Difference between revisions of "Noncommutative"

Latest revision as of 14:30, 26 December 2017

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:

$\begin{pmatrix}1&2\\3&4\end{pmatrix}\begin{pmatrix}5&6\\7&8\end{pmatrix}=\begin{pmatrix}19&22\\43&50\end{pmatrix}$

whereas

$\begin{pmatrix}5&6\\7&8\end{pmatrix}\begin{pmatrix}1&2\\3&4\end{pmatrix}=\begin{pmatrix}23&34\\31&46\end{pmatrix}$

Symmetries of a regular n-gon

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

@@ Line 7: / Line 7: @@
 ==Examples of noncommutative operations==
 ===Composition of functions===
-If <math>f(x)</math> and <math>g(x)</math> are functions, then usually, <math>(f\circ g)(x)\ne(g\circ f)(x)</math>. This can also be written <math>g(f(x))\ne f(g(x))</math>.
+If <math>f(x)</math> and <math>g(x)</math> are functions, then usually, <math>(f\circ g)(x)\ne(g\circ f)(x)</math>. This can also be written <math>f(g(x))\ne g(f(x))</math>.
-For example, suppose <math>f(x) = x^2</math> and <math>g(x) = x+1</math>.  Then <math>(f\circ g)(x)=g(f(x))=g(x^2)=x^2+1</math>, and <math>(g\circ f)(x)=f(g(x))=f(x+1)=(x+1)^2=x^2+2x+1</math>. Unless <math>x=0</math>, <math>(f\circ g)(x)</math> will not be the same as <math>(g\circ f)(x)</math>.
+For example, suppose <math>f(x) = x^2</math> and <math>g(x) = x+1</math>.  Then <math>(f \circ g)(x)=f(g(x))=f(x+1)=(x+1)^2=x^2+2x+1</math>, and <math>(g \circ f)(x)=g(f(x))=g(x^2)=x^2+1</math>. Unless <math>x=0</math>, <math>(f\circ g)(x)</math> will not be the same as <math>(g\circ f)(x)</math>.
 ===Matrix multiplication===

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