1. Real Numbers and Properties

2. Natural Numbers…… Known as “Counting” Numbers
Example: 1, 2, 3, 4, 5,…….

3. Whole Numbers…… You add the number 0 to the natural numbers.
Example: 0, 1, 2, 3, 4, 5…….

4. Integers…… Integers are made up of whole numbers and their opposites.
Example: …-4,-3,-2,-1,0,1,2,3,4….

5. Rational Numbers……
The set of rational numbers is made up of all of the following
a. Natural Numbers
b. Whole Numbers
c. Integers
d. Plus every repeating and
terminating decimal.

6. Examples of Rational Numbers……
A. ½ = 0.5 (Terminating Decimal)
B (Repeating Decimal)
C (Repeating Decimal)
D (Terminating Decimal)

7. Irrational Numbers….
Consists of Non-Terminating and Non-Repeating Decimals.
Example:

8. Real Numbers (ℝ) Rational Numbers (ℚ) Irrational Numbers Integers (ℤ)
Decimal form is non-terminating and non-repeating
Whole Numbers
Natural Numbers (ℕ)
1, 2, 3, …
0, 1, 2, 3, …
…-3, -2, -1, 0, 1, 2, 3, …
Decimal form either terminates or repeats
All rational and irrational numbers

9. The Number Line……
A number line consists of positive numbers (right of 0) and negative numbers (left of 0).
A real life example of a number line is a temperature thermometer.

10. For example….. -5 would represent 5 degrees below zero.
+4 would represent 4 degrees above zero.

11. Make the Comparison……
7 degrees below 0 is (warmer/colder) than 4 degrees above 0.
7 degrees below 0 is a (lower/higher) temperature than 4 degrees above 0.
Colder
Lower

12. Coordinates on a Graph….
Find the best estimate of the point.
a b c d
Answer:

13. Sets and Subsets…… A set is a group of numbers.
Example: Set A = {1,2,3,4,5}
A subset is a group of numbers in which every member is in another set.
Example: Set B = {1,2,3}
So, B is a subset of A.

14. Which of the following would represent a subset of integers?
States Sales Tax Rate
Amount of Gas in a Car
Number of Students in Class
A Dinner Receipt
Strategy: Eliminate those that are NOT integers.
7.5% - NO
6.5 Gallons – NO
12 – YES
$ NO

15. You Try…Which of the following would represent a subset of integers?
Costs of a TV
# of miles on the odometer of a car
A person’s weight
Number of residents in South Carolina No
Yes

16. Inequalities….. We use inequalities to compare numbers.
The following are inequalities:

17. Examples…….
“4 is less than 7” -
“9 is greater than or equal to 5” -

18. You Try……Insert the appropriate inequality sign.
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>

19. Least to Greatest……
This means to arrange numbers in the order from the smallest to the largest.
HINT: If there are fractions it might be easier to convert to decimals first.

20. Which Number is Smaller?

21. Which Number is Larger?

22. You Try…Compare
>
-0.8 >

23. Which Set is Ordered from Least to Greatest?
{-3/2, -3, 0, 2/3}
{-3, -3/2, 0, 2/3}
{0, 2/3, -3/2, -3}
{0, -3/2, -3, 2/3}
{-3/2, -3, 0, 2/3}
{-3, -3/2, 0, 2/3}
3. {0, 2/3, -3/2, -3}
4. {0, -3/2, -3, 2/3

24. What kinds of numbers are used to represent numbers below zero?
Answer:
NEGATIVE Numbers

25. Make -8 -4 a true statement.
Answer:
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26. Quick Review -400 -200 200 400 Coordinate of A:
Coordinate of A:
a) b) c) d) -500
2) Coordinate of B:
a) b) c) d) -50
3) Coordinate of C:
a) b) c) d) 275

27. Quick Review 4) Use , : -8 5 5) Which is smaller? or
6) Write from smallest to largest:
-3, -3.8, -5, 5.6, -5.6

28. Number Properties

29. Commutative Property- Changes Order
For Addition
A+B = B+A
Ex = 5
3+2 = 5
2+3=3+2
For Multiplication
AB = BA
Ex. 4(8) = 32
8(4) = 32
4(8) = 8(4)
THIS IS NOT TRUE FOR SUBTRACTION OR DIVISION!

30. Associative Property- Changes Grouping
For Multiplication
A(BC) = (AB)C
Ex.
2 (3 5) (2 3) 5
= 2(15) = (6)5
= = 30
2 x (3 x 5) = (2 x 3) x 5
For Addition
A + (B + C) = (A + B) + C
Ex.
5 + (2 + 4) (5 + 2) + 4
= = 7 + 4
= = 11
5 + (2 + 4) = (5 + 2) + 4
This is not true for subtraction or division!

31. Which Property? 3x 4 = 4 3x 6y + (7 + 3z) = (6y +7) +3z
(5x + 7) + 8y = 5x + (7 + 8y)
(3x)(2x + 5) = (2x + 5)(3x)
10x + 4y = 4y + 10x
(2x 5)(10y) = (2x)(5 10y)

32. Distributive Property
A (B + C) = AB + AC
(B + C) A = BA + CA
A (B – C) = AB – AC
(B – C) A = BA – CA
Ex. -3 (4 – 2x)
Strategy: Think -3 (4 – 2x) means -3 (4 + -2x)
= -3(4) + (-3)(-2x)
= x
TRY THESE:
A) 4 (6 +2a) B) -7 (-3m – 5)

33. Which Property? -3x(y + 2) + 4y = -3x(y) – 3x(2) + 4y

34. What is an example of the commutative prop. of addition?

35. Homework