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Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations, and any other mathematical objects and assembling them into expressions and formulas. Mathematical notation is widely used in mathematics, science, and engineering for representing complex concepts and properties in a concise, unambiguous, and accurate way.

For example, the physicist Albert Einstein's formula $E=mc^{2}$ is the quantitative representation in mathematical notation of mass–energy equivalence.^[1]

Mathematical notation was first introduced by François Viète at the end of the 16th century and largely expanded during the 17th and 18th centuries by René Descartes, Isaac Newton, Gottfried Wilhelm Leibniz, and overall Leonhard Euler.

Symbols

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

Letters as symbols

Letters are typically used for naming—in mathematical jargon, one says representing—mathematical objects. The Latin and Greek alphabets are used extensively, but a few letters of other alphabets are also used sporadically, such as the Hebrew ⁠ $\aleph$ ⁠, Cyrillic $Ш$ , and Hiragana $よ$ . Uppercase and lowercase letters are considered as different symbols. For Latin alphabet, different typefaces also provide different symbols. For example, $r,R,\mathbb {R} ,{\mathcal {R}},{\mathfrak {r}},$ and ${\mathfrak {R}}$ could theoretically appear in the same mathematical text with six different meanings. Normally, roman upright typeface is not used for symbols, except for symbols representing a standard function, such as the symbol " $\sin$ " of the sine function.^[2]

In order to have more symbols, and for allowing related mathematical objects to be represented by related symbols, diacritics, subscripts and superscripts are often used. For example, ${\hat {f'_{1}}}$ may denote the Fourier transform of the derivative of a function called $f_{1}.$

Other symbols

Symbols are not only used for naming mathematical objects. They can be used for operations $(+,-,/,\oplus ,\ldots ),$ for relations $(=,<,\leq ,\sim ,\equiv ,\ldots ),$ for logical connectives $(\implies ,\land ,\lor ,\ldots ),$ for quantifiers $(\forall ,\exists ),$ and for other purposes.

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

Expressions

An expression is a finite combination of symbols that is 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 operators, and may therefore be evaluated by the action of the operators in it. For example, $3+2$ is an expression in which the operator $+$ can be evaluated for giving the result $5.$ So, $3+2$ and $5$ are two different expressions that represent the same number. This is the meaning of the equality $3+2=5.$

A more complicated example is given by the expression ${\textstyle \int _{a}^{b}xdx}$ that can be evaluated to ${\textstyle {\frac {b^{2}}{2}}-{\frac {a^{2}}{2}}.}$ Although the resulting expression contains the operators of division, subtraction and exponentiation, it cannot be evaluated further because $a$ and $b$ denote unspecified numbers.

History

Numbers

It is believed that a notation to represent numbers was first developed at least 50,000 years ago.^[3] Early mathematical ideas such as finger counting^[4] 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 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 Babylonians and Greek Egyptians, and then as an integer by the Mayans, Indians and Arabs (see the history of zero).

Modern notation

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

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

Later, René Descartes (17th century) introduced the modern notation for variables and equations; in particular, the use of $x,y,z$ for unknown quantities and $a,b,c$ for known ones (constants). He introduced also the notation $i$ 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 $f(x),$ $e$ for the base of the natural logarithm, ${\textstyle \sum }$ for summation, etc.^[5] He also popularized the use of $π$ for the Archimedes constant (proposed by William Jones, based on an earlier notation of William Oughtred).^[6]

Since then many new notations have been introduced, often specific to a particular area of mathematics. Some notations are named after their inventors, such as Leibniz's notation, Legendre symbol, the Einstein summation convention, etc.

Typesetting

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

\sum _{n=0}^{\infty }{\frac {{\begin{bmatrix}a&b\\c&d\end{bmatrix}}^{n}}{n!}}.

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 = mc²) 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 letters to denote a wide variety of mathematical objects and variables. On some occasions, certain Hebrew letters are also used (such as in the context of infinite cardinals).

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

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

References

^ Einstein, Albert (1905). "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?". Annalen der Physik (in German). 323 (13): 639–641. Bibcode:1905AnP...323..639E. doi:10.1002/andp.19053231314. ISSN 0003-3804.
^ ISO 80000-2:2019
^ Eves, Howard (1990). An Introduction to the History of Mathematics (6 ed.). Saunders College Pub. p. 9. ISBN 978-0-03-029558-4.
^ Ifrah, Georges (2000). The Universal History of Numbers: From prehistory to the invention of the computer. Translated by Bellos, David; Harding, E. F.; Wood, Sophie; Monk, Ian. John Wiley and Sons. p. 48. ISBN 0-471-39340-1. (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.)
^ Boyer, Carl Benjamin; Merzbach, Uta C. (1991). A History of Mathematics. John Wiley & Sons. pp. 442–443. ISBN 978-0-471-54397-8.
^ Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. p. 166. ISBN 978-3-540-66572-4.

External links

Earliest Uses of Various Mathematical Symbols
Mathematical ASCII Notation how to type math notation in any text editor.
Mathematics as a Language at Cut-the-Knot
Stephen Wolfram: Mathematical Notation: Past and Future. October 2000. Transcript of a keynote address presented at MathML and Math on the Web: MathML International Conference.

[1] Einstein, Albert (1905). "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?". Annalen der Physik (in German). 323 (13): 639–641. Bibcode:1905AnP...323..639E. doi:10.1002/andp.19053231314. ISSN 0003-3804.

[2] ISO 80000-2:2019

[Eves_1990-3] Eves, Howard (1990). An Introduction to the History of Mathematics (6 ed.). Saunders College Pub. p. 9. ISBN 978-0-03-029558-4.

[Ifrah_2000-4] Ifrah, Georges (2000). The Universal History of Numbers: From prehistory to the invention of the computer. Translated by Bellos, David; Harding, E. F.; Wood, Sophie; Monk, Ian. John Wiley and Sons. p. 48. ISBN 0-471-39340-1. (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.)

[Boyer-Merzbach_1991-5] Boyer, Carl Benjamin; Merzbach, Uta C. (1991). A History of Mathematics. John Wiley & Sons. pp. 442–443. ISBN 978-0-471-54397-8.

[Arndt-Haenel_2006-6] Arndt, Jörg; Haenel, Christoph (2006). Pi Unleashed. Springer-Verlag. p. 166. ISBN 978-3-540-66572-4.

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[3]

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[6]

@@ Line 1: / Line 1: @@
 {{Short description|System of symbolic representation}}
-{{For|information on rendering mathematical formulae|Help:Displaying a formula|Wikipedia:Manual of Style/Mathematics}}[[File:01 Kreiszahl.svg|thumb|An unusual display of {{pi}} allowed by [[TeX]] (European style, with a comma as a [[decimal separator]])]]
+{{For|information on rendering mathematical formulae|Help:Displaying a formula|Wikipedia:Manual of Style/Mathematics}}
 {{More citations needed|date=June 2022}}
+{{Use dmy dates|date=November 2023|cs1-dates=y}}
+{{Use list-defined references|date=November 2023}}
 '''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, [[Albert Einstein]]'s equation <math>E=mc^2</math> is the quantitative representation in mathematical notation of the [[mass–energy equivalence]].
+For example, the physicist [[Albert Einstein]]'s formula <math>E=mc^2</math> is the quantitative representation in mathematical notation of [[mass–energy equivalence]].<ref>{{Cite journal |last=Einstein |first=Albert |date=1905 |title=Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? |url=https://onlinelibrary.wiley.com/doi/10.1002/andp.19053231314 |journal=Annalen der Physik |language=de |volume=323 |issue=13 |pages=639–641 |doi=10.1002/andp.19053231314 |bibcode=1905AnP...323..639E |issn=0003-3804}}</ref>
 Mathematical notation was first introduced by [[François Viète]] at the end of the 16th century and largely expanded during the 17th and 18th centuries by [[René Descartes]], [[Isaac Newton]], [[Gottfried Wilhelm Leibniz]], and overall [[Leonhard Euler]].
-==Symbols==
+== Symbols ==
 {{Main|Glossary of mathematical symbols}}
@@ Line 17: / Line 19: @@
 {{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. This is typically the [[Latin alphabet|Latin]] and [[Greek alphabet|Greek]] alphabets that are used, but some letters of [[Hebrew alphabet]] <math>(\aleph, \beth)</math> are sometimes used. Uppercase and lowercase letters are considered as different symbols. For Latin alphabet, different typefaces provide also 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 that are formed of several letters, such as the symbol "<math>\sin</math>" of the [[sine function]].
+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>
@@ Line 26: / Line 28: @@
 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==
+== Expressions ==
 {{Unsourced section|date=June 2022}}
@@ Line 35: / Line 37: @@
 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==
+== History ==
 {{Main|History of mathematical notation}}
-===Numbers===
+=== Numbers ===
-It is believed that a notation to represent [[number]]s was first developed at least 50,000 years ago{{sfn|Eves|1990|p=9}}—early mathematical ideas such as [[finger counting]]<ref>[[Georges Ifrah]] notes that humans learned to count on their hands. Ifrah shows, for example, a picture of [[Boethius]] (who lived 480–524 or 525) reckoning on his fingers in {{harvnb|Ifrah|2000|p=48}}.</ref> 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.
+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===
+=== 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.
@@ Line 50: / Line 52: @@
 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.{{sfn|Boyer|Merzbach|1991|p=442&ndash;443}} 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]]).{{sfn|Arndt|Haenel|2006|p=166}}
+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.
+Since then many new notations have been introduced, often specific to a particular area of mathematics. Some notations are named after their inventors, such as [[Leibniz's notation]], [[Legendre symbol]], the [[Einstein summation convention]], etc.
-===Typesetting===
+=== 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>
+: <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.
+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==
+== 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).
+The international standard [[ISO 80000-2]] (previously, [[ISO 31-11]]) specifies symbols for use in mathematical equations. The standard requires use of italic fonts for variables (e.g., {{nowrap|1=''E'' = ''mc''<sup>2</sup>}}) and roman (upright) fonts for mathematical constants (e.g., e or π).
-==Non-Latin-based mathematical notation==
+== 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]].
+[[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.)
-(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).
@@ Line 76: / Line 76: @@
 Braille-based mathematical notations used by blind people include [[Nemeth Braille]] and [[GS8 Braille]].
-==See also==
+== See also ==
 * [[Abuse of notation]]
 * [[Begriffsschrift]]
@@ Line 83: / Line 83: @@
 * [[History of mathematical notation]]
 * [[ISO 31-11]]
-* [[ISO/IEC_80000#Part_2:_Mathematics|ISO 80000-2]]
+* [[ISO/IEC 80000#Part 2: Mathematics|ISO 80000-2]]
 * [[Knuth's up-arrow notation]]
-* [[List of mathematical symbols]]
 * [[Mathematical Alphanumeric Symbols]]
 * [[Mathematical formula]]
@@ Line 96: / Line 95: @@
 * [[Modern Arabic mathematical notation]]
-==Notes==
+== References ==
+{{reflist|refs=
-<references/>
+<ref name="Ifrah_2000">{{cite book |author-last=Ifrah |author-first=Georges |author-link=Georges Ifrah |title=The Universal History of Numbers: From prehistory to the invention of the computer. |language=en |publisher=[[John Wiley and Sons]] |date=2000 |page=48 |isbn=0-471-39340-1 |translator-first1=David |translator-last1=Bellos |translator-first2=E. F. |translator-last2=Harding |translator-first3=Sophie |translator-last3=Wood |translator-first4=Ian |translator-last4=Monk}} (NB. Ifrah supports his thesis by quoting idiomatic phrases from languages across the entire world. He notes that humans learned to count on their hands. He shows, for example, a picture of [[Boethius]] (who lived 480–524 or 525) reckoning on his fingers.)</ref>
+<ref name="Boyer-Merzbach_1991">{{cite book |author-last1=Boyer |author-first1=Carl Benjamin |author-link1=Carl Benjamin Boyer |author-last2=Merzbach |author-first2=Uta C. |author-link2=Uta Merzbach |title=A History of Mathematics |date=1991 |publisher=[[John Wiley & Sons]] |isbn=978-0-471-54397-8 |pages=442&ndash;443 |url=https://archive.org/details/historyofmathema00boye/page/442}}</ref>
+<ref name="Eves_1990">{{cite book |author-last=Eves |author-first=Howard |author-link=Howard Eves |title=An Introduction to the History of Mathematics |date=1990 |edition=6 |isbn=978-0-03-029558-4 |page=9|publisher=Saunders College Pub. }}</ref>
+<ref name="Arndt-Haenel_2006">{{cite book |author-last1=Arndt |author-first1=Jörg |author-last2=Haenel |author-first2=Christoph |title=Pi Unleashed |publisher=[[Springer-Verlag]] |date=2006 |isbn=978-3-540-66572-4 |page=166 |url=https://books.google.com/books?id=QwwcmweJCDQC&pg=PA166}}</ref>
+}}
+== Further reading ==
-==References==
-* {{cite book
- | last1 = Arndt | first1 = Jörg
- | last2 = Haenel | first2 = Christoph
- | title = Pi Unleashed
- | publisher = [[Springer Science+Business Media|Springer-Verlag]]
- | year = 2006
- | isbn = 978-3-540-66572-4
- | page = 166
- | url = https://books.google.com/books?id=QwwcmweJCDQC&pg=PA166
- }}
-* {{citation
- | last1 = Boyer | first1 = Carl B. | author-link = Carl Benjamin Boyer
- | last2 = Merzbach | first2 = Uta C.| author2-link = Uta Merzbach
- | title = A History of Mathematics
- | year = 1991
- | publisher = [[John Wiley & Sons]]
- | isbn = 978-0-471-54397-8
- | url = https://archive.org/details/historyofmathema00boye/page/442
-}}
-* {{citation
- | last = Eves | first = Howard | author-link = Howard Eves
- | title = An Introduction to the History of Mathematics
- | year = 1990
- | edition = 6th
- | isbn = 978-0-03-029558-4
-}}
 * [[Florian Cajori]], [https://books.google.com/books?id=7juWmvQSTvwC ''A History of Mathematical Notations''] (1929), 2 volumes. {{isbn|0-486-67766-4}}
-* {{Citation
-  | last = Ifrah
-  | first = Georges
-  | author-link = Georges Ifrah
-  | title = The Universal History of Numbers: From prehistory to the invention of the computer.
-  | publisher = [[John Wiley and Sons]]
-  | year= 2000
-  | page = 48
-  | isbn = 0-471-39340-1
-}}. Translated from the French by David Bellos, E.F. Harding, Sophie Wood and Ian Monk. Ifrah supports his thesis by quoting idiomatic phrases from languages across the entire world.
 * Mazur, Joseph (2014), [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==
+== External links ==
 {{Commons category}}
-*[http://jeff560.tripod.com/mathsym.html Earliest Uses of Various Mathematical Symbols]
+* [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.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]]
+* [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.
+* [[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}}
@@ Line 151: / Line 119: @@
 [[Category:Mathematical notation| ]]
+[[Category:16th-century inventions]]