Algebra 1

Chapter 1 Section 6

1-6 Properties of Real Numbers

The commutative and associate properties of addition and multiplication allow you to rearrange an expression.

Commutative Property - You can add or multiply real numbers in any order.

a + b = b + a a · b = b · a

Associative Property – When you add or multiply 3 or more numbers, changing the grouping will not change the answer.

a + (b + c) = (a + b) + c

a · (b · c) = (a · b) · c

Example 1: Identifying Properties

Name the property that is illustrated in each equation.

A) (4 + x) + y = 4 + (x + y)

B) -5 · b = b · (-5)

C) 2 + (6 + m) = 2 + (m + 6)

The commutative and associative properties are true for addition and multiplication. They may not be true for other operations. A counterexample is an example that disproves a statement, or shows that it is false. One counterexample is enough to disprove a statement.

Example 2: Finding Counterexamples to Statements about Properties

Find a counterexample to disprove the statement “ The Associative Property is true for subtraction.”

Find three real numbers, a, b, c, such that

a - (b - c) ≠ (a - b) – c.

Let’s try some numbers.

The Distributive Property

For real numbers a, b, and c,

a(b + c) = ab + ac

Ex.)3(4 + 8) = 3(4) + 3(8)

3(12) = 12 + 24

36 = 36

Example 3:

A) 3(x + 2)

B) 6(x - 8)

c) x(x + 5)