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×b)×c=a×(b×c). 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 (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.

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associative law

Two closely related laws of number operations. In symbols, they are stated: a + (b + c) = (a + b) + c, and a(bc) = (ab)c. Stated in words: The terms or factors may be associated in any way desired and the result will be the same. This holds for the numbers generally encountered: positive and negative, integral and fractional, rational and irrational, real and imaginary. Exceptions occur (e.g., in nonassociative algebras and divergent infinite series). See also commutative law, distributive law.