1. NUMBERS
Different Kinds
and Properties
Part 1 (introduction)

2. Set
A collection of objects.
Set Notation
{ }
Natural
numbers
Counting numbers {1,2,3, …}
Whole
Numbers
Natural numbers and 0.
{0,1,2,3, …}
Integers
Positive and negative natural numbers and zero {… -2, -1, 0, 1, 2, 3, …}
Vocabulary
Rational
Number
A real number that can be expressed as a ratio of integers (fraction)
Irrational
Number
Any real number that is not rational.
Real Numbers
All numbers associated with the number line.

3. Essential Questions: How do you know if a number is a rational number?
What are the properties used to evaluate rational numbers?

4. Two Kinds of Real Numbers
Rational Numbers
Irrational Numbers

5. Rational Numbers
A rational number is a real number that can be written as a ratio of two integers.
A rational number written in decimal form is terminating or repeating.
EXAMPLES OF RATIONAL NUMBERS 16
1/2
3.56 -8
1.3333…
-3/4

6. Irrational Numbers Square roots of non-perfect “squares” Pi- īī
An irrational number is a number that cannot be written as a ratio of two integers.
Irrational numbers written as decimals are non-terminating and non-repeating.
Square roots of non-perfect “squares”
Pi- īī 17

7. Real Numbers Irrational numbers Rational numbers Integers Whole

8. Rational Numbers Natural Numbers - Natural counting numbers.
1, 2, 3, 4 …
Whole Numbers -
Natural counting numbers and zero.
0, 1, 2, 3 …
Integers -
Whole numbers and their opposites.
… -3, -2, -1, 0, 1, 2, 3 …
Rational Numbers -
Integers, fractions, and decimals.
Ex:
-0.76, -6/13, 0.08, 2/3

9. Making Connections
Biologists classify animals based on shared characteristics. The horned lizard is an animal, a reptile, a lizard, and a gecko. Rational Numbers are classified this way as well!
Animal
Reptile
Lizard
Gecko

10. Real Numbers Venn Diagram: Naturals, Wholes, Integers, Rationals

11. Reminder
Real numbers are all the positive, negative, fraction, and decimal numbers you have heard of.
They are also called Rational Numbers.
IRRATIONAL NUMBERS are usually decimals that do not terminate or repeat. They go on forever.
Examples: π

12. Properties
A property is something that is true for all situations.

13. Four Properties Distributive Commutative Associative
Identity properties of one and zero

15. Distributive Property
A(B + C) = AB + BC
4(3 + 5) = 4x3 + 4x5

16. Commutative Property of addition and multiplication
Order doesn’t matter
A x B = B x A
A + B = B + A

17. Associative Property of multiplication and Addition
Associative Property  (a · b) · c = a · (b · c)
Example: (6 · 4) · 3 = 6 · (4 · 3)
Associative Property  (a + b) + c = a + (b + c)
Example: (6 + 4) + 3 = 6 + (4 + 3)

18. Identity Properties
If you add 0 to any number, the number stays the same.
A + 0 = A or = 5
If you multiply any number times 1, the number stays the same.
A x 1 = A or 5 x 1 = 5

19. Example 1: Identifying Properties of Addition and Multiplication
Name the property that is illustrated in each equation.
A. (–4)  9 = 9  (–4) B.
(–4)  9 = 9  (–4)
The order of the numbers changed.
Commutative Property of Multiplication
The factors are grouped differently.
Associative Property of Addition

20. Example 2: Using the Commutative and Associate Properties
Simplify each expression. Justify each step.
Commutative Property of Addition =
Associative Property of Addition
= (29 + 1) + 37 =
Add.
= 67

21. Exit Slip! Name the property that is illustrated in each equation.
1. (–3 + 1) + 2 = –3 + (1 + 2)
2. 6  y  7 = 6 ● 7 ● y
Simplify the expression. Justify each step. 3.
Write each product using the Distributive Property. Then simplify
4. 4(98)
5. 7(32)
Associative Property of Add.
Commutative Property of Multiplication 22
392
224