1.2
57. E F
1, 2, 3, 4, 5, 6, 8
59. (D E ) F
2, 3, 4, 5
61. D (E F)
2, 3, 4, 5, 7
63. (D F) (E F)
2, 3, 4, 5
65. (D E) (D F)
2, 3, 4, 5, 7
58. E F
2, 4
60. (D F) E
2, 4
62. D (F E)
2, 3, 4, 5, 7
64. (D E ) (F E)
2, 4
66. (D F) (D E)
2, 3, 4, 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
69. 3 _____ D 70. 3 _____ D 71. D _____ E 72. D E _____ D 73. D F _____ F 74. 3 E _____ F 75. E E _____ F 76. E E _____ F 77. D _____ F F D 78. E _____ F F E List the elements in each set.
79. x x is an even natural number less than 20
2, 4, 6, . . . , 18
80. x x is a natural number greater than 6
7, 8, 9, . . .
81. x x is an odd natural number greater than 11
13, 15, 17, . . .
82. x x is an odd natural number less than 14
1, 3, 5, . . . , 13
83. x x is an even natural number between 4 and 79
6, 8, 10, . . . , 78
84. x x is an odd natural number between 12 and 57
13, 15, 17, . . . , 55
1.2
In this
section
●
The Rational Numbers
●
Graphing on the Number
Line
●
The Irrational Numbers
The Real Numbers
(1-7) 7
Write each set using set-builder notation.
85. 3, 4, 5, 6
x x is a natural number between 2 and 7
86. 1, 3, 5, 7
x x is an odd natural number less than 8
87. 5, 7, 9, 11, . . .
x x is an odd natural number greater than 4
88. 4, 5, 6, 7, . . .
x x is a natural number greater than 3
89. 6, 8, 10, 12, . . . , 82
x x is an even natural number between 5 and 83
90. 9, 11, 13, 15, . . . , 51
x x is an odd natural number between 8 and 52
GET TING MORE INVO LVED
91. Discussion. If A and B are finite sets, could A B be
infinite? Explain. No
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? A B, A B
93. Discussion. What is wrong with each statement? Explain.
a) 3 1, 2, 3
a) 3 1, 2, 3
b) 3 1, 2, 3
b) 3 1, 2, 3
c) 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? , 1, 2, 1, 2
b) List all possible subsets of 1, 2, 3. How many are
there? 8
c) Guess how many subsets there are of 1, 2, 3, 4. Verify
your guess by listing all the possible subsets. 16
d) How many subsets are there for 1, 2, 3, . . . , n? 2n
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
8
(1-8)
helpful
Chapter 1
hint
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.
The Real Numbers
integers. We use the letters N, W, and J to name these sets:
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:
a
Q 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. The
following 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.
1.2
(1-9) 9
The Real Numbers
Graphing on the Number Line
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.
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
study
2
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 …
■
10 (1-10)
Chapter 1
The Real Numbers
When we draw a number line, we might label only the integers. But every point
on the number line corresponds to a number. The set of all of these numbers is called
the set of real numbers, R. It is easy to locate rational numbers on the number line.
For example, the number 1 corresponds to a point halfway between 0 and 1, and 3
2
2
corresponds to a point halfway between 1 and 2, as shown in Fig. 1.9. However, on the
that do not correspond to rational numnumber line, there are points such as 2
bers. These points correspond to the set of irrational numbers, I. The rational
numbers Q and the irrational numbers I have no numbers in common, and together
they form the set of real numbers R.
1
—
2
––
–√ 2
–3
–2
–1
0
3
—
2
1
2
3
FIGURE 1.9
calculator
The Irrational Numbers
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.
Irrational numbers are those real numbers that cannot be expressed as a ratio of
integers. To see an example of an irrational number, consider the positive square root
). The square root of 2 is a number that you can multiply by
of 2 (in symbols 2
itself to get 2. So we can write (using a raised dot for times)
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.
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.
π=
Circumference
Diameter
π=
Circumference
Diameter
FIGURE 1.10
C
D
is not equal to 1.414 (in symbols, 2
1.414). The square root of 2 is apSo 2
1.414). There is no terminating or repeating
proximately 1.414 (in symbols, 2
is an irrational
decimal that will give exactly 2 when multiplied by itself. So 2
, 5, and 7
are also
number. It can be shown that other square roots such as 3
irrational numbers.
It is easy to write examples of irrational numbers in decimal form because as
decimal numbers they have digits that neither repeat nor terminate. Each of the following numbers 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. 1.10. 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 .
Figure 1.11 illustrates the subsets of the real numbers that we have been
discussing.
1.2
study
The Real Numbers
(1-11) 11
Real numbers (R)
tip
Rational numbers (Q)
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.
2 —,
–5 —
155
—,
—,
3
7
13
Irrational numbers (I)
5.26, 0.37373737…
–– –– ––
√2 , √6 , √7 , π
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 the following 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 the following statements is true or false.
b) J W
c) I Q d) 3 N
a) 7 Q
e) J I f) Q R
g) R N
h) R
Solution
a) False
e) True
WARM-UPS
■
b) False
f) True
c) True
g) False
d) False
h) True
True or false? Explain your answer.
1.
2.
3.
4.
The number is a rational number. False
The set of rational numbers is a subset of the set of real numbers. True
Zero is the only number that is a member of both Q and I. False
The set of real numbers is a subset of the set of irrational numbers. False
■
12 (1-12)
Chapter 1
WARM-UPS
The Real Numbers
(continued)
5.
6.
7.
8.
9.
The decimal number 0.44444 . . . is a rational number. True
The decimal number 4.212112111211112 . . . is a rational number. False
Every irrational number corresponds to a point on the number line. True
If a real number is not irrational, then it is rational. True
The set of counting numbers from 1 through 5 trillion is an infinite set.
False
10. There are infinitely many rational numbers. True
1. 2
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What are the integers?
The integers consist of the positive and negative counting
numbers and zero.
2. What are the rational numbers?
The rational numbers consist of all numbers that can be expressed as a ratio of integers.
3. What kinds of decimal numbers are rational numbers?
The repeating or terminating decimal numbers are rational
numbers.
4. What kinds of decimal numbers are irrational?
Decimals that neither repeat nor terminate are irrational
numbers.
5. What are the real numbers?
The set of real numbers is the union of the rational and irrational numbers.
6. What is the ratio of the circumference and diameter of any
circle?
The ratio of the circumference and diameter of any circle
is .
Determine whether each statement is true or false. Explain your
answer. See Example 1.
2
7. 6 Q True
8. Q True
7
9. 0 Q False
10. 0 N True
11. 0.6666 . . . Q True
12. 0.00976 Q False
13. N Q True
14. Q J False
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
0, 1, 2, 3, 4, 5
16. x x is a natural number less than 7
1, 2, 3, 4, 5, 6
17. a a is an integer greater than 5
4, 3, 2, 1, 0, 1, …
18. z z is an integer between 2 and 12
3, 4, 5, 6, 7, 8, 9, 10, 11
19. w w is a natural number between 0 and 5
1, 2, 3, 4
20. y y is a whole number greater than 0
1, 2, 3, 4, 5, …
21. x x is an integer between 3 and 5
2, 1, 0, 1, 2, 3, 4
22. y y is an integer between 4 and 7
3, 2, 1, 0, 1, 2, 3, 4, 5, 6
Determine which elements of the set
5
1 8
A 1
0, 3, , 0.025, 0, 2
, 3 , 2
2 2
are members of the following sets. See Example 3.
2
8
8
25. Whole numbers 0, 26. Integers 3, 0, 2
2
5
1 8
27. Rational numbers 3, , 0.025, 0, 3, 2
2 2
23. Real numbers
All
24. Natural numbers
8
28. Irrational numbers 1
0, 2
Determine whether each statement is true or false. Explain. See
Example 4.
29. Q R True
30. I Q False
31. I Q 0 False
32. J Q True
1.3
33. I Q R True
35. 0.2121121112 . . . Q
False
37. 3.252525 . . . I
False
39. 0.999 . . . I False
41. I True
34. J Q False
36. 0.3333 . . . Q
True
38. 3.1010010001 . . . I
True
40. 0.666 . . . Q True
42. Q False
GET TING MORE INVO LVED
65. Writing. What is the difference between a rational and an
irrational number? Why is 9
rational and 3
irrational?
66. 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.
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 55. 7 ___ Q 56. 8 ___ Q 57. 2
___ R 58. 2 ___ I 59. 0 ___ I 60. 0 ___ Q 61. 2, 3 ___ Q 62. 0, 1 ___ N 63. 3, 2
___ R 64. 3, 2
___ Q 1.3
(1-13) 13
Operations on the Set of Real Numbers
67. Exploration. Find the decimal representations of
2
,
9
2
,
99
23
,
99
23
,
999
234
,
999
23
,
9999
and
1234
.
9999
a) What do these decimals have in common?
b) What is the relationship between each fraction and its
decimal representation?
OPERATIONS ON THE SET
OF REAL NUMBERS
Computations in algebra are performed with positive and negative numbers. In this
section we will extend the basic operations of arithmetic to the negative numbers.
In this
section
●
Absolute Value
●
Addition
●
Subtraction
●
Multiplication
●
Division
●
Division by Zero
Absolute Value
The real numbers are the coordinates of the points on the number line. However, we
often refer to the points as numbers. For example, the numbers 5 and 5 are both
five units away from 0 on the number line shown in Fig. 1.12. A number’s distance
from 0 on the number line is called the absolute value of the number. We write a for “the absolute value of a.” Therefore 5 5 and 5 5.
5 units
–5
–4
–3
–2
5 units
–1
0
1
2
3
4
5
FIGURE 1.12
E X A M P L E
1
Absolute value
Find the value of 4 , 4 , and 0 .
Solution
Because both 4 and 4 are four units from 0 on the number line, we have 4 4
and 4 4. Because the distance from 0 to 0 on the number line is 0, we have
■
0 0.