1. Number

Lesson

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

- $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.

- $\left(-4\right)\times5=-20$(−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.

- $\left(-4\right)\times\left(-5\right)=20$(−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.

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.

- $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.

- $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.

- $\left(-20\right)\div\left(-5\right)=4$(−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.

Evaluate $5\times\left(-9\right)$5×(−9)

Evaluate $\left(-20\right)\div4$(−20)÷4

Evaluate $\left(-6\right)^3$(−6)3

Apply an understanding of integers to describe location, direction, amount, and changes in any of these, in various contexts.