Old page wikitext, before the edit (old_wikitext) | '{{Short description|System of symbolic representation}}
{{More citations needed|date=June 2022}}
{{For|information on rendering mathematical formulae|Help:Displaying a formula|Wikipedia:Manual of Style/Mathematics}}
{{Use dmy dates|date=November 2023|cs1-dates=y}}
{{Use list-defined references|date=November 2023}}
'''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 |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]].
==Symbolsof not being a member ==
{{Main|Glossary of mathematical symbols}}
The use of many symbols is the basis of mathematical notation. SdfhshshshsdfshsjsgsjajscjsksvsjsskhshsksbssshskgsajsbzfeuqpqdwuehaakavsvbqwertyuiopasdfghjklzxcvbnmThey 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==
{{Unsourced section|date=June 2022}}
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]], [[Einstein's 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., ''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=442–443 |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}}</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]]' |
New page wikitext, after the edit (new_wikitext) | '<nowiki>{qwertyuiopasdfghjklzxcvbnmabcdefghijklmnopqrstuv{Short description|System of symbolic representation}}blah blah</nowiki>
{{More citations needed|date=June 2022}}
{{For|information on rendering mathematical formulae|Help:Displaying a formula|Wikipedia:Manual of Style/Mathematics}}
{{Use dmy dates|date=November 2023|cs1-dates=y}}
{{Use list-defined references|date=November 2023}}
'''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 |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]].
==Symbolsof not being a member ==
{{Main|Glossary of mathematical symbols}}
The use of many symbols is the basis of mathematical notation. SdfhshshshsdfshsjsgsjajscjsksvsjsskhshsksbssshskgsajsbzfeuqpqdwuehaakavsvbqwertyuiopasdfghjklzxcvbnmThey 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==
{{Unsourced section|date=June 2022}}
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]], [[Einstein's 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., ''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=442–443 |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}}</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]]' |
Parsed HTML source of the new revision (new_html) | '<div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>{qwertyuiopasdfghjklzxcvbnmabcdefghijklmnopqrstuv{Short description|System of symbolic representation}}blah blah
</p>
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<style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For information on rendering mathematical formulae, see <a href="/enwiki/wiki/Help:Displaying_a_formula" title="Help:Displaying a formula">Help:Displaying a formula</a> and <a href="/enwiki/wiki/Wikipedia:Manual_of_Style/Mathematics" title="Wikipedia:Manual of Style/Mathematics">Wikipedia:Manual of Style/Mathematics</a>.</div>
<p>
<b>Mathematical notation</b> consists of using <a href="/enwiki/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">symbols</a> for representing <a href="/enwiki/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a>, unspecified <a href="/enwiki/wiki/Number" title="Number">numbers</a>, <a href="/enwiki/wiki/Relation_(mathematics)" title="Relation (mathematics)">relations</a>, and any other <a href="/enwiki/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> and assembling them into <a href="/enwiki/wiki/Expression_(mathematics)" title="Expression (mathematics)">expressions</a> and <a href="/enwiki/wiki/Formula" title="Formula">formulas</a>. Mathematical notation is widely used in <a href="/enwiki/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/enwiki/wiki/Science" title="Science">science</a>, and <a href="/enwiki/wiki/Engineering" title="Engineering">engineering</a> for representing complex <a href="/enwiki/wiki/Concept" title="Concept">concepts</a> and <a href="/enwiki/wiki/Property_(philosophy)" title="Property (philosophy)">properties</a> in a concise, unambiguous, and accurate way.
</p><p>For example, the physicist <a href="/enwiki/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=mc^{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>E</mi>
<mo>=</mo>
<mi>m</mi>
<msup>
<mi>c</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
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<annotation encoding="application/x-tex">{\displaystyle E=mc^{2}}</annotation>
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</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/9f73dbd37a0cac34406ee89057fa1b36a1e6a18e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.976ex; height:2.676ex;" alt="{\displaystyle E=mc^{2}}"></span> is the quantitative representation in mathematical notation of <a href="/enwiki/wiki/Mass%E2%80%93energy_equivalence" title="Mass–energy equivalence">mass–energy equivalence</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>Mathematical notation was first introduced by <a href="/enwiki/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a> at the end of the 16th century and largely expanded during the 17th and 18th centuries by <a href="/enwiki/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a>, <a href="/enwiki/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>, <a href="/enwiki/wiki/Gottfried_Wilhelm_Leibniz" title="Gottfried Wilhelm Leibniz">Gottfried Wilhelm Leibniz</a>, and overall <a href="/enwiki/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a>.
</p>
<div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="en" dir="ltr"><h2 id="mw-toc-heading">Contents</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div>
<ul>
<li class="toclevel-1 tocsection-1"><a href="#Symbolsof_not_being_a_member"><span class="tocnumber">1</span> <span class="toctext">Symbolsof not being a member</span></a>
<ul>
<li class="toclevel-2 tocsection-2"><a href="#Letters_as_symbols"><span class="tocnumber">1.1</span> <span class="toctext">Letters as symbols</span></a></li>
<li class="toclevel-2 tocsection-3"><a href="#Other_symbols"><span class="tocnumber">1.2</span> <span class="toctext">Other symbols</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-4"><a href="#Expressions"><span class="tocnumber">2</span> <span class="toctext">Expressions</span></a></li>
<li class="toclevel-1 tocsection-5"><a href="#History"><span class="tocnumber">3</span> <span class="toctext">History</span></a>
<ul>
<li class="toclevel-2 tocsection-6"><a href="#Numbers"><span class="tocnumber">3.1</span> <span class="toctext">Numbers</span></a></li>
<li class="toclevel-2 tocsection-7"><a href="#Modern_notation"><span class="tocnumber">3.2</span> <span class="toctext">Modern notation</span></a></li>
<li class="toclevel-2 tocsection-8"><a href="#Typesetting"><span class="tocnumber">3.3</span> <span class="toctext">Typesetting</span></a></li>
</ul>
</li>
<li class="toclevel-1 tocsection-9"><a href="#International_standard_mathematical_notation"><span class="tocnumber">4</span> <span class="toctext">International standard mathematical notation</span></a></li>
<li class="toclevel-1 tocsection-10"><a href="#Non-Latin-based_mathematical_notation"><span class="tocnumber">5</span> <span class="toctext">Non-Latin-based mathematical notation</span></a></li>
<li class="toclevel-1 tocsection-11"><a href="#See_also"><span class="tocnumber">6</span> <span class="toctext">See also</span></a></li>
<li class="toclevel-1 tocsection-12"><a href="#References"><span class="tocnumber">7</span> <span class="toctext">References</span></a></li>
<li class="toclevel-1 tocsection-13"><a href="#Further_reading"><span class="tocnumber">8</span> <span class="toctext">Further reading</span></a></li>
<li class="toclevel-1 tocsection-14"><a href="#External_links"><span class="tocnumber">9</span> <span class="toctext">External links</span></a></li>
</ul>
</div>
<div class="mw-heading mw-heading2"><h2 id="Symbolsof_not_being_a_member">Symbolsof not being a member</h2><span class="mw-editsection">
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href="/enwiki/w/index.php?title=Mathematical_notation&action=edit&section=1"title="Edit section: Symbolsof not being a member"
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<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/enwiki/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary of mathematical symbols</a></div>
<p>The use of many symbols is the basis of mathematical notation. SdfhshshshsdfshsjsgsjajscjsksvsjsskhshsksbssshskgsajsbzfeuqpqdwuehaakavsvbqwertyuiopasdfghjklzxcvbnmThey play a similar role as words in <a href="/enwiki/wiki/Natural_language" title="Natural language">natural languages</a>. They may play different roles in mathematical notation similarly as verbs, adjective and nouns play different roles in a sentence.
</p>
<div class="mw-heading mw-heading3"><h3 id="Letters_as_symbols">Letters as symbols</h3><span class="mw-editsection">
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href="/enwiki/w/index.php?title=Mathematical_notation&action=edit&section=2"title="Edit section: Letters as symbols"
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<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/enwiki/wiki/List_of_letters_used_in_mathematics,_science,_and_engineering" title="List of letters used in mathematics, science, and engineering">List of letters used in mathematics, science, and engineering</a></div>
<p>Letters are typically used for naming—in <a href="/enwiki/wiki/List_of_mathematical_jargon" class="mw-redirect" title="List of mathematical jargon">mathematical jargon</a>, one says <i>representing</i>—<a href="/enwiki/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a>. The <a href="/enwiki/wiki/Latin_alphabet" title="Latin alphabet">Latin</a> and <a href="/enwiki/wiki/Greek_alphabet" title="Greek alphabet">Greek</a> alphabets are used extensively, but a few letters of other alphabets are also used sporadically, such as the <a href="/enwiki/wiki/Hebrew_alphabet" title="Hebrew alphabet">Hebrew</a> <span class="nowrap">⁠<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \aleph }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">ℵ<!-- ℵ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \aleph }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/306c55e6bc96d94db729ff5821c8f45a34c72bce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \aleph }"></span>⁠</span>, <a href="/enwiki/wiki/Cyrillic_script" title="Cyrillic script">Cyrillic</a> <span class="texhtml">Ш</span>, and <a href="/enwiki/wiki/Hiragana" title="Hiragana">Hiragana</a> <span class="texhtml">よ</span>. Uppercase and lowercase letters are considered as different symbols. For Latin alphabet, different typefaces also provide different symbols. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r,R,\mathbb {R} ,{\mathcal {R}},{\mathfrak {r}},}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo>,</mo>
<mi>R</mi>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
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<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-tex-caligraphic" mathvariant="script">R</mi>
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<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">r</mi>
</mrow>
</mrow>
<mo>,</mo>
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</mrow>
<annotation encoding="application/x-tex">{\displaystyle r,R,\mathbb {R} ,{\mathcal {R}},{\mathfrak {r}},}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/6628084b4eca01231c8b54ea09de51eb22591411" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.149ex; height:2.509ex;" alt="{\displaystyle r,R,\mathbb {R} ,{\mathcal {R}},{\mathfrak {r}},}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {R}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="fraktur">R</mi>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {R}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/b5d31f64c0e02f0dc73d5aeb636a1caf2011dce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {R}}}"></span> 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 "<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>sin</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sin }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/ee55beec18afd710e7ab767964b915b020c65093" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.856ex; height:2.176ex;" alt="{\displaystyle \sin }"></span>" of the <a href="/enwiki/wiki/Sine_function" class="mw-redirect" title="Sine function">sine function</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p><p>In order to have more symbols, and for allowing related mathematical objects to be represented by related symbols, <a href="/enwiki/wiki/Diacritic" title="Diacritic">diacritics</a>, <a href="/enwiki/wiki/Subscript" class="mw-redirect" title="Subscript">subscripts</a> and <a href="/enwiki/wiki/Superscript" class="mw-redirect" title="Superscript">superscripts</a> are often used. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {f'_{1}}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<msubsup>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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<mo>′</mo>
</msubsup>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\hat {f'_{1}}}}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/ca08b9d46d5a855ce0a075bc9372a4d13ac992cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.193ex; height:3.509ex;" alt="{\displaystyle {\hat {f'_{1}}}}"></span> may denote the <a href="/enwiki/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> of the <a href="/enwiki/wiki/Derivative" title="Derivative">derivative</a> of a <a href="/enwiki/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> called <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>.</mo>
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<annotation encoding="application/x-tex">{\displaystyle f_{1}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/15589063e81f3e5fa2699321756b83cab936bb91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.84ex; height:2.509ex;" alt="{\displaystyle f_{1}.}"></span>
</p>
<div class="mw-heading mw-heading3"><h3 id="Other_symbols">Other symbols</h3><span class="mw-editsection">
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<p>Symbols are not only used for naming mathematical objects. They can be used for <a href="/enwiki/wiki/Operation_(mathematics)" title="Operation (mathematics)">operations</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (+,-,/,\oplus ,\ldots ),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mo>+</mo>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mo>,</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mo>,</mo>
<mo>⊕<!-- ⊕ --></mo>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (+,-,/,\oplus ,\ldots ),}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/1838a740b7cd4f8586f00a1520699dcc093b5d0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.902ex; height:2.843ex;" alt="{\displaystyle (+,-,/,\oplus ,\ldots ),}"></span> for <a href="/enwiki/wiki/Relation_(mathematics)" title="Relation (mathematics)">relations</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (=,<,\leq ,\sim ,\equiv ,\ldots ),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mo>=</mo>
<mo>,</mo>
<mo><</mo>
<mo>,</mo>
<mo>≤<!-- ≤ --></mo>
<mo>,</mo>
<mo>∼<!-- ∼ --></mo>
<mo>,</mo>
<mo>≡<!-- ≡ --></mo>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (=,<,\leq ,\sim ,\equiv ,\ldots ),}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/7301a985a49e8c74b68584afefe74722a8f87d1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.39ex; height:2.843ex;" alt="{\displaystyle (=,<,\leq ,\sim ,\equiv ,\ldots ),}"></span> for <a href="/enwiki/wiki/Logical_connective" title="Logical connective">logical connectives</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\implies ,\land ,\lor ,\ldots ),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mspace width="thickmathspace" />
<mo stretchy="false">⟹<!-- ⟹ --></mo>
<mspace width="thickmathspace" />
<mo>,</mo>
<mo>∧<!-- ∧ --></mo>
<mo>,</mo>
<mo>∨<!-- ∨ --></mo>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (\implies ,\land ,\lor ,\ldots ),}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/997ff31b3b9371a8becedb25fb044f7bc6120838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.478ex; height:2.843ex;" alt="{\displaystyle (\implies ,\land ,\lor ,\ldots ),}"></span> for <a href="/enwiki/wiki/Quantifier_(logic)" title="Quantifier (logic)">quantifiers</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\forall ,\exists ),}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mo>,</mo>
<mi mathvariant="normal">∃<!-- ∃ --></mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
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<annotation encoding="application/x-tex">{\displaystyle (\forall ,\exists ),}</annotation>
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</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/0a07b94cb636c83b580262f3b01af09b31eed3a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.075ex; height:2.843ex;" alt="{\displaystyle (\forall ,\exists ),}"></span> and for other purposes.
</p><p>Some symbols are similar to Latin or Greek letters, some are obtained by deforming letters, some are traditional <a href="/enwiki/wiki/Typographic_symbol" class="mw-redirect" title="Typographic symbol">typographic symbols</a>, but many have been specially designed for mathematics.
</p>
<div class="mw-heading mw-heading2"><h2 id="Expressions">Expressions</h2><span class="mw-editsection">
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<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236091366"><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/enwiki/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="/upwiki/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="/upwiki/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, /upwiki/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>does not <a href="/enwiki/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/enwiki/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a></b>.<span class="hide-when-compact"> Please help <a href="/enwiki/wiki/Special:EditPage/Mathematical_notation" title="Special:EditPage/Mathematical notation">improve this section</a> by <a href="/enwiki/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/enwiki/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>.</span> <span class="date-container"><i>(<span class="date">June 2022</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/enwiki/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table>
<p>An <a href="/enwiki/wiki/Expression_(mathematics)" title="Expression (mathematics)">expression</a> is a finite combination of <a href="/enwiki/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">symbols</a> that is <a href="/enwiki/wiki/Well-formed_formula" title="Well-formed formula">well-formed</a> according to rules that depend on the context. In general, an expression denotes or names a <a href="/enwiki/wiki/Mathematical_object" title="Mathematical object">mathematical object</a>, and plays therefore in the <a href="/enwiki/wiki/Language_of_mathematics" title="Language of mathematics">language of mathematics</a> the role of a <a href="/enwiki/wiki/Noun_phrase" title="Noun phrase">noun phrase</a> in the natural language.
</p><p>An expression contains often some <a href="/enwiki/wiki/Operator_(mathematics)" title="Operator (mathematics)">operators</a>, and may therefore be <i>evaluated</i> by the action of the operators in it. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3+2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>3</mn>
<mo>+</mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 3+2}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/a82be7d97da38d489ec230d0b3217453e8400e36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.165ex; height:2.343ex;" alt="{\displaystyle 3+2}"></span> is an expression in which the operator <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>+</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle +}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}"></span> can be evaluated for giving the result <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>5.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 5.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/62cfda82378d02d6ff65a09e66873314c7013888" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 5.}"></span> So, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3+2}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>3</mn>
<mo>+</mo>
<mn>2</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 3+2}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/a82be7d97da38d489ec230d0b3217453e8400e36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.165ex; height:2.343ex;" alt="{\displaystyle 3+2}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>5</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 5}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}"></span> are two different expressions that represent the same number. This is the meaning of the equality <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3+2=5.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>3</mn>
<mo>+</mo>
<mn>2</mn>
<mo>=</mo>
<mn>5.</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 3+2=5.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/fc4011f9c8012f01a64b37e67426926efd6a9790" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.073ex; height:2.343ex;" alt="{\displaystyle 3+2=5.}"></span>
</p><p>A more complicated example is given by the expression<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int _{a}^{b}xdx}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>a</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>b</mi>
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</msubsup>
<mi>x</mi>
<mi>d</mi>
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \int _{a}^{b}xdx}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/dbca8c9bb5a7c1b8b1fa995605b9e0852ffdde22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.714ex; height:3.676ex;" alt="{\textstyle \int _{a}^{b}xdx}"></span> that can be evaluated to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {b^{2}}{2}}-{\frac {a^{2}}{2}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>b</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mn>2</mn>
</mfrac>
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<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msup>
<mn>2</mn>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle {\frac {b^{2}}{2}}-{\frac {a^{2}}{2}}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/e31d9aed0d9638bb60a893b6253874b9b13389d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.398ex; height:4.009ex;" alt="{\textstyle {\frac {b^{2}}{2}}-{\frac {a^{2}}{2}}.}"></span> Although the resulting expression contains the operators of <a href="/enwiki/wiki/Division_(mathematics)" title="Division (mathematics)">division</a>, <a href="/enwiki/wiki/Subtraction" title="Subtraction">subtraction</a> and <a href="/enwiki/wiki/Exponentiation" title="Exponentiation">exponentiation</a>, it cannot be evaluated further because <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> denote unspecified numbers.
</p>
<div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection">
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<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/enwiki/wiki/History_of_mathematical_notation" title="History of mathematical notation">History of mathematical notation</a></div>
<div class="mw-heading mw-heading3"><h3 id="Numbers">Numbers</h3><span class="mw-editsection">
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<p>It is believed that a notation to represent <a href="/enwiki/wiki/Number" title="Number">numbers</a> was first developed at least 50,000 years ago<sup id="cite_ref-Eves_1990_3-0" class="reference"><a href="#cite_note-Eves_1990-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>—early mathematical ideas such as <a href="/enwiki/wiki/Finger_counting" class="mw-redirect" title="Finger counting">finger counting</a><sup id="cite_ref-Ifrah_2000_4-0" class="reference"><a href="#cite_note-Ifrah_2000-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> have also been represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The <a href="/enwiki/wiki/Tally_stick" title="Tally stick">tally stick</a> is a way of counting dating back to the <a href="/enwiki/wiki/Upper_Paleolithic" title="Upper Paleolithic">Upper Paleolithic</a>. Perhaps the oldest known mathematical texts are those of ancient <a href="/enwiki/wiki/Sumer" title="Sumer">Sumer</a>. The <a href="/enwiki/wiki/Census_quipu" class="mw-redirect" title="Census quipu">Census Quipu</a> of the Andes and the <a href="/enwiki/wiki/Ishango_Bone" class="mw-redirect" title="Ishango Bone">Ishango Bone</a> from Africa both used the <a href="/enwiki/wiki/Tally_mark" class="mw-redirect" title="Tally mark">tally mark</a> method of accounting for numerical concepts.
</p><p>The concept of <a href="/enwiki/wiki/Zero" class="mw-redirect" title="Zero">zero</a> 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 <a href="/enwiki/wiki/Babylonian_numerals" class="mw-redirect" title="Babylonian numerals">Babylonians</a> and <a href="/enwiki/wiki/Greek_numerals" title="Greek numerals">Greek Egyptians</a>, and then as an <a href="/enwiki/wiki/Integer" title="Integer">integer</a> by the <a href="/enwiki/wiki/Maya_numerals" title="Maya numerals">Mayans</a>, <a href="/enwiki/wiki/Indian_numerals" class="mw-redirect" title="Indian numerals">Indians</a> and <a href="/enwiki/wiki/Arabic_numerals" title="Arabic numerals">Arabs</a> (see <a href="/enwiki/wiki/History_of_zero" class="mw-redirect" title="History of zero">the history of zero</a>).
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<div class="mw-heading mw-heading3"><h3 id="Modern_notation">Modern notation</h3><span class="mw-editsection">
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<p>Until the 16th century, mathematics was essentially <a href="/enwiki/wiki/Rhetorical_algebra" class="mw-redirect" title="Rhetorical algebra">rhetorical</a>, in the sense that everything but explicit numbers was expressed in words. However, some authors such as <a href="/enwiki/wiki/Diophantus" title="Diophantus">Diophantus</a> used some symbols as abbreviations.
</p><p>The first systematic use of formulas, and, in particular the use of symbols (<a href="/enwiki/wiki/Variable_(mathematics)" title="Variable (mathematics)">variables</a>) for unspecified numbers is generally attributed to <a href="/enwiki/wiki/Fran%C3%A7ois_Vi%C3%A8te" title="François Viète">François Viète</a> (16th century). However, he used different symbols than those that are now standard.
</p><p>Later, <a href="/enwiki/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> (17th century) introduced the modern notation for variables and <a href="/enwiki/wiki/Equation" title="Equation">equations</a>; in particular, the use of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo>,</mo>
<mi>z</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x,y,z}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/bbeca34b28f569a407ef74a955d041df9f360268" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.641ex; height:2.009ex;" alt="{\displaystyle x,y,z}"></span> for <a href="/enwiki/wiki/Unknown_(mathematics)" class="mw-redirect" title="Unknown (mathematics)">unknown</a> quantities and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b,c}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>,</mo>
<mi>b</mi>
<mo>,</mo>
<mi>c</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a,b,c}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/f13f068df656c1b1911ae9f81628c49a6181194d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.302ex; height:2.509ex;" alt="{\displaystyle a,b,c}"></span> for known ones (<a href="/enwiki/wiki/Constant_(mathematics)" title="Constant (mathematics)">constants</a>). He introduced also the notation <span class="texhtml mvar" style="font-style:italic;">i</span> and the term "imaginary" for the <a href="/enwiki/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a>.
</p><p>The 18th and 19th centuries saw the standardization of mathematical notation as used today. <a href="/enwiki/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> was responsible for many of the notations currently in use: the <a href="/enwiki/wiki/Functional_notation" class="mw-redirect" title="Functional notation">functional notation</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f(x),}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/10535d1a7a971ffeeb216605cb846099fab2e653" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.064ex; height:2.843ex;" alt="{\displaystyle f(x),}"></span> <span class="texhtml"><i>e</i></span> for the base of the natural logarithm, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \sum }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mo>∑<!-- ∑ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\textstyle \sum }</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/8e2b0b7618be940f4e8c0d27f05ab75fbc13e83c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.454ex; height:2.843ex;" alt="{\textstyle \sum }"></span> for <a href="/enwiki/wiki/Summation" title="Summation">summation</a>, etc.<sup id="cite_ref-Boyer-Merzbach_1991_5-0" class="reference"><a href="#cite_note-Boyer-Merzbach_1991-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> He also popularized the use of <span class="texhtml mvar" style="font-style:italic;">π</span> for the <a href="/enwiki/wiki/Archimedes_constant" class="mw-redirect" title="Archimedes constant">Archimedes constant</a> (proposed by <a href="/enwiki/wiki/William_Jones_(mathematician)" title="William Jones (mathematician)">William Jones</a>, based on an earlier notation of <a href="/enwiki/wiki/William_Oughtred" title="William Oughtred">William Oughtred</a>).<sup id="cite_ref-Arndt-Haenel_2006_6-0" class="reference"><a href="#cite_note-Arndt-Haenel_2006-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>
</p><p>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 <a href="/enwiki/wiki/Leibniz%27s_notation" title="Leibniz's notation">Leibniz's notation</a>, <a href="/enwiki/wiki/Legendre_symbol" title="Legendre symbol">Legendre symbol</a>, <a href="/enwiki/wiki/Einstein%27s_summation_convention" class="mw-redirect" title="Einstein's summation convention">Einstein's summation convention</a>, etc.
</p>
<div class="mw-heading mw-heading3"><h3 id="Typesetting">Typesetting</h3><span class="mw-editsection">
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<p>General <a href="/enwiki/wiki/Typesetting_system" class="mw-redirect" title="Typesetting system">typesetting systems</a> 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:
</p>
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n=0}^{\infty }{\frac {{\begin{bmatrix}a&b\\c&d\end{bmatrix}}^{n}}{n!}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">∞<!-- ∞ --></mi>
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</munderover>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>[</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mi>d</mi>
</mtd>
</mtr>
</mtable>
<mo>]</mo>
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mrow>
<mi>n</mi>
<mo>!</mo>
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</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {{\begin{bmatrix}a&b\\c&d\end{bmatrix}}^{n}}{n!}}.}</annotation>
</semantics>
</math></span><img src="https://wikimedia.org/enwiki/api/rest_v1/media/math/render/svg/00f05c2431187190062e84cae7cc75ac50b68ee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.418ex; height:10.176ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {{\begin{bmatrix}a&b\\c&d\end{bmatrix}}^{n}}{n!}}.}"></span></dd></dl>
<p><a href="/enwiki/wiki/TeX" title="TeX">TeX</a> is a mathematically oriented typesetting system that was created in 1978 by <a href="/enwiki/wiki/Donald_Knuth" title="Donald Knuth">Donald Knuth</a>. It is widely used in mathematics, through its extension called <a href="/enwiki/wiki/LaTeX" title="LaTeX">LaTeX</a>, and is a <i>de facto</i> standard. (The above expression is written in LaTeX.)
</p><p>More recently, another approach for mathematical typesetting is provided by <a href="/enwiki/wiki/MathML" title="MathML">MathML</a>. However, it is not well supported in web browsers, which is its primary target.
</p>
<div class="mw-heading mw-heading2"><h2 id="International_standard_mathematical_notation">International standard mathematical notation</h2><span class="mw-editsection">
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<p>The international standard <a href="/enwiki/wiki/ISO_80000-2" class="mw-redirect" title="ISO 80000-2">ISO 80000-2</a> (previously, <a href="/enwiki/wiki/ISO_31-11" title="ISO 31-11">ISO 31-11</a>) specifies symbols for use in mathematical equations. The standard requires use of italic fonts for variables (e.g., <i>E</i>=<i>mc</i><sup>2</sup>) and roman (upright) fonts for mathematical constants (e.g., e or π).
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<div class="mw-heading mw-heading2"><h2 id="Non-Latin-based_mathematical_notation">Non-Latin-based mathematical notation</h2><span class="mw-editsection">
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<p><a href="/enwiki/wiki/Modern_Arabic_mathematical_notation" title="Modern Arabic mathematical notation">Modern Arabic mathematical notation</a> is based mostly on the <a href="/enwiki/wiki/Arabic_alphabet" title="Arabic alphabet">Arabic alphabet</a> and is used widely in the <a href="/enwiki/wiki/Arab_world" title="Arab world">Arab world</a>, especially in pre-<a href="/enwiki/wiki/Tertiary_education" title="Tertiary education">tertiary education</a>. (Western notation uses <a href="/enwiki/wiki/Arabic_numerals" title="Arabic numerals">Arabic numerals</a>, but the Arabic notation also replaces Latin letters and related symbols with Arabic script.)
</p><p>In addition to Arabic notation, mathematics also makes use of <a href="/enwiki/wiki/Greek_alphabet" title="Greek alphabet">Greek letters</a> to denote a wide variety of mathematical objects and variables. On some occasions, certain <a href="/enwiki/wiki/Hebrew_alphabet" title="Hebrew alphabet">Hebrew letters</a> are also used (such as in the context of <a href="/enwiki/wiki/Infinite_cardinal" class="mw-redirect" title="Infinite cardinal">infinite cardinals</a>).
</p><p>Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are <a href="/enwiki/wiki/Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a> and <a href="/enwiki/wiki/Coxeter%E2%80%93Dynkin_diagram" title="Coxeter–Dynkin diagram">Coxeter–Dynkin diagrams</a>.
</p><p>Braille-based mathematical notations used by blind people include <a href="/enwiki/wiki/Nemeth_Braille" title="Nemeth Braille">Nemeth Braille</a> and <a href="/enwiki/wiki/GS8_Braille" class="mw-redirect" title="GS8 Braille">GS8 Braille</a>.
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection">
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<ul><li><a href="/enwiki/wiki/Abuse_of_notation" title="Abuse of notation">Abuse of notation</a></li>
<li><a href="/enwiki/wiki/Begriffsschrift" title="Begriffsschrift">Begriffsschrift</a></li>
<li><a href="/enwiki/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary of mathematical symbols</a>
<ul><li><a href="/enwiki/wiki/Bourbaki_dangerous_bend_symbol" title="Bourbaki dangerous bend symbol">Bourbaki dangerous bend symbol</a></li></ul></li>
<li><a href="/enwiki/wiki/History_of_mathematical_notation" title="History of mathematical notation">History of mathematical notation</a></li>
<li><a href="/enwiki/wiki/ISO_31-11" title="ISO 31-11">ISO 31-11</a></li>
<li><a href="/enwiki/wiki/ISO/IEC_80000#Part_2:_Mathematics" title="ISO/IEC 80000">ISO 80000-2</a></li>
<li><a href="/enwiki/wiki/Knuth%27s_up-arrow_notation" title="Knuth's up-arrow notation">Knuth's up-arrow notation</a></li>
<li><a href="/enwiki/wiki/Mathematical_Alphanumeric_Symbols" title="Mathematical Alphanumeric Symbols">Mathematical Alphanumeric Symbols</a></li>
<li><a href="/enwiki/wiki/Mathematical_formula" class="mw-redirect" title="Mathematical formula">Mathematical formula</a></li>
<li><a href="/enwiki/wiki/Notation_in_probability_and_statistics" title="Notation in probability and statistics">Notation in probability and statistics</a></li>
<li><a href="/enwiki/wiki/Language_of_mathematics" title="Language of mathematics">Language of mathematics</a></li>
<li><a href="/enwiki/wiki/Scientific_notation" title="Scientific notation">Scientific notation</a></li>
<li><a href="/enwiki/wiki/Semasiography" title="Semasiography">Semasiography</a></li>
<li><a href="/enwiki/wiki/Table_of_mathematical_symbols" class="mw-redirect" title="Table of mathematical symbols">Table of mathematical symbols</a></li>
<li><a href="/enwiki/wiki/Vector_notation" title="Vector notation">Vector notation</a></li>
<li><a href="/enwiki/wiki/Modern_Arabic_mathematical_notation" title="Modern Arabic mathematical notation">Modern Arabic mathematical notation</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection">
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<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("/upwiki/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("/upwiki/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("/upwiki/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("/upwiki/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFEinstein1905" class="citation journal cs1 cs1-prop-foreign-lang-source">Einstein, Albert (1905). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/10.1002/andp.19053231314">"Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?"</a>. <i>Annalen der Physik</i> (in German). <b>323</b> (13): 639–641. <a href="/enwiki/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fandp.19053231314">10.1002/andp.19053231314</a>. <a href="/enwiki/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-3804">0003-3804</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annalen+der+Physik&rft.atitle=Ist+die+Tr%C3%A4gheit+eines+K%C3%B6rpers+von+seinem+Energieinhalt+abh%C3%A4ngig%3F&rft.volume=323&rft.issue=13&rft.pages=639-641&rft.date=1905&rft_id=info%3Adoi%2F10.1002%2Fandp.19053231314&rft.issn=0003-3804&rft.aulast=Einstein&rft.aufirst=Albert&rft_id=https%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2F10.1002%2Fandp.19053231314&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+notation" class="Z3988"></span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">ISO 80000-2:2019</span>
</li>
<li id="cite_note-Eves_1990-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Eves_1990_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEves1990" class="citation book cs1"><a href="/enwiki/wiki/Howard_Eves" title="Howard Eves">Eves, Howard</a> (1990). <i>An Introduction to the History of Mathematics</i> (6 ed.). p. 9. <a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/978-0-03-029558-4" title="Special:BookSources/978-0-03-029558-4"><bdi>978-0-03-029558-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+History+of+Mathematics&rft.pages=9&rft.edition=6&rft.date=1990&rft.isbn=978-0-03-029558-4&rft.aulast=Eves&rft.aufirst=Howard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+notation" class="Z3988"></span></span>
</li>
<li id="cite_note-Ifrah_2000-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ifrah_2000_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIfrah2000" class="citation book cs1"><a href="/enwiki/wiki/Georges_Ifrah" title="Georges Ifrah">Ifrah, Georges</a> (2000). <i>The Universal History of Numbers: From prehistory to the invention of the computer</i>. Translated by Bellos, David; Harding, E. F.; Wood, Sophie; Monk, Ian. <a href="/enwiki/wiki/John_Wiley_and_Sons" class="mw-redirect" title="John Wiley and Sons">John Wiley and Sons</a>. p. 48. <a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/0-471-39340-1" title="Special:BookSources/0-471-39340-1"><bdi>0-471-39340-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Universal+History+of+Numbers%3A+From+prehistory+to+the+invention+of+the+computer.&rft.pages=48&rft.pub=John+Wiley+and+Sons&rft.date=2000&rft.isbn=0-471-39340-1&rft.aulast=Ifrah&rft.aufirst=Georges&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+notation" class="Z3988"></span> (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 <a href="/enwiki/wiki/Boethius" title="Boethius">Boethius</a> (who lived 480–524 or 525) reckoning on his fingers.)</span>
</li>
<li id="cite_note-Boyer-Merzbach_1991-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Boyer-Merzbach_1991_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoyerMerzbach1991" class="citation book cs1"><a href="/enwiki/wiki/Carl_Benjamin_Boyer" title="Carl Benjamin Boyer">Boyer, Carl Benjamin</a>; <a href="/enwiki/wiki/Uta_Merzbach" title="Uta Merzbach">Merzbach, Uta C.</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/historyofmathema00boye/page/442"><i>A History of Mathematics</i></a>. <a href="/enwiki/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. pp. 442–443. <a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/978-0-471-54397-8" title="Special:BookSources/978-0-471-54397-8"><bdi>978-0-471-54397-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Mathematics&rft.pages=442-443&rft.pub=John+Wiley+%26+Sons&rft.date=1991&rft.isbn=978-0-471-54397-8&rft.aulast=Boyer&rft.aufirst=Carl+Benjamin&rft.au=Merzbach%2C+Uta+C.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fhistoryofmathema00boye%2Fpage%2F442&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+notation" class="Z3988"></span></span>
</li>
<li id="cite_note-Arndt-Haenel_2006-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Arndt-Haenel_2006_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArndtHaenel2006" class="citation book cs1">Arndt, Jörg; Haenel, Christoph (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QwwcmweJCDQC&pg=PA166"><i>Pi Unleashed</i></a>. <a href="/enwiki/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. p. 166. <a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/978-3-540-66572-4" title="Special:BookSources/978-3-540-66572-4"><bdi>978-3-540-66572-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pi+Unleashed&rft.pages=166&rft.pub=Springer-Verlag&rft.date=2006&rft.isbn=978-3-540-66572-4&rft.aulast=Arndt&rft.aufirst=J%C3%B6rg&rft.au=Haenel%2C+Christoph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQwwcmweJCDQC%26pg%3DPA166&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+notation" class="Z3988"></span></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection">
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<ul><li><a href="/enwiki/wiki/Florian_Cajori" title="Florian Cajori">Florian Cajori</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7juWmvQSTvwC"><i>A History of Mathematical Notations</i></a> (1929), 2 volumes. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/0-486-67766-4" title="Special:BookSources/0-486-67766-4">0-486-67766-4</a></li>
<li>Mazur, Joseph (2014), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YZLzjwEACAAJ&q=enlightening+symbols"><i>Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers</i></a>. Princeton, New Jersey: Princeton University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/enwiki/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/enwiki/wiki/Special:BookSources/978-0-691-15463-3" title="Special:BookSources/978-0-691-15463-3">978-0-691-15463-3</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection">
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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Mathematical_notation" class="extiw" title="commons:Category:Mathematical notation">Mathematical notation</a></span>.</div></div>
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<ul><li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/mathsym.html">Earliest Uses of Various Mathematical Symbols</a></li>
<li><a rel="nofollow" class="external text" href="http://www.apronus.com/math/mrwmath.htm">Mathematical ASCII Notation</a> how to type math notation in any text editor.</li>
<li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/language/index.shtml">Mathematics as a Language</a> at <a href="/enwiki/wiki/Alexander_Bogomolny#Cut-the-Knot" title="Alexander Bogomolny">Cut-the-Knot</a></li>
<li><a href="/enwiki/wiki/Stephen_Wolfram" title="Stephen Wolfram">Stephen Wolfram</a>: <a rel="nofollow" class="external text" href="http://www.stephenwolfram.com/publications/mathematical-notation-past-future/">Mathematical Notation: Past and Future</a>. October 2000. Transcript of a keynote address presented at <a href="/enwiki/wiki/MathML" title="MathML">MathML</a> and Math on the Web: MathML International Conference.</li></ul>
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id="Common_mathematical_notation,_symbols,_and_formulas" style="font-size:114%;margin:0 4em">Common <a href="/enwiki/wiki/Mathematics" title="Mathematics">mathematical</a> <a class="mw-selflink selflink">notation</a>, <a href="/enwiki/wiki/List_of_mathematical_symbols_by_subject" class="mw-redirect" title="List of mathematical symbols by subject">symbols</a>, and <a href="/enwiki/wiki/Help:Displaying_a_formula" title="Help:Displaying a formula">formulas</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible expanded navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Lists_of_Unicode_and_LaTeX_mathematical_symbols" style="font-size:114%;margin:0 4em">Lists of <a href="/enwiki/wiki/Unicode" title="Unicode">Unicode</a> and <a href="/enwiki/wiki/LaTeX" title="LaTeX">LaTeX</a> mathematical symbols</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/List_of_mathematical_symbols_by_subject" class="mw-redirect" title="List of mathematical symbols by subject">List of mathematical symbols by subject</a></li>
<li><a href="/enwiki/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary of mathematical symbols</a></li>
<li><a href="/enwiki/wiki/List_of_logic_symbols" title="List of logic symbols">List of logic symbols</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible expanded navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Lists_of_Unicode_symbols" style="font-size:114%;margin:0 4em">Lists of <a href="/enwiki/wiki/Unicode" title="Unicode">Unicode</a> symbols</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/List_of_Unicode_characters" title="List of Unicode characters">List of Unicode characters</a></li>
<li><a href="/enwiki/wiki/Unicode_block" title="Unicode block">Unicode block</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Alphanumeric</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Mathematical_Alphanumeric_Symbols" title="Mathematical Alphanumeric Symbols">Mathematical Alphanumeric Symbols</a></li>
<li><a href="/enwiki/wiki/Blackboard_bold#Usage" title="Blackboard bold">Blackboard bold</a></li>
<li><a href="/enwiki/wiki/Letterlike_Symbols" title="Letterlike Symbols">Letterlike Symbols</a></li>
<li><a href="/enwiki/wiki/Symbols_for_zero" title="Symbols for zero">Symbols for zero</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/enwiki/wiki/Arrow_(symbol)" title="Arrow (symbol)">Arrows</a> and <a href="/enwiki/wiki/Geometric_Shapes_(Unicode_block)" title="Geometric Shapes (Unicode block)">Geometric Shapes</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Arrow_(symbol)" title="Arrow (symbol)">Arrows</a></li>
<li><a href="/enwiki/wiki/Miscellaneous_Symbols_and_Arrows" title="Miscellaneous Symbols and Arrows">Miscellaneous Symbols and Arrows</a></li>
<li><a href="/enwiki/wiki/Geometric_Shapes_(Unicode_block)" title="Geometric Shapes (Unicode block)">Geometric Shapes (Unicode block)</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Operators</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Mathematical_operators_and_symbols_in_Unicode" title="Mathematical operators and symbols in Unicode">Mathematical operators and symbols</a></li>
<li><a href="/enwiki/wiki/Mathematical_Operators_(Unicode_block)" title="Mathematical Operators (Unicode block)">Mathematical Operators (Unicode block)</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/enwiki/wiki/Supplemental_Mathematical_Operators" title="Supplemental Mathematical Operators">Supplemental Math Operators</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Supplemental_Mathematical_Operators" title="Supplemental Mathematical Operators">Supplemental Mathematical Operators</a></li>
<li><a href="/enwiki/wiki/Number_Forms" title="Number Forms">Number Forms</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Miscellaneous_Mathematical_Symbols-A" title="Miscellaneous Mathematical Symbols-A">A</a></li>
<li><a href="/enwiki/wiki/Miscellaneous_Mathematical_Symbols-B" title="Miscellaneous Mathematical Symbols-B">B</a></li>
<li><a href="/enwiki/wiki/Miscellaneous_Technical" title="Miscellaneous Technical">Technical</a></li>
<li><a href="/enwiki/wiki/ISO_31-11" title="ISO 31-11">ISO 31-11</a> (Mathematical signs and symbols for use in physical sciences and technology)</li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible expanded navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Typographical_conventions_and_notations" style="font-size:114%;margin:0 4em">Typographical conventions and notations</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/enwiki/wiki/Language_of_mathematics" title="Language of mathematics">Language</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/APL_syntax_and_symbols#Monadic_functions" title="APL syntax and symbols">APL syntax and symbols</a></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Letters</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Diacritic" title="Diacritic">Diacritic</a></li>
<li><a href="/enwiki/wiki/List_of_letters_used_in_mathematics,_science,_and_engineering" title="List of letters used in mathematics, science, and engineering">Letters in STEM</a>
<ul><li><a href="/enwiki/wiki/Greek_letters_used_in_mathematics,_science,_and_engineering" title="Greek letters used in mathematics, science, and engineering">Greek letters in STEM</a></li>
<li><a href="/enwiki/wiki/Latin_letters_used_in_mathematics,_science,_and_engineering" title="Latin letters used in mathematics, science, and engineering">Latin letters in STEM</a></li></ul></li></ul>
</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a class="mw-selflink selflink">Mathematical notation</a></li>
<li><a href="/enwiki/wiki/List_of_mathematical_abbreviations" title="List of mathematical abbreviations">Abbreviations</a></li>
<li><a href="/enwiki/wiki/Notation_in_probability_and_statistics" title="Notation in probability and statistics">Notation in probability and statistics</a></li>
<li><a href="/enwiki/wiki/List_of_common_physics_notations" title="List of common physics notations">List of common physics notations</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible expanded navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Meanings_of_symbols" style="font-size:114%;margin:0 4em">Meanings of symbols</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">
<ul><li><a href="/enwiki/wiki/Glossary_of_mathematical_symbols" title="Glossary of mathematical symbols">Glossary of mathematical symbols</a></li>
<li><a href="/enwiki/wiki/List_of_mathematical_constants" title="List of mathematical constants">List of mathematical constants</a></li>
<li><a href="/enwiki/wiki/Physical_constant" title="Physical constant">Physical constants</a></li>
<li><a href="/enwiki/wiki/Table_of_mathematical_symbols_by_introduction_date" title="Table of mathematical symbols by introduction date">Table of mathematical symbols by introduction date</a></li>
<li><a href="/enwiki/wiki/List_of_typographical_symbols_and_punctuation_marks" title="List of typographical symbols and punctuation marks">List of typographical symbols and punctuation marks</a></li></ul>
</div></td></tr></tbody></table><div></div></td></tr></tbody></table></div></div>' |
