Any binary operation ° that satisfies the law
x ° (y ° z) = (x ° y) ° z
for all x, y, and z in the domain of °. The law is
known as the associative law. An expression involving several adjacent instances
of an associative operation can be interpreted unambiguously; the order in which
the operations are performed is irrelevant since the effects of different
evaluations are identical, though the work involved may differ. Consequently
parentheses are unnecessary, even in more complex expressions.
The arithmetic
operations of addition and multiplication are associative, though subtraction is
not. On a computer the associative law of addition of real numbers fails to hold
because of the inherent inaccuracy in the way real numbers are usually
represented (see floating-point notation), and the
addition of integers fails to hold because of the possibility of overflow.
