Properties of Real Numbers

Unit 2, Lesson 1Online Algebra 1VHS@pwcsCami Craig

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.

Properties of Real Numbers

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

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

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.

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.

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.

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

11 + 14 = 16 + 9

25 = 25

Multiplication

a(bc) = (ab)c

Or

4 x (6 x 2) = (4 x 6) x 2

…..Is this true? Try it!

4 x (6 x 2) = (4 x 6) x 2

4 x 12 = 24 x 2

48 = 48

Associative and Commutative Properties

How can you tell these properties apart?

Commutative Properties

8 + 11 = 11 + 8

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.

Distributive Property

The distributive property is:a(b + c) = ab + ac

Or 2( 4 + 5) = 2 x 4 + 2 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.

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 + 0 = -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.

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.

Let’s see what you have learned so far!What property does each example represent?

1. -2(3 + 4) = -2 x 3 + -2 x 4

2. 5 + (3 + 6) = (3 + 6) + 5

3. 5(1) = 5

4. 17 x (8 x 2) = (17 x 8) x 2

5. 9 + 0 = 9

6. 4 x ¼ = 1

1. Distributive Property

2. Commutative Property Notice that although there

are parentheses, it is the order that changes not the grouping.

3. Identity Property of Multiplication

4. Associative Property

5. Identity Property of Addition

6. Inverse Property of Mult.

Properties of Equality

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

Properties of Equality

1. Reflexive Property of Equality a = a -9 = -9

2. Symmetric Property of Equality If a = b, then b = a. If 15x = 45, then 45 = 15x

3. Transitive Property of Equality If a = b and b = c, then a = c If d = 3y and 3y = 6, then d = 6.

ReviewWhat property is each of the following an example of?

1. 9 = 9

2. a + 8 = 8 + a

3. If x + 8 = 9, and 9 = 4 + 5, then x + 8 = 4 + 5

4. 3(x – 7) = 3x – 21

5. 5 x 1 = 5

6. If 16 = 4x, then 4x = 16

7. 7 + -7 = 0

1. Reflexive

2. Commutative

3. Transitive

4. Distributive

5. Identity

6. Symmetric

7. Inverse