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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 pr ...
, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
. 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 the o ...
. 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
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 pr ...
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 we ...
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
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
s, 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 list of type ...
is
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 tex ...
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
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 pr ...
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 set ...
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), and ...
; 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 char ...
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 A glossary (from grc, γλῶσσα, ''glossa''; language, speech, wording) also known as a vocabulary or clavis, is an alphabetical list of Term (language), terms in a particular domain of knowledge with the definitions for those terms. Tradi ...
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 Technical standard, standard for the consistent character encoding, encoding, representation, and handling of Character (computing), text expre ...
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 engine A search engine is a software system designed to carry out web searches. They search the World Wide Web in a systematic way for particular information specified in a textual web search query. The search results are generally presented in a ...
s 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 ἄγ ...
, 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 logic, 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 for ...
, 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
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 ...
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 list of type ...
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 Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
, 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 real ...
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 Edge (geometry), edges and three Vertex (geometry), 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, an ...
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 ...
; 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 Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...
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 A blackboard (also known as a chalkboard) is a reusable writing surface on which text or drawings are made with sticks of calcium sulphate or calcium carbonate, known, when used for this purpose, as chalk. Blackboards were originally made of ...
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


See also

* List of mathematical symbols (Unicode and LaTeX) ** List of mathematical symbols by subject **
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 ...
*
Mathematical Alphanumeric Symbols (Unicode block) Mathematical Alphanumeric Symbols is a Unicode block comprising styled forms of Latin and Greek letters and decimal digits that enable mathematicians to denote different notions with different letter styles. The letters in various fonts ofte ...
**
Mathematical constants and functions 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 e ...
**
Table of mathematical symbols by introduction date The following table lists many specialized symbols commonly used in modern mathematics, ordered by their introduction date. Note that the table can also be ordered alphabetically by clicking on the relevant header title. See also * History of ...
*
List of Unicode characters As of Unicode version 15.0, there are 149,186 characters with code points, covering 161 modern and historical scripts, as well as multiple symbol sets. This article includes the 1062 characters in the Multilingual European Character Set 2 (ME ...
** Blackboard bold#Usage **
Letterlike Symbols Letterlike Symbols is a Unicode block containing 80 characters which are constructed mainly from the glyphs of one or more letters. In addition to this block, Unicode includes full styled mathematical alphabets, although Unicode does not expli ...
**
Unicode block A Unicode block is one of several contiguous ranges of numeric character codes (code points) of the Unicode character set that are defined by the Unicode Consortium for administrative and documentation purposes. Typically, proposals such as the ad ...
* Lists of
Mathematical operators and symbols in Unicode The Unicode Standard encodes almost all standard characters used in mathematics. Unicode Technical Report #25 provides comprehensive information about the character repertoire, their properties, and guidelines for implementation. Mathematical ope ...
**
Mathematical Operators Mathematical Operators is a Unicode block containing characters for mathematical, logical, and set notation. Notably absent are the plus sign (+), greater than sign (>) and less than sign (<), due to them already appearing in the Basi ...
and
Supplemental Mathematical Operators Supplemental Mathematical Operators is a Unicode block containing various mathematical symbols, including N-ary operators, summations and integrals, intersections and unions, logical and relational operators, and subset/superset relations. Block ...
** Miscellaneous Math Symbols: A, B,
Technical Technical may refer to: * Technical (vehicle), an improvised fighting vehicle * Technical analysis, a discipline for forecasting the future direction of prices through the study of past market data * Technical drawing, showing how something is co ...
**
Arrow (symbol) An arrow is a graphical symbol, such as ← or →, or a pictogram, used to point or indicate direction. In its simplest form, an arrow is a triangle, chevron, or concave kite, usually affixed to a line segment or rectangle, and in more co ...
and
Miscellaneous Symbols and Arrows Miscellaneous Symbols and Arrows is a Unicode block A Unicode block is one of several contiguous ranges of numeric character codes ( code points) of the Unicode character set that are defined by the Unicode Consortium for administrative and ...
an
arrow symbols
**
ISO 31-11 ISO 31-11:1992 was the part of international standard ISO 31 that defines ''mathematical signs and symbols for use in physical sciences and technology''. It was superseded in 2009 by ISO 80000-2:2009 and subsequently revised in 2019 as ISO-800 ...
(Mathematical signs and symbols for use in physical sciences and technology) ** Number Forms **
Geometric Shapes Geometric Shapes is a Unicode block of 96 symbols at code point range U+25A0–25FF. U+25A0–U+25CF The BLACK CIRCLE is displayed when typing in a password field, in order to hide characters from a screen recorder or shoulder surfing. U+2 ...
*
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 ...
*
Language of mathematics The language of mathematics or mathematical language is an extension of the natural language (for example English) that is used in mathematics and in science for expressing results (scientific laws, theorems, proofs, logical deductions, etc) with ...
**
Mathematical notation Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathematic ...
* Typographical conventions and common meanings of symbols: ** APL syntax and symbols **
Greek letters used in mathematics, science, and engineering Greek letters are used in mathematics, science, engineering, and other areas where mathematical notation is used as symbols for constants, special functions, and also conventionally for variables representing certain quantities. In these cont ...
**
Latin letters used in mathematics Many letters of the Latin alphabet, both capital and small, are used in mathematics, science, and engineering to denote by convention specific or abstracted constants, variables of a certain type, units, multipliers, or physical entities. Certain ...
**
List of common physics notations This is a list of common physical constants and variables, and their notations. Note that bold text indicates that the quantity is a vector. Latin characters Greek characters Other characters See also * List of letters used in mathema ...
**
List of letters used in mathematics and science :''This list is about the meanings of the letters used in mathematics, science and engineering. SI units are indicated in parentheses. For the Unicode blocksee Mathematical Alphanumeric Symbols.'' Latin and Greek letters are used in mathematics ...
**
List of mathematical abbreviations This following list features abbreviated names of mathematical functions, function-like operators and other mathematical terminology. :''This list is limited to abbreviations of two or more letters. The capitalization of some of these abbreviation ...
**
Mathematical notation Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathematic ...
**
Notation in probability and statistics Probability theory and statistics have some commonly used conventions, in addition to standard mathematical notation and mathematical symbols. Probability theory * Random variables are usually written in upper case roman letters: ''X'', ''Y'', ...
**
Physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature and have constant value in time. It is contrasted with a mathematical constant, ...
s **
Typographical conventions in mathematical formulae Typographical conventions in mathematical formulae provide uniformity across mathematical texts and help the readers of those texts to grasp new concepts quickly. Mathematical notation includes letters from various alphabets, as well as special ma ...


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 files, ...
{{DEFAULTSORT:Mathematical Symbols *
Mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
Symbols A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
Symbols A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...
Symbols A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different conc ...