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Signed Numbers: Addition 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 add signed numbers. By the end of the module students should be able to add numbers with like signs and with unlike signs and be able to use the calculator for addition of signed numbers.

Section Overview

  • Addition of Numbers with Like Signs
  • Addition with Zero
  • Addition of Numbers with Unlike Signs
  • Calculators

Addition of Numbers with Like Signs

The addition of the two positive numbers 2 and 3 is performed on the number line as follows.

Begin at 0, the origin.

Since 2 is positive, move 2 units to the right.

Since 3 is positive, move 3 more units to the right.

We are now located at 5.

A number line with has marks for the numbers -2 to 7. An arrow is drawn from 0 to 2, and from 2 to 5.

Thus, 2 + 3 = 52 + 3 = 5 size 12{"2 "+" 3 "=" 5"} {}.

Summarizing, we have

( 2 positive units ) + ( 3 positive units ) = ( 5 positive units ) ( 2 positive units ) + ( 3 positive units ) = ( 5 positive units ) size 12{ \( "2 positive units" \) + \( "3 positive units" \) = \( "5 positive units" \) } {}

The addition of the two negative numbers -2 and -3 is performed on the number line as follows.

Begin at 0, the origin.

Since -2 is negative, move 2 units to the left.

Since -3 is negative, move 3 more units to the left.

We are now located at -5.

A number line with has marks for the numbers -7 to 2. An arrow is drawn from 0 to -2, and from -2 to -5.

Thus, (2)+(3)=5(2)+(3)=5 size 12{ \( - 2 \) + \( - 3 \) = - 5} {}.

Summarizing, we have

( 2 negative units ) + ( 3 negative units ) = ( 5 negative units ) ( 2 negative units ) + ( 3 negative units ) = ( 5 negative units ) size 12{ \( "2 negative units" \) + \( "3 negative units" \) = \( "5 negative units" \) } {}

Observing these two examples, we can suggest these relationships:

( positive number ) + ( positive number ) = ( positive number ) ( positive number ) + ( positive number ) = ( positive number ) size 12{ \( "positive number" \) + \( "positive number" \) = \( "positive number" \) } {}

( negative number ) + ( negative number ) = ( negative number ) ( negative number ) + ( negative number ) = ( negative number ) size 12{ \( "negative number" \) + \( "negative number" \) = \( "negative number" \) } {}

Adding Numbers with the Same Sign

Addition of numbers with like sign:
To add two real numbers that have the same sign, add the absolute values of the numbers and associate with the sum the common sign.

Sample Set A

Find the sums.

Example 1

3+73+7 size 12{3+7} {}

|3| = 3 |7| = 7 |3| = 3 |7| = 7 Add these absolute values.

3 + 7 = 10 3+7=10

The common sign is “+.”

Thus, 3 + 7 =+103 + 7 =+10 size 12{"3 "+" 7 "= +"10"} {}, or 3 + 7 = 103 + 7 = 10 size 12{"3 "+" 7 "=" 10"} {}.

Example 2

(4)+(9)(4)+(9) size 12{ \( - 4 \) + \( - 9 \) } {}

|- 4| = 4 |- 9| = 9 |- 4| = 4 |- 9| = 9 Add these absolute values.

4 + 9 = 13 4+9=13

The common sign is “ - -.“

Thus, (4)+(9)=13(4)+(9)=13 size 12{ \( - 4 \) + \( - 9 \) = - "13"} {}.

Practice Set A

Find the sums.

Exercise 1

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

Solution

14

Exercise 2

41 + 1141 + 11 size 12{"41 "+" 11"} {}

Solution

52

Exercise 3

(- 4)+(- 8)(- 4)+(- 8) size 12{ \( "-4" \) + \( "–8" \) } {}

Solution

-12

Exercise 4

36+936+9 size 12{ left ( - "36" right )+ left ( - 9 right )} {}

Solution

-45

Exercise 5

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

Solution

-34

Exercise 6

23+5323+53 size 12{ - { {2} over {3} } + left ( - { {5} over {3} } right )} {}

Solution

7373 size 12{- { {7} over {3} } } {}

Exercise 7

- 2.8+(- 4.6)- 2.8+(- 4.6) size 12{"–2" "." 8+ \( " –4" "." 6 \) } {}

Solution

7.47.4 size 12{-7 "." 4} {}

Exercise 8

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

Solution

1616 size 12{-"16"} {}

Addition With Zero

Addition with Zero

Notice that
(0)+(a positive number)=(that same positive number)(0)+(a positive number)=(that same positive number) size 12{ \( 0 \) + \( "a positive number" \) = \( "that same positive number" \) } {}.
(0)+(a negative number)=(that same negative number)(0)+(a negative number)=(that same negative number) size 12{ \( 0 \) + \( "a negative number" \) = \( "that same negative number" \) } {}.

The Additive Identity Is Zero

Since adding zero to a real number leaves that number unchanged, zero is called the additive identity.

Addition of Numbers with Unlike Signs

The addition 2 +(- 6)2 +(- 6) size 12{"2 "+ \( " –6" \) } {}, two numbers with unlike signs, can also be illustrated using the number line.

Begin at 0, the origin.

Since 2 is positive, move 2 units to the right.

Since -6 is negative, move, from 2, 6 units to the left.

We are now located at -4.

A number line with has marks for the numbers -5 to 4. An arrow is drawn from 2 to -4, and from 0 to 2.

We can suggest a rule for adding two numbers that have unlike signs by noting that if the signs are disregarded, 4 can be obtained by subtracting 2 from 6. But 2 and 6 are precisely the absolute values of 2 and -6. Also, notice that the sign of the number with the larger absolute value is negative and that the sign of the resulting sum is negative.

Adding Numbers with Unlike Signs

Addition of numbers with unlike signs: To add two real numbers that have unlike signs, subtract the smaller absolute value from the larger absolute value and associate with this difference the sign of the number with the larger absolute value.

Sample Set B

Find the following sums.

Example 3

7 +(- 2)7 +(- 2) size 12{"7 "+ \( "–2" \) } {}

| 7 | = 7 Larger absolute value. Sign is positive. | 7 | = 7 Larger absolute value. Sign is positive.       | - 2 | = 2 Smaller absolute value. | - 2 | = 2 Smaller absolute value.

Subtract absolute values: 7 - 2= 57 - 2= 5 size 12{"7 – 2 "=" 5"} {}.

Attach the proper sign: "+."

Thus, 7 +(- 2)=+57 +(- 2)=+5 size 12{"7 "+ \( "–2" \) = +5} {} or 7 +(- 2)= 57 +(- 2)= 5 size 12{"7 "+ \( " –2" \) =" 5"} {}.

Example 4

3 +(- 11 )3 +(- 11 ) size 12{"3 "+ \( "–11" \) } {}

| 3 | = 3 Smaller absolute value. | 3 | = 3 Smaller absolute value.       | - 11 | = 11 Larger absolute value. Sign is negative. | - 11 | = 11 Larger absolute value. Sign is negative.

Subtract absolute values: 11 - 3= 811 - 3= 8 size 12{"11 – 3 "=" 8"} {}.

Attach the proper sign: " - -."

Thus, 3+(- 11)=- 83+(- 11)=- 8 size 12{3+ \( "–11" \) =" –8"} {}.

Example 5

The morning temperature on a winter's day in Lake Tahoe was -12 degrees. The afternoon temperature was 25 degrees warmer. What was the afternoon temperature?

We need to find - 12 + 25- 12 + 25 size 12{"–12 "+" 25"} {}.

| - 12 | = 12 Smaller absolute value. | - 12 | = 12 Smaller absolute value.       | 25 | = 25 Larger absolute value. Sign is positive. | 25 | = 25 Larger absolute value. Sign is positive.

Subtract absolute values: 25 - 12= 1625 - 12= 16 size 12{"25 – 12 "=" 16"} {}.

Attach the proper sign: "+."

Thus, - 12+ 25 = 13- 12+ 25 = 13 size 12{"–12 "+" 25 "=" 13"} {}.

Practice Set B

Find the sums.

Exercise 9

4 +(- 3)4 +(- 3) size 12{"4 "+ \( "–3" \) } {}

Solution

1

Exercise 10

3+53+5 size 12{ - 3+5} {}

Solution

2

Exercise 11

15+(18)15+(18) size 12{"15"+ \( - "18" \) } {}

Solution

-3

Exercise 12

0+(6)0+(6) size 12{0+ \( - 6 \) } {}

Solution

-6

Exercise 13

26+1226+12 size 12{ - "26"+"12"} {}

Solution

-14

Exercise 14

35+(78)35+(78) size 12{"35"+ \( - "78" \) } {}

Solution

-43

Exercise 15

15 +(- 10)15 +(- 10) size 12{"15 "+ \( "–10" \) } {}

Solution

5

Exercise 16

1.5+(2)1.5+(2) size 12{1 "." 5+ \( - 2 \) } {}

Solution

-0.5

Exercise 17

8+08+0 size 12{ - 8+0} {}

Solution

-8

Exercise 18

0 +(0.57)0 +(0.57) size 12{"0 "+ \( 0 "." "57" \) } {}

Solution

0.57

Exercise 19

879+454879+454 size 12{ - "879"+"454"} {}

Solution

-425

Calculators

Calculators having the A square with a plus and minus sign. key can be used for finding sums of signed numbers.

Sample Set C

Use a calculator to find the sum of -147 and 84.

Table 1
    Display Reads  
Type 147 147  
Press A square with a plus and minus sign. -147 This key changes the sign of a number. It is different than - -.
Press + -147  
Type 84 84  
Press = -63  

Practice Set C

Use a calculator to find each sum.

Exercise 20

673 +(- 721)673 +(- 721) size 12{"673 "+ \( "–721" \) } {}

Solution

-48

Exercise 21

- 8,261 + 2,206- 8,261 + 2,206 size 12{"–8,261 "+" 2,206"} {}

Solution

-6,085

Exercise 22

- 1,345.6+(- 6,648.1)- 1,345.6+(- 6,648.1) size 12{"–1,345" "." 6+ \( "–6,648" "." 1 \) } {}

Solution

-7,993.7

Exercises

Find the sums in the following 27 problems. If possible, use a calculator to check each result.

Exercise 23

4+124+12 size 12{4+"12"} {}

Solution

16

Exercise 24

8+68+6 size 12{8+6} {}

Exercise 25

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

Solution

-15

Exercise 26

6+206+20 size 12{ left (-6 right )+ left (-"20" right )} {}

Exercise 27

10+210+2 size 12{"10"+ left (-2 right )} {}

Solution

8

Exercise 28

8+158+15 size 12{8+ left (-"15" right )} {}

Exercise 29

16+916+9 size 12{-"16"+ left (-9 right )} {}

Solution

-25

Exercise 30

22+122+1 size 12{-"22"+ left (-1 right )} {}

Exercise 31

0+120+12 size 12{0+ left (-"12" right )} {}

Solution

-12

Exercise 32

0+40+4 size 12{0+ left (-4 right )} {}

Exercise 33

0+240+24 size 12{0+ left ("24" right )} {}

Solution

24

Exercise 34

6+1+76+1+7 size 12{-6+1+ left (-7 right )} {}

Exercise 35

5+12+45+12+4 size 12{-5+ left (-"12" right )+ left (-4 right )} {}

Solution

-21

Exercise 36

5+55+5 size 12{-5+5} {}

Exercise 37

7+77+7 size 12{-7+7} {}

Solution

0

Exercise 38

14+1414+14 size 12{-"14"+"14"} {}

Exercise 39

4+44+4 size 12{4+ left (-4 right )} {}

Solution

0

Exercise 40

9+99+9 size 12{9+ left (-9 right )} {}

Exercise 41

84+6184+61 size 12{"84"+ left (-"61" right )} {}

Solution

23

Exercise 42

13+5613+56 size 12{"13"+ left (-"56" right )} {}

Exercise 43

452+124452+124 size 12{"452"+ left (-"124" right )} {}

Solution

328

Exercise 44

636+989636+989 size 12{"636"+ left (-"989" right )} {}

Exercise 45

1,811+(935)1,811+(935) size 12{1,"811"+ \( -"935" \) } {}

Solution

876

Exercise 46

373+14373+14 size 12{-"373"+ left (-"14" right )} {}

Exercise 47

1,211+441,211+44 size 12{-1,"211"+ left (-"44" right )} {}

Solution

-1,255

Exercise 48

47.03+22.7147.03+22.71 size 12{-"47" "." "03"+ left (-"22" "." "71" right )} {}

Exercise 49

1.998+4.0861.998+4.086 size 12{-1 "." "998"+ left (-4 "." "086" right )} {}

Solution

-6.084

Exercise 50

In order for a small business to break even on a project, it must have sales of $21,000. If the amount of sales was $15,000, by how much money did this company fall short?

Exercise 51

Suppose a person has $56 in his checking account. He deposits $100 into his checking account by using the automatic teller machine. He then writes a check for $84.50. If an error causes the deposit not to be listed into this person’s account, what is this person’s checking balance?

Solution

-$28.50

Exercise 52

A person borrows $7 on Monday and then $12 on Tuesday. How much has this person borrowed?

Exercise 53

A person borrows $11 on Monday and then pays back $8 on Tuesday. How much does this person owe?

Solution

$3.00

Exercises for Review

Exercise 54

((Reference)) Find the reciprocal of 856856 size 12{8 { {5} over {6} } } {}.

Exercise 55

((Reference)) Find the value of 512+71813512+71813 size 12{ { {5} over {"12"} } + { {7} over {"18"} } - { {1} over {3} } } {}.

Solution

17361736 size 12{ { {"17"} over {"36"} } } {}

Exercise 56

((Reference)) Round 0.01628 to the nearest tenth.

Exercise 57

((Reference)) Convert 62% to a fraction.

Solution

62100=315062100=3150 size 12{ { {"62"} over {"100"} } = { {"31"} over {"50"} } } {}

Exercise 58

((Reference)) Find the value of 1212 size 12{ lline -"12" rline } {}.

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