Operations with rational numbers

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 \displaystyle -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 on the number line), 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 \displaystyle \frac{5}{4} to the left is moving one and a quarter units to the left.

Subtracting is still adding the opposite, for every pair of rational numbers p and q:

The reason is the one from Subtracting integers. The difference p - q is the number that has to be added to q to reach p, and p + (-q) is that number:

The first step is the associative property of addition, and the second uses the fact that a number and its opposite add to zero (Opposites and zero pairs). In the second number line above, \displaystyle -\frac{1}{4} - \left(-\frac{3}{2}\right) became \displaystyle -\frac{1}{4} + \frac{3}{2}, a move of \displaystyle \frac{3}{2} to the right.

Sign and size from absolute values. The integer rules carry over word for word, with |p|, the absolute value of p, as its distance from zero.

Where the minus sign of a fraction goes. For integers a and b with b \neq 0,

A fraction is a quotient, and a quotient of two numbers with different signs is negative, with the same size as if both were positive (Multiplying and dividing integers). So all three are the same number. For adding, use the middle form: keep the denominator positive and let the sign ride in the numerator, where integer addition knows what to do with it.

Adding over a common denominator. For integers a and b and a positive integer c,

This holds with negative numerators too. By the distributive property, \displaystyle \frac{a}{c} + \frac{b}{c} = a \cdot \frac{1}{c} + b \cdot \frac{1}{c} = (a + b) \cdot \frac{1}{c}. In words: \displaystyle \frac{a}{c} is a count of pieces of size \displaystyle \frac{1}{c}, a negative count meaning pieces of size \displaystyle -\frac{1}{c}, and counts of the same piece combine by integer addition.

So two signed fractions with different denominators are renamed first, exactly as in Adding and subtracting fractions: over a common denominator, usually the least common multiple of the two (Greatest common factor and least common multiple). The signs go into the numerators, the numerators are added as integers, and the result is simplified. For example,

The table agrees: \displaystyle \frac{5}{6} is farther from zero than \displaystyle \frac{3}{4}, so the sum is negative, and its size is \displaystyle \frac{10}{12} - \frac{9}{12} = \frac{1}{12}.

Negative mixed numbers. A mixed number is a sum: \displaystyle 2\frac{1}{3} means \displaystyle 2 + \frac{1}{3}. Then \displaystyle -2\frac{1}{3} is the opposite of that whole sum, and the minus sign belongs to both parts:

The opposite of a sum is the sum of the opposites, because \displaystyle (-2) + \left(-\frac{1}{3}\right) added to \displaystyle 2 + \frac{1}{3} gives 0 once the commutative and associative properties pair each part with its opposite. On the number line, \displaystyle -2\frac{1}{3} is two units left of zero and then a third of a unit further left.

That gives two ways to add or subtract mixed numbers. Turning each into a fraction always works. Adding the whole parts and the fraction parts separately also works, because a mixed number is a sum and the commutative and associative properties let the parts be regrouped, provided each part keeps the sign of its mixed number.

Part 2: decimals, mixed forms, distance and change

A terminating decimal is a fraction whose denominator is a power of ten. -2.85 is \displaystyle -\frac{285}{100}. Lining up the decimal points is the same as writing both numbers over the same power of ten, so it is the common denominator step done in place-value columns. Everything in Part 1 therefore applies: decide the sign from the absolute values, then add or subtract the absolute values with the decimal points lined up, writing zeros on the right where a number has fewer decimal places.

Fractions and decimals together. Put both numbers into one form first. A terminating decimal always becomes an exact fraction. A fraction becomes an exact, terminating decimal only when its denominator in lowest terms has no prime factors other than twos and fives (Which fractions give terminating decimals). \displaystyle \frac{3}{8} = 0.375 exactly, but \displaystyle \frac{1}{3} = 0.333\ldots never ends (Terminating and repeating decimals), and any rounded version of it makes the answer approximate. When a fraction like that is involved, work in fractions.

Rearranging a long sum. Addition of rational numbers keeps its commutative and associative properties: the order of the numbers may change, and so may which two are added first. Subtraction keeps neither: \displaystyle \frac{1}{2} - \frac{1}{4} = \frac{1}{4} but \displaystyle \frac{1}{4} - \frac{1}{2} = -\frac{1}{4}. So in a string of additions and subtractions, first rewrite every subtraction as adding the opposite. Then each number carries its own sign, the string is all additions, and the numbers may be reordered and grouped freely. Good groups are opposites (they add to 0), fractions with the same denominator, and decimals with decimals.

Distance and change. The distance between two rational numbers p and q on the number line is

as for integers (Distance on a number line). The order does not matter, since p - q and q - p are opposites and have the same absolute value. A change is the end value minus the start value, and unlike a distance it keeps its sign: a negative change means the quantity went down.

Worked examples

Examples 1 to 4 go with Part 1. Examples 5 to 9 go with Part 2.

Boundaries

A common denominator is needed to add or subtract, and no denominator may be zero. Adding numerators only makes sense when they count pieces of the same size. Multiplying fractions does not need one; that is another lesson.

The sign belongs in the numerator before adding. A denominator such as the -3 in \displaystyle \frac{2}{-3} is allowed, but the rule \displaystyle \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} is used here with a positive c, so move the sign up first: \displaystyle \frac{2}{-3} = \frac{-2}{3}.

The minus sign of a mixed number covers both parts. \displaystyle -2\frac{1}{3} = -2 - \frac{1}{3}, which is \displaystyle -\frac{7}{3}. The number \displaystyle -2 + \frac{1}{3} = -\frac{5}{3} is a different number. Splitting a mixed number into parts is allowed only if each part keeps the sign.

Changing form can lose exactness. Every terminating decimal is an exact fraction, but a fraction is an exact terminating decimal only when its denominator in lowest terms has no prime factor other than 2 and 5. Otherwise a decimal version is rounded and so is anything computed from it.

Only addition may be reordered. The commutative and associative properties are properties of addition. A string with subtractions must be rewritten as a sum of signed numbers before its terms are moved or regrouped.

A distance is never negative; a change can be. |p - q| is the same in either order and is 0 only when p = q. A change, end minus start, depends on the order and its sign says which way the quantity went.

Where this goes next

The same two moves, a common denominator and the sign rules, carry the rest of arithmetic with rational numbers. Multiplying and dividing them uses the sign rules without any common denominator. Rational numbers in real situations takes the changes and distances of Part 2 into balances, elevations and temperatures at length.

In algebra the numbers in an expression stop being whole almost at once. Combining like terms such as \displaystyle \frac{2}{3}x - \frac{5}{6}x is Example 2 with an x attached, and solving equations with negatives uses Example 4 at every step. Later, adding rational expressions such as \displaystyle \frac{1}{x} - \frac{2}{x+1} is the same procedure again, with polynomials in place of integers and a least common multiple of polynomials as the common denominator.

Exercises

Part 1: signed fractions

1. \displaystyle -\frac{5}{9} + \frac{2}{9}
2. \displaystyle -\frac{3}{4} + \left(-\frac{7}{10}\right)
3. \displaystyle \frac{7}{10} + \left(-\frac{4}{15}\right)
4. \displaystyle \frac{3}{8} - \frac{5}{6}
5. \displaystyle -\frac{1}{2} - \left(-\frac{5}{7}\right)
6. \displaystyle -\frac{4}{9} - \frac{1}{6}
7. \displaystyle \frac{2}{-3} + \frac{5}{6}

Part 1: mixed numbers

8. \displaystyle -3\frac{1}{2} + 1\frac{2}{3}
9. \displaystyle 2\frac{1}{4} - 5\frac{3}{8}
10. \displaystyle -1\frac{5}{6} - \left(-4\frac{1}{3}\right)
11. \displaystyle -2\frac{2}{5} - 1\frac{3}{4}

Part 1: missing numbers and moves

12. What number must be added to \displaystyle \frac{2}{3} to give \displaystyle -\frac{1}{2}?
13. Start at \displaystyle -\frac{1}{4} on a number line, move \displaystyle \frac{5}{8} to the left, then move \displaystyle \frac{3}{2} to the right. Where do you finish?

Part 2: signed decimals

14. -6.3 + 2.75
15. -1.08 - 4.9
16. 3.4 - 7.15
17. -12.5 - (-8.76)

Part 2: fractions and decimals together

18. \displaystyle 0.55 + \left(-\frac{7}{8}\right)
19. \displaystyle -\frac{5}{6} + 1.25
20. Find \displaystyle \frac{1}{3} + 0.25 exactly. Then find how far 0.33 + 0.25 is from the exact answer.

Part 2: rearranging

21. \displaystyle -4.35 + \frac{5}{6} + 4.35 + \left(-\frac{1}{3}\right)
22. \displaystyle \frac{3}{8} - 2.8 - 1\frac{3}{8} + 0.8

Part 2: distance and change

23. Find the distance between -3.2 and \displaystyle 1\frac{3}{4}.
24. Find the distance between \displaystyle -\frac{5}{6} and \displaystyle -2\frac{1}{4}.
25. A bank account is overdrawn, with a balance of -23.40 dollars. A deposit of 50 dollars is paid in, then a payment of 31.75 dollars goes out. What is the balance now?
26. The temperature was 1.8 degrees Celsius at dusk and -6.45 degrees Celsius at dawn. Find the change from dusk to dawn.

Reasoning

27. Find the value of \displaystyle -2\frac{1}{3} and of \displaystyle -2 + \frac{1}{3} as fractions, and explain from the meaning of a mixed number why they differ.
28. Show that for any rational numbers p and q, q - p is the opposite of p - q. Explain why this makes the distance |p - q| the same whichever number is written first.
29. Is the sum of two negative rational numbers always negative? Is the difference of two negative rational numbers always negative? Give a reason for each, or an example that shows it fails.
30. Explain why \displaystyle \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} still holds when a or b is a negative integer and c is a positive integer.
31. Show with examples that subtraction of rational numbers is neither commutative nor associative. Then explain why rewriting each subtraction as adding the opposite lets you rearrange the terms.