Commutative property

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Binary_operation/CYtow0bz">binary operation</a> is commutative if changing the order of the <a href="/facts/Operand/HSuUzrKz">operands</a> does not change the result. It is a fundamental property of many binary operations, and many <a href="/facts/Mathematical_proof/7ECDrU80">mathematical proofs</a> depend on it. Perhaps most familiar as a property of arithmetic, e.g. "3 + 4 = 4 + 3" or "2 × 5 = 5 × 2", the property can also be used in more advanced settings. The name is needed because there are operations, such as <a href="/facts/Division_(mathematics)/FqLKNkmq">division</a> and <a href="/facts/Subtraction/dEUrp6k0">subtraction</a>, that do not have it (for example, "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 <a href="/facts/Multiplication_(mathematics)/VtcUjpNv">multiplication</a> and <a href="/facts/Addition/apFWJBFB">addition</a> of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for <a href="/facts/Binary_relation/r9BwGgaK">binary relations</a>; a binary relation is said to be <a href="/facts/Symmetric_relation/ekCWDIAH">symmetric</a> if the relation applies regardless of the order of its operands; for example, <a href="/facts/Equality_(mathematics)/AQCuumLg">equality</a> is symmetric as two equal mathematical objects are equal regardless of their order.

Commutative property open-in-new

Commutative property