Intermediate Algebra (11th edition)

A rational number written as a fraction, such as or can also be expressed as a

decimal by dividing the numerator by the denominator.

0.125 Terminating decimal Repeating decimal

(rational number) (rational number)

20 20

40 20

0 Remainder is 0. 2 Remainder is never 0.

2

3

= 0.6

1

8

= 0.125

40 18

16 18

8 18

8 1.000 3 2.000Á

0.666Á

2

3 ,

1

8

4 CHAPTER 1 Review of the Real Number System

A bar is written over

the repeating digit(s).

Thus, terminating decimals, such as and repeating

decimals, such as and are rational numbers.

Decimal numbers that neither terminate nor repeat, which include many square

roots, are irrational numbers.

and Irrational numbers

NOTE Some square roots, such as and , are rational.

Another irrational number is the ratio of the circumference of a circle to its diam-

eter. See FIGURE 3.

Some rational and irrational numbers are graphed on the number line in FIGURE 4.

The rational numbers together with the irrational numbers make up the set of real

numbers.Every point on a number line corresponds to a real number, and every

real number corresponds to a point on the number line.

p,

2

9

25 =

3

216 = (^45)

22 = 1.414213562Á - 27 =-2.6457513Á

0.6=^23 0.27 = 113 ,

0.125=

1

8 , 0.8=

4

5 , and 2.75=

11

4 ,

d

 =Cd

 is approximately

3.141592653....

FIGURE 3

3

0.27^5

Real numbers

Irrational

numbers

Rational

numbers

2.75

• 4 – 3 – 2 – 101234
  
  
  • √ 7 √ 2 
  
  
  
  

√ 16

FIGURE 4

The fractions and graphed on the number line in FIGURE 2, are rational

numbers. A rational numbercan be expressed as the quotient of two integers, with

denominator not 0. The set of all rational numbers is written as follows.

Rational numbers

The set of rational numbers includes the natural numbers, whole numbers, and in-

tegers,since these numbers can be written as fractions. For example,

and 0 =

0

1

- 3 =.

- 3

1

14 = ,

14

1

,

e

p

q

` p and q are integers, q 0 f

3

- 4 ,

1

2