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Addition 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 add numbers with like signs and unlike signs, understand addition with zero.

Overview

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

Addition of Numbers with Like Signs

Let us add the two positive numbers 2 and 3. We perform this addition on the number line as follows.

We begin at 0, the origin.
Since 2 is positive, we move 2 units to the right.
Since 3 is positive, we move 3 more units to the right.
We are now located at 5.
Thus, 2+3=5 2+3=5 .

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

Summarizing, we have

(2 positive units)+(3 positive units)=(5 positive units) (2 positive units)+(3 positive units)=(5 positive units)

Now let us add the two negative numbers −2 −2 and −3 −3 . We perform this addition on the number line as follows.

We begin at 0, the origin.
Since −2 −2 is negative, we move 2 units to the left.
Since −3 −3 is negative, we move 3 more units to the left.
We are now located at −5 −5 .

Thus, (−2)+(−3)=−5 (−2)+(−3)=−5 .

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

Summarizing, we have

(2 negative units)+(3 negative units)=(5 negative units) (2 negative units)+(3 negative units)=(5 negative units)

These two examples suggest that

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

Adding Numbers with the Same Sign

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

Sample Set A

Find the sums.

Example 1

3+7 3+7

                                Add these absolute values. | 3 |=3 | 7 |=7 }        3+7=10                                 The common sign is "+."                                 Add these absolute values. | 3 |=3 | 7 |=7 }        3+7=10                                 The common sign is "+."

3+7=+10           or           3+7=10 3+7=+10           or           3+7=10

Example 2

(−4)+(−9) (−4)+(−9)

                                   Add these absolute values. | −4 |=4 | −9 |=9 }        4+9=13                                    The common sign is "−."                                    Add these absolute values. | −4 |=4 | −9 |=9 }        4+9=13                                    The common sign is "−."

(−4)+(−9)=−13 (−4)+(−9)=−13

Practice Set A

Find the sums.

Exercise 1

8+6 8+6

Solution

14

Exercise 2

41+11 41+11

Solution

52

Exercise 3

(−4)+(−8) (−4)+(−8)

Solution

−12 −12

Exercise 4

(−36)+(−9) (−36)+(−9)

Solution

−45 −45

Exercise 5

−14+(−20) −14+(−20)

Solution

−34 −34

Exercise 6

− 2 3 +( − 5 3 ) − 2 3 +( − 5 3 )

Solution

− 7 3 − 7 3

Exercise 7

−2.8+(−4.6) −2.8+(−4.6)

Solution

−7.4 −7.4

Addition with Zero

Notice that

Addition with 0

(0) + (a positive number) = (that same ​positive number) (0) + (a negative number) = (that same ​negative number) (0) + (a positive number) = (that same ​positive number) (0) + (a negative number) = (that same ​negative number)

The Additive Identity Is 0

Since adding 0 to a real number leaves that number unchanged, 0 is called the additive identity.

Addition of Numbers with Unlike Signs

Now let us perform the addition 2+(−6) 2+(−6) . These two numbers have unlike signs. This type of addition can also be illustrated using the number line.

We begin at 0, the origin.
Since 2 is positive, we move 2 units to the right.
Since −6 −6 is negative, we move, from the 2, 6 units to the left.
We are now located at −4 −4 .

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

A rule for adding two numbers that have unlike signs is suggested by noting that if the signs are disregarded, 4 can be obtained from 2 and 6 by subtracting 2 from 6. But 2 and 6 are precisely the absolute values of 2 and −6 −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

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

Sample Set B

Find the following sums.

Example 3

7+(−2) 7+(−2)

| 7 |=7 ︸ Larger absolute value. Sign  is "+".     | −2 |=2 ︸ Smaller absolute value. | 7 |=7 ︸ Larger absolute value. Sign  is "+".     | −2 |=2 ︸ Smaller absolute value.

Subtract absolute values: 7−2=5. Attach the proper sign: "+". Subtract absolute values: 7−2=5. Attach the proper sign: "+".

7+(−2)=+5          or          7+(−2)=5 7+(−2)=+5          or          7+(−2)=5

Example 4

3+(−11) 3+(−11)

| 3 |=3 ︸ Smaller absolute value.     | −11 |=11 ︸ Larger absolute value. Sign is "−". | 3 |=3 ︸ Smaller absolute value.     | −11 |=11 ︸ Larger absolute value. Sign is "−".

Subtract absolute values: 11−3=8. Attach the proper sign: "−". Subtract absolute values: 11−3=8. Attach the proper sign: "−".

3+(−11)=−8 3+(−11)=−8

Example 5

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

We need to find −12+25 −12+25 .

| −12 |=12 ︸ Smaller absolute value.     | 25 |=25 ︸ Larger​ absolute value. Sign is "+". | −12 |=12 ︸ Smaller absolute value.     | 25 |=25 ︸ Larger​ absolute value. Sign is "+".

Subtract absolute values: 25−12=13. Attach the proper sign: "+". Subtract absolute values: 25−12=13. Attach the proper sign: "+".

−12+25=13 −12+25=13

Thus, the afternoon temperature is 13 degrees.

Example 6

Use a calculator. Add −147+84 −147+84 .                                                     Display Reads

Type      147         147 Press     +/−   −147 Press    +           −147 Type        84          84 Press    =           −63 Type      147         147 Press     +/−   −147 Press    +           −147 Type        84          84 Press    =           −63

Practice Set B

Find the sums.

Exercise 8

4+(−3) 4+(−3)

Solution

1

Exercise 9

−3+5 −3+5

Solution

2

Exercise 10

15+(−18) 15+(−18)

Solution

−3 −3

Exercise 11

0+(−6) 0+(−6)

Solution

−6 −6

Exercise 12

−26+12 −26+12

Solution

−14 −14

Exercise 13

35+(−78) 35+(−78)

Solution

−43 −43

Exercise 14

15+(−10) 15+(−10)

Solution

5

Exercise 15

1.5+(−2) 1.5+(−2)

Solution

−0.5 −0.5

Exercise 16

−8+0 −8+0

Solution

−8 −8

Exercise 17

0+(0.57) 0+(0.57)

Solution

0.57 0.57

Exercise 18

−879+454 −879+454

Solution

−425 −425

Exercise 19

−1345.6+(−6648.1) −1345.6+(−6648.1)

Solution

−7993.7 −7993.7

Exercises

Find the sums for the the following problems.

Exercise 20

4+12 4+12

Solution

16

Exercise 21

8+6 8+6

Exercise 22

6+2 6+2

Solution

8

Exercise 23

7+9 7+9

Exercise 24

(−3)+(−12) (−3)+(−12)

Solution

−15−15

Exercise 25

(−6)+(−20) (−6)+(−20)

Exercise 26

(−4)+(−8) (−4)+(−8)

Solution

−12−12

Exercise 27

(−11)+(−8) (−11)+(−8)

Exercise 28

(−16)+(−8) (−16)+(−8)

Solution

−24−24

Exercise 29

(−2)+(−15) (−2)+(−15)

Exercise 30

14+(−3) 14+(−3)

Solution

11

Exercise 31

21+(−4) 21+(−4)

Exercise 32

14+(−6) 14+(−6)

Solution

8

Exercise 33

18+(−2) 18+(−2)

Exercise 34

10+(−8) 10+(−8)

Solution

2

Exercise 35

40+(−31) 40+(−31)

Exercise 36

(−3)+(−12) (−3)+(−12)

Solution

−15−15

Exercise 37

(−6)+(−20) (−6)+(−20)

Exercise 38

10+(−2) 10+(−2)

Solution

8

Exercise 39

8+(−15) 8+(−15)

Exercise 40

−2+(−6) −2+(−6)

Solution

−8−8

Exercise 41

−11+(−14) −11+(−14)

Exercise 42

−9+(−6) −9+(−6)

Solution

−15−15

Exercise 43

−1+(−1) −1+(−1)

Exercise 44

−16+(−9) −16+(−9)

Solution

−25−25

Exercise 45

−22+(−1) −22+(−1)

Exercise 46

0+(−12) 0+(−12)

Solution

−12−12

Exercise 47

0+(−4) 0+(−4)

Exercise 48

0+(24) 0+(24)

Solution

24

Exercise 49

−6+1+(−7) −6+1+(−7)

Exercise 50

−5+(−12)+(−4) −5+(−12)+(−4)

Solution

−21−21

Exercise 51

−5+5 −5+5

Exercise 52

−7+7 −7+7

Solution

0

Exercise 53

−14+14 −14+14

Exercise 54

4+(−4) 4+(−4)

Solution

0

Exercise 55

9+(−9) 9+(−9)

Exercise 56

84+(−61) 84+(−61)

Solution

23

Exercise 57

13+(−56) 13+(−56)

Exercise 58

452+(−124) 452+(−124)

Solution

328

Exercise 59

636+(−989) 636+(−989)

Exercise 60

1811+(−935) 1811+(−935)

Solution

876

Exercise 61

−373+(−14) −373+(−14)

Exercise 62

−1221+(−44) −1221+(−44)

Solution

−1265−1265

Exercise 63

−47.03+(−22.71) −47.03+(−22.71)

Exercise 64

−1.998+(−4.086) −1.998+(−4.086)

Solution

−6.084−6.084

Exercise 65

[(−3)+(−4)]+[(−6)+(−1)] [(−3)+(−4)]+[(−6)+(−1)]

Exercise 66

[(−2)+(−8)]+[(−3)+(−7)] [(−2)+(−8)]+[(−3)+(−7)]

Solution

−20−20

Exercise 67

[(−3)+(−8)]+[(−6)+(−12)] [(−3)+(−8)]+[(−6)+(−12)]

Exercise 68

[(−8)+(−6)]+[(−2)+(−1)] [(−8)+(−6)]+[(−2)+(−1)]

Solution

−17−17

Exercise 69

[4+(−12)]+[12+(−3)] [4+(−12)]+[12+(−3)]

Exercise 70

[5+(−16)]+[4+(−11)] [5+(−16)]+[4+(−11)]

Solution

−18−18

Exercise 71

[2+(−4)]+[17+(−19)] [2+(−4)]+[17+(−19)]

Exercise 72

[10+(−6)]+[12+(−2)] [10+(−6)]+[12+(−2)]

Solution

14

Exercise 73

9+[(−4)+7] 9+[(−4)+7]

Exercise 74

14+[(−3)+5] 14+[(−3)+5]

Solution

16

Exercise 75

[2+(−7)]+(−11) [2+(−7)]+(−11)

Exercise 76

[14+(−8)]+(−2) [14+(−8)]+(−2)

Solution

4

Exercise 77

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

Exercise 78

Suppose a person has $56.00 $56.00 in his checking account. He deposits $100.00 $100.00 into his checking account by using the automatic teller machine. He then writes a check for $84.50 $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 −$28.50

Exercise 79

A person borrows $7.00 $7.00 on Monday and then $12.00 $12.00 on Tuesday. How much has this person borrowed?

Exercise 80

A person borrows $11.00 $11.00 on Monday and then pays back $8.00 $8.00 on Tuesday. How much does this person owe?

Solution

$3.00 $3.00

Exercises for Review

Exercise 81

((Reference)) Simplify 4( 7 2 −6⋅ 2 3 ) 2 2 4( 7 2 −6⋅ 2 3 ) 2 2 .

Exercise 82

((Reference)) Simplify 35 a 6 b 2 c 5 7 b 2 c 4 35 a 6 b 2 c 5 7 b 2 c 4 .

Solution

5 a 6 c 5 a 6 c

Exercise 83

((Reference)) Simplify ( 12 a 8 b 5 4 a 5 b 2 ) 3 ( 12 a 8 b 5 4 a 5 b 2 ) 3 .

Exercise 84

((Reference)) Determine the value of | −8 | | −8 | .

Solution

8

Exercise 85

((Reference)) Determine the value of (| 2 |+ | 4 | 2 )+ | −5 | 2 (| 2 |+ | 4 | 2 )+ | −5 | 2 .

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