Classification of Real Numbers
Warm-Up
Numbers can be Classified
All the numbers that we know – positives, negatives, radicals, decimals, and fractions –
are called real numbers and they can be placed in different groups. Using a Venn
diagram, we can see how these sets are related.
Real Numbers
1
Naturals, Wholes, and Integers

Natural numbers (ℕ), or counting numbers, are the numbers we count with: 1, 2, 3, …

Whole numbers (𝑊) are the natural numbers plus zero: 0, 1, 2, 3, …
The natural and whole numbers are examples of infinite sets. They go on forever.

Integers (ℤ) are whole numbers and their opposites.
For example: …-2, -1, 0, 1, 2….
None of these sets include fractions or decimals.
Model Problem
Natural Numbers
Whole Numbers
Integers
Using the word bank above, tell all the sets to which each number belongs.
1) 2 ________________
2) 0 ___________________ 3) -1 ___________________
Exercise
Using the word bank above, tell all the sets to which each number belongs.
4) -3 ________________
5) 8 ___________________ 6) 0 ___________________
Rational and Irrational Numbers

Rational numbers can be expressed in the form of a fraction (or ratio).

Irrational numbers cannot be expressed in the form of a fraction.
You can tell whether numbers are rational or irrational just by looking at them.
2
Examples of Rational Numbers





All fractions
All integers
Terminating decimals
Repeating decimals
Square roots of PERFECT
SQUARES
1) Why are all fractions rational?
Examples of Irrational Numbers



1) A) Why are square roots of
nonperfect squares irrational?
2) Why are all integers rational?
3) A) Why are square roots of perfect
squares rational?
Decimals that don’t terminate or
repeat
Square roots of nonperfect squares
Pi (π)
B) Give an example: __________
2) Why is 𝜋 irrational?
B) Give an example:
Why Terminating or Repeating Decimals are Rational


Terminating decimals are decimals that end.
Examples: 0.5, 2.67
Repeating decimals repeat the same digit or set of digits: 0.33333…., 2.131313….
All fractions can be written as either terminating or repeating decimals and vice versa. Try it:
Because these decimals can be written as fractions, they are rational.
Example 1:
Example 2:
Example 3:
3
Model Problem
Reminder!
Tell whether each number is rational or irrational.
1) 4 ________________________
2) √3 ______________________
3) √4 ______________________
3
4) − 4
_____________________________
5) 2𝜋 ______________________
Examples of
Rational Numbers





Exercise
Tell whether each number is rational or irrational.
All fractions
All integers
Terminating
decimals
Repeating
decimals
Square roots of
PERFECT
SQUARES
Examples of
Irrational Numbers



Decimals that
don’t terminate
or repeat
Square roots of
nonperfect
squares
Pi (π)
1) 6 ________________________
2) √16 ______________________
3) −0.41414141 …. ______________________
4)
2
3
_____________________________
5) 0.121121112 …. ______________________
Putting it All Together
Let’s recall some of the number sets we did today.
4
Closing Activity
Using the terms above, tell ALL the number sets to which each number belongs.
Many of them will have more than one answer.
1) 0
2) ½
3) √2
5
Homework
Vocabulary – Using the definitions in your packet, write and define these terms in your
vocabulary notebook.
1)
2)
3)
4)
Natural numbers
Whole numbers
Integers
Rational Numbers
5) Irrational Numbers
6) Real Numbers
7) Terminating decimal
8) Repeating decimal
1) Place each number in the correct column in the table below. Many numbers will be
written in more than one column. The circled number has been done for you.
10
2
25
π
–32
0.123123…
–12
–1.5
 2
0

22,222
2.14
Natural
Numbers
10
2
22
5
10
2
3
–2.45
 0.33
33
0.000
1
2
3.1
Whole
Numbers
10
2
Integers
10
2
Rational
Numbers
Irrational
Numbers
10
2
6