1. Properties of Real Numbers
Unit 2, Lesson 1
Online Algebra 1
Cami Craig

2. Properties
In this lesson we are going to look at properties, rules of mathematics that can be proven.
We will be looking at properties of equality, and real number properties.

3. Properties of Real Numbers
Properties of Real Numbers include:
Commutative Property
Associative Property
Identity Property
Distributive Property
Inverse Property
Closure Property

4. Commutative Properties
Commutative Properties deal with order. Order in multiplication and addition do not matter!
Addition
a + b = b + a Or
5 + 9 = 9 + 5
…..Is this true? Try it!
Of course it is 14 = 14
Multiplication
ab = ba Or
5(9) = 9(5)
…..Is this true? Try it!
Of course it is 45 = 45

5. Commutative Properties and Subtraction
Does the Commutative Property hold for subtraction and division?
Let’s try:
If the commutative property holds for subtraction then the following should be true:
6 – 3 = 3 – 6
But we know it isn’t true
6 – 3 = 3 and 3 – 6 = -3
So the commutative property does NOT work for subtraction.

6. Commutative Properties and Division
Does the Commutative Property hold for division?
Let’s try:
If the commutative property holds for division then the following should be true:
10 ÷ 5 = 5 ÷ 10
But we know it isn’t true
10 ÷ 5 = 2 and 5 ÷ 10 = 0.5
So the commutative property does NOT work for division.

7. Associative Properties
Associative Properties deal with Grouping. Grouping in multiplication and addition do not matter! But just like the commutative properties the associative property does not apply to subtraction.
Multiplication
a(bc) = (ab)c Or
4 x (6 x 2) = (4 x 6) x 2
…..Is this true? Try it!
4 x 12 = 24 x 2
48 = 48
Addition
a + (b + c) = (a + b) + c Or
11 +(5 + 9) = (11 + 5) + 9
…..Is this true? Try it!
11 + ( 5 + 9) = (11 + 5) + 9 =
25 = 25

8. Associative and Commutative Properties
How can you tell these properties apart?
Commutative Properties =
4 x 5 = 5 x 4
Associative Properties
3 + ( 6 + 4) = (3 + 6) + 4
7 x (3 x 5) = (7 x 3) x 5
Notice that in the commutative property the order of the numbers change, while in the associative properties the order stays the same, but the grouping changes.

9. Distributive Property
The distributive property is:
a(b + c) = ab + ac Or
2( 4 + 5) = 2 x x 5
I like to call the Distributive Property the fair share property, because the number on the outside of the parentheses is multiplied to both numbers in the parentheses.

10. Identity Properties
The Identity Properties deal with getting back the same thing.
Addition
When we add 0 to a number, we get that original number back:
For example:
A + 0 = A
= -4
We actually call 0 the Additive Identity Element.
Multiplication
When we multiply 1 to a number, we get that original number back:
For example:
1a = a
-15(1) = -15
We actually call 1 the Multiplicative Identity Element.

11. Inverse and Closure Properties
The inverse property for addition states that a+ -a = 0.
The inverse property for multiplication states that a x 1/a = 1.
The closure property for addition states that a + b is a real number.
The closure property for multiplication states that a x b is a real number.

12. Let’s see what you have learned so far
Let’s see what you have learned so far! What property does each example represent?
-2(3 + 4) = -2 x x 4
5 + (3 + 6) = (3 + 6) + 5
5(1) = 5
17 x (8 x 2) = (17 x 8) x 2
9 + 0 = 9
4 x ¼ = 1
Distributive Property
Commutative Property
Notice that although there are parentheses, it is the order that changes not the grouping.
Identity Property of Multiplication
Associative Property
Identity Property of Addition
Inverse Property of Mult.

13. Properties of Equality
Properties of equality include the following:
The Reflexive Property
The Symmetric Property
The Transitive Property

14. Properties of Equality
Reflexive Property of Equality
a = a
-9 = -9
Symmetric Property of Equality
If a = b, then b = a.
If 15x = 45, then 45 = 15x
Transitive Property of Equality
If a = b and b = c, then a = c
If d = 3y and 3y = 6, then d = 6.

15. Review What property is each of the following an example of?
9 = 9
a + 8 = 8 + a
If x + 8 = 9, and 9 = 4 + 5, then x + 8 = 4 + 5
3(x – 7) = 3x – 21
5 x 1 = 5
If 16 = 4x, then 4x = 16
= 0
Reflexive
Commutative
Transitive
Distributive
Identity
Symmetric
Inverse