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 very different co ...

for representing

operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...

s, unspecified

number 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 ...

s, relations and any other

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 ...

s, and assembling them into

expression Expression may refer to: Linguistics * Expression (linguistics), a word, phrase, or sentence * Fixed expression, a form of words with a specific meaning * Idiom, a type of fixed expression * Metaphorical expression, a particular word, phrase, ...

s and formulas. Mathematical notation is widely used in mathematics,

science Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earli ...

, and

engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...

for representing complex

concept Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs. They play an important role in all aspects of cognition. As such, concepts are studied by s ...

s and

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

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

Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...

's equation

E=mc^2

is the quantitative representation in mathematical notation of the

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

. Mathematical notation was first introduced by

François Viète François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...

at the end of the 16th century, and largely expanded during the 17th and 18th century by

René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 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 science. Mathe ...

Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the g ...

Gottfried Wilhelm Leibniz Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mat ...

, and overall

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

Symbols

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

natural language In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languag ...

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 vast 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 i ...

, one says ''representing''—

s. This is typically the

Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power ...

and

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

alphabets that are used, but some letters of

Hebrew alphabet 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 ...

(\aleph, \beth)

are sometimes used. Uppercase and lowercase letters are considered as different symbols. For Latin alphabet, different typefaces provide also 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 that are formed of several letters, such as the symbol "

\sin

" of the

sine function In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is ...

. 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 ''diacriti ...

subscript 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, whil ...

s 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, whil ...

s are often used. For example,

\hat

may denote the

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

of the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

of a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

called

f_1.

Other symbols

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

(+, -, /, \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. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...

(\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.

Expressions

is a finite combination of

that is well-formed according to rules that depend on the context. In general, an expression denotes or names a

, and plays therefore in the

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) wi ...

the role of a

noun phrase In linguistics, a noun phrase, or nominal (phrase), is a phrase that has a noun or pronoun as its head or performs the same grammatical function as a noun. Noun phrases are very common cross-linguistically, and they may be the most frequently o ...

in the natural language. An expression contains often some

operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...

s, and may therefore be ''evaluated'' by the action of the operators in it. For example,

3+2

is an expression in which the operator

+

can be evaluated for giving the result

5.

So,

3+2

and

5

are two different expressions that represent the same number. This is the meaning of the equality

3+2=5.

A more complicated example is given by the expression

\int_a^b xdx

that can be evaluated to

\frac 2-\frac 2.

Although the resulting expression contains the operators of

division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military * Division (military), a formation typically consisting ...

subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

and

exponentiation Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...

, it cannot be evaluated further because and denote unspecified numbers.

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 n ...

Georges Ifrah Georges Ifrah (1947 – 1 November 2019) was a teacher of mathematics, a French author and a self-taught historian of mathematics, especially numerals. His work, ''From One to Zero: A Universal History of Numbers'' (1985, 1994) was translated into ...

notes that humans learned to count on their hands. Ifrah shows, for example, a picture of

Boethius Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, '' magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the t ...

(who lived 480–524 or 525) reckoning on his fingers in . have also been represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The

tally stick A tally stick (or simply tally) was an ancient memory aid device used to record and document numbers, quantities and messages. Tally sticks first appear as animal bones carved with notches during the Upper Palaeolithic; a notable example is the ...

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 coi ...

. Perhaps the oldest known mathematical texts are those of ancient Sumer. 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 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 10 ce ...

from Africa both used the

tally mark Tally marks, also called hash marks, are a unary numeral system (arguably). They are a form of numeral used for counting. They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate r ...

method of accounting for numerical concepts. The concept of

zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usu ...

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 Babylonia (; Akkadian: , ''māt Akkadī'') was an ancient Akkadian-speaking state and cultural area based in the city of Babylon in central-southern Mesopotamia (present-day Iraq and parts of Syria). It emerged as an Amorite-ruled state c. ...

and Greek Egyptians, and then as an

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

by the

Mayans The Maya peoples () are an ethnolinguistic group of indigenous peoples of Mesoamerica. The ancient Maya civilization was formed by members of this group, and today's Maya are generally descended from people who lived within that historical reg ...

Indians Indian or Indians may refer to: Peoples South Asia * Indian people, people of Indian nationality, or people who have an Indian ancestor ** Non-resident Indian, a citizen of India who has temporarily emigrated to another country * South Asia ...

and

Arabs The Arabs (singular: Arab; singular ar, عَرَبِيٌّ, DIN 31635: , , plural ar, عَرَب, DIN 31635: , Arabic pronunciation: ), also known as the Arab people, are an ethnic group mainly inhabiting the Arab world in Western Asia, ...

(see the history of zero).

Modern notation

Until the 16th century, mathematics was essentially

rhetorical Rhetoric () is the art of persuasion, which along with grammar and logic (or dialectic), is one of the three ancient arts of discourse. Rhetoric aims to study the techniques writers or speakers utilize to inform, persuade, or motivate part ...

, in the sense that everything but explicit numbers was expressed in words. However, some authors such as

Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the aut ...

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

(16th century). However, he used different symbols than those that are now standard. Later,

(17th century) introduced the modern notation for variables and equations; in particular, the use of

x,y,z

for

unknown Unknown or The Unknown may refer to: Film * ''The Unknown'' (1915 comedy film), a silent boxing film * ''The Unknown'' (1915 drama film) * ''The Unknown'' (1927 film), a silent horror film starring Lon Chaney * ''The Unknown'' (1936 film), a ...

quantities and

a,b,c

for known ones (

constant Constant or The Constant may refer to: Mathematics * Constant (mathematics), a non-varying value * Mathematical constant, a special number that arises naturally in mathematics, such as or Other concepts * Control variable or scientific const ...

s). He introduced also the notation and the term "imaginary" for the

imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition a ...

. 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 often used synonymously. The set is called the domain of the fun ...

f(x),

for the base of the natural logarithm,

\sum

for

summation In mathematics, summation is the addition of a sequence of any kind 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, m ...

, etc. He also popularized the use of for the Archimedes constant (proposed by William Jones, based on an earlier notation of

William Oughtred William Oughtred ( ; 5 March 1574 – 30 June 1660), also Owtred, Uhtred, etc., was an English mathematician and Anglican clergyman.'Oughtred (William)', in P. Bayle, translated and revised by J.P. Bernard, T. Birch and J. Lockman, ''A Genera ...

). 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 In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols and to represent infinitely small (or infinitesimal) increments of and , respectively, just ...

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

Einstein's summation convention In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...

, etc.

Typesetting

General

typesetting system Typesetting is the composition of text by means of arranging physical ''type'' (or ''sort'') in mechanical systems or ''glyphs'' in digital systems representing ''characters'' (letters and other symbols).Dictionary.com Unabridged. Random Ho ...

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 may refer to: People and fictional characters * Tex (nickname), a list of people and fictional characters with the nickname * Joe Tex (1933–1982), stage name of American soul singer Joseph Arrington Jr. Entertainment * ''Tex'', the Italian ...

is a mathematically oriented typesetting system that was created in

1978 Events January * January 1 – Air India Flight 855, a Boeing 747 passenger jet, crashes off the coast of Bombay, killing 213. * January 5 – Bülent Ecevit, of CHP, forms the new government of Turkey (42nd government). * January 6 – ...

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

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

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 (angiosper ...

, and is a ''de facto'' standard. (The above expression is written in LaTeX.) More recently, another approach for mathematical typesetting is provided by

MathML Mathematical Markup Language (MathML) is a mathematical markup language, an application of XML for describing mathematical notations and capturing both its structure and content. It aims at integrating mathematical formulae into World Wide We ...

. However, it is not well-supported in web browsers, which is its primary target. 01 Kreiszahl

Non-Latin-based mathematical notation

Modern Arabic mathematical notation Modern Arabic mathematical notation is a mathematical notation based on the Arabic script, used especially at pre-university levels of education. Its form is mostly derived from Western notation, but has some notable features that set it apart fro ...

is based mostly on the Arabic alphabet and is used widely in the

Arab world The Arab world ( ar, اَلْعَالَمُ الْعَرَبِيُّ '), formally the Arab homeland ( '), also known as the Arab nation ( '), the Arabsphere, or the Arab states, refers to a vast group of countries, mainly located in Western A ...

, especially in pre-

tertiary education Tertiary education, also referred to as third-level, third-stage or post-secondary education, is the educational level following the completion of secondary education. The World Bank, for example, defines tertiary education as including univer ...

. (Western notation uses

Arabic numerals Arabic numerals are the ten numerical digits: , , , , , , , , and . They are the most commonly used symbols to write decimal numbers. They are also used for writing numbers in other systems such as octal, and for writing identifiers such as ...

, 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 BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as ...

s to denote a wide variety of mathematical objects and variables. In some occasions, certain

Hebrew letter The Hebrew alphabet ( he, wikt:אלפבית, 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 languag ...

s are also used (such as in the context of

infinite cardinal In mathematics, transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. These include the transfinite cardinals, which are cardinal numbers used to qua ...

s). Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are

Penrose graphical notation In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971. A diagram in the notation consists of several sha ...

and

Coxeter–Dynkin diagram In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes). It describes ...

s. Braille-based mathematical notations used by blind people include

Nemeth Braille The Nemeth Braille Code for Mathematics is a Braille code for encoding mathematical and scientific notation linearly using standard six-dot Braille cells for tactile reading by the visually impaired. The code was developed by Abraham Nemeth. The ...

and GS8 Braille.

Notes

References

Florian Cajori Florian Cajori (February 28, 1859 – August 14 or 15, 1930) was a Swiss-American historian of mathematics. Biography Florian Cajori was born in Zillis, Switzerland, as the son of Georg Cajori and Catherine Camenisch. He attended schools firs ...

''A History of Mathematical Notations''
(1929), 2 volumes. * . Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk. Ifrah supports his thesis by quoting idiomatic phrases from languages across the entire world. * Mazur, Joseph (2014)
''Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers''
Princeton, New Jersey: Princeton University Press.

External links

how to type math notation in any text editor.
Mathematics as a Language
at Cut-the-Knot *

Stephen Wolfram Stephen Wolfram (; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer science, mathematics, and theoretical physics. In 2012, he was named a fellow of the American Ma ...

Mathematical Notation: Past and Future
October 2000. Transcript of a keynote address presented at

and Math on the Web: MathML International Conference. {{DEFAULTSORT:Mathematical Notation