College Algebra 9th txtbk

SECTION R–1 Algebra and Real Numbers 5
MATCHED PROBLEM 1 *
Perform the indicated operations.
(A) (52 73)
(C)
21
20 15
14
(B) (817) 1
(D) 5 (12 13)
Rational numbers have decimal expansions that are repeating or terminating. For example,
using long division,
2
3 0.666
22
7 3.142857
13
8 1.625
The number 6 repeats indefinitely.
The block 142857 repeats indefinitely.
Terminating expansion
Conversely, any decimal expansion that is repeating or terminating represents a rational
number (see Problems 49 and 50 in Exercise R-1).
The number 12 is irrational because it cannot be written in the form ab, where a and
b are integers, b 0 (for an explanation, see Problem 89 in Section R-3). Similarly, 13 is
irrational. But 14, which is equal to 2, is a rational number. In fact, if n is a positive integer,
then 1n is irrational unless n belongs to the sequence of perfect squares 1, 4, 9, 16, 25, . . .
(see Problem 90 in Section R-3).
We now return to our original question: how do you add or multiply two real numbers
that have nonrepeating and nonterminating decimal expansions? Although we will
not give a detailed answer to this question, the key idea is that every real number can
be approximated to any desired precision by rational numbers. For example, the irrational
number
is approximated by the rational numbers
12 1.414 213 562 . . .
14
10 1.4
141
100 1.41
1,414
1,000 1.414
14,142
10,000 1.4142
141,421
100,000 1.41421
.
Using the idea of approximation by rational numbers, we can extend the definitions of
rational number operations to include real number operations. The following box summarizes
the basic properties of real number operations.
*Answers to matched problems in a given section are found near the end of the section, before the exercise set.