Chapter 1: Sets, Operations and Algebraic Language
1.2: Classifying Real Numbers
All Numbers Can be Located on the Number Line
-3
-2
-1
0
1
Lesser
2
3
Greater
The set of real numbers includes two major classifications of numbers: Irrational and
Rational.
Another name for natural numbers is counting numbers.
Rational Number: any number that can be expresses as the ratio of two integers. This
includes fractions, repeating decimals, and terminating decimals.
Irrational Number: any number that cannot be expresses as the ratio of two integers.
Determining if a Number is rational or irrational
 If a number is an integer, it is rational, since it can be expressed as a ratio with the
integer as the numerator and 1 as the denominator.
 If a decimal is a repeating decimal, it is a rational number.
 If a decimal terminates, it is a rational number.
 If a decimal does not repeat or terminate, it is an irrational number.
 Numbers with names, such a  and e are irrational. They are given names because it
is impossible to state their infinitely long values.
 The square roots of all numbers (that are not perfect squares) are irrational.
 If a term reduced to simplest form contains an irrational number, the term is
irrational.
1
Chapter 1: Sets, Operations and Algebraic Language
1.2: Classifying Real Numbers
Examples:
Determine if the following numbers are rational or irrational:
Number
Rational or Irrational?
3
irrational, since the irrationality of  is transferred to the entire term

3


5 5
irrational, since the irrationality of  is transferred to the entire term

 1 and the irrationality of  cancels out. The result

of the cancellation is 1, which is a rational number, because it can be
1
expressed as the ratio of two integers, 1  .
1
is not irrational since
5  5  5  5  25  5 Note that any square root times itself
eliminates the radicand.
Testing Tips:
When distinguishing between rational and irrational numbers, look for these clues:
 Fractions – these are rational
 Terminating (finite) decimals – these are rational
 Repeating patterns in decimals – these are rational
 Square roots of perfect squares – these are rational
 Square roots of non-perfect squares – these are irrational
 Numbers with names, such as  - these are irrational
Do Now: Sample Regents Problems
Which number is rational?
(1) 
(3) 7
(2)
5
4
(4)
3
2
One Solution
This one is easy. The fraction is the rational number, because it is the ratio of two
integers.  is irrational because it never ends and never repeats. The square roots are
irrational because they are not square roots of perfect squares.
2
Chapter 1: Sets, Operations and Algebraic Language
1.2: Classifying Real Numbers
Another Regents Problem
The number 0.14114111411114 . . . is
(1) integral
(3) irrational
(2) rational
(4) whole
One Solution
This one is a little tricky. We know it is not a whole number or an integer because it has
a decimal, so our choice is between rational and irrational. There is a recognizable
pattern to this decimal, which is what is tricky. Non-repeating decimals are irrational.
Repeating decimals are rational. Does a recognizable pattern mean that the decimal
repeats. The answer is no. This pattern is not a repeating pattern. Thus,
0.14114111411114 . . . is an irrational number.
Homework:
Read p. 6-7
Do: Questions 1-3, p. 8 (yes, the answers are in the back of the book, will it help you to
just circle them?)
Additional Homework Questions:
The number 0.14114111411114 . . . is
(1) integral
(3) irrational
(2)
rational
(4) whole
080208a
1
Which number below is irrational?
4
, 20, 121
069923a
9
Why is the number you chose an irrational number?
2
3
99
, 164, 196
11
080432a Identify the expression that is a rational number and explain why it is rational.
4
Given:
Which number is rational?
(1) π
(3) 7
060003a
(2)
5
4
(4)
3
2
Which is an irrational number?
(1) 9
(3) 3
010219a
3
(2) 3.14
(4)
4
5
3
Chapter 1: Sets, Operations and Algebraic Language
1.2: Classifying Real Numbers
Homework Solutions
REGENTS QUESTIONS
SOLUTIONS
1
080208a
The number 0.14114111411114 . . . is
(1) integral
(3) irrational
(2) rational
(4) whole
2
069923a
Which number below is irrational?
4
, 20, 121
9
Why is the number you chose an
irrational number?
3
010219a
Which is an irrational number?
(1) 9
(3) 3
3
(2) 3.14
(4)
4
expressed as the ratio of two integers.
4 2

9 3
121 
11
1
9
99
, 164, 196
11
Identify the expression that is a rational
number and explain why it is rational.
Given:
5
20 is irrational because it may not be
(3)
3 is irrational because it may not be
expressed as the ratio of two integers.
080432a
4
The number 0.14114111411114 . . . is
irrational because it may not be
expressed as the ratio of two integers.
It is not a repeating decimal.
060003a
Which number is rational?
3
314
3.14 
1
100
14
, which is rational, because
1
it is the ratio of two integers.
196 
99
 3,
11
99
3
11
(2)
4
Chapter 1: Sets, Operations and Algebraic Language
1.2: Classifying Real Numbers
(1) π
(2)
5
4
5
is rational because it is the ratio of
4
two integers.
(3) 7
(4)
3
2
REGENTS QUESTIONS
SOLUTIONS
1
080208a
The number 0.14114111411114 . . . is
(1) integral
(3) irrational
(2) rational
(4) whole
2
010632a
π is an irrational number because it
Write an irrational number and explain may not be expressed as the ratio of
why it is irrational.
two integers.
3
069923a
Which number below is irrational?
4
, 20, 121
9
Why is the number you chose an
irrational number?
4
5
010416a
Which number is irrational?
(1) 9
(3) 0.3333
2
(2) 8
(4)
3
060303a
Which expression represents
irrational number?
(1) 2
(3) 0.17
The number 0.14114111411114 . . . is
irrational because it may not be
expressed as the ratio of two integers.
It is not a repeating decimal.
20 is irrational because it may not be
expressed as the ratio of two integers.
4 2

9 3
121 
11
1
(2)
8 is irrational because it may not be
expressed as the ratio of two integers.
9
an
5
3
1
0.3333 
3333
10000
(1)
2 is irrational because it may not be
expressed as the ratio of two integers.
Chapter 1: Sets, Operations and Algebraic Language
1.2: Classifying Real Numbers
(2)
1
2
0.17 
(4) 0
17
100
0
0
1
6
010219a
Which is an irrational number?
(1) 9
(3) 3
3
(2) 3.14
(4)
4
(3)
3 is irrational because it may not be
expressed as the ratio of two integers.
060211a
Which is an irrational number?
1
(1) 0
(3) 
3
(2)  
(4) 9
(2)
π is an irrational number because it
may not be expressed as the ratio of
two integers.
9
3
314
3.14 
1
100
7
0
0
1
9
3
1

1 1

3 3
8
080523a
Which is an irrational number?
(1) 0.3
(3) 49
3
(2)
(4) 
8
(4)
π is an irrational number because it
may not be expressed as the ratio of
two integers.
080718a
Which number is irrational?
5
(1)
(3) 121
4
(2) 0. 3
(4) π
(4)
π is an irrational number because it
may not be expressed as the ratio of
two integers.
0.3 
1
3
49 
7
1
9
0.3 
10
080432a
99
, 164, 196
11
Identify the expression that is a rational
Given:
6
1
3
121 
11
1
Chapter 1: Sets, Operations and Algebraic Language
1.2: Classifying Real Numbers
number and explain why it is rational.
14
, which is rational, because
1
it is the ratio of two integers.
196 
99
 3,
11
99
3
11
11
060813b
(4)
The value of x 2  9 is a real and
irrational number when x is equal to
(1) 5
(3) -3
(2) 0
(4) 4
x 2  9  42  9  7
12
060003a
Which number is rational?
(1) π
(3) 7
(2)
5
4
(4)
(2)
5
is rational because it is the ratio of
4
two integers.
3
2
13
060120a
Which is a rational number?
(1) 8
(3) 5 9
(2) 
(4) 6 2
(3)
5 9 is rational because it is the ratio
15
of two integers,
.
1
080102a
Which expression is rational?
(1)  
(3) 3
(4)
14
(2)
1
2
(4)
1 1
 , the ratio of two integers.
4 2
1
4
7