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[[File:lighterstill.jpg]][[File:Associativity_colors.jpg|right|frame]]

[[File:lighterstill.jpg]][[File:Associativity_colors.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]]

In [[mathematics]], '''associativity''' is a property of some [https://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 [https://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]]

[[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."

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 [https://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,

Associativity is not to be confused with [https://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]]

[[File:Associativity2.jpg]]

is not an example of associativity because the operand sequence changed when the 2 and 5 switched places.

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.

Associative operations are [[abundant]] in [[mathematics]]; in [[fact]], many [https://en.wikipedia.org/wiki/Algebraic_structure algebraic structures] (such as [https://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]

However, many important and interesting operations are non-associative; one common example would be the [https://en.wikipedia.org/wiki/Vector_cross_product vector cross product].[https://en.wikipedia.org/wiki/Associativity]

==See also==

==See also==

*[[Association]]

*[[Association]]

==References==

==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 .

# Dudek, W.A. (2001), "On some old problems in n-ary groups", Quasigroups and Related Systems 8: 15–36, https://www.quasigroups.eu/contents/contents8.php?m=trzeci .

[[Category: Mathematics]]

[[Category: Mathematics]]