Math Formula Guide

In Markdown, wrap LaTeX formulas with a single `$` symbol to create **inline formulas**.

Einstein's mass-energy equation $E = mc^2$ is one of the most famous formulas in physics.
The Pythagorean theorem can be expressed as $a^2 + b^2 = c^2$.
The area of a circle is $A = \pi r^2$.

Inline formulas display within the same line of text, perfect for embedding short mathematical expressions in paragraphs.

Use double `$$` symbols to create **block formulas** that are centered on their own line:

$$
E = mc^2
$$

$$
\sum_{i=1}^{n} i = \frac{n(n+1)}{2}
$$

Block formulas are ideal for displaying longer or more prominent mathematical expressions.

LaTeX supports common mathematical operators:

**Addition/Subtraction**: $a + b$, $a - b$

**Multiplication**: $a \times b$, $a \cdot b$

**Division**: $a \div b$

**Power**: $x^2$, $x^3$, $2^{10}$

**Modulo**: $a \mod b$

Combined example:

$$
(x + y)^2 = x^2 + 2xy + y^2
$$

Common comparison operators:

- Equal: $a = b$
- Not equal: $a \neq b$
- Greater than: $a > b$
- Less than: $a < b$
- Greater than or equal: $a \geq b$
- Less than or equal: $a \leq b$
- Approximately equal: $a \approx b$
- Equivalent to: $a \equiv b$

Use `\frac{numerator}{denominator}` to create fractions:

Simple fractions: $\frac{1}{2}$, $\frac{3}{4}$

Complex fractions: $\frac{a+b}{c-d}$

Nested fractions:

$$
\frac{1}{1+\frac{1}{1+\frac{1}{2}}}
$$

Use `\dfrac` for larger inline fractions: $\dfrac{a}{b}$ vs $\frac{a}{b}$

Use `\sqrt{}` for square roots and `\sqrt[n]{ }` for nth roots:

Square roots: $\sqrt{2}$, $\sqrt{x^2 + y^2}$

nth roots: $\sqrt[3]{8}$, $\sqrt[n]{x}$

Combined example:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

This is the famous **quadratic formula**.

Use `^` to create superscripts:

$x^2$, $x^3$, $x^{n+1}$

Multi-character superscripts require braces: $x^{10}$ (not `x^10`)

Common uses:
- Exponents: $e^{x}$
- Powers: $2^{32}$
- Transpose: $A^{T}$
- Derivatives: $f^{\prime}(x)$, $f^{\prime\prime}(x)$

Use `_` to create subscripts:

$x_1$, $x_2$, $x_{i,j}$

Multi-character subscripts require braces: $a_{max}$

Common uses:
- Indices: $x_1, x_2, \ldots, x_n$
- Matrix elements: $a_{ij}$
- Partial derivatives: $f_{x}$

Combined subscripts and superscripts: $x_1^2$, $a_{i}^{n}$

LaTeX provides full Greek letter support:

**Lowercase letters**:
$\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$, $\zeta$, $\eta$, $\theta$, $\lambda$, $\mu$, $\nu$, $\xi$, $\pi$, $\rho$, $\sigma$, $\tau$, $\phi$, $\chi$, $\psi$, $\omega$

**Uppercase letters**:
$\Gamma$, $\Delta$, $\Theta$, $\Lambda$, $\Xi$, $\Pi$, $\Sigma$, $\Phi$, $\Psi$, $\Omega$

**Variant forms**:
$\varepsilon$ (variant epsilon), $\vartheta$ (variant theta), $\varphi$ (variant phi)

Greek letters are widely used in mathematical formulas:

**Physics**:
$F = ma$ (force), $\omega = 2\pi f$ (angular frequency), $\lambda$ (wavelength), $\rho$ (density)

**Calculus**:
$\delta$ (variation symbol), $\epsilon$-$\delta$ definition

**Probability & Statistics**:
$\mu$ (mean), $\sigma$ (standard deviation), $\Omega$ (sample space)

$$
\Delta x = x_2 - x_1
$$

**Summation** `\sum`:

$$
\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}
$$

**Product** `\prod`:

$$
n! = \prod_{i=1}^{n} i
$$

**Limit** `\lim`:

$$
\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e
$$

Inline versions: $\sum_{i=1}^{n} i$, $\prod_{k=1}^{n} k$, $\lim_{x \to 0} f(x)$

**Indefinite integral**:

$$
\int x^2 \, dx = \frac{x^3}{3} + C
$$

**Definite integral**:

$$
\int_{0}^{1} x^2 \, dx = \frac{1}{3}
$$

**Multiple integrals**:

$$
\iint_D f(x,y) \, dx \, dy
$$

$$
\iiint_V f(x,y,z) \, dx \, dy \, dz
$$

**Line/surface integrals**:

$$
\oint_C \vec{F} \cdot d\vec{r}
$$

Use `\begin{matrix}` for unbracketed matrices:

$$
\begin{matrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{matrix}
$$

Use `\begin{pmatrix}` for parentheses:

$$
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
$$

Use `\begin{bmatrix}` for square brackets:

$$
\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}
$$

**Curly brace matrix** `\begin{Bmatrix}`:

$$
\begin{Bmatrix}
a & b \\
c & d
\end{Bmatrix}
$$

**Determinant** `\begin{vmatrix}`:

$$
\begin{vmatrix}
a & b \\
c & d
\end{vmatrix} = ad - bc
$$

**Matrix with ellipsis**:

$$
\begin{pmatrix}
a_{11} & a_{12} & \cdots & a_{1n} \\
a_{21} & a_{22} & \cdots & a_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
a_{m1} & a_{m2} & \cdots & a_{mn}
\end{pmatrix}
$$

Use the `align` environment to align multi-line equations. `&` specifies alignment points:

$$
\begin{align}
f(x) &= x^2 + 2x + 1 \\
     &= (x+1)^2
\end{align}
$$

Multi-step derivation:

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

Use the `cases` environment for piecewise functions or equation systems:

**Piecewise function**:

$$
f(x) = \begin{cases}
x^2 & \text{if } x \geq 0 \\
-x^2 & \text{if } x < 0
\end{cases}
$$

**Equation system**:

$$
\begin{cases}
x + y = 5 \\
2x - y = 1
\end{cases}
$$

**Absolute value function**:

$$
|x| = \begin{cases}
x & x \geq 0 \\
-x & x < 0
\end{cases}
$$

**Set symbols**:
- Element of: $x \in A$
- Not an element of: $x \notin A$
- Subset: $A \subset B$, $A \subseteq B$
- Proper subset: $A \subsetneq B$
- Union: $A \cup B$
- Intersection: $A \cap B$
- Empty set: $\emptyset$ or $\varnothing$
- Universal set: $\mathbb{U}$

**Logic symbols**:
- And: $p \land q$
- Or: $p \lor q$
- Not: $\neg p$
- Implies: $p \Rightarrow q$
- If and only if: $p \Leftrightarrow q$
- For all: $\forall$
- There exists: $\exists$

**Arrows**:
- Right arrow: $\rightarrow$ or $\to$
- Left arrow: $\leftarrow$
- Bidirectional arrow: $\leftrightarrow$
- Long arrow: $\Longrightarrow$
- Maps to: $\mapsto$

**Special symbols**:
- Infinity: $\infty$
- Partial derivative: $\partial$
- Nabla (gradient operator): $\nabla$
- Plus-minus: $\pm$, $\mp$
- Ellipsis: $\ldots$, $\cdots$, $\vdots$, $\ddots$
- Degrees: $30^\circ$
- Angle: $\angle ABC$

LaTeX provides various brackets. Use `\left` and `\right` for auto-sizing:

**Manual vs auto-sizing**:
$(\frac{a}{b})$ vs $\left(\frac{a}{b}\right)$

**Bracket types**:
- Parentheses: $\left( \frac{a}{b} \right)$
- Square brackets: $\left[ \frac{a}{b} \right]$
- Curly braces: $\left\{ \frac{a}{b} \right\}$
- Angle brackets: $\left\langle \frac{a}{b} \right\rangle$
- Absolute value: $\left| \frac{a}{b} \right|$
- Norm: $\left\| \vec{v} \right\|$

**One-sided brackets**:
$\left. \frac{dy}{dx} \right|_{x=1}$

**Math fonts**:
- Bold: $\mathbf{A}$ (vectors/matrices)
- Blackboard bold: $\mathbb{R}$ (real numbers), $\mathbb{N}$ (natural numbers)
- Italic: $\mathit{x}$ (variables)
- Roman: $\mathrm{d}x$ (differential symbol)
- Fraktur: $\mathfrak{g}$

**Common function names** (upright style):
$\sin x$, $\cos x$, $\tan x$
$\log x$, $\ln x$, $\exp(x)$
$\max(a,b)$, $\min(a,b)$
$\det(A)$, $\dim(V)$

Vectors:
$\vec{a}$, $\mathbf{a}$, $\hat{x}$

Spacing control in LaTeX math mode:

- No space: $ab$
- Thin space: $a\,b$
- Medium space: $a\;b$
- Thick space: $a\quad b$
- Extra thick space: $a\qquad b$

**Common layout tips**:

Adding space in integrals:
$$
\int_0^1 x^2 \, dx \quad \text{vs} \quad \int_0^1 x^2 dx
$$

Using `\text{}` to insert text in formulas:
$$
\text{When } x > 0, \text{ we have } f(x) = \sqrt{x}
$$

Using `\color{}` to change color (partial support):
$$
\textcolor{red}{E} = mc^2
$$

**Newton's Second Law**:

$$
\vec{F} = m\vec{a} = m\frac{d\vec{v}}{dt}
$$

**Schrödinger Equation**:

$$
i\hbar\frac{\partial}{\partial t}\Psi(\vec{r},t) = \hat{H}\Psi(\vec{r},t)
$$

**Euler's Identity**:

$$
e^{i\pi} + 1 = 0
$$

**Taylor Series**:

$$
f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n
$$

**Gaussian Integral**:

$$
\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}
$$

**1. Special character escaping**

In LaTeX, these characters need escaping: `# $ % & _ { }`
Use backslash to escape: $\$$, $\%$, $\&$, $\_$

**2. Line breaks in long formulas**

Use `\\` to break lines in the align environment:
$$
\begin{align}
y &= a_0 + a_1x + a_2x^2 \\
  &\quad + a_3x^3 + a_4x^4
\end{align}
$$

**3. Custom equation numbering**

Use `\tag{}` for custom numbering:
$$
a^n + b^n = c^n \tag{1}
$$

**4. Inline formula tips**

For complex inline formulas, consider using block formulas for better readability.

Want to learn more Markdown syntax? Check out the **[Syntax Guide](/en/syntax)** page for complete Markdown syntax documentation including headings, lists, code blocks, tables, diagrams, and more with live rendering previews.

Ready to edit math formulas? Open the **[Markdown Editor](/en/)** to start writing with real-time LaTeX/KaTeX formula rendering.

> **Tip**: This editor uses the KaTeX engine for math formula rendering, supporting most LaTeX math syntax. For more LaTeX symbols, refer to the [KaTeX documentation](https://katex.org/docs/supported).