1. The Real Number System
TEK: 8.2A Extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of real numbers

2. Vocabulary Real number Rational number Irrational number
Terminating decimal
Repeating Decimal
Non-terminating decimal
Integer
Whole number
Natural number

3. Guiding Questions
What is the difference between a rational and irrational number?
What is the difference between terminating and non- terminating decimals?
What are the subsets under rational numbers?
What is the difference between an integer and a whole number?
How do you classify square roots?

4. Real Numbers
Real numbers consist of all the rational and irrational numbers.
The real number system has many subsets:
Natural Numbers
Whole Numbers
Integers

5. Natural Numbers Natural numbers are the set of counting numbers.
{1, 2, 3,…}

6. Whole Numbers
Whole numbers are the set of numbers that include 0 plus the set of natural numbers.
{0, 1, 2, 3, 4, 5,…}

7. Integers Integers are the set of whole numbers and their opposites.
{…,-3, -2, -1, 0, 1, 2, 3,…}

8. Rational Numbers
Rational numbers are any numbers that can be expressed in the form of , where a and b are integers, and b ≠ 0.
They can always be expressed by using terminating decimals or repeating decimals.

9. Terminating Decimals
Terminating decimals are decimals that contain a finite number of digits.
(the decimal ends with a remainder of 0)
Examples:
36.8
0.125
4.5

10. Repeating Decimals
Repeating decimals are decimals that contain an infinite number of digits. (the decimal never ends with a remainder of 0)
Examples:
0.333… …
FYI…The line above the decimals indicate that number
repeats. The dot, dot, dot means the decimal continues

11. Irrational Numbers
Irrational numbers are any numbers that cannot be expressed as .
They are expressed as non-terminating, non-repeating decimals; decimals that go on forever without repeating a pattern.
Examples of irrational numbers: … …
(pi)

12. Venn Diagram of the Real Number System
Rational Numbers
Irrational Numbers
Integers
Whole Numbers
Natural
Numbers

13. Example
Classify all the following numbers as natural, whole, integer, rational, or irrational. List all that apply.
117 … -½
6.36 -3

14. To show how these number are classified, use the Venn diagram
To show how these number are classified, use the Venn diagram. Place the number where it belongs on the Venn diagram.
Rational Numbers
Irrational Numbers
Integers
Whole Numbers
6.36 …
Natural
Numbers -3
117
-1/2

15. Solution -3 is an integer and a rational number.
Now that all the numbers are placed where they belong in the Venn diagram, you can classify each number:
117 is a natural number, a whole number, an integer, and a rational number.
is a rational number.
0 is a whole number, an integer, and a rational number.
… is an irrational number.
-3 is an integer and a rational number.
6.36 is a rational number.
is an irrational number.

16. FYI…For Your Information
When taking the square root of any number that is not a perfect square, the resulting decimal will be non-terminating and non-repeating. Therefore, those numbers are always irrational.

17. Guiding Questions (you should now be able to answer)
What is the difference between a rational and irrational number?
What is the difference between terminating and non- terminating decimals?
What are the subsets under rational numbers?
What is the difference between an integer and a whole number?
How do you classify square roots?