Mathematical notation consists of using

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 different concep ...

for representing operations, unspecified

number A number is a mathematical object used to count, measure, and label. The most basic 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, relations, and any other

mathematical object A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a Glossary of mathematical symbols, symbol, and therefore can be involved in formulas. Commonly encounter ...

s and assembling them into expressions and

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 betwe ...

s. Mathematical notation is widely used in

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

science Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...

, and

engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...

for representing complex

concept A concept is an abstract idea that serves as a foundation for more concrete principles, thoughts, and beliefs. Concepts play an important role in all aspects of cognition. As such, concepts are studied within such disciplines as linguistics, ...

s and

properties Property is the ownership of land, resources, improvements or other tangible objects, or intellectual property. Property may also refer to: Philosophy and science * Property (philosophy), in philosophy and logic, an abstraction characterizing an ...

in a concise, unambiguous, and accurate way. For example, the physicist

Albert Einstein Albert Einstein (14 March 187918 April 1955) was a German-born theoretical physicist who is best known for developing the theory of relativity. Einstein also made important contributions to quantum mechanics. His mass–energy equivalence f ...

's formula

E=mc^2

is the quantitative representation in mathematical notation of

mass–energy equivalence In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame. The two differ only by a multiplicative constant and the units of measurement. The principle is described by the physicist Albert Einstei ...

. Mathematical notation was first introduced by François Viète at the end of the 16th century and largely expanded during the 17th and 18th centuries by

René Descartes René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...

Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...

Gottfried Wilhelm Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to ...

, and overall

Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...

Symbols and typeface

The use of many symbols is the basis of mathematical notation. They play a similar role as words in

natural language A natural language or ordinary language is a language that occurs naturally in a human community by a process of use, repetition, and change. It can take different forms, typically either a spoken language or a sign language. Natural languages ...

s. They may play different roles in mathematical notation similarly as verbs, adjective and nouns play different roles in a sentence.

Letters as symbols

Letters are typically used for naming—in

mathematical jargon The language of mathematics has a wide vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in ...

, one says ''representing''—

s. The

Latin Latin ( or ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken by the Latins (Italic tribe), Latins in Latium (now known as Lazio), the lower Tiber area aroun ...

and Greek alphabets are used extensively, but a few letters of other alphabets are also used sporadically, such as the

Hebrew Hebrew (; ''ʿÎbrit'') is a Northwest Semitic languages, Northwest Semitic language within the Afroasiatic languages, Afroasiatic language family. A regional dialect of the Canaanite languages, it was natively spoken by the Israelites and ...

Cyrillic The Cyrillic script ( ) is a writing system used for various languages across Eurasia. It is the designated national script in various Slavic, Turkic, Mongolic, Uralic, Caucasian and Iranic-speaking countries in Southeastern Europe, Ea ...

, and

Hiragana is a Japanese language, Japanese syllabary, part of the Japanese writing system, along with ''katakana'' as well as ''kanji''. It is a phonetic lettering system. The word ''hiragana'' means "common" or "plain" kana (originally also "easy", ...

. Uppercase and lowercase letters are considered as different symbols. For Latin alphabet, different typefaces also provide different symbols. For example,

r, R, \R, \mathcal R, \mathfrak r,

and

\mathfrak R

could theoretically appear in the same mathematical text with six different meanings. Normally, roman upright typeface is not used for symbols, except for symbols representing a standard function, such as the symbol "

\sin

" of the sine function. In order to have more symbols, and for allowing related mathematical objects to be represented by related symbols,

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 ''diacrit ...

s, subscripts and

superscript A subscript or superscript is a character (such as a number or letter) that is set slightly below or above the normal line of type, respectively. It is usually smaller than the rest of the text. Subscripts appear at or below the baseline, wh ...

s are often used. For example,

\hat

may denote the

Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...

of the

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

of a function called

f_1.

Other symbols

Symbols are not only used for naming mathematical objects. They can be used for operations

(+, -, /, \oplus, \ldots),

for relations

(=, <, \le, \sim, \equiv, \ldots),

for

logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the ...

(\implies, \land, \lor, \ldots),

for quantifiers

(\forall, \exists),

and for other purposes. Some symbols are similar to Latin or Greek letters, some are obtained by deforming letters, some are traditional typographic symbols, but many have been specially designed for mathematics.

International standard mathematical notation

The

International Organization for Standardization The International Organization for Standardization (ISO ; ; ) is an independent, non-governmental, international standard development organization composed of representatives from the national standards organizations of member countries. M ...

(ISO) is an

international standard An international standard is a technical standard developed by one or more international standards organizations. International standards are available for consideration and use worldwide. The most prominent such organization is the International O ...

development organization composed of representatives from the national standards organizations of member countries. The international standard ISO 80000-2 (previously,

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-80000 ...

) specifies symbols for use in mathematical equations. The standard requires use of italic fonts for variables (e.g., ) and roman (upright) fonts for mathematical constants (e.g., e or π).

Expressions and formulas

An expression is a written arrangement of

following the context-dependent,

syntactic In linguistics, syntax ( ) is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency ...

conventions of mathematical notation. Symbols can denote

numbers A number is a mathematical object used to count, measure, and label. The most basic 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 ...

, variables, operations, and functions. Other symbols include

punctuation Punctuation marks are marks indicating how a piece of writing, written text should be read (silently or aloud) and, consequently, understood. The oldest known examples of punctuation marks were found in the Mesha Stele from the 9th century BC, c ...

marks and

brackets A bracket is either of two tall fore- or back-facing punctuation marks commonly used to isolate a segment of text or data from its surroundings. They come in four main pairs of shapes, as given in the box to the right, which also gives their n ...

, used for grouping where there is not a well-defined

order of operations In mathematics and computer programming, the order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression. These rules are formalized with a ...

. Expressions are commonly distinguished from '' formulas'': expressions are a kind of

, whereas formulas are statements ''about'' mathematical objects. This is analogous to

, where a

noun phrase A noun phrase – or NP or nominal (phrase) – is a phrase that usually has a noun or pronoun as its head, and has the same grammatical functions as a noun. Noun phrases are very common cross-linguistically, and they may be the most frequently ...

refers to an object, and a whole sentence refers to a

fact A fact is a truth, true data, datum about one or more aspects of a circumstance. Standard reference works are often used to Fact-checking, check facts. Science, Scientific facts are verified by repeatable careful observation or measurement by ...

. For example,

8x-5

is an expression, while the inequality

8x-5 \geq 3

is a formula. To ''evaluate'' an expression means to find a numerical value equivalent to the expression.

Oxford English Dictionary The ''Oxford English Dictionary'' (''OED'') is the principal historical dictionary of the English language, published by Oxford University Press (OUP), a University of Oxford publishing house. The dictionary, which published its first editio ...

, s.v. “ Simplify (v.), sense 4.a”, "''To express (an equation or other mathematical expression) in a form that is easier to understand, analyse, or work with, e.g. by collecting like terms or substituting variables.''" Expressions can be ''evaluated'' or ''simplified'' by replacing operations that appear in them with their result. For example, the expression

8\times 2-5

simplifies to

16-5

, and evaluates to

11.

History

Numbers

It is believed that a notation to represent

s was first developed at least 50,000 years ago. Early mathematical ideas such as

finger counting Finger-counting, also known as dactylonomy, is the act of counting using one's fingers. There are multiple different systems used across time and between cultures, though many of these have seen a decline in use because of the spread of Arabic nu ...

have also been represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The tally stick is a way of counting dating back to the

Upper Paleolithic The Upper Paleolithic (or Upper Palaeolithic) is the third and last subdivision of the Paleolithic or Old Stone Age. Very broadly, it dates to between 50,000 and 12,000 years ago (the beginning of the Holocene), according to some theories ...

. Perhaps the oldest known mathematical texts are those of ancient

Sumer Sumer () is the earliest known civilization, located in the historical region of southern Mesopotamia (now south-central Iraq), emerging during the Chalcolithic and Early Bronze Age, early Bronze Ages between the sixth and fifth millennium BC. ...

. The Census Quipu of the Andes and the

Ishango Bone The Ishango bone, discovered at the "Fisherman Settlement" of Ishango in the Democratic Republic of the Congo, is a bone tool and possible mathematical device that dates to the Upper Paleolithic era. The curved bone is dark brown in color, about ...

from Africa both used the

tally mark Tally marks, also called hash marks, are a form of numeral used for counting. They can be thought of as a unary numeral system. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no interm ...

method of accounting for numerical concepts. The concept of

zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...

and the introduction of a notation for it are important developments in early mathematics, which predates for centuries the concept of zero as a number. It was used as a placeholder by the Babylonians and Greek Egyptians, and then as an

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

by the Mayans, Indians and

Arabs Arabs (, , ; , , ) are an ethnic group mainly inhabiting the Arab world in West Asia and North Africa. A significant Arab diaspora is present in various parts of the world. Arabs have been in the Fertile Crescent for thousands of yea ...

(see the history of zero).

Modern notation

Until the 16th century, mathematics was essentially rhetorical, in the sense that everything but explicit numbers was expressed in words. However, some authors such as

Diophantus Diophantus of Alexandria () (; ) was a Greek mathematician who was the author of the '' Arithmetica'' in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations. Although Jose ...

used some symbols as abbreviations. The first systematic use of formulas, and, in particular the use of symbols ( variables) for unspecified numbers is generally attributed to François Viète (16th century). However, he used different symbols than those that are now standard. Later,

(17th century) introduced the modern notation for variables and

equation In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...

s; in particular, the use of

x,y,z

for

unknown Unknown or The Unknown may refer to: Film and television Film * The Unknown (1915 comedy film), ''The Unknown'' (1915 comedy film), Australian silent film * The Unknown (1915 drama film), ''The Unknown'' (1915 drama film), American silent drama ...

quantities and

a,b,c

for known ones ( constants). He introduced also the notation and the term "imaginary" for the

imaginary unit The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...

. The 18th and 19th centuries saw the standardization of mathematical notation as used today.

was responsible for many of the notations currently in use: the

functional notation In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called ...

f(x),

for the base of the

natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...

\sum

for

summation In mathematics, summation is the addition of a sequence of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, pol ...

, etc. He also popularized the use of for the Archimedes constant (proposed by William Jones, based on an earlier notation of William Oughtred). Since then many new notations have been introduced, often specific to a particular area of mathematics. Some notations are named after their inventors, such as Leibniz's notation,

Legendre symbol In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo of an odd prime number ''p'': its value at a (nonzero) quadratic residue mod ''p'' is 1 and at a non-quadratic re ...

, the Einstein summation convention, etc.

Typesetting

General

typesetting system Typesetting is the composition of Written language, text for publication, display, or distribution by means of arranging metal type, physical ''type'' (or ''sort'') in mechanical systems or ''glyphs'' in digital systems representing ''char ...

s are generally not well suited for mathematical notation. One of the reasons is that, in mathematical notation, the symbols are often arranged in two-dimensional figures, such as in: :

\sum_^\infty \frac .

TeX Tex, TeX, TEX, may refer to: People and fictional characters * Tex (nickname), a list of people and fictional characters with the nickname * Tex Earnhardt (1930–2020), U.S. businessman * Joe Tex (1933–1982), stage name of American soul singer ...

is a mathematically oriented typesetting system that was created in 1978 by

Donald Knuth Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist and mathematician. He is a professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of comp ...

. It is widely used in mathematics, through its extension called

LaTeX Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latices are found in nature, but synthetic latices are common as well. In nature, latex is found as a wikt:milky, milky fluid, which is present in 10% of all floweri ...

, and is a ''de facto'' standard. (The above expression is written in LaTeX.) More recently, another approach for mathematical typesetting is provided by MathML. However, it is not well supported in web browsers, which is its primary target.

Non-Latin-based mathematical notation

Modern Arabic mathematical notation is based mostly on the

Arabic alphabet The Arabic alphabet, or the Arabic abjad, is the Arabic script as specifically codified for writing the Arabic language. It is a unicase, unicameral script written from right-to-left in a cursive style, and includes 28 letters, of which most ...

and is used widely in the

Arab world The Arab world ( '), formally the Arab homeland ( '), also known as the Arab nation ( '), the Arabsphere, or the Arab states, comprises a large group of countries, mainly located in West Asia and North Africa. While the majority of people in ...

, especially in pre-

tertiary education Tertiary education (higher education, or post-secondary education) is the educational level following the completion of secondary education. The World Bank defines tertiary education as including universities, colleges, and vocational schools ...

. (Western notation uses

Arabic numerals The ten Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9) are the most commonly used symbols for writing numbers. The term often also implies a positional notation number with a decimal base, in particular when contrasted with Roman numera ...

, but the Arabic notation also replaces Latin letters and related symbols with Arabic script.) In addition to Arabic notation, mathematics also makes use of

Greek letter The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC. It was derived from the earlier Phoenician alphabet, and is the earliest known alphabetic script to systematically write vowels as wel ...

s to denote a wide variety of mathematical objects and variables. On some occasions, certain Hebrew letters are also used (such as in the context of infinite cardinals). Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are Penrose graphical notation and Coxeter–Dynkin diagrams. Braille-based mathematical notations used by blind people include Nemeth Braille and GS8 Braille.

Meaning and interpretation

The

syntax In linguistics, syntax ( ) is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituenc ...

of notation defines how symbols can be combined to make well-formed expressions, without any given meaning or interpretation. The

semantics Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...

of notation interprets what the symbols represent and assigns a meaning to the expressions and formulas. The reverse process of taking a statement and writing it in logical or mathematical notation is called

translation Translation is the communication of the semantics, meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The English la ...

Interpretation

Given a

formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...

, an interpretation assigns a

domain of discourse In the formal sciences, the domain of discourse or universe of discourse (borrowing from the mathematical concept of ''universe'') is the set of entities over which certain variables of interest in some formal treatment may range. It is also ...

to the language. Specifically, it assigns each of the constant symbols to objects of the domain, function letters to functions within the domain, predicate letters to statments, and vairiables are assumed to range over the domain.

Map–territory relation

The

map–territory relation The map–territory relation is the relationship between an object and a representation of that object, as in the relation between a geographical territory and a map of it. Mistaking the map for the territory is a logical fallacy that occurs wh ...

describes the relationship between an object and the representation of that object, such as the

Earth Earth is the third planet from the Sun and the only astronomical object known to Planetary habitability, harbor life. This is enabled by Earth being an ocean world, the only one in the Solar System sustaining liquid surface water. Almost all ...

and a map of it. In mathematics, this is how the number 4 relates to its representation "4". The quotation marks are the formally correct usage, distinguishing the number from its name. However, it is fairly common practice in math to commit this falacy saying "Let x denote...", rather than "Let "x" denote..." which is generally harmless.

Software for mathematical typesetting

* AUCTeX * Authorea * Apache OpenOffice Math * AsciiMath * Calligra Words - Formula editor * CoCalc * GNOME LaTeX * GNU TeXmacs * Gummi * KaTeX * Kile *

* LibreOffice Math * LyX * MacTeX *

MathJax MathJax is a cross-browser JavaScript library that displays mathematical notation in web browsers, using MathML, LaTeX, and ASCIIMathML markup. MathJax is released as open-source software under the Apache License. The MathJax project started in ...

* MathML * MathType * Notepad++ * Overleaf * Scientific WorkPlace *

TeX Live TeX Live is a cross-platform, free software, free software distribution for the TeX typesetting system that includes major TeX-related programs, macro packages, and fonts. It is the replacement of its no-longer supported counterpart teTeX. It is ...

* Texmaker * TeXnicCenter * TeXShop * TeXstudio * TeXworks * Verbosus * Vim *

Visual Studio Code Visual Studio Code, commonly referred to as VS Code, is an integrated development environment developed by Microsoft for Windows, Linux, macOS and web browsers. Features include support for debugging, syntax highlighting, intelligent code comp ...

LaTeX Workshop
* WinEdt * WinFIG * WinShell

References

External links

how to type math notation in any text editor.
Mathematics as a Language
at

Cut-the-Knot Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet Union, Soviet-born Israeli Americans, Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow ...

* Stephen Wolfram
Mathematical Notation: Past and Future
October 2000. Transcript of a keynote address presented at MathML and Math on the Web: MathML International Conference. {{DEFAULTSORT:Mathematical Notation 16th-century inventions