# 10.6: Multiplication and Division of Signed Numbers

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Learning Objectives

• be able to multiply and divide signed numbers
• be able to multiply and divide signed numbers using a calculator

## Multiplication of Signed Numbers

Let us consider first, the product of two positive numbers. Multiply: $$3 \cdot 5$$.

$$3 \cdot 5$$ means $$5 + 5 + 5 = 15$$

This suggests that (In later mathematics courses, the word "suggests" turns into the word "proof." One example does not prove a claim. Mathematical proofs are constructed to validate a claim for all possible cases.)

$$\text{(positive number)} \cdot \text{(positive number)} = \text{(positive number)}$$

More briefly,

(+) (+) = (+)

Now consider the product of a positive number and a negative number. Multiply: (3)(-5)

(3)(-5) means (-5) + (-5) + (-5) = -15

This suggests that

$$\text{(positive number)} \cdot \text{(negative number)} = \text{(negative number)}$$

More briefly,

(+) (-) = (-)

By the commutative property of multiplication, we get

$$\text{(negative number)} \cdot \text{(positive number)} = \text{(negative number)}$$

More briefly,

(-) (+) = (-)

The sign of the product of two negative numbers can be suggested after observing the following illustration.

Multiply -2 by, respectively, 4, 3, 2, 1, 0, -1, -2, -3, -4. We have the following rules for multiplying signed numbers.

Rules for Multiplying Signed Numbers
Multiplying signed numbers:

1. To multiply two real numbers that have the same sign, multiply their absolute values. The product is positive.
(+) (+) = (+)
(-) (-) = (+)
2. To multiply two real numbers that have opposite signs, multiply their abso­lute values. The product is negative.
(+) (-) = (-)
(-) (+) = (-)

Sample Set A

Find the following products.

$$8 \cdot 6$$

Solution

$$\begin{array} {ccl} {|8|} & = & {8} \\ {|6|} & = & {6} \end{array} \big \}$$ Multiply these absolute values.

$$8 \cdot 6 = 48$$

Since the numbers have the same sign, the product is positive.

Thus, $$8 \cdot 6 = +48$$, or $$8 \cdot 6 = 48$$.

Sample Set A

(-8)(-6)

Solution

$$\begin{array} {ccl} {|-8|} & = & {8} \\ {|-6|} & = & {6} \end{array} \big \}$$ Multiply these absolute values.

$$8 \cdot 6 = 48$$

Since the numbers have the same sign, the product is positive.

Thus, $$(-8)(-6) = +48$$, or $$(-8)(-6) = 48$$.

Sample Set A

(-4)(7)

Solution

$$\begin{array} {ccl} {|-4|} & = & {4} \\ {|7|} & = & {7} \end{array} \big \}$$ Multiply these absolute values.

$$4 \cdot 7 = 28$$

Since the numbers have opposite signs, the product is negative.

Thus, (-4)(7) = -28.

Sample Set A

6(-3)

Solution

$$\begin{array} {ccl} {|6|} & = & {6} \\ {|-3|} & = & {3} \end{array} \big \}$$ Multiply these absolute values.

$$6 \cdot 3 = 18$$

Since the numbers have opposite signs, the product is negative.

Thus, 6(-3) = -18.

Practice Set A

Find the following products.

3(-8)

-24

Practice Set A

4(16)

64

Practice Set A

(-6)(-5)

30

Practice Set A

(-7)(-2)

14

Practice Set A

(-1)(4)

-4

Practice Set A

(-7)7

-49

## Division of Signed Numbers

To determine the signs in a division problem, recall that

$$\dfrac{12}{3} = 4$$ since $$12 = 3 \cdot 4$$

This suggests that

$$\dfrac{(+)}{(+)} = (+)$$

$$\dfrac{(+)}{(+)} = (+)$$ since (+) = (+) (+)

What is $$\dfrac{12}{-3}$$?

12 = (-3)(-4) suggets that $$\dfrac{12}{-3} = -4$$. That is,

$$\dfrac{(+)}{(-)} = (-)$$

(+) = (-) (-) suggets that $$\dfrac{(+)}{(-)} = (-)$$

What is $$\dfrac{-12}{3}$$?

-12 = (3)(-4) suggests that $$\dfrac{-12}{3} = -4$$. That is,

$$\dfrac{(-)}{(+)} = (-)$$

(-) = (+) (-) suggets that $$\dfrac{(-)}{(+)} = (-)$$

What is $$\dfrac{-12}{-3}$$?

-12 = (-3)(4) suggets that $$\dfrac{-12}{-3} = 4$$. That is,

$$\dfrac{(-)}{(-)} = (+)$$

(-) = (-)(+) suggests that $$\dfrac{(-)}{(-)} = (+)$$

We have the following rules for dividing signed numbers.

Rules for Dividing Signed Numbers
Dividing signed numbers:

1. To divide two real numbers that have the same sign, divide their absolute values. The quotient is positive.
$$\dfrac{(+)}{(+)} = (+)\dfrac{(-)}{(-)} = (+)$$
2. To divide two real numbers that have opposite signs, divide their absolute values. The quotient is negative.
$$\dfrac{(-)}{(+)} = (-)\dfrac{(+)}{(-)} = (-)$$

Sample Set B

Find the following quotients.

$$\dfrac{-10}{2}$$

Solution

$$\begin{array} {ccc} {|-10|} & = & {10} \\ {|2|} & = & {2} \end{array} \big \}$$ Divide these absolute values.

$$\dfrac{10}{2} = 5$$

Since the numbers have opposite signs, the quotient is negative.

Thus $$\dfrac{-10}{2} = -5$$.

Sample Set B

$$\dfrac{-35}{-7}$$

Solution

$$\begin{array} {ccc} {|-35|} & = & {35} \\ {|-7|} & = & {7} \end{array} \big \}$$ Divide these absolute values.

$$\dfrac{35}{7} = 5$$

Since the numbers have the same signs, the quotient is positive.

Thus $$\dfrac{-35}{-7} = 5$$.

Sample Set B

$$\dfrac{18}{-9}$$

Solution

$$\begin{array} {ccc} {|18|} & = & {18} \\ {|-9|} & = & {9} \end{array} \big \}$$ Divide these absolute values.

$$\dfrac{18}{9} = 2$$

Since the numbers have opposite signs, the quotient is negative.

Thus $$\dfrac{18}{-9} = -2$$.

Practice Set B

Find the following quotients.

$$\dfrac{-24}{-6}$$

4

Practice Set B

$$\dfrac{30}{-5}$$

-6

Practice Set B

$$\dfrac{-54}{27}$$

-2

Practice Set B

$$\dfrac{51}{17}$$

3

Sample Set C

Find the value of $$\dfrac{-6(4 - 7) - 2(8 - 9)}{-(4 + 1) + 1}$$.

Solution

Using the order of operations and what we know about signed numbers, we get,

$$\begin{array} {rcl} {\dfrac{-6(4 - 7) - 2(8 - 9)}{-(4 + 1) + 1}} & = & {\dfrac{-6(-3) - 2(-1)}{-(5) + 1}} \\ {} & = & {\dfrac{18 + 2}{-5 + 1}} \\ {} & = & {\dfrac{20}{-4}} \\ {} & = & {-5} \end{array}$$

Practice Set C

Find the value of $$\dfrac{-5(2 - 6) - 4(-8 - 1)}{2(3 - 10) - 9(-2)}$$.

14

## Calculators

Calculators with the key can be used for multiplying and dividing signed numbers.

Sample Set D

Use a calculator to find each quotient or product.

$$(-186) \cdot (-43)$$

Solution

Since this product involves a $$\text{(negative)} \cdot \text{(negative)}$$, we know the result should be a positive number. We'll illustrate this on the calculator.

 Display Reads Type 186 186 Press -186 Press $$\times$$ -186 Type 43 43 Press -43 Press = 7998

Thus, $$(-186) \cdot (-43) = 7,998$$

Sample Set D

$$\dfrac{158.64}{-54.3}$$. Round to one decimal place.

Solution

Since this product involves a $$\text{(negative)} \cdot \text{(negative)}$$, we know the result should be a positive number. We'll illustrate this on the calculator.

 Display Reads Type 158.64 158.64 Press $$\div$$ 158.64 Type 54.3 54.3 Press -54.3 Press = -2.921546961

Rounding to one decimal place we get -2.9.

Practice Set D

Use a calculator to find each value.

$$(-51.3) \cdot (-21.6)$$

1,108.08

Practice Set D

$$-2.5746 \div -2.1$$

1.226

Practice Set D

$$(0.006) \cdot (-0.241)$$. Round to three decimal places.

-0.001

## Exercises

Find the value of each of the following. Use a calculator to check each result.

Exercise $$\PageIndex{1}$$

(-2)(-8)

16

Exercise $$\PageIndex{2}$$

(-3)(-9)

Exercise $$\PageIndex{3}$$

(-4)(-8)

32

Exercise $$\PageIndex{4}$$

(-5)(-2)

Exercise $$\PageIndex{5}$$

(3)(-12)

-36

Exercise $$\PageIndex{6}$$

(4)(-18)

Exercise $$\PageIndex{7}$$

(10)(-6)

-60

Exercise $$\PageIndex{8}$$

(-6)(4)

Exercise $$\PageIndex{9}$$

(-2)(6)

-12

Exercise $$\PageIndex{10}$$

(-8)(7)

Exercise $$\PageIndex{11}$$

$$\dfrac{21}{7}$$

3

Exercise $$\PageIndex{12}$$

$$\dfrac{42}{6}$$

Exercise $$\PageIndex{13}$$

$$\dfrac{-39}{3}$$

-13

Exercise $$\PageIndex{14}$$

$$\dfrac{-20}{10}$$

Exercise $$\PageIndex{15}$$

$$\dfrac{-45}{-5}$$

9

Exercise $$\PageIndex{16}$$

$$\dfrac{-16}{-8}$$

Exercise $$\PageIndex{17}$$

$$\dfrac{25}{-5}$$

-5

Exercise $$\PageIndex{18}$$

$$\dfrac{36}{-4}$$

Exercise $$\PageIndex{19}$$

8 - (-3)

11

Exercise $$\PageIndex{20}$$

14 - (-20)

Exercise $$\PageIndex{21}$$

20 - (-8)

28

Exercise $$\PageIndex{22}$$

(-4) - (-1)

Exercise $$\PageIndex{23}$$

0 - 4

-4

Exercise $$\PageIndex{24}$$

0 - (-1)

Exercise $$\PageIndex{25}$$

-6 + 1 - 7

-12

Exercise $$\PageIndex{26}$$

15 - 12 - 20

Exercise $$\PageIndex{27}$$

1 - 6 - 7 + 8

-4

Exercise $$\PageIndex{28}$$

2 + 7 - 10 + 2

Exercise $$\PageIndex{29}$$

3(4 - 6)

-6

Exercise $$\PageIndex{30}$$

8(5 - 12)

Exercise $$\PageIndex{31}$$

-3(1 - 6)

15

Exercise $$\PageIndex{32}$$

-8(4 - 12) + 2

Exercise $$\PageIndex{33}$$

-4(1 - 8) + 3(10 - 3)

49

Exercise $$\PageIndex{34}$$

-9(0 - 2) + 4(8 - 9) + 0(-3)

Exercise $$\PageIndex{35}$$

6(-2 - 9) - 6(2 + 9) + 4(-1 - 1)

-140

Exercise $$\PageIndex{36}$$

$$\dfrac{3(4 + 1) - 2 (5)}{-2}$$

Exercise $$\PageIndex{37}$$

$$\dfrac{4(8 + 1) - 3 (-2)}{-4 - 2}$$

-7

Exercise $$\PageIndex{38}$$

$$\dfrac{-1(3 + 2) + 5}{-1}$$

Exercise $$\PageIndex{39}$$

$$\dfrac{-3(4 - 2) + (-3)(-6)}{-4}$$

-3

Exercise $$\PageIndex{40}$$

-1(4 + 2)

Exercise $$\PageIndex{41}$$

-1(6 - 1)

-5

Exercise $$\PageIndex{42}$$

-(8 + 21)

Exercise $$\PageIndex{43}$$

-(8 - 21)

13

#### Exercises for Review

Exercise $$\PageIndex{44}$$

Use the order of operations to simplify $$(5^2 + 3^2 + 2) \div 2^2$$.

Exercise $$\PageIndex{45}$$

Find $$\dfrac{3}{8}$$ of $$\dfrac{32}{9}$$.

$$\dfrac{4}{3} = 1 \dfrac{1}{3}$$

Exercise $$\PageIndex{46}$$

Write this number in decimal form using digits: “fifty-two three-thousandths”

Exercise $$\PageIndex{47}$$

The ratio of chlorine to water in a solution is 2 to 7. How many mL of water are in a solution that contains 15 mL of chlorine?

$$52 \dfrac{1}{2}$$
Exercise $$\PageIndex{48}$$