Noncommutative

Informally, noncommutative means that "order matters".

More formally, if $\displaystyle\star$ is some binary operation on a set, and x and y are elements of the set, then noncommutative means that $\displaystyle x \star y$ doesn't necessarily equal $\displaystyle y \star x$ .

Most common operations, such as addition or multiplication of numbers, are of course commutative: for example, 3+2 = 2+3 and 6x8 = 8x6.

Examples of noncommutative operations

Composition of functions

If $f(x)$ and $g(x)$ are functions, then usually $(f \circ g)(x) \not= (g \circ f)(x)$ , or to write it another way, $f(g(x)) \not= g(f(x))$ .

For example, suppose $f(x) = x^2$ and $g(x) = x+1$ . Then $(f \circ g)(x) = g(f(x)) = g(x^2) = x^2 + 1$ , whereas $(g \circ f)(x) = f(g(x)) = f(x+1) = (x+1)^2 = x^2+2x+1$ , and these are clearly not equal!