associative law

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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.

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associative law Rule of combination in mathematics, in which the result of two or more operations on terms does not depend on the way in which they are grouped. Thus, normal addition and multiplication of numbers follows the associative law, since a + (b + c) = (a + b) + c, and a × (b × c) = (a × b) × c.

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associative law See associative operation.