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{{Short description|System of symbolic representation}}
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'''Mathematical notation''' consists of using [[glossary of mathematical symbols|symbols]] for representing [[operation (mathematics)|operation]]s, unspecified [[number]]s, [[relation (mathematics)|relation]]s, and any other [[mathematical object]]s and assembling them into [[expression (mathematics)|expression]]s and [[formula]]s. Mathematical notation is widely used in [[mathematics]], [[science]], and [[engineering]] for representing complex [[concept]]s and [[property (philosophy)|properties]] in a concise, unambiguous, and accurate way.

For example, the physicist [[Albert Einstein]]'s formula <math>E=mc^2</math> is the quantitative representation in mathematical notation of [[mass–energy equivalence]].<ref>{{Cite journal |last=Einstein |first=Albert |date=1905 |title=Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.19053231314 |journal=Annalen der Physik |language=de |volume=323 |issue=13 |pages=639–641 |doi=10.1002/andp.19053231314 |bibcode=1905AnP...323..639E |issn=0003-3804}}</ref>

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]], [[Isaac Newton]], [[Gottfried Wilhelm Leibniz]], and overall [[Leonhard Euler]].

== Symbols ==
{{Main|Glossary of mathematical symbols}}

The use of many symbols is the basis of mathematical notation. They play a similar role as words in [[natural language]]s. They may play different roles in mathematical notation similarly as verbs, adjective and nouns play different roles in a sentence.

=== Letters as symbols===
{{main|List of letters used in mathematics, science, and engineering}}

Letters are typically used for naming—in [[list of mathematical jargon|mathematical jargon]], one says ''representing''—[[mathematical object]]s. The [[Latin alphabet|Latin]] and [[Greek alphabet|Greek]] alphabets are used extensively, but a few letters of other alphabets are also used sporadically, such as the [[Hebrew alphabet|Hebrew]] {{tmath|\aleph}}, [[Cyrillic script|Cyrillic]] {{math|Ш}}, and [[Hiragana]] {{math|よ}}. Uppercase and lowercase letters are considered as different symbols. For Latin alphabet, different typefaces also provide different symbols. For example, <math>r, R, \R, \mathcal R, \mathfrak r,</math> and <math>\mathfrak R</math> 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 "<math>\sin</math>" of the [[sine function]].<ref>ISO 80000-2:2019</ref>

In order to have more symbols, and for allowing related mathematical objects to be represented by related symbols, [[diacritic]]s, [[subscript]]s and [[superscript]]s are often used. For example, <math>\hat {f'_1}</math> may denote the [[Fourier transform]] of the [[derivative]] of a [[function (mathematics)|function]] called <math>f_1.</math>

=== Other symbols ===
Symbols are not only used for naming mathematical objects. They can be used for [[operation (mathematics)|operation]]s <math>(+, -, /, \oplus, \ldots),</math> for [[relation (mathematics)|relation]]s <math>(=, <, \le, \sim, \equiv, \ldots),</math> for [[logical connective]]s <math>(\implies, \land, \lor, \ldots),</math> for [[quantifier (logic)|quantifier]]s <math>(\forall, \exists),</math> and for other purposes.

Some symbols are similar to Latin or Greek letters, some are obtained by deforming letters, some are traditional [[typographic symbol]]s, but many have been specially designed for mathematics.

== Expressions ==
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An [[expression (mathematics)|expression]] is a finite combination of [[glossary of mathematical symbols|symbols]] that is [[well-formed formula|well-formed]] according to rules that depend on the context. In general, an expression denotes or names a [[mathematical object]], and plays therefore in the [[language of mathematics]] the role of a [[noun phrase]] in the natural language.

An expression contains often some [[operator (mathematics)|operator]]s, and may therefore be ''evaluated'' by the action of the operators in it. For example, <math>3+2</math> is an expression in which the operator <math>+</math> can be evaluated for giving the result <math>5.</math> So, <math>3+2</math> and <math>5</math> are two different expressions that represent the same number. This is the meaning of the equality <math>3+2=5.</math>

A more complicated example is given by the expression<math display="inline">\int_a^b xdx</math> that can be evaluated to <math display="inline">\frac {b^2}2-\frac {a^2}2.</math> Although the resulting expression contains the operators of [[division (mathematics)|division]], [[subtraction]] and [[exponentiation]], it cannot be evaluated further because {{mvar|a}} and {{mvar|b}} denote unspecified numbers.

== History ==
{{Main|History of mathematical notation}}

=== Numbers ===
It is believed that a notation to represent [[number]]s was first developed at least 50,000 years ago.<ref name="Eves_1990"/> Early mathematical ideas such as [[finger counting]]<ref name="Ifrah_2000"/> 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]]. Perhaps the oldest known mathematical texts are those of ancient [[Sumer]]. The [[census quipu|Census Quipu]] of the Andes and the [[Ishango Bone]] from Africa both used the [[tally mark]] method of accounting for numerical concepts.

The concept of [[zero]] 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 [[Babylonian numerals|Babylonians]] and [[Greek numerals|Greek Egyptians]], and then as an [[integer]] by the [[Maya numerals|Mayans]], [[Indian numerals|Indians]] and [[Arabic numerals|Arabs]] (see [[History of zero|the history of zero]]).

=== Modern notation ===
Until the 16th century, mathematics was essentially [[rhetorical algebra|rhetorical]], in the sense that everything but explicit numbers was expressed in words. However, some authors such as [[Diophantus]] used some symbols as abbreviations. 

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

Later, [[René Descartes]] (17th century) introduced the modern notation for variables and [[equation]]s; in particular, the use of <math>x,y,z</math> for [[unknown (mathematics)|unknown]] quantities and <math>a,b,c</math> for known ones ([[constant (mathematics)|constant]]s). He introduced also the notation {{mvar|i}} and the term "imaginary" for the [[imaginary unit]].

The 18th and 19th centuries saw the standardization of mathematical notation as used today. [[Leonhard Euler]] was responsible for many of the notations currently in use: the [[functional notation]] <math>f(x),</math> {{math|''e''}} for the base of the natural logarithm, <math display="inline">\sum</math> for [[summation]], etc.<ref name="Boyer-Merzbach_1991"/> He also popularized the use of {{pi}} for the [[Archimedes constant]] (proposed by [[William Jones (mathematician)|William Jones]], based on an earlier notation of [[William Oughtred]]).<ref name="Arndt-Haenel_2006"/>

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]], the [[Einstein summation convention]], etc.

=== Typesetting ===
General [[typesetting system]]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:
: <math>\sum_{n=0}^\infty \frac {\begin{bmatrix}a&b\\c&d\end{bmatrix}^n}{n!}.</math>

[[TeX]] is a mathematically oriented typesetting system that was created in 1978 by [[Donald Knuth]]. It is widely used in mathematics, through its extension called [[LaTeX]], 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.

== International standard mathematical notation ==
The international standard [[ISO 80000-2]] (previously, [[ISO 31-11]]) specifies symbols for use in mathematical equations. The standard requires use of italic fonts for variables (e.g., {{nowrap|1=''E'' = ''mc''<sup>2</sup>}}) and roman (upright) fonts for mathematical constants (e.g., e or π).

== Non-Latin-based mathematical notation ==
[[Modern Arabic mathematical notation]] is based mostly on the [[Arabic alphabet]] and is used widely in the [[Arab world]], especially in pre-[[tertiary education]].  (Western notation uses [[Arabic numerals]], 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 alphabet|Greek letter]]s to denote a wide variety of mathematical objects and variables. On some occasions, certain [[Hebrew alphabet|Hebrew letter]]s are also used (such as in the context of [[infinite cardinal]]s).

Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are [[Penrose graphical notation]] and [[Coxeter–Dynkin diagram]]s.

Braille-based mathematical notations used by blind people include [[Nemeth Braille]] and [[GS8 Braille]].

== See also ==
* [[Abuse of notation]]
* [[Begriffsschrift]]
* [[Glossary of mathematical symbols]]
** [[Bourbaki dangerous bend symbol]]
* [[History of mathematical notation]]
* [[ISO 31-11]]
* [[ISO/IEC 80000#Part 2: Mathematics|ISO 80000-2]]
* [[Knuth's up-arrow notation]]
* [[Mathematical Alphanumeric Symbols]]
* [[Mathematical formula]]
* [[Notation in probability and statistics]]
* [[Language of mathematics]]
* [[Scientific notation]]
* [[Semasiography]]
* [[Table of mathematical symbols]]
* [[Vector notation]]
* [[Modern Arabic mathematical notation]]

== References ==
{{reflist|refs=
<ref name="Ifrah_2000">{{cite book |author-last=Ifrah |author-first=Georges |author-link=Georges Ifrah |title=The Universal History of Numbers: From prehistory to the invention of the computer. |language=en |publisher=[[John Wiley and Sons]] |date=2000 |page=48 |isbn=0-471-39340-1 |translator-first1=David |translator-last1=Bellos |translator-first2=E. F. |translator-last2=Harding |translator-first3=Sophie |translator-last3=Wood |translator-first4=Ian |translator-last4=Monk}} (NB. Ifrah supports his thesis by quoting idiomatic phrases from languages across the entire world. He notes that humans learned to count on their hands. He shows, for example, a picture of [[Boethius]] (who lived 480–524 or 525) reckoning on his fingers.)</ref>
<ref name="Boyer-Merzbach_1991">{{cite book |author-last1=Boyer |author-first1=Carl Benjamin |author-link1=Carl Benjamin Boyer |author-last2=Merzbach |author-first2=Uta C. |author-link2=Uta Merzbach |title=A History of Mathematics |date=1991 |publisher=[[John Wiley & Sons]] |isbn=978-0-471-54397-8 |pages=442&ndash;443 |url=https://archive.org/details/historyofmathema00boye/page/442}}</ref>
<ref name="Eves_1990">{{cite book |author-last=Eves |author-first=Howard |author-link=Howard Eves |title=An Introduction to the History of Mathematics |date=1990 |edition=6 |isbn=978-0-03-029558-4 |page=9|publisher=Saunders College Pub. }}</ref>
<ref name="Arndt-Haenel_2006">{{cite book |author-last1=Arndt |author-first1=Jörg |author-last2=Haenel |author-first2=Christoph |title=Pi Unleashed |publisher=[[Springer-Verlag]] |date=2006 |isbn=978-3-540-66572-4 |page=166 |url=https://books.google.com/books?id=QwwcmweJCDQC&pg=PA166}}</ref>
}}

== Further reading ==
* [[Florian Cajori]], [https://books.google.com/books?id=7juWmvQSTvwC ''A History of Mathematical Notations''] (1929), 2 volumes. {{isbn|0-486-67766-4}}
* Mazur, Joseph (2014), [https://books.google.com/books?id=YZLzjwEACAAJ&q=enlightening+symbols ''Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers'']. Princeton, New Jersey: Princeton University Press. {{isbn|978-0-691-15463-3}}

== External links ==
{{Commons category}}
* [http://jeff560.tripod.com/mathsym.html Earliest Uses of Various Mathematical Symbols]
* [http://www.apronus.com/math/mrwmath.htm Mathematical ASCII Notation] how to type math notation in any text editor.
* [http://www.cut-the-knot.org/language/index.shtml Mathematics as a Language] at [[Alexander Bogomolny#Cut-the-Knot|Cut-the-Knot]]
* [[Stephen Wolfram]]: [http://www.stephenwolfram.com/publications/mathematical-notation-past-future/ Mathematical Notation: Past and Future]. October 2000. Transcript of a keynote address presented at [[MathML]] and Math on the Web: MathML International Conference.

{{Mathematical symbols notation language}}

{{DEFAULTSORT:Mathematical Notation}}

[[Category:Mathematical notation| ]]
[[Category:16th-century inventions]]