Download the real numbers

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58. E F
59. (D E ) F
60. (D F) E
61. D (E F)
62. D (F E)
63. (D F) (E F)
64. (D E ) (F E)
65. (D E) (D F)
66. (D F) (D E)
86. 1, 3, 5, 7
Use one of the symbols , , , , or in the blank of
each statement to make it correct.
67. D _____ x x is an odd natural number
68. E _____ x x is an even natural number smaller than 9
3 _____ D
D _____ E D F _____ F
E E _____ F
D _____ F F D
70.
72.
74.
76.
78.
3 _____ D
D E _____ D
3 E _____ F
E E _____ F
E _____ F F E
List the elements in each set.
79. x x is an even natural number less than 20
80. x x is a natural number greater than 6
81. x x is an odd natural number greater than 11
82. x x is an odd natural number less than 14
83. x x is an even natural number between 4 and 79
84. x x is an odd natural number between 12 and 57
In This Section
●
The Rational Numbers
●
Graphing on the Number Line
●
The Irrational Numbers
●
Intervals of Real Numbers
1.2
(1-7) 7
Write each set using set-builder notation. Answers may vary.
85. 3, 4, 5, 6
57. E F
69.
71.
73.
75.
77.
The Real Numbers
87. 5, 7, 9, 11, . . .
88. 4, 5, 6, 7, . . .
89. 6, 8, 10, 12, . . . , 82
90. 9, 11, 13, 15, . . . , 51
GET TING MORE INVOLVED
91. Discussion. If A and B are finite sets, could A B be
infinite? Explain.
92. Cooperative learning. Work with a small group to answer
the following questions. If A B and B A, then what can
you conclude about A and B? If (A B) (A B), then
what can you conclude about A and B?
93. Discussion. What is wrong with each statement?
Explain.
a) 3 1, 2, 3
b) 3 1, 2, 3
c) 94. Exploration. There are only two possible subsets of
1, namely, and 1.
a) List all possible subsets of 1, 2. How many are
there?
b) List all possible subsets of 1, 2, 3. How many are
there?
c) Guess how many subsets there are of 1, 2, 3, 4. Verify your guess by listing all the possible subsets.
d) How many subsets are there for 1, 2, 3, . . . , n?
THE REAL NUMBERS
The set of real numbers is the basic set of numbers used in algebra. There are many
different types of real numbers. To understand better the set of real numbers, we will
study some of the subsets of numbers that make up this set.
The Rational Numbers
We have already discussed the set of counting or natural numbers. The set of natural
numbers together with the number 0 is called the set of whole numbers. The whole
numbers together with the negatives of the counting numbers form the set of
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Helpful Hint
Page 8
The Real Numbers
integers. We use the letters N, W, and J to name these sets:
A negative number can be used to
represent a loss or a debt.The number 10 could represent a debt of
$10, a temperature of 10° below
zero, or an altitude of 10 feet below
sea level.
N 1, 2, 3, . . .
W 0, 1, 2, 3, . . .
J . . . , 3, 2, 1, 0, 1, 2, 3, . . .
The natural numbers
The whole numbers
The integers
Rational numbers are numbers that are written as ratios or as quotients of integers. We use the letter Q (for quotient) to name the set of rational numbers and write
the set in set-builder notation as follows:
Q a a and b are integers, with b 0
b
The rational numbers
Examples of rational numbers are
7,
9
,
4
17
,
10
0,
0
,
4
3
,
1
47
,
3
and
2
.
6
Note that the rational numbers are the numbers that can be expressed as a ratio (or
7
quotient) of integers. The integer 7 is rational because we can write it as .
1
Another way to describe rational numbers is by using their decimal form. To
obtain the decimal form, we divide the denominator into the numerator. For some rational numbers the division terminates, and for others it continues indefinitely. These
examples show some rational numbers and their equivalent decimal forms:
Calculator Close-Up
Display a fraction on a graphing
calculator, then press ENTER to convert to a decimal. The fraction feature converts a repeating decimal
into a fraction. Try this with your
calculator.
26
0.26
100
4
4.0
1
1
0.25
4
2
0.6666 . . .
3
25
0.252525 . . .
99
4177
4.2191919 . . .
990
Terminating decimal
Terminating decimal
Terminating decimal
The single digit 6 repeats.
The pair of digits 25 repeats.
The pair of digits 19 repeats.
Rational numbers are defined as ratios of integers, but they can be described also by
their decimal form. The rational numbers are those decimal numbers whose digits
either repeat or terminate.
E X A M P L E
1
Subsets of the rational numbers
Determine whether each statement is true or false.
a) 0 W
b) N J
c) 0.75 J
d) J Q
Solution
a) True, because 0 is a whole number.
b) True, because every natural number is also a member of the set of integers.
c) False, because the rational number 0.75 is not an integer.
■
d) True, because the rational numbers include the integers.
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Calculator Close-Up
These Calculator Close-ups are designed to help reinforce the concepts of algebra, not replace them.
Do not rely too heavily on your calculator or use it to replace the algebraic methods taught in this course.
(1-9) 9
The Real Numbers
Graphing on the Number Line
To construct a number line, we draw a straight line and label any convenient point with
the number 0. Now we choose any convenient length and use it to locate points to the
right of 0 as points corresponding to the positive integers and points to the left of 0 as
points corresponding to the negative integers. See Fig. 1.4. The numbers corresponding to the points on the line are called the coordinates of the points. The distance
between two consecutive integers is called a unit, and it is the same for any two consecutive integers. The point with coordinate 0 is called the origin. The numbers on the
number line increase in size from left to right. When we compare the size of any two
numbers, the larger number lies to the right of the smaller one on the number line.
1 unit
4
1 unit
Origin
3
2
1
0
1
2
3
4
FIGURE 1.4
It is often convenient to illustrate sets of numbers on a number line. The set of
integers, J, is illustrated or graphed as in Fig. 1.5. The three dots to the right and
left on the number line indicate that the integers go on indefinitely in both
directions.
…4
3
2
1
0
1
2
3
4 …
FIGURE 1.5
E X A M P L E
2
Study Tip
Take notes in class. Write down
everything you can. As soon as possible after class, rewrite your notes.
Fill in details and make corrections.
Make a note of examples and exercises in the text that are similar to examples in your notes. If your instructor takes the time to work an
example in class,it is a“good bet”that
your instructor expects you to understand the concepts involved.
Graphing on the number line
List the elements of each set and graph each set on a number line.
a) x x is a whole number less than 4
b) a a is an integer between 3 and 9
c) y y is an integer greater than 3
Solution
a) The whole numbers less than 4 are 0, 1, 2, and 3. Figure 1.6 shows the graph of
this set.
3
2
1
0
1
2
3
4
5
FIGURE 1.6
b) The integers between 3 and 9 are 4, 5, 6, 7, and 8. The graph is shown in
Fig. 1.7.
1
2
3
4
5
6
7
8
9
FIGURE 1.7
c) The integers greater than 3 are 2, 1, 0, 1, and so on. To indicate the continuing pattern, we use a series of dots on the graph in Fig. 1.8.
5
4
3
2
1
0
FIGURE 1.8
1
2
3 …
■
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The Real Numbers
The Irrational Numbers
Some numbers can be expressed as ratios of integers and some cannot. Numbers
that cannot be expressed as a ratio of integers are called irrational numbers. To
better understand irrational numbers consider the positive square root of 2 (in sym). The square root of 2 is a number that you can multiply by itself to get 2.
bols 2
So we can write (using a raised dot for times)
Calculator Close-Up
A calculator gives a 10-digit rational
approximation for 2. Note that if
the approximate value is squared,
you do not get 2.
The screen shot that appears on
this page and in succeeding pages
may differ from the display on your
calculator. You may have to consult
your manual to get the desired
results.
␲
Circumference
Diameter
␲
Circumference
Diameter
FIGURE 1.9
C
D
2
2.
2
If we look for 2 on a calculator or in Appendix B, we find 1.414. But if we
multiply 1.414 by itself, we get
(1.414)(1.414) 1.999396.
is not equal to 1.414 (in symbols, 2
1.414). The square root of 2 is apSo 2
proximately 1.414 (in symbols, 2 1.414). There is no terminating or repeating
decimal that will give exactly 2 when multiplied by itself. So 2 is an irrational
, 5, and 7
are also
number. It can be shown that other square roots such as 3
irrational numbers.
In decimal form the rational numbers either repeat or terminate. The irrational
numbers neither repeat nor terminate. Examine each of these numbers to see that
it has a continuing pattern that guarantees that its digits will neither repeat nor
terminate:
0.606000600000600000006 . . .
0.15115111511115 . . .
3.12345678910111213 . . .
So each of these numbers is an irrational number.
Since we generally work with rational numbers, the irrational numbers may
seem to be unnecessary. However, irrational numbers occur in some very real
situations. Over 2000 years ago people in the Orient and Egypt observed that the
ratio of the circumference and diameter is the same for any circle. This constant
value was proven to be an irrational number by Johann Heinrich Lambert in 1767.
Like other irrational numbers, it does not have any convenient representation as a
decimal number. This number has been given the name (Greek letter pi). See
Fig. 1.9. The value of rounded to nine decimal places is 3.141592654. When
using in computations, we frequently use the rational number 3.14 as an approximate value for .
The set of irrational numbers I and the set of rational numbers Q have no numbers in common and together form the set of real numbers R. The set of real numbers can be visualized as the set of all points on the number line. Two real numbers
are equal if they correspond to the same point on the number line. See Fig. 1.10.
2.99
3
––
√ 2
2
1
1
—
3
1
—
2
0
––
√2
1
––
√5
2
3
FIGURE 1.10
Figure 1.11 illustrates the relationship between the set of real numbers and the
various subsets that we have been discussing.
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Study Tip
The Real Numbers
1-11) 11
Real numbers (R)
Start a personal library. This book as
well as other books from which you
study should be the basis for your
library. You can also add books to
your library at garage sale prices
when your bookstore sells its old
texts. If you need to reference some
material in the future, it is much easier to use a book with which you are
familiar.
Rational numbers (Q)
2, —
5
155
—
—, —
—,
3
7
13
Irrational numbers (I)
5.26, 0.37373737…
–– –– ––
√2 , √6 , √7 , Integers (J)
0.5656656665…
Whole numbers (W)
Counting numbers (N)
…, 3, 2, 1, 0, 1, 2, 3, …
FIGURE 1.11
E X A M P L E
3
Classifying real numbers
Determine which elements of the set
7, 4, 0, 5, , 4.16, 12
1
are members of each of these sets.
a) Real numbers
b) Rational numbers
c) Integers
Solution
a) All of the numbers are real numbers.
1
b) The numbers 4, 0, 4.16, and 12 are rational numbers.
c) The only integers in this set are 0 and 12.
E X A M P L E
4
■
Subsets of the real numbers
Determine whether each of these statements is true or false.
Q
b) J W
c) I Q d) 3 N
a) 7
e) J I f) Q R
g) R N
h) R
Solution
a) False
e) True
b) False
f) True
c) True
g) False
d) False
h) True
■
Intervals of Real Numbers
(2, 3)
0
1
2
3
4
5
FIGURE 1.12
An interval of real numbers is the set of real numbers that lie between two real
numbers, which are called the endpoints of the interval. Interval notation is used
to represent intervals. For example, the interval notation (2, 3) is used to represent
the real numbers that lie between 2 and 3 on the number line. The graph of (2, 3) is
shown in Fig. 1.12. Parentheses are used to indicate that the endpoints do not belong
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The Real Numbers
to the interval, whereas brackets are used to indicate that the endpoints do belong to
the interval. The interval [2, 3] consists of the real numbers between 2 and 3 including the endpoints. It is graphed in Fig. 1.13.
The infinity symbol may also be used as an endpoint, even though it does not
represent a number. The interval (3, ) consists of the real numbers greater than 3
and extending infinitely far to the right on the number line. See Fig. 1.14. The interval (, 3) consists of the real numbers less than 3, as shown in Fig. 1.15. The
entire set of real numbers is written in interval notation as (, ) and graphed as
in Fig. 1.16. Parentheses are always used next to and .
[2, 3]
0
1
2
3
4
Page 12
5
FIGURE 1.13
(3, )
(, 3)
0
1
2
3
4
5
0
FIGURE 1.14
E X A M P L E
5
1
2
3
4
(, )
32 1 0 1 2 3
5
FIGURE 1.15
FIGURE 1.16
Interval notation
Write each interval of real numbers in interval notation and graph it.
a) The set of real numbers greater than or equal to 2
b) The set of real numbers less than 3
c) The set of real numbers between 1 and 5 inclusive
d) The set of real numbers greater than or equal to 2 and less than 4
Solution
a) The set of real numbers greater than or equal to 2 includes 2. It is written as
[2, ) and graphed in Fig. 1.17.
b) The set of real numbers less than 3 does not include 3. It is written as
(, 3) and graphed in Fig. 1.18.
c) The set of real numbers between 1 and 5 inclusive includes both 1 and 5. It is
written as [1, 5] and graphed in Fig. 1.19.
d) The set of real numbers greater than or equal to 2 and less than 4 includes 2 but
not 4. It is written as [2, 4) and graphed in Fig. 1.20.
(, 3)
[2, )
0
1
2
3
4
5
FIGURE 1.17
5 4 3 2 1
FIGURE 1.18
[2, 4)
[1, 5]
0 1 2 3 4 5 6
FIGURE 1.19
0
1
2
3
4
5
6
FIGURE 1.20
■
Intervals are sometimes graphed using a hollow circle to indicate that an endpoint is not in the interval and a solid circle to indicate that an endpoint is in the interval. The advantage of using parentheses and brackets on the graph is that they
match the interval notation and the interval notation looks like an abbreviated version of the graph.
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(1-13) 13
The Real Numbers
The intersection of two intervals is the set of real numbers that belong to both intervals. The union of two intervals is the set of real numbers that belong to one, or
the other, or both of the intervals.
E X A M P L E
6
Combining intervals
Write each union or intersection as a single interval.
a) (2, 4) (3, 6)
b) (2, 4) (3, 6)
c) (1, 2) [0, )
d) (1, 2) [0, )
Solution
a) Graph (2, 4) and (3, 6) as in Fig. 1.21. The union of the two intervals consists of
the real numbers between 2 and 6, which is written as (2, 6).
b) Examining the graphs in Fig. 1.21, we see that only the real numbers between 3
and 4 belong to both (2, 4) and (3, 6). So the intersection of (2, 4) and (3, 6) is
(3, 4).
c) Graph (1, 2) and [0, ) as in Fig. 1.22. The union of these intervals consists of
the real numbers greater than 1, which is written as (1, ).
d) Examining the graphs in Fig. 1.22, we see that the real numbers between 0 and
2 belong to both intervals. Note that 0 also belongs to both intervals but 2 does
not. So the intersection is [0, 2).
Intersection: (3, 4)
Intersection: [0, 2)
1
2
3
4
5
6
7
2
1
0
1
2
3
4
1
2
3
4
5
6
7
2
1
0
1
2
3
4
Union: (2, 6)
FIGURE 1.21
Union: (1, )
FIGURE 1.22
WARM-UPS
True or false? Explain your answer.
1. The number is a rational number.
2. The set of rational numbers is a subset of the set of real numbers.
3. Zero is the only number that is a member of both Q and I.
4. The set of real numbers is a subset of the set of irrational numbers.
5. The decimal number 0.44444 . . . is a rational number.
6. The decimal number 4.212112111211112 . . . is a rational number.
7. Every irrational number corresponds to a point on the number line.
8. The intervals (2, 6) and (3, 9) both contain the number 6.
9. (1, 3) [3, 4) (1, 4)
10. (1, 5) [2, 8) (2, 8)
■
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The Real Numbers
EXERCISES
Reading and Writing After reading this section, write out
the answers to these questions. Use complete sentences.
1. What are the integers?
20. y y is a whole number greater than 0
21. x x is an integer between 3 and 5
2. What are the rational numbers?
22. y y is an integer between 4 and 7
3. What kinds of decimal numbers are rational numbers?
4. What kinds of decimal numbers are irrational?
5. What are the real numbers?
Determine which elements of the set
5
1 8
A 10
, 3, , 0.025, 0, 2
, 3 , 2
2 2
are members of these sets. See Example 3.
23. Real numbers
6. What is the ratio of the circumference and diameter of
any circle?
24. Natural numbers
25. Whole numbers
Determine whether each statement is true or false. Explain
your answer. See Example 1.
2
7. 6 Q
8. Q
7
9. 0 Q
10. 0 N
11. 0.6666 . . . Q
12. 0.00976 Q
13. N Q
14. Q J
List the elements in each set and graph each set on a number
line. See Example 2.
15. x x is a whole number smaller than 6
26. Integers
27. Rational numbers
28. Irrational numbers
Determine whether each statement is true or false. Explain.
See Example 4.
29. Q R
30. I Q
31. I Q 0
32. J Q
33. I Q R
34. J Q 35. 0.2121121112 . . . Q 36. 0.3333 . . . Q
16. x x is a natural number less than 7
37. 3.252525 . . . I
38. 3.1010010001 . . . I
17. a a is an integer greater than 5
39. 0.999 . . . I
41. I
40. 0.666 . . . Q
42. Q
18. z z is an integer between 2 and 12
19. w w is a natural number between 0 and 5
Place one of the symbols , , , or in each blank so
that each statement is true.
43. N ___ W
44. J ___ Q
45. J ___ N
46. Q ___ W
47. Q ___ R
48. I ___ R
49. ___ I
50. ___ Q
51. N ___ R
52. W ___ R
53. 5 ___ J
54. 6 ___ J
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55.
57.
59.
61.
63.
7 ___ Q
2 ___ R
0 ___ I
2, 3 ___ Q
3, 2 ___ R
56.
58.
60.
62.
64.
8 ___ Q
2 ___ I
0 ___ Q
0, 1 ___ N
3, 2 ___ Q
Write each interval of real numbers in interval notation and
graph it. See Example 5.
65. The set of real numbers greater than 1
66. The set of real numbers greater than 2
(1-15) 15
The Real Numbers
77. 4321 0 1 2
78.
79.
3 4 5 6 7 8 9
40 50 60 70 80 90 100
80.
81.
3 4 5 6 7 8 9
9 8 7 6 5 4 3
82. 121110 9 8 7 6
67. The set of real numbers less than 1
68. The set of real numbers less than 5
69. The set of real numbers between 3 and 4
70. The set of real numbers between 1 and 3
71. The set of real numbers between 0 and 2 inclusive
Write each union or intersection as a single interval. See
Example 6.
83. (1, 5) (4, 9)
84. (1, 2) (0, 8)
85. (0, 3) (2, 8)
86. (1, 8) (2, 10)
87. (2, 4) (0, )
88. (, 4) (1, 5)
89. (, 2) (0, 6)
90. (3, 6) (0, )
91. [2, 5) (4, 9]
92. [2, 2] [2, 6)
93. [2, 6) [2, 8)
94. [1, 5] [2, 9]
72. The set of real numbers between 1 and 1 inclusive
GET TING MORE INVOLVED
73. The set of real numbers greater than or equal to 1 and
less than 3
95. Writing. What is the difference between a rational and
an irrational number? Why is 9 rational and 3
irrational?
74. The set of real numbers greater than 2 and less than or
equal to 5
96. Cooperative learning. Work in a small group to make a
list of the real numbers of the form n
, where n is a natural number between 1 and 100 inclusive. Decide on a
method for determining which of these numbers are rational and find them. Compare your group’s method and
results with other groups’ work.
Write the interval notation for the interval of real numbers
shown in each graph.
75.
76.
97. Exploration. Find the decimal representations of
2
,
9
2
,
99
23
,
99
23
,
999
234
,
999
23
,
9999
and
1234
.
9999
3 4 5 6 7 8 9
3 4 5 6 7 8 9
a) What do these decimals have in common?
b) What is the relationship between each fraction and its
decimal representation?