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Informally, noncommutative means "order matters".

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

Most common operations, such as addition and multiplication of numbers, are commutative. For example, $4\cdot3=3\cdot4=12$, and $2+3=3+2=5$.

Examples of noncommutative operations

Composition of functions

If $f(x)$ and $g(x)$ are functions, then usually, $(f\circ g)(x)\ne(g\circ f)(x)$. This can also be written $f(g(x))\ne g(f(x))$.

For example, suppose $f(x) = x^2$ and $g(x) = x+1$. Then $(f \circ g)(x)=f(g(x))=f(x+1)=(x+1)^2=x^2+2x+1$, and $(g \circ f)(x)=g(f(x))=g(x^2)=x^2+1$. Unless $x=0$, $(f\circ g)(x)$ will not be the same as $(g\circ f)(x)$.

Matrix multiplication

If $A$ and $B$ are both $n\times n$ matrices, then usually, $AB\ne BA$. For example:




Symmetries of a regular n-gon

The symmetries of a regular n-gon form a noncommutative group called a dihedral group.

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