1. 3-10 Warm Up Problem of the Day Lesson Presentation The Real Numbers
Pre-Algebra

2. 3-10 Warm Up The Real Numbers
Pre-Algebra
3-10
The Real Numbers
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. Learn to determine if a number is rational or irrational.

5. Vocabulary
irrational number
real number
Density Property

6. Biologists classify animals based on shared characteristics
Biologists classify animals based on shared characteristics. The gray lesser mouse lemur is an animal, a mammal, a primate, and a lemur.
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.
Animals
Mammals
Primates
Lemurs

7. 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

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.
Helpful Hint

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

10. Additional Examples 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.
Try This: 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

12. Additional Examples 2: Determining the Classification of All Numbers
State if the number is rational, irrational, or not a real number. A. 15
15 is a whole number that is not a perfect square.
irrational
0 3
0 3
= 0 B.
rational

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

14. State if the number is rational, irrational, or not a real number.
Try This: Examples 2
State if the 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

15. State if the number is rational, irrational, or not a real number.
Try This: Examples 2
State if the number is rational, irrational, or not a real number. C. –7
not a real number
64 81
8 9 =
64 81 D.
rational

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 Examples 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. Find a real number between 4 and 4 . 4 7 3 7
Try This: 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

19. 1. 2. – 3. 4. 5. Find a real number between –2 and –2 .
Lesson Quiz
Write all names that apply to each number. 1. 2
2. –
16 2
real, irrational
real, integer, rational
State if the number is rational, irrational, or not a real number. 3.
25 0 4.
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