1. Rational and Irrational Numbers

2. Rational and Irrational Numbers Essential Question
How do I distinguish between rational and irrational numbers?

3. Vocabulary
real number
irrational number

4. The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers.
Irrational numbers
Rational numbers
Real Numbers
Integers
Whole
numbers

5. Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals.
4 5 23
3 = 3.8
= 0.6
1.44 = 1.2

6. Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so is irrational.
A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.
Caution!

7. Reals Make a Venn Diagram that displays the following sets of numbers:
Reals, Rationals, Irrationals, Integers, Wholes, and Naturals.
Reals
Rationals
-2.65
Integers -3
-19
Wholes
Irrationals
Naturals
1, 2, 3...

8. Additional Example 1: Classifying Real Numbers
Write all classifications that apply to each number. A. 5
5 is a whole number that is not a perfect square.
irrational, real B.
–12.75
–12.75 is a terminating decimal.
rational, real
16 2
= = 2
4 2
16 2 C.
whole, integer, rational, real

9. whole, integer, rational, real
Check It Out! Example 1
Write all classifications that apply to each number. A. 9
9 = 3
whole, integer, rational, real B.
–35.9
–35.9 is a terminating decimal.
rational, real
81 3
= = 3
9 3
81 3 C.
whole, integer, rational, real

10. A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

11. Additional Example 2: Determining the Classification of All Numbers
State if each number is rational, irrational, or not a real number. A. 21
irrational
0 3
0 3
= 0 B.
rational

12. Additional Example 2: Determining the Classification of All Numbers
State if each number is rational, irrational, or not a real number.
4 0 C.
not a real number

13. State if each number is rational, irrational, or not a real number.
Check It Out! Example 2
State if each number is rational, irrational, or not a real number. A. 23
23 is a whole number that is not a perfect square.
irrational
9 0 B.
undefined, so not a real number

14. State if each number is rational, irrational, or not a real number.
Check It Out! Example 2
State if each number is rational, irrational, or not a real number.
64 81
8 9 =
64 81 C.
rational