Mathematical notation: Difference between revisions

From HandWiki
imported>StanislovAI
correction
 
Carolyn (talk | contribs)
0 edits
add
 
Line 1: Line 1:
{{Short description|System of symbolic representation}}
{{Short description|System of symbolic representation}}
{{Self-reference|For information on rendering mathematical formulae, see Help:Displaying a formula and Wikipedia:Manual of Style/Mathematics.}}
{{Use list-defined references|date=November 2023}}
{{Use list-defined references|date=November 2023}}
[[File:Highlighted LaTeX example.webp|thumb|Highlighted [[Software:LaTeX|LaTeX]] mathematical notation]]
'''Mathematical notation''' consists of using [[Glossary of mathematical symbols|symbols]] for representing [[Operation (mathematics)|operation]]s, unspecified [[Number|number]]s, [[Relation (mathematics)|relation]]s, and any other [[Philosophy:Mathematical object|mathematical object]]s and assembling them into [[Expression (mathematics)|expression]]s and [[Formula|formula]]s. Mathematical notation is widely used in [[Mathematics|mathematics]], [[Science|science]], and [[Engineering:Engineering|engineering]] for representing complex [[Philosophy:Concept|concept]]s and [[Philosophy:Property|properties]] in a concise, unambiguous, and accurate way.
'''Mathematical notation''' consists of using [[Glossary of mathematical symbols|symbols]] for representing [[Operation (mathematics)|operation]]s, unspecified [[Number|number]]s, [[Relation (mathematics)|relation]]s, and any other [[Philosophy:Mathematical object|mathematical object]]s and assembling them into [[Expression (mathematics)|expression]]s and [[Formula|formula]]s. Mathematical notation is widely used in [[Mathematics|mathematics]], [[Science|science]], and [[Engineering:Engineering|engineering]] for representing complex [[Philosophy:Concept|concept]]s and [[Philosophy:Property|properties]] in a concise, unambiguous, and accurate way.


For example, [[Biography:Albert Einstein|Albert Einstein]]'s equation <math>E=mc^2</math> is the quantitative representation in mathematical notation of the [[Mass–energy equivalence|mass–energy equivalence]].
For example, the physicist [[Biography:Albert Einstein|Albert Einstein]]'s formula <math>E=mc^2</math> is the quantitative representation in mathematical notation of [[Mass–energy equivalence|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 [[Biography:François Viète|François Viète]] at the end of the 16th century and largely expanded during the 17th and 18th centuries by [[Biography:René Descartes|René Descartes]], [[Biography:Isaac Newton|Isaac Newton]], [[Biography:Gottfried Wilhelm Leibniz|Gottfried Wilhelm Leibniz]], and overall [[Biography:Leonhard Euler|Leonhard Euler]].
Mathematical notation was first introduced by [[Biography:François Viète|François Viète]] at the end of the 16th century and largely expanded during the 17th and 18th centuries by [[Biography:René Descartes|René Descartes]], [[Biography:Isaac Newton|Isaac Newton]], [[Biography:Gottfried Wilhelm Leibniz|Gottfried Wilhelm Leibniz]], and overall [[Biography:Leonhard Euler|Leonhard Euler]].


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


Line 14: Line 17:
=== Letters as symbols===
=== Letters as symbols===


Letters are typically used for naming—in [[List of mathematical jargon|mathematical jargon]], one says ''representing''—[[Philosophy:Mathematical object|mathematical object]]s. This is typically the Latin and Greek alphabets that are used, but some letters of [[Social:Hebrew alphabet|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''—[[Philosophy:Mathematical object|mathematical object]]s. The [[Latin alphabet|Latin]] and Greek alphabets are used extensively, but a few letters of other alphabets are also used sporadically, such as the [[Social:Hebrew alphabet|Hebrew]] {{tmath|\aleph}}, [[Social: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, diacritics, subscripts and superscripts are often used. For example, <math>\hat {f'_1}</math> may denote the [[Fourier transform]] of the [[Derivative|derivative]] of a [[Function (mathematics)|function]] called <math>f_1.</math>
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, <math>\hat {f'_1}</math> may denote the [[Fourier transform]] of the [[Derivative|derivative]] of a [[Function (mathematics)|function]] called <math>f_1.</math>
Line 23: Line 26:
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.
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==
=== International standard mathematical notation ===
{{Unsourced section|date=June 2022}}
The [[Organization:International Organization for Standardization|International Organization for Standardization]] (ISO) is an [[International standard|international standard]] development organization composed of representatives from the national [[Engineering:Standards organization|standards organizations]] of member countries. The international standard ISO 80000-2 (previously, [[ISO 31-11]]) specifies symbols for use in mathematical equations. The standard requires use of italic fonts for variables (e.g., {{nowrap|1=''E'' = ''mc''<sup>2</sup>}}) and roman (upright) fonts for mathematical constants (e.g., e or π).


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


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


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|subtraction]] and [[Exponentiation|exponentiation]], it cannot be evaluated further because {{mvar|a}} and {{mvar|b}} denote unspecified numbers.
To ''evaluate'' an expression means to find a numerical [[Value (mathematics)|value]] equivalent to the expression.<ref>Oxford English Dictionary, s.v. "Evaluate (v.), sense a", "''Mathematics. To work out the ‘value’ of (a quantitative expression); to find a numerical expression for (any quantitative fact or relation).''"</ref><ref>Oxford English Dictionary, s.v. “Simplify (v.), sense 4.a”, "''To express (an equation or other mathematical expression) in a form that is easier to understand, analyse, or work with, e.g. by collecting like terms or substituting variables.''"</ref> Expressions can be ''evaluated'' or ''simplified'' by replacing [[Operation (mathematics)|operations]] that appear in them with their result. For example, the expression <math>8\times 2-5</math> simplifies to <math>16-5</math>, and evaluates to <math>11.</math>


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


===Numbers===
=== Numbers ===
It is believed that a notation to represent [[Number|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 [[Social:Tally stick|tally stick]] is a way of counting dating back to the [[Earth:Upper Paleolithic|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.
It is believed that a notation to represent [[Number|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 [[Social:Tally stick|tally stick]] is a way of counting dating back to the [[Earth:Upper Paleolithic|Upper Paleolithic]]. Perhaps the oldest known mathematical texts are those of ancient [[Social:Sumer|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 [[Babylonian numerals|Babylonians]] and [[Greek numerals|Greek Egyptians]], and then as an [[Integer|integer]] by the [[Maya numerals|Mayans]], [[Indian numerals|Indians]] and [[Arabic numerals|Arabs]] (see the history of zero).
The concept of [[Zero|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|integer]] by the [[Maya numerals|Mayans]], [[Indian numerals|Indians]] and [[Arabic numerals|Muslims]] (see the history of zero).


===Modern notation===
=== 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 [[Biography:Diophantus|Diophantus]] used some symbols as abbreviations.  
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 [[Biography:Diophantus|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 [[Biography:François Viète|François Viète]] (16th century). However, he used different symbols than those that are now standard.
[[Biography:Michael Stifel|Michael Stifel]]'s (1487–1567) most important work, {{lang|la|Arithmetica integra}} (1544), contains important innovations in mathematical notation.  It has the first use of multiplication by juxtaposition (with no symbol between the terms) in Europe. He is the first to use the term ''exponent''.


Later, [[Biography:René Descartes|René Descartes]] (17th century) introduced the modern notation for variables and [[Equation|equation]]s; in particular, the use of <math>x,y,z</math> for 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|imaginary unit]].
The first systematic use of formulas, and, in particular, the use of symbols ([[Variable (mathematics)|variables]]) for unspecified numbers, is generally attributed to [[Biography:François Viète|François Viète]] (1540–1603). However, he used different symbols than those that are now standard.


The 18th and 19th centuries saw the standardization of mathematical notation as used today. [[Biography:Leonhard Euler|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|summation]], etc.<ref name="Boyer-Merzbach_1991"/> He also popularized the use of {{pi}} for the Archimedes constant (proposed by [[Biography:William Jones (mathematician)|William Jones]], based on an earlier notation of William Oughtred).<ref name="Arndt-Haenel_2006"/>
Later, [[Biography:René Descartes|René Descartes]] (1596–1650) introduced the modern notation for variables and [[Equation|equation]]s; in particular, the use of <math>x,y,z</math> for 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|imaginary unit]].


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.
The 18th and 19th centuries saw the standardization of mathematical notation as used today. [[Biography:Leonhard Euler|Leonhard Euler]] (1707–1783) 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|natural logarithm]], <math display="inline">\sum</math> for [[Summation|summation]], etc.<ref name="Boyer-Merzbach_1991"/> He also popularized the use of {{pi}} for the Archimedes constant (proposed by [[Biography:William Jones (mathematician)|William Jones]], based on an earlier notation of William Oughtred).<ref name="Arndt-Haenel_2006"/>


===Typesetting===
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:
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:
:<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>
 
[[Software:TeX|TeX]] ({{IPAc-en|t|ɛ|x}}) is a mathematically oriented typesetting system that was created in 1978 by [[Biography:Donald Knuth|Donald Knuth]]. It is widely used in mathematics, through its extension called [[Software:LaTeX|LaTeX]] ({{IPAc-en|ˈ|l|ɑː|t|ɛ|x}} or {{IPAc-en|ˈ|l|e:|t|ɛ|x}}), 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 are its primary target.


[[Software:TeX|TeX]] is a mathematically oriented typesetting system that was created in 1978 by [[Biography:Donald Knuth|Donald Knuth]]. It is widely used in mathematics, through its extension called [[Software:LaTeX|LaTeX]], and is a ''de facto'' standard. (The above expression is written in LaTeX.)
== Non-Latin-based mathematical notation ==
[[Modern Arabic mathematical notation]] is based mostly on the Arabic alphabet and is used widely in the [[Place:Arab world|Arab world]], especially in pre-[[Social:Tertiary education|tertiary education]].  (Western notation uses [[Arabic numerals]], but the Arabic notation also replaces Latin letters and related symbols with Arabic script.)


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.
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 [[Social:Hebrew alphabet|Hebrew letter]]s are also used (such as in the context of infinite cardinals).


==International standard mathematical notation==
Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are [[Physics:Penrose graphical notation|Penrose graphical notation]] and [[Coxeter–Dynkin diagram]]s.
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==
Braille-based mathematical notations used by blind people include [[Nemeth Braille]] and GS8 Braille.
[[Modern Arabic mathematical notation]] is based mostly on the Arabic alphabet and is used widely in the [[Place:Arab world|Arab world]], especially in pre-[[Social:Tertiary education|tertiary education]].


(Western notation uses [[Arabic numerals]], but the Arabic notation also replaces Latin letters and related symbols with Arabic script.)
== Meaning and interpretation ==
The [[Syntax|syntax]] of notation defines how symbols can be combined to make [[Expression (mathematics)#Well-defined expressions|well-formed expressions]], without any given meaning or interpretation. The [[Semantics|semantics]] of notation interprets what the symbols represent and assigns a meaning to the expressions and formulas. The reverse process of taking a statement and writing it in logical or mathematical notation is called [[Philosophy:Logic translation|translation]].  


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 [[Social:Hebrew alphabet|Hebrew letter]]s are also used (such as in the context of infinite cardinals).
=== Interpretation ===
Given a [[Formal language|formal language]], an [[Philosophy:Interpretation (logic)|interpretation]] assigns a [[Philosophy:Domain of discourse|domain of discourse]] to the language. Specifically, it assigns each of the constant symbols to objects of the domain, function letters to functions within the domain, predicate letters to statements, and vairiables are assumed to range over the domain.


Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are [[Physics:Penrose graphical notation|Penrose graphical notation]] and Coxeter–Dynkin diagrams.
=== Map–territory relation ===
The [[Map–territory relation|map–territory relation]] describes the relationship between an object and the representation of that object, such as the [[Astronomy:Earth|Earth]] and a [[Earth:Map|map]] of it. In mathematics, this is how the number 4 relates to its representation "4". The quotation marks are the formally correct usage, distinguishing the number from its name. However, it is fairly common practice in math to commit this fallacy saying "Let x denote...", rather than "Let "x" denote..." which is generally harmless.


Braille-based mathematical notations used by blind people include [[Nemeth Braille]] and GS8 Braille.
== Software for mathematical typesetting ==
{{col-begin|width=70%}}
{{col-2}}
* [[Software:AUCTeX|AUCTeX]]
* [[Software:Authorea|Authorea]]
* [[Software:Apache OpenOffice#Components|Apache OpenOffice Math]] 
* [[AsciiMath]] 
* [[Software:Calligra Words#Formula editor|Calligra Words - Formula editor]]
* [[Software:CoCalc|CoCalc]]
* [[Software:GNOME LaTeX|GNOME LaTeX]]
* [[Software:GNU TeXmacs|GNU TeXmacs]] 
* [[Software:Gummi|Gummi]]
* [[Software:KaTeX|KaTeX]] 
* [[Software:Kile|Kile]]
* [[Software:LaTeX|LaTeX]] 
* LibreOffice Math 
* [[Software:LyX|LyX]]
* [[Software:MacTeX|MacTeX]]
* [[Software:MathJax|MathJax]] 
* [[MathML]] 
{{col-2}}
* [[Software:MathType|MathType]]
* [[Software:Notepad++|Notepad++]]
* [[Software:Overleaf|Overleaf]]
* [[Software:Scientific WorkPlace|Scientific WorkPlace]]
* [[Software:TeX|TeX]]
* [[Software:TeX Live|TeX Live]]
* [[Software:Texmaker|Texmaker]]
* [[Software:TeXnicCenter|TeXnicCenter]]
* [[Software:TeXShop|TeXShop]]
* [[Software:TeXstudio|TeXstudio]]
* [[Software:TeXworks|TeXworks]]
* [[Software:Verbosus|Verbosus]]
* [[Software:Vim (text editor)|Vim]]
* [[Software:Visual Studio Code|Visual Studio Code]] - [https://github.com/James-Yu/latex-workshop/wiki LaTeX Workshop]
* [[Software:WinEdt|WinEdt]]
* [[Software:WinFIG|WinFIG]]
* [[Software:WinShell|WinShell]]
{{col-end}}


==See also==
== See also ==
* [[Abuse of notation]]
* [[Abuse of notation]]
* [[Begriffsschrift]]
* [[Associative property]]
* [[Glossary of mathematical symbols]]
* Chemistry notation
** [[Bourbaki dangerous bend symbol]]
* [[Philosophy:Denotation|Denotation]]
* [[History of mathematical notation]]
* [[ISO 31-11]]
* ISO 80000-2
* [[Knuth's up-arrow notation]]
* [[Knuth's up-arrow notation]]
* [[List of mathematical symbols]]
* [[Language of mathematics]]
* [[Software:List of open-source software for mathematics|List of open-source software for mathematics]]
* [[Mathematical Alphanumeric Symbols]]
* [[Mathematical Alphanumeric Symbols]]
* Mathematical formula
* [[Modern Arabic mathematical notation]]
* [[Notation in probability and statistics]]
* [[Notation in probability and statistics]]
* [[Language of mathematics]]
* [[Philosophy:Principle of compositionality|Principle of compositionality]]
* [[Scientific notation]]
* [[Scientific notation]]
* [[Social:Semasiography|Semasiography]]
* [[Social:Semasiography|Semasiography]]
* Table of mathematical symbols
* [[Syntactic sugar]]
* [[Vector notation]]
* [[Vector notation]]
* [[Modern Arabic mathematical notation]]
* List of letters used in mathematics, science, and engineering


==References==
== References ==
{{reflist|refs=
<references>
<ref name="Ifrah_2000">{{cite book |author-last=Ifrah |author-first=Georges |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 [[Biography:Boethius|Boethius]] (who lived 480–524 or 525) reckoning on his fingers.)</ref>
<ref name="Ifrah_2000">{{cite book |author-last=Ifrah |author-first=Georges |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 [[Biography:Boethius|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-last2=Merzbach |author-first2=Uta C. |author-link2=Uta Merzbach |title=A History of Mathematics |date=1991 |publisher={{wipe|John Wiley & Sons}} |isbn=978-0-471-54397-8 |pages=442&ndash;443 |url=https://archive.org/details/historyofmathema00boye/page/442}}</ref>
<ref name="Boyer-Merzbach_1991">{{cite book |author-last1=Boyer |author-first1=Carl Benjamin |author-last2=Merzbach |author-first2=Uta C. |author-link2=Uta Merzbach |title=A History of Mathematics |date=1991 |publisher=[[Company:John Wiley & Sons|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 |title=An Introduction to the History of Mathematics |date=1990 |edition=6 |isbn=978-0-03-029558-4 |page=9}}</ref>
<ref name="Eves_1990">{{cite book |author-last=Eves |author-first=Howard |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=[[Physics:Springer-Verlag|Springer-Verlag]] |date=2006 |isbn=978-3-540-66572-4 |page=166 |url=https://books.google.com/books?id=QwwcmweJCDQC&pg=PA166}}</ref>
<ref name="Arndt-Haenel_2006">{{cite book |author-last1=Arndt |author-first1=Jörg |author-last2=Haenel |author-first2=Christoph |title=Pi Unleashed |publisher=[[Physics:Springer-Verlag|Springer-Verlag]] |date=2006 |isbn=978-3-540-66572-4 |page=166 |url=https://books.google.com/books?id=QwwcmweJCDQC&pg=PA166}}</ref>
}}
</references>


==Further reading==
== Further reading ==
* [[Biography:Florian Cajori|Florian Cajori]], [https://books.google.com/books?id=7juWmvQSTvwC ''A History of Mathematical Notations''] (1929), 2 volumes. {{isbn|0-486-67766-4}}
* [[Biography:Florian Cajori|Florian Cajori]], ''[[A History of Mathematical Notations]]'' (1929), [https://archive.org/details/b29980343_0001/page/32/mode/2up Vol. 1], [https://archive.org/details/b29980343_0002/page/n3/mode/2up Vol. 2]. (Dover reprint 2011, {{isbn|0-486-67766-4}})
* Mazur, Joseph (2014), [https://books.google.com/books?id=YZLzjwEACAAJ&q=enlightening+symbols ''Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers'']. Princeton, New Jersey: Princeton University Press. {{isbn|978-0-691-15463-3}}
* 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 ==
* [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.
* [https://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 [[Biography:Alexander Bogomolny#Cut-the-Knot|Cut-the-Knot]]
* [https://www.cut-the-knot.org/language/index.shtml Mathematics as a Language] at [[Biography:Alexander Bogomolny#Cut-the-Knot|Cut-the-Knot]]
* [[Biography:Stephen Wolfram|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.
* [[Biography:Stephen Wolfram|Stephen Wolfram]]: [https://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}}
{{DEFAULTSORT:Mathematical Notation}}

Latest revision as of 05:15, 23 May 2026

Short description: System of symbolic representation

Template:Self-reference

File:Highlighted LaTeX example.webp
Highlighted LaTeX mathematical notation

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=mc2 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 and typeface

Main article: Glossary of mathematical 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 ℵ, 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,ℝ,ℛ,𝔯, and 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, f'1^ may denote the Fourier transform of the derivative of a function called f1.

Other symbols

Symbols are not only used for naming mathematical objects. They can be used for operations (+,−,/,⊕,…), for relations (=,<,≤,∼,≡,…), for logical connectives (⟹,∧,∨,…), for quantifiers (∀,∃), 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.

International standard mathematical notation

The International Organization for Standardization (ISO) is an international standard development organization composed of representatives from the national standards organizations of member countries. The international standard ISO 80000-2 (previously, ISO 31-11) specifies symbols for use in mathematical equations. The standard requires use of italic fonts for variables (e.g., E = mc2) and roman (upright) fonts for mathematical constants (e.g., e or π).

Expressions and formulas

An expression is a written arrangement of symbols following the context-dependent, syntactic conventions of mathematical notation. Symbols can denote numbers, variables, operations, and functions.[3] Other symbols include punctuation marks and brackets, used for grouping where there is not a well-defined order of operations.

Expressions are commonly distinguished from formulas: expressions are a kind of mathematical object, whereas formulas are statements about mathematical objects.[4] This is analogous to natural language, where a noun phrase refers to an object, and a whole sentence refers to a fact. For example, 8x−5 is an expression, while the inequality 8x−5≥3 is a formula.

To evaluate an expression means to find a numerical value equivalent to the expression.[5][6] Expressions can be evaluated or simplified by replacing operations that appear in them with their result. For example, the expression 8×2−5 simplifies to 16−5, and evaluates to 11.

History

Main article: History of mathematical notation

Numbers

It is believed that a notation to represent numbers was first developed at least 50,000 years ago.[7] Early mathematical ideas such as finger counting[8] 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 Muslims (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.

Michael Stifel's (1487–1567) most important work, Arithmetica integra (1544), contains important innovations in mathematical notation. It has the first use of multiplication by juxtaposition (with no symbol between the terms) in Europe. He is the first to use the term exponent.

The first systematic use of formulas, and, in particular, the use of symbols (variables) for unspecified numbers, is generally attributed to François Viète (1540–1603). However, he used different symbols than those that are now standard.

Later, René Descartes (1596–1650) 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 (1707–1783) was responsible for many of the notations currently in use: the functional notation f(x), e for the base of the natural logarithm, ∑ for summation, etc.[9] He also popularized the use of π for the Archimedes constant (proposed by William Jones, based on an earlier notation of William Oughtred).[10]

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:

∑n=0∞[abcd]nn!.

TeX (/tɛx/) 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 (/ˈlɑːtɛx/ or /ˈleɪtɛx/), 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 are its primary target.

Non-Latin-based mathematical notation

Modern Arabic mathematical notation is based mostly on the Arabic alphabet and is used widely in the Arab world, especially in pre-tertiary education. (Western notation uses Arabic numerals, but the Arabic notation also replaces Latin letters and related symbols with Arabic script.)

In addition to Arabic notation, mathematics also makes use of Greek 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.

Meaning and interpretation

The syntax of notation defines how symbols can be combined to make well-formed expressions, without any given meaning or interpretation. The semantics of notation interprets what the symbols represent and assigns a meaning to the expressions and formulas. The reverse process of taking a statement and writing it in logical or mathematical notation is called translation.

Interpretation

Given a formal language, an interpretation assigns a domain of discourse to the language. Specifically, it assigns each of the constant symbols to objects of the domain, function letters to functions within the domain, predicate letters to statements, and vairiables are assumed to range over the domain.

Map–territory relation

The map–territory relation describes the relationship between an object and the representation of that object, such as the Earth and a map of it. In mathematics, this is how the number 4 relates to its representation "4". The quotation marks are the formally correct usage, distinguishing the number from its name. However, it is fairly common practice in math to commit this fallacy saying "Let x denote...", rather than "Let "x" denote..." which is generally harmless.

Software for mathematical typesetting

See also

References

  1. ↑ Einstein, Albert (1905). "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?" (in de). Annalen der Physik 323 (13): 639–641. doi:10.1002/andp.19053231314. ISSN 0003-3804. Bibcode: 1905AnP...323..639E. https://onlinelibrary.wiley.com/doi/10.1002/andp.19053231314. 
  2. ↑ ISO 80000-2:2019
  3. ↑ Oxford English Dictionary, s.v. “Expression (n.), sense II.7,” "A group of symbols which together represent a numeric, algebraic, or other mathematical quantity or function."
  4. ↑ Stoll, Robert R. (1963). Set Theory and Logic. San Francisco, CA: Dover Publications. ISBN 978-0-486-63829-4. 
  5. ↑ Oxford English Dictionary, s.v. "Evaluate (v.), sense a", "Mathematics. To work out the ‘value’ of (a quantitative expression); to find a numerical expression for (any quantitative fact or relation)."
  6. ↑ Oxford English Dictionary, s.v. “Simplify (v.), sense 4.a”, "To express (an equation or other mathematical expression) in a form that is easier to understand, analyse, or work with, e.g. by collecting like terms or substituting variables."
  7. ↑ An Introduction to the History of Mathematics (6 ed.). Saunders College Pub.. 1990. p. 9. ISBN 978-0-03-029558-4. 
  8. ↑ (in en) The Universal History of Numbers: From prehistory to the invention of the computer.. John Wiley and Sons. 2000. 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.)
  9. ↑ A History of Mathematics. John Wiley & Sons. 1991. pp. 442–443. ISBN 978-0-471-54397-8. https://archive.org/details/historyofmathema00boye/page/442. 
  10. ↑ Pi Unleashed. Springer-Verlag. 2006. p. 166. ISBN 978-3-540-66572-4. https://books.google.com/books?id=QwwcmweJCDQC&pg=PA166. 

Further reading



Retrieved from "https://handwiki.org/wiki/index.php?title=Mathematical_notation&oldid=4347806"