1. How Do I Distinguish Between Rational And Irrational Numbers?
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How Do I Distinguish Between Rational And Irrational Numbers?
Course 3
Lesson Presentation

2. Learn to determine if a number is rational or irrational.

3. Vocabulary
real number
rational number
irrational number

4. Recall that rational numbers can be written as fractions
Recall that rational numbers can be written as fractions. Rational numbers can also be written as decimals that either terminate or repeat.
4 5 23
3 = 3.8
= 0.6
1.44 = 1.2

5. Irrational numbers cannot be written as a fraction & can only be written as decimals that do not terminate or repeat (NON-TERMINATING/NON-REPEATING DECIMALS. The square root of a non-perfect square is an irrational number.
For example,
2 ≈ …
A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.
Caution!

6. The set of real numbers consists of the set of rational numbers and the set of irrational numbers.
Integers
Whole
numbers

7. Example 1: Classifying Real Numbers
Write all names 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

8. Write all names that apply to each number.
Check It Out: Example 1
Write all names 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

9. 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. Example 2: Determining the Classification of All Numbers
State if each number is rational, irrational, or not a real number. C. –4
not a real number
4 9
2 3 =
4 9 D.
rational

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.
not a number, 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. C. –7
not a real number
64 81
8 9 =
64 81 D.
rational

13. The Density Property of real numbers states that between any two real numbers is another real number. This property is also true for rational numbers, but not for whole numbers or integers. For instance, there is no integer between –2 and –3.

14. Example 3: Applying the Density Property of Real Numbers
Find a real number between and
3 5
2 5
There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.
2 5
÷ 2
3 5
5 5
= ÷ 2
1 2
= 7 ÷ 2 = 3 3
1 5
2 5 4
3 5
4 5 3
1 2
A real number between and is 3 .
3 5
2 5
1 2

15. Find a real number between 4 and 4 . 4 7 3 7
Check It Out: Example 3
Find a real number between and
4 7
3 7
There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.
3 7
÷ 2
4 7
7 7
= ÷ 2
1 2
= 9 ÷ 2 = 4 4
2 7
3 7
4 7
5 7
1 7
6 7 4
1 2
A real number between and is
4 7
3 7
1 2

16. Lesson Quiz
Write all names that apply to each number. 1. 2
2. –
16 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 5.
Find a real number between –2 and –2 .
3 8
3 4
Possible answer –2 .
5 8