A mathematical symbol is a figure or a combination of figures that is used to represent a

mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical p ...

, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entirely constituted with symbols of various types, many symbols are needed for expressing all mathematics. The most basic symbols are the

decimal digit A numerical digit (often shortened to just digit) is a single symbol used alone (such as "2") or in combinations (such as "25"), to represent numbers in a positional numeral system. The name "digit" comes from the fact that the ten digits (Latin ...

s (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and the letters of the

Latin alphabet The Latin alphabet or Roman alphabet is the collection of letters originally used by the ancient Romans to write the Latin language. Largely unaltered with the exception of extensions (such as diacritics), it used to write English and th ...

. The decimal digits are used for representing numbers through the

Hindu–Arabic numeral system The Hindu–Arabic numeral system or Indo-Arabic numeral system Audun HolmeGeometry: Our Cultural Heritage 2000 (also called the Hindu numeral system or Arabic numeral system) is a positional decimal numeral system, and is the most common syste ...

. Historically, upper-case letters were used for representing points in geometry, and lower-case letters were used for variables and constants. Letters are used for representing many other sorts of

s. As the number of these sorts has remarkably increased in modern mathematics, the

Greek alphabet The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as w ...

and some

Hebrew letters The Hebrew alphabet ( he, אָלֶף־בֵּית עִבְרִי, ), known variously by scholars as the Ktav Ashuri, Jewish script, square script and block script, is an abjad script used in the writing of the Hebrew language and other Jewis ...

are also used. In mathematical formulas, the standard

typeface A typeface (or font family) is the design of lettering that can include variations in size, weight (e.g. bold), slope (e.g. italic), width (e.g. condensed), and so on. Each of these variations of the typeface is a font. There are thousands o ...

italic type In typography, italic type is a cursive font based on a stylised form of calligraphic handwriting. Owing to the influence from calligraphy, italics normally slant slightly to the right. Italics are a way to emphasise key points in a printed ...

for Latin letters and lower-case Greek letters, and upright type for upper case Greek letters. For having more symbols, other typefaces are also used, mainly

boldface In typography, emphasis is the strengthening of words in a text with a font in a different style from the rest of the text, to highlight them. It is the equivalent of prosody stress in speech. Methods and use The most common methods in W ...

\mathbf ,\ldots

script typeface Script typefaces are based upon the varied and often fluid stroke created by handwriting. They are generally used for display or trade printing, rather than for extended body text in the Latin alphabet. Some Greek alphabet typefaces, especially ...

\mathcal ,\ldots

(the lower-case script face is rarely used because of the possible confusion with the standard face), German fraktur

\mathfrak ,\ldots

, and

blackboard bold Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol (usually vertical or near-vertical lines) are doubled. The symbols usually denote number sets. One way of pro ...

\mathbb _q

(the other letters are rarely used in this face, or their use is unconventional). The use of Latin and Greek letters as symbols for denoting

s is not described in this article. For such uses, see

Variable (mathematics) In mathematics, a variable (from Latin '' variabilis'', "changeable") is a symbol that represents a mathematical object. A variable may represent a number, a vector, a matrix, a function, the argument of a function, a set, or an element of a ...

and

List of mathematical constants A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. For exa ...

. However, some symbols that are described here have the same shape as the letter from which they are derived, such as

\textstyle\prod

and

\textstyle\sum

. These letters alone are not sufficient for the needs of mathematicians, and many other symbols are used. Some take their origin in

punctuation mark Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, whether read silently or aloud. A ...

s and

diacritic A diacritic (also diacritical mark, diacritical point, diacritical sign, or accent) is a glyph added to a letter or to a basic glyph. The term derives from the Ancient Greek (, "distinguishing"), from (, "to distinguish"). The word ''diacriti ...

s traditionally used in

typography Typography is the art and technique of arranging type to make written language legible, readable and appealing when displayed. The arrangement of type involves selecting typefaces, point sizes, line lengths, line-spacing ( leading), ...

; others by deforming

letter form A letterform, letter-form or letter form, is a term used especially in typography, palaeography, calligraphy and epigraphy to mean a letter's shape. A letterform is a type of glyph, which is a specific, concrete way of writing an abstract charac ...

s, as in the cases of

\in

and

\forall

. Others, such as and , were specially designed for mathematics.

Layout of this article

Normally, entries of a glossary are structured by topics and sorted alphabetically. This is not possible here, as there is no natural order on symbols, and many symbols are used in different parts of mathematics with different meanings, often completely unrelated. Therefore, some arbitrary choices had to be made, which are summarized below. The article is split into sections that are sorted by an increasing level of technicality. That is, the first sections contain the symbols that are encountered in most mathematical texts, and that are supposed to be known even by beginners. On the other hand, the last sections contain symbols that are specific to some area of mathematics and are ignored outside these areas. However, the long section on brackets has been placed near to the end, although most of its entries are elementary: this makes it easier to search for a symbol entry by scrolling. Most symbols have multiple meanings that are generally distinguished either by the area of mathematics where they are used or by their ''syntax'', that is, by their position inside a formula and the nature of the other parts of the formula that are close to them. As readers may not be aware of the area of mathematics to which is related the symbol that they are looking for, the different meanings of a symbol are grouped in the section corresponding to their most common meaning. When the meaning depends on the syntax, a symbol may have different entries depending on the syntax. For summarizing the syntax in the entry name, the symbol

\Box

is used for representing the neighboring parts of a formula that contains the symbol. See for examples of use. Most symbols have two printed versions. They can be displayed as

Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, wh ...

characters, or in

LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well. In nature, latex is found as a milky fluid found in 10% of all flowering plants (angiosperms ...

format. With the Unicode version, using search engines and copy-pasting are easier. On the other hand, the LaTeX rendering is often much better (more aesthetic), and is generally considered a standard in mathematics. Therefore, in this article, the Unicode version of the symbols is used (when possible) for labelling their entry, and the LaTeX version is used in their description. So, for finding how to type a symbol in LaTeX, it suffices to look at the source of the article. For most symbols, the entry name is the corresponding Unicode symbol. So, for searching the entry of a symbol, it suffices to type or copy the Unicode symbol into the search textbox. Similarly, when possible, the entry name of a symbol is also an

anchor An anchor is a device, normally made of metal , used to secure a vessel to the bed of a body of water to prevent the craft from drifting due to wind or current. The word derives from Latin ''ancora'', which itself comes from the Greek ἄ� ...

, which allows linking easily from another Wikipedia article. When an entry name contains special characters such as and , , there is also an anchor, but one has to look at the article source to know it. Finally, when there is an article on the symbol itself (not its mathematical meaning), it is linked to in the entry name.

Arithmetic operators

Equality, equivalence and similarity

Comparison

Set theory

Basic logic

Several logical symbols are widely used in all mathematics, and are listed here. For symbols that are used only in

mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

, or are rarely used, see

List of logic symbols In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics. Additionally, the subs ...

Blackboard bold

The

is widely used for denoting the basic

number system A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...

s. These systems are often also denoted by the corresponding uppercase bold letter. A clear advantage of blackboard bold is that these symbols cannot be confused with anything else. This allows using them in any area of mathematics, without having to recall their definition. For example, if one encounters

\mathbb R

in combinatorics, one should immediately know that this denotes the

real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s, although combinatorics does not study the real numbers (but it uses them for many proofs).

Calculus

(Capital Greek letter

delta Delta commonly refers to: * Delta (letter) (Δ or δ), a letter of the Greek alphabet * River delta, at a river mouth * D ( NATO phonetic alphabet: "Delta") * Delta Air Lines, US * Delta variant of SARS-CoV-2 that causes COVID-19 Delta may also ...

—not to be confused with

\triangle

, which may denote a geometric

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colline ...

or, alternatively, the

symmetric difference In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \. Th ...

of two sets. (Note: the notation

\Box

is not recommend for the four-gradient since both

\Box

and

^2

are used to denote the

d'Alembertian In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of Mi ...

; see below.) (here an actual box, not a placeholder)

Linear and multilinear algebra

Advanced group theory

Infinite numbers

Brackets

Many sorts of brackets are used in mathematics. Their meanings depend not only on their shapes, but also on the nature and the arrangement of what is delimited by them, and sometimes what appears between or before them. For this reason, in the entry titles, the symbol is used as a placeholder for schematizing the syntax that underlies the meaning.

Parentheses

Square brackets

Braces

} }

Other brackets

Symbols that do not belong to formulas

In this section, the symbols that are listed are used as some sorts of punctuation marks in mathematical reasoning, or as abbreviations of English phrases. They are generally not used inside a formula. Some were used in classical logic for indicating the logical dependence between sentences written in plain English. Except for the first two, they are normally not used in printed mathematical texts since, for readability, it is generally recommended to have at least one word between two formulas. However, they are still used on a black board for indicating relationships between formulas. , "as was to be shown") is often used for the same purpose, either in its upper-case form or in lower case.

Miscellaneous

References

External links

Jeff Miller: ''Earliest Uses of Various Mathematical Symbols''

; Some Unicode charts of mathematical operators and symbols:
Index of Unicode symbols

Range 2100–214F: Unicode Letterlike Symbols

Range 2190–21FF: Unicode Arrows

Range 2200–22FF: Unicode Mathematical Operators

Range 27C0–27EF: Unicode Miscellaneous Mathematical Symbols–A

Range 2980–29FF: Unicode Miscellaneous Mathematical Symbols–B

Range 2A00–2AFF: Unicode Supplementary Mathematical Operators
; Some Unicode cross-references:
Short list of commonly used LaTeX symbols
an
Comprehensive LaTeX Symbol List

MathML Characters
- sorts out Unicode, HTML and MathML/TeX names on one page

Unicode values and Postscript names
from the source code for

Ghostscript Ghostscript is a suite of software based on an interpreter for Adobe Systems' PostScript and Portable Document Format (PDF) page description languages. Its main purposes are the rasterization or rendering of such page description language file ...

{{DEFAULTSORT:Mathematical Symbols * Mathematics Symbols Symbols Symbols