Editing Commutative property (section)

{{Short description|Property of some mathematical operations}}
{{redirect|Commutative|other uses}}
{{Use dmy dates|date=June 2023}}
{{Infobox mathematical statement
| name = {{PAGENAMEBASE}}
| image = [[File:Commutativity of binary operations (without question mark).svg|220px|class=skin-invert-image]]
| type = [[Property (mathematics)|Property]]
| field = [[Algebra]]
| statement = A [[binary operation]] is ''commutative'' if changing the order of the [[operand]]s does not change the result.
| symbolic statement = <math>x * y = y * x \quad\forall x,y\in S.</math>
}}

In [[mathematics]], a [[binary operation]] is '''commutative''' if changing the order of the [[operand]]s does not change the result. It is a fundamental property of many binary operations, and many [[mathematical proof]]s depend on it. Perhaps most familiar as a property of arithmetic, e.g. {{nowrap|1="3 + 4 = 4 + 3"}} or {{nowrap|1="2 × 5 = 5 × 2"}}, the property can also be used in more advanced settings. The name is needed because there are operations, such as [[division (mathematics)|division]] and [[subtraction]], that do not have it (for example, {{nowrap|"3 − 5 ≠ 5 − 3"}}); such operations are ''not'' commutative, and so are referred to as '''noncommutative operations'''.

The idea that simple operations, such as the [[multiplication (mathematics)|multiplication]] and [[addition]] of numbers, are commutative was for many centuries implicitly assumed. Thus, this property was not named until the 19th century, when new [[algebraic structure]]s started to be studied.{{sfn|Rice|2011|p=[https://books.google.com/books?id=YruifIx88AQC&pg=PA4 4]}}