude** is the radius of the wheel, - the **midline** is the height of the hub, - the **period** is the time for one turn, - the **phase**, the choice of sine or cosine and any sideways shift, comes from where the point is when the clock starts. Given a time, the model gives a height; given a height, inverse trigonometry gives the times. ## The procedure ### Height from the angle turned Put the hub at height $k$ above the ground and let the wheel have radius $r$. Measure the angle a spoke makes the way the unit circle does: from the horizontal ray pointing right from the hub, counterclockwise as you watch. A point on the rim at angle $\theta$ sits $r\sin\theta$ above the hub (below it when $\sin\theta$ is negative), because on a circle of radius $r$ the vertical coordinate of the point at angle $\theta$ is $r$ times the one on the unit circle. So its height above the ground is $$h = k + r\sin\theta$$ The wheel turns at a steady angular-speed $\omega$ radians per unit of time, so after time $t$ a point that started at angle $\theta_0$ has angle $\theta = \theta_0 + \omega t$ (when the wheel turns counterclockwise as you watch it). Substituting, ### [Heron's formula](https://math.wiki/area-of-a-triangle-from-three-sides/) **Summary:** Taking half the perimeter and combining it with each side to get the area. **Question:** You know the three sides of a triangle, but no angle and no height. Can the area come straight from the three lengths? ## When you reach for it Three lengths are often the easiest thing to measure. A tape stretched along each edge of a triangular field gives its sides; the angles at the corners need an instrument, and the height needs a line dropped at a right angle to a side, which is awkward when the field is in the way. So the question comes up in practice: with all three sides known (SSS) and nothing else, what is the area $K$? There is already a route. The law of cosines turns three sides into an angle, and the area from two sides and the included angle, $K = \tfrac{1}{2}ab\sin C$, turns that angle into the area. Heron's formula does both steps once, in general, so that the angle never has to be found. It is named for Heron of Alexandria, a Greek mathematician and engineer usually dated to the first century. As in those lessons, the triangle is $ABC$, and each side is named by the lowercase letter of the angle across from it. ## The procedure ### The semiperimeter The perimeter of the triangle is $a + b + c$. Half of it is the **semiperimeter**, $$s = \frac{a + b + c}{2}.$$ The formula uses $s$ together with the three differences $s - a$, $s - b$ and $s - c$. Each difference has a meaning. Since $2s = a + b + c$, $$s - a = \frac{b + c - a}{2},$$ which is half the amount by which the two other sides together exceed $a$. The triangle-inequality says $b + c > a$ in every triangle, so $s - a > 0$, and in the same way $s - b > 0$ and $s - c > 0$. ### [Inverse trig functions](https://math.wiki/restricting-the-domain-for-an-inverse/) **Summary:** A repeating function fails the one to one test, so only a chosen slice of it can be inverted. **Question:** A sine key turns an angle into a number. Can a key turn the number back into the angle, when infinitely many angles share every sine? ## Why this exists Mirjana runs a spotlight from a theater balcony. A rangefinder on the light reads how far the actor is along the beam, and she knows how far the light sits above the actor. Their ratio is the sine of the tilt, so she knows $\sin\theta$ and wants $\theta$. That is sine run backward: not "what is the sine of this angle" but "which angle has this sine". Running a function backward is what an inverse function does: if $f(a) = b$, then $f^{-1}(b) = a$. That only works when no two inputs share an output, and sine is periodic, so every value it takes once it takes again in every cycle. The sine of $-\frac{\pi}{3}$, of $\frac{4\pi}{3}$ and of $\frac{5\pi}{3}$ is $-\frac{\sqrt{3}}{2}$ every time, and infinitely many more angles join them. A backward key would not know which to give. The cure is the one from restricting a domain to get an inverse: keep one piece of the graph on which the function is one to one, and invert only that piece. This lesson makes that choice for sine, cosine and tangent, once, so that every book and every calculator gives the same answer. The inverse ratio keys of right triangle trigonometry are these same functions; there they only ever met acute angles, where nothing could go wrong. ## Definition ### Part 1: the sine slice and arcsine On the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$, sine climbs steadily from $-1$ to $1$ and takes every value in $[-1, 1]$ exactly once. The **restricted sine** is $\sin x$ with its domain cut down to that interval. ### [Inverse trigonometric ratios](https://math.wiki/inverse-trig-ratios/) **Summary:** From a ratio back to the angle. **Question:** You know two sides of a right triangle but neither of its acute angles. How do you get the angle back from the sides? ## When you reach for it Solving right triangles starts from an angle and a side and finds the other sides. Very often the situation is the other way round. A ramp rises $0.5$ meters over $6$ meters of floor; a rectangle is $9$ wide and $4$ high; a boat has gone $3$ km east and $7$ km north. Lengths are easy to measure, and two of them are known. The angle is what is wanted. Two sides of a right triangle fix it completely: two right triangles with the same two legs, or the same leg and hypotenuse, are congruent. So the acute angles are fixed too, and the only question is how to compute them. The two known sides give a ratio, and the ratio belongs to the angle, as sine, cosine and tangent showed. This lesson runs that link backwards: from the ratio to the angle. ## The procedure In the figure the right angle is at $C$, the leg $AC$ is $8$, the leg $BC$ is $5$, and the angle $\theta$ at $A$ is wanted. ### Name the two known sides from the wanted angle The names opposite, adjacent and hypotenuse depend on which angle you stand at, as opposite-adjacent-hypotenuse set out, so stand at the angle you want. From $A$, the leg $BC = 5$ is opposite and the leg $AC = 8$ is adjacent. The hypotenuse $AB$ is not known. ### Choose the ratio that holds both Each ratio uses exactly two of the three sides: sine uses opposite and hypotenuse, cosine uses adjacent and hypotenuse, tangent uses opposite and adjacent. Pick the one whose two sides are the two you know, so that the equation has no unknown in it except the angle. Here that is the tangent: ### [Law of cosines](https://math.wiki/derive-the-law-of-cosines/) **Summary:** Placing the triangle on the coordinate plane and applying the distance formula to the third vertex gives the law. **Question:** The law of sines needs a side and the angle across from it. What do you do with two sides and the angle between them, or with three sides and no angle at all? ## When you reach for it The law of sines works on a triangle once you know one side together with the angle across from it. Two kinds of triangle never hand you such a pair: - **SAS**: two sides and the angle between them, the **included** angle. - **SSS**: all three sides, and no angle. Both arrangements fix the triangle (SAS and SSS are congruence criteria), so the missing parts are determined. The **law of cosines** computes them. It also answers a question the other tools handle badly: given only lengths, how big is an angle? As before, the triangle is $ABC$ and each side is named by the lowercase letter of the angle across from it. ## The procedure ### Day 1: the law, from coordinates Place the triangle on the coordinate plane with $C$ at the origin and side $CB$ along the positive $x$-axis, so $B = (a, 0)$. Side $CA$ has length $b$ and leaves the origin at the angle $C$ measured from the positive $x$-axis. The point on the unit-circle at that angle is $(\cos C, \sin C)$; scaling it by $b$ gives the point at distance $b$ in the same direction: $$A = (b\cos C,\ b\sin C).$$ Now $c$ is just the distance from $A$ to $B$, and the distance-formula gives its square: $$\begin{aligned} c^2 &= (b\cos C - a)^2 + (b\sin C - 0)^2 \\ &= b^2\cos^2 C - 2ab\cos C + a^2 \\ &\qquad + b^2\sin^2 C \\ &= a^2 + b^2(\cos^2 C + \sin^2 C) \\ &\qquad - 2ab\cos C. \end{aligned}$$ The Pythagorean identity $\cos^2 C + \sin^2 C = 1$ collapses the parentheses: ### [Law of sines](https://math.wiki/derive-the-law-of-sines/) **Summary:** An altitude splits the triangle into two right triangles sharing a height, and equating the heights gives the law. **Question:** Sine, cosine and tangent need a right angle. What can you say about a triangle that has none? ## When you reach for it The sine, cosine and tangent ratios solve right triangles. Most triangles have no right angle: two lookout towers and a smoke plume, three stakes in a field, a triangle in a bridge truss. A triangle with no right angle is called an **oblique triangle**, and **solving** a triangle means finding all three of its angles and all three of its sides. The **law of sines** solves an oblique triangle whenever you know two angles and one side. That happens in two arrangements: - **AAS** (angle, angle, side): two angles and a side that is not between them. - **ASA** (angle, side, angle): two angles and the side between them, the **included** side. Either way the triangle is fixed: AAS and ASA are congruence criteria, so only one triangle has those parts, and the law of sines computes the rest of it. Throughout, the triangle is $ABC$, and each side is named by the lowercase letter of the angle across from it: side $a$ is opposite $A$, side $b$ is opposite $B$, side $c$ is opposite $C$. ## The procedure ### The law, from an altitude Start with a triangle whose angles are all acute. Drop the altitude from $C$ straight down to side $AB$, meeting it at $D$, and call its length $h$. The altitude splits the triangle into two right triangles, $ADC$ and $BDC$, and $h$ is a leg of both. In right triangle $ADC$ the hypotenuse is $b$ and the side opposite $A$ is $h$, so by the sine ratio $\sin A = \dfrac{h}{b}$, which gives $h = b \sin A$. In right triangle $BDC$ the hypotenuse is $a$ and the side opposite $B$ is $h$, so $\sin B = \dfrac{h}{a}$ and $h = a \sin B$. ### [Laws of sines and cosines](https://math.wiki/laws-of-sines-and-cosines/) **Summary:** Trig for triangles that are not right. **Question:** Sine, cosine and tangent ratios need a right angle. How do you solve a triangle that has none? ## What this area is about Most triangles have no right angle. Two laws extend trigonometry to every triangle $ABC$, with side $a$ across from angle $A$, and so on. Which law applies depends on which three parts you know, and once enough parts are known, the area follows too. ## How these fit together **The two laws.** The law of sines, $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$, solves a triangle from two angles and a side (AAS or ASA). The law of cosines, $c^2 = a^2 + b^2 - 2ab\cos C$, handles two sides and the included angle (SAS) or three sides (SSS), and contains the Pythagorean theorem as the case $C = 90^\circ$. **A warning case.** The ambiguous case: two sides and an angle not between them (SSA) can give zero, one or two triangles. For an acute angle $A$ the height $h = b\sin A$ decides the count (for a right or obtuse $A$, whether $a > b$), and because $\sin\theta = \sin(180^\circ - \theta)$, an angle found by inverse sine may have an obtuse partner. **Area.** Area from two sides and the included angle, $K = \frac{1}{2}ab\sin C$, and Heron's formula, from three sides alone. **Use.** Applications of oblique triangles chooses among all of these for surveying and navigation. ## A path through it The law of sines first: the ambiguous case, the sine area formula and the law of cosines all build on it. Heron's formula comes after the law of cosines, and the applications come last. Next, the law of cosines leads to the dot product, and the sine area formula to the cross product as area. ### [Limacons and cardioids](https://math.wiki/cardioid/) **Summary:** A heart shaped curve that appears when the constant term and the varying term are equal in size. **Question:** What happens to the circle $r = a$ when its distance from the pole is pushed out on one side and pulled in on the other by $b\cos\theta$, and why does the result sometimes have a dent, a point, or a loop inside a loop? ## Why this exists The circles of polar-graph-of-a-circle come from $r = a$, a fixed distance, and $r = b\cos\theta$, a distance swinging between $b$ and $-b$. What curve does their sum trace? The answer is a family of curves called **limaçons**. They are easy in polar form, awkward in rectangular form, and show plainly what a negative $r$ does (plot-a-polar-point). ## Definition A **limaçon** is the graph of a polar equation $$r = a + b\cos\theta \qquad \text{or} \qquad r = a + b\sin\theta$$ with $a$ and $b$ nonzero real numbers. A **cardioid** is the limaçon with $|a| = |b|$; its name means heart shaped. Its shape is named by the ratio $\frac{|a|}{|b|}$: | Ratio | Shape | |:-:|---| | $0 < \frac{\vert a \vert}{\vert b \vert} < 1$ | limaçon with an inner loop | | $\frac{\vert a \vert}{\vert b \vert} = 1$ | cardioid | | $1 < \frac{\vert a \vert}{\vert b \vert} < 2$ | dimpled limaçon (one loop with a dent) | | $\frac{\vert a \vert}{\vert b \vert} \ge 2$ | convex limaçon (one loop, no dent) | ## Intuition Think of $r = a + b\cos\theta$, with $a, b > 0$, as the circle $r = a$ bent: the distance grows to $a + b$ at $\theta = 0$, shrinks to $a - b$ at $\theta = \pi$, and stays $a$ straight up and down. - When $a$ is much larger than $b$, the circle becomes an egg, wider on the right. - As $a$ shrinks toward $b$, the left side flattens and then dents inward. - At $a = b$ the left side reaches the pole ($r = 0$ at $\theta = \pi$) and the dent closes into a point, the cusp of the cardioid. - When $a < b$, $r$ near $\theta = \pi$ is negative. Plotted backward, those points land right of the pole, inside the big loop: the inner loop. ### [Magnitude and direction of a vector](https://math.wiki/magnitude-of-a-vector/) **Summary:** Finding a vector's length from its components with the distance formula. **Question:** A vector is a trip of so much across and so much up. How long is the trip, which way does it point, and how do you get the two components back from a length and a direction? ## When you reach for it Emi, Juho and Lotta drive hovercraft for a club on a large frozen lake. The club's map puts the launch ramp at the origin, with $x$ in meters east and $y$ in meters north, and a logger on each craft records a run in component form: so many meters east, so many north. A driver, though, asks different questions. How far did I go? Which way was I pointed? Those are the vector's **magnitude** and **direction**. The traffic also runs the other way. A course card says "drive $150$ meters at $64^\circ$", and to put that on the map, or to add it to another run later, you need its components. Component form is the name to compute with; magnitude and direction are the name to describe and to steer by. This skill turns either name into the other. ## The procedure **Magnitude.** The magnitude of $\mathbf{v} = \langle a, b \rangle$, written $\|\mathbf{v}\|$, is the length of any of its drawings. All drawings of one vector have the same length, so measure the easiest, the position drawing from $O(0, 0)$ to $(a, b)$. The distance formula gives $\sqrt{(a - 0)^2 + (b - 0)^2}$: $$\|\mathbf{v}\| = \sqrt{a^2 + b^2}$$ In Figure 1 this is the Pythagorean theorem on a right triangle with legs $|a|$ and $|b|$. Squaring removes the signs, so the formula is the same in every quadrant. For the vector from $P$ to $Q$ it is simply the distance $PQ$. **Direction angle.** The direction angle $\theta$ of a nonzero vector is the angle in standard position whose terminal side runs along the position drawing: measured counterclockwise from the positive $x$-axis, with $0^\circ \le \theta < 360^\circ$. The point $(a, b)$ sits on that terminal side at distance $\|\mathbf{v}\|$ from $O$, so by the definition of the trig functions of any angle ### [Midline and vertical shift](https://math.wiki/vertical-shift-of-a-sinusoid/) **Summary:** Moving the whole wave up or down, which moves the midline without changing the amplitude. **Question:** A fishing bobber rides a pond's swell, rising and falling around a level that is not zero. How do you read that resting level, and the highest and lowest points, straight from the formula, and how do you rebuild the formula from them? ## When you reach for it The graphs of $y = \sin x$ and $y = \cos x$ (graph-sine-from-the-unit-circle) balance on the $x$-axis: they rise $1$ above it and fall $1$ below it. Most waves in the world balance somewhere else. Nodira marks a dock post and measures the height of her fishing bobber above the mark as the pond's swell lifts and drops it. The bobber bobs around $7$ inches, never around $0$. A wave like that is written $y = a\sin x + k$ or $y = a\cos x + k$. The number $a$ sets how far the wave swings; its size $|a|$ is the **amplitude** (amplitude), and this lesson uses that one fact without reteaching it. The number $k$ sets where the wave balances. You reach for this lesson when you need the balance line, the highest value or the lowest value of such a wave from its formula, or when you know the highest and lowest values and need $a$ and $k$. Throughout, the input is $x$ itself: no squeeze (period-from-the-b-value) and no sideways slide (factor-before-reading-a-phase-shift). One cycle is $2\pi$ wide. ## The procedure **Adding $k$ lifts the whole wave.** In $y = \sin x + 3$, the output at every $x$ is $3$ more than $\sin x$ at the same $x$. Each point of the sine graph moves straight up $3$ units, which is the vertical shift of any graph by a number added to the output. Nothing moves sideways, so the wave keeps its shape and its cycle; it simply balances on the line $y = 3$ now. ### [Modeling things that repeat](https://math.wiki/modeling-periodic-phenomena/) **Summary:** Using a sine or cosine model for any quantity that rises and falls on a steady cycle. **Question:** A wheel turns, a tide comes in, a spring bobs. How do a few measured facts about a steady cycle become a formula that predicts it? ## What this area is about Many quantities rise and fall on a steady cycle: a seat on a turning wheel, the water at a harbor, a weight on a spring. A sine or cosine describes each, and every number in the formula is a fact about the situation: amplitude, midline, period and where the cycle stands when the clock starts. With the input in radians, a period $P$ means $b = \frac{2\pi}{P}$. ## How these fit together All three apply one model, $y = A\cos\bigl(b(t - c)\bigr) + k$ or its sine form, and differ in where the numbers come from. Height on a turning wheel reads them off the wheel: radius, hub height, time per turn, starting position. Tides, daylight and sound builds them from facts in words: highest and lowest values, period and a peak time. Simple harmonic motion is the concept behind a spring, or a pendulum on small swings, $d(t) = a\cos(\omega t)$ or $d(t) = a\sin(\omega t)$ with $\omega$ in radians per second, read as the height of a point turning on a circle. ## A path through it No lesson here is a prerequisite of another, but they point to each other. The wheel is the most concrete and the other two refer back to it, so start there; tides and simple harmonic motion follow. The wheel and tide lessons ask a model for the value at a given time and for the times it reaches a given level. The second needs inverse trigonometry, whose answer lies in a restricted range, so the other times come from symmetry and from adding whole periods. ### [Modulus, argument and polar form](https://math.wiki/modulus-and-argument/) **Summary:** The distance from the origin is the modulus and the direction angle is the argument. **Question:** A nonzero complex number is fixed by how far it sits from 0 and in which direction. Can the number be written with those two facts instead of its two parts, and how do you pass between the two ways of writing it? ## Why this exists The form $a + bi$ says where a complex number sits by going across and then up. That suits adding, which works part by part, but it hides the two facts that the complex plane makes visible: how far the number is from $0$, its modulus, and which way it points, its argument. For points in the plane this choice is old news. A point can be named by $(x, y)$ or by a distance and an angle, and converting between polar and rectangular passes from one to the other with cosine and sine. Complex numbers get the same second name. Writing $z$ by its length and angle costs nothing when adding, but it turns multiplying, powers and roots from long algebra into short arithmetic. This lesson sets up that form and the conversions in both directions. ## Definition Let $z = a + bi$ be a nonzero complex number, plotted at the point $(a, b)$. Its **modulus** is $r = |z| = \sqrt{a^2 + b^2}$, the distance from $0$. An **argument** of $z$ is any angle $\theta$ in standard position whose terminal side runs from $0$ through $z$. Both were introduced in plotting complex numbers. The point $(a, b)$ lies at distance $r$ along the terminal side of $\theta$, so by the definition of cosine and sine, $a = r\cos\theta$ and $b = r\sin\theta$. Putting these into $a + bi$: $$z = r(\cos\theta + i\sin\theta)$$ This is the **polar form**, also called the **trigonometric form**, of $z$. The factor in parentheses is often shortened to $\operatorname{cis}\theta$ (for "cosine, i sine"), so $z = r\operatorname{cis}\theta$. ### [Multiplying, powers and De Moivre's theorem](https://math.wiki/powers-of-a-complex-number/) **Summary:** Using the theorem to work out a high power without repeated multiplication. **Question:** Multiplying two complex numbers in the form a + bi takes four products and a sign change. Raising one to the twelfth power by hand would take eleven of those. Is there a shorter road? ## When you reach for it In the form $a + bi$, multiplying complex numbers means distributing and replacing $i^2$ by $-1$. That is fine once. It is slow for a product of several factors and hopeless for $(1 + i)^{12}$, and it hides what multiplication does to a point in the plane. Polar form fixes all of this. Once each factor is written as $r\operatorname{cis}\theta = r(\cos\theta + i\sin\theta)$, a product, a quotient or a power comes from multiplying lengths and adding angles. Reach for it when the numbers are already in polar form, when a power is high or negative, when a point is to be turned about $0$, and, turned around, when you want a formula for $\cos 3\theta$ or $\sin 4\theta$. The lesson runs over two days. Day 1 is multiplying and dividing in polar form, and the picture of a product as a stretch and a turn. Day 2 is De Moivre's theorem for powers, positive and negative, and its use for multiple-angle formulas. ## The procedure Throughout, $\operatorname{Arg} z$ is the principal argument in $(-\pi, \pi]$, as in modulus-and-argument, and every modulus is positive. ### Day 1: multiplying in polar form Let $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$. The moduli are real numbers, so they come out to the front (real numbers commute with everything), and the two factors in parentheses are multiplied by distributing: $$\begin{aligned} &(\cos\theta_1 + i\sin\theta_1)(\cos\theta_2 + i\sin\theta_2) \\ &= \cos\theta_1\cos\theta_2 + i\cos\theta_1\sin\theta_2 \\ &\quad + i\sin\theta_1\cos\theta_2 + i^2\sin\theta_1\sin\theta_2 \end{aligned}$$ ### [Opposite, adjacent and hypotenuse](https://math.wiki/opposite-adjacent-hypotenuse/) **Summary:** Naming the three sides of a right triangle relative to whichever acute angle is being looked at. **Question:** In a right triangle, how do you name each side by where it sits relative to one acute angle, so that a rule about that angle can say exactly which side it means? ## Why this exists A right triangle has one square corner and two sharp ones. Right-triangle trigonometry is built from statements about one of the sharp corners, each of the form "this side compared with that side". Such a statement needs a way to point at a side that works in every right triangle, however it is drawn, turned or lettered. "The bottom side" fails as soon as the triangle is turned over, and calling it $\overline{BC}$ fails as soon as the letters change. The way out is to name each side by where it sits relative to the angle being looked at: across from it, beside it, or across from the right angle. The price is that two of those names belong to the angle rather than to the side: look from the other sharp corner and the same side gets a different name. ## Definition Start from the words of parts of a triangle. In a right triangle the two sides that form the right angle are the **legs**, and the side opposite the right angle is the **hypotenuse**. A triangle has at most one right angle (classifying triangles by angles), so the hypotenuse is one fixed side. The other two angles are acute: the three angles add to $180^\circ$ (the triangle angle sum) and the right angle takes $90^\circ$, so the other two add to $90^\circ$ and each is less than $90^\circ$. Now choose one of the two acute angles. Let $\triangle ABC$ have its right angle at $C$, and choose $\angle A$. Relative to $\angle A$, the three sides are named as follows. ### [Orthogonal vectors](https://math.wiki/orthogonal-vectors/) **Summary:** Two vectors are perpendicular exactly when their dot product is zero. **Question:** Two arrows are given by their components. Without drawing them or finding a single angle, how can you tell for certain that they meet at a right angle? ## Why this exists Right angles are everywhere: the corner of a rectangle, a wall standing on a floor, the radius meeting a tangent line. Showing that two directions meet at exactly $90^\circ$ used to mean finding angles with inverse tangents, or comparing slopes and setting vertical lines aside as a special case. A picture is no proof: two arrows at $87^\circ$ look exactly like two at $90^\circ$. The dot product settles it with one line of arithmetic. Its sign table already had a row for zero: a right angle gives $\mathbf{u} \cdot \mathbf{v} = 0$. This lesson turns that row into a test that runs both ways and has no exceptions, then uses it to build perpendicular vectors, perpendicular lines and right triangles straight from coordinates. ## Definition Two vectors $\mathbf{u}$ and $\mathbf{v}$ are **orthogonal**, written $\mathbf{u} \perp \mathbf{v}$, when their dot product is zero: $$\mathbf{u} \perp \mathbf{v} \quad\text{means}\quad \mathbf{u} \cdot \mathbf{v} = 0.$$ For nonzero vectors this is exactly the same as being **perpendicular**: the angle between them is $90^\circ$ (proved below). The **zero vector** is the one exception to the picture. Since $\mathbf{0} \cdot \mathbf{v} = 0$ for every $\mathbf{v}$, the definition makes $\mathbf{0}$ orthogonal to every vector, including itself. That is a convention adopted on purpose, so that statements like "every multiple of $\langle -2, 3 \rangle$ is orthogonal to $\langle 3, 2 \rangle$" need no exception for the multiple $0$. The zero vector has no direction, so it is never called perpendicular to anything. ### [Parametric equations](https://math.wiki/parameter-and-orientation/) **Summary:** The parameter acts like time, so a parametric curve is traced in a definite direction. **Question:** An equation in $x$ and $y$ tells you where a path lies. How do you also say where a moving object is at each moment, and which way it is going? ## Why this exists The equation $(x - 3)^2 + (y + 1)^2 = 4$ describes a circle all at once. It cannot say where a point on the circle is after $5$ seconds, where it started, or which way it goes around. It is not even a function of $x$, since a vertical line can cross a circle twice (see function-concept). A moving object raises exactly these questions. A robot mower crossing a lawn or a ball flying through the air has an $x$-coordinate that changes with time and a $y$-coordinate that changes with time. The natural description is two functions of time, one for each coordinate. That pair carries what a single equation in $x$ and $y$ throws away: the order in which the points are visited. The lesson takes two days. Part 1 defines parametric equations, plots them from a table, reads their direction and writes a line through two points. Part 2 traces circles and ellipses with cosine and sine, and models a thrown ball. ## Definition Let $t$ run through an interval $I$ of real numbers. Two functions $x = f(t)$ and $y = g(t)$ on $I$ are **parametric equations**, and $t$ is the **parameter**. Each value of $t$ gives one point $(x(t), y(t))$, and the set of all these points is a **plane curve**. The **orientation** of the curve is the direction in which its points are traced as $t$ increases, shown by arrows along the curve. When $I$ is $a \le t \le b$, the point at $t = a$ is the **initial point** and the point at $t = b$ is the **terminal point**. The parameter is often time, but it need not be; it is the input both coordinates depend on. ### [Periodic functions](https://math.wiki/period-and-cycle/) **Summary:** One full repetition of the pattern is a cycle and its width along the input axis is the period. **Question:** A swing ride at a carnival goes out and back, out and back, all evening. If you know one trip exactly, do you know every moment of the ride? ## Why this exists Many quantities come back to where they were and then do the same thing again: the position of a carnival swing ride, the color of a stoplight, the height of a point on a turning wheel. For such a quantity a long table is wasteful. One full repetition, together with its length, already tells everything. periodicity-on-the-unit-circle showed that each trigonometric value comes back after a full turn. This lesson names the idea for any function: what it means to repeat, what one repetition is, and how long it lasts. ## Definition A function $f$ is **periodic** when some positive number $P$ shifts the input without changing the output: $$f(x + P) = f(x)$$ for every $x$ in the domain, with $x + P$ in the domain exactly when $x$ is. Such a $P$ is a **repeat shift** of $f$. The **period** of a periodic function is its smallest repeat shift, when a smallest one exists. On a graph it is a horizontal length, measured along the input axis. A **cycle** is the part of the graph over any input interval exactly one period long, such as $a \le x < a + P$. Any starting point $a$ gives a cycle. The outputs need not be numbers. A stoplight's color as a function of time is periodic too. ## Intuition Think of a cycle as a tile. Cut out the graph over one period, make copies, and lay them end to end in both directions. The result is the whole graph, and nothing in it is new. Sliding the graph right by $P$ moves every tile onto the next one, so the shifted graph sits exactly on the original. That is what $f(x + P) = f(x)$ says. Any shorter slide leaves some point out of place. ### [Periodicity and the periodicity identities](https://math.wiki/periodicity-on-the-unit-circle/) **Summary:** Adding a full turn returns to the same point, so every trig value repeats after one revolution. **Question:** A paddle wheel can turn through $\\frac{29\\pi}{3}$ radians, many times around. Does finding the sine of an angle that large need anything new, or has the circle already answered it? ## Why this exists A paddle wheel on a riverboat turns all day. If a paint mark on one paddle starts level with the axle and the wheel has turned through $\frac{29\pi}{3}$ radians, how high is the mark? The angle is more than four full turns, and none of the special angles in a table is that large. The circle has already answered the question. After every full turn the mark is back where it started, so the wheel's position after $\frac{29\pi}{3}$ radians is its position after whatever is left once the full turns are taken away. Since sine and cosine are the coordinates of the terminal point, they come back too. This lesson states that fact as six identities, one for each trigonometric function, and finds one surprise: tangent and cotangent repeat after only half a turn. ## Definition Two angles in standard position are **coterminal** when they differ by an integer number of full turns, as in standard-position-angle. In radians a full turn is $2\pi$, so $t$ and $t + 2\pi n$ are coterminal for every integer $n$ (positive, negative or zero; angles-and-their-measure has radians and degrees side by side). A function $f$ **repeats after a shift** $p > 0$ when $$f(t + p) = f(t)$$ for every $t$ in its domain, with $t + p$ in the domain exactly when $t$ is. The **periodicity identities** say which shifts the six trigonometric functions repeat after: $$\begin{aligned} \sin(t + 2\pi n) &= \sin t \\ \cos(t + 2\pi n) &= \cos t \\ \csc(t + 2\pi n) &= \csc t \\ \sec(t + 2\pi n) &= \sec t \\ \tan(t + \pi n) &= \tan t \\ \cot(t + \pi n) &= \cot t \end{aligned}$$ ### [Phase shift](https://math.wiki/factor-before-reading-a-phase-shift/) **Summary:** The sideways slide can only be read once the input is written as the coefficient times a shifted variable. **Question:** The graph of $y = \sin\left(3x - \frac{\pi}{2}\right)$ is a sine wave slid sideways. How far did it slide, and why is the answer not $\frac{\pi}{2}$? ## When you reach for it Reach for this move whenever a sine or cosine has a number added to or subtracted from the input after the input has been multiplied, as in $$y = \sin\left(3x - \frac{\pi}{2}\right)$$ and you want to know where its cycles start: which way the graph slid, and how far. The number you can see, $\frac{\pi}{2}$, is not the answer. The input has to be rewritten as the coefficient times a shifted variable, $3\left(x - \frac{\pi}{6}\right)$, and only then can the slide be read. Here it is $\frac{\pi}{6}$. ## The procedure ### The sliding rule you already have From shifting a graph: if $g$ is any function and $h$ is a number, the graph of $y = g(x - h)$ is the graph of $y = g(x)$ moved $h$ to the right (to the left when $h$ is negative). The rule has one condition that is easy to pass over. It is about **$x$ itself** being replaced by $x - h$. Nothing else about the formula may change. ### Why the subtracted number is not the slide Start from $g(x) = \sin 3x$, a sine wave squeezed so that one cycle takes $\frac{2\pi}{3}$ instead of $2\pi$ (see period-from-the-b-value). Is $y = \sin\left(3x - \frac{\pi}{2}\right)$ the same as $g\left(x - \frac{\pi}{2}\right)$? Replace $x$ by $x - \frac{\pi}{2}$ and see: $$\begin{aligned} g\left(x - \frac{\pi}{2}\right) &= \sin\left(3\left(x - \frac{\pi}{2}\right)\right) \\ &= \sin\left(3x - \frac{3\pi}{2}\right) \end{aligned}$$ That is a different function. At $x = \frac{\pi}{3}$ the given function is $\sin\left(\pi - \frac{\pi}{2}\right) = \sin\frac{\pi}{2} = 1$, while $\sin\left(3 \cdot \frac{\pi}{3} - \frac{3\pi}{2}\right) = \sin\left(-\frac{\pi}{2}\right) = -1$. One is at a peak where the other is at a trough. ### [Plotting polar points](https://math.wiki/plot-a-polar-point/) **Summary:** Locating a point by turning through an angle and then walking out a distance. **Question:** A sonar screen shows each contact as a distance and a direction from the ship at its center. How do you name a point that way, and why does the same point end up with many names? ## When you reach for it Yuina watches a sonar screen on a ship. The ship sits at the center, rings mark distances $1$ km apart, and a contact shows up as a blip: so many kilometers out, in some direction. To report it as $(x, y)$ she would need how far east and how far north it is, which the screen does not show. What it shows directly is a **distance** and a **direction**. Polar coordinates name points exactly that way: a direction, given as an angle, and a distance along it. Reach for them whenever a picture is built around one center: a sonar screen, an air traffic control display, a wheel, and later whole curves whose polar equations are short. This lesson names and plots single points; turning a polar name into $x$ and $y$ is the next step, polar-to-rectangular-conversion. ## The procedure ### The pole, the polar axis and the grid Fix a point $O$, called the **pole**, and a ray starting at $O$, called the **polar axis**. On paper the pole is placed where the origin would be and the polar axis along the positive $x$-axis, pointing right. Every direction out of the pole is then an angle in standard position: the turn from the polar axis, counterclockwise when the angle is positive and clockwise when it is negative. Angles here are in radians unless a degree sign is written (angles-and-their-measure). A **polar grid** draws the two kinds of line this system needs: circles centered at the pole, one unit apart, and rays out of the pole at evenly spaced angles. Distance from the pole is read off the circles; direction is read off the rays. ### [Polar circles and symmetry tests](https://math.wiki/polar-graph-of-a-circle/) **Summary:** Equations that trace a circle centered at the origin or passing through it. **Question:** Which polar equations draw a circle, how do you sketch one without plotting a hundred points, and how can you tell from the equation alone that its graph is the same on both sides of a line? ## When you reach for it A polar equation $r = f(\theta)$ is graphed by choosing directions $\theta$ and walking out the distance $r$ that the equation gives (plot-a-polar-point). That works for any equation but takes many points. Four families of equations are worth knowing on sight, because their graphs are a circle or a line and can be drawn at once: $$r = a, \qquad \theta = c, \qquad r = a\cos\theta, \qquad r = a\sin\theta$$ Each can be confirmed by converting to rectangular form (convert-an-equation-between-polar-and-rectangular). For other polar curves, three quick symmetry tests on the equation tell you which half, or which quarter, of the graph is enough to compute. Throughout, the pole is the origin, the polar axis is the positive $x$-axis, $a \ne 0$, and $r$ may be negative: $(r, \theta)$ with $r < 0$ is the point $|r|$ units out in the direction $\theta + \pi$. ## The procedure ### Constant r, constant theta The graph of $r = a$ is every point at distance $|a|$ from the pole: the circle of radius $|a|$ centered at the pole, $x^2 + y^2 = a^2$. A negative $a$ gives the same circle, since $(a, \theta)$ and $(-a, \theta + \pi)$ name the same point. The graph of $\theta = c$ is every point $(r, c)$ with $r$ any real number: the ray at angle $c$ ($r > 0$), the opposite ray ($r < 0$) and the pole. Together they make the whole line through the pole at angle $c$. ### The circles r = a cos theta and r = a sin theta ### [Polar coordinates](https://math.wiki/polar-coordinates/) **Summary:** Locate a point by distance from the origin and angle from the x-axis. **Question:** Some pictures are built around one center. Can a point be named by how far out it is and which way, instead of how far across and how far up? ## What this area is about Polar coordinates name a point $(r, \theta)$: turn through the angle $\theta$ from the polar axis, then walk the distance $r$. Here the pole sits at the origin, the polar axis along the positive $x$-axis, and angles are in radians unless a degree sign is written. $r$ may be negative (walk $|r|$ the opposite way), so every point has many polar names, and the pole is $(0, \theta)$ for every $\theta$. ## How these fit together **A procedure for single points.** Plotting polar points sets up the pole, the polar axis and the polar grid, and shows why one point has many names. **Moving between systems.** Converting between polar and rectangular moves one point at a time with $x = r\cos\theta$, $y = r\sin\theta$ and $x^2 + y^2 = r^2$, in both directions. Converting an equation applies the same relations to a whole curve, so a circle like $x^2 + y^2 = 49$ becomes $r = 7$. ## A path through it Plot points first, then convert single points, then convert whole equations: each lesson uses the one before it. This opens polar equations and their graphs, such as polar circles and cardioids, and the polar form of a complex number through modulus and argument. In calculus it leads to the slope of a polar curve. ### [Polar coordinates and vectors](https://math.wiki/polar-and-vectors/) **Summary:** Direction and distance instead of x and y; quantities with a direction. **Question:** Some quantities have a direction as well as a size, and some points are easier to find by distance and angle than by across and up. How does trigonometry describe them? ## What this area is about Three descriptions that $x$ and $y$ alone handle badly. A vector is a length with a direction, such as a velocity. Polar coordinates name a point by a distance and an angle, $(r, \theta)$. Parametric equations give $x$ and $y$ each as a function of a parameter $t$, so a curve is traced in a definite direction. ## How these fit together **Vectors.** Vectors defines the arrow, its components $\langle a, b \rangle$ and its length. Vector addition and scalar multiplication adds and scales. The dot product gives the angle between two nonzero vectors. Vectors in space carries these tools to $\langle a, b, c \rangle$, and the cross product returns a vector perpendicular to two space vectors, as long as the area of their parallelogram. **Curves traced in order.** Parametric equations define the curve and its orientation; eliminating the parameter recovers one equation in $x$ and $y$. **Polar.** Polar coordinates plot and convert points and equations. Polar equations graph $r = f(\theta)$. Polar form writes a nonzero complex number as $r(\cos\theta + i\sin\theta)$, so multiplying multiplies lengths and adds angles. ## A path through it Vectors first: every other child needs it. Then three strands. Vector operations, the dot product, vectors in space, and last the cross product, which needs the dot product and vectors in space. Parametric equations, then eliminating the parameter. Polar coordinates, then polar equations and polar form. ### [Polar equations and their graphs](https://math.wiki/polar-equations/) **Summary:** Curves described by giving the distance from the origin as a function of the direction. **Question:** When the distance from the pole is a rule in the angle, what curves come out, and how can you sketch them without plotting a hundred points? ## What this area is about A polar equation $r = f(\theta)$ draws a curve: for each direction $\theta$, walk out the distance $r$, where a negative $r$ means walking $|r|$ in the direction $\theta + \pi$. A point lies on the graph when at least one of its polar names satisfies the equation. Some curves that are awkward in $x$ and $y$ have short polar equations, and three families are worth knowing on sight. ## How these fit together **A procedure to start with.** Polar circles and symmetry tests covers $r = a$, $\theta = c$, $r = a\cos\theta$ and $r = a\sin\theta$ (with $a \ne 0$), and gives symmetry tests on the equation that cut the work of sketching other curves. **Two families of curves.** Limacons and cardioids are $r = a + b\cos\theta$ or $r = a + b\sin\theta$ ($a, b \ne 0$); the ratio $\frac{|a|}{|b|}$ decides between an inner loop, a cardioid, a dent, or a plain loop. Rose curves are $r = a\cos n\theta$ or $r = a\sin n\theta$ ($a \ne 0$, $n$ a positive integer), whose petal count depends on whether $n$ is odd or even. ## A path through it Circles and the symmetry tests come first, since limacons and roses are each introduced by changing a circle's equation. Then take the two families; each points to the other where their habits overlap. These curves are where polar coordinates pay off: each is a short equation in $r$ and $\theta$ and a long one in $x$ and $y$. ### [Polar form of a complex number](https://math.wiki/polar-form-of-a-complex-number/) **Summary:** Writing a complex number by how far it sits from the origin and in which direction. **Question:** Multiplying complex numbers in the form a + bi is slow, and high powers and roots are worse. Is there a way of writing them that makes this easy? ## What this area is about A nonzero complex number $z = a + bi$ sits at the point $(a, b)$, so it can also be named by its distance from $0$ and its direction, just as a point gets polar coordinates. The polar form $z = r(\cos\theta + i\sin\theta)$, with $r = |z| > 0$, does not help with adding, but it turns multiplying, powers and roots into multiplying lengths and adding angles. ## How these fit together **The definition.** Modulus, argument and polar form defines the modulus $|z| = \sqrt{a^2 + b^2}$ and an argument $\theta$, and converts between $a + bi$ and polar form. **A result and its procedure.** Multiplying, powers and De Moivre's theorem shows that a product multiplies moduli and adds arguments, and that $\left(r\operatorname{cis}\theta\right)^n = r^n\operatorname{cis}n\theta$ for every integer $n$. **A procedure run backwards.** Roots of a complex number and roots of unity solves $z^n = w$ for $w \ne 0$ and a positive integer $n$: there are exactly $n$ roots, evenly spaced on a circle about $0$. ## A path through it Modulus and argument first, then De Moivre's theorem, then roots, which runs De Moivre's theorem in reverse. Afterward, complex eigenvalues use the same picture for rotation-scaling matrices. A later page, the complex plane, revisited (not written yet), is planned to treat complex numbers as points and arrows, with multiplication as rotating and scaling. ### [Power reducing and half angle formulas](https://math.wiki/power-reducing-formulas/) **Summary:** Rewriting a squared trig function as a first power of a double angle. **Question:** A squared sine or cosine is awkward to graph and to add up. Can the square be traded for a plain cosine of a doubled angle, and does the same trade give the sine and cosine of half an angle? ## Why this exists A square such as $\sin^2 x$ is harder to work with than a first power. Its graph is not one of the familiar waves at sight, and a sum of squares and fourth powers resists simplifying. The double angle formulas already link squares to a doubled angle: $\cos 2x = 1 - 2\sin^2 x = 2\cos^2 x - 1$. Solved for the square, they turn every $\sin^2 x$ or $\cos^2 x$ into a first power of $\cos 2x$. Read the other way, with $x$ standing for half an angle, the same equations give the sine and cosine of half an angle from the cosine of the whole one. That is how exact values such as $\cos 112.5^\circ$ are found, and the only new difficulty is choosing a sign. ## Definition The **power reducing formulas** write a square as a first power of a cosine of the doubled angle: $$\begin{aligned} \sin^2 x &= \frac{1 - \cos 2x}{2} \\ \cos^2 x &= \frac{1 + \cos 2x}{2} \\ \tan^2 x &= \frac{1 - \cos 2x}{1 + \cos 2x} \end{aligned}$$ The first two hold for every real $x$; the third holds wherever $\tan x$ exists, that is, wherever $\cos x \ne 0$. The **half angle formulas** are the same facts with $\frac{x}{2}$ in place of $x$: $$\begin{aligned} \sin\frac{x}{2} &= \pm\sqrt{\frac{1 - \cos x}{2}} \\ \cos\frac{x}{2} &= \pm\sqrt{\frac{1 + \cos x}{2}} \end{aligned}$$ Each $\pm$ means one sign, not both: it is the sign that $\sin\frac{x}{2}$ or $\cos\frac{x}{2}$ actually has, read from the quadrant where the angle $\frac{x}{2}$ ends, by the rules of determine-a-quadrant-from-two-signs. The quadrant of $x$ itself does not decide it. The tangent of half an angle needs no sign choice at all: ### [Product to sum and sum to product formulas](https://math.wiki/product-to-sum-formulas/) **Summary:** Turning a product of two trig functions into a sum, which adding the sum and difference formulas produces. **Question:** Neither $52.5^\\circ$ nor $7.5^\\circ$ is a special angle. Can $\\sin 52.5^\\circ\\cos 7.5^\\circ$ still be found exactly from special-angle values, and can a sum of two waves be read as one wave times another? ## Why this exists A product of two trig functions is awkward. To find $\sin 52.5^\circ\cos 7.5^\circ$ exactly you would need each factor, and neither angle is on the unit circle's list of special angles. To graph $\cos 5x\cos 2x$ you cannot simply add two familiar waves. A sum has the opposite problem: the graph of $\cos 9t + \cos 7t$ is hard to picture, and an equation such as $\sin 5x + \sin x = 0$ cannot be split into simpler equations until the left side is a product. The cosine sum and difference formulas and the sine versions already contain the fix. Each pair has the same two products on the right with opposite signs, so adding or subtracting a pair isolates one product. That gives formulas that turn a product into a sum, and read backward, formulas that turn a sum into a product. ## Definition For all real numbers $a$ and $b$ (or angles in degrees), the **product to sum formulas** are $$\begin{aligned} \cos a\cos b &= \tfrac{1}{2}\left[\cos(a - b) + \cos(a + b)\right] \\ \sin a\sin b &= \tfrac{1}{2}\left[\cos(a - b) - \cos(a + b)\right] \\ \sin a\cos b &= \tfrac{1}{2}\left[\sin(a + b) + \sin(a - b)\right] \end{aligned}$$ A product of two cosines, of two sines, or of a sine and a cosine becomes half a sum or difference of single functions, with no products left. For all real numbers $u$ and $v$, the **sum to product formulas** are $$\begin{aligned} \sin u + \sin v &= 2\sin\tfrac{u + v}{2}\cos\tfrac{u - v}{2} \\ \sin u - \sin v &= 2\cos\tfrac{u + v}{2}\sin\tfrac{u - v}{2} \\ \cos u + \cos v &= 2\cos\tfrac{u + v}{2}\cos\tfrac{u - v}{2} \\ \cos u - \cos v &= -2\sin\tfrac{u + v}{2}\sin\tfrac{u - v}{2} \end{aligned}$$ ### [Radian measure](https://math.wiki/why-radians-measure-arc-length/) **Summary:** A radian is the angle whose arc equals the radius, so radian measure is arc length divided by radius. **Question:** Degrees cut a full turn into 360 parts for no reason found inside the circle. Can an angle be measured with the circle itself? ## Why this exists A degree is $\frac{1}{360}$ of a full turn, and the number $360$ is a choice made long ago, not something the circle forces. It works well for drawing and for naming angles, but it puts a stray factor into every formula that joins an angle to a length. In arc length the arc of an $n^\circ$ angle is $s = \frac{\pi n r}{180}$, and that $\frac{\pi}{180}$ is there only because degrees were chosen. Radian measure removes it. It measures an angle by the one length the circle already has, its radius: how many radii of arc does the angle cut off? With that unit the arc is simply the radius times the angle, and the formulas of trigonometry and, later, calculus take their simplest form. Degrees stay in use, so you need both and the bridge between them. Degrees also have their own finer units, minutes and seconds, used wherever an angle has to be read precisely: a surveyor's instrument, a map's latitude, or the joint of a robotic arm. This lesson covers all three. ## Definition **Degrees, minutes and seconds.** One degree is split into $60$ **minutes**, and one minute into $60$ **seconds**: $$1^\circ = 60' \qquad 1' = 60'' \qquad 1^\circ = 3600''$$ An angle written $47^\circ 21' 36''$ means $47$ degrees plus $21$ minutes plus $36$ seconds. These minutes and seconds measure angle, not time; they only borrow the same base of $60$. **Radian measure.** Put the vertex of an angle at the center of a circle of radius $r$. The angle cuts off an arc of length $s$. The **radian measure** of the angle is the arc length divided by the radius: ### [Reciprocal and quotient identities](https://math.wiki/quotient-identities/) **Summary:** Tangent equals sine over cosine, and cotangent equals cosine over sine. **Question:** Secant, cosecant, tangent and cotangent are all built from the two coordinates that sine and cosine already name. Can every one of them be written in sine and cosine alone, and when is the rewritten form allowed to replace the original? ## Why this exists The lesson on the six functions on the unit circle defined secant, cosecant, tangent and cotangent as fractions built from the terminal point $(x, y)$, and sine and cosine as the point's coordinates. Every one of the six is therefore made from the same two numbers. Saying so as equations between the functions themselves, such as $\tan t = \frac{\sin t}{\cos t}$, turns a definition into a tool: any expression in the six functions can be rewritten in sine and cosine, and two expressions that look different can be compared on equal terms. A tool used to replace one expression by another has to come with the set where the replacement is honest. That is what makes these statements identities rather than loose abbreviations: each holds on its common domain, and the common domain is part of the statement. ## Definition The **reciprocal identities** say that three of the functions are one over the other three. The **quotient identities** say that tangent and cotangent are quotients of sine and cosine. For every angle $t$ at which both sides are defined, $$\begin{gathered} \csc t = \frac{1}{\sin t} \qquad \sec t = \frac{1}{\cos t} \\ \cot t = \frac{1}{\tan t} \end{gathered}$$ $$\tan t = \frac{\sin t}{\cos t} \qquad \cot t = \frac{\cos t}{\sin t}$$ These five, with the Pythagorean identities that come next, are often called the **fundamental identities**. In right triangles the same reciprocal pairs appear as ratios of sides; the identities extend them to every angle at which the functions exist. ### [Reference angles](https://math.wiki/reference-angles-for-negative-and-large-angles/) **Summary:** Bringing an angle inside one turn first, then taking its acute angle to the x axis. **Question:** A skater spins through $1{,}020^\\circ$, or turns $\\frac{17\\pi}{6}$ the other way. Is there a short road from an angle that size to its sine and cosine, without drawing every turn? ## When you reach for it On the unit circle, $\cos\theta$ and $\sin\theta$ are the coordinates of the terminal point of $\theta$ (cosine-and-sine-as-coordinates). That definition works for every angle, but it does not say how to find the coordinates when the angle is $1{,}020^\circ$, $-\frac{17\pi}{6}$ or $5$ radians. The exact values you know by heart belong to acute angles: $30^\circ$, $45^\circ$ and $60^\circ$ from the special right triangles. A reference angle is the bridge. Any angle whose terminal side is off the axes makes an acute angle with the $x$-axis, and the terminal point sits at the same distances from the two axes as the terminal point of that acute angle. So the values of the big angle are the values of the acute one, up to sign. You reach for it whenever a trig value is wanted at an angle that is negative, larger than a quarter turn, or more than a full turn, and whenever you need every angle that shares a value. ## The procedure **The reference angle.** Let $\theta$ be an angle in standard position whose terminal side is not on an axis. Its **reference angle** $\theta'$ is the acute angle between the terminal side and the $x$-axis. It is always measured to the $x$-axis, the positive half or the negative half, whichever is nearer, and never to the $y$-axis. **Bring the angle inside one turn.** Adding or subtracting a full turn does not move the terminal side, so $\theta$ and $\theta \pm 360^\circ$ (or $\theta \pm 2\pi$) are coterminal and have the same terminal point, as standard-position-angle showed. Add or subtract whole turns until the angle lands in $0^\circ \le \theta_0 < 360^\circ$ (or $0 \le \theta_0 < 2\pi$). Nothing is lost: $\theta_0$ and $\theta$ share every trig value. For a negative angle, add turns; for a large one, subtract them. ### [Right triangle trigonometry](https://math.wiki/right-triangle-trig/) **Summary:** Ratios of sides in a right triangle, named by an angle. **Question:** Knowing that a ladder leans at 70 degrees says nothing yet about how high it reaches. How can an angle be turned into lengths, and lengths back into an angle? ## What this area is about In a right triangle, nothing so far links a side to an angle. Similar triangles supply the link: every right triangle with the same acute angle has the same shape, so the ratio of any two sides is a number that belongs to the angle. Here the ratios are defined for an acute angle, between $0^\circ$ and $90^\circ$; the unit circle later extends them to every angle. ## How these fit together **Definitions.** Opposite, adjacent and hypotenuse names each side relative to one acute angle. Sine, cosine and tangent are three ratios of those sides, and secant, cosecant and cotangent are their reciprocals. **Exact values.** Trig values of the special right triangles reads exact values for $30^\circ$, $45^\circ$ and $60^\circ$ off two familiar triangles. **Procedures.** Solving right triangles goes from an angle and a side to the other sides. Inverse trigonometric ratios go from two sides back to an acute angle. Angles of elevation and depression apply both to heights and distances you cannot reach. ## A path through it Opposite, adjacent and hypotenuse first, then sine, cosine and tangent. After that come the reciprocal ratios, the special triangles, solving right triangles and inverse ratios; the inverse lesson refers back to solving right triangles, and elevation and depression build on solving right triangles. Next: the unit circle carries these ratios to every angle, and the laws of sines and cosines carry them to triangles with no right angle. ### [Roots of a complex number and roots of unity](https://math.wiki/roots-of-a-complex-number/) **Summary:** Every nonzero complex number has n different nth roots, spaced evenly around a circle. **Question:** The real number 8 has one real cube root, 2. Among the complex numbers, how many numbers cube to 8, where are they, and how do you find them all? ## When you reach for it Over the real numbers, $x^3 = 8$ has one solution and $x^4 = -81$ has none. Over the complex numbers both equations have exactly as many solutions as their degree: three cube roots of $8$ and four fourth roots of $-81$. De Moivre's theorem raises a number to a power by raising its modulus and multiplying its argument; this lesson runs it backwards. Reach for it whenever an equation has the form $z^n = w$, or can be brought to that form: finding all square, cube or higher roots of a complex number, solving $z^n - w = 0$, solving an equation that is quadratic in $z^n$, and working with the roots of unity, the solutions of $z^n = 1$, which turn up wherever something repeats in $n$ equal steps around a circle. ## The procedure Throughout, $\operatorname{cis}\theta$ means $\cos\theta + i\sin\theta$, as in modulus-and-argument, $n$ is a positive integer, and $w \ne 0$. **Setting up.** Write $w = R\operatorname{cis}\varphi$ with $R = |w| > 0$ and $\varphi$ any argument of $w$. Look for $z$ in the same form, $z = r\operatorname{cis}\theta$ with $r > 0$ ($z = 0$ cannot work, since $0^n = 0 \ne w$). By De Moivre's theorem the equation $z^n = w$ becomes $$r^n\operatorname{cis}n\theta = R\operatorname{cis}\varphi$$ Two numbers in polar form with positive moduli are equal exactly when their moduli are equal and their angles differ by a whole number of turns. So $r^n = R$ and $n\theta = \varphi + 2\pi k$ for some integer $k$. The first says $r = \sqrt[n]{R}$, the one positive real number whose $n$th power is $R$. The second says $\theta = \frac{\varphi + 2\pi k}{n}$. ### [Rose curves](https://math.wiki/rose-curves/) **Summary:** Petal shaped curves whose petal count depends on whether the multiplier inside is odd or even. **Question:** What does a polar graph look like when the distance from the pole swings out and back several times on one trip around, and how can you tell from the equation how many petals it has and where they point? ## Why this exists In polar-graph-of-a-circle the equation $r = a\cos\theta$ gives a distance that swings once from $a$ to $-a$ as the direction turns half way around, and the graph is a circle. Now make the swing faster. In $r = a\cos 3\theta$ the distance reaches its largest size, falls to the pole and turns negative three times as often. Each time the distance leaves the pole and comes back, the point draws a loop, and the loops together look like a flower. The rectangular equations are hard to use ($r = \cos 2\theta$ becomes $(x^2 + y^2)^3 = (x^2 - y^2)^2$), but the polar ones can be read at a glance: how many petals, how long, which way they point, and how much turning draws them all. The petal count depends on whether the multiplier of $\theta$ is odd or even. ## Definition A **rose curve** is the graph of $$r = a\cos n\theta \qquad \text{or} \qquad r = a\sin n\theta,$$ where $a \ne 0$ and $n$ is a positive integer. As in plot-a-polar-point, $r$ may be negative: $(r, \theta)$ with $r < 0$ is the point $|r|$ units from the pole in the direction $\theta + \pi$. A **petal** is the piece of the curve drawn while $\theta$ runs between two neighboring zeros of $r$: the point leaves the pole and returns to it. The **tip** of a petal is its point farthest from the pole, and the **petal length** is the distance from the pole to the tip. ## Intuition Graph $r$ against $\theta$ on rectangular axes, as in graph-sine-from-the-unit-circle. For $r = 4\cos 3\theta$ this is a cosine wave of height $4$ and period $\frac{2\pi}{3}$ (period-from-the-b-value). Its zeros are $\frac{\pi}{3}$ apart. ### [Scaling a vector](https://math.wiki/scalar-multiple-of-a-vector/) **Summary:** Multiplying by a number stretches or shrinks a vector, and a negative number also reverses it. **Question:** A trip of so much across and so much up can be taken twice, taken halfway, or taken backward. What does multiplying a vector by a number do to its components, its length and its direction? ## Why this exists Merey, Nursultan and Sanzhar are building a pen plotter, a drawing machine that drags a pen across a sheet of paper. Its controller moves the pen by a vector each second, measured in centimeters: so much across, so much up, in component form. Running the motor for three seconds moves the pen three times as far the same way. Running it at half speed moves it half as far. Running it in reverse sends the pen back the way it came. Each of these is a vector multiplied by a number. Numbers and vectors are different kinds of objects, so the product has to be defined, and then its effect on the two things a vector carries, its magnitude and direction, has to be worked out. The answer is simple: the length is multiplied by the size of the number, and the direction is kept or reversed by its sign. ## Definition In vector work an ordinary real number is called a **scalar**, because multiplying by it scales. For a scalar $k$ and a vector $\mathbf{v} = \langle a, b \rangle$, the **scalar multiple** $k\mathbf{v}$ is the vector whose components are $k$ times those of $\mathbf{v}$: $$k\mathbf{v} = k\langle a, b \rangle = \langle ka, kb \rangle$$ In $\mathbf{i}$, $\mathbf{j}$ form the same rule reads $k(a\mathbf{i} + b\mathbf{j}) = ka\,\mathbf{i} + kb\,\mathbf{j}$. Three multiples get names. The **opposite** of $\mathbf{v}$ is $-\mathbf{v} = (-1)\mathbf{v} = \langle -a, -b \rangle$. The **zero vector** is $\mathbf{0} = \langle 0, 0 \rangle$, and $0\mathbf{v} = \mathbf{0}$ for every $\mathbf{v}$. Dividing a vector by a nonzero number $k$ means multiplying it by $\frac{1}{k}$. ### [Secant, cosecant and cotangent on the unit circle](https://math.wiki/undefined-values-of-tangent-and-secant/) **Summary:** A zero coordinate makes the matching reciprocal or quotient undefined, so some angles have no value at all. **Question:** Sine and cosine have a value at every angle. Tangent, secant, cosecant and cotangent each divide by a coordinate, and a coordinate can be zero. Which angles are left without a value, and which values can these functions never take? ## Why this exists In a right triangle the secant, cosecant and cotangent are the cosine, sine and tangent turned upside down. A triangle never has a side of length $0$, so the triangle versions are never in danger. Once cosine and sine become coordinates on the unit circle, every angle has a terminal point, and that point may sit on an axis with a coordinate of $0$. Turning a $0$ upside down is impossible. So the circle raises a question the triangle never asked: at which angles does each function have no value, and which outputs can it produce? The answers are the domain and range of the six functions. They decide where graphs break and which equations have no solution. ## Definition Let $\theta$ be an angle in standard position with terminal point $(x, y)$ on the unit circle. The six **trigonometric functions** of $\theta$ are $$\begin{aligned} \sin\theta &= y & \cos\theta &= x \\ \csc\theta &= \frac{1}{y} & \sec\theta &= \frac{1}{x} \\ \cot\theta &= \frac{x}{y} & \tan\theta &= \frac{y}{x} \end{aligned}$$ Sine and cosine always exist. Cosecant and cotangent divide by $y$, so they need $y \ne 0$; secant and tangent divide by $x$, so they need $x \ne 0$. In words: the **secant** is one over the cosine, the **cosecant** is one over the sine, and the **cotangent** is the cosine divided by the sine. When its condition fails, the function is **undefined** at $\theta$: it has no value there, not the value $0$. For acute angles these match the triangle ratios. ### [Secant, cosecant and cotangent ratios](https://math.wiki/secant-ratio/) **Summary:** The ratio of the hypotenuse to the side next to an acute angle, the reciprocal of cosine. **Question:** Sine, cosine and tangent each divide one side of a right triangle by another. What happens when the same two sides are divided the other way up, and why would anyone want those numbers? ## Why this exists A right triangle has three sides, and two sides can be divided in two orders. Sine, cosine and tangent use three of the six possible quotients. The other three are their reciprocals, and they get names too: **secant**, **cosecant** and **cotangent**. They carry no new information, since each is $1$ divided by a ratio already known. They exist because the side being found is often on top. A crane operator who knows the horizontal reach of the boom and its angle wants the boom's length, which is the hypotenuse over the reach. That quotient is the secant of the angle, and with a name it can be computed and written in one step instead of as a fraction with a cosine in its denominator. The same three ratios turn up again in the identities of trigonometry and, later, in calculus, so they are worth knowing by name. ## Definition Take an acute angle $A$ in a right triangle and name the sides from $A$, as in opposite, adjacent and hypotenuse. The three **reciprocal trigonometric ratios** of $A$ are $$\begin{aligned} \sec A &= \frac{\text{hypotenuse}}{\text{adjacent}} \\ \csc A &= \frac{\text{hypotenuse}}{\text{opposite}} \\ \cot A &= \frac{\text{adjacent}}{\text{opposite}} \end{aligned}$$ read "the secant of $A$", "the cosecant of $A$" and "the cotangent of $A$". In the usual lettering of Figure 1, with the right angle at $C$, $$\sec A = \frac{c}{b} \qquad \csc A = \frac{c}{a} \qquad \cot A = \frac{b}{a}$$ ### [Signs of the trig functions by quadrant](https://math.wiki/determine-a-quadrant-from-two-signs/) **Summary:** Pinning down where an angle ends when told, for instance, that sine is negative and tangent is positive. **Question:** You are told only that $\\cos\\theta$ is negative and $\\tan\\theta$ is negative. Where on the circle does the angle end, and what does that fix about the other four values? ## When you reach for it On the unit circle, cosine and sine are coordinates: if the terminal point of $\theta$ is $(x, y)$, then $\cos\theta = x$ and $\sin\theta = y$. Knowing one value is rarely enough. The circle equation $x^2 + y^2 = 1$ hands back the other coordinate only up to sign, because a vertical or horizontal line meets the circle twice. Something else has to choose. That something is usually a second sign. A problem says "$\cos\theta = -\frac{55}{73}$ and $\tan\theta < 0$", or "$\sin\theta < 0$ and $\tan\theta > 0$". Reach for this move whenever a value comes with a condition like that, whenever you need the sign of a trig value at an angle you cannot evaluate exactly, and whenever an angle is described only by the region it lies in. It turns two signs into a quadrant, and a quadrant into the sign of all six functions. ## The procedure ### Where each sign comes from Write the six functions in terms of the terminal point $(x, y)$. The first three come from the unit circle; the other three are their reciprocals, the secant, cosecant and cotangent, which carry over from right triangles to every angle the same way: $$\begin{aligned} \sin\theta &= y & \csc\theta &= \frac{1}{y} \\ \cos\theta &= x & \sec\theta &= \frac{1}{x} \\ \tan\theta &= \frac{y}{x} & \cot\theta &= \frac{x}{y} \end{aligned}$$ A number and its reciprocal always have the same sign, since their product is $1$, which is positive. So the six functions fall into three pairs that share a sign: ### [Simple harmonic motion](https://math.wiki/simple-harmonic-motion/) **Summary:** The steady back and forth of a spring or a pendulum, written as a sine or cosine of time. **Question:** A weight on a spring, pulled down and let go, bobs up and down on its own. Where will it be a second later, how long does each bounce take, and where along its path is it moving fastest? ## Why this exists Hang a weight on a spring and it settles at a resting place, its **equilibrium**. Pull it down and let go, and it bobs up and down past that place again and again. The tip of a diving board after a jump does the same, and so does a car body riding on its springs after a bump. In each case the spring pulls the object back toward its resting place, harder the farther away it is; OpenStax describes this restoring force as directly proportional to the displacement. Such motion repeats on a steady cycle, so a sine or cosine of time can describe it, and questions about the object become questions about a function: where it is at a given moment, how long a bounce lasts, and where it moves fastest. This lesson builds on amplitude, period-from-the-b-value and write-a-sinusoid-from-a-graph. Its neighbors ferris-wheel-model and tide-models use the same functions for a turning wheel and a harbor. ## Definition Measure the object's position by its **displacement** $d$ from equilibrium, positive on one side (here, up) and negative on the other, and let $t$ be time in seconds. The object is in **simple harmonic motion** when $$d(t) = a\cos(\omega t) \quad \text{or} \quad d(t) = a\sin(\omega t)$$ for constants $a \ne 0$ and $\omega > 0$. - $|a|$ is the **amplitude**, the greatest distance from equilibrium. - $\omega$ (omega) is the **angular frequency**, in radians per second. - The **period** $P$ is the time for one full cycle, out and back. - The **frequency** $f$ is the number of cycles per second, measured in **hertz** ($\text{Hz}$). ### [Simplifying trig expressions](https://math.wiki/simplify-trig-expressions/) **Summary:** Using the basic identities to rewrite a messy expression as a short one. **Question:** An expression like $\frac{\cos x}{1 - \sin x} - \frac{\cos x}{1 + \sin x}$ is a tangle of functions and fractions. Which identities untangle it, and on which inputs is the short form a faithful stand-in for the long one? ## When you reach for it Reach for these moves when an expression carries more than it needs: several different functions of the same angle, a fraction inside a fraction, two fractions waiting to be combined, or a square such as $1 - \sin^2 x$ that one identity collapses. A shorter form is easier to evaluate, graph and solve, which is why trigonometric equations usually start with a step like this. The same identities also run the other way, into algebra. A radical such as $\sqrt{9 - x^2}$ turns into a single trigonometric function when the variable is replaced by $3\sin\theta$, because the identity $1 - \sin^2\theta = \cos^2\theta$ swallows the subtraction under the root. That replacement is called a trigonometric substitution. To simplify is not to prove. Here the task is to start from one expression and find a shorter equal one. Showing that two given expressions are equal is verifying an identity, the next lesson. ## The procedure Everything rests on four families of identities, each from an earlier lesson. - Reciprocal and quotient, holding wherever the left side exists: $\sec x = \frac{1}{\cos x}$, $\csc x = \frac{1}{\sin x}$, $\tan x = \frac{\sin x}{\cos x}$, $\cot x = \frac{\cos x}{\sin x}$. - Pythagorean, holding wherever both sides exist: $\sin^2 x + \cos^2 x = 1$, $1 + \tan^2 x = \sec^2 x$, $1 + \cot^2 x = \csc^2 x$. - Cofunction: $\sin\left(\frac{\pi}{2} - x\right) = \cos x$, $\cos\left(\frac{\pi}{2} - x\right) = \sin x$, $\cot\left(\frac{\pi}{2} - x\right) = \tan x$. - Even and odd: $\cos(-x) = \cos x$ and $\sec(-x) = \sec x$, while $\sin(-x) = -\sin x$, and likewise $\tan$, $\csc$ and $\cot$ are odd. ### [Sine and tangent of a sum or difference](https://math.wiki/sum-and-difference-formulas-for-sine/) **Summary:** The sine versions, which follow from the cosine versions through the cofunction identities. **Question:** Cosine already has a rule for the sum of two angles. What are the matching rules for sine and tangent, and what do they say about the angle at which two lines cross? ## Why this exists The sum-formula-for-cosine turns the cosine of $a + b$ into the sines and cosines of $a$ and $b$. That gives exact cosines at angles such as $15^\circ$ and $75^\circ$, but a sine is just as often the thing wanted: the height of a point on a wheel, the $y$-coordinate of a terminal point, the sine in the law of sines. A rule for $\sin(a + b)$ is the missing half. The rule does not need a new figure or a new proof from scratch. The cofunction-identities say a sine is the cosine of the complementary angle, and the complement of $a + b$ is a difference, which the cosine formula already handles. Dividing the sine rule by the cosine rule then gives a rule for tangent. Tangent is the slope of a line through the origin, so that last rule answers a question from coordinate geometry: at what angle do two lines with known slopes cross? ## Definition For all real numbers $a$ and $b$ (or angles in degrees), the **sum and difference formulas for sine** are the identities $$\begin{aligned} \sin(a + b) &= \sin a\cos b + \cos a\sin b \\ \sin(a - b) &= \sin a\cos b - \cos a\sin b \end{aligned}$$ Read in words: "sine cosine plus cosine sine". The sign between the two products matches the sign between the angles, unlike cosine, where it flips. The **sum and difference formulas for tangent** are $$\begin{aligned} \tan(a + b) &= \frac{\tan a + \tan b}{1 - \tan a\tan b} \\[4pt] \tan(a - b) &= \frac{\tan a - \tan b}{1 + \tan a\tan b} \end{aligned}$$ ### [Sine, cosine and tangent](https://math.wiki/sine-ratio/) **Summary:** The ratio of the side opposite an acute angle to the hypotenuse. **Question:** Two right triangles share the same sharp angle but are different sizes. What do their sides have in common, and how can that be turned into a number that belongs to the angle? ## Why this exists A right triangle has three sides and three angles, and before this lesson they are linked only among themselves. The Pythagorean theorem ties the three sides to each other, and the triangle angle sum ties the three angles to each other. But nothing yet joins a side to an angle. Knowing that a ladder leans at $70^\circ$ to the ground says nothing, so far, about how high up the wall it reaches. The join comes from similar triangles. Every right triangle with a $70^\circ$ angle has the same shape, whatever its size, so the ratio of any two of its sides is the same number in all of them. That number belongs to the angle. Three of these ratios get names, **sine**, **cosine** and **tangent**, and with them a question about an angle can be turned into a question about lengths. The lesson is taught in two parts. The first part defines the three ratios, reads them off triangles, and shows why each one depends only on the angle. The second part shows how the three are tied to each other: the sine of one acute angle is the cosine of the other, the tangent is the sine divided by the cosine, and the squares of the sine and the cosine add to $1$. ## Definition Take an acute angle $A$ and any right triangle that has $A$ as one of its two acute angles. Name the sides from $A$, as in opposite, adjacent and hypotenuse. The three **trigonometric ratios** of $A$ are $$\begin{aligned} \sin A &= \frac{\text{opposite}}{\text{hypotenuse}} \\ \cos A &= \frac{\text{adjacent}}{\text{hypotenuse}} \\ \tan A &= \frac{\text{opposite}}{\text{adjacent}} \end{aligned}$$ ### [Sine, cosine and tangent as functions of a real number](https://math.wiki/circular-functions-of-a-real-number/) **Summary:** Once angles are in radians, the input to a trig function can be read as a distance along the circle. **Question:** A calculator gives $\\sin 10 \\approx -0.544$ even though nobody said degrees, or said angle at all. What is the $10$ the input of? ## Why this exists In cosine-and-sine-as-coordinates, the cosine and sine belonged to an angle: turn the terminal side, find where it meets the unit circle, read the coordinates. Many of the places sine and cosine appear have no angle in sight. A runner's position depends on the distance run, a tide's height on the hour, a sound's pressure on the time. To graph $y = \sin t$ or to write $h(t) = 3 + 2\cos t$ in function-notation, the input has to be a plain real number, like the $x$ in $f(x) = x^2$. Radian measure is what makes the step possible. On a circle of radius $1$, an arc is exactly as long as the angle it spans, measured in radians. So a real number $t$ can be read as a distance traveled around the circle, with no angle mentioned, and the cosine and sine become functions whose input is any real number and whose output is a real number. ## Definition Picture the number line standing upright, touching the unit circle at $(1, 0)$, with its $0$ at that point and its positive numbers going up. Now wrap the line around the circle: the positive half winds counterclockwise and the negative half winds clockwise, with no stretching. Each real number $t$ lands on one point of the circle, called $P(t)$. Stated without the picture: for a real number $t$, the point $P(t)$ is reached by starting at $(1, 0)$ and traveling a distance $|t|$ along the unit circle, counterclockwise if $t > 0$ and clockwise if $t < 0$; $P(0) = (1, 0)$. If $P(t) = (x, y)$, then ### [Solving basic trig equations](https://math.wiki/solve-a-basic-trig-equation/) **Summary:** Finding the angles whose sine, cosine or tangent equals a given number. **Question:** An inverse key hands back one angle with a given sine. How do you find every angle with that sine, and pick out the ones in the stretch you care about? ## When you reach for it An equation such as $\tan x = -\sqrt{3}$ or $3\cos\theta + 2 = 0$ asks for every angle that makes it true. It is evaluation run backward: the unit circle turns an angle into a value, and here the value is given and the angles are wanted. Two facts shape every answer. A trig function is periodic, so once one angle works, infinitely many do: sine and cosine repeat every $2\pi$, tangent every $\pi$. And within a single turn, sine and cosine usually take each value twice, so even one turn holds two answers. A problem therefore asks for one of two things: - **all solutions**, an infinite list written as families such as $x = \frac{\pi}{6} + 2\pi k$, with $k$ any integer; - **the solutions in an interval**, such as $[0, 2\pi)$, a finite list picked out of those families. Part 1 (the first day) handles values read straight off the unit circle. Part 2 handles any other value, where an inverse trig function supplies one angle and the symmetry of the circle supplies the second. The same moves find when a Ferris wheel rider reaches a given height or a tide reaches a given level; those lessons put them to work in context. ## The procedure ### Part 1: get the function alone In $5\cos x - 1 = 3$ the expression $\cos x$ stands for a single unknown number, so the ordinary moves of algebra apply to it: add $1$ to both sides, divide by $5$, and $\cos x = \frac{4}{5}$. Every equation in this lesson is first brought to one of the forms $\sin x = c$, $\cos x = c$ or $\tan x = c$. ### [Solving equations with a multiple angle](https://math.wiki/solve-multiple-angle-equations/) **Summary:** When the unknown is multiplied inside the function, the extra cycles create extra solutions. **Question:** When the angle inside is $3x$ or $\\frac{x}{3}$ instead of $x$, how many solutions are there, and how do you find every one? ## When you reach for it Solving a basic trig equation finds every $x$ with $\sin x = c$, $\cos x = c$ or $\tan x = c$. Often the unknown sits inside the function multiplied by a number: $2\sin(3x) = \sqrt{2}$, $\cos\left(\frac{x}{3}\right) = -\frac{1}{2}$, $\tan(2x) = -1$. Such equations come from waves whose period is not $2\pi$, and they are what a model such as a Ferris wheel or a tide produces when you ask when it reaches a given level. The method is the basic one applied to the inner angle, with one new fact: on a fixed interval of $x$, the inner angle covers a stretched interval. For $\sin(3x)$ it runs through three turns while $x$ runs through one, so there are three times as many solutions. For $\cos\left(\frac{x}{3}\right)$ it runs through a third of a turn, and there may be fewer solutions, or none. ## The procedure ### Name the inner angle In $\sin(3x) = c$, the angle the sine acts on is $3x$, not $x$. Call it $u = 3x$. Then the equation reads $\sin u = c$, a basic equation, and everything known about basic equations applies to $u$. This is allowed because $u = bx$ with $b \neq 0$ matches each $x$ with exactly one $u$ and each $u$ with exactly one $x = \frac{u}{b}$. So $x$ solves $\sin(bx) = c$ exactly when $u = bx$ solves $\sin u = c$: finding all the $u$ and dividing each by $b$ finds all the $x$, no more and no fewer. ### Stretch the interval along with the angle When $x$ is restricted, $u$ is restricted too. For $b > 0$, multiplying each part of $0 \le x < 2\pi$ by $b$ keeps the inequalities pointing the same way: ### [Solving right triangles](https://math.wiki/find-a-missing-side-with-a-trig-ratio/) **Summary:** Choosing the ratio that links the known angle, the known side and the side being found. **Question:** You know one acute angle of a right triangle and the length of one side. How do you find the other sides without measuring them? ## When you reach for it A right triangle has six parts: three sides and three angles. One of the angles is already known, because it is the right angle. Tell someone one of the acute angles, which side you know (the hypotenuse, or the leg opposite or next to that angle) and its length, and the triangle is fixed: any two right triangles that share those facts are congruent. So the other two sides are not up for grabs. They have definite lengths, and this lesson is how to compute them. What makes that possible is similarity. Every right triangle with a $35^\circ$ angle has the same shape, so in every one of them the side opposite the $35^\circ$ angle divided by the hypotenuse is the same number. That number is $\sin 35^\circ$, and a calculator knows it. The ratios of sine, cosine and tangent carry the shape; the one side you know carries the size. Put them together in one equation and the unknown side drops out. **Solving a right triangle** means finding every side and angle that was not given. With one acute angle and one side, the steps below find all of them. With two sides and no acute angle, the third side comes from the Pythagorean theorem and the angles come from the inverse ratios, which is a different move. ## The procedure Take the triangle in the figure: the right angle is at $C$, $\angle A = 35^\circ$, and the hypotenuse $AB$ is $12$. The legs $BC = x$ and $AC = y$ are wanted. ### [Solving trig equations by factoring, quadratics and identities](https://math.wiki/solve-trig-equations-by-factoring/) **Summary:** Getting one side to zero and factoring so each piece can be set equal to zero on its own. **Question:** An equation holds two trig terms, or a square, or a double angle, so no single sine, cosine or tangent can be got alone. How do you break it into basic equations without losing a solution or picking up a false one? ## When you reach for it A basic trig equation has one sine, cosine or tangent of $x$, and algebra gets it alone. Many equations will not allow that. In $2\sin^2 x = \sin x$ the sine appears twice, once squared. In $2\cos^2 x + 3\sin x = 3$ two different functions appear. In $\cos 2x + \cos x = 0$ the angles differ. The move for all of them is the one that solves polynomial equations: get $0$ on one side, factor, and use the zero-product-property. A product is zero only when one of its factors is zero, so one hard equation becomes several basic ones, each finished as before. When the equation is not yet in a form that factors, an identity first rewrites it in one function of one angle. Part 1 (the first day) factors equations that are already in one or two functions of $x$, including those that are quadratic in $\sin x$ or $\cos x$. Part 2 (the second day) uses a Pythagorean or double angle identity first, and ends with squaring both sides, a last resort that needs every answer checked. ## The procedure ### Part 1: zero on one side, then factor The zero product property only speaks about a product equal to $0$. So every term goes to one side first. Then the expression is factored just as a polynomial would be, treating $\sin x$ and $\cos x$ as unknown numbers: a common factor comes out by the distributive property, four terms may factor by grouping, and a quadratic factors into two binomials. Each factor set equal to $0$ is a basic equation. ### [Solving trigonometric equations](https://math.wiki/trig-equations/) **Summary:** Find every angle that works, on an interval or in general. **Question:** An inverse key gives back one angle. When a sine or cosine equals some number, how do you find every angle that works? ## What this area is about A trig equation runs evaluation backward: the value is given and the angles are wanted. Two facts shape every answer. The functions are periodic (sine and cosine every $2\pi$, tangent every $\pi$, in radians), so one solution brings infinitely many. And an inverse function returns only the one angle in its restricted range, so the circle's symmetry has to supply the rest. An answer is either every solution, written as families with $k$ any integer, or the finite list inside a stated interval such as $[0, 2\pi)$. ## How these fit together **The core procedure.** Solving basic trig equations gets $\sin x$, $\cos x$ or $\tan x$ alone, finds the solutions in one turn, then extends them by the period. Everything else reduces to it. **Harder equations.** Equations with a multiple angle such as $\sin(3x) = c$ solve for the inner angle, stretch the interval to match, and divide at the end. Factoring, quadratics and identities breaks an equation with two trig terms, a square or a double angle into basic ones; squaring both sides can add false solutions, so after squaring every candidate is checked. **Beyond equality.** Trigonometric inequalities find whole stretches of angles where a function stays above or below a level, cut at the crossings of the matching equation and at any point where the function is undefined. ## A path through it Basic equations first: the multiple-angle lesson builds on them, and the factoring and inequality lessons refer back to them. The other three follow; none of them is a prerequisite of another. The factoring lesson also needs the double angle formulas. ### [The ambiguous case](https://math.wiki/height-test-for-ssa/) **Summary:** The altitude from the given angle decides whether zero, one or two triangles can close. **Question:** Two sides and an angle that is not between them sometimes build no triangle, sometimes one, sometimes two. How do you tell which before you start, and how do you find them all? ## When you reach for it You know two sides of a triangle and an angle that is **not** between them: side–side–angle, or **SSA**. In triangle $ABC$ that means you know the angle $A$, the side $a$ across from it, and a second side $b$. (why-ssa-is-not-a-criterion showed with compass and ruler that these parts do not pin down one triangle.) This lesson settles the question with trigonometry: how many triangles fit, zero, one or two, and what each one is. The law of sines does the solving, but on its own it hides the answer to "how many", because it finds the angle $B$ from its sine, and one sine belongs to two angles. The **height test** answers "how many" first, from one number. Because the count can be two, SSA is called the **ambiguous case**. ## The procedure The lesson runs in two parts. Part 1 is the height test with an acute given angle: zero, one or two triangles. Part 2 adds an obtuse given angle, solves both triangles when there are two, and reads the same count straight off the law of sines. ### Part 1: the side that swings Draw what is known. The angle $A$ is two rays from the vertex $A$. Side $b$ runs from $A$ along one ray to the vertex $C$, so $C$ is fixed. The vertex $B$ must lie on the other ray, and it must be exactly $a$ from $C$. Picture side $a$ as a rod hinged at $C$, swinging; every place its free end touches the ray is a triangle. In the language of circles: the circle of radius $a$ around $C$ meets the ray zero, one or two times. ### [The cross product](https://math.wiki/cross-product-magnitude-as-area/) **Summary:** Its length is the area of the parallelogram the two vectors form. **Question:** Two arrows in space span a flat parallelogram. Is there a single calculation that returns an arrow standing straight up from it, as long as the parallelogram's area? ## Why this exists The dot product turns two vectors into a number that measures how much they point the same way. Many questions in space ask for something else. Which direction is straight out of a tilted roof panel? How large is a triangle whose corners are three points in space? Is a third point on the line through two others? One operation, which exists only in three dimensions, answers all three. The **cross product** takes vectors $\mathbf{u}$ and $\mathbf{v}$ in space and returns a vector $\mathbf{u} \times \mathbf{v}$ perpendicular to both, whose length is the area of the parallelogram they span and whose direction is fixed by the right-hand rule of three-dimensional-vectors. ## Definition For $\mathbf{u} = \langle u_1, u_2, u_3 \rangle$ and $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$, the cross product is the vector $$\begin{aligned} \mathbf{u} \times \mathbf{v} = \langle\, &u_2 v_3 - u_3 v_2, \\ &u_3 v_1 - u_1 v_3, \\ &u_1 v_2 - u_2 v_1 \,\rangle. \end{aligned}$$ The pattern is easiest to carry as a $3 \times 3$ determinant with $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ in the first row, $\mathbf{u}$ in the second and $\mathbf{v}$ in the third: $$\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}$$ Expanding along the first row, with the checkerboard signs $+, -, +$ of the cofactor expansion, gives three $2 \times 2$ determinants: ### [The dot product](https://math.wiki/dot-product/) **Summary:** Multiply matching parts and add; it measures how aligned two vectors are. **Question:** Adding and scaling vectors never says how far apart two directions are. Can one number, taken straight from the components, say it? ## What this area is about For $\mathbf{u} = \langle a_1, b_1 \rangle$ and $\mathbf{v} = \langle a_2, b_2 \rangle$, the dot product $\mathbf{u} \cdot \mathbf{v} = a_1 a_2 + b_1 b_2$ is a single number, not a vector. For nonzero vectors it equals $\|\mathbf{u}\|\,\|\mathbf{v}\|\cos\theta$, with $\theta$ the angle between them ($0 \le \theta \le \pi$). So it measures how well two directions line up, and that one fact drives everything in this area. ## How these fit together **The definition and its meaning.** The dot product and the angle between vectors defines the number from components and proves the cosine formula, which recovers the angle with one inverse cosine. **Two results built on it.** Orthogonal vectors are those with $\mathbf{u} \cdot \mathbf{v} = 0$; for nonzero vectors that means exactly a right angle. Vector and scalar projection divide the dot product by $\|\mathbf{v}\|$ or $\|\mathbf{v}\|^2$ (needing $\mathbf{v} \ne \mathbf{0}$) to measure how much of one vector runs along another and to split a vector into perpendicular pieces; the same lesson finds the work a constant force does as the dot product $\mathbf{F} \cdot \mathbf{d}$. ## A path through it The cosine formula comes first, since both other lessons rest on it. Take orthogonal vectors next: projection uses orthogonality to split a vector into a piece along $\mathbf{v}$ and a piece perpendicular to it. After this, the same product carries into higher dimensions in dot product, length and orthogonality in n dimensions, and orthogonality sets up the cross product. ### [The dot product and the angle between vectors](https://math.wiki/dot-product-as-magnitudes-and-cosine/) **Summary:** It equals the two lengths times the cosine of the angle between them, so it measures how aligned they are. **Question:** Two arrows leave the same point. Can one number, computed straight from their components, say how closely they point the same way, and recover the angle between them? ## Why this exists Vectors can be added, subtracted and stretched by a number. None of those operations answers a question that comes up as soon as two vectors share a starting point: how far apart are their directions? You could find each direction angle and subtract, but that means two arctangents, a quadrant check for each, and a correction whenever the difference comes out above $180^\circ$. There is a better tool. Multiply the matching components of the two vectors and add. The result is a single number, and that number turns out to be the two lengths times the cosine of the angle between the vectors. So one line of arithmetic on components reports how well two vectors line up, and the angle comes out with one inverse cosine and no quadrant check at all. ## Definition The lesson runs over two days. Day 1 is the dot product computed from components and the algebra it obeys. Day 2 is what the number means: the cosine formula, the angle between two vectors, and what the sign of the dot product says. For $\mathbf{u} = \langle a_1, b_1 \rangle$ and $\mathbf{v} = \langle a_2, b_2 \rangle$ in component form, the **dot product** is $$\mathbf{u} \cdot \mathbf{v} = a_1 a_2 + b_1 b_2.$$ Multiply the horizontal components, multiply the vertical components, add. The answer is a real number (a *scalar*), not a vector, which is why the dot product is also called the **scalar product**. In i-and-j-notation the same rule reads ### [The Pythagorean identities](https://math.wiki/derive-the-pythagorean-identity/) **Summary:** The identity is the circle equation with the coordinates renamed as cosine and sine. **Question:** Sine and cosine are the two coordinates of a point on a circle of radius one. What does the circle itself force on them, and how much of the other five values does one value then decide? ## When you reach for it The reciprocal and quotient identities write tangent, cotangent, secant and cosecant in terms of sine and cosine. They say nothing about how sine and cosine relate to each other. Yet the two are not free: they are the coordinates of one point, and that point lies on a circle. The equation of that circle, written in the new names, is the **Pythagorean identity**, and dividing it through gives two more. Reach for these identities whenever one trigonometric value of an angle is known and another is wanted, especially when the angle itself is never found. Reach for them also whenever an expression contains a square such as $\sin^2\theta$ or $\sec^2\theta$, since each identity lets one square be traded for another, and whenever you need to test whether two numbers could be the sine and cosine of the same angle. ## The procedure ### Where $\sin^2\theta + \cos^2\theta = 1$ comes from Let the terminal side of $\theta$ meet the unit circle at $(x, y)$. Every point of that circle is at distance $1$ from the origin, so by the distance formula, which is the Pythagorean theorem applied to the coordinates, $$x^2 + y^2 = 1$$ On the unit circle $x = \cos\theta$ and $y = \sin\theta$. Renaming the coordinates gives $(\cos\theta)^2 + (\sin\theta)^2 = 1$. The square of a sine is written $\sin^2\theta$, with the exponent after the name, so that it is not mistaken for $\sin(\theta^2)$: ### [The unit circle](https://math.wiki/unit-circle/) **Summary:** Trig for any angle, read as coordinates on a circle of radius one. **Question:** A right triangle only has room for an acute angle. What could the cosine of 140 degrees, or the sine of a negative angle, possibly mean? ## What this area is about Put a circle of radius $1$ at the origin. Every angle in standard position, of any size or sign, has a terminal side that meets the circle at one point $(x, y)$, and that point defines $\cos\theta = x$ and $\sin\theta = y$. For acute angles this agrees with the right triangle ratios; for every other angle it gives a value where the triangle gave none. ## How these fit together **The definition.** Trig functions on the unit circle makes cosine and sine coordinates, and tangent the slope of the terminal side. Secant, cosecant and cotangent on the unit circle adds the other three and finds where a zero coordinate leaves no value, such as $\tan\theta$ at $\theta = \frac{\pi}{2}$. **Structure of the circle.** Symmetry and the even-odd identities and periodicity ($2\pi$ for sine and cosine, $\pi$ for tangent) relate the values at different angles. Sine, cosine and tangent as functions of a real number reads a radian input as distance along the circle. **Procedures.** Signs by quadrant turns two signs into a quadrant. Reference angles reduce any angle off the axes to an acute one, up to sign. Exact values on the unit circle collects the exact values at multiples of $\frac{\pi}{6}$ and $\frac{\pi}{4}$. ## A path through it The coordinate definition first: every other lesson here builds on it. Reference angles and the reciprocal functions come before exact values, which need both. Symmetry, periodicity, signs by quadrant and the real-number view follow the definition. ### [Tides, daylight and sound](https://math.wiki/tide-models/) **Summary:** Describing the height of water at a harbor as a wave with a period of about half a day. **Question:** The water in a harbor rises and falls twice a day, the hours of daylight grow and shrink over a year, and a pure tone pushes the air back and forth hundreds of times a second. How do a few facts about such a cycle become a formula that predicts it? ## When you reach for it Reach for this move when a quantity rises and falls on a steady cycle and you know four facts about it: its highest value, its lowest value, how long one cycle lasts, and when it peaks. The depth of water at a harbor, the hours of daylight over a year and the pressure of the air carrying a pure tone all behave this way. A cosine with those four facts built in describes the whole cycle, and then it answers two kinds of question: what the value is at a given time, and when the value reaches a given level. The move puts together four pieces you already have: the amplitude, the midline, the period from the value of b and the phase shift. write-a-sinusoid-from-a-graph reads the same four numbers off a picture, and ferris-wheel-model builds them from a wheel; here they arrive as facts in words. The harbor and the town in this lesson are made up, and their numbers are illustrative, not measurements of a real place. ## The procedure **Building the model.** Call the highest value $M$, the lowest value $m$, the period $P$, and a time at which the quantity peaks $c$. Then the model is $$y = A\cos\bigl(b(t - c)\bigr) + k$$ with $$\begin{aligned} A &= \frac{M - m}{2}, \quad k = \frac{M + m}{2}, \\ b &= \frac{2\pi}{P} \end{aligned}$$ Each piece is forced. The cosine of an angle is largest, $1$, when the angle is $0$; here the angle is $b(t - c)$, which is $0$ exactly when $t = c$, so the model peaks at time $c$. That is why a cosine is the natural choice when a peak time is known: the peak time goes straight in as the phase shift, and $A$ stays positive. As $\cos$ runs between $-1$ and $1$, the output runs between $k - A$ and $k + A$, and the formulas give ### [Trig functions on the unit circle](https://math.wiki/cosine-and-sine-as-coordinates/) **Summary:** On the unit circle the x coordinate of the terminal point is the cosine and the y coordinate is the sine. **Question:** A right triangle gives an angle a sine and a cosine only when the angle is between $0^\\circ$ and $90^\\circ$. What could $\\cos 140^\\circ$ or $\\sin(-200^\\circ)$ possibly mean? ## Why this exists The sine, cosine and tangent of an angle were defined from a right triangle: opposite over hypotenuse, adjacent over hypotenuse, opposite over adjacent. A right triangle only has room for an acute angle, so those definitions say nothing about $140^\circ$, about $-200^\circ$, or about the $410^\circ$ a turning arm can sweep through. Yet angles in standard position can be any size, and anything that turns (a wheel, a pendulum, a merry-go-round) needs a sine and a cosine for all of them. The fix is to stop asking for a triangle and ask for a point instead. Put a circle of radius $1$ at the origin. Every angle in standard position, whatever its size or sign, has a terminal side, and that ray meets the circle at exactly one point. The coordinates of that point become the cosine and the sine. For acute angles this gives back the triangle ratios exactly, so nothing already learned is lost; for every other angle it gives a value where there was none. The lesson runs over two days. Part 1 (Examples 1 to 4) defines the terminal point and reads the three functions from it. Part 2 (Examples 5 to 7) shows why the new definition extends the old one, moves to circles of any radius, and reads the tangent as a slope. ## Definition The **unit circle** is the circle of radius $1$ centered at the origin. By the equation of a circle its points are exactly those with $$x^2 + y^2 = 1$$ Let $\theta$ be an angle in standard position. The **terminal point** of $\theta$ is the point where its terminal side meets the unit circle. If the terminal point is $(x, y)$, then ### [Trig values of the special right triangles](https://math.wiki/trig-values-of-special-right-triangles/) **Summary:** Reading exact sine, cosine and tangent values off the thirty sixty ninety and forty five triangles. **Question:** The sine, cosine and tangent of most angles come out of a calculator as decimals that have to be cut off somewhere. Are there angles whose values can be written down exactly, by hand? ## When you reach for it Ask a calculator for $\sin 40^\circ$ and it answers $0.6427876\ldots$, a decimal it has to cut off somewhere. For three angles there is nothing to cut off. The sine, cosine and tangent of $30^\circ$, $45^\circ$ and $60^\circ$ are exact numbers built from $1$, $\sqrt{2}$ and $\sqrt{3}$, and they can be read straight off two triangles you already know from the special right triangles: half a square, and half an equilateral triangle. Reach for this whenever one of these three angles turns up and the answer should be exact: an exact length, slope or area, an expression to simplify, or a value to recognize and turn back into an angle. The same three angles come back all the way round the unit circle, so this small table ends up carrying most of the exact values in trigonometry. Many people learn the nine values by heart. You never need to. Anyone who can draw the two triangles can rebuild the whole table in a minute, and the drawing also tells you why each value is what it is. ## The procedure ### Why one triangle is enough The sine, cosine and tangent of an acute angle are ratios of two sides of a right triangle that contains it. Every right triangle with a $30^\circ$ angle has angles $30^\circ$, $60^\circ$ and $90^\circ$, since the angles of a triangle add to $180^\circ$. So any two such triangles have two angles equal and are similar by AA similarity. Similar triangles have their sides in proportion, which means the ratio of two sides is the same in both: a triangle twice as big has both sides doubled, and the doubling cancels. ### [Trigonometric functions and their graphs](https://math.wiki/trig-functions/) **Summary:** Sine and cosine as waves, tangent as something wilder. **Question:** Once an angle can be any real number, sine and cosine become functions with graphs. What do those graphs look like, and what can they model? ## What this area is about With the angle measured in radians, sine and cosine take any real input and give an output in $[-1, 1]$, repeating every $2\pi$. Their graphs are waves. Stretched and shifted, a wave fits anything that rises and falls on a steady cycle; cut down to one slice, it can run backward. ## How these fit together **Definitions.** Periodic functions defines a period and a cycle for any function. **The graphs.** Graphing sine and cosine builds $y = \sin x$ and $y = \cos x$ from the unit circle. Graphing tangent and the reciprocal functions draws the four functions that divide by a coordinate, with vertical asymptotes where that coordinate is $0$. **Changing the wave.** Amplitude, period and phase shift reads each number of $y = a\sin\big(b(x - h)\big) + k$, with $a \ne 0$ and $b \ne 0$, off the formula. Writing an equation from a wave runs that backward, and modeling things that repeat builds the formula from facts about a wheel, a tide or a spring. **Inverses.** Inverse trig functions keeps one slice of each graph so arcsine, arccosine and arctangent exist; composing trig and inverse trig functions evaluates expressions like $\cos\left(\arcsin\frac{5}{13}\right)$ exactly. ## A path through it Two strands. In one, graphing sine and cosine comes first. After it, the tangent family, and separately amplitude, period and phase, then writing an equation from a wave, then modeling. In the other, periodic functions, then inverse trig functions, then compositions. ### [Trigonometric identities](https://math.wiki/trig-identities/) **Summary:** Equations true for every angle, and how to use them. **Question:** Some trig equations hold for every angle where both sides are defined. Which ones, where do they come from, and what are they good for? ## What this area is about An identity is an equation true at every input where both sides are defined, such as $\sin^2 x + \cos^2 x = 1$. Either side may then replace the other. That swap is how trig expressions are simplified, how new formulas are proved from old ones, and how equations are brought to a form that can be solved. ## How these fit together **The idea.** What an identity is separates an identity from an equation to be solved. **Basic identities.** Reciprocal and quotient identities, cofunction identities and the Pythagorean identities. **Procedures.** Simplifying trig expressions and verifying identities. **Formulas for combined angles.** Cosine of a sum or difference comes first; sine and tangent of a sum follow from it. From those come double angle formulas, power reducing and half angle formulas, product to sum and sum to product formulas and combining a sine and a cosine into one wave. **Using them.** Solving trigonometric equations finds every angle that works. ## A path through it What an identity is, then quotient identities, then the Pythagorean identities, which open simplifying and verifying. Cofunction identities need only the first lesson. Cosine of a sum needs the Pythagorean identities; sine of a sum needs cosine of a sum and the cofunctions. Product to sum needs the cosine formula; double angle and one wave need the sine formula; power reducing needs double angle. Trig equations come after the Pythagorean and double angle formulas. ### [Trigonometric inequalities](https://math.wiki/trig-inequalities/) **Summary:** Finding the stretches of angles where a trig function stays above or below a given value. **Question:** An equation such as sin x = 0.6 picks out single angles. Where does the sine stay above 0.6, and how do you write down every one of those angles at once? ## Why this exists Solving a trig equation finds the angles where a function hits a value exactly. Many questions ask something wider: for which angles does the quantity stay above a level, or below it? How long a Ferris wheel rider stays above a given height, or when a tide leaves a harbor deep enough, are questions of this kind, and those lessons answer them in context. This lesson sets out the method behind them. The answer to such a question is not a list of angles but whole stretches of them, written in interval-notation. The same task for polynomials is done in polynomial and rational inequalities: find where the expression equals zero or is undefined, cut the line there, and test each piece. Trig inequalities work the same way, with one new feature: the cut points repeat with the period, so the stretches repeat too. ## Definition A **trigonometric inequality** is an inequality whose unknown appears inside a trig function, such as $2\cos x + 1 < 0$ or $\tan x \ge 1$. Its **solution set** is the set of every real $x$ that makes it true, either inside a stated interval such as $[0, 2\pi)$ or on the whole line. Angles are in radians unless degrees are named. The equation made by putting $=$ in place of the inequality sign is the **matching equation**. Its solutions are the **crossings**, the angles where the graph meets the level. A strict inequality ($<$ or $>$) leaves the crossings out, so a stretch ends there with an **open end**, a parenthesis. A non-strict one ($\le$ or $\ge$) keeps them, a **closed end**, a square bracket. A point where the function is undefined, such as an asymptote of tangent, is never a solution, so an end there is always open. ### [Trigonometry](https://math.wiki/trigonometry/) **Summary:** Angles and triangles, then the waves they turn into. **Question:** Trigonometry starts with the sides of a right triangle and ends with waves, vectors and polar curves. How does one idea stretch that far? ## What this area is about Trigonometry links angles to lengths. In a right triangle, the ratio of two sides depends only on an acute angle. Placing the angle on a circle of radius $1$ extends sine and cosine to every real input, so they become periodic functions whose graphs are waves. From there come identities, triangles with no right angle, and new ways to describe direction and position. ## How these fit together **Measuring angles.** Angles and their measure lets an angle turn any amount in either direction and introduces radians. **Definitions of the functions.** Right triangle trigonometry defines the six ratios for an acute angle; the unit circle extends them to angles of any size and sign, from the coordinates of a point on the circle. **Results and their uses.** The laws of sines and cosines solve any triangle. Trigonometric functions and their graphs draw, transform and invert the waves. Trigonometric identities are equations true at every input where both sides are defined, and solve trig equations. **Applications.** Polar coordinates and vectors describe direction and distance: vectors, parametric curves, polar graphs and the polar form of a complex number. ## A path through it Angles and their measure, and right triangle trigonometry, first; then the unit circle. After it, the laws of sines and cosines, and trig functions; trig identities follow trig functions. Polar coordinates and vectors come last, because they draw on all six. ### [Unit circle symmetry and even-odd identities](https://math.wiki/unit-circle-symmetry/) **Summary:** Reflecting a terminal point across an axis or the origin changes only the signs of its coordinates. **Question:** Once you know where one angle lands on the unit circle, what do you already know about the angles that land on its mirror images? ## Why this exists In cosine-and-sine-as-coordinates, every angle $t$ in standard position got a terminal point $(\cos t, \sin t)$ on the unit circle $x^2 + y^2 = 1$. That circle is an unusually symmetric figure. As in test-symmetry-algebraically, replacing $x$ by $-x$, $y$ by $-y$, or both, leaves its equation unchanged, so the circle is its own mirror image across the $x$-axis, across the $y$-axis, and through the origin (line-symmetry). So each mirror image of a terminal point is again a point of the circle, and therefore the terminal point of some other angle. Knowing one point gives three more for free. Reading their coordinates turns the cosine or sine of a negative angle, or of an angle on the left half of the circle, into the cosine or sine of $t$ itself, with at most a change of sign. ## Definition Let $t$ be an angle in standard position with terminal point $P = (x, y) = (\cos t, \sin t)$. The three **mirror images** of $P$ are: | Move | Image of $(x, y)$ | Terminal point of | In degrees | |---|:-:|:-:|:-:| | reflect across the $x$-axis | $(x, -y)$ | $-t$ | $-\theta$ | | reflect across the $y$-axis | $(-x, y)$ | $\pi - t$ | $180^\circ - \theta$ | | half turn about the origin | $(-x, -y)$ | $\pi + t$ | $180^\circ + \theta$ | A function $f$ is **even** when $f(-t) = f(t)$ for every $t$ in its domain and **odd** when $f(-t) = -f(t)$, exactly as in test-symmetry-algebraically. The **even and odd identities** below say which of the six trig functions are which. ### [Unit vectors and i, j notation](https://math.wiki/i-and-j-notation/) **Summary:** Expressing any plane vector as a multiple of the horizontal unit vector plus a multiple of the vertical one. **Question:** A direction alone has no length. How do you write down "this way" as a vector, and then build a vector of any length you like pointing that way? ## Why this exists A vector carries two pieces of information, a length and a direction, and magnitude and direction showed how to read each one off the components $\langle a, b \rangle$. Often the two pieces arrive separately. Ghada, Jamila and Karim pilot tugboats, and a tug's compass says which way the bow points while its log says how far it went. The heading has no length of its own. To treat "this way" as a vector at all, you fix its length at the one value that changes nothing when you multiply by it: $1$. A vector of length $1$ is a pure direction. Every other vector in that direction is a length times it, and the two unit vectors along the axes, $\mathbf{i}$ and $\mathbf{j}$, give a second way to write every vector: so much of "one right" plus so much of "one up". ## Definition A **unit vector** is a vector with magnitude $1$. For a nonzero vector $\mathbf{v} = \langle a, b \rangle$, the **unit vector in the direction of $\mathbf{v}$** is $\mathbf{v}$ with each component divided by its magnitude: $$\frac{\mathbf{v}}{\|\mathbf{v}\|} = \left\langle \frac{a}{\|\mathbf{v}\|},\ \frac{b}{\|\mathbf{v}\|} \right\rangle$$ The two **standard unit vectors** point along the positive axes: $$\mathbf{i} = \langle 1, 0 \rangle, \qquad \mathbf{j} = \langle 0, 1 \rangle$$ **$\mathbf{i}$, $\mathbf{j}$ notation.** The vector $\langle a, b \rangle$ is also written $$a\mathbf{i} + b\mathbf{j}$$ read "$a$ of $\mathbf{i}$ plus $b$ of $\mathbf{j}$": $a$ steps one to the right and $b$ steps one up. For now, $a\mathbf{i} + b\mathbf{j}$ is simply another name for $\langle a, b \rangle$. Multiplying a vector by a number is a scalar multiple and adding two vectors is vector addition; with those in hand, $a\mathbf{i}$ really is $\langle a, 0 \rangle$, $b\mathbf{j}$ really is $\langle 0, b \rangle$, and their sum really is $\langle a, b \rangle$. A term with a zero coefficient is left out, so $\langle 4, 0 \rangle = 4\mathbf{i}$ and $\langle 0, -1 \rangle = -\mathbf{j}$. ### [Vector addition and scalar multiplication](https://math.wiki/vector-operations/) **Summary:** Tip to tail, and scaling. **Question:** One trip follows another, or a trip is taken twice, halfway, or backward. What vector describes the result? ## What this area is about Vectors can be combined. Two of them add tip to tail, and a vector can be multiplied by a real number, a scalar, to stretch it, shrink it, or reverse it. Both operations have a picture with arrows and a calculation on components, and the two always agree: for $\mathbf{u} = \langle a, b \rangle$ and $\mathbf{v} = \langle c, d \rangle$, $\mathbf{u} + \mathbf{v} = \langle a + c, b + d \rangle$ and $k\mathbf{u} = \langle ka, kb \rangle$. ## How these fit together **Two definitions.** Adding and subtracting vectors defines the sum by the triangle rule and the difference as the arrow that joins two heads. Scaling a vector defines $k\mathbf{v}$: the length is multiplied by $|k|$, and the direction reverses when $k < 0$. Each lesson refers to the other (the difference uses the opposite $-\mathbf{v}$), so they are best taken side by side. **An application.** Forces and velocities as vectors adds velocities, such as a plane's air velocity and the wind, and adds several forces acting on one object. ## A path through it Addition and scaling together, then forces and velocities, which need the triangle rule. After this come the dot product, adding and scaling vectors in n dimensions, and motion in the plane. ### [Vector and scalar projection](https://math.wiki/scalar-projection/) **Summary:** The signed length of that shadow, with no direction attached. **Question:** A force pulls at a slant, but the object can only move along one line. How much of the pull actually points along that line, and how much work does it do? ## Why this exists A rope pulls a raft at a slant to the riverbank, but the raft can only travel along the river. A person pushes a loaded dolly partly sideways, yet the dolly rolls in a straight line. In both cases only part of the force goes along the motion, and the question is how big that part is. The dot product already holds the answer. It equals $\|\mathbf{u}\|\,\|\mathbf{v}\|\cos\theta$, and $\|\mathbf{u}\|\cos\theta$ is the length of the side next to $\theta$ in a right triangle with hypotenuse $\|\mathbf{u}\|$. Divide the dot product by one length instead of two and that side appears: the part of $\mathbf{u}$ that runs along $\mathbf{v}$. This lesson names that number, turns it into a vector, uses it to split any vector into two perpendicular pieces, and uses it to measure the work a force does. ## Definition Let $\mathbf{u}$ be any vector and $\mathbf{v}$ a nonzero vector, with $\theta$ the angle between them when $\mathbf{u}$ is nonzero. The **scalar projection** of $\mathbf{u}$ onto $\mathbf{v}$, also called the **component of $\mathbf{u}$ in the direction of $\mathbf{v}$**, is the number $$\operatorname{comp}_{\mathbf{v}}\mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|}.$$ It is the signed length of the shadow of $\mathbf{u}$ on the line of $\mathbf{v}$: positive when the shadow points the same way as $\mathbf{v}$, negative when it points the opposite way, zero when $\mathbf{u}$ is perpendicular to $\mathbf{v}$. ### [Vectors](https://math.wiki/vectors/) **Summary:** An arrow with length and direction; the same arrow anywhere is the same vector. **Question:** A segment says how far apart two points are. How do you also say which way you went, and turn that into numbers you can compute with? ## What this area is about A vector carries a length and a direction. Every arrow that makes the same trip is the same vector, wherever it is drawn. This area builds that idea step by step: an arrow, then a pair of numbers $\langle a, b \rangle$, then the length and direction read back off those numbers, then a second way of writing the same vector. ## How these fit together **Definitions.** Directed line segments put an arrow on a segment, so it has a start, a finish and a direction. Component form names a vector by its change across and up, $\langle x_2 - x_1, y_2 - y_1 \rangle$, terminal minus initial. Unit vectors and i, j notation writes $\langle a, b \rangle$ as $a\mathbf{i} + b\mathbf{j}$ and builds the unit vector $\frac{\mathbf{v}}{\|\mathbf{v}\|}$ for $\mathbf{v} \ne \mathbf{0}$. **A procedure.** Magnitude and direction finds $\|\mathbf{v}\| = \sqrt{a^2 + b^2}$ and the direction angle of a nonzero vector, and goes back from length and angle to components. ## A path through it Directed segments, then component form, then magnitude and direction, then unit vectors: each needs the one before. This opens vector addition and scalar multiplication, and with them the dot product, converting between polar and rectangular coordinates and vectors in space. ### [Vectors in space](https://math.wiki/three-dimensional-vectors/) **Summary:** Extending components, magnitude and the dot product to three coordinates. **Question:** An arrow in the plane needs two numbers. A cable running from a lamp up to a ceiling hook needs three. Do length, adding, scaling and the angle test all carry over when the third number arrives? ## Why this exists Every vector so far has lived on a flat page: a run and a rise, $\langle a, b \rangle$. Real directions are rarely that flat. A cable from a lamp to a ceiling hook goes across, along and up at once, and a force pulling along that cable has a part in each of the three directions. Two numbers cannot describe it. The fix costs one more number. A vector in space is written $\langle a, b, c \rangle$, and every tool of the plane survives the change: components from two points, length from the Pythagorean theorem, adding and scaling part by part, the dot product and the angle it gives, and the test for a right angle. This lesson recalls each 2D idea in one line and then does the 3D version. ## Definition **Axes.** Space gets three perpendicular number lines through an origin $O$, as in three-dimensional-coordinates. They are set up as a **right-hand system**: curl the fingers of your right hand from the positive $x$-axis toward the positive $y$-axis, and your thumb points along the positive $z$-axis. In the figures here the $y$-axis runs right, the $z$-axis runs up, and the $x$-axis comes toward you, drawn slanting down and to the left. A point $(a, b, c)$ is plotted by drawing the box with one corner at $O$ and the opposite corner at the point: $a$ along $x$, $b$ along $y$, $c$ up. **Components.** In the plane, the component form lists the change in $x$ and the change in $y$ from tail to head. In space a vector $\mathbf{v}$ lists three changes: ### [Verifying identities](https://math.wiki/verify-by-working-one-side/) **Summary:** Changing only one side of a proposed identity until it matches the other side. **Question:** Someone claims that $\frac{\sec x + \csc x}{\tan x + \cot x}$ and $\sin x + \cos x$ are the same function. Checking a few angles cannot settle it. What does an argument look like that covers every angle at once, and how do you show that a claim like this is false? ## When you reach for it An identity claims that two expressions agree at every input where both are defined, the inputs called the **common domain** in what-an-identity-is. To **verify** an identity is to prove that claim. A few matching values prove nothing, because checks cover only the inputs checked. The proof has to be a chain of equalities in which every link is an identity you already trust, and then it covers every input of the common domain at once. Reach for this when an identity is stated and you are asked to show it holds, when a formula in a later lesson is derived from earlier ones, and when you want to know whether a rewrite you found while simplifying really is safe. Reach for its partner, the counterexample, when you suspect the claim is false: one input where both sides exist and disagree settles it. The tools are the ones already in hand. The reciprocal and quotient identities turn every function into sine and cosine, the Pythagorean identities trade a square for its partner, and the algebra of rational-expressions combines and cancels fractions, with $\sin x$ and $\cos x$ in the place of the variable. Throughout, $k$ stands for any integer. ## The procedure The lesson runs over three days. Day 1 sets the rule that makes a proof a proof and does the most common move, rewriting in sine and cosine. Day 2 adds fractions and conjugates. Day 3 turns to claims that are false, and to choosing a strategy before starting. ### [What an identity is](https://math.wiki/what-an-identity-is/) **Summary:** An equation true for every value where both sides are defined, rather than for a few solutions. **Question:** The equals sign in $2\\cos t = 1$ asks which angles work. The equals sign in $\\cos(-t) = \\cos t$ claims something about every angle at once. How do you tell the two apart, and what does \"every angle\" mean when one side has no value at some angles? ## Why this exists The unit circle lessons used one equals sign for two different kinds of statement. One kind asks a question: $2\cos t = 1$ holds at $t = \frac{\pi}{3}$, at $t = -\frac{\pi}{3}$ and at every angle that differs from those by full turns in either direction, and fails everywhere else, so the work is to find those angles. The other kind states a fact: $\cos(-t) = \cos t$ and $\sin(t + 2\pi) = \sin t$ hold for every angle, so there is nothing to solve. Instead, either side may be swapped for the other wherever it appears. This unit is built on the second kind. Simplifying, verifying and solving all work by replacing one side of a known identity with the other, which is safe only if the statement really holds for every input. The idea is not new. A linear equation whose variable cancels into a true statement holds for every number, and polynomial identities are proved by expanding. Trigonometry adds one feature: tangent, secant, cosecant and cotangent have no value at some angles, so the two sides of a statement need not exist at the same inputs. ## Definition An equation in a variable claims that two expressions have the same value. Only a value at which both sides are defined can be tested, and the set of those values is the equation's **common domain**. - An equation is an **identity** if it is true for every value in its common domain. - An equation is a **conditional equation** if it is true for some values in its common domain and false for others. The values where it is true are its **solutions**. - An equation true for no value at all has **no solution**. ### [Writing an equation from a wave](https://math.wiki/write-a-sinusoid-from-a-graph/) **Summary:** Reading the amplitude, midline, period and shift off a picture and assembling the matching formula. **Question:** A wave on a screen rises to a peak, falls to a trough and repeats. Which formula draws exactly that wave, and how much of the picture do you need to find it? ## When you reach for it The four earlier lessons read a wave off its formula: the amplitude from $A$, the period from $b$, the phase shift from $c$ and the midline from $k$ in $$\begin{aligned} y &= A\sin\bigl(b(x - c)\bigr) + k \quad\text{or} \\ y &= A\cos\bigl(b(x - c)\bigr) + k \end{aligned}$$ This lesson runs the reading backward. You have a picture of a wave, or a sentence about one, or a table of its values, and you want a formula that produces it. You need one whenever a wave has to be computed rather than looked at: its height at a time the picture does not show, or a formula to hand to a calculator. The lesson runs over two days. Part 1 (Examples 1 to 3) works from a graph. Part 2 (Examples 4 to 7) works from a description or a table and settles which of sine or cosine to use. ## The procedure ### Part 1: from a graph **Midline and amplitude.** Find the highest value $M$ and the lowest value $m$. The wave swings the same distance above and below its midline, so the midline sits halfway between and the amplitude is half the gap: $$k = \frac{M + m}{2}, \qquad |A| = \frac{M - m}{2}$$ **Period.** The period $P$ is the horizontal distance from one peak to the next. When only a peak and the next trough are visible, that distance is half a period, and a peak to the next midline crossing is a quarter period. Then, taking $b$ positive, $$b = \frac{2\pi}{P}$$ because a cycle of sine or cosine is $2\pi$ wide and $b$ squeezes it to $P$.