1. Real Numbers Lesson 1.1

2. Introduction…. What are real numbers?
Real Numbers: are any number except for variables
Variables– letters that represent numbers
Where do we see real numbers in life (outside of math class)?

3. Definitions
Integers: any whole number that is positive, negative, or zero
Non-integers: fractions that CAN be written as terminating or repeating decimals
Rational Numbers: any number that can be written as a ratio in the form of 𝑎 𝑏 where a and b are both integers and b is NOT zero

4. Types of rational numbers
- Rational numbers either terminate or repeat
A terminating decimal come to a complete stop
A repeating decimal continues the same digit or block of digits forever
Examples

5. Example 1 write each fraction as a decimal
Write as a decimal…SHOW YOUR WORK
Remember that the fraction bar means “divided by”
Divide the numerator by the denominator
Divide until the remainder is zero, adding zeros after the decimal point in the dividend as needed
Write as a decimal… SHOW YOUR WORK
Divide until the remainder is zero or until the digits in the quotient begin to repeat
Add zeros after the decimal point in the dividend as needed
When a decimal has one or more digits that repeat indefinitely, write the decimal with a bar over the repeating digits.
Are these two fractions rational? Why or why not?
Turn to page 8 in your textbook and do “Your Turn”

6. Example 2 Expressing decimals as rational numbers
Write each fraction in simplest form
0.825
The decimal means “825 thousandths” so put 825 over 1000
Simplify: divide the numerator and denominator by 25
0.37
How many repeating decimals are there?
The decimal 0.37…. Means 37 hundredths. BUT there is a REPEAT SIGN….
Because of the repeating sign, you have to subtract the denominator by 1 (100-1)
Simplify if necessary
Turn to page 9 in your textbook and do “Your Turn”

7. Finding Square Roots &Cube Roots Lesson 1.1

8. Definitions Square root : is used to unsquare a number
Symbol for square root is √
There are two square roots for every positive number
EXAMPLE: the square root of 36 is ……..
HINT: what two same numbers go into 36?
Correct 6 x 6 = 36
The square root for 36 is both 6 and -6
Principle Square Root: the positive number
EXAMPLE: what is the principle square root of 36?
Is it 6 or -6
The answer is 6. It is the positive number

9. Square root examples 1. 49 7 and -7 2. 144 12 and -12 3. 1 25
What is the square root(s) for …..
7 and -7
2. 144
12 and -12
and −1 5

10. Perfect squares What is a perfect square?
Let’s Explore Perfect Squares
12 = √
22 = √
32 = √
42 = √
52 = √
62 = √
72 = √
82 = √
92 = √
102 = √
Perfect squares
What is a perfect square?
Perfect Square: has two square roots that are integers.
The numbers are the same.
What are examples?
Example: the perfect square of 81 is 9 and -9

11. Example 3: Square Roots X2 = 121
1. solve for x by taking the square root of both sides (since square roots and squaring are inverse operations)
√ X2 = √121
X= +11
X2 =
1. take the square root of the numerator and the denominator
√ X2 = √16 √169 X=

12. Cube roots What is a cube root? Cube Root: is to uncube a number
Let’s Explore Perfect Cubes
13 = 3 √
23 = √
33 = √
43 = 3 √
53 = √
63 = √
73 = √
83 = √
93 = √
103 = 3 √
Cube roots
What is a cube root?
Cube Root: is to uncube a number
There is only one cube root for every positive number
Symbol 3 √
Example:
The cube root of 8 is 2 because 2 x 2 x 2 = 8
The cube root of 27 is 3 because 3 x 3 x 3 =27
Perfect Cube: has a cube root that is an integer.
The examples above are perfect cubes

13. Turn to page 10 and do problems 7-10 on “Your Turn”
Example 4: Cube Roots
X3 = 729
X3 = 8 125
Solve for x by taking the cube root of both sides
3√ X3 = 3√ 729
Apply the definition of cube roots
Why can the number NOT be negative?
When a number is cubed, the answer can only be positive
1. take the cube root of the numerator and the denominator
3√ X3 = 3√8 3√125
X= 2 5
Turn to page 10 and do problems 7-10 on “Your Turn”

14. Exploring irrational numbers Lesson 1.1

15. Irrational numbers
Irrational Numbers are numbers that have a decimal expansion that go on forever (infinity) without a repeating pattern
Examples
…..
Π = ….

16. Estimating irrational numbers
Square roots of numbers that are not perfect squares are irrational
Example √3
to estimate the √3, first find the two consecutive perfect squares that 3 is between
√1 = 1 √4 = 2
Simplify the square roots of perfect squares
√3 is between 1 and 2
Find the square root of 3 on your calculator…. Plot on a number line
Turn to page in the textbook and complete those pages together!

17. Practice activity: rational vs. irrational
Tell whether each real number is rational or irrational
-23.7
Rational----- decimal terminate
…..
Irrational---- decimal doesn’t terminate or repeat
5/9
Rational-----number is in fraction for
√15
Irrational---- decimal form does not terminate