Constant (mathematics) - Wikipedia

Constant (mathematics) · A fixed and well-defined number or other non-changing mathematical object. The terms mathematical constant or physical constant are ... Constant(mathematics) FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Functionorvaluewhichdoesnotchange Foranarrowertreatmentrelatedtothissubject,seeMathematicalconstant. Inmathematics,thewordconstantconveysmultiplemeanings.Asanadjective,itreferstonon-variance(i.e.unchangingwithrespecttosomeothervalue);asanoun,ithastwodifferentmeanings: Afixedandwell-definednumberorothernon-changingmathematicalobject.Thetermsmathematicalconstantorphysicalconstantaresometimesusedtodistinguishthismeaning.[1] Afunctionwhosevalueremainsunchanged(i.e.,aconstantfunction).[2]Suchaconstantiscommonlyrepresentedbyavariablewhichdoesnotdependonthemainvariable(s)inquestion. Forexample,ageneralquadraticfunctioniscommonlywrittenas: a x 2 + b x + c , {\displaystyleax^{2}+bx+c\,,} wherea,bandcareconstants(orparameters),andxavariable—aplaceholderfortheargumentofthefunctionbeingstudied.Amoreexplicitwaytodenotethisfunctionis x ↦ a x 2 + b x + c , {\displaystylex\mapstoax^{2}+bx+c\,,} whichmakesthefunction-argumentstatusofx(andbyextensiontheconstancyofa,bandc)clear.Inthisexamplea,bandcarecoefficientsofthepolynomial.Sincecoccursinatermthatdoesnotinvolvex,itiscalledtheconstanttermofthepolynomialandcanbethoughtofasthecoefficientofx0.Moregenerally,anypolynomialtermorexpressionofdegreezero(novariable)isaconstant.[3]: 18  Contents 1Constantfunction 2Context-dependence 3Notablemathematicalconstants 4Constantsincalculus 4.1Examples 5Seealso 6References 7Externallinks Constantfunction[edit] Mainarticles:ConstantfunctionandNullary Aconstantmaybeusedtodefineaconstantfunctionthatignoresitsargumentsandalwaysgivesthesamevalue.[4]Aconstantfunctionofasinglevariable,suchas f ( x ) = 5 {\displaystylef(x)=5} ,hasagraphofahorizontallineparalleltothex-axis.[5]Suchafunctionalwaystakesthesamevalue(inthiscase5),becausethevariabledoesnotappearintheexpressiondefiningthefunction. Graphof f ( x ) = 5 {\displaystylef(x)=5} . Context-dependence[edit] Thecontext-dependentnatureoftheconceptof"constant"canbeseeninthisexamplefromelementarycalculus: d d x 2 x = lim h → 0 2 x + h − 2 x h = lim h → 0 2 x 2 h − 1 h = 2 x lim h → 0 2 h − 1 h since  x  isconstant(i.e.doesnotdependon  h ) = 2 x ⋅ c o n s t a n t , where  c o n s t a n t  meansnotdependingon  x . {\displaystyle{\begin{aligned}{\frac{d}{dx}}2^{x}&=\lim_{h\to0}{\frac{2^{x+h}-2^{x}}{h}}=\lim_{h\to0}2^{x}{\frac{2^{h}-1}{h}}\\[8pt]&=2^{x}\lim_{h\to0}{\frac{2^{h}-1}{h}}&&{\text{since}}x{\text{isconstant(i.e.doesnotdependon}}h{\text{)}}\\[8pt]&=2^{x}\cdot\mathbf{constant,}&&{\text{where}}\mathbf{constant}{\text{meansnotdependingon}}x.\end{aligned}}} "Constant"meansnotdependingonsomevariable;notchangingasthatvariablechanges.Inthefirstcaseabove,itmeansnotdependingon h;inthesecond,itmeansnotdependingon x.Aconstantinanarrowercontextcouldberegardedasavariableinabroadercontext. Notablemathematicalconstants[edit] Mainarticle:Mathematicalconstant Somevaluesoccurfrequentlyinmathematicsandareconventionallydenotedbyaspecificsymbol.Thesestandardsymbolsandtheirvaluesarecalledmathematicalconstants.Examplesinclude: 0(zero). 1(one),thenaturalnumberafterzero. π(pi),theconstantrepresentingtheratioofacircle'scircumferencetoitsdiameter,approximatelyequalto3.141592653589793238462643.[6] e,approximatelyequalto2.718281828459045235360287.[7] i,theimaginaryunitsuchthati2=−1.[8] 2 {\displaystyle{\sqrt{2}}} (squarerootof2),thelengthofthediagonalofasquarewithunitsides,approximatelyequalto1.414213562373095048801688.[9] φ(goldenratio),approximatelyequalto1.618033988749894848204586,oralgebraically, 1 + 5 2 {\displaystyle1+{\sqrt{5}}\over2} .[10] Constantsincalculus[edit] Incalculus,constantsaretreatedinseveraldifferentwaysdependingontheoperation.Forexample,thederivative(rateofchange)ofaconstantfunctioniszero.Thisisbecauseconstants,bydefinition,donotchange.Theirderivativeishencezero. Conversely,whenintegratingaconstantfunction,theconstantismultipliedbythevariableofintegration. Duringtheevaluationofalimit,aconstantremainsthesameasitwasbeforeandafterevaluation. Integrationofafunctionofonevariableofteninvolvesaconstantofintegration.Thisarisesduetothefactthattheintegralistheinverse(opposite)ofthederivativemeaningthattheaimofintegrationistorecovertheoriginalfunctionbeforedifferentiation.Thederivativeofaconstantfunctioniszero,asnotedabove,andthedifferentialoperatorisalinearoperator,sofunctionsthatonlydifferbyaconstanttermhavethesamederivative.Toacknowledgethis,aconstantofintegrationisaddedtoanindefiniteintegral;thisensuresthatallpossiblesolutionsareincluded.Theconstantofintegrationisgenerallywrittenas'c',andrepresentsaconstantwithafixedbutundefinedvalue. Examples[edit] Iffistheconstantfunctionsuchthat f ( x ) = 72 {\displaystylef(x)=72} foreveryxthen f ′ ( x ) = 0 ∫ f ( x ) d x = 72 x + c lim x → 0 f ( x ) = 72 {\displaystyle{\begin{aligned}f'(x)&=0\\\intf(x)\,dx&=72x+c\\\lim_{x\rightarrow0}f(x)&=72\end{aligned}}} Seealso[edit] Constant(disambiguation) Expression Levelset Listofmathematicalconstants Physicalconstant References[edit] ^"DefinitionofCONSTANT".www.merriam-webster.com.Retrieved2021-11-09. ^Weisstein,EricW."Constant".mathworld.wolfram.com.Retrieved2020-08-08. ^Foerster,PaulA.(2006).AlgebraandTrigonometry:FunctionsandApplications,Teacher'sEdition(Classics ed.).UpperSaddleRiver,NJ:PrenticeHall.ISBN 0-13-165711-9. ^Tanton,James(2005).Encyclopediaofmathematics.NewYork:FactsonFile.ISBN 0-8160-5124-0.OCLC 56057904. ^"Algebra".tutorial.math.lamar.edu.Retrieved2021-11-09. ^Arndt,Jörg;Haenel,Christoph(2001).Pi–Unleashed.Springer.p. 240.ISBN 978-3540665724. ^Weisstein,EricW."e".mathworld.wolfram.com.Retrieved2021-11-09. ^Weisstein,EricW."i".mathworld.wolfram.com.Retrieved2021-11-09. ^Weisstein,EricW."Pythagoras'sConstant".mathworld.wolfram.com.Retrieved2021-11-09. ^Weisstein,EricW."GoldenRatio".mathworld.wolfram.com.Retrieved2021-11-09. Externallinks[edit] MediarelatedtoConstantsatWikimediaCommons Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Constant_(mathematics)&oldid=1091159181" Categories:AlgebraElementarymathematicsConstantsHiddencategories:ArticleswithshortdescriptionShortdescriptionisdifferentfromWikidataCommonscategorylinkisonWikidata Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Inotherprojects WikimediaCommons Languages AfrikaansالعربيةAzərbaycancaBân-lâm-gúБашҡортсаБеларускаяБеларуская(тарашкевіца)БългарскиCatalàЧӑвашлаČeštinaCymraegDanskEestiEspañolEsperantoEuskaraفارسیFøroysktFrançaisGàidhligGalegoХальмг한국어HrvatskiBahasaIndonesiaÍslenskaItalianoҚазақшаKreyòlayisyenLatinaLietuviųMagyarМакедонскиമലയാളംNederlands日本語Oʻzbekcha/ўзбекчаਪੰਜਾਬੀPortuguêsРусскийSimpleEnglishSlovenčinaSlovenščinaکوردیСрпски/srpskiSuomiSvenskaTagalogதமிழ்ไทยTürkçeУкраїнськаاردوTiếngViệt粵語中文 Editlinks