Mathematics Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! == Symbolic notation and terminology == {{main|Mathematical notation|Language of mathematics|Glossary of mathematics}} [[File:Sigma summation notation.svg|thumb|An explanation of the sigma (Σ) [[summation]] notation]] Mathematical notation is widely used in science and [[engineering]] for representing complex [[concept]]s and [[property (philosophy)|properties]] in a concise, unambiguous, and accurate way. This notation consists of [[glossary of mathematical symbols|symbols]] used for representing [[operation (mathematics)|operation]]s, unspecified numbers, [[relation (mathematics)|relation]]s and any other mathematical objects, and then assembling them into [[expression (mathematics)|expression]]s and formulas.<ref>{{cite conference |last=Wolfram |first=Stephan |date=October 2000 |author-link=Stephen Wolfram |title=Mathematical Notation: Past and Future |conference=MathML and Math on the Web: MathML International Conference 2000, Urbana Champaign, USA |url=https://www.stephenwolfram.com/publications/mathematical-notation-past-future/ |url-status=live |archive-url=https://web.archive.org/web/20221116150905/https://www.stephenwolfram.com/publications/mathematical-notation-past-future/ |archive-date=November 16, 2022 |access-date=February 3, 2024}}</ref> More precisely, numbers and other mathematical objects are represented by symbols called variables, which are generally [[Latin alphabet|Latin]] or [[Greek alphabet|Greek]] letters, and often include [[subscript]]s. Operation and relations are generally represented by specific [[Glossary of mathematical symbols|symbols]] or [[glyph]]s,<ref>{{cite journal |last1=Douglas |first1=Heather |last2=Headley |first2=Marcia Gail |last3=Hadden |first3=Stephanie |last4=LeFevre |first4=Jo-Anne |author4-link=Jo-Anne LeFevre |date=December 3, 2020 |title=Knowledge of Mathematical Symbols Goes Beyond Numbers |journal=Journal of Numerical Cognition |volume=6 |issue=3 |pages=322–354 |doi=10.5964/jnc.v6i3.293 |doi-access=free |eissn=2363-8761 |s2cid=228085700}}</ref> such as {{math|+}} ([[plus sign|plus]]), {{math|×}} ([[multiplication sign|multiplication]]), <math display =inline>\int</math> ([[integral sign|integral]]), {{math|1==}} ([[equals sign|equal]]), and {{math|<}} ([[less-than sign|less than]]).<ref name=AMS>{{cite web |last1=Letourneau |first1=Mary |last2=Wright Sharp |first2=Jennifer |date=October 2017 |title=AMS Style Guide |page=75 |publisher=[[American Mathematical Society]] |url=https://www.ams.org/publications/authors/AMS-StyleGuide-online.pdf |url-status=live |archive-url=https://web.archive.org/web/20221208063650/https://www.ams.org//publications/authors/AMS-StyleGuide-online.pdf |archive-date=December 8, 2022 |access-date=February 3, 2024}}</ref> All these symbols are generally grouped according to specific rules to form expressions and formulas.<ref>{{cite journal |last1=Jansen |first1=Anthony R. |last2=Marriott |first2=Kim |last3=Yelland |first3=Greg W. |year=2000 |title=Constituent Structure in Mathematical Expressions |journal=Proceedings of the Annual Meeting of the Cognitive Science Society |volume=22 |publisher=[[University of California Merced]] |eissn=1069-7977 |oclc=68713073 |url=https://escholarship.org/content/qt35r988q9/qt35r988q9.pdf |url-status=live |archive-url=https://web.archive.org/web/20221116152222/https://escholarship.org/content/qt35r988q9/qt35r988q9.pdf |archive-date=November 16, 2022 |access-date=February 3, 2024}}</ref> Normally, expressions and formulas do not appear alone, but are included in sentences of the current language, where expressions play the role of [[noun phrase]]s and formulas play the role of [[clause]]s. Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous [[Technical definition|definitions]] that provide a standard foundation for communication. An axiom or [[postulate]] is a mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a [[conjecture]]. Through a series of rigorous arguments employing [[deductive reasoning]], a statement that is [[formal proof|proven]] to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a [[Lemma (mathematics)|lemma]]. A proven instance that forms part of a more general finding is termed a [[corollary]].<ref>{{cite book |last=Rossi |first=Richard J. |year=2006 |title=Theorems, Corollaries, Lemmas, and Methods of Proof |series=Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts |publisher=[[John Wiley & Sons]] |pages=1–14, 47–48 |isbn=978-0-470-04295-3 |lccn=2006041609 |oclc=64085024}}</ref> Numerous technical terms used in mathematics are [[neologism]]s, such as ''[[polynomial]]'' and ''[[homeomorphism]]''.<ref>{{cite web |url=https://mathshistory.st-andrews.ac.uk/Miller/mathword/ |title=Earliest Uses of Some Words of Mathematics |website=MacTutor |publisher=[[University of St. Andrews]] |publication-place=Scotland, UK |url-status=live |archive-url=https://web.archive.org/web/20220929032236/https://mathshistory.st-andrews.ac.uk/Miller/mathword/ |archive-date=September 29, 2022 |access-date=February 3, 2024}}</ref> Other technical terms are words of the common language that are used in an accurate meaning that may differ slightly from their common meaning. For example, in mathematics, "[[logical disjunction|or]]" means "one, the other or both", while, in common language, it is either ambiguous or means "one or the other but not both" (in mathematics, the latter is called "[[exclusive or]]"). Finally, many mathematical terms are common words that are used with a completely different meaning.<ref>{{cite journal |last=Silver |first=Daniel S. |date=November–December 2017 |title=The New Language of Mathematics |journal=The American Scientist |volume=105 |number=6 |pages=364–371 |publisher=[[Sigma Xi]] |doi=10.1511/2017.105.6.364 |doi-access=free |issn=0003-0996 |lccn=43020253 |oclc=1480717 |s2cid=125455764}}</ref> This may lead to sentences that are correct and true mathematical assertions, but appear to be nonsense to people who do not have the required background. For example, "every [[free module]] is [[flat module|flat]]" and "a [[field (mathematics)|field]] is always a [[ring (mathematics)|ring]]". 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