Download Chapter 1: Sets, Operations and Algebraic Language 1.2: Classifying Real Numbers

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
The number 0.14114111411114 . . . is
irrational because it may not be
expressed as the ratio of two integers.
It is not a repeating decimal.
2
069923a
Which number below is irrational?
4
, 20, 121
9
Why is the number you chose an
irrational number?
20 is irrational because it may not be
expressed as the ratio of two integers.
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.
3
99
, 164, 196
11
Identify the expression that is a rational
number and explain why it is rational.
Given:
5
121 
11
1
9
080432a
4
4 2

9 3
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?
20 is irrational because it may not be
expressed as the ratio of two integers.
010416a
Which number is irrational?
(1) 9
(3) 0.3333
2
(2) 8
(4)
3
(2)
8 is irrational because it may not be
expressed as the ratio of two integers.
4
5
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.
4 2

9 3
121 
11
1
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