Download 1.2 Algebraic Expressions and Sets of Numbers

Section 1.2
1.2
Algebraic Expressions and Sets of Numbers 16A
Algebraic Expressions and Sets of Numbers
(continued)
OBJECTIVE
OBJECTIVES
6 Express Numbers as
Decimals.
7 Classify Sums and Products of
Real Numbers.
Expressing Numbers as Decimals
6
Earlier, we mentioned that every real number can be written as a decimal. If the decimal repeats or terminates, then the number is a rational number. If the decimal does
not terminate or repeat, then the number is an irrational number. You can use long
division to demonstrate that a rational number can be expressed as a repeating or terminating decimal.
EXAMPLE 9
Write
14
as a repeating decimal.
33
Solution
0.4242
33冄 14.0000
132
80
66
140
132
80
66
14
14
= 0.42
33
PRACTICE
a.
9
Write the following numbers in decimal form.
5
16
b.
0.3125
21
25
0.84
2
9
c.
d.
0.2
6
11
e.
0.54
17
22
0.772
f.
48
55
0.872
To demonstrate that a decimal is a rational number, you can express the decimal as a
fraction. For a terminating decimal, this is done by using the place value of the final
digit in the decimal to form a fraction. For repeating decimals, the digits that repeat
can be expressed as a fraction with a denominator of 9, 99, 999, etc., depending upon
the number of repeating digits.
E X A M P L E 1 0 Write the following decimals as fractions in simplest terms.
a. 0.61
b. 0.36
c. 0.198
d. 0.472
Solution
61
100
36
4
b. 0.36 =
=
99
11
198
22
c. 0.198 =
=
999
111
a. 0.61 =
d. 0.472 = 0.4 + 0.072 = 0.4 +
1
4
72
2
4
26
(0.72) =
+
= +
=
10
10
990
5
55
55
PRACTICE
10
a. 0.0625
1
16
b. 0.782
391
500
c. 0.16
16
99
d. 0.297
11
37
e. 0.52
47
90
f. 0.654
36
55
16B CHAPTER 1
Real Numbers and Algebraic Expressions
OBJECTIVE
Classify Sums and Products of Real Numbers
7
Rational numbers and irrational numbers are subsets of real numbers. One of the
properties of the set of real numbers is called the closure property. It can be stated as
follows:
If a and b are real numbers, then a ⫹ b and ab are real numbers.
The concept of closure can also be applied to subsets of real numbers. For example, the
set of whole numbers is closed under addition and multiplication, because the sum of
any two numbers is a whole number and the product of any two whole numbers is also
a whole number.
E X A M P L E 1 1 Determine whether the following statements are true or false..
a.
b.
c.
d.
The set of whole numbers is closed under subtraction.
The set of irrational numbers is closed under addition and multiplication.
The sum of a rational number and an integer is a rational number.
The sum of a rational number and an irrational number is a rational number.
Solution
a. False. The difference of two whole numbers may be a negative number. For
example, 5 ⫺ 8 ⫽ −3.
b. False. The sum of two irrational numbers that are additive inverses of each other,
such as p and -p, is the rational number zero. The product of two irrational numbers could be a rational number. For example, 22 · 22 = 2.
a
c. True. Consider the rational number . Let c be an integer such that a, b, and c are
b
bc
all integers. The number c can be expressed as the rational number
. The sum of
b
these rational numbers is also a rational number, because the numerator a + bc is
an integer, as integers are closed under addition.
d. False. The sum of a rational number and an irrational number is irrational. Consider
the decimal expansion of an irrational number, such as 22 = 1.414213. . . . The
decimal expansion extends to infinity and does not repeat. The sum of this number
and a rational number, which has a terminating or repeating decimal, would yield a
result that is still without pattern or termination.
PRACTICE
11
a.
b.
c.
d.
The set of whole numbers is closed under division.
The set of rational numbers is closed under addition and multiplication.
The product of a rational number and an integer is a rational number.
The product of a nonzero rational number and an irrational number is an irrational
number.
a. False. 1 , 2 = 0.5
b. True. Consider the rational numbers ba and dc where a, b, c, and d are integers. The sum of the
(ad + bc)
ac
two numbers is bd and the product is bd
. Because integers are closed under addition and
multiplication, ad+bc, ac, and bd are integers. Therefore, both the sum and product of the two
rational numbers are rational numbers.
c. True. Consider the rational number ba where a and b are integers, and the integer c. The product of the two numbers is ac
b . Because integers are closed under multiplication, ac is an integer.
Therefore, the product of the two numbers is a rational number.
d. True. Consider the rational number ba , where a and b are integers, and the irrational number
az
z. The product of the two numbers is b . The product of an integer and a number whose decimal
expansion extends to infinity has no pattern will still be a number whose decimal expansion has
no pattern. Therefore, the product is not a rational number, so it is irrational.