Objective 10 Properties of addition and multiplication © 2002 by R. Villar All Rights Reserved

Properties of addition and multiplication Properties of Real Numbers: these are statements that are true for all real numbers. Consider the following: = = 4 3 Order doesn’t matter when adding or multiplying. The properties that state this are called the Commutative Properties Commutative Properties: for all real numbers a and b: for addition a + b = b + a for multiplication a b = b a

Consider the following: 5 + (3 + 4) = (5 + 3) + 42 (3 4) = (2 3) (7) = (8) (12) = (6) 4 12 = = 24 These examples show that when adding or multiplying, grouping (where you place the parentheses) does not matter. The properties that state this are called the Associative Properties Associative Properties: for all real numbers a, b and c: for addition (a + b) + c = a + (b + c) for multiplication(a b) c = a (b c)

The Distributive Property allows you to multiply each term inside a set of parentheses by a factor outside the parentheses. Distributive Property: For all real numbers a, b and c, a(b + c) = ab + ac (b + c)a = ab + ac a(b – c) = ab – ac (b – c)a = ab – ac = 12(23) = 276 Here is another way to simplify the same problem… 12(20 + 3) = = = 276 Consider the following: 12( )

Here are a couple of other properties you should know... Identity Properties: for all real numbers a: for addition a + 0 = a for multiplication a 1 = a Closure Properties: for all real numbers a and b: for addition a + b is a real number for multiplication a b is a real number

Examples: Identify the property that is being illustrated by the following statements... a.(2 + 4) + 6 = (4 + 2) + 6 b = 71 c.w(z + 38) = wz + w(38) Although the statement contains parentheses, the only part of the statement that has been changed is the order of the addition inside of the parentheses. Commutative Property of Addition Identity Property of Addition Notice that the w is being multiplied (or distributed) by each term in the parentheses. Distributive Property