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

2. Objective IDENTIFY RATIONAL AND IRRATIONAL NUMBERS
EXPLORE DECIMAL REPRESENTATIONS OF RATIONAL AND IRRATIONAL NUMBERS

3. What does it Mean? The number line goes on forever.
Every point on the line is a REAL number.
There are no gaps on the number line.
Between the whole numbers and the fractions there are numbers that are decimals but they don’t terminate and are not repeating decimals.

4. Real Numbers REAL NUMBERS 154,769,852,354 1.333
-5, … -8 61 π
49%

5. Two Kinds of Real Numbers
Rational Numbers
Irrational Numbers

6. Rational Numbers
A rational number is a real number that can be written as a fraction or a ratio.
A rational number written in decimal form is terminating or repeating.

7. Examples of Rational Numbers16
1/2
3.56 -8
1.3333…
- 3/4

8. REAL NUMBERS
NATURAL
Numbers
WHOLE
Numbers
IRRATIONAL
Numbers
INTEGERS
RATIONAL
Numbers

9. Irrational Numbers
An irrational number is a number that cannot be written as a fraction of two integers.
Irrational numbers written as decimals are non-terminating and non-repeating. They go on forever ∞.

10. 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.
A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.
Caution!

11. Examples of Irrational NumbersPi

12. Try this! a) Irrational b) Irrational c) Rational d) Rational
e) Irrational

13. 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

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

15. 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

16. 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

17. Comparing Rational and Irrational Numbers
When comparing different forms of rational and irrational numbers, convert the numbers to the same form.
Compare and
(convert -3 to …
… > 3 7 3 7

18. Practice

19. Ordering Rational and Irrational Numbers
To order rational and irrational numbers, convert all of the numbers to the same form.
You can also find the approximate locations of rational and irrational numbers on a number line.

20. Example Order these numbers from least to greatest.
¹/₄, 75%, .04, 10%, ⁹/₇
¹/₄ becomes 0.25
75% becomes 0.75
0.04 stays 0.04
10% becomes 0.10
⁹/₇ becomes …
Answer: , 10%, ¹/₄, 75%, ⁹/₇

21. Practice
Order these from least to greatest: