Common Formulas

Inline and Block Formulas

Inline formulas: Insert formulas within the line using the symbols: $formula content$, e.g., $xyz$

Block formulas: Insert formulas on a new line, centered, using the symbols: $$formula content$$, e.g., $$xyz$$

Superscripts, Subscripts, and Combinations

1. Superscript symbol: ^, e.g., $x^4$
2. Subscript symbol: _, e.g., $x_1$
3. Combination symbol: {}, e.g., ${16}_{8}O{2+}_{2}$

Chinese Characters, Font, and Format

Chinese character form: \mbox{}, e.g., $V_{\mbox{initial}}$

Font control: \displaystyle, e.g., $\displaystyle \frac{x+y}{y+z}$

Underline symbol: \underline, e.g., $\underline{x+y}$

Tag: \tag{number}, e.g., $\tag{11}$

Overbrace: \overbrace{expression}, e.g., $\overbrace{a+b+c+d}^{2.0}$

Underbrace: \underbrace{expression}, e.g., $a+\underbrace{b+c}_{1.0}+d$

Above symbol: \stackrel{above}{base}, e.g., $\vec{x}\stackrel{\mathrm{def}}{=}{x_1,\dots,x_n}$

Placeholders

Double quad space: \qquad, e.g., $x \qquad y$

Quad space: \quad, e.g., $x \quad y$

Large space: \, e.g., $x \ y$

Medium space: \:, e.g., $x : y$

Small space: \,, e.g., $x , y$

No space: , e.g., $xy$

Tight space: \!, e.g., $x \! y$

Delimiters and Combinations

Parentheses: \big(\big) \Big(\Big) \bigg(\bigg) \Bigg(\Bigg), e.g., $\big(\big) \Big(\Big) \bigg(\bigg) \Bigg(\Bigg)$

Brackets: [], e.g., $[x+y]$

Braces: \{ \}, e.g., ${x+y}$

Adaptive brackets: \left \right, e.g., $\left(x\right)$, $\left(x{yz}\right)$

Combination formula: {upper formula \choose lower formula}, e.g., ${n+1 \choose k}={n \choose k}+{n \choose k-1}$

Combination formula: {upper formula \atop lower formula}, e.g., $\sum_{k_0,k_1,\ldots>0 \atop k_0+k_1+\cdots=n}A_{k_0}A_{k_1}\cdots$

Arithmetic Operations

Addition: +, e.g., $x+y=z$

Subtraction: -, e.g., $x-y=z$

Addition/Subtraction: \pm, e.g., $x \pm y=z$

Subtraction/Addition: \mp, e.g., $x \mp y=z$

Multiplication: \times, e.g., $x \times y=z$

Dot multiplication: \cdot, e.g., $x \cdot y=z$

Star multiplication: \ast, e.g., $x \ast y=z$

Division: \div, e.g., $x \div y=z$

Slash division: /, e.g., $x/y=z$

Fraction: \frac{numerator}{denominator}, e.g., $\frac{x+y}{y+z}$

Fraction: {numerator} \over {denominator}, e.g., ${x+y} \over {y+z}$

Absolute value: ||, e.g., $|x+y|$

Advanced Operations

Mean operation: \overline{expression}, e.g., $\overline{xyz}$

Square root: \sqrt, e.g., $\sqrt x$

Root: \sqrt[n]{radicand}, e.g., $\sqrt[3]{x+y}$

Logarithm: \log, e.g., $\log(x)$

Limit: \lim, e.g., $\lim^{x \to \infty}_{y \to 0}{\frac{x}{y}}$

Limit: \displaystyle \lim, e.g., $\displaystyle \lim^{x \to \infty}_{y \to 0}{\frac{x}{y}}$

Summation: \sum, e.g., $\sum^{x \to \infty}_{y \to 0}{\frac{x}{y}}$

Summation: \displaystyle \sum, e.g., $\displaystyle \sum^{x \to \infty}_{y \to 0}{\frac{x}{y}}$

Integration: \int, e.g., $\int^{\infty}_{0}{xdx}$

Integration: \displaystyle \int, e.g., $\displaystyle \int^{\infty}_{0}{xdx}$

Partial derivative: \partial, e.g., $\frac{\partial x}{\partial y}$

Matrix representation: \begin{matrix} \end{matrix}, e.g., $\left[ \begin{matrix} 1 &2 &\cdots &4\5 &6 &\cdots &8\\vdots &\vdots &\ddots &\vdots\13 &14 &\cdots &16\end{matrix} \right]$

Logical Operations

Equal: =, e.g., $x+y=z$

Greater than: >, e.g., $x+y>z$

Less than: <, e.g., $x+y

Greater than or equal to: \geq, e.g., $x+y \geq z$

Less than or equal to: \leq, e.g., $x+y \leq z$

Not equal: \neq, e.g., $x+y \neq z$

Not greater than or equal to: \ngeq, e.g., $x+y \ngeq z$

Not greater than or equal to: \not\geq, e.g., $x+y \not\geq z$

Not less than or equal to: \nleq, e.g., $x+y \nleq z$

Not less than or equal to: \not\leq, e.g., $x+y \not\leq z$

Approximately equal: \approx, e.g., $x+y \approx z$

Identically equal: \equiv, e.g., $x+y \equiv z$

Set Operations

Element of: \in, e.g., $x \in y$

Not an element of: \notin, e.g., $x \notin y$

Not an element of: \not\in, e.g., $x \not\in y$

Subset: \subset, e.g., $x \subset y$

Superset: \supset, e.g., $x \supset y$

Proper subset: \subseteq, e.g., $x \subseteq y$

Not a proper subset: \subsetneq, e.g., $x \subsetneq y$

Proper superset: \supseteq, e.g., $x \supseteq y$

Not a proper superset: \supsetneq, e.g., $x \supsetneq y$

Not a subset: \not\subset, e.g., $x \not\subset y$

Not a superset: \not\supset, e.g., $x \not\supset y$

Union: \cup, e.g., $x \cup y$

Intersection: \cap, e.g., $x \cap y$

Set difference: \setminus, e.g., $x \setminus y$

XOR: \bigodot, e.g., $x \bigodot y$

AND: \bigotimes, e.g., $x \bigotimes y$

Real number set: \mathbb{R}, e.g., \mathbb{R}

Integer set: \mathbb{Z}, e.g., \mathbb{Z}

Empty set: \emptyset, e.g., $\emptyset$

Mathematical Symbols

Infinity: \infty, e.g., $\infty$

Imaginary number: \imath, e.g., $\imath$

Imaginary number: \jmath, e.g., $\jmath$

Math symbol: \hat{a}, e.g., $\hat{a}$

Math symbol: \check{a}, e.g., $\check{a}$

Math symbol: \breve{a}, e.g., $\breve{a}$

Math symbol: \tilde{a}, e.g., $\tilde{a}$

Math symbol: \bar{a}, e.g., $\bar{a}$

Vector symbol: \vec{a}, e.g., $\vec{a}$

Math symbol: \acute{a}, e.g., $\acute{a}$

Math symbol: \grave{a}, e.g., $\grave{a}$

Math symbol: \mathring{a}, e.g., $\mathring{a}$

First derivative: \dot{a}, e.g., $\dot{a}$

Second derivative: \ddot{a}, e.g., $\ddot{a}$

Up arrow: \uparrow, e.g., $\uparrow$

Up arrow: \Uparrow, e.g., $\Uparrow$

Down arrow: \downarrow, e.g., $\downarrow$

Down arrow: \Downarrow, e.g., $\Downarrow$

Left arrow: \leftarrow, e.g., $\leftarrow$

Left arrow: \Leftarrow, e.g., $\Leftarrow$

Right arrow: \rightarrow, e.g., $\rightarrow$

Right arrow: \Rightarrow, e.g., $\Rightarrow$

Bottom-aligned ellipsis: \ldots, e.g., $1,2,\ldots,n$

Middle-aligned ellipsis: \cdots, e.g., $x_1^2 + x_2^2 + \cdots + x_n^2$

Vertical ellipsis: \vdots, e.g., $\vdots$

Diagonal ellipsis: \ddots, e.g., $\ddots$

Greek Letters

Letter	Symbol	Letter	Symbol
A	A	α	\alpha
B	B	β	\beta
Γ	\Gamma	γ	\gamma
Δ	\Delta	δ	\delta
E	E	ϵ	\epsilon
Z	Z	ζ	\zeta
H	H	η	\eta
Θ	\Theta	θ	\theta
I	I	ι	\iota
K	K	κ	\kappa
Λ	\Lambda	λ	\lambda
M	M	μ	\mu
N	N	ν	\nu
Ξ	\Xi	ξ	\xi
O	O	ο	\omicron
Π	\Pi	π	\pi
P	P	ρ	\rho
Σ	\Sigma	σ	\sigma
T	T	τ	\tau
Υ	\Upsilon	υ	\upsilon
Φ	\Phi	ϕ	\phi
X	X	χ	\chi
Ψ	\Psi	ψ	\psi
Ω	\Omega	ω	\omega