Download Positive and Negative Integers

0.4
Positive and Negative
Integers
0.4
OBJECTIVES
1. Represent integers on a number line
2. Order signed numbers
3. Evaluate numerical expressions involving
absolute value
When numbers are used to represent physical quantities (altitudes, temperatures, and
amounts of money are examples), it may be necessary to distinguish between positive and
negative quantities. It is convenient to represent these quantities with plus () or minus
() signs. For instance,
The altitude of Mount Whitney is 14,495 feet (ft) above sea level (14,495).
14,495 ft
Mount Whitney
The altitude of Death Valley is 282 ft below sea level (282).
282 ft
Death Valley
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The temperature in Chicago is 10° below zero (10°).
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CHAPTER 0
AN ARITHMETIC REVIEW
An account could show a gain of $100 (100), or a loss of $100 (100).
These numbers suggest the need to extend the whole numbers to include both positive numbers (like 100) and negative numbers (like 282).
To represent the negative numbers, we extend the number line to the left of zero and
name equally spaced points.
Numbers used to name points to the right of zero are positive numbers. They are written
with a positive () sign or with no sign at all.
+6 and 9 are positive numbers
Numbers used to name points to the left of zero are negative numbers. They are always
written with a negative () sign.
3 and 20 are negative numbers
Read “negative 3.”
Positive and negative numbers considered together are signed numbers.
Here is the number line extended to include both positive and negative numbers.
NOTE 0 is not considered a
Zero is neither positive
nor negative
3 2 1
Negative numbers
0
1
2
3
Positive numbers
The numbers used to name the points shown on the number line above are called the
integers. The integers consist of the natural numbers, their negatives, and the number 0. We
can represent the set of integers by
NOTE The dots are called
ellipses and indicate that the
pattern continues.
{. . . , 3, 2, 1, 0, 1, 2, 3, . . .}
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signed number.
POSITIVE AND NEGATIVE INTEGERS
SECTION 0.4
37
Example 1
Representing Integers on the Number Line
Represent the following integers on the number line shown.
3, 12, 8, 15, 7
12
7 3
15 10 5
8
0
5
15
10
15
CHECK YOURSELF 1
Represent the following integers on a number line.
1, 9, 4, 11, 8, 20
15 10 5
0
5
10
15
20
The set of numbers on the number line is ordered. The numbers get smaller moving to
the left on the number line and larger moving to the right.
4 3 2
1
0
1
2
3
4
When a set of numbers is written from smallest to largest, the numbers are said to be in
ascending order.
Example 2
Ordering Signed Numbers
Place each set of numbers in ascending order.
(a) 9, 5, 8, 3, 7
From smallest to largest, the numbers are
8, 5, 3, 7, 9
Note that this is the order in which the numbers appear on a number line
as we move from left to right.
(b) 3, 2, 18, 20, 13
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From smallest to largest, the numbers are
20, 13, 2, 3, 18
CHECK YOURSELF 2
Place each set of numbers in ascending order.
(a) 12, 13, 15, 2, 8, 3
(b) 3, 6, 9, 3, 8
CHAPTER 0
AN ARITHMETIC REVIEW
The least and greatest numbers in a set are called the extreme values. The least element is
called the minimum and the greatest element is called the maximum.
Example 3
Labeling Extreme Values
For each set of numbers, determine the minimum and maximum values.
(a) 9, 5, 8, 3, 7
From our previous ordering of these numbers, we see that 8, the least element, is the minimum, and 9, the greatest element, is the maximum.
(b) 3, 2, 18, 20, 13
20 is the minimum and 18 is the maximum.
CHECK YOURSELF 3
For each set of numbers, determine the minimum and maximum values.
(a) 12, 13, 15, 2, 8, 3
(b) 3, 6, 9, 3, 8
Integers are not the only kind of signed numbers. Decimals and fractions can also be
thought of as signed numbers.
Example 4
Identifying Signed Numbers that are Integers
Which of the following signed numbers are also integers?
(a) 145 is an integer.
(b) 28 is an integer.
(c) 0.35 is not an integer.
2
(d) is not an integer.
3
CHECK YOURSELF 4
Which of the following signed numbers are also integers?
23
1054
0.23
0
500
4
5
Sometimes we refer to the negative of a number as its “opposite.” But what is the opposite
of the opposite of a number? It is the number itself. The next example illustrates.
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POSITIVE AND NEGATIVE INTEGERS
SECTION 0.4
39
Example 5
Find the Opposite for Each Number
The opposite of 5 is 5.
The opposite of 9 is 9.
(a) 5
(b) 9
CHECK YOURSELF 5
Find the opposite for each number.
(b) 12
(a) 17
An important idea for our work in this chapter is the absolute value of a number. This
represents the distance of the point named by the number from the origin on the number
line.
5 units
5
5 units
0
5
The absolute value of 5 is 5. The absolute value of 5 is also 5.
The absolute value of a positive number or zero is itself. The absolute value of a negative number is its opposite.
In symbols we write
5 5
and
Read “the absolute
value of 5.”
5 5
Read “the absolute
value of negative 5.”
The absolute value of a number does not depend on whether the number is to the right
or to the left of the origin, but on its distance from the origin.
Example 6
© 2001 McGraw-Hill Companies
Simplifying Absolute Value Expressions
(a) 7 7
(b) 7 7
(c) 7 7
This is the negative, or opposite, of the absolute value of negative 7.
(d) 10 10 10 10 20
Absolute value bars serve as another set of grouping symbols, so do the operation inside
first.
(e) 8 3 5 5
(f) 8 3 8 3 5
Here, evaluate the absolute values, then subtract.
CHAPTER 0
AN ARITHMETIC REVIEW
CHECK YOURSELF 6
Evaluate.
(a) 8
(d) 94
(b) 8
(e) 9 4
(c) 8
(f) 9 4
CHECK YOURSELF ANSWERS
1.
119
20 15 10 5
1
0
4
8
5
20
10
15
20
2. (a) 13, 8, 3, 2, 12, 15
(b) 9, 3, 3, 6, 8
3. (a) minimum is 13; maximum is 15 (b) minimum is 9; maximum is 8
4. 23, 1054, 0, and 500
5. (a) 17; (b) 12
6. (a) 8; (b) 8; (c) 8; (d) 13; (e) 5; (f ) 5
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40
Name
0.4 Exercises
Section
Date
Represent each quantity with a signed number.
1. An altitude of 400 feet (ft) above sea level
ANSWERS
2. An altitude of 80 ft below sea level
3. A loss of $200
1.
4. A profit of $400
2.
5. A decrease in population of 25,000
3.
6. An increase in population of 12,500
4.
Represent the integers on the number lines shown.
5.
7. 5, 15, 18, 8, 3
8. 18, 4, 5, 13, 9
20
10
0
10
20
6.
7.
20
10
0
10
20
8.
Which numbers in the following sets are integers?
9.
10.
2
5, , 175, 234, 0.64
9
3
45, 0.35, , 700, 26
5
Place each of the following sets in ascending order.
11. 3, 5, 2, 0, 7, 1, 8
12. 2, 7, 1, 8, 6, 1, 0
13. 9, 2, 11, 4, 6, 1, 5
14. 23, 18, 5, 11, 15, 14, 20
9.
10.
11.
12.
13.
14.
15.
15. 6, 7, 7, 6, 3, 3
16. 12, 13, 14, 14, 15, 15
For each set, determine the maximum and minimum values.
17. 5, 6, 0, 10, 3, 15, 1, 8
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18. 9, 1, 3, 11, 4, 2, 5, 2
19. 21, 15, 0, 7, 9, 16, 3, 11
16.
17.
18.
19.
20. 22, 0, 22, 31, 18, 5, 3
20.
21. 3, 0, 1, 2, 5, 4, 1
21.
22. 2, 7, 3, 5, 10, 5
22.
Find the opposite of each number.
23. 15
24. 18
23.
24.
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ANSWERS
25.
25. 11
26. 34
27. 19
28. 5
29. 7
30. 54
26.
27.
28.
29.
30.
31.
Evaluate.
32.
31. 17
32. 28
33. 10
34. 7
35. 3
36. 5
37. 8
38. 13
39. 23
40. 43
41. 99
42. 1111
43. 44
44. 55
45. 158
46. 113
47. 15 8
48. 11 3
49. 92
50. 74
51. 87
52. 94
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
48.
49.
50.
51.
52.
42
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47.
ANSWERS
53.
Label each statement as true or false.
54.
53. All whole numbers are integers.
54. All nonzero integers are signed numbers.
55.
55. All integers are whole numbers.
56.
56. All signed numbers are integers.
57. All negative integers are whole numbers.
58. Zero is neither positive nor negative.
57.
58.
Place absolute value bars in the proper location on the left side of the expression so that
the equation is true.
59.
60.
59. 6 (2) 4
61.
60. 8 (3) 5
62.
61. 6 (2) 8
63.
62. 8 (3) 11
Represent each quantity with a signed number.
63. Soil erosion. The erosion of 5 centimeters (cm) of topsoil from an Iowa corn field.
64. Soil formation. The formation of 2.5 cm of new topsoil on the African savanna.
64.
65.
66.
65. Checking accounts. The withdrawal of $50 from a checking account.
66. Saving accounts. The deposit of $200 in a savings account.
67. Temperature. The temperature change pictured.
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60°F
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© 2001 McGraw-Hill Companies
1:00 P.M.
67.
68.
50°F
2:00 P.M.
68. Stocks. An increase of 75 points in the Dow-Jones average.
JUNE
JULY
43
ANSWERS
69.
69. Baseball. An eight-game losing streak by the local baseball team.
70.
70. Population. An increase of 25,000 in the population of the city.
71.
71. Positive trade balance. A country exported $90,000,000 more than it imported,
72.
72. Negative trade balance. A country exported $60,000,000 less than it imported,
creating a positive trade balance.
creating a negative trade balance.
73.
For each collection of numbers given in exercises 73 to 76, answer the following:
74.
(a)
(b)
(c)
(d)
75.
76.
Which number is smallest?
Which number lies farthest from the origin?
Which number has the largest absolute value?
Which number has the smallest absolute value?
73. 6, 3, 8, 7, 2
74. 8, 3, 5, 4, 9
77.
75. 2, 6, 1, 0, 2, 5
76. 9, 0, 2, 3, 6
77. Simplify each of the following:
(7)
((7))
(((7)))
Based on your answers, generalize your results.
Answers
3. 200
1. 400 or (400)
15
7.
10
35
0
5. 25,000
18
10
20
9. 5, 175, 234
7, 5, 1, 0, 2, 3, 8
13. 11, 6, 2, 1, 4, 5, 9
7, 6, 3, 3, 6, 7
17. Max: 15; Min: 6
19. Max: 21, Min: 15
Max: 5; Min: 2
23. 15
25. 11
27. 19
29. 7
31. 17
10
35. 3
37. 8
39. 5
41. 18
43. 0
45. 7
7
49. 11
51. 1
53. True
55. False
57. False
6 (2) 4
61. 62 8
63. 5
65. 50
10°F
69. 8
71. 90,000,000
6; 8; 8; 2
75. 2; 6; 6; 0
77.
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11.
15.
21.
33.
47.
59.
67.
73.
20
8
44