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

2. Learning Goal
Decompose the Real Number system into parts to see the differences between sets of numbers.

3. Success Criteria I can identify subsets of the real number system
I can sort rational and irrational numbers
I can locate rational and irrational numbers on the number line

4. 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

5. 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

6. 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

7. 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!

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. 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

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

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. A. 23
23 is a whole number that is not a perfect square.
irrational
9 0 B.
undefined, so 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.
64 81
8 9 =
64 81 C.
rational

14. 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

15. 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

16. The Density Property of real numbers states that between any two real numbers is another real number. This property is not true when you limit yourself to whole numbers or integers. For instance, there is no integer between –2 and –3.

17. Additional 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
A real number between and is 3 .
3 5
2 5
1 2
Check: Use a graph. 3
1 5
2 5 4
3 5
4 5 3
1 2

18. 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
A real number between and is
4 7
3 7
1 2
Check: Use a graph. 4
2 7
3 7
4 7
5 7
1 7
6 7 4
1 2

19. 1. 2. – 4. 3. 5. Lesson Quiz 2 16 2 real, irrational
Write all classifications 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

20. Sorting Activity
Sort the cards from the envelope into rational numbers and irrational numbers
Can the rational numbers be broken into smaller categories?

21. Number Line Activity Draw a number line on the construction paper
Use the colored stickers to label each number from the set
Green-Rational only
Pink-Rational, but could be a whole number or an integer also
Yellow-Irrational

22. How’d we do with our Success Criteria?
I can identify subsets of the real number system
I can sort rational and irrational numbers
I can locate rational and irrational numbers on the number line

23. So, did we meet our Learning Goal?
Decompose the Real Number system into parts to see the differences between sets of numbers.