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'{{Short description|System of symbolic representation}} {{For|information on rendering mathematical formulae|Help:Displaying a formula|Wikipedia:Manual of Style/Mathematics}} {{More citations needed|date=June 2022}} '''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]]. 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== {{Main|Glossary of mathematical symbols}} The use of many symbols is the basis of mathematical notation. They play a similar role as words in [[natural language]]s. They may play different roles in mathematical notation similarly as verbs, adjective and nouns play different roles in a sentence. === Letters as symbols=== 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]]. In order to have more symbols, and for allowing related mathematical objects to be represented by related symbols, [[diacritic]]s, [[subscript]]s and [[superscript]]s are often used. For example, <math>\hat {f'_1}</math> may denote the [[Fourier transform]] of the [[derivative]] of a [[function (mathematics)|function]] called <math>f_1.</math> === Other symbols === Symbols are not only used for naming mathematical objects. They can be used for [[operation (mathematics)|operation]]s <math>(+, -, /, \oplus, \ldots),</math> for [[relation (mathematics)|relation]]s <math>(=, <, \le, \sim, \equiv, \ldots),</math> for [[logical connective]]s <math>(\implies, \land, \lor, \ldots),</math> for [[quantifier (logic)|quantifier]]s <math>(\forall, \exists),</math> and for other purposes. Some symbols are similar to Latin or Greek letters, some are obtained by deforming letters, some are traditional [[typographic symbol]]s, but many have been specially designed for mathematics. ==Expressions== {{Unsourced section|date=June 2022}} An [[expression (mathematics)|expression]] is a finite combination of [[glossary of mathematical symbols|symbols]] that is [[well-formed formula|well-formed]] according to rules that depend on the context. In general, an expression denotes or names a [[mathematical object]], and plays therefore in the [[language of mathematics]] the role of a [[noun phrase]] in the natural language. An expression contains often some [[operator (mathematics)|operator]]s, and may therefore be ''evaluated'' by the action of the operators in it. For example, <math>3+2</math> is an expression in which the operator <math>+</math> can be evaluated for giving the result <math>5.</math> So, <math>3+2</math> and <math>5</math> are two different expressions that represent the same number. This is the meaning of the equality <math>3+2=5.</math> A more complicated example is given by the expression<math display = inline>\int_a^b xdx</math> that can be evaluated to <math display=inline>\frac {b^2}2-\frac {a^2}2.</math> Although the resulting expression contains the operators of [[division (mathematics)|division]], [[subtraction]] and [[exponentiation]], it cannot be evaluated further because {{mvar|a}} and {{mvar|b}} denote unspecified numbers. Mathematical notation is a system of symbols and conventions used to represent mathematical concepts, expressions, equations, and relationships in a concise and standardized way. It allows mathematicians, scientists, and engineers to communicate complex ideas and calculations effectively. In mathematical notation, various symbols, letters, and characters are used to denote variables, constants, operations, functions, sets, and other mathematical entities. For example, the symbol "+" represents addition, "x" often denotes a variable, and "π" represents the mathematical constant pi. Common mathematical notations include: Arithmetic Operations: "+" for addition "-" for subtraction "*" or "×" for multiplication "/" or "÷" for division Exponents and Powers: "^" or "**" for exponentiation (e.g., x^2 means x raised to the power of 2) Functions: "f(x)" represents a function of x "sin(x)" represents the sine function of x "log(x)" represents the logarithm function of x, usually base 10 Greek Letters: α (alpha), β (beta), γ (gamma), δ (delta), etc. are often used as variables in mathematics. Set Notation: "{}" for denoting sets (e.g., {1, 2, 3} represents the set of integers from 1 to 3) Equations and Inequalities: "=" for equality (e.g., x = 3 means x is equal to 3) "<" for less than (e.g., x < 5 means x is less than 5) ">" for greater than (e.g., x > 2 means x is greater than 2) These are just a few examples of the numerous symbols and notations used in mathematics. Different branches of mathematics often have their own specialized notations, but they all serve the purpose of making mathematical ideas more precise and easier to communicate.<ref>https://www.calculatestudy.com/scientific-notation-calculator</ref> ==History== {{Main|History of mathematical notation}} ===Numbers=== It is believed that a notation to represent [[number]]s was first developed at least 50,000 years ago{{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. The concept of [[zero]] and the introduction of a notation for it are important developments in early mathematics, which predates for centuries the concept of zero as a number. It was used as a placeholder by the [[Babylonian numerals|Babylonians]] and [[Greek numerals|Greek Egyptians]], and then as an [[integer]] by the [[Maya numerals|Mayans]], [[Indian numerals|Indians]] and [[Arabic numerals|Arabs]] (see [[History of zero|the history of zero]]). ===Modern notation=== Until the 16th century, mathematics was essentially [[rhetorical algebra|rhetorical]], in the sense that everything but explicit numbers was expressed in words. However, some authors such as [[Diophantus]] used some symbols as abbreviations. The first systematic use of formulas, and, in particular the use of symbols ([[variable (mathematics)|variables]]) for unspecified numbers is generally attributed to [[François Viète]] (16th century). However, he used different symbols than those that are now standard. Later, [[René Descartes]] (17th century) introduced the modern notation for variables and [[equation]]s; in particular, the use of <math>x,y,z</math> for [[unknown (mathematics)|unknown]] quantities and <math>a,b,c</math> for known ones ([[constant (mathematics)|constant]]s). He introduced also the notation {{mvar|i}} and the term "imaginary" for the [[imaginary unit]]. The 18th and 19th centuries saw the standardization of mathematical notation as used today. [[Leonhard Euler]] was responsible for many of the notations currently in use: the [[functional notation]] <math>f(x),</math> {{math|''e''}} for the base of the natural logarithm, <math display=inline>\sum</math> for [[summation]], etc.{{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]]). Since then many new notations have been introduced, often specific to a particular area of mathematics. Some notations are named after their inventors, such as [[Leibniz's notation]], [[Legendre symbol]], [[Einstein's summation convention]], etc. ===Typesetting=== General [[typesetting system]]s are generally not well suited for mathematical notation. One of the reasons is that, in mathematical notation, the symbols are often arranged in two-dimensional figures such as in :<math>\sum_{n=0}^\infty \frac {\begin{bmatrix}a&b\\c&d\end{bmatrix}^n}{n!}.</math> [[TeX]] is a mathematically oriented typesetting system that was created in [[1978]] by [[Donald Knuth]]. It is widely used in mathematics, through its extension called [[LaTeX]], and is a ''de facto'' standard. (The above expression is written in LaTeX.) More recently, another approach for mathematical typesetting is provided by [[MathML]]. However, it is not well supported in web browsers, which is its primary target. [[File:01 Kreiszahl.svg|thumb|center|An unusual display of {{pi}} allowed by [[TeX]] (European style, with a comma as a [[decimal separator]])]] ==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''²) 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. In some occasions, certain [[Hebrew alphabet|Hebrew letter]]s are also used (such as in the context of [[infinite cardinal]]s). Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are [[Penrose graphical notation]] and [[Coxeter–Dynkin diagram]]s. Braille-based mathematical notations used by blind people include [[Nemeth Braille]] and [[GS8 Braille]]. ==See also== * [[Abuse of notation]] * [[Begriffsschrift]] * [[Glossary of mathematical symbols]] ** [[Bourbaki dangerous bend symbol]] * [[History of mathematical notation]] * [[ISO 31-11]] * [[Knuth's up-arrow notation]] * [[List of mathematical symbols]] * [[Mathematical Alphanumeric Symbols]] * [[Mathematical formula]] * [[Notation in probability and statistics]] * [[Language of mathematics]] * [[Scientific notation]] * [[Semasiography]] * [[Table of mathematical symbols]] * [[Vector notation]] * [[Modern Arabic mathematical notation]] ==Notes== <references/> ==References== * {{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== {{Commons category}} *[http://jeff560.tripod.com/mathsym.html Earliest Uses of Various Mathematical Symbols] *[http://www.apronus.com/math/mrwmath.htm Mathematical ASCII Notation] how to type math notation in any text editor. *[http://www.cut-the-knot.org/language/index.shtml Mathematics as a Language] at [[Alexander Bogomolny#Cut-the-Knot|Cut-the-Knot]] *[[Stephen Wolfram]]: [http://www.stephenwolfram.com/publications/mathematical-notation-past-future/ Mathematical Notation: Past and Future]. October 2000. Transcript of a keynote address presented at [[MathML]] and Math on the Web: MathML International Conference. {{Mathematical symbols notation language}} {{DEFAULTSORT:Mathematical Notation}} [[Category:Mathematical notation| ]]'