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Textbook by: Denny Burzynski, Wade Ellis. E-mail the authors

# Multiplication and Division of Signed Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to multiply and divide signed numbers. By the end of the module students should be able to multiply and divide signed numbers and be able to multiply and divide signed numbers using a calculator.

## Section Overview

• Multiplication of Signed Numbers
• Division of Signed Numbers
• Calculators

## Multiplication of Signed Numbers

Let us consider first, the product of two positive numbers. Multiply: 3 53 5 size 12{"3 " cdot " 5"} {}.

3 53 5 size 12{"3 " cdot " 5"} {} means 5+5+5=155+5+5=15 size 12{5+5+5="15"} {}

This suggests1 that

( positive number ) ( positive number ) = ( positive number ) ( positive number ) ( positive number ) = ( positive number ) size 12{ $$"positive number"$$ cdot $$"positive number"$$ = $$"positive number"$$ } {}

More briefly,

( + ) ( + ) = ( + ) ( + ) ( + ) = ( + ) size 12{ $$+$$ $$+$$ = $$+$$ } {}

Now consider the product of a positive number and a negative number. Multiply: (3)(5)(3)(5) size 12{ $$3$$ $$- 5$$ } {}.

(3)(5)(3)(5) size 12{ $$3$$ $$- 5$$ } {} means (5)+(5)+(5)=15(5)+(5)+(5)=15 size 12{ $$- 5$$ + $$- 5$$ + $$- 5$$ = - "15"} {}

This suggests that

( positive number ) ( negative number ) = ( negative number ) ( positive number ) ( negative number ) = ( negative number ) size 12{ $$"positive number"$$ cdot $$"negative number"$$ = $$"negative number"$$ } {}

More briefly,

( + ) ( - ) = ( - ) ( + ) ( - ) = ( - ) size 12{ $$+$$ $$-$$ = $$-$$ } {}

By the commutative property of multiplication, we get

( negative number ) ( positive number ) = ( negative number ) ( negative number ) ( positive number ) = ( negative number ) size 12{ $$"negative number"$$ cdot $$"positive number"$$ = $$"negative number"$$ } {}

More briefly,

( ) ( + ) = ( ) ( ) ( + ) = ( ) size 12{ $$-$$ $$+$$ = $$-$$ } {}

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.
( + ) ( + ) = ( + ) ( + ) ( + ) = ( + ) size 12{ $$+$$ $$+$$ = $$+$$ } {}
( ) ( ) = ( + ) ( ) ( ) = ( + ) size 12{ $$-$$ $$-$$ = $$+$$ } {}
2. To multiply two real numbers that have opposite signs, multiply their abso­lute values. The product is negative.
( + ) ( ) = ( ) ( + ) ( ) = ( ) size 12{ $$+$$ $$-$$ = $$-$$ } {}
( ) ( + ) = ( ) ( ) ( + ) = ( ) size 12{ $$-$$ $$+$$ = $$-$$ } {}

### Sample Set A

Find the following products.

#### Example 1

8 68 6 size 12{"8 " cdot " 6"} {}

|8| = 8 |6| = 6 |8| = 8 |6| = 6 Multiply these absolute values.

8 6 = 48 8 6 = 48 size 12{8 cdot 6="48"} {}

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

Thus, 86=+4886=+48 size 12{8 cdot 6"=+""48"} {}, or 86=4886=48 size 12{8 cdot 6="48"} {}.

#### Example 2

(8)(6)(8)(6) size 12{ $$- 8$$ $$- 6$$ } {}

|- 8| = 8 |- 6| = 6 |- 8| = 8 |- 6| = 6 Multiply these absolute values.

8 6 = 48 8 6 = 48 size 12{8 cdot 6="48"} {}

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

Thus, (8)(6)=+48(8)(6)=+48 size 12{ $$- 8$$ $$- 6$$ "=+""48"} {}, or (8)(6)=48(8)(6)=48 size 12{ $$- 8$$ $$- 6$$ ="48"} {}.

#### Example 3

(4)(7)(4)(7) size 12{ $$- 4$$ $$7$$ } {}

|- 4| = 4 |7| = 7 |- 4| = 4 |7| = 7 Multiply these absolute values.

4 7 = 28 4 7 = 28 size 12{4 cdot 7="28"} {}

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

Thus, (4)(7)=28(4)(7)=28 size 12{ $$- 4$$ $$7$$ = - "28"} {}.

#### Example 4

6(3)6(3) size 12{6 $$- 3$$ } {}

|6| = 6 |- 3| = 3 |6| = 6 |- 3| = 3 Multiply these absolute values.

6 3 = 18 6 3 = 18 size 12{6 cdot 3="18"} {}

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

Thus, 6(3)=186(3)=18 size 12{6 $$- 3$$ = - "18"} {}.

### Practice Set A

Find the following products.

#### Exercise 1

3(8)3(8) size 12{3 $$- 8$$ } {}

-24

#### Exercise 2

4(16)4(16) size 12{4 $$"16"$$ } {}

64

#### Exercise 3

(6)(5)(6)(5) size 12{ $$- 6$$ $$- 5$$ } {}

30

#### Exercise 4

(7)(2)(7)(2) size 12{ $$- 7$$ $$- 2$$ } {}

14

#### Exercise 5

(1)(4)(1)(4) size 12{ $$- 1$$ $$4$$ } {}

-4

#### Exercise 6

(7)7(7)7 size 12{ $$- 7$$ $$7$$ } {}

-49

## Division of Signed Numbers

To determine the signs in a division problem, recall that

123=4123=4 size 12{ { {"12"} over {3} } =4} {} since 12=3412=34 size 12{"12"=3 cdot 4} {}

This suggests that

### ( + ) ( + ) = ( + ) ( + ) ( + ) = ( + ) size 12{ { { $$+$$ } over { $$+$$ } } = $$+$$ } {}

(+)(+)=(+)(+)(+)=(+) size 12{ { { $$+$$ } over { $$+$$ } } = $$+$$ } {} since (+)=(+)(+)(+)=(+)(+) size 12{ $$+$$ = $$+$$ $$+$$ } {}

What is 123123 size 12{ { {"12"} over { - 3} } } {}?

12=(3)(4)12=(3)(4) size 12{ - "12"= $$- 3$$ $$- 4$$ } {} suggests that 123=4123=4 size 12{ { {"12"} over { - 3} } = - 4} {}. That is,

### ( + ) ( − ) = ( − ) ( + ) ( − ) = ( − ) size 12{ { { $$+$$ } over { $$-$$ } } = $$-$$ } {}

(+)=()()(+)=()() size 12{ $$+$$ = $$-$$ $$-$$ } {} suggests that (+)()=()(+)()=() size 12{ { { $$+$$ } over { $$-$$ } } = $$-$$ } {}

What is 123123 size 12{ { { - "12"} over {3} } } {}?

12=(3)(4)12=(3)(4) size 12{ - "12"= $$3$$ $$- 4$$ } {} suggests that 123=4123=4 size 12{ { { - "12"} over {3} } = - 4} {}. That is,

### ( − ) ( + ) = ( − ) ( − ) ( + ) = ( − ) size 12{ { { $$-$$ } over { $$+$$ } } = $$-$$ } {}

()=(+)()()=(+)() size 12{ $$-$$ = $$+$$ $$-$$ } {} suggests that ()(+)=()()(+)=() size 12{ { { $$-$$ } over { $$+$$ } } = $$-$$ } {}

What is 123123 size 12{ { { - "12"} over { - 3} } } {}?

12=(3)(4)12=(3)(4) size 12{ - "12"= $$- 3$$ $$4$$ } {} suggests that 123=4123=4 size 12{ { { - "12"} over { - 3} } =4} {}. That is,

### ( − ) ( − ) = ( + ) ( − ) ( − ) = ( + ) size 12{ { { $$-$$ } over { $$-$$ } } = $$+$$ } {}

()=()(+)()=()(+) size 12{ $$-$$ = $$-$$ $$+$$ } {} suggests that ()()=(+)()()=(+) size 12{ { { $$-$$ } over { $$-$$ } } = $$+$$ } {}

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.
( + ) ( + ) = ( + ) ( + ) ( + ) = ( + ) size 12{ { { $$+$$ } over { $$+$$ } } = $$+$$ } {} ( ) ( ) = ( + ) ( ) ( ) = ( + ) size 12{ { { $$-$$ } over { $$-$$ } } = $$+$$ } {}
2. To divide two real numbers that have opposite signs, divide their absolute values. The quotient is negative.
( ) ( + ) = ( ) ( ) ( + ) = ( ) size 12{ { { $$-$$ } over { $$+$$ } } = $$-$$ } {} ( + ) ( ) = ( ) ( + ) ( ) = ( ) size 12{ { { $$+$$ } over { $$-$$ } } = $$-$$ } {}

### Sample Set B

Find the following quotients.

#### Example 5

102102 size 12{ { { - "10"} over {2} } } {}

|- 10| = 10 |2| = 2 |- 10| = 10 |2| = 2 Divide these absolute values.

10 2 = 5 10 2 = 5 size 12{ { {"10"} over {2} } =5} {}

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

Thus 102=5102=5 size 12{ { { - "10"} over {2} } = - 5} {}.

#### Example 6

357357 size 12{ { { - "35"} over { - 7} } } {}

|- 35| = 35 |- 7| = 7 |- 35| = 35 |- 7| = 7 Divide these absolute values.

35 7 = 5 35 7 = 5 size 12{ { {"35"} over {7} } =5} {}

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

Thus, 357=5357=5 size 12{ { { - "35"} over { - 7} } =5} {}.

#### Example 7

189189 size 12{ { {"18"} over { - 9} } } {}

|18| = 18 |- 9| = 9 |18| = 18 |- 9| = 9 Divide these absolute values.

18 9 = 2 18 9 = 2 size 12{ { {"18"} over {9} } =2} {}

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

Thus, 189=2189=2 size 12{ { {"18"} over { - 9} } =2} {}.

### Practice Set B

Find the following quotients.

#### Exercise 7

246246 size 12{ { { - "24"} over { - 6} } } {}

4

#### Exercise 8

305305 size 12{ { {"30"} over { - 5} } } {}

-6

#### Exercise 9

54275427 size 12{ { { - "54"} over {"27"} } } {}

-2

#### Exercise 10

51175117 size 12{ { {"51"} over {"17"} } } {}

3

### Sample Set C

#### Example 8

Find the value of 6(47)2(89)(4+1)+16(47)2(89)(4+1)+1 size 12{ { { - 6 $$4 - 7$$ - 2 $$8 - 9$$ } over { - $$4+1$$ +1} } } {}.

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

- 6(4-7)-2(8-9) -(4+1)+1 = - 6(- 3)-2(- 1) -(5)+1 = 18+2 - 5+1 = 20- 4 = - 5 - 6(4-7)-2(8-9) -(4+1)+1 = - 6(- 3)-2(- 1) -(5)+1 = 18+2 - 5+1 = 20- 4 = - 5

### Practice Set C

#### Exercise 11

Find the value of 5(26)4(81)2(310)9(2)5(26)4(81)2(310)9(2) size 12{ { { - 5 $$2 - 6$$ - 4 $$- 8 - 1$$ } over {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.

#### Example 9

(186)(43)(186)(43) size 12{ $$- "186"$$ cdot $$- "43"$$ } {}

Since this product involves a (negative)(negative)(negative)(negative) size 12{ $$"negative"$$ cdot $$"negative"$$ } {}, we know the result should be a positive number. We'll illustrate this on the calculator.

 Display Reads Type 186 186 Press -186 Press × -186 Type 43 43 Press -43 Press = 7998

Thus, (186)(43)=7,998(186)(43)=7,998 size 12{ $$- "186"$$ cdot $$- "43"$$ =7,"998"} {}.

#### Example 10

158.6454.3158.6454.3 size 12{ { {"158" "." "64"} over { - "54" "." 3} } } {}. Round to one decimal place.

 Display Reads Type 158.64 158.64 Press ÷ 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.

#### Exercise 12

(- 51.3)(- 21.6)(- 51.3)(- 21.6) size 12{ $$"–51" "." 3$$ cdot $$"–21" "." 6$$ } {}

1,108.08

#### Exercise 13

- 2.5746÷- 2.1- 2.5746÷- 2.1 size 12{"–2" "." "5746" div " –2" "." 1} {}

1.226

#### Exercise 14

(0.006)(-0.241)(0.006)(-0.241) size 12{ $$0 "." "006"$$ 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 15

2828 size 12{ left (-2 right ) left (-8 right )} {}

16

### Exercise 16

3939 size 12{ left (-3 right ) left (-9 right )} {}

### Exercise 17

4848 size 12{ left (-4 right ) left (-8 right )} {}

32

### Exercise 18

5252 size 12{ left (-5 right ) left (-2 right )} {}

### Exercise 19

312312 size 12{ left (3 right ) left (-"12" right )} {}

-36

### Exercise 20

418418 size 12{ left (4 right ) left (-"18" right )} {}

### Exercise 21

106106 size 12{ left ("10" right ) left (-6 right )} {}

-60

### Exercise 22

6464 size 12{ left (-6 right ) left (4 right )} {}

### Exercise 23

2626 size 12{ left (-2 right ) left (6 right )} {}

-12

### Exercise 24

8787 size 12{ left (-8 right ) left (7 right )} {}

### Exercise 25

217217 size 12{ { {"21"} over {7} } } {}

3

### Exercise 26

426426 size 12{ { {"42"} over {6} } } {}

### Exercise 27

393393 size 12{ { {-"39"} over {3} } } {}

-13

### Exercise 28

20102010 size 12{ { {-"20"} over {"10"} } } {}

### Exercise 29

455455 size 12{ { {-"45"} over {-5} } } {}

9

### Exercise 30

168168 size 12{ { {-"16"} over {-8} } } {}

### Exercise 31

255255 size 12{ { {"25"} over {-5} } } {}

-5

### Exercise 32

364364 size 12{ { {"36"} over {-4} } } {}

### Exercise 33

8383 size 12{8- left (-3 right )} {}

11

### Exercise 34

14201420 size 12{"14"- left (-"20" right )} {}

### Exercise 35

208208 size 12{"20"- left (-8 right )} {}

28

### Exercise 36

4141 size 12{-4- left (-1 right )} {}

### Exercise 37

0404 size 12{0-4} {}

-4

### Exercise 38

0101 size 12{0- left (-1 right )} {}

### Exercise 39

6+176+17 size 12{-6+1-7} {}

-12

### Exercise 40

151220151220 size 12{"15"-"12"-"20"} {}

### Exercise 41

167+8167+8 size 12{1-6-7+8} {}

-4

### Exercise 42

2+710+22+710+2 size 12{2+7-"10"+2} {}

### Exercise 43

346346 size 12{3 left (4-6 right )} {}

-6

### Exercise 44

85128512 size 12{8 left (5-"12" right )} {}

### Exercise 45

316316 size 12{-3 left (1-6 right )} {}

15

### Exercise 46

8412+28412+2 size 12{-8 left (4-"12" right )+2} {}

### Exercise 47

418+3103418+3103 size 12{-4 left (1-8 right )+3 left ("10"-3 right )} {}

49

### Exercise 48

902+489+03902+489+03 size 12{-9 left (0-2 right )+4 left (8-9 right )+0 left (-3 right )} {}

### Exercise 49

62962+9+41162962+9+411 size 12{6 left (-2-9 right )-6 left (2+9 right )+4 left (-1-1 right )} {}

-140

### Exercise 50

34+125234+1252 size 12{ { {3 left (4+1 right )-2 left (5 right )} over {-2} } } {}

### Exercise 51

48+1324248+13242 size 12{ { {4 left (8+1 right )-3 left (-2 right )} over {-4-2} } } {}

-7

### Exercise 52

13+2+5113+2+51 size 12{ { {-1 left (3+2 right )+5} over {-1} } } {}

### Exercise 53

342+364342+364 size 12{ { {-3 left (4-2 right )+ left (-3 right ) left (-6 right )} over {-4} } } {}

-3

### Exercise 54

14+214+2 size 12{-1 left (4+2 right )} {}

### Exercise 55

161161 size 12{-1 left (6-1 right )} {}

-5

### Exercise 56

8+218+21 size 12{- left (8+"21" right )} {}

### Exercise 57

821821 size 12{- left (8-"21" right )} {}

13

### Exercises for Review

#### Exercise 58

((Reference)) Use the order of operations to simplify 52+32+2÷2252+32+2÷22 size 12{ left (5 rSup { size 8{2} } +3 rSup { size 8{2} } +2 right )¸2 rSup { size 8{2} } } {}.

#### Exercise 59

((Reference)) Find 38 of 32938 of 329 size 12{ { {3} over {8} } " of " { {"32"} over {9} } } {}.

##### Solution

43=11343=113 size 12{ { {4} over {3} } =1 { {1} over {3} } } {}

#### Exercise 60

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

#### Exercise 61

((Reference)) 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?

##### Solution

52125212 size 12{"52" { {1} over {2} } } {}

#### Exercise 62

((Reference)) Perform the subtraction 820820 size 12{-8- left (-"20" right )} {}

## Footnotes

1. 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.

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