Multiplying and dividing integers
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?
Where you are 6 in Integers
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.
If either factor is 0, the product is 0.
In symbols, for counting numbers a and b:
Dividing two integers. For integers a and b with b \neq 0, a \div b is the number q with q \cdot b = a. Division undoes multiplication, so it follows the same sign rule: the quotient is positive when the signs are the same and negative when they are different.
There are several ways to write a product with a negative factor. (-3)(4), -3 \cdot 4, (-3) \times 4 and -3(4) all mean the same product. The brackets around a negative number keep its minus sign from being read as subtraction: 5(-2) is a product, while 5 - 2 is a difference.
Intuition
Read the first factor as an instruction, starting from zero on the number line. A positive first factor says “add this many copies of the second factor”. A negative first factor says “take away this many copies”. Figure 1 carries out all four sign combinations of 3 and 4.
The first two rows are ordinary repeated addition. The third row takes 4 away three times: 0 - 4 - 4 - 4 = -12. The fourth takes -4 away three times, and taking away -4 is the same as adding 4 (subtracting is adding the opposite):
So a negative first factor does whatever the positive one does, and then lands on the other side of zero. In particular, multiplying by -1 turns a number into its opposite: (-1)(7) = -7 and (-1)(-7) = 7.
A pattern shows the same thing without any picture. Multiply -3 by numbers that go down one at a time. Each step down takes one fewer copy of -3, so each product is 3 more than the one before. The pattern does not stop at zero, and the products keep growing once the second factor is negative.
| b | (-3) \cdot b |
|---|---|
| 3 | -9 |
| 2 | -6 |
| 1 | -3 |
| 0 | 0 |
| -1 | 3 |
| -2 | 6 |
| -3 | 9 |
Results
The sign of a product of two. Same signs give a positive product; different signs give a negative product. The size of the product is the product of the sizes: |ab| = |a| \cdot |b|.
Multiplying by negative one. For every integer a, (-1) \cdot a = -a, the opposite of a.
Many factors. In a product with no zero factor, the sign is decided by how many factors are negative. An even number of negative factors gives a positive product; an odd number gives a negative product. If any factor is 0, the product is 0.
Powers of a negative number. For counting numbers a and n, (-a)^n is positive when n is even and negative when n is odd. The brackets matter: -a^n means -(a^n), the opposite of the power, so -7^2 = -49 while (-7)^2 = 49.
The sign of a quotient. With a divisor that is not zero, same signs give a positive quotient and different signs a negative one. Dividing by -1 gives the opposite: a \div (-1) = -a.
Why it is true
Two facts about zero are used throughout. A number plus its opposite is 0, and any number times 0 is 0.
Multiplying by negative one gives the opposite. Take any integer a. By the distributive property,
So (-1)a is the number that adds to a to give 0. Only one number does that, the opposite of a. Therefore (-1)a = -a.
A positive times a negative. For counting numbers, 3 \cdot (-4) is three copies of -4, which is -12. The same comes out of the distributive property, which does not need repeated addition:
So 3 \cdot (-4) is the opposite of 3 \cdot 4 = 12, which is -12.
A negative times a positive. The commutative property says the order of factors does not matter, so (-3)(4) = 4 \cdot (-3) = -12.
A negative times a negative. Distribute -3 over 4 + (-4):
The first product is -12, so -12 + (-3)(-4) = 0. The only number that adds to -12 to give 0 is 12. Therefore (-3)(-4) = 12. Nothing about 3 and 4 was special: the same three lines, with any counting numbers a and b, give (-a)(-b) = ab.
This is the sense in which the rule is forced. If (-3)(-4) were anything other than 12, the distributive property would fail for integers, and every later step of algebra that expands brackets would be unsafe.
Many factors. Multiply from left to right. Each negative factor switches the sign of the running product and each positive factor leaves it alone. Starting from a positive sign, an even number of switches comes back to positive and an odd number ends on negative.
Quotients. Take -12 \div (-3). It is the number q with q \cdot (-3) = -12. If q were negative, q \cdot (-3) would be positive, and if q were 0 the product would be 0; so q is positive, and its size must be 12 \div 3 = 4. Therefore -12 \div (-3) = 4. The same reasoning gives every other case: a quotient has whatever sign makes “quotient times divisor equals dividend” true, and that is the product rule run backwards.
Worked examples
Examples 1 to 4 belong to Part 1, products. Examples 5 to 10 belong to Part 2, powers, quotients and mixed expressions.
Four products of two integers
Multiply: (a) -7 \cdot 6, (b) (-8)(-5), (c) 9(-4), (d) (-1)(-15).
Solution(a) The signs are different, so the product is negative. The sizes give 7 \cdot 6 = 42, so -7 \cdot 6 = -42.
(b) Both factors are negative. Same signs give a positive product, and 8 \cdot 5 = 40, so (-8)(-5) = 40.
(c) A positive times a negative: different signs, so 9(-4) = -36. As a check, nine copies of -4 are a debt of 36.
(d) Multiplying by -1 gives the opposite, and the opposite of -15 is 15. So (-1)(-15) = 15. The sign rule agrees: same signs, positive product.
A product of several factors
Find (a) (-2)(5)(-3)(-1) and (b) (-4)(-5)(3)(-2)(-1).
Solution(a) Multiply from left to right, using the rule for two factors at each step:
The quicker route counts the negative factors. There are three, an odd number, so the product is negative. The sizes multiply to 2 \cdot 5 \cdot 3 \cdot 1 = 30, so the product is -30.
(b) There are four negative factors, an even number, so the product is positive. The commutative and associative properties let us reorder and regroup the factors, so pair the negatives:
Each pair of negatives makes a positive, which is why an even count of them leaves the product positive. The product is 120.
A diver, forward and back in time
A diver sinks at a steady rate, so her height changes by -6 meters each minute. Find the change in her height as (number of minutes) times (change per minute), counting minutes into the future as positive and minutes into the past as negative.
(a) Compared with where she is now, where will she be 4 minutes from now?
(b) Compared with where she is now, where was she 3 minutes ago?
Solution(a) Four minutes from now is +4 minutes. The change is 4(-6) = -24, so she will be 24 meters lower than she is now.
(b) Three minutes ago is -3 minutes. The change is (-3)(-6) = 18, so she was 18 meters higher than she is now. This is what the sign rule says, and it is also plain sense: she has been sinking, so earlier she was higher up.
Check. Start 18 meters above her present height and let 3 minutes pass: 18 + 3(-6) = 18 + (-18) = 0. She arrives exactly where she is now.
Distributing a negative number
Find -6(7 - 10) in two ways: by working out the brackets first, and by the distributive property.
SolutionBrackets first: 7 - 10 = -3, and -6(-3) = 18, since the signs are the same.
By the distributive property, write the subtraction as adding the opposite, 7 - 10 = 7 + (-10), and multiply each part by -6:
The two ways agree, and they agree only because (-6)(-10) is positive. Had it been -60, the second way would give -42 + (-60) = -102, which disagrees with the first. This is the argument of “Why it is true” seen in numbers.
Powers of a negative number
Find (a) (-3)^4, (b) (-2)^5, (c) -3^4, (d) -(-2)^5.
SolutionAn exponent counts how many times the base is used as a factor, and the base is exactly what the exponent touches.
(a) The base is -3, so (-3)^4 = (-3)(-3)(-3)(-3). Four negative factors is an even number, so the result is positive, and 3^4 = 81. So (-3)^4 = 81.
(b) (-2)^5 has five factors of -2, an odd number, so it is negative: (-2)^5 = -32.
(c) Without brackets the exponent touches only the 3. So -3^4 means -(3^4) = -81, the opposite of 81.
(d) First (-2)^5 = -32 from part (b). Its opposite is -(-32) = 32.
Figure 2 shows why the signs alternate. Each further factor of -2 doubles the distance from zero and lands on the other side.
Four quotients
Divide, and check each answer by multiplying: (a) -72 \div 8, (b) -90 \div (-6), (c) 65 \div (-13), (d) 0 \div (-9).
Solution(a) The signs are different, so the quotient is negative, and 72 \div 8 = 9. So -72 \div 8 = -9. Check: (-9)(8) = -72.
(b) Same signs, so the quotient is positive: -90 \div (-6) = 15. Check: 15(-6) = -90.
(c) Different signs: 65 \div (-13) = -5. Check: (-5)(-13) = 65.
(d) The quotient is the number that gives 0 when multiplied by -9, and that number is 0. So 0 \div (-9) = 0. Check: 0 \cdot (-9) = 0. Zero has no sign, and the sign rule has nothing to decide here.
Finding a missing number
Find n: (a) (-7)n = 56, (b) n \div (-4) = -9.
Solution(a) Some number times -7 gives 56. Division undoes multiplication, so n = 56 \div (-7). The signs differ, so n = -8. Check: (-7)(-8) = 56.
(b) Some number divided by -4 gives -9. Multiplication undoes division, so n = (-9)(-4) = 36. Check: 36 \div (-4) = -9.
All four operations
Find (a) -4(-3)^2 - 18 \div (-6) and (b) \dfrac{(-6)(-10)}{-4 - 1}.
Solution(a) Follow the order of operations: the power first, then multiplying and dividing, then subtracting.
The power (-3)^2 is 9, positive, because its two factors are both negative.
(b) A fraction bar groups everything above it and everything below it, so work out the top and the bottom first. The top is (-6)(-10) = 60. The bottom is -4 - 1 = -5. Then 60 \div (-5) = -12.
Substituting negative values
Find the value of x^2 - 3xy when x = -2 and y = 5. Then find -x^2 when x = -2.
SolutionPut each value in its own brackets, so that every minus sign stays attached to its number:
Here 3(-2)(5) has one negative factor, so it is -30, and subtracting -30 is adding 30.
For -x^2, the exponent touches only x, and the minus sign is applied afterwards: -x^2 = -\bigl((-2)^2\bigr) = -4.
The sign before the size
Decide the sign of (-2)^7 (-5)^2 (-1)^{12} (-3) before working out its value. Then find the value.
SolutionWritten out in full, the product has 7 factors of -2, 2 factors of -5, 12 factors of -1 and one -3. That is 7 + 2 + 12 + 1 = 22 negative factors, an even number, so the product is positive.
The sizes are 2^7 = 128, 5^2 = 25, 1^{12} = 1 and 3, so the value is
Check. Work piece by piece: (-2)^7 = -128 and (-5)^2 = 25 give -3200; times (-1)^{12} = 1 is still -3200; times -3 is 9600.
Boundaries
Division by zero has no answer. -8 \div 0 would be a number q with q \cdot 0 = -8, but every number times 0 is 0, so there is no such q. And 0 \div 0 would be a q with q \cdot 0 = 0, which every number satisfies, so there is no single answer. Either way, dividing by 0 is left undefined. Dividing 0 by a number that is not zero is fine and gives 0.
A quotient of integers need not be an integer. -7 \div 2 has no integer answer, since no integer times 2 is -7. The answer is the fraction -\tfrac{7}{2}, and the same sign rule still gives its sign. Quotients like this belong to multiplying and dividing rational numbers.
The rule is for multiplying and dividing only. Sums and differences follow the rules of Adding and subtracting integers: -3 + (-4) = -7, a negative number, even though the two numbers have the same sign.
Counting negative factors decides the sign only when no factor is zero. A product with a zero factor is 0, which is neither positive nor negative.
The even and odd rule for powers needs the brackets. It describes (-a)^n. In -a^n the base is a, so the result is the opposite of a^n whatever n is.
Where this goes next
The same sign rules run unchanged through fractions and decimals, in Multiplying and dividing rational numbers and alongside sums in Operations with rational numbers. Expressions that mix powers, negative numbers and the order of operations are taken further in Order of operations with powers.
The fact that a product of two numbers with the same sign is positive has a long reach. It says that the square of any number is 0 or positive, which later decides when an equation such as x^2 = -4 has no solution among the numbers on the line, and why distances computed from squares are never negative.
Exercises
Part 1: products of integers
Multiply.
- -6 \cdot 9
- (-7)(-8)
- 11(-5)
- (-1)(-23)
- (-3)(4)(-5)
- (-2)(-2)(-2)(-5)
- (-1)(6)(-1)(-7)(0)
Part 1: products that mean something
- The water in a tank drops 5 centimeters every hour. Counting hours into the future as positive, by how much does the level change in the next 7 hours?
- For the same tank, counting hours into the past as negative, how did the level 4 hours ago compare with the level now?
- Find -8(3 - 9) in two ways: brackets first, and by the distributive property.
- Find the value of -5x when x = -12.
Part 2: powers of a negative number
- (-4)^3
- (-1)^{10}
- -10^2 and (-10)^2
- (-3)^5
Part 2: quotients and missing numbers
- -84 \div 7
- -120 \div (-8)
- 96 \div (-12)
- -54 \div (-1)
- Find n when (-9)n = -63.
- Find n when n \div (-6) = 8.
Part 2: all four operations
- 5(-4) - 36 \div (-9)
- (-2)^4 - 3(-5)
- \dfrac{(-8)(-9)}{-2 - 4}
- Find the value of a^2 - 4ab when a = -3 and b = 2.
Reasoning
- A product of 9 integers, none of them zero, is positive. How many of the factors could be negative? Explain.
- Use the distributive property, and the fact that (-5)(2) = -10, to explain why (-5)(-2) = 10.
- Without working out the values, decide whether (-7)^{13} is positive or negative, and whether -7^{12} is positive or negative. Explain each.
- Use the link between multiplication and division to explain why -15 \div 0 has no value.
- Explain why the square of an integer is never negative.
Answers
- -54. Different signs give a negative product, and 6 \cdot 9 = 54.
- 56. Same signs give a positive product.
- -55. Different signs.
- 23. Multiplying by -1 gives the opposite of -23.
- 60. Two negative factors, an even number, so positive; 3 \cdot 4 \cdot 5 = 60.
- 40. Four negative factors, an even number, so positive; 2 \cdot 2 \cdot 2 \cdot 5 = 40.
- 0. One factor is 0, so the product is 0 whatever the other factors are.
- The change is 7(-5) = -35: the level ends 35 centimeters lower.
- The change is (-4)(-5) = 20: four hours ago the level was 20 centimeters higher than now.
- 48 both ways. Brackets first: 3 - 9 = -6 and -8(-6) = 48. Distributing: (-8)(3) + (-8)(-9) = -24 + 72 = 48.
- 60. -5(-12) = 60, since the signs are the same.
- -64. Three factors of -4, an odd number, so negative; 4^3 = 64.
- 1. Ten factors of -1, an even number, so positive.
- -10^2 = -100 and (-10)^2 = 100. In -10^2 the exponent touches only the 10, so it is the opposite of 100; in (-10)^2 the base is -10 and two negative factors give a positive.
- -243. Five factors of -3, an odd number, so negative; 3^5 = 243.
- -12. Different signs; check (-12)(7) = -84.
- 15. Same signs; check 15(-8) = -120.
- -8. Different signs; check (-8)(-12) = 96.
- 54. Dividing by -1 gives the opposite.
- n = 7. n = -63 \div (-9) = 7; check (-9)(7) = -63.
- n = -48. n = 8(-6) = -48; check -48 \div (-6) = 8.
- -16. 5(-4) = -20 and 36 \div (-9) = -4, so -20 - (-4) = -20 + 4 = -16.
- 31. (-2)^4 = 16 and 3(-5) = -15, so 16 - (-15) = 31.
- -12. The top is (-8)(-9) = 72, the bottom is -2 - 4 = -6, and 72 \div (-6) = -12.
- 33. (-3)^2 - 4(-3)(2) = 9 - (-24) = 33.
- 0, 2, 4, 6 or 8. A product with no zero factor is positive exactly when the number of negative factors is even, and the even numbers from 0 to 9 are 0, 2, 4, 6 and 8.
- By the distributive property, (-5)(2) + (-5)(-2) = (-5)\bigl(2 + (-2)\bigr), and the right side is (-5) \cdot 0 = 0. So -10 + (-5)(-2) = 0, and the only number that adds to -10 to make 0 is 10. Hence (-5)(-2) = 10.
- (-7)^{13} is negative: it has 13 factors of -7, an odd number. -7^{12} is also negative: the exponent touches only the 7, so 7^{12} is positive and the minus sign makes its opposite.
- -15 \div 0 would be a number q with q \cdot 0 = -15. Every number times 0 is 0, never -15, so no such q exists and the quotient has no value.
- The square of a is a \cdot a, two factors with the same sign. If a is positive the product is positive; if a is negative, two negatives also give a positive; if a = 0 the square is 0. In no case is it negative.