Beginning Algebra, 11th Edition

(c) Integers: 0, and 5

(d)Rational numbers: 5, 0, , 5, and 5.8

Each of these numbers can be written as the quotient of two integers.

(e) Irrational numbers:

(f )Real numbers: All the numbers in the set are real numbers. NOW TRY

OBJECTIVE 2 Tell which of two real numbers is less than the other. Given

any two positive integers, you probably can tell which number is less than the other.

Positive numbers decrease as the corresponding points on the number line go to the

left. For example, because 8 is to the left of 12 on the number line. This or-

dering is extended to all real numbers by definition.

8612

 2

Aor

58

(^3 10) B
1

4 Aor

13

0.6 (^) Aor 4 B,
2

• (^3) B
  2

- 3 ,

- 5,

SECTION 1.4 Real Numbers and the Number Line 31

NOW TRY
EXERCISE 2
List the numbers in the
following set that belong
to each set of numbers.

(a)Whole numbers

(b)Integers

(c) Rational numbers

(d)Irrational numbers

E-7, - 45 , 0,  3 , 2.7, p, 13F

NOW TRY ANSWERS

2. (a)0, 13 (b)
   (c)
   (d) 3 , p
   
   
   • 7, - 45 , 0, 2.7, 13
     
     
     • 7, 0, 13
     
     
     
     
   
   
   
   

Ordering of Real Numbers

For any two real numbers aand b, ais less thanbif alies to the left of bon the

number line. See FIGURE 8.

This means that any negative number is less than 0, and any negative number is

less than any positive number. Also, 0 is less than any positive number.

Determining the Order of Real Numbers

Is the statement trueor false?

Locate and on a number line, as shown in FIGURE 9. Since lies to the

left of - 1 on the number line, - 3 is less than -1.The statement - 3 6- 1 is true.

- 3 - 1 - 3

- 3 6- 1

EXAMPLE 3

ab

alies to the left of b,

or

FIGURE 8

a 6 b.

–3 lies to the left of –1, so –3 < –1.

–4 –3 –2 –1 0 1 2 3

FIGURE 9 NOW TRY

We can also say that, for any two real numbers aand b, ais greater thanbif a

lies to the right of bon the number line. See FIGURE 10.

OBJECTIVE 3 Find the additive inverse of a real number.By a property of

the real numbers, for any real number x(except 0), there is exactly one number on

the number line the same distance from 0 as x, but on the oppositeside of 0.

See FIGURE 11. Such pairs of numbers are called additive inverses,or opposites,of

each other.

ba

alies to the right of b,

or

FIGURE 10

a 7 b.

–3 –1.5–1 0 1 1.5 3

Pairs of additive inverses, or opposites

• √ 5 √ 5

FIGURE 11

NOW TRY
EXERCISE 3
Determine whether the state-
ment is trueor false.

• 8 ...- 9

3. false