View source for Mathematical notation

{{Short description|System of symbolic representation}}
{{For|information on rendering mathematical formulae|Help:Displaying a formula|Wikipedia:Manual of Style/Mathematics}}
{{More citations needed|date=June 2022}}
{{Use dmy dates|date=November 2023|cs1-dates=y}}
{{Use list-defined references|date=November 2023}}
[[File:Highlighted LaTeX example.webp|thumb|Highlighted [[LaTeX|{{stylized LaTeX}}]] mathematical notation]]

'''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 and typeface ==
{{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.

=== International standard mathematical notation ===
The [[International Organization for Standardization]] (ISO) is an [[international standard]] development organization composed of representatives from the national [[Standards organization|standards organizations]] of member countries. 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).

== Expressions and formulas ==
An expression is a written arrangement of [[Symbol (mathematics)|symbols]] following the context-dependent, [[Syntax (logic)|syntactic]] conventions of mathematical notation. Symbols can denote [[numbers]], [[Variable (mathematics)|variables]], [[Operation (mathematics)|operations]], and [[Function (mathematics)|functions]].<ref>[[Oxford English Dictionary]], s.v. “[[doi:10.1093/OED/4555505636|Expression (n.), sense II.7]],” "''A group of symbols which together represent a numeric, algebraic, or other mathematical quantity or function.''"</ref> Other symbols include [[punctuation]] marks and [[Bracket (mathematics)|brackets]], used for [[Symbols of grouping|grouping]] where there is not a well-defined [[order of operations]].

Expressions are commonly distinguished from ''[[Mathematical formula|formulas]]'': expressions are a kind of [[mathematical object]], whereas formulas are statements ''about'' mathematical objects.<ref>{{cite book |last=Stoll |first=Robert R. |title=Set Theory and Logic |publisher=Dover Publications |year=1963 |isbn=978-0-486-63829-4 |location=San Francisco, CA}}</ref> This is analogous to [[natural language]], where a [[noun phrase]] refers to an object, and a whole [[Sentence (linguistics)|sentence]] refers to a [[fact]]. For example, <math>8x-5</math> is an expression, while the [[Inequality (mathematics)|inequality]] <math>8x-5 \geq 3 </math> is a formula.

To ''evaluate'' an expression means to find a numerical [[Value (mathematics)|value]] equivalent to the expression.<ref>[[Oxford English Dictionary]], s.v. "[[doi:10.1093/OED/3423541985|Evaluate (v.), sense a]]", "''Mathematics. To work out the ‘value’ of (a quantitative expression); to find a numerical expression for (any quantitative fact or relation).''"</ref><ref>[[Oxford English Dictionary]], s.v. “[[doi:10.1093/OED/1018661347|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.''"</ref> Expressions can be ''evaluated'' or ''simplified'' by replacing [[Operation (mathematics)|operations]] that appear in them with their result. For example, the expression <math>8\times 2-5</math> simplifies to <math>16-5</math>, and evaluates to <math>11.</math>

== 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.

== 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]].

== Meaning and interpretation ==
The [[syntax]] of notation defines how symbols can be combined to make [[Expression (mathematics)#Well-defined expressions|well-formed expressions]], without any given meaning or interpretation. The [[semantics]] 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 [[Logic translation|translation]]. 

=== Interpretation ===
Given a [[formal language]], an [[Interpretation (logic)|interpretation]] assigns a [[domain of discourse]] 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]] describes the relationship between an object and the representation of that object, such as the [[Earth]] 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 ==
{{See also|Typesetting|Comparison of TeX editors|Mathematical markup language}}
{{col-begin|width=70%}}
{{col-2}}
* [[AUCTeX]]
* [[Authorea]]
* [[Apache OpenOffice#Components|Apache OpenOffice Math]]  
* [[AsciiMath]]  
* [[Calligra Words#Formula editor|Calligra Words - Formula editor]]
* [[CoCalc]]
* [[GNOME LaTeX]]
* [[GNU TeXmacs]]  
* [[Gummi (software)|Gummi]]
* [[KaTeX]]  
* [[Kile]]
* [[LaTeX]]  
* [[LibreOffice Math#Included applications in LibreOffice|LibreOffice Math]]  
* [[LyX]]
* [[MacTeX]]
* [[MathJax]]  
* [[MathML]]  
{{col-2}}
* [[MathType]]
* [[Notepad++]]
* [[Overleaf]]
* [[Scientific WorkPlace]]
* [[TeX]]
* [[TeX Live]]
* [[Texmaker]]
* [[TeXnicCenter]]
* [[TeXShop]]
* [[TeXstudio]]
* [[TeXworks]]
* [[Verbosus]]
* [[Vim (text editor)|Vim]]
* [[Visual Studio Code]] - [https://github.com/James-Yu/latex-workshop/wiki LaTeX Workshop] 
* [[WinEdt]]
* [[WinFIG]]
* [[WinShell]]
{{col-end}}

== See also ==
* [[Abuse of notation]]
* [[Chemistry notation]]
* [[Denotation]]
* [[Knuth's up-arrow notation]]
* [[Language of mathematics]]
* [[List of open-source software for mathematics]]
* [[Mathematical Alphanumeric Symbols]]
* [[Modern Arabic mathematical notation]]
* [[Notation in probability and statistics]]
* [[Principle of compositionality]]
* [[Scientific notation]]
* [[Semasiography]]
* [[Syntactic sugar]]
* [[Vector 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]], ''[[A History of Mathematical Notations]]'' (1929), [https://archive.org/details/b29980343_0001/page/32/mode/2up Vol. 1], [https://archive.org/details/b29980343_0002/page/n3/mode/2up Vol. 2]. (Dover reprint 2011, {{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]]