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In layman's terms, an equation demonstrates commutativity when the constants or variables can be moved around an operation without changing the answer (e.g. 1 + 2 = 2 + 1  or  2 * 3 = 3 * 2). It is as if the numbers are "commuting" from one place to another.  Thus, using real numbers, both multiplication and addition are commutative, and subtraction and division are not.  For [[vector]]s, the [[dot product]] is commutative, while the [[cross product]] is anticommutative.

In layman's terms, an equation demonstrates commutativity when the constants or variables can be moved around an operation without changing the answer (e.g. 1 + 2 = 2 + 1  or  2 * 3 = 3 * 2). It is as if the numbers are "commuting" from one place to another.  Thus, using real numbers, both multiplication and addition are commutative, and subtraction and division are not.  For [[vector]]s, the [[dot product]] is commutative, while the [[cross product]] is anticommutative.

The commutative property does not imply the [[associative property]], nor vice versa.  For example, matrix multiplication is associative but not commutative.  Also, define the function <math>\Box</math> on integers by <math>a \Box b = 2(a + b)</math>.  Although, <math>\Box</math> is commutative, it is not associative (as <math>((1 \Box 0) \Box 0) = 2 \Box 0 = 4</math> but <math>(1 \Box (0 \Box 0)) = 1 \Box 0 = 2</math>.

The commutative property does not imply the [[associative property]], nor vice versa.  For example, matrix multiplication is associative but not commutative.  Also, define the function <math>\Box</math> on integers by <math>a \Box b = 2(a + b)</math>.  Although, <math>\Box</math> is commutative, it is not associative, as <math>((1 \Box 0) \Box 0) = 2 \Box 0 = 4</math> but <math>(1 \Box (0 \Box 0)) = 1 \Box 0 = 2</math>.

[[Category:Mathematics]]

[[Category:Mathematics]]