3.4 Multiply and divide integers

Multiply integers

Since multiplication is mathematical shorthand for repeated addition, our counter model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction.

We remember that $a·b$ means add $a,b$ times. Here, we are using the model shown in [link] just to help us discover the pattern.

Now consider what it means to multiply $5$ by $\mathrm{-3}.$ It means subtract $5,3$ times. Looking at subtraction as taking away , it means to take away $5,3$ times. But there is nothing to take away, so we start by adding neutral pairs as shown in [link] .

In both cases, we started with $\mathbf{\text{15}}$ neutral pairs. In the case on the left, we took away $\mathbf{\text{5}},\mathbf{\text{3}}$ times and the result was $-\mathbf{\text{15}}.$ To multiply $\left(\mathrm{-5}\right)\left(\mathrm{-3}\right),$ we took away $-\mathbf{\text{5}},\mathbf{\text{3}}$ times and the result was $\mathbf{\text{15}}.$ So we found that

Notice that for multiplication of two signed numbers , when the signs are the same, the product    is positive, and when the signs are different, the product is negative.

When we multiply a number by $1,$ the result is the same number. What happens when we multiply a number by $\mathrm{-1}?$ Let’s multiply a positive number and then a negative number by $\mathrm{-1}$ to see what we get.

Each time we multiply a number by $\mathrm{-1},$ we get its opposite.

Divide integers

Division is the inverse operation of multiplication. So, $15÷3=5$ because $5·3=15$ In words, this expression says that $\mathbf{\text{15}}$ can be divided into $\mathbf{\text{3}}$ groups of $\mathbf{\text{5}}$ each because adding five three times gives $\mathbf{\text{15}}.$ If we look at some examples of multiplying integers , we might figure out the rules for dividing integers .