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Basic Operations with Real Numbers: Subtraction 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: understand the definition of subtraction, be able to subtract signed numbers.

Overview

  • Definition of Subtraction
  • Subtraction of Signed Numbers

Definition of Subtraction

We know from our experience with arithmetic that the subtraction 5−2 5−2 produces 3, that is, 5−2=3 5−2=3 . Illustrating this process on the number line suggests a rule for subtracting signed numbers.

A number line with arrows on each end, labeled from negative one to seven in increments of one. There is a curved arrow starting from zero, and pointing towards five. There is another curved arrow starting from five, and pointing towards three.

We begin at 0, the origin.
 Since 5 is positive, we move 5 units to the right.
 Then, we move 2 units to the left to get to 3. (This reminds us of addition with a negative number.)

This illustration suggests that 5−2 5−2 is the same as 5+(−2) 5+(−2) .
This leads us directly to the definition of subtraction.

Definition of Subtraction

If a a and b b are real numbers, a−b a−b is the same as a+(−b) a+(−b) , where −b −b is the opposite of b b .

Subtraction of Signed Numbers

The preceding definition suggests the rule for subtracting signed numbers.

Subtraction of Signed Numbers

To perform the subtraction a−b a−b , add the opposite of b b to a a , that is, change the sign of b b and add.

Sample Set A

Perform the subtractions.

Example 1

5−3=5+(−3)=2 5−3=5+(−3)=2

A number line with arrows on each end, labeled from negative three to six in increments of one. There is a curved arrow starting from zero, and pointing towards five. There is another curved arrow starting from five, and pointing towards two.

Example 2

4−9=4+(−9)=−5 4−9=4+(−9)=−5

A number line with arrows on each end, labeled from negative six to six in increments of one. There is a curved arrow starting from zero, and pointing towards four. There is another curved arrow starting from four, and pointing towards negative five.

Example 3

−4−6=−4+(−6)=−10 −4−6=−4+(−6)=−10

A number line with arrows on each end, labeled from negative twelve to three in increments of one. There is curved arrow starting from zero, and pointing towards negative four. There is another curved arrow starting from negative four, and pointing towards negative ten.

Example 4

−3−(−12)=−3+12=9 −3−(−12)=−3+12=9

A number line with arrows on each end, labeled from negative five to twelve in increments of one. There is a curved arrow starting from zero, and pointing towards negative three. There is another curved arrow starting from negative three, and pointing towards nine.

Example 5

0−(−15)=0+15=15 0−(−15)=0+15=15

A number line with arrows on each end, labeled from negative two to eighteen in increments of one. There is a curved arrow starting from zero, and pointing towards fifteen.

Example 6

The high temperature today in Lake Tahoe was 26 ∘ F 26 ∘ F . The low temperature tonight is expected to be − 7 ∘ F − 7 ∘ F . How many degrees is the temperature expected to drop?
We need to find the difference between 26 and −7 −7 .

26−(−7)=26+7=33 26−(−7)=26+7=33

Thus, the expected temperature drop is 33 ∘ F 33 ∘ F .

Example 7

−6−(−5)−10 = −6+5+(−10) = (−6+5)+(−10) = −1+(−10) = −11 −6−(−5)−10 = −6+5+(−10) = (−6+5)+(−10) = −1+(−10) = −11

Practice Set A

Perform the subtractions.

Exercise 1

9−6 9−6

Solution

3

Exercise 2

6−9 6−9

Solution

−3 −3

Exercise 3

0−7 0−7

Solution

−7 −7

Exercise 4

1−14 1−14

Solution

−13 −13

Exercise 5

−8−12 −8−12

Solution

−20 −20

Exercise 6

−21−6 −21−6

Solution

−27 −27

Exercise 7

−6−(−4) −6−(−4)

Solution

−2 −2

Exercise 8

8−(−10) 8−(−10)

Solution

18

Exercise 9

1−(−12) 1−(−12)

Solution

13

Exercise 10

86−(−32) 86−(−32)

Solution

118

Exercise 11

0−16 0−16

Solution

−16 −16

Exercise 12

0−(−16) 0−(−16)

Solution

16

Exercise 13

0−(8) 0−(8)

Solution

−8 −8

Exercise 14

5−(−5) 5−(−5)

Solution

10

Exercise 15

24−(−(−24)) 24−(−(−24))

Solution

0

Exercises

For the following exercises, perform the indicated operations.

Exercise 16

8−3 8−3

Solution

5

Exercise 17

12−7 12−7

Exercise 18

5−6 5−6

Solution

−1−1

Exercise 19

14−30 14−30

Exercise 20

2−15 2−15

Solution

−13−13

Exercise 21

5−18 5−18

Exercise 22

1−7 1−7

Solution

−6−6

Exercise 23

4−11 4−11

Exercise 24

−6−5 −6−5

Solution

−11−11

Exercise 25

−8−14 −8−14

Exercise 26

−1−12 −1−12

Solution

−13−13

Exercise 27

−4−4 −4−4

Exercise 28

−6−8 −6−8

Solution

−14−14

Exercise 29

−1−12 −1−12

Exercise 30

−5−(−3) −5−(−3)

Solution

−2−2

Exercise 31

−11−(−8) −11−(−8)

Exercise 32

−7−(−12) −7−(−12)

Solution

5

Exercise 33

−2−(−10) −2−(−10)

Exercise 34

−4−(−15) −4−(−15)

Solution

11

Exercise 35

−11−(−16) −11−(−16)

Exercise 36

−1−(−6) −1−(−6)

Solution

5

Exercise 37

−8−(−14) −8−(−14)

Exercise 38

−15−(−10) −15−(−10)

Solution

−5−5

Exercise 39

−11−(−4) −11−(−4)

Exercise 40

−16−(−8) −16−(−8)

Solution

−8−8

Exercise 41

−12−(−11) −12−(−11)

Exercise 42

0−6 0−6

Solution

−6−6

Exercise 43

0−15 0−15

Exercise 44

0−(−7) 0−(−7)

Solution

7

Exercise 45

0−(−10) 0−(−10)

Exercise 46

67−38 67−38

Solution

29

Exercise 47

142−85 142−85

Exercise 48

816−1140 816−1140

Solution

−324−324

Exercise 49

105−421 105−421

Exercise 50

−550−(−121) −550−(−121)

Solution

−429−429

Exercise 51

−15.016−(4.001) −15.016−(4.001)

Exercise 52

−26+7−52 −26+7−52

Solution

−71−71

Exercise 53

−15−21−(−2) −15−21−(−2)

Exercise 54

−104−(−216)−(−52) −104−(−216)−(−52)

Solution

164

Exercise 55

−0.012−(−0.111)−(0.035) −0.012−(−0.111)−(0.035)

Exercise 56

[ 5+(−6) ]−[ 2+(−4) ] [ 5+(−6) ]−[ 2+(−4) ]

Solution

1

Exercise 57

[ 2+(−8) ]−[ 5+(−7) ] [ 2+(−8) ]−[ 5+(−7) ]

Exercise 58

[ 4+(−11) ]−[ 2+(−10) ] [ 4+(−11) ]−[ 2+(−10) ]

Solution

1

Exercise 59

[ 9+(−6) ]−[ 4+(−12) ] [ 9+(−6) ]−[ 4+(−12) ]

Exercise 60

(11−8)−(1−6) (11−8)−(1−6)

Solution

8

Exercise 61

(5−12)−(4−10) (5−12)−(4−10)

Exercise 62

(1−10)−(2−15) (1−10)−(2−15)

Solution

4

Exercise 63

(0−8)−(4−12) (0−8)−(4−12)

Exercise 64

(−4+7)−(2−5) (−4+7)−(2−5)

Solution

6

Exercise 65

(−6+2)−(5−11) (−6+2)−(5−11)

Exercise 66

[ −8+(−5+3) ]−[ 9−(−3−5) ] [ −8+(−5+3) ]−[ 9−(−3−5) ]

Solution

−27−27

Exercise 67

[ −4+(−1+6) ]−[ 7−(−6−1) ] [ −4+(−1+6) ]−[ 7−(−6−1) ]

Exercise 68

[ 2−(−6+10) ]−[ 1−(2−11) ] [ 2−(−6+10) ]−[ 1−(2−11) ]

Solution

−12−12

Exercise 69

[ 5−(−2−5) ]−[ 2−(−1−4) ] [ 5−(−2−5) ]−[ 2−(−1−4) ]

Exercise 70

When a particular machine is operating properly, its meter will read 34. If a broken bearing in the machine causes the meter reading to drop by 45 units, what is the meter reading?

Solution

−11−11

Exercise 71

The low temperature today in Denver was − 4 ∘ F − 4 ∘ F and the high was 42 ∘ F 42 ∘ F . What is the temperature difference?

Exercises for Review

Exercise 72

((Reference)) Use the distributive property to expand 4x(5y+11) 4x(5y+11) .

Solution

20xy+44x 20xy+44x

Exercise 73

((Reference)) Simplify 2 (3 x 2 y 2 ) 3 (2 x 4 y 3 ) 0 27 x 4 y 3 2 (3 x 2 y 2 ) 3 (2 x 4 y 3 ) 0 27 x 4 y 3 . Assume x≠0,y≠0 x≠0,y≠0 .

Exercise 74

((Reference)) Simplify | −( 4 2 + 2 2 − 3 2 ) | | −( 4 2 + 2 2 − 3 2 ) | .

Solution

11

Exercise 75

((Reference)) Find the sum. −8+(−14) −8+(−14) .

Exercise 76

((Reference)) Find the sum. 3+(−6) 3+(−6) .

Solution

−3−3

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