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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

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Integers .

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 a · b means add a , b times. Here, we are using the model just to help us discover the pattern.

Two images are shown side-by-side. The image on the left has the equation five times three at the top. Below this it reads “add 5, 3 times.” Below this depicts three rows of blue counters, with five counters in each row. Under this, it says “15 positives.” Under thisis the equation“5 times 3 equals 15.” The image on the right reads “negative 5 times three. The three is in parentheses. Below this it reads, “add negative five, three times.” Under this are fifteen red counters in three rows of five. Below this it reads” “15 negatives”. Below this is the equation negative five times 3 equals negative 15.”

The next two examples are more interesting.

What does it mean 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 on the workspace. Then we take away 5 three times.

This figure has two columns. In the top row, the left column contains the expression 5 times negative 3. This means take away 5, three times. Below this, there are three groups of five red negative counters, and below each group of red counters is an identical group of five blue positive counters. What are left are fifteen negatives, represented by 15 red counters. Underneath the counters is the equation 5 times negative 3 equals negative 15. In the top row, the right column contains the expression negative 5 times negative 3. This means take away negative 5, three times. Below this, there are three groups of five blue positive counters, and below each group of blue counters is an identical group of five red negative counters. What are left are fifteen positives, represented by 15 blue counters. Underneath the blue counters is the equation negative 5 times negative 3 equals 15.

In summary:

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 .
  • 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
7 · 4 = 28 −8 ( −6 ) = 48
Different signs Product Example
Positive · negative
Negative · positive
Negative
Negative
7 ( −9 ) = −63 −5 · 10 = −50

Multiply: −9 · 3 −2 ( −5 ) 4 ( −8 ) 7 · 6 .

Solution


−9 · 3 Multiply, noting that the signs are different so the product is negative. −27


−2 ( −5 ) Multiply, noting that the signs are the same so the product is positive. 10


4 ( −8 ) Multiply, with different signs. −32


7 · 6 Multiply, with same signs. 42

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Multiply: −6 · 8 −4 ( −7 ) 9 ( −7 ) 5 · 12 .

−48 28 −63 60

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Multiply: −8 · 7 −6 ( −9 ) 7 ( −4 ) 3 · 13 .

−56 54 −28 39

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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 ) Multiply. −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

−1 a = a

Multiplying a number by −1 gives its opposite.

Multiply: −1 · 7 −1 ( −11 ) .

Solution


  1. −1 · 7 Multiply, noting that the signs are different −7 so the product is negative. −7 is the opposite of 7 .


  2. −1 ( −11 ) Multiply, noting that the signs are the same 11 so the product is positive. 11 is the opposite of −11 .
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Multiply: −1 · 9 −1 · ( −17 ) .

−9 17

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Multiply: −1 · 8 −1 · ( −16 ) .

−8 16

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Divide integers

What about division ? Division is the inverse operation of multiplication. So, 15 ÷ 3 = 5 because 15 · 3 = 5 . 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.

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

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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