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# Basic Operations with Real Numbers: 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 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 .

Summarizing, we have

(2positiveunits)+(3positiveunits)=(5positiveunits) (2positiveunits)+(3positiveunits)=(5positiveunits)

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 .

Summarizing, we have

(2negativeunits)+(3negativeunits)=(5negativeunits) (2negativeunits)+(3negativeunits)=(5negativeunits)

These two examples suggest that

(positivenumber)+(positivenumber)=(positivenumber) (negativenumber)+(negativenumber)=(negativenumber) (positivenumber)+(positivenumber)=(positivenumber) (negativenumber)+(negativenumber)=(negativenumber)

### 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

Addtheseabsolutevalues. | 3 |=3 | 7 |=7 } 3+7=10 Thecommonsignis"+." Addtheseabsolutevalues. | 3 |=3 | 7 |=7 } 3+7=10 Thecommonsignis"+."

3+7=+10or3+7=10 3+7=+10or3+7=10

### Example 2

(4)+(9) (4)+(9)

Addtheseabsolutevalues. | 4 |=4 | 9 |=9 } 4+9=13 Thecommonsignis"." Addtheseabsolutevalues. | 4 |=4 | 9 |=9 } 4+9=13 Thecommonsignis"."

(4)+(9)=13 (4)+(9)=13

## Practice Set A

Find the sums.

8+6 8+6

14

41+11 41+11

52

(4)+(8) (4)+(8)

12 12

### Exercise 4

(36)+(9) (36)+(9)

45 45

14+(20) 14+(20)

34 34

### Exercise 6

2 3 +( 5 3 ) 2 3 +( 5 3 )

7 3 7 3

### Exercise 7

2.8+(4.6) 2.8+(4.6)

#### Solution

7.4 7.4

Notice that

(0)+(apositivenumber)=(thatsamepositivenumber) (0)+(anegativenumber)=(thatsamenegativenumber) (0)+(apositivenumber)=(thatsamepositivenumber) (0)+(anegativenumber)=(thatsamenegativenumber)

### 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 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 Largerabsolutevalue. Signis"+". | 2 |=2 Smallerabsolutevalue. | 7 |=7 Largerabsolutevalue. Signis"+". | 2 |=2 Smallerabsolutevalue.

Subtractabsolutevalues: 72=5. Attachthepropersign: "+". Subtractabsolutevalues: 72=5. Attachthepropersign: "+".

7+(2)=+5or7+(2)=5 7+(2)=+5or7+(2)=5

### Example 4

3+(11) 3+(11)

| 3 |=3 Smallerabsolutevalue. | 11 |=11 Largerabsolutevalue. Signis"". | 3 |=3 Smallerabsolutevalue. | 11 |=11 Largerabsolutevalue. Signis"".

Subtractabsolutevalues: 113=8. Attachthepropersign: "". Subtractabsolutevalues: 113=8. Attachthepropersign: "".

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 Smallerabsolutevalue. | 25 |=25 Largerabsolutevalue. Signis"+". | 12 |=12 Smallerabsolutevalue. | 25 |=25 Largerabsolutevalue. Signis"+".

Subtractabsolutevalues: 2512=13. Attachthepropersign: "+". Subtractabsolutevalues: 2512=13. Attachthepropersign: "+".

12+25=13 12+25=13

Thus, the afternoon temperature is 13 degrees.

### Example 6

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.

4+(3) 4+(3)

1

3+5 3+5

2

15+(18) 15+(18)

3 3

0+(6) 0+(6)

6 6

26+12 26+12

14 14

35+(78) 35+(78)

43 43

15+(10) 15+(10)

5

1.5+(2) 1.5+(2)

0.5 0.5

8+0 8+0

8 8

### Exercise 17

0+(0.57) 0+(0.57)

0.57 0.57

879+454 879+454

425 425

### Exercise 19

1345.6+(6648.1) 1345.6+(6648.1)

7993.7 7993.7

## Exercises

Find the sums for the the following problems.

4+12 4+12

16

8+6 8+6

6+2 6+2

8

7+9 7+9

### Exercise 24

(3)+(12) (3)+(12)

1515

### Exercise 25

(6)+(20) (6)+(20)

(4)+(8) (4)+(8)

1212

### Exercise 27

(11)+(8) (11)+(8)

### Exercise 28

(16)+(8) (16)+(8)

2424

### Exercise 29

(2)+(15) (2)+(15)

14+(3) 14+(3)

11

21+(4) 21+(4)

14+(6) 14+(6)

8

18+(2) 18+(2)

10+(8) 10+(8)

2

40+(31) 40+(31)

### Exercise 36

(3)+(12) (3)+(12)

1515

### Exercise 37

(6)+(20) (6)+(20)

10+(2) 10+(2)

8

8+(15) 8+(15)

2+(6) 2+(6)

88

11+(14) 11+(14)

9+(6) 9+(6)

1515

1+(1) 1+(1)

16+(9) 16+(9)

2525

22+(1) 22+(1)

0+(12) 0+(12)

1212

0+(4) 0+(4)

0+(24) 0+(24)

24

6+1+(7) 6+1+(7)

### Exercise 50

5+(12)+(4) 5+(12)+(4)

2121

5+5 5+5

7+7 7+7

0

14+14 14+14

4+(4) 4+(4)

0

9+(9) 9+(9)

84+(61) 84+(61)

23

13+(56) 13+(56)

### Exercise 58

452+(124) 452+(124)

328

### Exercise 59

636+(989) 636+(989)

### Exercise 60

1811+(935) 1811+(935)

876

### Exercise 61

373+(14) 373+(14)

### Exercise 62

1221+(44) 1221+(44)

12651265

### Exercise 63

47.03+(22.71) 47.03+(22.71)

### Exercise 64

1.998+(4.086) 1.998+(4.086)

6.0846.084

### Exercise 65

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

### Exercise 66

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

2020

### Exercise 67

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

### Exercise 68

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

1717

### Exercise 69

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

### Exercise 70

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

1818

### Exercise 71

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

### Exercise 72

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

14

### Exercise 73

9+[(4)+7] 9+[(4)+7]

### Exercise 74

14+[(3)+5] 14+[(3)+5]

16

### Exercise 75

[2+(7)]+(11) [2+(7)]+(11)

### Exercise 76

[14+(8)]+(2) [14+(8)]+(2)

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 .

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 | .

8

### Exercise 85

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

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