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Basic Operations with Real Numbers: Multiplication and Division of Signed Numbers

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

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of |x|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples. Objectives of this module: be able to multiply and divide signed numbers.

Overview

  • Multiplication of Signed Numbers
  • Division of Signed Numbers

Multiplication of Signed Numbers

Let us consider first the product of two positive numbers.

Multiply: 3⋅5 3⋅5 .
 3⋅5 3⋅5 means 5+5+5=15 5+5+5=15 .

This suggests that

(positive number)⋅(positive number)=positive number (positive number)⋅(positive number)=positive number .

More briefly, (+)(+)=+ (+)(+)=+ .

Now consider the product of a positive number and a negative number.

Multiply: (3)(−5) (3)(−5) .
 (3)(−5) (3)(−5) means (−5)+(−5)+(−5)=−15 (−5)+(−5)+(−5)=−15 .

This suggests that

(positive number)⋅(negative number)=negative number (positive number)⋅(negative number)=negative number

More briefly, (+)(-)=- (+)(-)=- .

By the commutative property of multiplication, we get

(negative number)⋅(positive number)=negative number (negative number)⋅(positive number)=negative number

More briefly, (-)(+)=- (-)(+)=- .

The sign of the product of two negative numbers can be determined using the following illustration: Multiply −2 −2 by, respectively, 4, 3, 2, 1, 0, −1, −2, −3, −4 4, 3, 2, 1, 0, −1, −2, −3, −4 . Notice that when the multiplier decreases by 1, the product increases by 2.

4(−2)=−8 3(−2)=−6 2(−2)=−4 1(−2)=−2 }→ As we know, (+)(−)=−. 0(−2)=0        → As we know,0⋅(any number)=0. 4(−2)=−8 3(−2)=−6 2(−2)=−4 1(−2)=−2 }→ As we know, (+)(−)=−. 0(−2)=0        → As we know,0⋅(any number)=0.

−1(−2)=2 −2(−2)=4 −3(−2)=6 −4(−2)=8 }→ This pattern suggests   (−)(−)=+. −1(−2)=2 −2(−2)=4 −3(−2)=6 −4(−2)=8 }→ This pattern suggests   (−)(−)=+.

We have the following rules for multiplying signed numbers.

Rules for Multiplying Signed Numbers

To multiply two real numbers that have

  1. the same sign, multiply their absolute values. The product is positive.
     (+)(+)=+ (−)(−)=+ (+)(+)=+ (−)(−)=+
  2. opposite signs, multiply their absolute values. The product is negative.
     (+)(−)=− (−)(+)=− (+)(−)=− (−)(+)=−

Sample Set A

Find the following products.

Example 1

8⋅6 8⋅6

Multiply these absolute values. | 8 |=8 | 6 |=6 } 8⋅6=48 Since the numbers have the same sign, the product is positive. 8⋅6=+48 ​​or         8⋅6=48 Multiply these absolute values. | 8 |=8 | 6 |=6 } 8⋅6=48 Since the numbers have the same sign, the product is positive. 8⋅6=+48 ​​or         8⋅6=48

Example 2

(−8)(−6) (−8)(−6)

Multiply these absolute values. | −8 |=8 | −6 |=6 } 8⋅6=48 Since the numbers have the same sign, the product is positive. (−8)(−6)=+48     or       (−8)(−6)=48 Multiply these absolute values. | −8 |=8 | −6 |=6 } 8⋅6=48 Since the numbers have the same sign, the product is positive. (−8)(−6)=+48     or       (−8)(−6)=48

Example 3

(−4)(7) (−4)(7)

Multiply these absolute values. | −4 |     =4 | 7 | =7 } 4⋅7=28 Since the numbers have opposite signs, the product is negative. (−4)(7)=−28 Multiply these absolute values. | −4 |     =4 | 7 | =7 } 4⋅7=28 Since the numbers have opposite signs, the product is negative. (−4)(7)=−28

Example 4

6(−3) 6(−3)

Multiply these absolute values. | 6 |     =6 | −3 | =3 } 6⋅3=18 Since the numbers have opposite signs, the product is negative. 6(−3)=−18 Multiply these absolute values. | 6 |     =6 | −3 | =3 } 6⋅3=18 Since the numbers have opposite signs, the product is negative. 6(−3)=−18

Practice Set A

Find the following products.

Exercise 1

3(−8) 3(−8)

Solution

−24 −24

Exercise 2

4(16) 4(16)

Solution

64

Exercise 3

(−6)(−5) (−6)(−5)

Solution

30

Exercise 4

(−7)(−2) (−7)(−2)

Solution

14

Exercise 5

(−1)(4) (−1)(4)

Solution

−4 −4

Exercise 6

(−7)7 (−7)7

Solution

−49 −49

Division of Signed Numbers

We can determine the sign pattern for division by relating division to multiplication. Division is defined in terms of multiplication in the following way.

If b⋅c=a b⋅c=a , then a b =c, b≠0 a b =c, b≠0 .

For example, since 3⋅4=12 3⋅4=12 , it follows that 12 3 =4 12 3 =4 .

Notice the pattern:

Since 3⋅4 ︸ b⋅c=a =12 3⋅4 ︸ b⋅c=a =12 , it follows that 12 3 ︸ a b =c =4 12 3 ︸ a b =c =4

The sign pattern for division follows from the sign pattern for multiplication.

  1. Since (+)(+) ︸ b⋅c=a =+ (+)(+) ︸ b⋅c=a =+ , it follows that (+) (+) ︸ a b =c =+ (+) (+) ︸ a b =c =+ , that is,

    (positive number) (positive number) =positive number (positive number) (positive number) =positive number

  2. Since (−)(−) ︸ b⋅c=a =+ (−)(−) ︸ b⋅c=a =+ , it follows that (+) (−) ︸ a b =c =− (+) (−) ︸ a b =c =− , that is,

    (positive number) (negative number) =negative number (positive number) (negative number) =negative number

  3. Since (+)(−) ︸ b⋅c=a =− (+)(−) ︸ b⋅c=a =− , it follows that (−) (+) ︸ a b =c =− (−) (+) ︸ a b =c =− , that is,

    (negative​ number) (positive number) =negative number (negative​ number) (positive number) =negative number

  4. Since (−)(+) ︸ b⋅c=a =− (−)(+) ︸ b⋅c=a =− , it follows that (−) (−) ︸ a b =c =+ (−) (−) ︸ a b =c =+ , that is

    (negative number) (negative number) =positive number (negative number) (negative number) =positive number

We have the following rules for dividing signed numbers.

Rules for Dividing Signed Numbers

To divide two real numbers that have

  1. the same sign, divide their absolute values. The quotient is positive.
     (+) (+) =+ (−) (−) =+ (+) (+) =+ (−) (−) =+
  2. opposite signs, divide their absolute values. The quotient is negative.
     (−) (+) =− (+) (−) =− (−) (+) =− (+) (−) =−

Sample Set B

Find the following quotients.

Example 5

−10 2 −10 2

| -10 |= 10 | 2 |= 2 } Divide these absolute values. 10 2 =5 -10 2      =-5 Since the numbers have opposite signs, the quotient is negative. | -10 |= 10 | 2 |= 2 } Divide these absolute values. 10 2 =5 -10 2      =-5 Since the numbers have opposite signs, the quotient is negative.

Example 6

−35 −7 −35 −7

| -35 |= 35 | -7 |= 7 } Divide these absolute values. 35 7 =5 -35 -7      =5 Since the numbers have same signs, the quotient is positive. | -35 |= 35 | -7 |= 7 } Divide these absolute values. 35 7 =5 -35 -7      =5 Since the numbers have same signs, the quotient is positive.

Example 7

18 −9 18 −9

| 18 |= 18 | -9 |= 9 } Divide these absolute values. 18 9 =2 18 -9      =-2 Since the numbers have opposite signs, the quotient is negative. | 18 |= 18 | -9 |= 9 } Divide these absolute values. 18 9 =2 18 -9      =-2 Since the numbers have opposite signs, the quotient is negative.

Practice Set B

Find the following quotients.

Exercise 7

−24 −6 −24 −6

Solution

4

Exercise 8

30 −5 30 −5

Solution

−6 −6

Exercise 9

−54 27 −54 27

Solution

−2 −2

Exercise 10

51 17 51 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 .

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

Example 9

Find the value of z= x−u s z= x−u s if x=57, u=51, and s=2 x=57, u=51, and s=2 .

Substituting these values we get

z= 57−51 2 = 6 2 =3 z= 57−51 2 = 6 2 =3

Practice Set C

Exercise 11

Find the value of −7(4−8)+2(1−11) −5(1−6)−17 −7(4−8)+2(1−11) −5(1−6)−17 .

Solution

1

Exercise 12

Find the value of P= n(n−3) 2n P= n(n−3) 2n , if n=5 n=5 .

Solution

1

Exercises

Find the value of each of the following expressions.

Exercise 13

(−2)(−8) (−2)(−8)

Solution

16

Exercise 14

(−3)(−9) (−3)(−9)

Exercise 15

(−4)(−8) (−4)(−8)

Solution

32

Exercise 16

(−5)(−2) (−5)(−2)

Exercise 17

(−6)(−9) (−6)(−9)

Solution

54

Exercise 18

(−3)(−11) (−3)(−11)

Exercise 19

(−8)(−4) (−8)(−4)

Solution

32

Exercise 20

(−1)(−6) (−1)(−6)

Exercise 21

(3)(−12) (3)(−12)

Solution

−36−36

Exercise 22

(4)(−18) (4)(−18)

Exercise 23

8(−4) 8(−4)

Solution

−32−32

Exercise 24

5(−6) 5(−6)

Exercise 25

9(−2) 9(−2)

Solution

−18−18

Exercise 26

7(−8) 7(−8)

Exercise 27

(−6)4 (−6)4

Solution

−24−24

Exercise 28

(−7)6 (−7)6

Exercise 29

(−10)9 (−10)9

Solution

−90−90

Exercise 30

(−4)12 (−4)12

Exercise 31

(10)(−6) (10)(−6)

Solution

−60−60

Exercise 32

(−6)(4) (−6)(4)

Exercise 33

(−2)(6) (−2)(6)

Solution

−12−12

Exercise 34

(−8)(7) (−8)(7)

Exercise 35

21 7 21 7

Solution

3

Exercise 36

42 6 42 6

Exercise 37

−39 3 −39 3

Solution

−13−13

Exercise 38

−20 10 −20 10

Exercise 39

−45 −5 −45 −5

Solution

9

Exercise 40

−16 −8 −16 −8

Exercise 41

25 −5 25 −5

Solution

−5−5

Exercise 42

36 −4 36 −4

Exercise 43

8−(−3) 8−(−3)

Solution

11

Exercise 44

14−(−20) 14−(−20)

Exercise 45

20−(−8) 20−(−8)

Solution

28

Exercise 46

−4−(−1) −4−(−1)

Exercise 47

0−4 0−4

Solution

−4−4

Exercise 48

0−(−1) 0−(−1)

Exercise 49

−6+1−7 −6+1−7

Solution

−12−12

Exercise 50

15−12−20 15−12−20

Exercise 51

1−6−7+8 1−6−7+8

Solution

−4−4

Exercise 52

2+7−10+2 2+7−10+2

Exercise 53

3(4−6) 3(4−6)

Solution

−6−6

Exercise 54

8(5−12) 8(5−12)

Exercise 55

−3(1−6) −3(1−6)

Solution

15

Exercise 56

−8(4−12)+2 −8(4−12)+2

Exercise 57

−4(1−8)+3(10−3) −4(1−8)+3(10−3)

Solution

49

Exercise 58

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

Exercise 59

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

Solution

−140−140

Exercise 60

3(4+1)−2(5) −2 3(4+1)−2(5) −2

Exercise 61

4(8+1)−3(−2) −4−2 4(8+1)−3(−2) −4−2

Solution

−7−7

Exercise 62

−1(3+2)+5 −1 −1(3+2)+5 −1

Exercise 63

−3(4−2)+(−3)(−6) −4 −3(4−2)+(−3)(−6) −4

Solution

−3−3

Exercise 64

−1(4+2) −1(4+2)

Exercise 65

−1(6−1) −1(6−1)

Solution

−5−5

Exercise 66

−(8+21) −(8+21)

Exercise 67

−(8−21) −(8−21)

Solution

13

Exercise 68

−(10−6) −(10−6)

Exercise 69

−(5−2) −(5−2)

Solution

−3−3

Exercise 70

−(7−11) −(7−11)

Exercise 71

−(8−12) −(8−12)

Solution

4

Exercise 72

−3[(−1+6)−(2−7)] −3[(−1+6)−(2−7)]

Exercise 73

−2[(4−8)−(5−11)] −2[(4−8)−(5−11)]

Solution

−4−4

Exercise 74

−5[(−1+5)+(6−8)] −5[(−1+5)+(6−8)]

Exercise 75

−[(4−9)+(−2−8)] −[(4−9)+(−2−8)]

Solution

15

Exercise 76

−3[−2(1−5)−3(−2+6)] −3[−2(1−5)−3(−2+6)]

Exercise 77

−2[−5(−10+11)−2(5−7)] −2[−5(−10+11)−2(5−7)]

Solution

2

Exercise 78

P=R−C. Find P if R=2000 and C=2500. P=R−C. Find P if R=2000 and C=2500.

Exercise 79

z= x−u s . Find z if x=23, u=25, and s=1. z= x−u s . Find z if x=23, u=25, and s=1.

Solution

−2−2

Exercise 80

z= x−u s . Find z if x=410, u=430, and s=2.5. z= x−u s . Find z if x=410, u=430, and s=2.5.

Exercise 81

m= 2s+1 T . Find m if s=−8 and T=5. m= 2s+1 T . Find m if s=−8 and T=5.

Solution

−3−3

Exercise 82

m= 2s+1 T . Find m if s=−10 and T=−5. m= 2s+1 T . Find m if s=−10 and T=−5.

Exercise 83

Use a calculator. F=( p 1 − p 2 ) r 4 ⋅9. Find F if  p 1 =10,   p 2 =8,r=3. F=( p 1 − p 2 ) r 4 ⋅9. Find F if  p 1 =10,   p 2 =8,r=3.

Solution

1458

Exercise 84

Use a calculator. F=( p 1 − p 2 ) r 4 ⋅9. Find F if  p 1 =12,   p 2 =7,r=2. F=( p 1 − p 2 ) r 4 ⋅9. Find F if  p 1 =12,   p 2 =7,r=2.

Exercise 85

P=n(n−1)(n−2). Find P if n=−4. P=n(n−1)(n−2). Find P if n=−4.

Solution

−120−120

Exercise 86

P=n(n−1)(n−2)(n−3). Find P if n=−5. P=n(n−1)(n−2)(n−3). Find P if n=−5.

Exercise 87

P= n(n−2)(n−4) 2n . Find P if n=−6. P= n(n−2)(n−4) 2n . Find P if n=−6.

Solution

40

Exercises for Review

Exercise 88

((Reference)) What natural numbers can replace x x so that the statement −4<x≤3 −4<x≤3 is true?

Exercise 89

((Reference)) Simplify (x+2y) 5 (3x−1) 7 (x+2y) 3 (3x−1) 6 (x+2y) 5 (3x−1) 7 (x+2y) 3 (3x−1) 6 .

Solution

( x+2y ) 2 ( 3x−1 ) ( x+2y ) 2 ( 3x−1 )

Exercise 90

((Reference)) Simplify ( x n y 3t ) 5 ( x n y 3t ) 5 .

Exercise 91

((Reference)) Find the sum. −6+(−5) −6+(−5) .

Solution

−11−11

Exercise 92

((Reference)) Find the difference. −2−(−8) −2−(−8) .

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