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#REDIRECT [[Association]]
[[File:lighterstill.jpg]][[File:Associativitypicture.jpg|right|frame]]
In [[mathematics]], '''associativity''' is a property of some [http://en.wikipedia.org/wiki/Binary_operation binary operations]. It means that, within an [[expression]] containing two or more occurrences in a row of the same associative operator, the order in which the operations are [[performed]] does not matter as long as the [[sequence]] of the [http://en.wikipedia.org/wiki/Operand operands] is not changed. That is, rearranging the parentheses in such an expression will not [[change]] its [[value]]. Consider for instance the [[equation]]
[[File:Associativity1.jpg]]
Even though the parentheses were rearranged (the left side requires adding 5 and 2 first, then adding 1 to the result, whereas the right side requires adding 2 and 1 first, then 5), the [[value]] of the [[expression]] was not [[Change|altered]]. Since this holds true when [[performing]] addition on any [http://en.wikipedia.org/wiki/Real_number real numbers], we say that "addition of real numbers is an associative operation."
Associativity is not to be confused with [http://en.wikipedia.org/wiki/Commutativity commutativity]. Commutativity justifies changing the order or [[sequence]] of the operands within an expression while associativity does not. For example,
[[File:Associativity2.jpg]]
is an example of associativity because the parentheses were changed (and consequently the order of operations during evaluation) while the operands 5, 2, and 1 appeared in the exact same order from left to right in the expression.
[[File:Associativity3.jpg]]
is not an example of associativity because the operand sequence changed when the 2 and 5 switched places.
Associative operations are [[abundant]] in [[mathematics]]; in [[fact]], many [http://en.wikipedia.org/wiki/Algebraic_structure algebraic structures] (such as [http://en.wikipedia.org/wiki/Semigroup_(mathematics) semigroups] and categories) explicitly require their binary operations to be associative.
However, many important and interesting operations are non-associative; one common example would be the [http://en.wikipedia.org/wiki/Vector_cross_product vector cross product].[http://en.wikipedia.org/wiki/Associativity]
==References==
# Dudek, W.A. (2001), "On some old problems in n-ary groups", Quasigroups and Related Systems 8: 15–36, http://www.quasigroups.eu/contents/contents8.php?m=trzeci .
[[Category: Mathematics]]