CHAPTER 1 Whole Numbers, Integers, and Introduction to Algebra
CHAPTER 1 Whole Numbers, Integers, and Introduction to Algebra
1.5 Multiply and Divide Integers
Learning Objectives
By the end of this section, you will be able to:
- Multiply integers
- Divide integers
- Simplify expressions with integers
- Evaluate variable expressions with integers
- Translate English phrases to algebraic expressions
- Use integers in applications
Multiply Integers
Since multiplication is mathematical shorthand for repeated addition, our 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. Here, we will use the model just to help us discover the pattern.
We remember that means add a, b times. Here, we are using the model just to help us discover the pattern.
The next two examples are more interesting.
What does it mean to multiply 5 by 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 on the workspace. Then we take away 5 three times.
In summary:
Notice that for multiplication of two signed numbers, when the:
- signs are the same, the product is positive.
- signs are different, the product is negative.
We’ll put this all together in the chart below
Multiplication of Signed Numbers
For multiplication of two signed numbers:
| Same signs | Product | Example |
|---|---|---|
| Two positives Two negatives |
Positive Positive |
| Different signs | Product | Example |
| Positive \cdot negative Negative \cdot positive |
Negative Negative |
EXAMPLE 1
Multiply: a) b) c) d) .
| a) Multiply, noting that the signs are different so the product is negative. |
|
| b) Multiply, noting that the signs are the same so the product is positive. |
|
| c) Multiply, with different signs. |
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| d) Multiply, with same signs. |
TRY IT 1.1
Multiply: a) b) c) d) .
Show answer
a) b) 28 c) d) 60
TRY IT 1.2
Multiply: a) b) c) d) .
Show answer
a) b) 54 c) d) 39
When we multiply a number by 1, the result is the same number. What happens when we multiply a number by ? Let’s multiply a positive number and then a negative number by to see what we get.
Each time we multiply a number by , we get its opposite!
Multiplication by
Multiplying a number by gives its opposite.
EXAMPLE 2
Multiply: a) b) .
| a) Multiply, noting that the signs are different so the product is negative. |
|
| b) Multiply, noting that the signs are the same so the product is positive. |
TRY IT 2.1
Multiply: a) b) .
Show answer
a) b) 17
TRY IT 2.2
Multiply: a) b) .
Show answer
a) b) 16
Divide Integers
What about division? Division is the inverse operation of multiplication. So, because . In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.
Division follows the s