# MathJax

## Example

When $$a \ne 0$$, there are two solutions to $$ax^2 + bx + c = 0$$ and they are $x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$

## Modes

In inline mode:$$\sum_{i=0}^n i^2=\frac{(n^2+n)(2n+1)}{6}$$ becomes: $$\sum_{i=0}^n i^2=\frac{(n^2+n)(2n+1)}{6}$$.
In display mode:$\sum_{i=0}^n i^2=\frac{(n^2+n)(2n+1)}{6}$ becomes: $$\sum_{i=0}^n i^2=\frac{(n^2+n)(2n+1)}{6}$$

But the display style can be enforced with \displaystyle inside text and with \textstyle inside display mode:
\displaystyle \lim_{t \to 0} \int_t^1 f(t)\, dt versus \textstyle \lim_{t \to 0} \int_t^1 f(t)\, dt Compare $$\displaystyle \lim_{t \to 0} \int_t^1 f(t)\, dt$$ versus $$\lim_{t \to 0} \int_t^1 f(t)\, dt$$.

# Syntax

## Greek letters

\alpha \beta \Delta \Omega  = $$\alpha \beta \Delta \Omega$$.

## Parentheses

 (1) [2] \{3\} |x| \lvert{x}\rvert \lVert{x}\rVert  = $$(1) [2] \{3\} |x| \lvert{x}\rvert \lVert{x}\rVert$$.

\left( \right)  = $$\left( \frac{x}{y} \right)$$ \left[ \right]  = $$\left[ \frac{x}{y} \right]$$ \left\{ \right\}  = $$\left\{ \frac{x}{y} \right\}$$ \left| \right|  = $$\left| \frac{x}{y} \right|$$ \lvert \rvert  = $$\lvert \frac{x}{y} \rvert$$ \left\vert \right\vert  = $$\left\vert \frac{x}{y} \right\vert$$ \lVert \rVert  = $$\lVert \frac{x}{y} \rVert$$ \left\Vert \right\Vert  = $$\left\Vert \frac{x}{y} \right\Vert$$ \left\langle \right\rangle  = $$\left\langle \frac{x}{y} \right\rangle$$

Invisible parentheses with .: \left. \right\}  $$\left. \frac{x}{y} \right\}$$

## Sums, integrals, etc

\sum_1^ni^2 $$\sum_1^ni^2$$ \sum_{i=1}^\infty x^2 $$\sum_{i=1}^\infty x^2$$ \iint_{i=1}^\infty x^2 $$\iint_{i=1}^\infty x^2$$ \prod_{i=1}^n x^2 $$\prod_{i=1}^n x^2$$

## Fractions

\frac ab $$\frac ab$$ \frac{a+1}{b+1} $$\frac{a+1}{b+1}$$ \sqrt[3] \frac{x^3}{b} $$\sqrt[3] \frac{x^3}{b}$$ x^\frac 23 $$x^\frac 23$$

\lim_{x\to 0} \text{ is undefined} $$\lim_{x\to 0} \text{ is undefined}$$ $\lim_{x\to 0} \text{ is undefined}$

\hat x \widehat {xy} \bar x \overline {xyz} $$\hat x \widehat {xy} \bar x \overline {xyz}$$

\vec x \overrightarrow {xy} $$\vec x \overrightarrow {xy}$$ \lvert x \rvert \lVert x \rVert $$\lvert x \rvert \lVert x \rVert$$ _5C_3 $$_5C_3$$

## Matrices

### matrix

 $$\begin{matrix} 1 & x \\ 1 & y \end{matrix}$$ 
$$\begin{matrix} 1 & x \\ 1 & y \end{matrix}$$

### pmatrix

 $$\begin{pmatrix} 1 & x \\ 1 & y \end{pmatrix}$$ 
$$\begin{pmatrix} 1 & x \\ 1 & y \end{pmatrix}$$

### bmatrix

$$\begin{bmatrix} 1 & x \\ 1 & y \end{bmatrix}$$

### Bmatrix

$$\begin{Bmatrix} 1 & x \\ 1 & y \end{Bmatrix}$$

### vmatrix

$$\begin{vmatrix} 1 & x \\ 1 & y \end{vmatrix}$$

### Vmatrix

$$\begin{Vmatrix} 1 & x \\ 1 & y \end{Vmatrix}$$

### Arrays

 \left[ \begin{array}{cc|c} 1 & 2 & 3 \\ \hline 1 & x & y \\ \end{array} \right] 

$$\left[ \begin{array}{cc|c} 1 & 2 & 3 \\ \hline 1 & x & y \\ \end{array} \right]$$

 \bigl(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\bigr) 

$$\bigl(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\bigr)$$

 \begin{align} a^2-b^2 & = (a+b)(a-b) \\ &= a^2 + ab -ab +b^2 \end{align} 

\begin{align} a^2-b^2 & = (a+b)(a-b) \\ &= a^2 + ab -ab +b^2 \end{align}

 f(n) = \begin{cases} n/2, & \text{if $n$ is even} \\ 3n+1, & \text{if $n$ is odd} \end{cases} 

$$f(n) = \begin{cases} n/2, & \text{if n is even} \\ 3n+1, & \text{if n is odd} \end{cases}$$

 \begin{array}{c|lcr} n & \text{Left} & \text{Center} & \text{Right} \\ \hline 1 & 0.24 & 1 & 125 \\ 2 & -1 & 189 & -8 \\ 3 & -20 & 2000 & 1+10i \end{array} 

$$\begin{array}{c|lcr} n & \text{Left} & \text{Center} & \text{Right} \\ \hline 1 & 0.24 & 1 & \color{red}{125} \\ 2 & -1 & 189 & -8 \\ 3 & -20 & 2000 & 1+10i \end{array}$$

 $$\bbox[yellow,5px,border:2px solid red] { e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n \qquad (1) }$$ 

$$\bbox[#ffe,5px,border:2px solid #fee] { e^x=\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n \qquad (1) }$$

 $$\begin{array}{ |c|c|c| } \hline a & a^2 \pmod{5} & 2a^2 \pmod{5} \\ \hline 0 & 0 & 0 \\ 1 & 1 & 2 \\ 2 & 4 & 3 \\ \hline \end{array}$$ 

$$\begin{array}{ |c|c|c| } \hline a & a^2 \pmod{5} & 2a^2 \pmod{5} \\ \hline 0 & 0 & 0 \\ 1 & 1 & 2 \\ 2 & 4 & 3 \\ \hline \end{array}$$

 \implies  $$\implies$$  \impliedby  $$\impliedby$$  \iff  $$\iff$$  \mapsto  $$\mapsto$$  \to  $$\to$$  \gets  $$\gets$$  \rightarrow  $$\rightarrow$$  \leftarrow  $$\leftarrow$$  \Rightarrow  $$\Rightarrow$$  \Leftarrow  $$\Leftarrow$$

 \|\mathbf{v}\|  $$\|\mathbf{v}\|$$

 \int_{a}^{b} \! f(x)\,\mathrm{d}x  $$\int_{a}^{b} \! f(x)\,\mathrm{d}x$$

 \binom nk  $$\binom nk$$