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).

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## 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

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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