5.5 Real Numbers and Their Properties 261 Now we introduce the associative property. Associative Property Addition Multiplication + + = + + a b c a b c ( ) ( ) ⋅ ⋅ = ⋅ ⋅ a b c a b c ( ) ( ) for any real numbers a, b, and c. The associative property states that when adding or multiplying three real numbers, we may place parentheses around any two adjacent numbers. The result is the same regardless of the placement of parentheses. For example, + + = + + + = + = (34)5 3(45) 7 5 3 9 12 12 ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ = (34)5 3(45) 12 5 3 20 60 60 The associative property does not hold for the operations of subtraction or division. For example, − − ≠ − − ÷ ÷ ≠ ÷ ÷ (10 6) 2 10 (6 2) and (27 9) 3 27 (9 3) Note the difference between the commutative property and the associative property. The commutative property involves a change in order, whereas the associative property involves a change in grouping (or the association of numbers that are grouped together). Another property of the real numbers is the distributive property of multiplication over addition. Distributive Property of Multiplication over Addition ⋅ + = ⋅ + ⋅ a b c a b a c ( ) for any real numbers a, b, and c. For example, if = = a b 3, 4, and = c 5, then ⋅ + = ⋅ + ⋅ ⋅ = + = 3(4 5) (34) (35) 3 9 12 15 27 27 This result indicates that, when using the distributive property, you may either add the numbers within parentheses first and then multiply or multiply first and then add. Note that the distributive property involves two operations, addition and multiplication. Although positive integers were used in the example, any real numbers could have been used. We frequently use the commutative, associative, and distributive properties without realizing that we are doing so. To add + + 13 4 6, we may add the + 4 6 first to get 10. To this sum we then add 13 to get 23. Here we have done the equivalent of placing parentheses around the + 4 6. We can do so because of the associative property of addition. To multiply × 102 11 in our heads, we might multiply × = 100 11 1100 and × = 2 11 22 and add these two products to get 1122. We are permitted to do so because of the distributive property. × = + × = × + × = + = 102 11 (100 2) 11 (100 11) (2 11) 1100 22 1122
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