# associative

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.