What a letter stands for
A letter sits where a number should be. Is it one number you have not found yet, or every number at once?
Where you are 5 in Variables
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 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.
A quantity that changes. The letter names an amount that can take different values in different cases or at different times: a price, a count, a length of time. Suppose a cinema ticket costs 9 dollars. If t is the number of tickets bought, the cost in dollars is 9t. On one visit t might be 2, on another 5. The letter is not hiding one number. It stands for whichever value the quantity has in the case you are looking at.
A placeholder in a general rule. The letter marks a slot in a statement that is meant to hold for every number of some kind. The sentence
says that adding zero to a number leaves it unchanged, and it says so about every number at once: 3 + 0 = 3, 17 + 0 = 17, \displaystyle \frac{1}{2} + 0 = \frac{1}{2}, and so on. A formula such as A = lw for the area of a rectangle, and a pattern rule such as “stage n uses 3n + 1 sticks”, work the same way. Each is one sentence standing for infinitely many facts.
Some books keep the word variable for the second and third jobs and call a letter doing the first job an unknown. Both words are in common use. What matters is the job.
Whatever the job, the numbers a letter may stand for come from some collection, stated or understood from what the letter measures. A number of tickets is a whole number; a length is a positive number; in a + 0 = a the letter may be any number at all.
A constant is a number whose value stays the same throughout the problem, such as the 5 in {n + 5} or the 9 in 9t.
Any letter will do. x, y, n, a, b and c are common, and a letter that reminds you of the quantity, t for tickets or l for length, is often easier to read. The letter carries no meaning of its own: {n + 5 = 12} and {k + 5 = 12} say exactly the same thing.
Multiplication beside a letter. The sign \times looks like the letter x, so algebra writes multiplication another way. A number written against a letter means multiply: 9t is 9 times t, and lw is l times w. Between two numbers a raised dot or brackets are needed, since 34 is thirty-four: 3 \cdot 4 and 3(4) are 3 times 4.
Intuition
Think of a letter as a label on a box that holds a number.
For an unknown, one number is sealed inside and the problem gives clues to which. For a quantity that changes, the contents are swapped from one case to the next, and the label tells you what the contents measure. For a placeholder, the box sits inside a rule, and the rule promises to come out true whatever allowed number you put in.
Two boxes with the same label are the same box. Wherever n appears in one problem, it is that one box, holding that one number. Boxes with different labels are simply different boxes, and nothing stops them from holding the same number.
A pattern shows the placeholder job most clearly. In Figure 1, each new square in the row needs three more sticks.
The rule “stage n uses 3n + 1 sticks” does not describe stage 1 or stage 2 in particular. The n is a slot for the stage number, and whichever stage you choose, the same n fills both places: the stage number and the number in 3n + 1.
Results
One letter, one number. Within one problem, every appearance of the same letter stands for the same number.
So in x + x = 10 the two x’s are one number added to itself, which makes x = 5. They cannot be 3 and 7, even though 3 + 7 = 10.
Different letters promise nothing. Within one problem, two different letters may stand for different numbers or for the same number, unless the problem says which.
Renaming changes nothing. Replacing a letter everywhere it appears in a problem by a new letter not already used in that problem gives a problem that says exactly the same thing.
Every number, some, or none. A sentence that makes a claim about a letter, such as an equation, is true for every number the letter may stand for, or true for some of them only, or true for none of them. If it is true for every allowed number, the sentence is a general rule and the letter can be any of them. If it is true for some only, the letter’s number must be one of those. If it is true for none, no number fits. Which case it is depends on the numbers allowed: n + n = 5 is true only for \displaystyle n = 2\frac{1}{2}, but if n counts people it is true for none.
Why it is true
The first result is not a theorem proved from something earlier. It is the agreement that makes a letter useful at all, and nothing else would work.
A letter is a name, and a name picks out one thing each time it is used. If the x in x + x = 10 could be 3 in one place and 7 in the other, it could just as well be 1 and 9, or 4 and 6, and the sentence would no longer say anything about x. It would say only that two numbers add to 10. The way to say that is with two letters, x + y = 10, and that is why the second result holds: two different letters make no promise either way, so x = 4, y = 6 fits and so does x = 5, y = 5.
Renaming changes nothing for the same reason. What a problem says lies in which places must hold the same number and what is claimed about that number. Changing n to k everywhere keeps both. The new letter has to be one not already in use: renaming the y in x + y = 10 as x would give x + x = 10, which says something different, because two places that could hold different numbers now must hold the same one.
The last result follows from what a claim is. Put any allowed number in place of the letter and the sentence becomes either true or false. So either every allowed number makes it true, or some do and some do not, or none does. There is no fourth possibility.
A general rule claims something about every number in its collection, and there are infinitely many. Checking 3 + 0 = 3 and 17 + 0 = 17 checks two cases. That is evidence, not a reason. The reason a + 0 = a holds for every a is what adding zero means: adding nothing to an amount leaves it as it was. A rule with letters is only as good as the argument behind it, and a pattern that holds in the cases you tried is not yet a proof.
Worked examples
Which job is the letter doing?
Say whether the letter in each sentence is an unknown, a quantity that changes, or a placeholder in a general rule. If it is an unknown, say which number it stands for.
(a) y + 6 = 15 \qquad (b) b \cdot 1 = b
(c) A bus ride costs 2 dollars, so r rides cost 2r dollars.
Solution(a) The sentence makes a claim, and the claim is true for only one number: 9 + 6 = 15, while a number larger than 9 gives more than 15 and a smaller one gives less. So y is an unknown, and it stands for 9.
(b) Multiplying a number by 1 gives one lot of that number, which is the number itself. That holds whatever b is, so the sentence is a general rule and b is a placeholder for any number. This is the fact that multiplying by one changes nothing.
(c) Nothing is claimed here that could be true or false. An amount is described: 2r is the cost. The letter r is the number of rides, which could be 1 one day and 6 the next, so it is a quantity that changes. Since it counts rides, it stands for a whole number: 0, 1, 2, 3, \dots
The same letter, the same number
(a) In x + x + x = 12, what number is x? Could the three x’s be 2, 4 and 6, which also add to 12?
(b) In x + y + z = 12, could x, y and z be 2, 4 and 6? Could they all be 4?
Solution(a) All three x’s are the same number, so the sentence says that three copies of one number make 12. Three 4’s do, 4 + 4 + 4 = 12, while a larger number gives more and a smaller one less. So x = 4. The choice 2, 4, 6 does add to 12, but it gives one letter three different values, which one letter in one problem cannot have.
(b) Three different letters make no promise that their numbers differ or match. Since 2 + 4 + 6 = 12, the choice x = 2, y = 4, z = 6 fits. Since 4 + 4 + 4 = 12, so does x = 4, y = 4, z = 4: different letters may hold the same number. This one sentence does not fix any of the three letters; many choices fit it.
A letter for a quantity that changes
Two sisters share a birthday. Mia is 4 years older than Lena. Let L be Lena’s age in years, so that Mia’s age in years is L + 4.
What do L and L + 4 stand for? Which is the variable and which is the constant? Find Mia’s age when Lena is 3, 8 and 15. Does L stand for one number?
SolutionL stands for Lena’s age in years and L + 4 for Mia’s. Both ages grow year by year, so L is a variable; the difference between them is 4 in every year, so 4 is a constant.
| Lena’s age, L | Mia’s age, L + 4 |
|---|---|
| 3 | 7 |
| 8 | 12 |
| 15 | 19 |
At any one moment L is one number, Lena’s age then. As the years pass it takes value after value, and the single expression L + 4 gives Mia’s age at every one of them: one line in place of a table that never ends. Since we are counting whole years, L stands for a whole number.
A letter in a pattern
Figure 1 shows a pattern in which stage n uses 3n + 1 sticks.
(a) What does n stand for, and which numbers may it be?
(b) The three stages drawn have 4, 7 and 10 sticks, and the rule agrees with all three. Why does that not show that the rule holds for every stage, and what does?
Solution(a) n is the stage number, which is also the number of squares in that stage. The pattern starts at stage 1, so n may be 1, 2, 3, \dots and nothing else: there is no stage \displaystyle 2\frac{1}{2}. Both n’s in “stage n uses 3n + 1 sticks” are the same number, which is how the rule ties a stage to its count of sticks. With n = 2 in both places it says that stage 2 uses 3 \cdot 2 + 1 = 7 sticks, as drawn.
(b) Three stages are three cases, and they show nothing about stage 50, which nobody has drawn. The reason is in how the stages are built. Start with the one stick at the left end. Each square is then finished by three new sticks: its top, its bottom and its right side, since its left side is already there. Stage n has n squares, so it uses 1 stick plus n lots of 3, which is 3n + 1. Nothing in that argument depends on which stage it is, so it holds for every n at once.
A formula, and a letter that becomes an unknown
The area of a rectangle is given by the formula A = lw, with the letters as in Figure 2.
(a) What do the letters stand for, and which numbers may they be?
(b) A problem says that a rectangle is 7 meters long and has an area of 28 square meters. In this problem, which letters stand for known numbers, which letter is unknown, and what number does it stand for?
Solution(a) l is the length and w the width, measured in the same unit, and A is the area in square units of that unit. All three are positive numbers. The formula is a rule for every rectangle, so as it stands the letters are placeholders.
(b) Within this problem l stands for 7 and A for 28, everywhere they appear, so the formula says
Now w is an unknown: the one number that gives 28 when multiplied by 7. That number is 4, since 7 \cdot 4 = 28. The rectangle is 4 meters wide.
The formula did not change. The problem did. One formula holds placeholders when it is stated as a rule and an unknown once a problem fixes the other letters.
Every number, some, or none
Here n may be any number. For which numbers is each sentence true?
(a) n + n = 2n \qquad (b) n + n = n
(c) n + 1 = n \qquad (d) n \div n = 1
Solution(a) n + n is two lots of n, and 2n is two lots of n. The two sides are the same amount whatever n is, so the sentence is true for every number. It is a general rule.
(b) The sentence says that adding n to n leaves it unchanged. Adding a number leaves an amount unchanged only when the number added is 0. So the sentence is true only for n = 0. Here n is an unknown, and it stands for 0.
(c) Adding 1 to a number always gives a larger number, never the same one. The sentence is true for no number. No number fits.
(d) A number divided by itself is 1, since the number fits into itself exactly once: for example 6 \div 6 = 1. That holds for every number but one. At n = 0 the sentence reads 0 \div 0 = 1, and 0 \div 0 does not name a number: it would have to be the number that gives 0 when multiplied by 0, and every number does that, so there is no single answer. This is why division by zero is left undefined. So the sentence is true for every number except 0, and a careful statement of the rule says so:
The same letter did a different job in each sentence: a placeholder in (a), an unknown in (b), no number at all in (c), and a placeholder with one stated exception in (d).
Boundaries
One problem at a time. One letter, one number holds within a single problem. In the next problem x starts afresh and may stand for something else entirely. Two exercises on the same page each have their own x.
A formula used twice is used twice. If one problem involves two rectangles, A = lw applies to each, but l cannot be both lengths at once, so the second rectangle needs its own letters.
A rule covers only the numbers it is stated for. n \div n = 1 fails at 0, so the rule must leave 0 out, and it claims nothing about 0. A letter that counts people or tickets stands only for whole numbers, and a letter that measures a length stands only for positive numbers, even when nobody writes that down.
A letter stands for a number. In 9t, t is the number of tickets, not the tickets themselves. Saying what a letter stands for means naming a number, with its unit where it has one: “let t be the number of tickets”, “let L be Lena’s age in years”.
Not every letter is a variable. Some letters are names for one fixed number, such as \pi, the number that multiplies a circle’s diameter to give its circumference, the distance around it. Some are units: in 5\text{ m} the letter m means meters, and 5\text{ m} is 5 meters, not 5 times a number m. And in geometry a capital letter often names a point, not a number.
Where this goes next
Every lesson in algebra leans on the rule of one letter, one number.
Next come the parts of an expression, terms, coefficients and constants, and the difference between an expression and an equation, in expression, equation and inequality. Writing letters into sentences is translating words into expressions, and the reverse is reading an expression aloud.
Putting a number in place of a letter and working out the result is evaluating an expression, starting with substituting one value and going on to evaluating a formula. Finding the number an unknown stands for is solving an equation, from one-step equations onwards, and checking a solution is putting that number back in.
Letters for quantities that change grow into independent and dependent variables, and the question of every number, some or none returns in no solution, one solution, or every number.
Exercises
Which job is the letter doing?
For Exercises 1 to 5, say whether the letter is an unknown, a quantity that changes, or a placeholder in a general rule. If it is an unknown, say which number it stands for.
- m + 8 = 20
- 0 + c = c
- A bag of apples weighs 2 kilograms, so b bags weigh 2b kilograms.
- 12 - k = 5
- 6 \cdot d = d \cdot 6
One letter, one number
- In k + k = 18, what number is k? Could the two k’s be 8 and 10?
- Give two different choices of p and q that make p + q = 10 true, with p and q equal in one of them.
- In x + x + 4 = 20, could the two x’s be 6 and 10? Could they both be 8?
- One problem says that n = 3 and, later, that n + n = 8. Can both be true?
- One problem says that a + b = 9 and that a = 4. What number is b?
Letters in stories
- A train ticket costs t dollars, and a bus ticket costs 3 dollars less, so a bus ticket costs t - 3 dollars. Say what t and t - 3 stand for and which number is the constant. Then find the cost of a bus ticket when a train ticket costs 10, 15 and 24 dollars.
- What kind of number can each letter stand for? (a) s, the number of students on a bus. (b) h, the height of a tree in meters. (c) T, the temperature outside in degrees Celsius.
- A gym charges 5 dollars a visit. If v is the number of visits in a month, the cost in dollars is 5v. Could v be \displaystyle 3\frac{1}{2}? Could it be 0?
Rules that hold for every number
In Exercises 14 to 17 the letters may stand for any numbers.
- Check that a + b = b + a is true when a = 6 and b = 9. Does this check show that it is true for every pair of numbers?
- Find numbers a and b showing that {a \div b = b \div a} is not true for every pair of numbers.
- For which numbers is n \cdot 0 = 0 true? For which numbers is n \cdot 0 = n true?
- For which numbers is n + 2 = n + 5 true?
Formulas and patterns
- Triangles are made from sticks in a row, pointing up and down in turn, each new triangle sharing one side with the one before it. Stage n has n triangles, and the rule says it uses {2n + 1} sticks. What does n stand for, and which numbers may it be? Does the rule say anything about a stage 0?
- The perimeter of a square with side s is P = 4s. A square has a perimeter of 36 centimeters. Which letter is the unknown in this problem, and what number does it stand for?
Reasoning
- Explain why, within one problem, x cannot stand for 3 in one place and 5 in another.
- Explain the difference between the letter in n + 4 = 9 and the letter in n + 4 = 4 + n.
- Explain why the rule in Exercise 18 holds for every stage, not only for the few you could draw and count.
- Explain why the rule n \div n = 1 has to say “for every number n \neq 0”.
Answers
- An unknown, standing for 12, since 12 + 8 = 20, a larger number gives more than 20 and a smaller one gives less.
- A placeholder in a general rule. Adding a number to zero gives that number, whatever c is.
- A quantity that changes. b is the number of bags, which can differ from one purchase to the next, so it stands for a whole number.
- An unknown, standing for 7, since 12 - 7 = 5 and taking away any other number from 12 leaves something other than 5.
- A placeholder in a general rule. The order of a multiplication does not change the product, so the two sides agree whatever d is; this is the order property of multiplication.
- k = 9; no. Both k’s are the same number, so two copies of it make 18, and 9 + 9 = 18. The pair 8 and 10 would be two different numbers for one letter.
- For example p = 3, q = 7, and p = 5, q = 5. Any two numbers adding to 10 fit, since different letters may hold different numbers or the same one.
- Not 6 and 10: although 6 + 10 + 4 = 20, that gives one letter two numbers. Both 8, yes: 8 + 8 + 4 = 20, so x stands for 8.
- No. If n = 3 throughout the problem, then n + n = 3 + 3 = 6, not 8. One letter holds one number in one problem, so the two sentences cannot both be true.
- b = 5. The a in a + b = 9 is the same a, so 4 + b = 9, and 4 + 5 = 9.
- t is the cost of a train ticket in dollars and {t - 3} is the cost of a bus ticket in dollars; 3 is the constant. A bus ticket costs 7, 12 and 21 dollars.
- (a) A whole number, 0, 1, 2, \dots, since s counts. (b) A positive number, not necessarily whole. (c) A number that may be positive, zero or negative, and need not be whole: water freezes at 0 degrees Celsius, and on a cold day the outside temperature is below that.
- Not \displaystyle 3\frac{1}{2}, since v counts visits and so stands for a whole number. It can be 0, a month with no visits.
- 6 + 9 = 15 and 9 + 6 = 15, so it holds for that pair. No: one pair is one case out of infinitely many. The rule holds for every pair because putting two amounts together gives the same total in either order, not because of this check.
- For example a = 6 and b = 3: 6 \div 3 = 2 but \displaystyle 3 \div 6 = \frac{1}{2}. Any two different positive numbers will do.
- n \cdot 0 = 0 is true for every number: multiplying any number by 0 gives 0. n \cdot 0 = n is true only for n = 0: its left side is always 0, so it says 0 = n.
- No number. Adding 5 to a number always gives 3 more than adding 2 to it, so the two sides are never equal.
- n is the stage number, which is also the number of triangles, so it may be 1, 2, 3, \dots and nothing else. The pattern starts at stage 1, so the rule says nothing about a stage 0: it covers only the numbers it is stated for.
- s is the unknown, and it stands for 9: in this problem P stands for 36, so 36 = 4s, and 4 \cdot 9 = 36. The side is 9 centimeters.
- A letter is a name for one number throughout a problem. If x could be 3 in one place and 5 in another, it could equally be any two numbers, and a sentence such as x + x = 10 would say nothing about x at all. When two places may hold different numbers, the problem uses two different letters.
- n + 4 = 9 is true only when n = 5, so there n is an unknown standing for 5. n + 4 = 4 + n is true for every number, because the order of adding does not change a sum, so there n is a placeholder in a general rule.
- Start with the one stick at the left end. Each triangle is finished by two new sticks, because one of its sides is the last stick of the triangle before it. Stage n has n triangles, so it uses 1 stick plus n lots of 2, which is 2n + 1. Nothing in this depends on which stage it is, so it holds for every n.
- At n = 0 the sentence would read 0 \div 0 = 1, and 0 \div 0 is not a number: every number gives 0 when multiplied by 0, so no single number is the answer. Any other number fits into itself exactly once, so n \div n = 1 for every n \neq 0. The rule is true for every number except 0, and it must say so to be true as stated.