# 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.

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

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