Mathematical notation: Difference between revisions

Content deleted Content added

Inline

Latest revision as of 08:06, 16 December 2024

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.

[1]

[2]

[3]

[4]

[5]

[6]

@@ Line 1: / Line 1: @@
 {{Short description|System of symbolic representation}}
-{{for|information on rendering mathematical formulae|Help:Formula|Help:Displaying a formula|Wikipedia:Manual of Style/Mathematics}}
+{{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.
-'''Mathematical notation''' is a system of [[symbol]]ic representations of mathematical objects and ideas. Mathematical notations are used in [[mathematics]], the [[physical sciences]], [[engineering]], and [[economics]]. Mathematical notations include relatively simple symbolic representations, such as the numbers 0, 1 and 2; [[Variable (mathematics)|variables]] such as ''x'', ''y'' and ''z''; delimiters such as "(" and "|"; [[function (mathematics)|function]] symbols such as {{math|[[sine|sin]]}}; operator symbols such as "[[Plus and minus signs#Plus sign|+]]"; [[Relation (math)|relational symbols]] such as "<"; conceptual symbols such as {{math|[[limit (mathematics)|lim]]}} and [[derivative|''dy/dx'']]; [[equation]]s and complex diagrammatic notations such as [[Penrose graphical notation]] and [[Coxeter–Dynkin diagram]]s.<ref>{{Cite web|last=Helmenstine|first=Anne Marie|date=June 27, 2019|title=Why Mathematics Is a Language|url=https://www.thoughtco.com/why-mathematics-is-a-language-4158142|access-date=2020-08-08|website=ThoughtCo|language=en}}</ref>
+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>
-==Definition==
-Mathematical notation is a [[writing system]] used for recording concepts in mathematics.
-*The notation uses symbols or symbolic [[expression (mathematics)|expressions]] that are intended to have a precise semantic meaning.
-*In the [[history of mathematics]], these symbols have denoted numbers, shapes, patterns and change. The notation can also include symbols for parts of the conventional discourse between mathematicians, when viewing [[mathematics as a language]].
+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]].
-The media used for writing are recounted below, but common materials currently include paper and pencil, board and chalk (or dry-erase marker), and electronic media. Systematic adherence to mathematical concepts is a fundamental concept of mathematical notation. For related concepts, see [[logical argument]], [[mathematical logic]], and [[model theory]].
-==Expressions==
+== Symbols ==
+{{Main|Glossary of mathematical symbols}}
-A [[Expression (mathematics)|mathematical expression]] is a ''sequence'' of symbols that can be evaluated. For example, if the symbols represent numbers, then the expressions are evaluated according to a conventional [[order of operations]] which provides for calculation, if possible, of any expressions within parentheses, followed by any exponents and roots, then multiplications and divisions, and finally any additions or subtractions, all done from left to right.
+The use of many symbols is the basis of mathematical notation. They play a similar role as words in [[natural language]]s. They may play different roles in mathematical notation similarly as verbs, adjective and nouns play different roles in a sentence.
-In a [[computer language]], these rules are implemented by the [[compiler]]s. For more on expression evaluation, see the [[computer science]] topics: [[eager evaluation]], [[lazy evaluation]], [[shortcut evaluation]], and [[evaluation operator]].
+=== Letters as symbols===
-==Precise semantic meaning==
+{{main|List of letters used in mathematics, science, and engineering}}
-Modern mathematics needs to be precise, because [[ambiguous]] notations do not allow [[formal proof]]s. Suppose that we have [[Proposition (mathematics)|statement]]s, denoted by some formal [[sequence]] of symbols, about some objects (for example, numbers, shapes, patterns).  Until the statements can be shown to be valid, their meaning is not yet resolved. During the reasoning process, we might let the symbols refer to those denoted objects, perhaps in a [[model (abstract)|model]]. The [[semantics]] of that object has a [[heuristic]] side and a [[deductive]] side. In either case, we might want to know the properties of that object, which we might then list in an [[intensional definition]].
+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>
-Those properties might then be expressed by some well-known and agreed-upon symbols from a [[table of mathematical symbols]]. This mathematical notation might include annotations such as
-*"All ''x''", "No ''x''", "[[existential quantification|There is an]] ''x''" (or its equivalent, "Some ''x''"), "A [[set (mathematics)|set]]", "A [[function (mathematics)|function]]"
-*"A mapping from the [[real number]]s to the [[complex number]]s"
+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>
-In different contexts, the same symbol or notation can be used to represent different concepts (just as multiple symbols can be used to represent the same concept). Therefore, to fully understand a piece of mathematical writing, it is important to first check the definitions of the notations given by the author. This may be problematic, for instance, if the author assumes the reader is already familiar with the notation in use.
+=== Other symbols ===
-==History==
+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.
-{{main article|History of mathematical notation}}
+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.
-===Counting===
-It is believed that a mathematical notation to represent [[counting]] was first developed at least 50,000 years ago<ref>''An Introduction to the History of Mathematics'' (6th Edition) by [[Howard Eves]] (1990) p.9</ref>—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.
+== Expressions ==
-The development of [[zero]] as a number is one of the most important developments in early mathematics. 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]] for more information).
+{{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.
-===Geometry becomes analytic===
-The earliest mathematical viewpoints in [[geometry]] did not lend themselves well to counting. The [[natural number]]s, their relationship to [[fraction (mathematics)|fraction]]s, and the identification of [[Continuum (theory)|continuous]] quantities actually took millennia to take form, and even longer to allow for the development of notation.
+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>
-In fact, it was not until the invention of [[analytic geometry]] by [[René Descartes]] that geometry became more subject to a numerical notation.<ref>{{citation
- | last = Boyer | first = C. B.
- | journal = The American Mathematical Monthly
- | jstor = 2308751
- | mr = 0105335
- | quote = The great accomplishment of Descartes in mathematics invariably is described as the arithmetization of geometry.
- | pages = 390–393
- | title = Descartes and the geometrization of algebra
- | volume = 66
- | year = 1959
- | issue = 5
- | doi=10.2307/2308751}}</ref> Some symbolic shortcuts for mathematical concepts came to be used in the publication of geometric proofs. Moreover, the power and authority of geometry's theorem and proof structure greatly influenced non-geometric treatises, such as [[Philosophiae Naturalis Principia Mathematica|Principia Mathematica]] by [[Isaac Newton]] for instance.
+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.
-===Modern notation===
-The 18th and 19th centuries saw the creation and standardization of mathematical notation as used today. [[Leonhard Euler]] was responsible for many of the notations currently in use: the use of ''a'', ''b'', ''c'' for constants and ''x'', ''y'', ''z'' for unknowns, ''e'' for the base of the natural logarithm, sigma (Σ) for [[summation]], ''i'' for the [[imaginary unit]], and the functional notation ''f''(''x''). He also popularized the use of π for the [[Archimedes constant]] (due to [[William Jones (mathematician)|William Jones]]' proposal for the use of π in this way based on the earlier notation of [[William Oughtred]]).
+== History ==
-In addition, many fields of mathematics bear the imprint of their creators for notation: the differential operator of [[Gottfried Wilhelm Leibniz|Leibniz]],<ref>{{cite web|url=http://www.maths.tcd.ie/pub/HistMath/People/Leibniz/RouseBall/RB_Leibnitz.html|title=Gottfried Wilhelm Leibnitz|access-date=5 October 2014}}</ref> the [[cardinal number|cardinal]] infinities of [[Georg Cantor]] (in addition to the [[Infinity symbol|lemniscate]] (∞) of [[John Wallis]]), the [[modular arithmetic|congruence]] symbol (≡) of [[Carl Friedrich Gauss|Gauss]], and so forth.
+{{Main|History of mathematical notation}}
-===Computerized 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.
-Mathematically oriented markup languages such as [[TeX]], [[LaTeX]] and, more recently, [[MathML]], are powerful enough to express a wide variety of mathematical notations.
+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]]).
-Theorem-proving software comes with its own notations for mathematics;{{fact|date=July 2021}} the [http://www.omdoc.org/ OMDoc project] seeks to provide an open commons for such notations; and the [https://uniformal.github.io/doc/language/ MMT language] provides a basis for interoperability between other notations.{{fact|date=July 2021}}
-==Non-Latin-based mathematical 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.
-[[Modern Arabic mathematical notation]] is based mostly on the [[Arabic alphabet]] and is used widely in the [[Arab world]], especially in pre-[[tertiary education]].
+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.
-(Western notation uses [[Arabic numerals]], but the Arabic notation also replaces Latin letters and related symbols with Arabic script.)
+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]].
-In addition to Arabic notation, mathematics also makes use of [[Greek alphabet]]s to denote a wide variety of mathematical objects and variables. In some occasions, certain [[Hebrew alphabet]]s are also used (such as in the context of [[infinite cardinal]]s).
+The 18th and 19th centuries saw the standardization of mathematical notation as used today. [[Leonhard Euler]] was responsible for many of the notations currently in use: the [[functional notation]] <math>f(x),</math> {{math|''e''}} for the base of the natural logarithm, <math display="inline">\sum</math> for [[summation]], etc.<ref name="Boyer-Merzbach_1991"/> He also popularized the use of {{pi}} for the [[Archimedes constant]] (proposed by [[William Jones (mathematician)|William Jones]], based on an earlier notation of [[William Oughtred]]).<ref name="Arndt-Haenel_2006"/>
+Since then many new notations have been introduced, often specific to a particular area of mathematics. Some notations are named after their inventors, such as [[Leibniz's notation]], [[Legendre symbol]], the [[Einstein summation convention]], etc.
+=== Typesetting ===
+General [[typesetting system]]s are generally not well suited for mathematical notation. One of the reasons is that, in mathematical notation, the symbols are often arranged in two-dimensional figures, such as in:
+: <math>\sum_{n=0}^\infty \frac {\begin{bmatrix}a&b\\c&d\end{bmatrix}^n}{n!}.</math>
+[[TeX]] is a mathematically oriented typesetting system that was created in 1978 by [[Donald Knuth]]. It is widely used in mathematics, through its extension called [[LaTeX]], and is a ''de facto'' standard. (The above expression is written in LaTeX.)
+More recently, another approach for mathematical typesetting is provided by [[MathML]]. However, it is not well supported in web browsers, which is its primary target.
+== International standard mathematical notation ==
+The international standard [[ISO 80000-2]] (previously, [[ISO 31-11]]) specifies symbols for use in mathematical equations. The standard requires use of italic fonts for variables (e.g., {{nowrap|1=''E'' = ''mc''<sup>2</sup>}}) and roman (upright) fonts for mathematical constants (e.g., e or π).
+== Non-Latin-based mathematical notation ==
+[[Modern Arabic mathematical notation]] is based mostly on the [[Arabic alphabet]] and is used widely in the [[Arab world]], especially in pre-[[tertiary education]].  (Western notation uses [[Arabic numerals]], but the Arabic notation also replaces Latin letters and related symbols with Arabic script.)
+In addition to Arabic notation, mathematics also makes use of [[Greek alphabet|Greek letter]]s to denote a wide variety of mathematical objects and variables. On some occasions, certain [[Hebrew alphabet|Hebrew letter]]s are also used (such as in the context of [[infinite cardinal]]s).
 Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are [[Penrose graphical notation]] and [[Coxeter–Dynkin diagram]]s.
 Braille-based mathematical notations used by blind people include [[Nemeth Braille]] and [[GS8 Braille]].
-==Encoding==
-The codes for mathematical notation in the standard for names of scripts [[ISO 15924]] are {{code|Zmth}}<ref name="IETF">{{cite web |title=IETF Language Subtag Registry |url=https://www.iana.org/assignments/language-subtag-registry/language-subtag-registry |publisher=IANA |access-date=10 September 2021 |language=en |date=2021-08-06}}</ref> and {{code|995}}.
-==See also==
+== See also ==
 * [[Abuse of notation]]
 * [[Begriffsschrift]]
@@ Line 79: / Line 83: @@
 * [[History of mathematical notation]]
 * [[ISO 31-11]]
-* [[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 89: / Line 92: @@
 * [[Semasiography]]
 * [[Table of mathematical symbols]]
-* [[Typographical conventions in mathematical formulae]]
 * [[Vector notation]]
 * [[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==
-* [[Florian Cajori]], [https://books.google.com/books?id=7juWmvQSTvwC&printsec=frontcover ''A History of Mathematical Notations''] (1929), 2 volumes. {{isbn|0-486-67766-4}}
+* [[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}}
-*{{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&dq=enlightening+symbols&hl=en&sa=X&ved=0ahUKEwi_wNvAo_DhAhVOvFkKHW9kAOUQ6AEIMDAB ''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 [[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}}
-{{MathematicalSymbolsNotationLanguage}}
 {{DEFAULTSORT:Mathematical Notation}}
 [[Category:Mathematical notation| ]]
+[[Category:16th-century inventions]]