# Mathematical Symbols

A comprehensive collection of symbols used in mathematics — categorized by function, subject and type into tables along with each symbol's usage and meaning.

The language and vocabulary of mathematics contain a large amount of symbols — some being more technical than others. Like letters in the alphabet, they can be used to form words, phrases and sentences that would constitute a larger part of the mathematical lexicon. $\begin{gather*}x \longrightarrow x+1 \longrightarrow (x+1)^2 \longrightarrow (x+1)^2 \ge 0 \\ \longrightarrow \forall x \in \mathbb{R} [ (x+1)^2 \ge 0 ] \end{gather*}$ A math symbol can be used for different purposes from one mathematical subfield to another (e.g., $\sim$ as logical negation and similarity of triangle), just as multiple symbols can be used to delineate the same concept or relation (e.g., $\times$ and $\cdot$ in multiplication).

A basic understanding about mathematical terminology is essential to a solid foundation in higher mathematics. To that end, the following is a compilation of some of the most well-adapted, commonly-used symbols in mathematics.

Moreover, these symbols are further categorized by their function into tables. More comprehensive lists of symbols — as categorized by subject and type — can be also found in the relevant pages below (or in the navigational panel).

Get the complete, comprehensive list of mathematical symbols in eBook form — along with each symbol’s usage and LaTeX code.

## Constants

In mathematics, constants are symbols that are used to refer to non-varying objects. These can include key numbers, key mathematical sets, key mathematical infinities and other key mathematical objects (such as the identity matrix $I$).

Mathematical constants often take form of an alphabet letter — or a derivative of it. In some occasions, a constant might be regarded as a variable in the larger context. The following tables feature some of the most commonly-used constants, along with their name, meaning and usage.

### Key Mathematical Sets

For a more comprehensive list, see key mathematical sets in algebra.

### Key Mathematical Infinities

In mathematics, many different types of infinity exist. These include the purely notational use of the lemniscate symbol ($\infty$), and the use of the following symbols in the context of cardinal/ordinal infinities:

For a more comprehensive list, see cardinality-related symbols.

## Variables

A mathematical variable is a symbol that functions as a placeholder for varying expressions or quantities. The same variable can be used on a repeated basis to refer to the same thing — or quantified to form sentences that have a more definite meaning: \begin{gather*}x, y \longrightarrow x + e^x = y \longrightarrow \exists y \in \mathbb{R}\, (x + e^x = y) \\ \longrightarrow \forall x \in \mathbb{R} \, \exists y \in \mathbb{R}\, (x + e^x = y) \end{gather*} In some cases, variables can be thought of as constants in narrower contexts (e.g., as parameters), while in other cases, variables are used in conjunction with subscripts to make up for the lack of letters (e.g., $x_3$).

While variables in mathematics are often used to represent numbers, they can also be used to represent other objects such as vectors, functions and matrices. The following tables document some of the most common conventions for variables — along with the context where they are adopted and used.

### Variables in Geometry

For more symbols in geometry and trigonometry, see geometry and trigonometry symbols.

### Variables in Calculus

For a more comprehensive list, see constants and variables in calculus.

### Variables in Linear Algebra

For a more comprehensive list, see variables in algebra.

### Variables in Set Theory and Logic

For more comprehensive lists on the topics, see variables in logic and variables in set theory.

### Variables in Probability and Statistics

For a more comprehensive list, see variables in probability and statistics.

## Delimiters

Similar to punctuation marks in English, delimiters are a set of symbols which indicate the boundaries between independent mathematical expressions. They are often used to specify the scope for which an operation or rule would apply, and can occur both as an isolate symbol or as a pair of opposite-looking symbols.

In many scenarios, delimiters are used primarily for grouping purposes. The following table features some of the most commonly-used delimiters, along with their function and usage.

## Operators

An operator is a symbol used to denote an operation — a function which takes one or multiple objects to another similar object. Most of the operators are unary and binary in nature (i.e., taking one and two inputs to their intended target, respectively), with the most common ones being the arithmetic operators (e.g., $+$).

Much like the case in English, operators allow one to expand the lexicon of mathematics where only finitely many symbols exist. The following tables feature some of the most commonly-used operators in mathematics — along with their usage and intended meaning.

### Function-related Operators

For a more comprehensive list, see function-related symbols.

### Elementary Functions

For a more comprehensive list, see key functions in algebra.

### Algebra-related Operators

For a more comprehensive list, see operators in algebra.

### Geometry-related Operators

For more symbols in geometry and trigonometry, see geometry and trigonometry symbols.

### Logic-related Operators

For a more comprehensive list, see operators in logic.

### Set-related Operators

For a more comprehensive list, see operators in set theory.

### Vector-related Operators

For a more comprehensive list, see operators in linear algebra.

### Matrix-related Operators

For a more comprehensive list, see operators in linear algebra.

### Probability-related Operators

For a more comprehensive list, see operators in probability and statistics.

### Statistics-related Operators

For a more comprehensive list, see statistical operators.

### Key Probability Functions and Distributions

For a more comprehensive list, see probability-distribution-related operators.

### Calculus-related Operators

For a more comprehensive list, see calculus and analysis symbols.

## Relational Symbols

Relational symbols are used to express mathematical relations between multiple objects. Many relational symbols are binary in nature, in that they take two objects as inputs and turn them into complete, meaningful sentences (as in the case of the inequality symbol $<$).

Since relational symbols form the building blocks of mathematical sentences, they are of foundational importance in mathematics. The following tables document some of the most commonly-used relational symbols — along with their usage and meaning.

### Comparison-based Relational Symbols

For a more comprehensive list, see comparison-based relational symbols in algebra.

### Geometry-related Relational Symbols

For more symbols in geometry and trigonometry, see geometry and trigonometry symbols.

### Set-related Relational Symbols

For a more comprehensive list, see relational symbols in set theory.

### Logic-related Relational Symbols

For a more comprehensive list, see relational symbols in logic.

### Probability-related Relational Symbols

For a more comprehensive list, see relational symbols in probability and statistics.

### Calculus-related Relational Symbols

For a more comprehensive list, see relational symbols in asymptotic analysis.

## Notational Symbols

A notational symbol is a convention or shorthand whose role is different from that of a constant, variable, delimiter, operator or relational symbol. It often simply delineates the notational system being used, and might even refer to concepts that have little bearing to any definite mathematical object (e.g., $\infty$).

### Notational Symbols in Geometry and Trigonometry

For more symbols in geometry and trigonometry, see geometry and trigonometry symbols.

### Notational Symbols in Calculus

For more symbols in calculus, see calculus and analysis symbols.

### Notational Symbols in Probability and Statistics

For a more comprehensive list, see notational symbols in probability and statistics.

For lists of symbols categorized by type and subject, refer to the relevant pages below for more.

Get the master summary of mathematical symbols in eBook form — along with each symbol’s usage and LaTeX code.