Noncommutive

Noncommutive is a term used in mathematics to describe an operation or a system where the order of operands matters. In a commutive operation, performing the operation with elements a and b results in the same outcome regardless of whether it is done as a * b or b * a. However, in a noncommutive operation, a * b is generally not equal to b * a.

A common example of a commutive operation is addition. For instance, 5 + 3 is equal to 3 + 5, both resulting in 8. Multiplication of ordinary numbers is also commutive; 4 * 7 is the same as 7 * 4, both equaling 28.

In contrast, matrix multiplication is a prime example of a noncommutive operation. If you have two matrices A and B, the product AB will typically not be the same as the product BA. Another area where noncommitivity arises is in the study of groups in abstract algebra. The composition of certain transformations, like rotations in three-dimensional space, can also be noncommutive. This property is fundamental in various fields of mathematics and physics, including quantum mechanics, where operators often do not commute.