this, consider a floating point representation with a 4-bit mantissa: Even though most computers compute with 24 or 53 bits of mantissa, this is an important source of rounding error, and approaches such as the Kahan summation algorithm are ways to minimise the errors. It can be especially problematic in parallel computing. Notation for non-associative operations In general, parentheses must be used to indicate the order of evaluation if a non-associative operation appears more than once in an expression (unless the notation specifies the order in another way, like ). However, mathematicians agree on a particular order of evaluation for several common non-associative operations. This is simply a notational convention to avoid parentheses. 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: Subtraction and division of real numbers Function application This notation can be motivated by the currying isomorphism, which enables partial application. Right-associative operations include the following: Exponentiation of real numbers in superscript notation 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: Formatted correctly, the superscript inherently behaves as a set of parentheses; e.g. in the expression the addition is performed before the exponentiation despite there being no explicit parentheses wrapped around it. Thus given an expression such as , the full exponent of the base is evaluated first. However, in some contexts, especially in handwriting, the difference between , and can be hard to see. In such a case, right-associativity is usually implied. Function definition Using right-associative notation for these operations can be motivated by the Curry–Howard correspondence and by the currying isomorphism. Non-associative operations for which no conventional evaluation order is defined include the following. Exponentiation of real numbers in infix notation Knuth's up-arrow operators Taking the cross product of three vectors Taking the pairwise average of real numbers Taking the relative complement of sets .(Compare material nonimplication in logic.) History William Rowan Hamilton seems to have coined the term "associative property" around 1844, a time when he was contemplating the non-associative algebra of the octonions he had learned about from John T. Graves. See also Light's associativity test Telescoping series, the use of addition associativity for cancelling terms in