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associative law

associative law, in mathematics, law holding that for a given operation combining three quantities, two at a time, the initial pairing is arbitrary; e.g., using the operation of addition, the numbers 2, 3, and 4 may be combined (2+3)+4 = 5+4 = 9 or 2+(3+4) = 2+7 = 9. More generally, in addition, for any three numbers a, b, and c the associative law is expressed as ( a + b )+ c = a +( b + c ). Multiplication of numbers is also associative, i.e., ( a × bc = a ×( b × c ). In general, any binary operation, symbolized by ∘, joining mathematical entities A, B, and C obeys the associative law if ( AB )∘ C = A ∘( BC ) for all possible choices of A, B, and C. Not all operations are associative. For example, ordinary division is not, since (60÷12)÷3 = 5÷3 = 5/3, while 60÷(12÷3) = 60÷4 = 15. When an operation is associative, the parentheses indicating which quantities are first to be combined may be omitted, e.g., (2+3)+4 = 2+(3+4) = 2+3+4.

The Columbia Electronic Encyclopedia, 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

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