1.06 Multiplying and dividing integers

Multiplying integers

We know how to multiply and divide whole numbers when they are all positive. Now let's look at what happens when negative integers are included in questions. The product is the answer when two numbers are multiplied together.

Your knowledge of times tables will really help you through this chapter, we just need to know what to do when one or more of the numbers we are multiplying or dividing are negative.

The product of two positive integers is a positive integer

Worked example 1

• $4\times5=20$4×5=20
• $16\times10=160$16×10=160

We just need to learn a couple of extra rules to our existing knowledge, to account for negatives.

The product of a positive integer and a negative integer is a negative integer.

Worked example 2

• $\left(-4\right)\times5=-20$(−4)×5=−20
• $16\times\left(-10\right)=-160$16×(−10)=−160

In both of the above cases, we can take out a factor of $-1$−1, giving us $-1\times\left(4\times5\right)=-20$−1×(4×5)=−20, and $-1\times\left(16\times10\right)=-160$−1×(16×10)=−160 respectively.

We can see that is does not matter which term is negative when we are finding the product; the product of a negative number and a positive number will always be negative. But what if both numbers are negative?

The product of two negative integers is a positive integer.

Worked example 3

• $\left(-4\right)\times\left(-5\right)=20$(−4)×(−5)=20
• $\left(-16\right)\times\left(-10\right)=160$(−16)×(−10)=160

In these cases, we can think of it as taking out two factors of $-1$−1.

So $\left(-4\right)\times\left(-5\right)=\left(-1\times-1\right)\times\left(4\times5\right)=1\times20=20$(−4)×(−5)=(−1×−1)×(4×5)=1×20=20

Worked example 4

Find $\sqrt{25}$√25.

We know that $5^2=25$52=25, so it makes sense that $\sqrt{25}=5$√25=5. However using the above rule, we now know that: $-5\times-5=(-5)^2=25$−5×−5=(−5)2=25. So this means that $\sqrt{25}=-5$√25=−5 as well. So: 

Dividing integers

As division is just the opposite of multiplication, the rules for dividing by integers are the same as the rules for multiplying integers.

The quotient of two positive integers is a positive integer.

Worked example 5

• $20\div5=4$20÷​5=4
• $160\div10=16$160÷​10=16

To account for negatives, we can follow the same rules as we did when multiplying with integers.

The quotient of a negative and a positive integer is a negative integer.

Worked example 6

• $20\div\left(-5\right)=-4$20÷​(−5)=−4
• $\left(-160\right)\div10=-16$(−160)÷​10=−16

Again, in both cases, we can take out a negative factor of $-1$−1, giving us $\frac{1}{-1}\times\frac{20}{5}=-4$1−1​×205​=−4, and $\frac{-1}{1}\times\frac{160}{10}=-16$−11​×16010​=−16 respectively.

As before, we can see that is does not matter which term is negative when we are finding the quotient; the quotient of two numbers, when one is negative, will always be negative. But what if both numbers are negative?

The quotient of two negative integers is a positive integer.

Worked example 7

• $\left(-20\right)\div\left(-5\right)=4$(−20)÷​(−5)=4
• $\left(-160\right)\div\left(-10\right)=16$(−160)÷​(−10)=16

Like with multiplying two negatives, we can think of it as taking out two factors of $-1$−1.

So $\left(-20\right)\div\left(-5\right)=\frac{-1}{-1}\times\frac{20}{5}=1\times4=4$(−20)÷​(−5)=−1−1​×205​=1×4=4.

Practice questions

Question 1

Question 2

Question 3