1.8

Introduction Properties of Real Numbers allow you to

write equivalent expressions and to simplify expressions.

The following list of properties applies to addition and multiplication.

What about Subtraction and Division? Remember our rules for addition and

multiplication: We can think of all subtraction problems as

addition problems (add the opposite) and all division problems can be turned into multiplication problems (multiply by the reciprocal.)

The Commutative Property Think: Change the order of the numbers;

move the numbers around The Commutative Property of Addition

a + b = b + a Example: 6 + 2 = 2 + 6

Is that true? The Commutative Property of

Multiplication a b = b a Example: 4 5 = 5 4

Is that true?

The Associative Property Think: Move the parentheses to

associate (or combine) different numbers

Associative Property of Addition (a + b) + c = a + (b + c) Example: (1 + 2) + 3 = 1 + (2 + 3)

Is that true? Associative Property of Multiplication

(a b) c = a (b c) Example: (4 2) 3 = 4(2 3)

Is that true?

Identity Properties Identity Property of Addition

Think: What can I add to a number without changing its identity? Add 0 and I’ll get what I started with

a + 0 = a Example: 14 + 0 = 14

Identity Property of Multiplication Think: What can I multiply a number by without

changing its identity? Multiply by 1 and I’ll get what I started with

a 1 = a Example: 26 1 = 26

Inverse Properties

Inverse Property of Addition Think: adding opposites = 0 a + (-a) = 0 Example: 6 + -6 = 0

Inverse Property of Multiplication Think: multiplying reciprocals = 1 a ( ) = 1 Example: 7 ( ) = 1

The Distributive Property Think: Distribute your outside number to

each of your inside numbers Multiply both of your inside #’s by your outside

#’s, then add or subtract. a( b + c) = ab + ac a (b – c) = ab – ac Example: 10(20 – 2) = 10(20) – 10(2)

10(18) = 200 – 20 180 = 180

Multiplication Properties Multiplication Property of Zero

Think: Multiply anything by 0 and you’ll get 0 n 0 = 0 Example: 245.5 0 = 0

Multiplication Property of –1 Think: Multiply by -1 means you switch the

sign -1 n = –n -1 68 = – 68

Identifying These Properties

Think: What’s happening with the numbers? What operation is involved?

1.6 + 2 = 2 + 6 We’re switching the order Commutative Property of Addition

2.5 + 0 = 5 We’re adding zero. We get what we

started with Identity Property of Addition

Think: What’s happening with the numbers? What operation is involved?

3.-3 + (5 + 6) = (-3 + 5) + 6 We’re not changing the order, but we’re

moving around the parentheses. We’re adding.

Associative Property of Addition4. 3 1 = 3

We’re multiplying by 1, and we get what we started with.

Identity Property of Multiplication

Think: What’s happening with the numbers? What operation is involved?

5.- 8 + 8 = 0 We’re adding opposites and we get 0. Inverse Property of Addition

6.10 ( ) = 1 We’re multiplying reciprocals and we get

1. Inverse Property of Multiplication

Think: What’s happening with the numbers? What operation is involved?

7.6(2 – a) = 12 – 6a We’re multiplying both of our inside

numbers by our outside number. The Distributive Property

8. -1 5 = -5 We’re multiplying 5 by -1, and we get the

opposite of 5. Multiplication Property of -1

Think: What’s happening with the numbers? What operation is involved?

9.3(a 4) = (3 a) 4 We aren’t changing the order, but we’re

moving around the parentheses. We’re multiplying.

The Associative Property of Multiplication10. 8 0 = 0

We’re multiplying by 0 and we get 0. Multiplication Property of Zero

Before you Leave…

On your notes page, give me an example of the Identity Property of Addition.

Explain the difference between the Commutative Property of Multiplication and the Associative Property of Multiplication.