Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real numbers, it can be said that "addition and multiplication of real numbers are associative operations".
If the product operation is associative, the generalized associative law says that all these expressions will yield the same result. So unless the expression with omitted parentheses already has a different meaning (see below), the parentheses can be considered unnecessary and "the" product can be written unambiguously as
As the number of elements increases, the number of possible ways to insert parentheses grows quickly, but they remain unnecessary for disambiguation.
Joint denial is an example of a truth functional connective that is not associative.
For such an operation the order of evaluation does matter. For example:
Also although addition is associative for finite sums, it is not associative inside infinite sums (series). For example,
To illustrate this, consider a floating point representation with a 4-bit mantissa:
A left-associative operation is a non-associative operation that is conventionally evaluated from left to right, i.e.,
while a right-associative operation is conventionally evaluated from right to left:
Both left-associative and right-associative operations occur. Left-associative operations include the following:
This notation can be motivated by the currying isomorphism, which enables partial application.
Exponentiation is commonly used with brackets or right-associatively because a repeated left-associative exponentiation operation is of little use. Repeated powers would mostly be rewritten with multiplication:
Non-associative operations for which no conventional evaluation order is defined include the following.