Section 1.

1 Real Numbers

Learning Objectives

In this section, you will:

 Classify a real number as a natural, whole, integer, rational, or irrational number.

 Perform calculations using order of operations.
 Use the following properties of real numbers: commutative, associative, distributive,
inverse, and identity.
 Evaluate algebraic expressions.
 Simplify algebraic expressions.

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Classifying a Real Number
 The set of natural numbers is the set of counting numbers, starting from 1: N=¿{
1 , 2, 3 , … }
 The set of whole numbers is the set of natural numbers and zero:
W ={0 , 1 ,2 , 3 , … }.
 The set of integers includes the opposites of the natural numbers to the set of whole
numbers:
Z={… ,−3 ,−2 ,−1, 0 , 1 ,2 , 3 , … }.
 Rational numbers are numbers that can be written as fractions:

{|a
 Q= a and b are integers and b ≠ 0 }. Because they are fractions, any rational number
b
can be represented as either:
15
a) a terminating decimal: =1.875, or
8
4
b) a repeating decimal: =0.36363636 …=0. 36
11

 Irrational numbers are numbers that cannot be written as fractions and include never-
ending decimal numbers, like π: Q' ={h∨h is not a rational number}.

The set of all rational and irrational numbers together make up the set of real numbers,
denoted by R. The real numbers can be visualized on a horizontal line with an arbitrary point
chosen as 0, the negative numbers to the left of 0 and the positive numbers to the right of 0,
called the real number line.
Example 1

Write each of the following as a rational number.

a) −25
b) 37
c) 0.59
d) −0.8

Example 2
Write each of the following rational numbers as either a terminating or repeating decimal.
24
a)
4
16
b)
3
−17
c)
20
5
d)
12
82
e)
5
−2
f)
9

Example 3
Classify each number as either positive or negative and as either rational or irrational. Does the
number lie to the left or right of 0 on the number line?
6
a)
42
b) −√ 121
c) −6 π
9
d)
10
e) √ 53
 −20
f)
3

Think-Pair-Share 1:
True or False. Decide if the following statements are true or false. If a statement is true,
explain. If a statement is false, provide a counterexample.

a) Any rational number is a real number.

b) Any real number is a rational number.
c) Any natural number is a rational number.
d) Any rational number is a natural number.
e) √ 3 is a rational number.

f) The division of two natural numbers will always result in a rational number.

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Sets of Numbers as Subsets
Beginning with the natural numbers, we have expanded each set to form a larger set,
meaning that there is a subset relationship between the sets of numbers. These relationships
become more obvious when seen as a Venn diagram.
 Example 4

Classify each number as being a natural number (N), whole number (W), integer (Z), rational
number (Q), and/or irrational number (Q’).
a) √ 25
8
b)
3
c) √ 71
d) −7
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Order of Operations

Operations in mathematical expressions must be evaluated in a systematic order, which can

be simplified using the acronym PEMDAS:
Parentheses
Exponents
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

Example 5

Use the order of operations to evaluate each of the following expressions.

a) ¿

b) 3 ( 5 ∙2 ) −2 [ ( 4−2 )−3 2 ]+1

2
3 −4
c) −√ 7−3
5

10−2∙ 3
d) 2
13∙ 2−5

Think-Pair-Share 2:
1. What is the order of operations? What acronym is used to describe the order of
operations, and what does it stand for?
2. Use the order of operations to evaluate each of the following expressions.
a) ¿
 b) 4 ( 3∙ 2 )−3 [ (1−2 )−22 ]

3
2 −3
c) −√ 16−7
5
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Properties of Real Numbers

The following properties hold for real numbers a , b ,and c .

Addition Multiplication
Commutative Property a+ b=b+a a ∙ b=b ∙ a

Associative Property a+ ( b+c )= ( a+b )+ c a ∙(b∙ c )=(a ∙b)∙ c

Distributive Property a ∙ ( b+ c ) =a ∙ b+a ∙ c

Identity Property There exists a unique real There exists a unique real
number called the additive number called the
identity, 0, such that: multiplicative identity, 1,
such that:
a+ 0=0+ a=a a ∙ 1=1 ∙ a=a

Inverse Property Every real number a has an Every real number a has a
additive inverse or opposite, multiplicative inverse or
−a , such that: 1
reciprocal, , such that:
a
a+ (−a ) =(−a)+ a=0 1 1
a∙( )=( )∙ a=1
a a

Example 6

Use the properties of real numbers to rewrite and simplify each expression. State which
properties apply.
a) 3 ∙7 +3 ∙3
 b) ( 3+ 4 ) + (−4 )

c) 8−(12+3)

d) ( )
2 1 5
∙ ∙
5 3 2

e) 10[1.25+ (−0.75 ) ]

Think-Pair-Share 3:
a) Determine whether the statement is true or false: The multiplicative inverse of a rational
number is also rational.

( )
2 3 5
b) Evaluate the expression ∙ ∙ and state what properties applies.
3 2 7
c) What properties of real numbers would simplify the following expression: 4+7(x−1)?
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Evaluating Algebraic Expressions

An algebraic expression is a collection of constants and variables joined together by the
algebraic operations of addition, subtraction, multiplication, and division.
For example, in the expression x +5, 5 is called a constant because it does not vary and x is
called a variable because it does.
To evaluate an algebraic expression means to find the value of the expression when the variable
is replaced by a given number.

Example 7
a) List the constants and variables for each algebraic expression.
a) x +3
2 4
b) πr
5
c) √ 3 m 2 n5
b) Evaluate the expression 3 x−5for each value of x.
a) x=0
b) x=1
c) x=−2
1
d) x=
3

Think-Pair-Share 4:
Evaluate each expression for the given values.
t
a) for t=2
3t−1
b) a+ ab+b for a=10 , b=−3
4 3
c) π r for r =2
3
d) √ 2 m2 n3 for m=2 , n=3

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Simplifying Algebraic Expressions

To simplify algebraic expressions, we use the properties of real numbers.

Steps:

 Remove any grouping symbol such as brackets and parentheses by multiplying factors.
 Use the exponent rule to remove grouping if the terms contain exponents.

 Combine the like terms by addition or subtraction.

 Combine the constants.

Example 8

Simplify each algebraic expression.

a) 2 x−3 y + x−4 y+ 3
b) 2 q−3(5−q)+2
c) 5 mn−3 m+mn+ n
MEDIA

Access these online resources for additional instruction and practice with real numbers.

 Simplify an Expression.
 Evaluate an Expression 1.
 Evaluate an Expression 2.

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Source: Openstax College Algebra 2e.