1. Rational Vs. Irrational
Making sense of rational and Irrational numbers

2. Biologists classify animals based on shared characteristics
Biologists classify animals based on shared characteristics. The horned lizard is an animal, a reptile, a lizard, and a gecko.
You already know that some numbers can be classified as whole numbers, integers, or rational numbers. The number 2 is a whole number, an integer, and a rational number. It is also a real number.
Animal
Reptile
Lizard
Gecko

3. 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

4. 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

5. 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!

6. 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

7. 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

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

9. 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

10. 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

11. 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

12. 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

13. 1. 2. – 4. 3. Lesson Quiz 16 2 2 real, irrational
Write all classifications that apply to each number. 1.
16 2 2
2. –
real, irrational
real, integer, rational
State if each number is rational, irrational, or not a real number.
25 0 4. 3.
4 • 9
not a real number
rational