Download Adding Signed Numbers

9.2
Adding Signed Numbers
OBJECTIVES
9.2
1. Use a number line to find the sum of two signed
numbers
2. Add two numbers with the same sign
3. Add two numbers with opposite signs
In the previous section, we introduced the idea of signed numbers. Now we will examine
the four arithmetic operations (addition, subtraction, multiplication, and division) and
see how those operations are performed when signed numbers are involved. We start by
considering addition.
An application may help. As before, let’s represent a gain of money as a positive number
and a loss as a negative number.
If you gain $300 and then gain $400, the result is a
gain of $700:
300 400 700
If you loss $300 and then lose $400, the result is a
loss of $700:
300 (400) 700
If you gain $300 and then lose $400, the result is a
loss of $100:
300 (400) 100
If you lose $300 and then gain $400, the result is a
gain of $100:
300 400 100
The number line can be used to illustrate the addition of integers. Starting at the origin,
we move to the right for positive numbers and to the left for negative numbers.
Example 1
Adding Signed Numbers
Add
4
2
.
3
3
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4
3
0
2
3
1
2
4
2
Start at the origin and move units to the right. Then move more to the right to find the
3
3
sum. So we have
4
2
6
2
3
3
3
677
CHAPTER 9
THE REAL NUMBER SYSTEM
CHECK YOURSELF 1
Add.
(a) 5 6
(b)
7
5
4
4
The number line will also help you visualize the sum of two negative numbers.
Remember, we move to the left for negative numbers.
Example 2
Adding Signed Numbers
(a) Add (3) (4).
4
3
7
3
0
Start at the origin and move 3 units to the left. Then move 4 more units to the left to find the
sum. From the graph we see that the sum is
(3) (4) 7
2 2.
(b) Add 3
1
12
2
32
1
32
0
3
As before, we start at the origin. From that point move units left. Then move another
2
1
unit left to find the sum. In this case
2
2 2 2
3
1
CHECK YOURSELF 2
Add.
(a) (4) (5)
(c) (5) (15)
(b) (3) (7)
5
3
(d) 2
2
You have probably noticed some helpful patterns in the previous examples. These patterns will allow you to do the work mentally without having to use the number line. Look
at the following rule.
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678
ADDING SIGNED NUMBERS
SECTION 9.2
679
Rules and Properties: Adding Signed Numbers Case 1:
Same Sign
NOTE This means that the sum
of two positive numbers is
positive and the sum of two
negative numbers is negative.
If two numbers have the same sign, add their absolute values. Give the sum the
sign of the original numbers.
Example 3
Adding Signed Numbers
(a) (8) (5) 13
Add the absolute values (8 5 13),
and give the sum the sign () of the
original numbers.
(b) [(3) (4)] (6)
Add inside the brackets as your first step.
(7) (6) 13
CHECK YOURSELF 3
Add mentally.
(a) 7 9
(c) (5.8) (3.2)
(b) (7) (9)
(d) [(5) (2)] (3)
Let’s again use the number line to illustrate the addition of two numbers. This time the
numbers will have different signs.
Example 4
Adding Signed Numbers
(a) Add 3 (6).
6
3
3
0
3
First move 3 units to the right of the origin. Then move 6 units to the left.
3 (6) 3
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(b) Add 4 7.
7
4
4
0
3
This time move 4 units to the left of the origin as the first step. Then move 7 units to the
right.
4 7 3
680
CHAPTER 9
THE REAL NUMBER SYSTEM
CHECK YOURSELF 4
Add.
(a) 7 (5)
(c) 4 9
(b) 4 (8)
(d) 7 3
You have no doubt noticed that, in adding a positive number and a negative number,
sometimes the sum is positive and sometimes it is negative. This depends on which of the
numbers has the larger absolute value. This leads us to the second part of our addition rule.
Rules and Properties: Adding Signed Numbers Case 2:
Different Signs
If two numbers have different signs, subtract their absolute values, the smaller
from the larger. Give the result the sign of the number with the larger absolute
value.
Example 5
Adding Signed Numbers
(a) 7 (19) 12
Because the two numbers have different signs, subtract the absolute values (19 7 12).
The sum has the sign () of the number with the larger absolute value, 19.
(b) 13 7 6
Subtract the absolute values (13 7 6). The sum has the sign () of the number with the
larger absolute value, 13.
(c) 8.2 4.5 3.7
Subtract the absolute values (8.2 4.5 3.7). The sum has the sign () of the number
with the larger absolute value, 8.2.
CHECK YOURSELF 5
Add mentally.
(a) 5 (14)
(d) 7 (8)
(b) 7 (8)
7
2
(e) 3
3
(c) 8 15
(f) 5.3 (2.3)
There are two properties of addition that we should mention before concluding this
section. First, the sum of any number and 0 is always that number. In symbols,
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NOTE Remember, signed
numbers can be fractions and
decimals as well as integers.
ADDING SIGNED NUMBERS
Rules and Properties:
SECTION 9.2
681
Additive Identity Property
For any number a,
NOTE No number loses its
identity after addition with 0.
Zero is called the additive
identity.
a00aa
Example 6
Adding Signed Numbers
Add.
(a) 9 0 9
(b) 0 (8) 8
(c) (25) 0 25
CHECK YOURSELF 6
Add.
(a) 8 0
NOTE The opposite of a
number is also called the
additive inverse of that
number.
(b) 0 (7)
(c) (36) 0
We’ll need one further definition to state our second property. Every number has an
opposite. It corresponds to a point the same distance from the origin as the given number,
but in the opposite direction.
3
NOTE 3 and 3 are opposites.
3
3
0
3
The opposite of 9 is 9.
The opposite of 15 is 15.
Our second property states that the sum of any number and its opposite is 0.
Rules and Properties:
NOTE Here a represents the
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opposite of the number a. The
sum of any number and its
opposite, or additive inverse,
is 0.
Additive Inverse Property
For any number a, there exists a number a such that
a (a) (a) a 0
Example 7
Adding Inverses
(a) 9 (9) 0
(b) 15 15 0
(c) (2.3) 2.3 0
(d)
4
4
0
5
5
682
CHAPTER 9
THE REAL NUMBER SYSTEM
CHECK YOURSELF 7
Add.
(a) (17) 17
1
1
(c) 3
3
NOTE All properties of
addition from Section 1.2 apply
when negative numbers are
involved.
(b) 12 (12)
(d) (1.6) 1.6
We can now use the associative and commutative properties of addition, introduced in
Section 1.2, to find the sum when more than two signed numbers are involved. Example 8
illustrates these properties.
Example 8
Adding Signed Numbers
property to reverse the order of
addition for 3 and 5. We then
group 5 and 5. Do you see
why?
(5) (3) 5
(5) 5 (3)
[(5) 5] (3)
0 (3) 3
CHECK YOURSELF 8
Add.
(a) (4) 5 (3)
(b) (8) 4 8
CHECK YOURSELF ANSWERS
1.
3.
5.
7.
(a) 11; (b) 3
2. (a) 9; (b) 10; (c) 20; (d) 4
(a) 16; (b) 16; (c) 9; (d) 10
4. (a) 2; (b) 4; (c) 5; (d) 4
(a) 9; (b) 15; (c) 7; (d) 1; (e) 3; (f) 3
6. (a) 8; (b) 7; (c) 36
(a) 0; (b) 0; (c) 0; (d) 0
8. (a) 2; (b) 4
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NOTE We use the commutative
Name
Exercises
9.2
Section
Date
Add.
1. 3 6
ANSWERS
2. 5 9
3. 11 5
4. 8 7
1.
2.
3.
4.
5.
5.
3
5
4
4
6.
6.
7
8
3
3
7.
7.
1
4
2
5
8.
8.
2
5
3
9
9.
10.
9. (2) (3)
11.
13.
10. (1) (9)
12.
14.
3
7
5
5
1
3
2
8
3
12
5
5
4
3
7
14
11.
12.
13.
14.
15.
16.
15. (1.6) (2.3)
16. (3.5) (2.6)
17.
18.
17. 9 (3)
18. 10 (4)
19. 8 (14)
20. 7 (11)
19.
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20.
21.
21.
4 2
3
1
22.
3 6
2
1
22.
23.
23.
5 20
4
9
24.
6 12
11
5
24.
683
ANSWERS
25.
25. 11.4 13.4
26. 5.2 9.2
27. 3.6 7.6
28. 2.6 4.9
29. 9 0
30. 15 0
31. 18 0
32. 14 0
33. 7 (7)
34. 12 (12)
35. 14 14
36. 5 5
37. 9 (17) 9
38. 15 (3) (15)
39. 2 5 (11) 4
40. 7 (9) (5) 6
41. (4) 6 (3) 0
42. 7 (3) 5 (11)
43. 1 (2) 3 (4)
44. (9) 0 (2) 12
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
47.
45.
5
4
5
3
3
3
46. 13
4
6
5
5
5
48.
49.
47. 50.
3
7
1
2
4
4
49. 2.3 (5.4) (2.9)
684
48.
1
5
1
3
6
2
50. (5.4) (2.1) (3.5)
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46.
ANSWERS
Evaluate and round your answer to the nearest tenth.
51.
52.
51. (4.1967 5.2943) (3.1698)
53.
54.
52. 5.3297 4.1897 (3.2869)
55.
53. 7.19863 4.8629 3.2689 (5.7936)
56.
57.
54. (3.6829) 4.5687 7.28967 (5.1623)
58.
59.
60.
Evaluate each of the following expressions.
55. 3 (4)
56. (11) 9
61.
62.
57. 17 8
58. 27 14
63.
64.
59. 3 2 (4)
60. 2 7 (5)
65.
66.
61. 2 (3) (3) 2
62. 8 (10) 12 14
67.
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68.
Label each of the following statements as true or false.
69.
63. 10 6 6 10
64. 5 (9) 9 5
70.
65. 3 2 3 2
66. 8 3 8 3
Place absolute value bars in the proper location on the left side of the expression to make
the statement true.
67. 3 7 10
68. 5 9 14
69. 6 7 (4) 3
70. 10 15 (9) 4
685
Answers
1. 9
3. 16
15. 3.9
7.
19. 6
13
10
9. 5
21.
1
4
11. 2
23. 7
20
13. 25. 2
7
8
27. 4
9
31. 18
33. 0
35. 0
37. 17
39. 0
41. 1
2
45. 2
47. 3
49. 6
51. 2.1
53. 4.9
1
57. 9
59. 5
61. 2
63. True
65. False
3 7 10
69. 6 7 (4) 3
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29.
43.
55.
67.
17. 6
5. 2
686