An associative rule is one in which the result of two or more operations does not depend on the order in which the operations are carried out.

 

Three numbers, x, y, and z, are said to be associative under addition if

 

    x + (y + z) = (x + y) + z

 

and to be associative under multiplication if

 

    x × (y × z) = (x × y) × z

 

In general, three elements a, b, and c of a set S are associative under the binary operation (an operation that works on two elements at a time) * if

 

    a * (b * c) = (a * b) * c

 

The word incorporates the Greek root soci, from which we also get "social," and may have been first used in the modern mathematical sense by William Hamilton around 1850. Compare with distributive and commutative.