1. Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes

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

3. Problem of the Day
The circumference of a circle is approximately 3.14 times its diameter. A circular path 1 meter wide has an inner diameter of 100 meters. How much farther is it around the outer edge of the path than the inner edge?
6.28 m

4. Sunshine State Standards
MA.8.A.6.2 …Compare mathematical expressions involving real numbers and radical expressions.

5. Vocabulary
irrational number
real number
Density Property

6. Biologists classify animals based on shared characteristics
Biologists classify animals based on shared characteristics. The cardinal is an animal, a vertebrate, a bird, and a passerine.
Animals
You already know that some numbers can be classified as whole numbers, integers, or rational numbers.
Vertebrates
Birds
Passerines

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

8. 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!

9. The set of real numbers consists of the set of rational numbers and the set of irrational numbers.

10. Additional 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

11. 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;
natural, whole, integer, rational, real
3 5 B. –
rational, real C.
3.724
rational, real

12. Additional Example 2: Determining the Classification of All Numbers
State if each number is rational, irrational, or not a real number.
0 3
0 3
= 0 A.
rational B. –4
not a real number

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

14. Check It Out: Example 2
State if the number is rational, irrational, or not a real number. Justify your answer. A. –7
√–7 is not a real number because it is the square root of a negative number.
6 0 B. –
–6 0
is not a real number because division by 0 is not
defined.

15. Check It Out: Example 2
State if the number is rational, irrational, or not a real number. Justify your answer. C.
√59 + 5
√59 + 5
= √64 = 8 is a rational number.

16. 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.

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 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

18. Check It Out: Example 3
Find a real number between and
2 8
1 8
1 2
1 8
2 8
83 8 =
1 2
83 8 =
3 16
= 5
3 16
Possible answer: 5

19. Lesson Quiz for Student Response Systems
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems 19 19

20. 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. Justify your answer.
25 0 4. 3.
4 • 9
not a real number;  by 0
rational;
2  3 is 6 5.
Find a real number between –2 and –2 .
3 8
3 4
Possible answer –2 .
5 8

21. Lesson Quiz for Student Response Systems
1. Identify all names that apply to
A. real, irrational
B. real, rational C. not real, irrational
D. not real, rational 21 21

22. Lesson Quiz for Student Response Systems
2. Identify the name that applies to
A. irrational
B. rational C. not a real number
D. none 22 22

23. Lesson Quiz for Student Response Systems
3. Identify a real number between
A. –4
B. C. D. 23 23