Cut and paste maths examples

here is place to put some maths markup templates for inclusion

set notations
http://latex.wikia.com/wiki/Sum-class_symbol

$$\mathbb{N}$$

Natural number axioms http://math.wikia.com/wiki/Peano_axioms

equations
A linear equation is an equation with variables x1,...,xn (commonly x, y with two variables and x, y, z with three) that can be written in the form
 * a1x1 + a2x2 + ... + anxn = b,

where b and a1,...,an are real or complex numbers which are usually known beforehand.

Matrix Notation
http://fileformats.wikia.com/wiki/MathML http://www.mediawiki.org/wiki/Manual:Math http://www.mediawiki.org/wiki/Extension:Math

For a given system of equations, each equation can be written as a row in a matrix. For a coefficient matrix, each of the coefficients is written in aligned columns. For example, the system
 * $$2x_2 - 8x_3 = 8$$
 * $$x_1 - 2x_2 + x_3 = 0$$
 * $$-4x_1 + 5x_2 + 9x_3 = -9$$

would be written as the coefficient matrix
 * $$\begin{bmatrix}

0 & 2 & -8 \\ 1 & -2 & 3 \\ -4 & 5 & 9 \end{bmatrix}$$.

An augmented matrix shows the coefficient matrix with an additional column for the values of b. The augmented matrix for the above system is
 * $$\begin{bmatrix}

0 & 2 & -8 & 8\\ 1 & -2 & 1 & 0\\ -4 & 5 & 9 & -9\end{bmatrix}$$.

The goal of solving by a matrix is to produce a matrix of the form
 * $$\begin{bmatrix}

1 & 0 & \cdots & 0 & s_1 \\ 0 & 1 & \cdots & 0 & s_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & s_n \end{bmatrix}$$,

The \left[ and \right] are delimiters of adjustable size that make the brackets around the matrix. If you want parentheses instead of square brackets, use \left( and \right). To make vertical bars for determinants, use \left\vert and \right\vert. You can also make curly braces via \left\{ and \right\}. (For curly braces, you need to put a backslash in front of the braces so that LaTeX realizes they are not LaTeX grouping symbols.)


 * $$\begin{bmatrix}

0 & 2 & -8 & 8\\ 1 & -2 & 1 & 0\\ -4 & 5 & 9 & -9\end{bmatrix}$$.

Calculus notation
$$ \operatorname{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}\,dt = \frac{e^{-x^2}}{x\sqrt{\pi}}\sum_{n=0}^\infty (-1)^n \frac{(2n)!}{n!(2x)^{2n}} $$

$$\int_a^b \! f(x) \, \mathrm{d} x.$$

$$ \iint\limits_D \, \dif x\,dif y \quad \iiint\limits_E \, \dif x\,\dif y\,\mathrm{d}z $$

$$\oint \! \nabla f \, \dif t = 0$$

$$\int_a^b \! f(x) \, \mathrm{d}x.$$