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Signed Numbers: Subtraction 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 subtract signed numbers. By the end of the module students should understand the definition of subtraction, be able to subtract signed numbers and be able to use a calculator to subtract signed numbers.

Section Overview

  • Definition of Subtraction
  • The Process of Subtraction
  • Calculators

Definition of Subtraction

We know from experience with arithmetic that the subtraction 5 - 25 - 2 size 12{"5 – 2"} {} produces 3, that is 5 - 2= 35 - 2= 3 size 12{"5 – 2 "=" 3"} {}. We can suggest a rule for subtracting signed numbers by illus­trating this process on the number line.

A number line with hash marks from -1 to 7. There is an arrow from 0 to 5 and from 5 to 3.

Begin at 0, the origin.

Since 5 is positive, move 5 units to the right.

Then, move 2 units to the left to get to 6. (This reminds us of addition with a negative number.)

From this illustration we can see that 5 - 25 - 2 size 12{"5 – 2"} {} is the same as 5 +(-2)5 +(-2) size 12{"5 "+ \( "–2" \) } {}. This leads us directly to the definition of subtraction.

Definition of Subtraction

If aa size 12{a} {} and bb size 12{b} {} are real numbers, a - ba - b size 12{"a–b"} {} is the same as a +(-b)a +(-b) size 12{"a "+ \( "–b" \) } {}, where -b-b size 12{ –b} {} is the opposite of bb size 12{b} {}.

The Process of Subtraction

From this definition, we suggest the following rule for subtracting signed numbers.

Subtraction of Signed Numbers

To perform the subtraction a - ba - b size 12{"a–b"} {}, add the opposite of bb size 12{b} {} to aa size 12{a} {}, that is, change the sign of bb size 12{b} {} and add.

Sample Set A

Perform the indicated subtractions.

Example 1

5−3=5+(−3)=25−3=5+(−3)=2 size 12{5 - 3=5+ \( - 3 \) =2} {}

Example 2

4 - 9= 4 +(- 9)=- 54 - 9= 4 +(- 9)=- 5 size 12{"4–9 "=" 4 "+ \( "–9" \) =" –5"} {}

Example 3

−4−6=−4+(−6)=−10−4−6=−4+(−6)=−10 size 12{ - 4 - 6= - 4+ \( - 6 \) = - "10"} {}

Example 4

−3−(−12)=−3+12=9−3−(−12)=−3+12=9 size 12{ - 3 - \( - "12" \) = - 3+"12"=9} {}

Example 5

0 -(- 15)= 0 + 15 = 150 -(- 15)= 0 + 15 = 15 size 12{"0 – " \( "–15" \) =" 0 "+" 15 "=" 15"} {}

Example 6

The high temperature today in Lake Tahoe was 26°F. The low temperature tonight is expected to be -7°F. How many degrees is the temperature expected to drop?

We need to find the difference between 26 and -7.

26 - ( - 7 ) = 26 + 7 = 33 26 - ( - 7 ) = 26 + 7 = 33 size 12{"26 – " \( "–7" \) =" 26 "+" 7 "=" 33"} {}

Thus, the expected temperature drop is 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 indicated subtractions.

Exercise 1

9−69−6 size 12{9 - 6} {}

Solution

3

Exercise 2

6−96−9 size 12{6 - 9} {}

Solution

-3

Exercise 3

0−70−7 size 12{0 - 7} {}

Solution

-7

Exercise 4

1−141−14 size 12{1 - "14"} {}

Solution

-13

Exercise 5

−8−12−8−12 size 12{ - 8 - "12"} {}

Solution

-20

Exercise 6

−21−6−21−6 size 12{ - "21" - 6} {}

Solution

-27

Exercise 7

−6−(−4)−6−(−4) size 12{ - 6 - \( - 4 \) } {}

Solution

-2

Exercise 8

8−(−10)8−(−10) size 12{8 - \( - "10" \) } {}

Solution

18

Exercise 9

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

Solution

13

Exercise 10

86−(−32)86−(−32) size 12{"86" - \( - "32" \) } {}

Solution

118

Exercise 11

0−160−16 size 12{0 - "16"} {}

Solution

-16

Exercise 12

0−(−16)0−(−16) size 12{0 - \( - "16" \) } {}

Solution

16

Exercise 13

0−(8)0−(8) size 12{0 - \( 8 \) } {}

Solution

-8

Exercise 14

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

Solution

10

Exercise 15

24−[−(−24)]24−[−(−24)] size 12{"24" - \[ - \( - "24" \) \] } {}

Solution

0

Calculators

Calculators can be used for subtraction of signed numbers. The most efficient calculators are those with a A box with a plus and minus sign. key.

Sample Set B

Use a calculator to find each difference.

Example 8

3,187−8,7193,187−8,719 size 12{3,"187" - 8,"719"} {}

Table 1
Display Reads
Type 3187 3187
Press - 3187
Type 8719 8719
Press = -5532

Thus, 3,187−8,719=−5,5323,187−8,719=−5,532 size 12{3,"187" - 8,"719"= - 5,"532"} {}.

Example 9

−156−(−211)−156−(−211) size 12{ - "156" - \( - "211" \) } {}

Method A:

Table 2
Display Reads
Type 156 156
Press A box with a plus and minus sign. -156
Type - -156
Press 211 211
Type A box with a plus and minus sign. -211
Press = 55

Thus, −156−(−211)=55−156−(−211)=55 size 12{ - "156" - \( - "211" \) ="55"} {}.

Method B:

We manually change the subtraction to an addition and change the sign of the number to be subtracted.

−156−−211−156−−211 size 12{ - "156" - left ( - "211" right )} {} becomes −156+211−156+211 size 12{ - "156"+"211"} {}

Table 3
Display Reads
Type 156 156
Press A box with a plus and minus sign. -156
Press + -156
Type 211 211
Press = 55

Practice Set B

Use a calculator to find each difference.

Exercise 16

44−31544−315 size 12{"44" - "315"} {}

Solution

-271

Exercise 17

12.756−15.00312.756−15.003 size 12{"12" "." "756" - "15" "." "003"} {}

Solution

-2.247

Exercise 18

−31.89−44.17−31.89−44.17 size 12{ - "31" "." "89" - "44" "." "17"} {}

Solution

-76.06

Exercise 19

−0.797−(−0.615)−0.797−(−0.615) size 12{ - 0 "." "797" - \( - 0 "." "615" \) } {}

Solution

-0.182

Exercises

For the following 18 problems, perform each subtraction. Use a calculator to check each result.

Exercise 20

8−38−3 size 12{8-3} {}

Solution

5

Exercise 21

12−712−7 size 12{"12"-7} {}

Exercise 22

5−65−6 size 12{5-6} {}

Solution

-1

Exercise 23

14−3014−30 size 12{"14"-"30"} {}

Exercise 24

−6−8−6−8 size 12{-6-8} {}

Solution

-14

Exercise 25

−1−12−1−12 size 12{-1-"12"} {}

Exercise 26

−5−−3−5−−3 size 12{-5- left (-3 right )} {}

Solution

-2

Exercise 27

−11−−8−11−−8 size 12{-"11"- left (-8 right )} {}

Exercise 28

0−60−6 size 12{0-6} {}

Solution

-6

Exercise 29

0−150−15 size 12{0-"15"} {}

Exercise 30

0−−70−−7 size 12{0- left (-7 right )} {}

Solution

7

Exercise 31

0−−100−−10 size 12{0- left (-"10" right )} {}

Exercise 32

67−3867−38 size 12{"67"-"38"} {}

Solution

29

Exercise 33

142−85142−85 size 12{"142"-"85"} {}

Exercise 34

816−1140816−1140 size 12{"816"-"1140"} {}

Solution

-324

Exercise 35

105−421105−421 size 12{"105"-"421"} {}

Exercise 36

−550−−121−550−−121 size 12{-"550"- left (-"121" right )} {}

Solution

-429

Exercise 37

−15.016−4.001−15.016−4.001 size 12{-"15" "." "016"- left (4 "." "001" right )} {}

For the following 4 problems, perform the indicated operations.

Exercise 38

−26+7−52−26+7−52 size 12{-"26"+7-"52"} {}

Solution

-71

Exercise 39

−15−21−−2−15−21−−2 size 12{-"15"-"21"- left (-2 right )} {}

Exercise 40

−104−−216−−52−104−−216−−52 size 12{-"104"- left (-"216" right )- left (-"52" right )} {}

Solution

164

Exercise 41

−0.012−−0.111−0.035−0.012−−0.111−0.035 size 12{-0 "." "012"- left (-0 "." "111" right )- left (0 "." "035" right )} {}

Exercise 42

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

Exercise 43

The low temperature today in Denver was −4°−4° size 12{-4°} {}F and the high was −42°−42° size 12{-"42"°} {}F. What is the temperature difference?

Exercises for Review

Exercise 44

((Reference)) Convert 16.021516.0215 size 12{"16" "." "02" { {1} over {5} } } {} to a decimal.

Solution

16.022

Exercise 45

((Reference)) Find 4.01 of 6.2.

Exercise 46

((Reference)) Convert 516516 size 12{ { {5} over {"16"} } } {} to a percent.

Solution

31.25%

Exercise 47

((Reference)) Use the distributive property to compute the product: 15⋅8215⋅82 size 12{"15" cdot "82"} {}.

Exercise 48

((Reference)) Find the sum: 16+(−21)16+(−21) size 12{"16"+ \( -"21" \) } {}.

Solution

-5

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