Multiplying and Dividing Integers

Read this text. Pay special attention to the sections Multiplication of Signed Numbers and Division of Signed Numbers. These sections give summaries of the rules for multiplying and dividing positive and negative integers. Complete the practice questions and check your answers.

Multiplying and Dividing 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 Figure 3.19 just to help us discover the pattern.

This image has two columns. The first column has 5 times 3. Underneath, it states add 5, 3 times. Under this there are 3 rows

Figure 3.19

Now consider what it means to multiply 5 by −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 Figure 3.20.

This figure has 2 columns. The first column has 5 times negative 3. Underneath it states take away 5, 3 times. Under this the

Figure 3.20

In both cases, we started with 15 neutral pairs. In the case on the left, we took away $5, 3$ times and the result was −15. To multiply $(−5)(−3)$ , we took away $−5, 3$ times and the result was 15. So we found that

$5⋅3=15$	$−5(3)=−15$
$5(−3)=−15$	$(−5)(−3)=15$

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.

Multiplication of Signed Numbers

The sign of the product of two numbers depends on their signs.

Same signs	Product
•Two positives •Two negatives	Positive Positive

Different signs	Product
•Positive • negative •Negative • positive	Negative Negative

When we multiply a number by 1, the result is the same number. What happens when we multiply a number by −1? Let's multiply a positive number and then a negative number by −1 to see what we get.

$−1⋅4$	$−1(−3)$
$−4$	$3$
$−4$ is the opposite of $4$	$3$ is the opposite of $−3$

Each time we multiply a number by −1, we get its opposite.

Multiplication by −1

Multiplying a number by −1 gives its opposite.

$−1a=−a$

Divide Integers

Division is the inverse operation of multiplication. So,

$15÷3=5$ because

$5⋅3=15$ In words, this expression says that 15 can be divided into 3 groups of 5 each because adding five three times gives 15. If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.

$5⋅3=15$ so $15÷3=5$	$−5(3)=−15$ so $−15÷3=−5$
$(−5)(−3)=15$ so $15÷(−3)=−5$	$5(−3)=−15$ so $−15÷−3=5$

Division of signed numbers follows the same rules as multiplication. When the signs are the same, the quotient is positive, and when the signs are different, the quotient is negative.

Division of Signed Numbers

The sign of the quotient of two numbers depends on their signs.

Same signs	Quotient
•Two positives •Two negatives	Positive Positive

Different signs	Quotient
•Positive & negative •Negative & positive	Negative Negative

Remember, you can always check the answer to a division problem by multiplying.

Just as we saw with multiplication, when we divide a number by 1, the result is the same number. What happens when we divide a number by −1? Let's divide a positive number and then a negative number by −1 to see what we get.