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4-8 The Real Numbers
Evaluating Algebraic Expressions
Rational and
Irrational
Numbers
4-8 The Real Numbers
Rational and
Evaluating Algebraic Expressions
Irrational Numbers
Essential Question
How do I distinguish between
rational and irrational
numbers?
4-8 The Real Numbers
Vocabulary
Evaluating
Algebraic Expressions
real number
irrational number
4-8 The Real Numbers
The set of real numbers is all numbers that can be
written on a number line. It consists of the set of
Evaluating
Expressions
rational
numbers andAlgebraic
the set of irrational
numbers.
Real Numbers
Rational numbers
Integers
Whole
numbers
Irrational numbers
The Real Number System
 Natural Numbers - The set of natural numbers is
often referred to as the set of counting numbers,
because they are those numbers that we use to
count.

Also notice that 0 is not included in the natural
numbers.
 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …∞}
5
The Real Number System
 Whole Numbers - the natural numbers together
with 0.
 {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …∞}
6
The Real Number System
 Integers - If we include the negative numbers
with the whole numbers, we have a new set of
numbers that are called integers.
 {…, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …}
7
4-8 The Real Numbers
Our
next group of numbers
is called
the rational
Evaluating
Algebraic
Expressions
numbers. Rational numbers can be written as
the quotient of two integers (a fraction) or as
either terminating or repeating decimals.
3
4
= 3.8
5
23
= 0.6
1.44 = 1.2
So...rational numbers include fractions,
terminating decimals, and repeating decimals.
4-8 The Real Numbers
So, the rational numbers include the natural numbers,
whole numbers, and integers.
Evaluating Algebraic Expressions
Reals
Rationals
1/2
-2.65
Integers
-3
Wholes
-19
0
1, 2, 3...
Naturals
Rational Numbers
 {p/q : p and q are integers, q is not zero}
10
Number
As a fraction
5
5/1
.75
¾
.001
1/1000
.22222222…
2/9
.1245443782…
?
Rational?
4-8 The Real Numbers
Irrational numbers can be written only as
decimals that do not terminate or repeat. They
Evaluating Algebraic Expressions
cannot be written as the quotient of two integers
(a simple fraction). If a whole number is not a
perfect square, then its square root is an
irrational number. For example, 2 is not a
perfect square, so 2 is irrational.
Caution!
A repeating decimal may not appear to
repeat on a calculator, because calculators
show a finite number of digits.
4-8 The Real Numbers
What other examples of irrational
numbers
are there?
Evaluating
Algebraic
Expressions
12
4-8 The Real Numbers
Let’s Review all of our sets of numbers in the real
number system - Reals, Rationals, Irrationals,
Integers, Wholes,
and Naturals.
Evaluating
Algebraic
Expressions
Rationals
-3
-2.65
Integers
-19
Wholes
0
Naturals
1, 2, 3...
Reals
Irrationals
4-8 The Real Numbers
Additional Example 1: Classifying Real Numbers
Write all classifications that apply to each number.
Evaluating Algebraic Expressions
A.
5 is a whole number that is
not a perfect square.
irrational, real
5
B. –12.75 –12.75 is a terminating decimal.
rational, real
C.
16 2
16 2
4
=
=2
2
whole, integer, rational, real
4-8 The Real Numbers
Check It Out! Example 1
Write all classifications that apply to each number.
Evaluating Algebraic Expressions
A.
9
9
=3
whole, integer, rational, real
B.
C.
–35.9
–35.9 is a terminating decimal.
rational, real
81 3
81 3
9
=
=3
3
whole, integer, rational, real
4-8 The Real Numbers
Evaluating Algebraic Expressions
A fraction with a denominator of 0 is undefined
because you cannot divide by zero. So it is not a
number at all.
4-8 The Real Numbers
Additional Example 2: Determining the Classification
of All Numbers
Evaluating
Algebraic
Expressions
State
if each number
is rational,
irrational,
or not a real number.
A.
21
irrational
B.
0 3
rational
0 3
=0
4-8 The Real Numbers
Additional Example 2: Determining the Classification
of All Numbers
Evaluating
Algebraic
Expressions
State
if each number
is rational,
irrational,
or not a real number.
C. 4
0
not a real number
4-8 The Real Numbers
Check It Out! Example 2
State
if each number
is rational,
irrational,
Evaluating
Algebraic
Expressions
or not a real number.
A.
23
23 is a whole number that
is not a perfect square.
irrational
B.
9 0
undefined, so not a real number
4-8 The Real Numbers
Check It Out! Example 2
State
if each number
is rational,
irrational,
Evaluating
Algebraic
Expressions
or not a real number.
C.
64
81
rational
8 98 9
=
64
81