Multiplying and Dividing Integers

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Multiplying and dividing integers uses two sign rules. If the two numbers have the same sign, the answer is positive. If they have different signs, the answer is negative. So (-4) \times (-5) = 20 and (-20) \div 4 = -5. Multiply or divide the numbers as usual, then use the sign rule.

Before this lesson: You should know negative numbers and adding and subtracting integers.

Why the Sign Rules Work

Multiplication is repeated adding. So 3 \times (-4) means three groups of −4:

3 \times (-4) = (-4) + (-4) + (-4) = -12

That shows a positive times a negative is negative.

What about a negative times a negative? Look at this pattern. Each time the first number goes down by 1, the answer goes up by 2.

ProblemAnswer
3 \times (-2)−6
2 \times (-2)−4
1 \times (-2)−2
0 \times (-2)0
(-1) \times (-2)2
(-2) \times (-2)4

To keep the pattern going, a negative times a negative has to be positive.

Sign Rules for Multiplying and Dividing Integers

SignsAnswerMultiplyingDividing
Positive and positivePositive6 \times 3 = 1818 \div 3 = 6
Negative and negativePositive(-6) \times (-3) = 18(-18) \div (-3) = 6
Positive and negativeNegative6 \times (-3) = -1818 \div (-3) = -6
Negative and positiveNegative(-6) \times 3 = -18(-18) \div 3 = -6

Dividing follows the same rules because division undoes multiplication. For example, (-18) \div (-3) = 6 because 6 \times (-3) = -18.

Multiplying more than two numbers

Count the negative signs. An even number of negatives gives a positive answer. An odd number of negatives gives a negative answer.

Zero

Zero times any number is 0. Zero divided by any other number is also 0, so 0 \div (-5) = 0. But you can’t divide by 0. A problem like 8 ÷ 0 has no answer, so we call it undefined.

Worked Examples

Example 1: A negative times a positive

Find (-7) \times 8.

Step 1. Multiply the numbers without the signs: 7 × 8 = 56.

Step 2. The signs are different, so the answer is negative.

Answer: (-7) \times 8 = -56.

Example 2: A negative times a negative

Find (-9) \times (-6).

Step 1. Multiply the numbers: 9 × 6 = 54.

Step 2. The signs are the same, so the answer is positive.

Answer: (-9) \times (-6) = 54.

Example 3: Dividing integers

Find (-63) \div 7 and (-48) \div (-6).

Step 1. 63 ÷ 7 = 9. The signs are different, so the answer is negative: −9.

Step 2. 48 ÷ 6 = 8. The signs are the same, so the answer is positive: 8.

Check: Multiply back. (-9) \times 7 = -63 and 8 \times (-6) = -48, so both answers are right.

Answer: (-63) \div 7 = -9 and (-48) \div (-6) = 8.

Example 4: More than two numbers

Find (-2) \times 5 \times (-3) \times (-1).

Step 1. Multiply the numbers without the signs: 2 × 5 × 3 × 1 = 30.

Step 2. Count the negatives: −2, −3 and −1. That is 3 negatives, which is an odd number.

Step 3. An odd number of negatives gives a negative answer.

Answer: (-2) \times 5 \times (-3) \times (-1) = -30.

Example 5: Word problem

Over 6 hours, the temperature fell a total of 18 degrees. It fell by the same amount each hour. What was the change in temperature each hour?

Step 1. A drop of 18 degrees is a change of −18.

Step 2. Split the change evenly over 6 hours by dividing.

(-18) \div 6 = -3

Answer: The change was −3 degrees per hour. The temperature fell 3 degrees each hour.

Negative Numbers and Exponents

An exponent tells you how many times to multiply a number by itself. With negative numbers, watch the parentheses closely.

\begin{aligned}
(-5)^2 &= (-5) \times (-5) = 25 \\
-5^2 &= -(5 \times 5) = -25 \\
(-2)^3 &= (-2) \times (-2) \times (-2) = -8
\end{aligned}

In (-5)^2, the whole −5 is squared. In -5^2, only the 5 is squared, and the minus sign stays in front.

A negative number raised to an even power is positive. Raised to an odd power, it is negative. You’ll learn more about powers in the Exponents lesson.

A fraction bar also means divide. So \frac{-24}{6} = -4 uses the same sign rules. See Fractions for more on fractions.

Common Mistakes

MistakeWhat it looks likeHow to fix it
Using the adding rule(-3) \times (-4) = -12When you multiply, same signs give a positive: 12.
Treating -4^2 like (-4)^2-4^2 = 16Without parentheses, only the 4 is squared: -4^2 = -16.
Not counting every negative(-1) \times (-2) \times (-3) = 6Three negatives is an odd number, so the answer is -6.
Dividing by zero8 \div 0 = 0Dividing by 0 is undefined. (But 0 \div 8 = 0.)
Using the wrong rule for division(-20) \div (-5) = -4Same signs give a positive answer: 4.

Practice Problems

Find each product or quotient.

  1. 9 \times (-4)
  2. (-7) \times (-7)
  3. (-11) \times 3
  4. (-54) \div 9
  5. 72 \div (-8)
  6. (-45) \div (-5)
  7. (-3) \times (-2) \times (-5)
  8. (-1) \times 4 \times (-2) \times (-3) \times (-1)
  9. Find (-3)^2 and -3^2.
  10. A diver goes down 4 feet every second. What is her change in depth after 9 seconds?
Show answers
  1. 9 × 4 = 36. Different signs, so 9 \times (-4) = -36.
  2. 7 × 7 = 49. Same signs, so (-7) \times (-7) = 49.
  3. 11 × 3 = 33. Different signs, so (-11) \times 3 = -33.
  4. 54 ÷ 9 = 6. Different signs, so (-54) \div 9 = -6.
  5. 72 ÷ 8 = 9. Different signs, so 72 \div (-8) = -9.
  6. 45 ÷ 5 = 9. Same signs, so (-45) \div (-5) = 9.
  7. 3 × 2 × 5 = 30. There are 3 negatives (odd), so the answer is -30.
  8. 1 × 4 × 2 × 3 × 1 = 24. There are 4 negatives (even), so the answer is 24.
  9. (-3)^2 = (-3) \times (-3) = 9 and -3^2 = -(3 \times 3) = -9.
  10. Going down 4 feet each second is a change of −4 per second. 9 \times (-4) = -36, so her depth changes by −36 feet. She is 36 feet deeper.

Multiplying and Dividing Integers FAQ

Why is a negative times a negative positive?

Multiplying by a negative number flips the sign to the other side of 0. If you flip twice, you end up back on the positive side. The pattern table above shows the same thing.

What is a negative divided by a negative?

A negative divided by a negative is positive, because the signs are the same: (-30) \div (-6) = 5.

Is a negative times a positive always negative?

Yes, as long as neither number is 0. If one of the numbers is 0, the answer is 0.

What is (−1) squared?

(-1)^2 = (-1) \times (-1) = 1. But -1^2 means the opposite of 1^2, which is −1.

Summary

  • Multiply or divide the numbers as usual, then decide the sign.
  • Same signs give a positive answer. Different signs give a negative answer.
  • Division uses the same sign rules as multiplication.
  • With many numbers, count the negatives: even means positive, odd means negative.
  • Watch parentheses with exponents: (-5)^2 = 25, but -5^2 = -25.
  • You can’t divide by 0.

Previous lesson: Adding and Subtracting Integers

Next lesson: Coordinate Plane

Back to the Pre-Algebra (Math Basics) Index

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