CHAPTER 5
Number Theory and the
Real Number System
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5.5
Real Numbers and Their Properties
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2
Objectives
1. Recognize the subsets of the real numbers.
2. Recognize the properties of real numbers.
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3
The Set Real Numbers
• The union of the rational numbers and the irrational
numbers is the set of real numbers.
• The sets that make up the real numbers are called
subsets of the real numbers.
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4
Example 1: Classifying Real Numbers
Consider the following set of numbers:
3

7,  , 0,0.6,
4

5,  , 7.3,

81 

List the numbers in the set that are
a. natural numbers
b. whole numbers
c. Integers
d. rational numbers
e. irrational numbers f. real numbers
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Example 1: Classifying Real Numbers (continued)
Solution:
3

7,  , 0,0.6,
4

a. natural numbers
b. whole numbers
c. integers
d. rational numbers
e. irrational numbers
f. real numbers
5,  , 7.3,
Because
81
0,

81 

81
81 = 9
because whole numbers include 0
and the natural numbers
0, 81, -7
because integers include whole
numbers and the negative natural
numbers
0, 81, -7, -¾, 0.6, & 7.3
because these numbers can be
expressed as a quotient or as a
terminating or repeating decimal
5 ,π
0, 81, -7, -¾, 0.6, 5 , 7.3, & π
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because neither terminate nor
have blocks of repeating digits
because real numbers have all
the above numbers as subsets
6
Properties of the Real Numbers
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7
Properties of the Real Numbers
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8
Properties of the Real Numbers
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9
Example 2: Identifying Properties of Real
Numbers
Name the property illustrated:
a. 3  7  7  3
b. (4 + 7) + 6 = 4 + (7 + 6)


Commutative property of multiplication
Associative property of addition
c. 2 3  5  6  2 5
Distributive property of
multiplication over addition
d. 17  (17)  0
Inverse property of addition
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