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'{{for|information on rendering mathematical formulae|Help:Formula|Help:Displaying a formula|Wikipedia:Manual of Style/Mathematics}} {{See Also|List of mathematical symbols}} '''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 [[sine|{{math|sin}}]]; operator symbols such as "[[Plus and minus signs#Plus sign|+]]"; [[Relation (math)|relational symbols]] such as "<"; conceptual symbols such as [[limit (mathematics)|{{math|lim}}]] and [[derivative|''dy/dx'']]; [[equation]]s and complex diagrammatic notations such as [[Penrose graphical notation]] and [[Coxeter–Dynkin diagram]]s.<ref name=":0">{{Cite web|date=2020-03-01|title=Compendium of Mathematical Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/|access-date=2020-08-08|website=Math Vault|language=en-US}}</ref><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> ==Definition== A 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]]. 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== 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. 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]]. ==Precise semantic meaning== 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]]. 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''", "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 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).<ref name=":0" /> 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. ==History== {{main article|History of mathematical notation}} ===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. 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). ===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. 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. ===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 [[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]]). 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. ===Computerized notation=== Mathematically oriented markup languages such as [[TeX]], [[LaTeX]] and, more recently, [[MathML]], are powerful enough to express a wide variety of mathematical notations. Theorem-proving software naturally comes with its own notations for mathematics; 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. ==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 alphabets]] to denote a wide variety of mathematical objects and variables. In some occasions, certain [[Hebrew alphabet|Hebrew alphabets]] are also used (such as in the context of [[Infinite cardinal|infinite cardinals]]).<ref>{{Cite web|date=2020-03-20|title=Greek/Hebrew/Latin-based Symbols in Mathematics|url=https://mathvault.ca/hub/higher-math/math-symbols/greek-hebrew-latin-symbols/|access-date=2020-08-08|website=Math Vault|language=en-US}}</ref> Some mathematical notations are mostly diagrammatic, and so are almost entirely script independent. Examples are [[Penrose graphical notation]] and [[Coxeter–Dynkin diagram]]s. Braille-based mathematical notations used by blind people include [[Nemeth Braille]] and [[GS8 Braille]]. ==See also== * [[Abuse of notation]] * [[Begriffsschrift]] * [[Glossary of mathematical symbols]] ** [[Bourbaki dangerous bend symbol]] * [[History of mathematical notation]] * [[ISO 31-11]] * [[ISO 80000-2]] * [[Knuth's up-arrow notation]] * [[Mathematical Alphanumeric Symbols]] * [[Notation in probability and statistics]] * [[Language of mathematics]] * [[Scientific notation]] * [[Semasiography]] * [[Table of mathematical symbols]] * [[Typographical conventions in mathematical formulae]] * [[Vector notation]] * [[Modern Arabic mathematical notation]] ==Notes== <references/> ==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}} *{{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== {{Commonscat}} *[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 [[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. {{DEFAULTSORT:Mathematical Notation}} [[Category:Mathematical notation| ]]'