1. Warm Up
Each square root is between two integers. Name the two integers.
Estimate each value.
Round to the nearest tenth.
10 and 11
2. – 15
–4 and –3
1.4
4. – 123
–11.1

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

3. Vocabulary
irrational number
real number
Density Property

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. 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

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

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

8. What are the different types of numbers?
Real Numbers
Rationals
Irrationals
Integers
Naturals
Wholes

9. Fill In Your Real Number Chart
Counting “Natural” Numbers: 1, 2, 3, 4, 5, 6, . . .
Whole Numbers: 0, 1, 2, 3, 4, . . .
Integers: , -2, -1, 0, 1, 2, 3,
Rational Numbers: 0, …1/10, …1/5, …1/4, , …1/2, …1, perfect squares
Real Numbers: all numbers
Irrationals: π, non-repeating decimal, nonperfect squares

10. Classifying Real Numbers
Write all names that apply to each number (whole, integer, rational, irrational, real)

11. Example 1 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

12. Example 2 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

13. Determining the Classification of All Numbers
State if each number is rational, irrational, or not a real number.

14. Example 3 A. 21
irrational
0 3
0 3
= 0 B.
rational

15. Example 3 continued.. C. –4
not a real number
4 9
2 3 =
4 9 D.
rational

16. Example 4 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

17. Example 4 Continued… C. –7
not a real number
64 81
8 9 =
64 81 D.
rational

18. 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.

19. Find a real number between a set of numbers
There are many solutions. Let’s try to find the solution that is halfway between the two numbers

20. Find a real number between 3 and 3 . 3 5 2 5
Example 5
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

21. Find a real number between 4 and 4 . 4 7 3 7
Example 6
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

22. Lesson Summary
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