The field of mathematics customarily uses letters as symbols for key mathematical objects. As a result, it often draws upon alphabets from other languages — such as Greek, Hebrew and Latin — whenever native symbols are lacking.
In particular, the following is a comprehensive list of alphabets from these three languages, along with the mathematical context each letter finds itself in. Other comprehensive lists of symbols — as categorized by subject and type — can be also found in the relevant pages below (or in the navigational panel).
Table of Contents
Prefer the PDF version instead?
Get the master summary of mathematical symbols in eBook form — along with each symbol’s usage and LaTeX code.
Greek Symbols
Mathematics makes extensive use of Greek symbols to refer to key mathematical constants, variables, functions and other entities. The following table documents the complete 24 letters (plus 1 archaic letter) of the Greek alphabet, along with each symbol’s case, English equivalent, example and usage.
| Symbol Name | Used For | Example |
|---|---|---|
| $\alpha$ (Lowercase alpha, “a” in English) | Variable for angles, statistical significance level | At $\alpha=0.01$, the null hypothesis is rejected. |
| $\mathrm{B}$ (Uppercase beta, “B” in English) | Beta function | For all $x, y \in \mathbb{R}$, $\mathrm{B}(x, y) = \mathrm{B} (y, x)$. |
| $\beta$ (Lowercase beta, “b” in English) | Standardized regression coefficient, probability of type II error | $\beta$ denotes the probability that the null hypothesis is accepted — given that it’s false. |
| $\Gamma$ (Uppercase gamma, “G” in English) | Gamma function, Gamma distribution | For all $n \in \mathbb{N}_+$, $\Gamma(n) = (n-1)!$. |
| $\gamma$ (Lowercase gamma , “g” in English) | Euler–Mascheroni constant | $\displaystyle \gamma = \lim_{n \to \infty}$ $\left(\frac{1}{1}+\cdots+\frac{1}{n}\ – \ln n\right)$ |
| $\Delta$ (Uppercase delta, “D” in English) | Discriminant, finite difference operator, Laplace operator | $\Delta (k_1 f + k_2 g) =$ $k_1 \Delta f + k_2 \Delta g$ |
| $\delta$ (Lowercase delta, “d” in English) | Kronecker delta function, Dirac delta function | $\delta_{ij} = \begin{cases} 0 & i \ne j \\ 1 & i=j \end{cases}$ |
| $\epsilon$, $\varepsilon$ (Lowercase epsilon, “e” in English) | Variable in proofs involving limits | Given any $\varepsilon > 0$, there is an $n \in \mathbb{N}$ such that $\displaystyle \left|\frac{1}{n}\right| < \varepsilon$. |
| $\digamma$ (digamma, archaic letter) | Digamma function | $\digamma (x) = \dfrac{\Gamma^{\prime}(x)}{\Gamma (x)}$ |
| $\zeta$ (Lowercase zeta, “z” in English) | Riemann zeta function | $\zeta(0) = -\dfrac{1}{2}$ |
| $\eta$ (Lowercase eta, “h” in English) | Dirichlet eta function | $\eta(0) = \dfrac{1}{2}$ |
| $\Theta$ (Uppercase theta, “Th” in English) | Big-Theta notation | $f(n) \in \Theta (g(n))$ if $f(n)$ is eventually bounded between $k_1 g(n)$ and $k_2 g(n)$. |
| $\theta$, $\vartheta$ (Lowercase theta, “th” in English) | Variable for angles | $\sin (2\theta) = \\ 2 \sin\theta \cos\theta$ |
| $\iota$ (Lowercase iota, “i” in English) | Inclusion function in set theory | $\iota (x) = x$ |
| $\kappa$ (Lowercase kappa, “k” in English) | Curvature | $\kappa = \dfrac{1}{R}$ |
| $\Lambda$ (Uppercase lambda, “L” in English) | Set of all logical validities in first-order logic | $[ \forall x (x=x) ] \in \Lambda$ |
| $\lambda$ (Lowercase lambda, “l” in English) | Parameter in Poisson and exponential distribution, variable for eigenvalues | $A\mathbf{v}=\lambda \mathbf{v}$ |
| $\mu$ (Lowercase mu, “m” in English) | Population mean, Möbius function | $H_0\!:\!\mu_1 = \mu_2$ |
| $\nu$ (Lowercase nu, “n” in English) | Variable for degree of freedom | $\chi^2 (\nu) =$ $\mathrm{Gamma}\left(\nu/2, 1/2\right)$ |
| $\Xi$ (Uppercase xi, “X” in English) | Riemann’s original Xi function | $\Xi (-z)=\Xi (z)$ |
| $\xi$ (Lowercase xi, “x” in English) | Riemann Xi function | $\xi(2) = \dfrac{\pi}{6}$ |
| $\omicron$ (Lowercase omicron, “o” in English) | Little-o notation | $x \in \omicron (x^2)$ |
| $\Pi$ (Uppercase pi, “P” in English) | Pi product operator | $\prod_{i=1}^5 i = 5!$ |
| $\pi$ (Lowercase pi, “p” in English) | Archimedes’ constant, prime-counting function, population proportion | $A = \pi r^2$ |
| $\rho$ (Lowercase rho, “r” in English) | Population correlation | $H_a\!:\! \rho > 0$ |
| $\Sigma$ (Uppercase sigma, “S” in English) | Summation operator | $\sum_{i=1}^{10} i = 55$ |
| $\sigma$, $\varsigma$ (Lowercase sigma, “s” in English) | Population standard deviation, variable for permutations | $\sigma(1)=2, \sigma(2)=3,$ $\sigma(3)=1$ |
| $\tau$ (Lowercase tau, “t” in English) | Ratio between a circle’s circumference and radius | $\tau = 2 \pi$ |
| $\mathrm{Y}$, $\Upsilon$ (Uppercase upsilon, “U” in English) | Upsilon function | $\Upsilon(z) = \\ \displaystyle \sum_{i=1}^{\infty} \frac{1}{i^2 + z^2}$ |
| $\upsilon$ (Lowercase upsilon, “u” in English) | General variable | $\upsilon^{\prime}(t)+2\upsilon(t) = 3$ |
| $\Phi$ (Uppercase phi, “Ph” in English) | Golden ratio conjugate, cdf of standard normal distribution | $\Phi = \dfrac{1}{\varphi} \approx 0.618$ |
| $\phi$, $\varphi$ (Lowercase phi, “ph” in English) | Golden ratio, Euler’s totient function, variable for angles, pdf of Z-distribution | $\varphi = \dfrac{1+\sqrt{5}}{2}$ |
| $\chi$ (Lowercase chi, “ch” in English) | Chi-squared distribution, Euler characteristic | $\chi = V-E + F$ |
| $\Psi$ (Uppercase psi, “Ps” in English) | Variable for sets of sentences | $\Phi \cup \Psi$ proves sentence $\alpha$. |
| $\psi$ (Lowercase psi, “ps” in English) | Reciprocal Fibonacci constant | $\displaystyle \psi = \frac{1}{1} + \frac{1}{1} + \frac{1}{2} +$ $\displaystyle \frac{1}{3} + \frac{1}{5} + \frac{1}{8} + \cdots$ |
| $\Omega$ (Uppercase omega, “O” in English) | Big-Omega notation, Omega constant | $\Omega e^{\Omega} = 1$ |
| $\omega$ (Lowercase omega, “o” in English) | Smallest infinite ordinal, prime omega function | For all $n \in \mathbb{N}$, $n < \omega$. |
Among the 24 letters in the Greek alphabet, there are 14 letters whose uppercase shares the same appearance as their Latin counterpart. The following table documents an entire list of them.
| Symbol | Uppercase of | Symbol | Uppercase of |
|---|---|---|---|
| $\mathrm{A}$ | Alpha | $\mathrm{M}$ | Mu |
| $\mathrm{B}$ | Beta | $\mathrm{N}$ | Nu |
| $\mathrm{E}$ | Epsilon | $\mathrm{O}$ | Omicron |
| $\mathrm{Z}$ | Zeta | $\mathrm{P}$ | Rho |
| $\mathrm{H}$ | Eta | $\mathrm{T}$ | Tau |
| $\mathrm{I}$ | Iota | $\mathrm{Y}$ | Upsilon |
| $\mathrm{K}$ | Kappa | $\mathrm{X}$ | Chi |
Hebrew Letters
Mathematics occasionally makes use of a subset of the Hebrew alphabet to refer to key numbers and functions in the theory of transfinite cardinals. The following table documents these symbols — along with their usage and example.
| Symbol Name | Used For | Example |
|---|---|---|
| $\aleph$ (Aleph) | Aleph numbers | $|\mathbb{N}|=\aleph_0$ |
| $\beth$ (Beth) | Beth numbers | $\beth_{n+1}=2^{\beth_{n}}$ |
| $\gimel$ (Gimel) | Gimel function | For all infinite cardinals $\kappa$, $\gimel(\kappa) > \kappa$. |
| $\daleth$ (Daleth) | Originally intended as fourth transfinite cardinal | N/A |
For other symbols related to infinite cardinals and ordinals, see cardinality-related symbols.
Latin-based Alphabets
Since mathematical communications are often carried out in English (and other languages based on the Latin alphabet), mathematics often borrows and modifies these letters to refer to key sets, numbers, functions and other entities.
As a general rule of thumb, these symbols can be categorized into 4 typefaces: regular typeface, Fraktur, calligraphic/cursive and blackboard bold. The following table documents the most common of these — along with their name, usage, example and meaning.
| Symbol Name | Used For | Example |
|---|---|---|
| $\mathbb{A}$ (Blackboard bold A) | Set of algebraic numbers | $\sqrt{2} \in \mathbb{A}, e \notin \mathbb{A}$ |
| $\mathbb{B}$ (Blackboard bold B) | Boolean domain | $\mathbb{B}=\{0, 1\}$ |
| $\mathbb{C}$ (Blackboard bold C) | Set of complex numbers | $2+3i \in \mathbb{C}$ |
| $\mathfrak{c}$ (Fraktur c) | Cardinality of continuum | $|\mathbb{R}|=\mathfrak{c}$ |
| $e$ (Regular e) | Euler’s constant | $\displaystyle e = \lim_{n \to \infty} \left(1+ \frac{1}{n}\right)^n$ |
| $\mathbb{F}$ (Blackboard bold F) | Variable for fields | Given polynomials $p_1, p_2 \in \mathbb{F}[x]$, $p_1 p_2 \in \mathbb{F}[x]$. |
| $\mathbb{H}$ (Blackboard bold H) | Set of quaternions | $3 + 5i + 7j + 9k \in \mathbb{H}$ |
| $I$ (Regular I) | Identity matrix | $AI = IA = A$ |
| $\mathbb{I}$ (Blackboard bold I) | Set of imaginary numbers | $5i \in \mathbb{I}, 2+3i \notin \mathbb{I}$ |
| $i$ (Regular i) | Imaginary unit | $i^2 = -1$ |
| $\ell$ (Cursive l) | Variable for lines | $\ell_1 \parallel \ell_2$ |
| $\mathcal{L}$ (Calligraphic l) | Laplace transform | $\mathcal{L}\{e^t\} = \dfrac{1}{s-1}$ |
| $\mathbb{N}$ (Blackboard bold N) | Set of natural numbers | $0 \in \mathbb{N}_0, 0 \notin \mathbb{N}_1$ |
| $\mathbb{O}$ (Blackboard bold O) | Set of octonions | For all $x, y \in \mathbb{O}$, $\| xy \| = \|x\| \|y\|$. |
| $\mathbb{P}$ (Blackboard bold P) | Set of prime numbers | $51 \notin \mathbb{P}$ |
| $\mathcal{P}$ (Calligraphic P) | Power set | $|\mathcal{P}(\mathbb{N})| > |\mathbb{N}|$ |
| $p$ (Regular p) | Sample proportion | $p = \dfrac{X}{n}$ |
| $\mathbb{Q}$ (Blackboard bold Q) | Set of rational numbers | For all $x, y \in \mathbb{Q}$, $x+y \in \mathbb{Q}$. |
| $\mathbb{R}$ (Blackboard bold R) | Set of real numbers | $\pi \in \mathbb{R}, -3 \notin \mathbb{R}_+$ |
| $r$ (Regular r) | Sample correlation | $r_{xy}=r_{yx}$ |
| $s$ (Regular s) | Sample standard deviation | $s = \sqrt{\dfrac{\sum (X – \overline{X})^2}{n-1}}$ |
| $Z$ (Regular Z) | Standard normal distribution | $Z \sim N(0,1)$ |
| $\mathbb{Z}$ (Blackboard bold Z) | Set of integers | $0 \in \mathbb{Z}, 0 \notin \mathbb{Z}_+$ |
For the master list of symbols, see mathematical symbols. For lists of symbols categorized by type and subject, refer to the relevant pages below for more.
- Arithmetic and Common Math Symbols
- Geometry and Trigonometry Symbols
- Logic Symbols
- Set Theory Symbols
- Greek, Hebrew, Latin-based Symbols
- Algebra Symbols
- Probability and Statistics Symbols
- Calculus and Analysis Symbols
Prefer the PDF version instead?
Get the master summary of mathematical symbols in eBook form — along with each symbol’s usage and LaTeX code.
Additional Resources
- Ultimate LaTeX Reference Guide: Definitive reference guide to make the LaTeXing process more streamlined, more efficient and less painful
- Definitive Guide to Learning Higher Mathematics: A 10-principle framework for tackling higher mathematical learning, thinking and problem solving efficiently
- 10 Commandments of Higher Mathematical Learning: An illustrated web guide on 10 scalable rules for learning higher mathematics
- Definitive Glossary of Higher Mathematical Jargon: A tour around higher mathematics in 100 terms
