Math Formulas
Markdown supports embedding math formulas using LaTeX syntax, providing professional mathematical expression capabilities for technical documents, academic papers, and teaching materials.
Basic LaTeX Math Syntax
Inline Formulas
Use single dollar signs $ to enclose formulas:
This is an inline formula: $E = mc^2$, which is Einstein's mass-energy equation.
The area of a circle is $A = \pi r^2$, where $r$ is the radius.
The solution to the quadratic equation: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$Rendered Output:
This is an inline formula: $E = mc^2$, which is Einstein's mass-energy equation.
The area of a circle is $A = \pi r^2$, where $r$ is the radius.
The solution to the quadratic equation: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Block Formulas
Use double dollar signs $$ to enclose formulas, which will be displayed on a separate centered line:
$$
\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}
$$
$$
\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}
$$
$$
\lim_{x \to 0} \frac{\sin x}{x} = 1
$$Rendered Output:
$$ \int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi} $$
$$ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} $$
$$ \lim_{x \to 0} \frac{\sin x}{x} = 1 $$
Basic Math Elements
Superscripts and Subscripts
<!-- Superscripts -->
$x^2$, $e^{i\pi}$, $2^{10}$
<!-- Subscripts -->
$x_1$, $a_{ij}$, $\log_2 n$
<!-- Combined -->
$x_1^2$, $a_{i,j}^{(k)}$, $\sum_{i=1}^n x_i^2$Rendered Output:
$x^2$, $e^{i\pi}$, $2^{10}$
$x_1$, $a_{ij}$, $\log_2 n$
$x_1^2$, $a_{i,j}^{(k)}$, $\sum_{i=1}^n x_i^2$
Fractions
<!-- Basic fractions -->
$\frac{1}{2}$, $\frac{a}{b}$, $\frac{x+y}{x-y}$
<!-- Continued fractions -->
$\frac{1}{1 + \frac{1}{2 + \frac{1}{3 + \cdots}}}$
<!-- Complex fractions -->
$\frac{\partial^2 f}{\partial x^2}$, $\frac{d}{dx}\left(\frac{1}{x}\right)$Rendered Output:
$\frac{1}{2}$, $\frac{a}{b}$, $\frac{x+y}{x-y}$
$\frac{1}{1 + \frac{1}{2 + \frac{1}{3 + \cdots}}}$
$\frac{\partial^2 f}{\partial x^2}$, $\frac{d}{dx}\left(\frac{1}{x}\right)$
Square Roots
<!-- Square roots -->
$\sqrt{2}$, $\sqrt{x^2 + y^2}$
<!-- n-th roots -->
$\sqrt[3]{8}$, $\sqrt[n]{x}$
<!-- Complex roots -->
$\sqrt{\frac{a}{b}}$, $\sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}$Rendered Output:
$\sqrt{2}$, $\sqrt{x^2 + y^2}$
$\sqrt[3]{8}$, $\sqrt[n]{x}$
$\sqrt{\frac{a}{b}}$, $\sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}}$
Symbols and Operators
Greek Letters
<!-- Lowercase Greek letters -->
$\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, $\zeta$, $\eta$, $\theta$
$\iota$, $\kappa$, $\lambda$, $\mu$, $\nu$, $\xi$, $\pi$, $\rho$
$\sigma$, $\tau$, $\upsilon$, $\phi$, $\chi$, $\psi$, $\omega$
<!-- Uppercase Greek letters -->
$\Alpha$, $\Beta$, $\Gamma$, $\Delta$, $\Epsilon$, $\Zeta$, $\Eta$, $\Theta$
$\Lambda$, $\Xi$, $\Pi$, $\Sigma$, $\Phi$, $\Psi$, $\Omega$Rendered Output:
$\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, $\zeta$, $\eta$, $\theta$
$\iota$, $\kappa$, $\lambda$, $\mu$, $\nu$, $\xi$, $\pi$, $\rho$
$\sigma$, $\tau$, $\upsilon$, $\phi$, $\chi$, $\psi$, $\omega$
$\Alpha$, $\Beta$, $\Gamma$, $\Delta$, $\Epsilon$, $\Zeta$, $\Eta$, $\Theta$
$\Lambda$, $\Xi$, $\Pi$, $\Sigma$, $\Phi$, $\Psi$, $\Omega$
Operators
<!-- Basic operations -->
$+$, $-$, $\times$, $\div$, $\pm$, $\mp$
<!-- Relational operations -->
$=$, $\neq$, $<$, $>$, $\leq$, $\geq$, $\ll$, $\gg$
<!-- Logical operations -->
$\land$, $\lor$, $\lnot$, $\implies$, $\iff$
<!-- Set operations -->
$\in$, $\notin$, $\subset$, $\supset$, $\cup$, $\cap$, $\emptyset$
<!-- Other symbols -->
$\infty$, $\partial$, $\nabla$, $\propto$, $\approx$, $\equiv$Rendered Output:
$+$, $-$, $\times$, $\div$, $\pm$, $\mp$
$=$, $\neq$, $<$, $>$, $\leq$, $\geq$, $\ll$, $\gg$
$\land$, $\lor$, $\lnot$, $\implies$, $\iff$
$\in$, $\notin$, $\subset$, $\supset$, $\cup$, $\cap$, $\emptyset$
$\infty$, $\partial$, $\nabla$, $\propto$, $\approx$, $\equiv$
Advanced Math Structures
Summation and Integration
<!-- Summation -->
$$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$
$$\sum_{k=0}^{\infty} \frac{x^k}{k!} = e^x$$
<!-- Integration -->
$$\int_a^b f(x) dx$$
$$\oint_C \mathbf{F} \cdot d\mathbf{r}$$
$$\iint_D f(x,y) \, dx \, dy$$
$$\iiint_V f(x,y,z) \, dx \, dy \, dz$$
<!-- Limits -->
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$$
$$\lim_{x \to 0^+} \frac{1}{x} = +\infty$$Rendered Output:
$$\sum_{i=1}^{n} i = \frac{n(n+1)}{2}$$
$$\sum_{k=0}^{\infty} \frac{x^k}{k!} = e^x$$
$$\int_a^b f(x) dx$$
$$\oint_C \mathbf{F} \cdot d\mathbf{r}$$
$$\iint_D f(x,y) , dx , dy$$
$$\iiint_V f(x,y,z) , dx , dy , dz$$
$$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$$
$$\lim_{x \to 0^+} \frac{1}{x} = +\infty$$
Matrices and Determinants
<!-- Basic matrix -->
$$
\begin{matrix}
a & b \\
c & d
\end{matrix}
$$
<!-- Matrix with parentheses -->
$$
\begin{pmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{pmatrix}
$$
<!-- Determinant -->
$$
\begin{vmatrix}
a & b \\
c & d
\end{vmatrix} = ad - bc
$$
<!-- System of equations -->
$$
\begin{cases}
x + y = 1 \\
2x - y = 0
\end{cases}
$$
<!-- Large matrix -->
$$
\begin{bmatrix}
1 & 0 & \cdots & 0 \\
0 & 1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & 1
\end{bmatrix}
$$Rendered Output:
$$ \begin{matrix} a & b \ c & d \end{matrix} $$
$$ \begin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{pmatrix} $$
$$ \begin{vmatrix} a & b \ c & d \end{vmatrix} = ad - bc $$
$$ \begin{cases} x + y = 1 \ 2x - y = 0 \end{cases} $$
$$ \begin{bmatrix} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 \end{bmatrix} $$
Multi-line Formulas
<!-- Aligned multi-line formulas -->
$$
\begin{align}
f(x) &= ax^2 + bx + c \\
&= a(x^2 + \frac{b}{a}x) + c \\
&= a(x + \frac{b}{2a})^2 + c - \frac{b^2}{4a}
\end{align}
$$
<!-- Piecewise cases -->
$$
f(x) = \begin{cases}
x^2 & \text{if } x \geq 0 \\
-x^2 & \text{if } x < 0
\end{cases}
$$
<!-- Numbered formulas -->
$$
E = mc^2 \tag{1}
$$
$$
F = ma \tag{2}
$$Rendered Output:
$$ \begin{align} f(x) &= ax^2 + bx + c \ &= a(x^2 + \frac{b}{a}x) + c \ &= a(x + \frac{b}{2a})^2 + c - \frac{b^2}{4a} \end{align} $$
$$ f(x) = \begin{cases} x^2 & \text{if } x \geq 0 \ -x^2 & \text{if } x < 0 \end{cases} $$
$$ E = mc^2 \tag{1} $$
$$ F = ma \tag{2} $$
Fonts and Styles
Math Fonts
<!-- Bold -->
$\mathbf{A}$, $\mathbf{x}$, $\boldsymbol{\alpha}$
<!-- Italic (default) -->
$A$, $x$, $\alpha$
<!-- Blackboard bold -->
$\mathbb{R}$, $\mathbb{C}$, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$
<!-- Calligraphic -->
$\mathcal{A}$, $\mathcal{B}$, $\mathcal{F}$, $\mathcal{L}$
<!-- Script -->
$\mathscr{A}$, $\mathscr{B}$, $\mathscr{F}$, $\mathscr{L}$
<!-- Monospace -->
$\mathtt{text}$, $\mathtt{ABC}$
<!-- Roman -->
$\mathrm{d}x$, $\mathrm{sin}$, $\mathrm{cos}$Rendered Output:
$\mathbf{A}$, $\mathbf{x}$, $\boldsymbol{\alpha}$
$A$, $x$, $\alpha$
$\mathbb{R}$, $\mathbb{C}$, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$
$\mathcal{A}$, $\mathcal{B}$, $\mathcal{F}$, $\mathcal{L}$
$\mathscr{A}$, $\mathscr{B}$, $\mathscr{F}$, $\mathscr{L}$
$\mathtt{text}$, $\mathtt{ABC}$
$\mathrm{d}x$, $\mathrm{sin}$, $\mathrm{cos}$
Size Control
<!-- Different sizes -->
$\tiny{tiny}$ $\small{small}$ $\normalsize{normal}$ $\large{large}$ $\Large{Large}$ $\LARGE{LARGE}$ $\huge{huge}$
<!-- Usage in formulas -->
$$\Large \sum_{i=1}^{n} \small x_i = \normalsize X$$Rendered Output:
$\tiny{tiny}$ $\small{small}$ $\normalsize{normal}$ $\large{large}$ $\Large{Large}$ $\LARGE{LARGE}$ $\huge{huge}$
$$\Large \sum_{i=1}^{n} \small x_i = \normalsize X$$
Special Symbols and Marks
Arrows
<!-- Single arrows -->
$\leftarrow$, $\rightarrow$, $\uparrow$, $\downarrow$
<!-- Double arrows -->
$\leftrightarrow$, $\updownarrow$
<!-- Long arrows -->
$\longleftarrow$, $\longrightarrow$, $\longleftrightarrow$
<!-- Double line arrows -->
$\Leftarrow$, $\Rightarrow$, $\Leftrightarrow$
<!-- Special arrows -->
$\mapsto$, $\to$, $\gets$, $\hookrightarrow$, $\leadsto$Rendered Output:
$\leftarrow$, $\rightarrow$, $\uparrow$, $\downarrow$
$\leftrightarrow$, $\updownarrow$
$\longleftarrow$, $\longrightarrow$, $\longleftrightarrow$
$\Leftarrow$, $\Rightarrow$, $\Leftrightarrow$
$\mapsto$, $\to$, $\gets$, $\hookrightarrow$, $\leadsto$
Superscripts and Decorations
<!-- Hat -->
$\hat{a}$, $\widehat{abc}$
<!-- Tilde -->
$\tilde{a}$, $\widetilde{abc}$
<!-- Overline -->
$\bar{a}$, $\overline{abc}$
<!-- Underline -->
$\underline{abc}$
<!-- Vector arrow -->
$\vec{a}$, $\overrightarrow{AB}$
<!-- Dot -->
$\dot{a}$, $\ddot{a}$, $\dddot{a}$Rendered Output:
$\hat{a}$, $\widehat{abc}$
$\tilde{a}$, $\widetilde{abc}$
$\bar{a}$, $\overline{abc}$
$\underline{abc}$
$\vec{a}$, $\overrightarrow{AB}$
$\dot{a}$, $\ddot{a}$, $\dddot{a}$
Complex Formula Examples
Physics Formulas
<!-- Schrödinger equation -->
$$i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t)$$
<!-- Maxwell's equations -->
$$
\begin{align}
\nabla \cdot \mathbf{E} &= \frac{\rho}{\epsilon_0} \\
\nabla \cdot \mathbf{B} &= 0 \\
\nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\
\nabla \times \mathbf{B} &= \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}
\end{align}
$$
<!-- Lorentz transformation -->
$$
\begin{pmatrix}
ct' \\
x' \\
y' \\
z'
\end{pmatrix} =
\begin{pmatrix}
\gamma & -\gamma v/c & 0 & 0 \\
-\gamma v/c & \gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
ct \\
x \\
y \\
z
\end{pmatrix}
$$Rendered Output:
$$i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t)$$
$$ \begin{align} \nabla \cdot \mathbf{E} &= \frac{\rho}{\epsilon_0} \ \nabla \cdot \mathbf{B} &= 0 \ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \ \nabla \times \mathbf{B} &= \mu_0\mathbf{J} + \mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t} \end{align} $$
$$ \begin{pmatrix} ct' \ x' \ y' \ z' \end{pmatrix} = \begin{pmatrix} \gamma & -\gamma v/c & 0 & 0 \ -\gamma v/c & \gamma & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} ct \ x \ y \ z \end{pmatrix} $$
Mathematical Theorems
<!-- Fourier transform -->
$$\mathcal{F}\{f(t)\} = F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$$
<!-- Taylor expansion -->
$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$
<!-- Euler's formula -->
$$e^{i\theta} = \cos\theta + i\sin\theta$$
<!-- Gaussian integral -->
$$\int_{-\infty}^{\infty} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}} \quad (a > 0)$$
<!-- Bayes' theorem -->
$$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$Rendered Output:
$$\mathcal{F}{f(t)} = F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$$
$$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$
$$e^{i\theta} = \cos\theta + i\sin\theta$$
$$\int_{-\infty}^{\infty} e^{-ax^2} dx = \sqrt{\frac{\pi}{a}} \quad (a > 0)$$
$$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$
Algorithm Complexity
<!-- Time complexity -->
$$O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!)$$
<!-- Recurrence relation -->
$$T(n) = \begin{cases}
1 & \text{if } n = 1 \\
2T(n/2) + O(n) & \text{if } n > 1
\end{cases}$$
<!-- Master theorem -->
$$T(n) = aT(n/b) + f(n)$$
Where $a \geq 1$, $b > 1$, $f(n)$ is an asymptotically positive function.Rendered Output:
$$O(1) < O(\log n) < O(n) < O(n \log n) < O(n^2) < O(2^n) < O(n!)$$
$$T(n) = \begin{cases} 1 & \text{if } n = 1 \ 2T(n/2) + O(n) & \text{if } n > 1 \end{cases}$$
$$T(n) = aT(n/b) + f(n)$$ Where $a \geq 1$, $b > 1$, $f(n)$ is an asymptotically positive function.
Best Practices for Math Formulas
Writing Suggestions
✅ Recommended:
1. **Use semantic commands**:
- Use `\sin`, `\cos`, `\log` instead of `sin`, `cos`, `log`
- Use `\mathrm{d}x` for differentials
2. **Keep spacing reasonable**:
- Add appropriate spaces around operators: `\,` (thin space), `\;` (medium space), `\quad` (large space)
3. **Use matching brackets**:
- Auto-size: `\left(\right)`, `\left[\right]`, `\left\{\right\}`
4. **Align formulas**:
- Use the `align` environment to align equal signs
- Use `&` to mark alignment points
❌ Avoid:
1. Not breaking long formulas into lines
2. Missing necessary brackets
3. Inconsistent symbol usage
4. Ignoring syntax error checksCommon Error Corrections
<!-- ❌ Incorrect -->
$sin(x)$, $log(x)$, $max(a,b)$
<!-- ✅ Correct -->
$\sin(x)$, $\log(x)$, $\max(a,b)$
<!-- ❌ Incorrect -->
$(\frac{a}{b})$
<!-- ✅ Correct -->
$\left(\frac{a}{b}\right)$
<!-- ❌ Incorrect -->
$x=1+2+3+...+n$
<!-- ✅ Correct -->
$x = 1 + 2 + 3 + \cdots + n$Accessibility Considerations
To improve formula accessibility:
1. **Add text descriptions**:
$$E = mc^2$$
> This is Einstein's mass-energy equation, meaning energy equals mass times the speed of light squared.
2. **Use alternative text**:
Add simplified explanations after complex formulas
3. **Avoid using color alone for distinction**:
Use different symbols or styles to distinguish concepts
4. **Keep formulas concise**:
Break complex formulas into multiple stepsSupported Math Environments
Markdown Processor Support
| Processor | Math Support | Syntax | Configuration |
|---|---|---|---|
| GitHub | ✅ | $...$, $$...$$ | Automatic |
| GitLab | ✅ | $...$, $$...$$ | Needs to be enabled |
| VitePress | ✅ | $...$, $$...$$ | Plugin config |
| Jekyll | ✅ | $...$, $$...$$ | MathJax/KaTeX |
| Hugo | ✅ | $...$, $$...$$ | Theme support |
VitePress Configuration Example
// .vitepress/config.js
export default {
markdown: {
math: true
}
}Rendering Engines
Common math formula rendering engines:
1. **MathJax**:
- Most complete, supports full LaTeX
- High rendering quality, but slower loading
2. **KaTeX**:
- Fast rendering
- Supports common math syntax
- Smaller size
3. **MathML**:
- Native browser support
- Complex syntax, usually auto-generatedRelated Syntax
- HTML Embedding - HTML enhancements
- Diagrams - Chart drawing
- Best Practices - Writing recommendations
Tools and Resources
Online Editors
- LaTeX Live: Real-time preview of LaTeX formulas
- MathJax Demo: Test MathJax rendering
- KaTeX Demo: KaTeX formula testing
- Desmos: Graphical math expressions
Formula Editing Tools
- MathType: Professional math formula editor
- LaTeX Workshop (VS Code): LaTeX writing plugin
- MathQuill: Visual math editor
- Wiris: Online math formula editor
Reference Resources
- LaTeX Math Symbols: Math symbol reference table
- Detexify: Handwritten LaTeX symbol recognition
- MathJax Documentation: Official documentation
- KaTeX Supported Functions: Supported function list
By mastering math formula syntax, you can elegantly express complex mathematical concepts and formulas in technical documentation.