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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.

A list of equations. 4 times negative 2 equals negative 8. 3 times negative 2 equals negative 6. 2 times negative 2 equals negative 4. 1 times negative 2 equals negative 2. For all this, the following label is listed on the side: As we know, a negative times a positive equals a negative. The list continues. 0 times negative 2 equals 0. The following label is listed to the side: As we know, 0 times any number equals 0. The list continues further. Negative 1 times negative 2 equals 2. Negative 2 times negative 2 equals 4. Negative 3 times negative 2 equals 6. Negative 4 times negative 2 equals 8. The following label is listed to the side: The pattern suggested is a negative times a negative equals a positive. For the entire list, the label at the top says: when this number decreases by 1, the first factor in each multiplication problem, the product increases by 2.

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, 8⋅6=+488⋅6=+48 size 12{8 cdot 6"=+""48"} {}, or 8⋅6=488⋅6=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 \( - 8 \) } {}

Solution

-24

Exercise 2

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

Solution

64

Exercise 3

(−6)(−5)(−6)(−5) size 12{ \( - 6 \) \( - 5 \) } {}

Solution

30

Exercise 4

(−7)(−2)(−7)(−2) size 12{ \( - 7 \) \( - 2 \) } {}

Solution

14

Exercise 5

(−1)(4)(−1)(4) size 12{ \( - 1 \) \( 4 \) } {}

Solution

-4

Exercise 6

(−7)7(−7)7 size 12{ \( - 7 \) \( 7 \) } {}

Solution

-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=3⋅412=3⋅4 size 12{"12"=3 cdot 4} {}

This suggests that

( + ) ( + ) = ( + ) ( + ) ( + ) = ( + ) size 12{ { { \( + \) } over { \( + \) } } = \( + \) } {}

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

What is 12−312−3 size 12{ { {"12"} over { - 3} } } {}?

−12=(−3)(−4)−12=(−3)(−4) size 12{ - "12"= \( - 3 \) \( - 4 \) } {} suggests that 12−3=−412−3=−4 size 12{ { {"12"} over { - 3} } = - 4} {}. That is,

( + ) ( − ) = ( − ) ( + ) ( − ) = ( − ) size 12{ { { \( + \) } over { \( - \) } } = \( - \) } {}

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

What is −123−123 size 12{ { { - "12"} over {3} } } {}?

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

( − ) ( + ) = ( − ) ( − ) ( + ) = ( − ) size 12{ { { \( - \) } over { \( + \) } } = \( - \) } {}

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

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

−12=(−3)(4)−12=(−3)(4) size 12{ - "12"= \( - 3 \) \( 4 \) } {} suggests that −12−3=4−12−3=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

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

Example 6

−35−7−35−7 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, −35−7=5−35−7=5 size 12{ { { - "35"} over { - 7} } =5} {}.

Example 7

18−918−9 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, 18−9=218−9=2 size 12{ { {"18"} over { - 9} } =2} {}.

Practice Set B

Find the following quotients.

Exercise 7

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

Solution

4

Exercise 8

30−530−5 size 12{ { {"30"} over { - 5} } } {}

Solution

-6

Exercise 9

−5427−5427 size 12{ { { - "54"} over {"27"} } } {}

Solution

-2

Exercise 10

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

Solution

3

Sample Set C

Example 8

Find the value of −6(4−7)−2(8−9)−(4+1)+1−6(4−7)−2(8−9)−(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(2−6)−4(−8−1)2(3−10)−9(−2)−5(2−6)−4(−8−1)2(3−10)−9(−2) size 12{ { { - 5 \( 2 - 6 \) - 4 \( - 8 - 1 \) } over {2 \( 3 - "10" \) - 9 \( - 2 \) } } } {}.

Solution

14

Calculators

Calculators with the A box with a plus and minus sign. 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.

Table 1
    Display Reads
Type 186 186
Press A box with a plus and minus sign. -186
Press × -186
Type 43 43
Press A box with a plus and minus sign. -43
Press = 7998

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

Example 10

158.64−54.3158.64−54.3 size 12{ { {"158" "." "64"} over { - "54" "." 3} } } {}. Round to one decimal place.

Table 2
    Display Reads
Type 158.64 158.64
Press ÷ 158.64
Type 54.3 54.3
Press A box with a plus and minus sign. -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 \( "–21" "." 6 \) } {}

Solution

1,108.08

Exercise 13

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

Solution

1.226

Exercise 14

(0.006)⋅(-0.241)(0.006)⋅(-0.241) size 12{ \( 0 "." "006" \) cdot \( – 0 "." "241" \) } {}. Round to three decimal places.

Solution

-0.001

Exercises

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

Exercise 15

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

Solution

16

Exercise 16

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

Exercise 17

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

Solution

32

Exercise 18

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

Exercise 19

3−123−12 size 12{ left (3 right ) left (-"12" right )} {}

Solution

-36

Exercise 20

4−184−18 size 12{ left (4 right ) left (-"18" right )} {}

Exercise 21

10−610−6 size 12{ left ("10" right ) left (-6 right )} {}

Solution

-60

Exercise 22

−64−64 size 12{ left (-6 right ) left (4 right )} {}

Exercise 23

−26−26 size 12{ left (-2 right ) left (6 right )} {}

Solution

-12

Exercise 24

−87−87 size 12{ left (-8 right ) left (7 right )} {}

Exercise 25

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

Solution

3

Exercise 26

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

Exercise 27

−393−393 size 12{ { {-"39"} over {3} } } {}

Solution

-13

Exercise 28

−2010−2010 size 12{ { {-"20"} over {"10"} } } {}

Exercise 29

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

Solution

9

Exercise 30

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

Exercise 31

25−525−5 size 12{ { {"25"} over {-5} } } {}

Solution

-5

Exercise 32

36−436−4 size 12{ { {"36"} over {-4} } } {}

Exercise 33

8−−38−−3 size 12{8- left (-3 right )} {}

Solution

11

Exercise 34

14−−2014−−20 size 12{"14"- left (-"20" right )} {}

Exercise 35

20−−820−−8 size 12{"20"- left (-8 right )} {}

Solution

28

Exercise 36

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

Exercise 37

0−40−4 size 12{0-4} {}

Solution

-4

Exercise 38

0−−10−−1 size 12{0- left (-1 right )} {}

Exercise 39

−6+1−7−6+1−7 size 12{-6+1-7} {}

Solution

-12

Exercise 40

15−12−2015−12−20 size 12{"15"-"12"-"20"} {}

Exercise 41

1−6−7+81−6−7+8 size 12{1-6-7+8} {}

Solution

-4

Exercise 42

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

Exercise 43

34−634−6 size 12{3 left (4-6 right )} {}

Solution

-6

Exercise 44

85−1285−12 size 12{8 left (5-"12" right )} {}

Exercise 45

−31−6−31−6 size 12{-3 left (1-6 right )} {}

Solution

15

Exercise 46

−84−12+2−84−12+2 size 12{-8 left (4-"12" right )+2} {}

Exercise 47

−41−8+310−3−41−8+310−3 size 12{-4 left (1-8 right )+3 left ("10"-3 right )} {}

Solution

49

Exercise 48

−90−2+48−9+0−3−90−2+48−9+0−3 size 12{-9 left (0-2 right )+4 left (8-9 right )+0 left (-3 right )} {}

Exercise 49

6−2−9−62+9+4−1−16−2−9−62+9+4−1−1 size 12{6 left (-2-9 right )-6 left (2+9 right )+4 left (-1-1 right )} {}

Solution

-140

Exercise 50

34+1−25−234+1−25−2 size 12{ { {3 left (4+1 right )-2 left (5 right )} over {-2} } } {}

Exercise 51

48+1−3−2−4−248+1−3−2−4−2 size 12{ { {4 left (8+1 right )-3 left (-2 right )} over {-4-2} } } {}

Solution

-7

Exercise 52

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

Exercise 53

−34−2+−3−6−4−34−2+−3−6−4 size 12{ { {-3 left (4-2 right )+ left (-3 right ) left (-6 right )} over {-4} } } {}

Solution

-3

Exercise 54

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

Exercise 55

−16−1−16−1 size 12{-1 left (6-1 right )} {}

Solution

-5

Exercise 56

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

Exercise 57

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

Solution

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 −8−−20−8−−20 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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