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., (ab)c=a(bc). In general, any binary operation, symbolized by ?, joining mathematical entities A, B, and C obeys the associative law if (A?B)?C=A?(B?C) for all possible choices of A, B, and C. Not all operations are associative. For example, ordinary division is not, since (6012)3=53=5/3, while 60(123)=604=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.

See more Encyclopedia articles on: Mathematics