Number - Wikipedia

A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Number FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Mathematicalabstractionofthecommonconcept Forotheruses,seeNumber(disambiguation). Subsetsofthecomplexnumbers Anumberisamathematicalobjectusedtocount,measure,andlabel.Theoriginalexamplesarethenaturalnumbers1,2,3,4,andsoforth.[1]Numberscanberepresentedinlanguagewithnumberwords.Moreuniversally,individualnumberscanberepresentedbysymbols,callednumerals;forexample,"5"isanumeralthatrepresentsthenumberfive.Asonlyarelativelysmallnumberofsymbolscanbememorized,basicnumeralsarecommonlyorganizedinanumeralsystem,whichisanorganizedwaytorepresentanynumber.ThemostcommonnumeralsystemistheHindu–Arabicnumeralsystem,whichallowsfortherepresentationofanynumberusingacombinationoftenfundamentalnumericsymbols,calleddigits.[2][3]Inadditiontotheiruseincountingandmeasuring,numeralsareoftenusedforlabels(aswithtelephonenumbers),forordering(aswithserialnumbers),andforcodes(aswithISBNs).Incommonusage,anumeralisnotclearlydistinguishedfromthenumberthatitrepresents. Inmathematics,thenotionofanumberhasbeenextendedoverthecenturiestoinclude0,[4]negativenumbers,[5]rationalnumberssuchasonehalf ( 1 2 ) {\displaystyle\left({\tfrac{1}{2}}\right)} ,realnumberssuchasthesquarerootof2 ( 2 ) {\displaystyle\left({\sqrt{2}}\right)} andπ,[6]andcomplexnumbers[7]whichextendtherealnumberswithasquarerootof−1(anditscombinationswithrealnumbersbyaddingorsubtractingitsmultiples).[5]Calculationswithnumbersaredonewitharithmeticaloperations,themostfamiliarbeingaddition,subtraction,multiplication,division,andexponentiation.Theirstudyorusageiscalledarithmetic,atermwhichmayalsorefertonumbertheory,thestudyofthepropertiesofnumbers. Besidestheirpracticaluses,numbershaveculturalsignificancethroughouttheworld.[8][9]Forexample,inWesternsociety,thenumber13isoftenregardedasunlucky,and"amillion"maysignify"alot"ratherthananexactquantity.[8]Thoughitisnowregardedaspseudoscience,beliefinamysticalsignificanceofnumbers,knownasnumerology,permeatedancientandmedievalthought.[10]NumerologyheavilyinfluencedthedevelopmentofGreekmathematics,stimulatingtheinvestigationofmanyproblemsinnumbertheorywhicharestillofinteresttoday.[10] Duringthe19thcentury,mathematiciansbegantodevelopmanydifferentabstractionswhichsharecertainpropertiesofnumbers,andmaybeseenasextendingtheconcept.Amongthefirstwerethehypercomplexnumbers,whichconsistofvariousextensionsormodificationsofthecomplexnumbersystem.Inmodernmathematics,numbersystems(sets)areconsideredimportantspecialexamplesofmoregeneralcategoriessuchasringsandfields,andtheapplicationoftheterm"number"isamatterofconvention,withoutfundamentalsignificance.[11] Contents 1History 1.1Numerals 1.2Firstuseofnumbers 1.3Zero 1.4Negativenumbers 1.5Rationalnumbers 1.6Irrationalnumbers 1.7Transcendentalnumbersandreals 1.8Infinityandinfinitesimals 1.9Complexnumbers 1.10Primenumbers 2Mainclassification 2.1Naturalnumbers 2.2Integers 2.3Rationalnumbers 2.4Realnumbers 2.5Complexnumbers 3Subclassesoftheintegers 3.1Evenandoddnumbers 3.2Primenumbers 3.3Otherclassesofintegers 4Subclassesofthecomplexnumbers 4.1Algebraic,irrationalandtranscendentalnumbers 4.2Constructiblenumbers 4.3Computablenumbers 5Extensionsoftheconcept 5.1p-adicnumbers 5.2Hypercomplexnumbers 5.3Transfinitenumbers 5.4Nonstandardnumbers 6Seealso 7Notes 8References 9Externallinks History[edit] Thissection'sfactualaccuracyisdisputed.RelevantdiscussionmaybefoundonTalk:Number.Pleasehelptoensurethatdisputedstatementsarereliablysourced.(November2014)(Learnhowandwhentoremovethistemplatemessage) Numerals[edit] Mainarticle:Numeralsystem Numbersshouldbedistinguishedfromnumerals,thesymbolsusedtorepresentnumbers.TheEgyptiansinventedthefirstcipherednumeralsystem,andtheGreeksfollowedbymappingtheircountingnumbersontoIonianandDoricalphabets.[12]Romannumerals,asystemthatusedcombinationsoflettersfromtheRomanalphabet,remaineddominantinEuropeuntilthespreadofthesuperiorHindu–Arabicnumeralsystemaroundthelate14thcentury,andtheHindu–Arabicnumeralsystemremainsthemostcommonsystemforrepresentingnumbersintheworldtoday.[13]Thekeytotheeffectivenessofthesystemwasthesymbolforzero,whichwasdevelopedbyancientIndianmathematiciansaround500AD.[13] Firstuseofnumbers[edit] Mainarticle:Historyofancientnumeralsystems Bonesandotherartifactshavebeendiscoveredwithmarkscutintothemthatmanybelievearetallymarks.[14]Thesetallymarksmayhavebeenusedforcountingelapsedtime,suchasnumbersofdays,lunarcyclesorkeepingrecordsofquantities,suchasofanimals. Atallyingsystemhasnoconceptofplacevalue(asinmoderndecimalnotation),whichlimitsitsrepresentationoflargenumbers.Nonethelesstallyingsystemsareconsideredthefirstkindofabstractnumeralsystem. ThefirstknownsystemwithplacevaluewastheMesopotamianbase 60system(c. 3400 BC)andtheearliestknownbase 10systemdatesto3100 BCinEgypt.[15] Zero[edit] ThefirstknowndocumenteduseofzerodatestoAD628,andappearedintheBrāhmasphuṭasiddhānta,themainworkoftheIndianmathematicianBrahmagupta.Hetreated 0asanumberanddiscussedoperationsinvolvingit,includingdivision.Bythistime(the7th century)theconcepthadclearlyreachedCambodiaasKhmernumerals,anddocumentationshowstheidealaterspreadingtoChinaandtheIslamicworld. Thenumber605inKhmernumerals,fromaninscriptionfrom683AD.Earlyuseofzeroasadecimalfigure. Brahmagupta'sBrāhmasphuṭasiddhāntaisthefirstbookthatmentionszeroasanumber,henceBrahmaguptaisusuallyconsideredthefirsttoformulatetheconceptofzero.Hegaverulesofusingzerowithnegativeandpositivenumbers,suchas"zeroplusapositivenumberisapositivenumber,andanegativenumberpluszeroisthenegativenumber."TheBrāhmasphuṭasiddhāntaistheearliestknowntexttotreatzeroasanumberinitsownright,ratherthanassimplyaplaceholderdigitinrepresentinganothernumberaswasdonebytheBabyloniansorasasymbolforalackofquantityaswasdonebyPtolemyandtheRomans. Theuseof0asanumbershouldbedistinguishedfromitsuseasaplaceholdernumeralinplace-valuesystems.Manyancienttextsused 0.BabylonianandEgyptiantextsusedit.Egyptiansusedthewordnfrtodenotezero balanceindoubleentryaccounting.IndiantextsusedaSanskritwordShunyeorshunyatorefertotheconceptofvoid.Inmathematicstextsthiswordoftenreferstothenumberzero.[16]Inasimilarvein,Pāṇini(5thcenturyBC)usedthenull(zero)operatorintheAshtadhyayi,anearlyexampleofanalgebraicgrammarfortheSanskritlanguage(alsoseePingala). ThereareotherusesofzerobeforeBrahmagupta,thoughthedocumentationisnotascompleteasitisintheBrāhmasphuṭasiddhānta. RecordsshowthattheAncientGreeksseemedunsureaboutthestatusof 0asanumber:theyaskedthemselves"howcan'nothing'besomething?"leadingtointerestingphilosophicaland,bytheMedievalperiod,religiousargumentsaboutthenatureandexistenceof 0andthevacuum.TheparadoxesofZenoofEleadependinpartontheuncertaininterpretationof 0.(TheancientGreeksevenquestionedwhether 1wasanumber.) ThelateOlmecpeopleofsouth-centralMexicobegantouseasymbolforzero,ashellglyph,intheNewWorld,possiblybythe4thcenturyBCbutcertainlyby40 BC,whichbecameanintegralpartofMayanumeralsandtheMayacalendar.Mayaarithmeticusedbase 4andbase 5writtenasbase 20.GeorgeI.Sánchezin1961reportedabase 4,base 5"finger"abacus.[17][better source needed] By130AD,Ptolemy,influencedbyHipparchusandtheBabylonians,wasusingasymbolfor 0(asmallcirclewithalongoverbar)withinasexagesimalnumeralsystemotherwiseusingalphabeticGreeknumerals.Becauseitwasusedalone,notasjustaplaceholder,thisHellenisticzerowasthefirstdocumenteduseofatruezerointheOldWorld.InlaterByzantinemanuscriptsofhisSyntaxisMathematica(Almagest),theHellenisticzerohadmorphedintotheGreekletterOmicron(otherwisemeaning 70). AnothertruezerowasusedintablesalongsideRomannumeralsby525(firstknownusebyDionysiusExiguus),butasaword,nullameaningnothing,notasasymbol.Whendivisionproduced 0asaremainder,nihil,alsomeaningnothing,wasused.Thesemedievalzeroswereusedbyallfuturemedievalcomputists(calculatorsofEaster).Anisolateduseoftheirinitial,N,wasusedinatableofRomannumeralsbyBedeoracolleagueabout725,atruezerosymbol. Negativenumbers[edit] Furtherinformation:Historyofnegativenumbers Theabstractconceptofnegativenumberswasrecognizedasearlyas100–50BCinChina.TheNineChaptersontheMathematicalArtcontainsmethodsforfindingtheareasoffigures;redrodswereusedtodenotepositivecoefficients,blackfornegative.[18]ThefirstreferenceinaWesternworkwasinthe3rd centuryADinGreece.Diophantusreferredtotheequationequivalentto4x+20=0(thesolutionisnegative)inArithmetica,sayingthattheequationgaveanabsurdresult. Duringthe600s,negativenumberswereinuseinIndiatorepresentdebts.Diophantus'previousreferencewasdiscussedmoreexplicitlybyIndianmathematicianBrahmagupta,inBrāhmasphuṭasiddhāntain628,whousednegativenumberstoproducethegeneralformquadraticformulathatremainsinusetoday.However,inthe12th centuryinIndia,Bhaskaragivesnegativerootsforquadraticequationsbutsaysthenegativevalue"isinthiscasenottobetaken,foritisinadequate;peopledonotapproveofnegativeroots". Europeanmathematicians,forthemostpart,resistedtheconceptofnegativenumbersuntilthe17th century,althoughFibonacciallowednegativesolutionsinfinancialproblemswheretheycouldbeinterpretedasdebts(chapter 13ofLiberAbaci,1202)andlateraslosses(inFlos).RenéDescartescalledthemfalserootsastheycroppedupinalgebraicpolynomialsyethefoundawaytoswaptruerootsandfalserootsaswell.Atthesametime,theChinesewereindicatingnegativenumbersbydrawingadiagonalstrokethroughtheright-mostnon-zerodigitofthecorrespondingpositivenumber'snumeral.[19]ThefirstuseofnegativenumbersinaEuropeanworkwasbyNicolasChuquetduringthe15th century.Heusedthemasexponents,butreferredtothemas"absurdnumbers". Asrecentlyasthe18thcentury,itwascommonpracticetoignoreanynegativeresultsreturnedbyequationsontheassumptionthattheyweremeaningless. Rationalnumbers[edit] Itislikelythattheconceptoffractionalnumbersdatestoprehistorictimes.TheAncientEgyptiansusedtheirEgyptianfractionnotationforrationalnumbersinmathematicaltextssuchastheRhindMathematicalPapyrusandtheKahunPapyrus.ClassicalGreekandIndianmathematiciansmadestudiesofthetheoryofrationalnumbers,aspartofthegeneralstudyofnumbertheory.[citationneeded]ThebestknownoftheseisEuclid'sElements,datingtoroughly300 BC.OftheIndiantexts,themostrelevantistheSthanangaSutra,whichalsocoversnumbertheoryaspartofageneralstudyofmathematics. Theconceptofdecimalfractionsiscloselylinkedwithdecimalplace-valuenotation;thetwoseemtohavedevelopedintandem.Forexample,itiscommonfortheJainmathsutratoincludecalculationsofdecimal-fractionapproximationstopiorthesquarerootof2.[citationneeded]Similarly,Babylonianmathtextsusedsexagesimal(base 60)fractionswithgreatfrequency. Irrationalnumbers[edit] Furtherinformation:Historyofirrationalnumbers TheearliestknownuseofirrationalnumberswasintheIndianSulbaSutrascomposedbetween800and500 BC.[20][better source needed]ThefirstexistenceproofsofirrationalnumbersisusuallyattributedtoPythagoras,morespecificallytothePythagoreanHippasusofMetapontum,whoproduceda(mostlikelygeometrical)proofoftheirrationalityofthesquarerootof2.ThestorygoesthatHippasusdiscoveredirrationalnumberswhentryingtorepresentthesquarerootof2asafraction.However,Pythagorasbelievedintheabsolutenessofnumbers,andcouldnotaccepttheexistenceofirrationalnumbers.Hecouldnotdisprovetheirexistencethroughlogic,buthecouldnotacceptirrationalnumbers,andso,allegedlyandfrequentlyreported,hesentencedHippasustodeathbydrowning,toimpedespreadingofthisdisconcertingnews.[21][better source needed] The16thcenturybroughtfinalEuropeanacceptanceofnegativeintegralandfractionalnumbers.Bythe17th century,mathematiciansgenerallyuseddecimalfractionswithmodernnotation.Itwasnot,however,untilthe19thcenturythatmathematiciansseparatedirrationalsintoalgebraicandtranscendentalparts,andoncemoreundertookthescientificstudyofirrationals.IthadremainedalmostdormantsinceEuclid.In1872,thepublicationofthetheoriesofKarlWeierstrass(byhispupilE.Kossak),EduardHeine,[22]GeorgCantor,[23]andRichardDedekind[24]wasbroughtabout.In1869,CharlesMérayhadtakenthesamepointofdepartureasHeine,butthetheoryisgenerallyreferredtotheyear1872.Weierstrass'smethodwascompletelysetforthbySalvatorePincherle(1880),andDedekind'shasreceivedadditionalprominencethroughtheauthor'slaterwork(1888)andendorsementbyPaulTannery(1894).Weierstrass,Cantor,andHeinebasetheirtheoriesoninfiniteseries,whileDedekindfoundshisontheideaofacut(Schnitt)inthesystemofrealnumbers,separatingallrationalnumbersintotwogroupshavingcertaincharacteristicproperties.ThesubjecthasreceivedlatercontributionsatthehandsofWeierstrass,Kronecker,[25]andMéray. Thesearchforrootsofquinticandhigherdegreeequationswasanimportantdevelopment,theAbel–Ruffinitheorem(Ruffini1799,Abel1824)showedthattheycouldnotbesolvedbyradicals(formulasinvolvingonlyarithmeticaloperationsandroots).Henceitwasnecessarytoconsiderthewidersetofalgebraicnumbers(allsolutionstopolynomialequations).Galois(1832)linkedpolynomialequationstogrouptheorygivingrisetothefieldofGaloistheory. Continuedfractions,closelyrelatedtoirrationalnumbers(andduetoCataldi,1613),receivedattentionatthehandsofEuler,[26]andattheopeningofthe19th centurywerebroughtintoprominencethroughthewritingsofJosephLouisLagrange.OthernoteworthycontributionshavebeenmadebyDruckenmüller(1837),Kunze(1857),Lemke(1870),andGünther(1872).Ramus[27]firstconnectedthesubjectwithdeterminants,resulting,withthesubsequentcontributionsofHeine,[28]Möbius,andGünther,[29]inthetheoryofKettenbruchdeterminanten. Transcendentalnumbersandreals[edit] Furtherinformation:Historyofπ Theexistenceoftranscendentalnumbers[30]wasfirstestablishedbyLiouville(1844,1851).Hermiteprovedin1873thateistranscendentalandLindemannprovedin1882thatπistranscendental.Finally,Cantorshowedthatthesetofallrealnumbersisuncountablyinfinitebutthesetofallalgebraicnumbersiscountablyinfinite,sothereisanuncountablyinfinitenumberoftranscendentalnumbers. Infinityandinfinitesimals[edit] Furtherinformation:Historyofinfinity TheearliestknownconceptionofmathematicalinfinityappearsintheYajurVeda,anancientIndianscript,whichatonepointstates,"Ifyouremoveapartfrominfinityoraddaparttoinfinity,stillwhatremainsisinfinity."InfinitywasapopulartopicofphilosophicalstudyamongtheJainmathematiciansc.400 BC.Theydistinguishedbetweenfivetypesofinfinity:infiniteinoneandtwodirections,infiniteinarea,infiniteeverywhere,andinfiniteperpetually.Thesymbol ∞ {\displaystyle{\text{∞}}} isoftenusedtorepresentaninfinitequantity. AristotledefinedthetraditionalWesternnotionofmathematicalinfinity.Hedistinguishedbetweenactualinfinityandpotentialinfinity—thegeneralconsensusbeingthatonlythelatterhadtruevalue.GalileoGalilei'sTwoNewSciencesdiscussedtheideaofone-to-onecorrespondencesbetweeninfinitesets.ButthenextmajoradvanceinthetheorywasmadebyGeorgCantor;in1895hepublishedabookabouthisnewsettheory,introducing,amongotherthings,transfinitenumbersandformulatingthecontinuumhypothesis. Inthe1960s,AbrahamRobinsonshowedhowinfinitelylargeandinfinitesimalnumberscanberigorouslydefinedandusedtodevelopthefieldofnonstandardanalysis.Thesystemofhyperrealnumbersrepresentsarigorousmethodoftreatingtheideasaboutinfiniteandinfinitesimalnumbersthathadbeenusedcasuallybymathematicians,scientists,andengineerseversincetheinventionofinfinitesimalcalculusbyNewtonandLeibniz. Amoderngeometricalversionofinfinityisgivenbyprojectivegeometry,whichintroduces"idealpointsatinfinity",oneforeachspatialdirection.Eachfamilyofparallellinesinagivendirectionispostulatedtoconvergetothecorrespondingidealpoint.Thisiscloselyrelatedtotheideaofvanishingpointsinperspectivedrawing. Complexnumbers[edit] Furtherinformation:Historyofcomplexnumbers TheearliestfleetingreferencetosquarerootsofnegativenumbersoccurredintheworkofthemathematicianandinventorHeronofAlexandriainthe1stcenturyAD,whenheconsideredthevolumeofanimpossiblefrustumofapyramid.Theybecamemoreprominentwheninthe16th centuryclosedformulasfortherootsofthirdandfourthdegreepolynomialswerediscoveredbyItalianmathematicianssuchasNiccolòFontanaTartagliaandGerolamoCardano.Itwassoonrealizedthattheseformulas,evenifonewasonlyinterestedinrealsolutions,sometimesrequiredthemanipulationofsquarerootsofnegativenumbers. Thiswasdoublyunsettlingsincetheydidnotevenconsidernegativenumberstobeonfirmgroundatthetime.WhenRenéDescartescoinedtheterm"imaginary"forthesequantitiesin1637,heintendeditasderogatory.(Seeimaginarynumberforadiscussionofthe"reality"ofcomplexnumbers.)Afurthersourceofconfusionwasthattheequation ( − 1 ) 2 = − 1 − 1 = − 1 {\displaystyle\left({\sqrt{-1}}\right)^{2}={\sqrt{-1}}{\sqrt{-1}}=-1} seemedcapriciouslyinconsistentwiththealgebraicidentity a b = a b , {\displaystyle{\sqrt{a}}{\sqrt{b}}={\sqrt{ab}},} whichisvalidforpositiverealnumbersaandb,andwasalsousedincomplexnumbercalculationswithoneofa,bpositiveandtheothernegative.Theincorrectuseofthisidentity,andtherelatedidentity 1 a = 1 a {\displaystyle{\frac{1}{\sqrt{a}}}={\sqrt{\frac{1}{a}}}} inthecasewhenbothaandbarenegativeevenbedeviledEuler.Thisdifficultyeventuallyledhimtotheconventionofusingthespecialsymboliinplaceof − 1 {\displaystyle{\sqrt{-1}}} toguardagainstthismistake. The18thcenturysawtheworkofAbrahamdeMoivreandLeonhardEuler.DeMoivre'sformula(1730)states: ( cos ⁡ θ + i sin ⁡ θ ) n = cos ⁡ n θ + i sin ⁡ n θ {\displaystyle(\cos\theta+i\sin\theta)^{n}=\cosn\theta+i\sinn\theta} whileEuler'sformulaofcomplexanalysis(1748)gaveus: cos ⁡ θ + i sin ⁡ θ = e i θ . {\displaystyle\cos\theta+i\sin\theta=e^{i\theta}.} TheexistenceofcomplexnumberswasnotcompletelyaccepteduntilCasparWesseldescribedthegeometricalinterpretationin1799.CarlFriedrichGaussrediscoveredandpopularizeditseveralyearslater,andasaresultthetheoryofcomplexnumbersreceivedanotableexpansion.Theideaofthegraphicrepresentationofcomplexnumbershadappeared,however,asearlyas1685,inWallis'sDealgebratractatus. Alsoin1799,Gaussprovidedthefirstgenerallyacceptedproofofthefundamentaltheoremofalgebra,showingthateverypolynomialoverthecomplexnumbershasafullsetofsolutionsinthatrealm.ThegeneralacceptanceofthetheoryofcomplexnumbersisduetothelaborsofAugustinLouisCauchyandNielsHenrikAbel,andespeciallythelatter,whowasthefirsttoboldlyusecomplexnumberswithasuccessthatiswellknown.[peacock term] Gaussstudiedcomplexnumbersoftheforma+bi,whereaandbareintegral,orrational(andiisoneofthetworootsofx2+1=0).Hisstudent,GottholdEisenstein,studiedthetypea+bω,whereωisacomplexrootofx3−1=0.Othersuchclasses(calledcyclotomicfields)ofcomplexnumbersderivefromtherootsofunityxk−1=0forhighervaluesofk.ThisgeneralizationislargelyduetoErnstKummer,whoalsoinventedidealnumbers,whichwereexpressedasgeometricalentitiesbyFelixKleinin1893. In1850VictorAlexandrePuiseuxtookthekeystepofdistinguishingbetweenpolesandbranchpoints,andintroducedtheconceptofessentialsingularpoints.[clarificationneeded]Thiseventuallyledtotheconceptoftheextendedcomplexplane. Primenumbers[edit] Primenumbershavebeenstudiedthroughoutrecordedhistory.[citationneeded]EucliddevotedonebookoftheElementstothetheoryofprimes;initheprovedtheinfinitudeoftheprimesandthefundamentaltheoremofarithmetic,andpresentedtheEuclideanalgorithmforfindingthegreatestcommondivisoroftwonumbers. In240BC,EratosthenesusedtheSieveofEratosthenestoquicklyisolateprimenumbers.ButmostfurtherdevelopmentofthetheoryofprimesinEuropedatestotheRenaissanceandlatereras.[citationneeded] In1796,Adrien-MarieLegendreconjecturedtheprimenumbertheorem,describingtheasymptoticdistributionofprimes.OtherresultsconcerningthedistributionoftheprimesincludeEuler'sproofthatthesumofthereciprocalsoftheprimesdiverges,andtheGoldbachconjecture,whichclaimsthatanysufficientlylargeevennumberisthesumoftwoprimes.YetanotherconjecturerelatedtothedistributionofprimenumbersistheRiemannhypothesis,formulatedbyBernhardRiemannin1859.TheprimenumbertheoremwasfinallyprovedbyJacquesHadamardandCharlesdelaVallée-Poussinin1896.GoldbachandRiemann'sconjecturesremainunprovenandunrefuted. Mainclassification[edit] "Numbersystem"redirectshere.Forsystemsforexpressingnumbers,seeNumeralsystem. Seealso:Listoftypesofnumbers Numbersystems Complex : C {\displaystyle:\;\mathbb{C}} Real : R {\displaystyle:\;\mathbb{R}} Rational : Q {\displaystyle:\;\mathbb{Q}} Integer : Z {\displaystyle:\;\mathbb{Z}} Natural : N {\displaystyle:\;\mathbb{N}} Zero:0 One:1 Primenumbers Compositenumbers Negativeintegers Fraction Finitedecimal Dyadic(finitebinary) Repeatingdecimal Irrational Algebraic Transcendental Imaginary Numberscanbeclassifiedintosets,callednumbersystems,suchasthenaturalnumbersandtherealnumbers.[31]Themajorcategoriesofnumbersareasfollows: Mainnumbersystems N {\displaystyle\mathbb{N}} Natural 0,1,2,3,4,5,...or1,2,3,4,5,... N 0 {\displaystyle\mathbb{N}_{0}} or N 1 {\displaystyle\mathbb{N}_{1}} aresometimesused. Z {\displaystyle\mathbb{Z}} Integer ...,−5,−4,−3,−2,−1,0,1,2,3,4,5,... Q {\displaystyle\mathbb{Q}} Rational a/bwhereaandbareintegersandbisnot0 R {\displaystyle\mathbb{R}} Real Thelimitofaconvergentsequenceofrationalnumbers C {\displaystyle\mathbb{C}} Complex a+biwhereaandbarerealnumbersandiisaformalsquarerootof −1 Thereisgenerallynoprobleminidentifyingeachnumbersystemwithapropersubsetofthenextone(byabuseofnotation),becauseeachofthesenumbersystemsiscanonicallyisomorphictoapropersubsetofthenextone.[citationneeded]Theresultinghierarchyallows,forexample,totalk,formallycorrectly,aboutrealnumbersthatarerationalnumbers,andisexpressedsymbolicallybywriting N ⊂ Z ⊂ Q ⊂ R ⊂ C {\displaystyle\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\subset\mathbb{C}} . Naturalnumbers[edit] Mainarticle:Naturalnumber Thenaturalnumbers,startingwith1 Themostfamiliarnumbersarethenaturalnumbers(sometimescalledwholenumbersorcountingnumbers):1,2,3,andsoon.Traditionally,thesequenceofnaturalnumbersstartedwith 1(0wasnotevenconsideredanumberfortheAncientGreeks.)However,inthe19th century,settheoristsandothermathematiciansstartedincluding 0(cardinalityoftheemptyset,i.e.0 elements,where 0isthusthesmallestcardinalnumber)inthesetofnaturalnumbers.[32][33]Today,differentmathematiciansusethetermtodescribebothsets,including 0ornot.ThemathematicalsymbolforthesetofallnaturalnumbersisN,alsowritten N {\displaystyle\mathbb{N}} ,andsometimes N 0 {\displaystyle\mathbb{N}_{0}} or N 1 {\displaystyle\mathbb{N}_{1}} whenitisnecessarytoindicatewhetherthesetshouldstartwith0or1,respectively. Inthebase10numeralsystem,inalmostuniversalusetodayformathematicaloperations,thesymbolsfornaturalnumbersarewrittenusingtendigits:0,1,2,3,4,5,6,7,8,and9.Theradixorbaseisthenumberofuniquenumericaldigits,includingzero,thatanumeralsystemusestorepresentnumbers(forthedecimalsystem,theradixis10).Inthisbase 10system,therightmostdigitofanaturalnumberhasaplacevalueof 1,andeveryotherdigithasaplacevaluetentimesthatoftheplacevalueofthedigittoitsright. Insettheory,whichiscapableofactingasanaxiomaticfoundationformodernmathematics,[34]naturalnumberscanberepresentedbyclassesofequivalentsets.Forinstance,thenumber 3canberepresentedastheclassofallsetsthathaveexactlythreeelements.Alternatively,inPeanoArithmetic,thenumber 3isrepresentedassss0,wheresisthe"successor"function(i.e., 3isthethirdsuccessorof 0).Manydifferentrepresentationsarepossible;allthatisneededtoformallyrepresent 3istoinscribeacertainsymbolorpatternofsymbolsthreetimes. Integers[edit] Mainarticle:Integer Thenegativeofapositiveintegerisdefinedasanumberthatproduces 0whenitisaddedtothecorrespondingpositiveinteger.Negativenumbersareusuallywrittenwithanegativesign(aminussign).Asanexample,thenegativeof 7iswritten −7,and7+(−7)=0.Whenthesetofnegativenumbersiscombinedwiththesetofnaturalnumbers(including 0),theresultisdefinedasthesetofintegers,Zalsowritten Z {\displaystyle\mathbb{Z}} .HeretheletterZcomesfromGermanZahl 'number'.Thesetofintegersformsaringwiththeoperationsadditionandmultiplication.[35] Thenaturalnumbersformasubsetoftheintegers.Asthereisnocommonstandardfortheinclusionornotofzerointhenaturalnumbers,thenaturalnumberswithoutzeroarecommonlyreferredtoaspositiveintegers,andthenaturalnumberswithzeroarereferredtoasnon-negativeintegers. Rationalnumbers[edit] Mainarticle:Rationalnumber Arationalnumberisanumberthatcanbeexpressedasafractionwithanintegernumeratorandapositiveintegerdenominator.Negativedenominatorsareallowed,butarecommonlyavoided,aseveryrationalnumberisequaltoafractionwithpositivedenominator.Fractionsarewrittenastwointegers,thenumeratorandthedenominator,withadividingbarbetweenthem.Thefractionm/nrepresentsmpartsofawholedividedintonequalparts.Twodifferentfractionsmaycorrespondtothesamerationalnumber;forexample1/2and2/4areequal,thatis: 1 2 = 2 4 . {\displaystyle{1\over2}={2\over4}.} Ingeneral, a b = c d {\displaystyle{a\overb}={c\overd}} ifandonlyif a × d = c × b . {\displaystyle{a\timesd}={c\timesb}.} Iftheabsolutevalueofmisgreaterthann(supposedtobepositive),thentheabsolutevalueofthefractionisgreaterthan 1.Fractionscanbegreaterthan,lessthan,orequalto 1andcanalsobepositive,negative,or 0.Thesetofallrationalnumbersincludestheintegerssinceeveryintegercanbewrittenasafractionwithdenominator 1.Forexample −7canbewritten −7/1.ThesymbolfortherationalnumbersisQ(forquotient),alsowritten Q {\displaystyle\mathbb{Q}} . Realnumbers[edit] Mainarticle:Realnumber ThesymbolfortherealnumbersisR,alsowrittenas R . {\displaystyle\mathbb{R}.} Theyincludeallthemeasuringnumbers.Everyrealnumbercorrespondstoapointonthenumberline.Thefollowingparagraphwillfocusprimarilyonpositiverealnumbers.Thetreatmentofnegativerealnumbersisaccordingtothegeneralrulesofarithmeticandtheirdenotationissimplyprefixingthecorrespondingpositivenumeralbyaminussign,e.g.−123.456. Mostrealnumberscanonlybeapproximatedbydecimalnumerals,inwhichadecimalpointisplacedtotherightofthedigitwithplacevalue 1.Eachdigittotherightofthedecimalpointhasaplacevalueone-tenthoftheplacevalueofthedigittoitsleft.Forexample,123.456represents123456/1000,or,inwords,onehundred,twotens,threeones,fourtenths,fivehundredths,andsixthousandths.Arealnumbercanbeexpressedbyafinitenumberofdecimaldigitsonlyifitisrationalanditsfractionalparthasadenominatorwhoseprimefactorsare2or5orboth,becausethesearetheprimefactorsof10,thebaseofthedecimalsystem.Thus,forexample,onehalfis0.5,onefifthis0.2,one-tenthis0.1,andonefiftiethis0.02.Representingotherrealnumbersasdecimalswouldrequireaninfinitesequenceofdigitstotherightofthedecimalpoint.Ifthisinfinitesequenceofdigitsfollowsapattern,itcanbewrittenwithanellipsisoranothernotationthatindicatestherepeatingpattern.Suchadecimaliscalledarepeatingdecimal.Thus1/3canbewrittenas0.333...,withanellipsistoindicatethatthepatterncontinues.Foreverrepeating3sarealsowrittenas0.3.[36] Itturnsoutthattheserepeatingdecimals(includingtherepetitionofzeroes)denoteexactlytherationalnumbers,i.e.,allrationalnumbersarealsorealnumbers,butitisnotthecasethateveryrealnumberisrational.Arealnumberthatisnotrationaliscalledirrational.Afamousirrationalrealnumberisthenumberπ,theratioofthecircumferenceofanycircletoitsdiameter.Whenpiiswrittenas π = 3.14159265358979 … , {\displaystyle\pi=3.14159265358979\dots,} asitsometimesis,theellipsisdoesnotmeanthatthedecimalsrepeat(theydonot),butratherthatthereisnoendtothem.Ithasbeenprovedthatπisirrational.Anotherwell-knownnumber,proventobeanirrationalrealnumber,is 2 = 1.41421356237 … , {\displaystyle{\sqrt{2}}=1.41421356237\dots,} thesquarerootof2,thatis,theuniquepositiverealnumberwhosesquareis2.Boththesenumbershavebeenapproximated(bycomputer)totrillions(1trillion=1012=1,000,000,000,000)ofdigits. Notonlytheseprominentexamplesbutalmostallrealnumbersareirrationalandthereforehavenorepeatingpatternsandhencenocorrespondingdecimalnumeral.Theycanonlybeapproximatedbydecimalnumerals,denotingroundedortruncatedrealnumbers.Anyroundedortruncatednumberisnecessarilyarationalnumber,ofwhichthereareonlycountablymany.Allmeasurementsare,bytheirnature,approximations,andalwayshaveamarginoferror.Thus123.456isconsideredanapproximationofanyrealnumbergreaterorequalto1234555/10000andstrictlylessthan1234565/10000(roundingto3decimals),orofanyrealnumbergreaterorequalto123456/1000andstrictlylessthan123457/1000(truncationafterthe3.decimal).Digitsthatsuggestagreateraccuracythanthemeasurementitselfdoes,shouldberemoved.Theremainingdigitsarethencalledsignificantdigits.Forexample,measurementswitharulercanseldombemadewithoutamarginoferrorofatleast0.001m.Ifthesidesofarectanglearemeasuredas1.23 mand4.56 m,thenmultiplicationgivesanareafortherectanglebetween5.614591m2and5.603011m2.Sincenoteventheseconddigitafterthedecimalplaceispreserved,thefollowingdigitsarenotsignificant.Therefore,theresultisusuallyroundedto5.61. Justasthesamefractioncanbewritteninmorethanoneway,thesamerealnumbermayhavemorethanonedecimalrepresentation.Forexample,0.999...,1.0,1.00,1.000,...,allrepresentthenaturalnumber 1.Agivenrealnumberhasonlythefollowingdecimalrepresentations:anapproximationtosomefinitenumberofdecimalplaces,anapproximationinwhichapatternisestablishedthatcontinuesforanunlimitednumberofdecimalplacesoranexactvaluewithonlyfinitelymanydecimalplaces.Inthislastcase,thelastnon-zerodigitmaybereplacedbythedigitonesmallerfollowedbyanunlimitednumberof9's,orthelastnon-zerodigitmaybefollowedbyanunlimitednumberofzeros.Thustheexactrealnumber3.74canalsobewritten3.7399999999...and3.74000000000....Similarly,adecimalnumeralwithanunlimitednumberof0'scanberewrittenbydroppingthe0'stotherightofthedecimalplace,andadecimalnumeralwithanunlimitednumberof9'scanberewrittenbyincreasingtherightmost-9digitbyone,changingallthe9'stotherightofthatdigitto0's.Finally,anunlimitedsequenceof0'stotherightofthedecimalplacecanbedropped.Forexample,6.849999999999...=6.85and6.850000000000...=6.85.Finally,ifallofthedigitsinanumeralare0,thenumberis0,andifallofthedigitsinanumeralareanunendingstringof9's,youcandroptheninestotherightofthedecimalplace,andaddonetothestringof9stotheleftofthedecimalplace.Forexample,99.999...=100. Therealnumbersalsohaveanimportantbuthighlytechnicalpropertycalledtheleastupperboundproperty. Itcanbeshownthatanyorderedfield,whichisalsocomplete,isisomorphictotherealnumbers.Therealnumbersarenot,however,analgebraicallyclosedfield,becausetheydonotincludeasolution(oftencalledasquarerootofminusone)tothealgebraicequation x 2 + 1 = 0 {\displaystylex^{2}+1=0} . Complexnumbers[edit] Mainarticle:Complexnumber Movingtoagreaterlevelofabstraction,therealnumberscanbeextendedtothecomplexnumbers.Thissetofnumbersarosehistoricallyfromtryingtofindclosedformulasfortherootsofcubicandquadraticpolynomials.Thisledtoexpressionsinvolvingthesquarerootsofnegativenumbers,andeventuallytothedefinitionofanewnumber:asquarerootof −1,denotedbyi,asymbolassignedbyLeonhardEuler,andcalledtheimaginaryunit.Thecomplexnumbersconsistofallnumbersoftheform a + b i {\displaystyle\,a+bi} whereaandbarerealnumbers.Becauseofthis,complexnumberscorrespondtopointsonthecomplexplane,avectorspaceoftworealdimensions.Intheexpressiona+bi,therealnumberaiscalledtherealpartandbiscalledtheimaginarypart.Iftherealpartofacomplexnumberis 0,thenthenumberiscalledanimaginarynumberorisreferredtoaspurelyimaginary;iftheimaginarypartis 0,thenthenumberisarealnumber.Thustherealnumbersareasubsetofthecomplexnumbers.Iftherealandimaginarypartsofacomplexnumberarebothintegers,thenthenumberiscalledaGaussianinteger.ThesymbolforthecomplexnumbersisCor C {\displaystyle\mathbb{C}} . Thefundamentaltheoremofalgebraassertsthatthecomplexnumbersformanalgebraicallyclosedfield,meaningthateverypolynomialwithcomplexcoefficientshasarootinthecomplexnumbers.Likethereals,thecomplexnumbersformafield,whichiscomplete,butunliketherealnumbers,itisnotordered.Thatis,thereisnoconsistentmeaningassignabletosayingthatiisgreaterthan 1,noristhereanymeaninginsayingthatiislessthan 1.Intechnicalterms,thecomplexnumberslackatotalorderthatiscompatiblewithfieldoperations. Subclassesoftheintegers[edit] Evenandoddnumbers[edit] Mainarticle:Evenandoddnumbers Anevennumberisanintegerthatis"evenlydivisible"bytwo,thatisdivisiblebytwowithoutremainder;anoddnumberisanintegerthatisnoteven.(Theold-fashionedterm"evenlydivisible"isnowalmostalwaysshortenedto"divisible".)Anyoddnumbernmaybeconstructedbytheformulan=2k+1,forasuitableintegerk.Startingwithk=0,thefirstnon-negativeoddnumbersare{1,3,5,7,...}.Anyevennumbermhastheformm=2kwherekisagainaninteger.Similarly,thefirstnon-negativeevennumbersare{0,2,4,6,...}. Primenumbers[edit] Mainarticle:Primenumber Aprimenumber,oftenshortenedtojustprime,isanintegergreaterthan1thatisnottheproductoftwosmallerpositiveintegers.Thefirstfewprimenumbersare2,3,5,7,and11.Thereisnosuchsimpleformulaasforoddandevennumberstogeneratetheprimenumbers.Theprimeshavebeenwidelystudiedformorethan2000yearsandhaveledtomanyquestions,onlysomeofwhichhavebeenanswered.Thestudyofthesequestionsbelongstonumbertheory.Goldbach'sconjectureisanexampleofastillunansweredquestion:"Iseveryevennumberthesumoftwoprimes?" Oneansweredquestion,astowhethereveryintegergreaterthanoneisaproductofprimesinonlyoneway,exceptforarearrangementoftheprimes,wasconfirmed;thisprovenclaimiscalledthefundamentaltheoremofarithmetic.AproofappearsinEuclid'sElements. Otherclassesofintegers[edit] Manysubsetsofthenaturalnumbershavebeenthesubjectofspecificstudiesandhavebeennamed,oftenafterthefirstmathematicianthathasstudiedthem.ExampleofsuchsetsofintegersareFibonaccinumbersandperfectnumbers.Formoreexamples,seeIntegersequence. Subclassesofthecomplexnumbers[edit] Algebraic,irrationalandtranscendentalnumbers[edit] Algebraicnumbersarethosethatareasolutiontoapolynomialequationwithintegercoefficients.Realnumbersthatarenotrationalnumbersarecalledirrationalnumbers.Complexnumberswhicharenotalgebraicarecalledtranscendentalnumbers.Thealgebraicnumbersthataresolutionsofamonicpolynomialequationwithintegercoefficientsarecalledalgebraicintegers. Constructiblenumbers[edit] Motivatedbytheclassicalproblemsofconstructionswithstraightedgeandcompass,theconstructiblenumbersarethosecomplexnumberswhoserealandimaginarypartscanbeconstructedusingstraightedgeandcompass,startingfromagivensegmentofunitlength,inafinitenumberofsteps. Computablenumbers[edit] Mainarticle:Computablenumber Acomputablenumber,alsoknownasrecursivenumber,isarealnumbersuchthatthereexistsanalgorithmwhich,givenapositivenumbernasinput,producesthefirstndigitsofthecomputablenumber'sdecimalrepresentation.Equivalentdefinitionscanbegivenusingμ-recursivefunctions,Turingmachinesorλ-calculus.Thecomputablenumbersarestableforallusualarithmeticoperations,includingthecomputationoftherootsofapolynomial,andthusformarealclosedfieldthatcontainstherealalgebraicnumbers. Thecomputablenumbersmaybeviewedastherealnumbersthatmaybeexactlyrepresentedinacomputer:acomputablenumberisexactlyrepresentedbyitsfirstdigitsandaprogramforcomputingfurtherdigits.However,thecomputablenumbersarerarelyusedinpractice.Onereasonisthatthereisnoalgorithmfortestingtheequalityoftwocomputablenumbers.Moreprecisely,therecannotexistanyalgorithmwhichtakesanycomputablenumberasaninput,anddecidesineverycaseifthisnumberisequaltozeroornot. Thesetofcomputablenumbershasthesamecardinalityasthenaturalnumbers.Therefore,almostallrealnumbersarenon-computable.However,itisverydifficulttoproduceexplicitlyarealnumberthatisnotcomputable. Extensionsoftheconcept[edit] p-adicnumbers[edit] Mainarticle:p-adicnumber Thep-adicnumbersmayhaveinfinitelylongexpansionstotheleftofthedecimalpoint,inthesamewaythatrealnumbersmayhaveinfinitelylongexpansionstotheright.Thenumbersystemthatresultsdependsonwhatbaseisusedforthedigits:anybaseispossible,butaprimenumberbaseprovidesthebestmathematicalproperties.Thesetofthep-adicnumberscontainstherationalnumbers,butisnotcontainedinthecomplexnumbers. Theelementsofanalgebraicfunctionfieldoverafinitefieldandalgebraicnumbershavemanysimilarproperties(seeFunctionfieldanalogy).Therefore,theyareoftenregardedasnumbersbynumbertheorists.Thep-adicnumbersplayanimportantroleinthisanalogy. Hypercomplexnumbers[edit] Mainarticle:hypercomplexnumber Somenumbersystemsthatarenotincludedinthecomplexnumbersmaybeconstructedfromtherealnumbersinawaythatgeneralizetheconstructionofthecomplexnumbers.Theyaresometimescalledhypercomplexnumbers.TheyincludethequaternionsH,introducedbySirWilliamRowanHamilton,inwhichmultiplicationisnotcommutative,theoctonions,inwhichmultiplicationisnotassociativeinadditiontonotbeingcommutative,andthesedenions,inwhichmultiplicationisnotalternative,neitherassociativenorcommutative. Transfinitenumbers[edit] Mainarticle:transfinitenumber Fordealingwithinfinitesets,thenaturalnumbershavebeengeneralizedtotheordinalnumbersandtothecardinalnumbers.Theformergivestheorderingoftheset,whilethelattergivesitssize.Forfinitesets,bothordinalandcardinalnumbersareidentifiedwiththenaturalnumbers.Intheinfinitecase,manyordinalnumberscorrespondtothesamecardinalnumber. Nonstandardnumbers[edit] Hyperrealnumbersareusedinnon-standardanalysis.Thehyperreals,ornonstandardreals(usuallydenotedas*R),denoteanorderedfieldthatisaproperextensionoftheorderedfieldofrealnumbersRandsatisfiesthetransferprinciple.Thisprincipleallowstruefirst-orderstatementsaboutRtobereinterpretedastruefirst-orderstatementsabout*R. Superrealandsurrealnumbersextendtherealnumbersbyaddinginfinitesimallysmallnumbersandinfinitelylargenumbers,butstillformfields. Seealso[edit] Mathematicsportal Concretenumber Listofnumbers Listofnumbersinvariouslanguages Listoftypesofnumbers Mathematicalconstant –Fixednumberthathasreceivedaname Complexnumbers Numericalcognition Ordersofmagnitude Physicalconstant –Universalandunchangingphysicalquantity Pi –Euclideanratioofthecircumferenceofacircletoitsdiameter Positionalnotation –Methodforrepresentingorencodingnumbers Primenumber –Positiveintegerwithexactlytwodivisors,1anditself Scalar(mathematics) –Elementsofafield,e.g.realnumbers,inthecontextoflinearalgebra Subitizingandcounting Notes[edit] ^"number,n."OEDOnline.OxfordUniversityPress.Archivedfromtheoriginalon2018-10-04.Retrieved2017-05-16. ^"numeral,adj.andn."OEDOnline.OxfordUniversityPress. ^Inlinguistics,anumeralcanrefertoasymbollike5,butalsotoawordoraphrasethatnamesanumber,like"fivehundred";numeralsincludealsootherwordsrepresentingnumbers,like"dozen". ^Matson,John."TheOriginofZero".ScientificAmerican.Archivedfromtheoriginalon2017-08-26.Retrieved2017-05-16. 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^RichardDedekind,Stetigkeit&irrationaleZahlen(Braunschweig:FriedrichVieweg&Sohn,1872).Subsequentlypublishedin:———,GesammeltemathematischeWerke,ed.RobertFricke,EmmyNoether&ÖysteinOre(Braunschweig:FriedrichVieweg&Sohn,1932),vol.3,pp.315–334. ^L.Kronecker,"UeberdenZahlbegriff",[Crelle’s]JournalfürdiereineundangewandteMathematik,№101(1887):337–355. ^LeonhardEuler,"Conjecturacircanaturamaeris,proexplicandisphaenomenisinatmosphaeraobservatis",ActaAcademiaeScientiarumImperialisPetropolitanae,1779,1(1779):162–187. ^Ramus,"DeterminanternesAnvendelsetilatbestemmeLovenfordeconvergerendeBröker",in:DetKongeligeDanskeVidenskabernesSelskabsnaturvidenskabeligeogmathematiskeAfhandlinger(Kjoebenhavn:1855),p.106. ^EduardHeine,"EinigeEigenschaftenderLaméschenFunktionen",[Crelle’s]JournalfürdiereineundangewandteMathematik,№56(Jan.1859):87–99at97. 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References[edit] TobiasDantzig,Number,thelanguageofscience;acriticalsurveywrittenfortheculturednon-mathematician,NewYork,TheMacmillanCompany,1930.[ISBN missing] ErichFriedman,What'sspecialaboutthisnumber? StevenGalovich,IntroductiontoMathematicalStructures,HarcourtBraceJavanovich,1989,ISBN 0-15-543468-3. PaulHalmos,NaiveSetTheory,Springer,1974,ISBN 0-387-90092-6. MorrisKline,MathematicalThoughtfromAncienttoModernTimes,OxfordUniversityPress,1990.ISBN 978-0195061352 AlfredNorthWhiteheadandBertrandRussell,PrincipiaMathematicato*56,CambridgeUniversityPress,1910.[ISBN missing] LeoCory,ABriefHistoryofNumbers,OxfordUniversityPress,2015,ISBN 978-0-19-870259-7. Externallinks[edit] WikimediaCommonshasmediarelatedtoNumbers. Wikiquotehasquotationsrelatedto:Number LookupnumberinWiktionary,thefreedictionary. WikiversityhaslearningresourcesaboutPrimarymathematics:Numbers Nechaev,V.I.(2001)[1994],"Number",EncyclopediaofMathematics,EMSPress Tallant,Jonathan."DoNumbersExist?".Numberphile.BradyHaran.Archivedfromtheoriginalon2016-03-08.Retrieved2013-04-06. BBCRadio4,InOurTime:NegativeNumbers '4000YearsofNumbers',lecturebyRobinWilson,07/11/07,GreshamCollege(availablefordownloadasMP3orMP4,andasatextfile). Krulwich,Robert(2011-06-22)."What'stheWorld'sFavoriteNumber?".NPR.Retrieved2011-09-17.;"CuddlingWith9,SmoochingWith8,WinkingAt7".NPR.2011-08-11.Retrieved2011-09-17. On-LineEncyclopediaofIntegerSequences vteNumbersystemsCountablesets Naturalnumbers ( N {\displaystyle\mathbb{N}} ) Integers ( Z {\displaystyle\mathbb{Z}} ) Rationalnumbers ( Q {\displaystyle\mathbb{Q}} ) Constructiblenumbers Algebraicnumbers ( A {\displaystyle\mathbb{A}} ) Periods Computablenumbers Arithmeticalnumbers Set-theoreticallydefinablenumbers Gaussianintegers Compositionalgebras Divisionalgebras:Realnumbers ( R {\displaystyle\mathbb{R}} ) Complexnumbers ( C {\displaystyle\mathbb{C}} ) Quaternions ( H {\displaystyle\mathbb{H}} ) Octonions ( O {\displaystyle\mathbb{O}} ) Splittypes Over R {\displaystyle\mathbb{R}} : Split-complexnumbers Split-quaternions Split-octonionsOver C {\displaystyle\mathbb{C}} : Bicomplexnumbers Biquaternions Bioctonions Otherhypercomplex Dualnumbers Dualquaternions Dual-complexnumbers Hyperbolicquaternions Sedenions ( S {\displaystyle\mathbb{S}} ) Split-biquaternions Multicomplexnumbers Geometricalgebra Algebraofphysicalspace Spacetimealgebra Othertypes Cardinalnumbers Extendednaturalnumbers Irrationalnumbers Fuzzynumbers Hyperrealnumbers Levi-Civitafield Surrealnumbers Transcendentalnumbers Ordinalnumbers p-adicnumbers(p-adicsolenoids) Supernaturalnumbers Superrealnumbers Classification List vteNumbertheoryFields Algebraicnumbertheory Analyticnumbertheory Geometricnumbertheory Computationalnumbertheory Transcendentalnumbertheory Diophantinegeometry Arithmeticcombinatorics Arithmeticgeometry Arithmetictopology Arithmeticdynamics Keyconcepts Numbers Naturalnumbers Primenumbers Rationalnumbers Irrationalnumbers Algebraicnumbers Transcendentalnumbers p-adicnumbers Arithmetic Modulararithmetic Chineseremaindertheorem Arithmeticfunctions Advancedconcepts Quadraticforms Modularforms L-functions Diophantineequations Diophantineapproximation Continuedfractions Category Listoftopics Listofrecreationaltopics Wikibook Wikversity AuthoritycontrolGeneral IntegratedAuthorityFile(Germany) Nationallibraries Spain France(data) UnitedStates Japan Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Number&oldid=1061923130" Categories:GrouptheoryNumbersMathematicalobjectsHiddencategories:CS1errors:URLAllarticleslackingreliablereferencesArticleslackingreliablereferencesfromJanuary2017WebarchivetemplatewaybacklinksArticleswithshortdescriptionShortdescriptionisdifferentfromWikidataWikipediaindefinitelymove-protectedpagesAccuracydisputesfromNovember2014AllaccuracydisputesArticlescontainingSanskrit-languagetextArticleslackingreliablereferencesfromSeptember2020ArticlescontainingLatin-languagetextAllarticleswithunsourcedstatementsArticleswithunsourcedstatementsfromSeptember2020ArticlescontainingGerman-languagetextAllarticleswithpeacocktermsArticleswithpeacocktermsfromSeptember2020WikipediaarticlesneedingclarificationfromSeptember2020ArticleswithunsourcedstatementsfromJune2017PageswithmissingISBNsCommonslinkisonWikidataArticleswithGNDidentifiersArticleswithBNEidentifiersArticleswithBNFidentifiersArticleswithLCCNidentifiersArticleswithNDLidentifiersPagesthatuseadeprecatedformatofthemathtags Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk Variants expanded collapsed Views ReadEditViewhistory More expanded collapsed Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Inotherprojects WikimediaCommonsWikiquote Languages AfrikaansAlemannischአማርኛÆngliscالعربيةAragonésܐܪܡܝܐԱրեւմտահայերէնArmãneashtiArpetanঅসমীয়াAsturianuAtikamekwAvañe'ẽAzərbaycancaتۆرکجهবাংলাBanjarBân-lâm-gúБашҡортсаБеларускаяБеларуская(тарашкевіца)BikolCentralБългарскиBoarischབོད་ཡིགBosanskiBrezhonegБуряадCatalàЧӑвашлаČeštinaChiShonaChoctawCorsuCymraegDanskDeutschडोटेलीEestiΕλληνικάEmiliànerumagnòlEspañolEsperantoEstremeñuEuskaraفارسیFøroysktFrançaisFryskFulfuldeGaeilgeGàidhligGalego贛語Хальмг한국어HawaiʻiՀայերենहिन्दीHrvatskiIdoIlokanoBahasaIndonesiaInterlinguaИронIsiXhosaIsiZuluÍslenskaItalianoעבריתJawaKabɩyɛಕನ್ನಡქართულიҚазақшаKernowekKiswahiliKreyòlayisyenKriyòlgwiyannenKurdîЛаккуລາວLatinaLatviešuLëtzebuergeschLietuviųLimburgsLinguaFrancaNovaLa.lojban.LugandaMagyarमैथिलीМакедонскиMalagasyമലയാളംमराठीBahasaMelayuMirandésМонголမြန်မာဘာသာNāhuatlNaVosaVakavitiNederlandsNēhiyawēwin/ᓀᐦᐃᔭᐍᐏᐣनेपालीनेपालभाषा日本語NordfriiskNorskbokmålNorsknynorskNouormandNovialOccitanОлыкмарийOʻzbekcha/ўзбекчаਪੰਜਾਬੀPangasinanپنجابیپښتوPatoisPlattdüütschPolskiPortuguêsRomânăRunaSimiРусиньскыйРусскийСахатылаसंस्कृतम्SängöScotsSesothosaLeboaShqipSicilianuSimpleEnglishسنڌيSlovenčinaSlovenščinaŚlůnskiSoomaaligaکوردیСрпски/srpskiSrpskohrvatski/српскохрватскиSundaSuomiSvenskaTagalogதமிழ்TaclḥitTaqbaylitTarandíneТатарча/tatarçaతెలుగుไทยትግርኛТоҷикӣತುಳುTürkçeTürkmençeУдмуртУкраїнськаاردوVènetoTiếngViệtVõro文言Winaray吴语XitsongaייִדישYorùbá粵語Žemaitėška中文Dagbanli Editlinks