This document defines and provides examples of different types of real numbers: - Rational numbers can be written as ratios of integers and have terminating or repeating decimals. Irrational numbers cannot be written as ratios and have non-terminating, non-repeating decimals. - Real numbers include rational and irrational numbers. Rational numbers further include integers, whole numbers, and natural numbers. Integers are whole numbers and their opposites, whole numbers are positive integers and zero, and natural numbers are counting numbers. - Examples are provided to illustrate the different types of numbers and how to classify numbers like π, √4, and 3/4.

1. REAL NUMBERS
(as opposed to fake numbers?)

2. Two Kinds of Real Numbers
• Rational Numbers
• Irrational Numbers

3. Branching Real Numbers
Real Numbers
Rational Irrational
Now let’s break these
subsets down
further!

4. Rational Numbers
A rational number…
• is a real number that can be
written as a ratio (fraction) of
two integers.
• written in decimal form is
terminating (ends) or repeating
(same pattern of numbers).

5. Examples of Rational
Numbers
•16
•1/2
•3.56
•-8
•1.3333…
•- 3/4
Can you write 16 as a ratio?
Now try writing -8 as a ratio!

6. Irrational Numbers
• An irrational number
• is a number that cannot be
written as a ratio (fraction) of
two integers.
• written as decimals are
non-terminating (never end!) and
non-repeating (no pattern).

7. Examples of Irrational
Numbers
• Square roots of
non-perfect
“squares”
• Pi
17
What is the square root of 17?
What is another non-perfect square?

8. Integers
One of the subsets of rational
numbers
So what do integers branch off of?

9. Integers
• Integers are rational numbers
because they can be written as
fraction with 1 as the denominator.
*Remember*
Rational numbers can be written as a
ratio (or in other words, a fraction)

10. What are integers?
• Integers are the whole numbers and their
opposites.
• Examples of integers are
6
-12
0
186
-934
What do you notice about these integers?

11. Whole Numbers
One of the subsets of rational
numbers and integers
So what do whole numbers branch off of?

12. Whole Numbers
• All positive numbers that are
counting numbers plus zero.
• How can you remember that zero
goes with whole numbers?
• Examples: 0, 1, 2, 3, 4, 5 …

13. Natural Numbers
One of the subsets of rational
numbers, integers and whole
numbers
So what do natural numbers branch off of?

14. Natural Numbers
• All of your counting numbers
• What numbers do you count with?

15. Classify the following
Numbers
• What subset(s) do the following
numbers belong to? Remember they
could belong to more than one!
• 0
• Whole, Integers, Rational, and Real

16. Classify the following
Numbers (Con’t)
• 4
• Natural, Whole, Integer, Rational, Real
• -9
• Integer, Rational, Real
• Π
• Irrational, Real

17. Classify the following
Numbers (Con’t)
• 3/4 or 0.75
• Rational and Real
• √4
• Natural, Whole, Integer, Rational, Real
• √15
• Irrational, Real