超實(shí)數(shù)是“數(shù)”嗎

超實(shí)數(shù)是“數(shù)〃嗎？

超實(shí)數(shù)是“數(shù)”嗎？

在我們國(guó)內(nèi)，普遍認(rèn)為：超實(shí)數(shù)不是"數(shù)”，請(qǐng)見科普中國(guó)以及全部高等數(shù)學(xué)教科書。

在國(guó)外，超實(shí)數(shù)是“數(shù)”，請(qǐng)見本文附件。

注：超實(shí)數(shù)是非標(biāo)準(zhǔn)書。數(shù)，因而是“數(shù)”。

袁萌陳啟清7月14日

附件：

Anumberisamathematicalobjectusedtocount,measure,andlabel.Theoriginal

examplesarethenaturalnumbers1,2,3,4,andsoforth,[1]Awrittensymbollike

"5"thatrepresentsanumberiscalledanumeral.Anumeralsystemisanorganized

waytowriteandmanipulatethistypeofsymbol,forexampletheHindu-Arabic

numeralsystemallowscombinationsofnumericaldigitslike"5"and"0"to

representlargernumberslike50,[2]Anumeralinlinguisticscanrefertoasymbol

like5,thewordsorphrasethatnamesanumber,like"fivehundred",orotherwords

thatmeanaspecificnumber,like"dozen".Inadditiontotheiruseincountingand

measuring,numeralsareoftenusedforlabels(aswithtelephonenumbers],for

ordering(aswithserialnumbers],andforcodes(aswithISBNs].Incommonusage,

numbermayrefertoasymbol,awordorphrase,orthemathematicalobject.

Inmathematics,thenotionofnumberhasbeenextendedoverthecenturiesto

include0,[3]negativenumbers,[4]rationalnumberssuchas

1

/

2

and-

2

/

3

,realnumbers[5]suchasV2andIT,andcomplexnumbers,[6]whichextendthereal

numberswithasquarerootof-1(anditscombinationswithrealnumbersby

additionandmultiplication].[4]Calculationswithnumbersaredonewith

arithmeticaloperations,themostfamiliarbeingaddition,subtraction,multiplication,

division,andexponentiation.Theirstudyorusageiscalledarithmetic.Thesame

termmayalsorefertonumbertheory,thestudyofthepropertiesofnumbers.

Besidestheirpracticaluses,numbershaveculturalsignificancethroughoutthe

world.[7][8]Forexample,inWesternsociety,thenumber13isregardedasunlucky,

and"amillion"maysignify"alot,"[7]Thoughitisnowregardedaspseudoscience,

beliefinamysticalsignificanceofnumbers,knownasnumerology,permeated

ancientandmedievalthought,[9]Numerologyheavilyinfluencedthedevelopment

ofGreekmathematics,stimulatingtheinvestigationofmanyproblemsinnumber

theorywhicharestillofinteresttoday.[9]

Duringthe19thcentury,mathematiciansbegantodevelopmanydifferent

abstractionswhichsharecertainpropertiesofnumbersandmaybeseenas

extendingtheconcept.Amongthefirstwerethehypercomplexnumbers,which

consistofvariousextensionsormodificationsofthecomplexnumbersystem.Today,

numbersystemsareconsideredimportantspecialexamplesofmuchmoregeneral

categoriessuchasringsandfields,andtheapplicationoftheterm"number"isa

matterofconvention,withoutfundamentalsignificance.[10]

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

Numerals[edit]

Mainarticle:Numeralsystem

Numbersshouldbedistinguishedfromnumerals,thesymbolsusedtorepresent

numbers.TheEgyptiansinventedthefirstcipherednumeralsystem,andtheGreeks

followedbymappingtheircountingnumbersontoIonianandDoricalphabets.[11]

Romannumerals,asystemthatusedcombinationsoflettersfromtheRoman

alphabet,remaineddominantinEuropeuntilthespreadofthesuperiorHindu-

Arabicnumeralsystemaroundthelate14thcentury,andtheHindu-Arabicnumeral

systemremainsthemostcommonsystemforrepresentingnumbersintheworld

today.[12]Thekeytotheeffectivenessofthesystemwasthesymbolforzero,which

wasdevelopedbyancientIndianmathematiciansaround500AD.[12]

Firstuseofnumbers[edit]

Mainarticle:Historyofancientnumeralsystems

Bonesandotherartifactshavebeendiscoveredwithmarkscutintothemthatmany

believearetallymarks.[13]Thesetallymarksmayhavebeenusedforcounting

elapsedtime,suchasnumbersofdays,lunarcyclesorkeepingrecordsofquantities,

suchasofanimals.

Atallyingsystemhasnoconceptofplacevalue(asinmoderndecimalnotation),

whichlimitsitsrepresentationoflargenumbers.Nonethelesstallyingsystemsare

consideredthefirstkindofabstractnumeralsystem.

ThefirstknownsystemwithplacevaluewastheMesopotamianbase60system(ca.

3400BC)andtheearliestknownbase10systemdatesto3100BCinEgypt.[14]

Zero[edit]

ThefirstknowndocumenteduseofzerodatestoAD628,andappearedinthe

Brahmasphutasiddhanta,themainworkoftheIndianmathematicianBrahmagupta.

Hetreated0asanumberanddiscussedoperationsinvolvingit,includingdivision.

Bythistime(the7thcentury)theconcepthadclearlyreachedCambodiaasKhmer

numerals,anddocumentationshowstheidealaterspreadingtoChinaandthe

Islamicworld.

Thenumber605inKhmernumerals,fromaninscriptionfrom683AD.Earlyuseof

zeroasadecimalfigure.

Brahmagupta'sBrahmasphutasiddhantaisthefirstbookthatmentionszeroasa

number,henceBrahmaguptaisusuallyconsideredthefirsttoformulatetheconcept

ofzero.Hegaverulesofusingzerowithnegativeandpositivenumbers,suchas

'Zeroplusapositivenumberisapositivenumber,andanegativenumberpluszero

isthenegativenumber'.TheBrahmasphutasiddhantaistheearliestknowntextto

treatzeroasanumberinitsownright,ratherthanassimplyaplaceholderdigitin

representinganothernumberaswasdonebytheBabyloniansorasasymbolfora

lackofquantityaswasdonebyPtolemyandtheRomans.

Theuseof0asanumbershouldbedistinguishedfromitsuseasaplaceholder

numeralinplace-valuesystems.Manyancienttextsused0.Babylonianand

Egyptiantextsusedit.Egyptiansusedthewordnfrtodenotezerobalanceindouble

entryaccounting.IndiantextsusedaSanskritwordShunyeorshunyatorefertothe

conceptofvoid.Inmathematicstextsthiswordoftenreferstothenumberzero.[15]

Inasimilarvein,Panini(5thcenturyBC)usedthenull(zero)operatorinthe

Ashtadhyayi,anearlyexampleofanalgebraicgrammarfortheSanskritlanguage

(alsoseePingala].

ThereareotherusesofzerobeforeBrahmagupta,thoughthedocumentationisnot

ascompleteasitisintheBrahmasphutasiddhanta.

RecordsshowthattheAncientGreeksseemedunsureaboutthestatusof0asa

number:theyaskedthemselves"howcan'nothing'besomething?"leadingto

interestingphilosophicaland,bytheMedievalperiod,religiousargumentsaboutthe

natureandexistenceof0andthevacuum.TheparadoxesofZenoofEleadependin

partontheuncertaininterpretationof0.(TheancientGreeksevenquestioned

whether1wasanumber.]

ThelateOlmecpeopleofsouth-centralMexicobegantouseasymbolforzero,a

shellglyph,intheNewWorld,possiblybythe4thcenturyBCbutcertainlyby40BC,

whichbecameanintegralpartofMayanumeralsandtheMayacalendar.Mayan

arithmeticusedbase4andbase5writtenasbase20,Sanchezin1961reporteda

base4,base5"finger"abacus.

By130AD,Ptolemy,influencedbyHipparchusandtheBabylonians,wasusinga

symbolfor0(asmallcirclewithalongoverbar)withinasexagesimalnumeral

systemotherwiseusingalphabeticGreeknumerals.Becauseitwasusedalone,not

asjustaplaceholder,thisHellenisticzerowasthefirstdocumenteduseofatrue

zerointheOldWorld.InlaterByzantinemanuscriptsofhisSyntaxisMathematica

(Almagest),theHellenisticzerohadmorphedintotheGreekletterOmicron

(otherwisemeaning70).

AnothertruezerowasusedintablesalongsideRomannumeralsby525(first

knownusebyDionysiusExiguus),butasaword,nullameaningnothing,notasa

symbol.Whendivisionproduced0asaremainder,nihil,alsomeaningnothing,was

used.Thesemedievalzeroswereusedbyallfuturemedievalcomputists

(calculatorsofEaster),Anisolateduseoftheirinitial,N,wasusedinatableof

RomannumeralsbyBedeoracolleagueabout725,atruezerosymbol.

Negativenumbers[edit]

Furtherinformation:Historyofnegativenumbers

Theabstractconceptofnegativenumberswasrecognizedasearlyas100-50BCin

China.TheNineChaptersontheMathematicalArtcontainsmethodsforfindingthe

areasoffigures;redrodswereusedtodenotepositivecoefficients,blackfor

negative.[16]ThefirstreferenceinaWesternworkwasinthe3rdcenturyADin

Greece.Diophantusreferredtotheequationequivalentto4x+20=0(thesolution

isnegative]inArithmetica,sayingthattheequationgaveanabsurdresult.

Duringthe600s,negativenumberswereinuseinIndiatorepresentdebts.

Diophantus'previousreferencewasdiscussedmoreexplicitlybyIndian

mathematicianBrahmagupta,inBrahmasphutasiddhanta628,whousednegative

numberstoproducethegeneralformquadraticformulathatremainsinusetoday.

However,inthe12thcenturyinIndia,Bhaskaragivesnegativerootsforquadratic

equationsbutsaysthenegativevalue"isinthiscasenottobetaken,foritis

inadequate;peopledonotapproveofnegativeroots."

Europeanmathematicians,forthemostpart,resistedtheconceptofnegative

numbersuntilthe17thcentury,althoughFibonacciallowednegativesolutionsin

financialproblemswheretheycouldbeinterpretedasdebts(chapter13ofLiber

Abaci,1202)andlateraslosses(inFlos),Atthesametime,theChinesewere

indicatingnegativenumbersbydrawingadiagonalstrokethroughtheright-most

non-zerodigitofthecorrespondingpositivenumber'snumeral.[17]Thefirstuseof

negativenumbersinaEuropeanworkwasbyNicolasChuquetduringthe15th

century.Heusedthemasexponents,butreferredtothemas"absurdnumbers".

Asrecentlyasthe18thcentury,itwascommonpracticetoignoreanynegative

resultsreturnedbyequationsontheassumptionthattheyweremeaningless,justas

ReneDescartesdidwithnegativesolutionsinaCartesiancoordinatesystem.

Rationalnumbers[edit]

Itislikelythattheconceptoffractionalnumbersdatestoprehistorictimes.The

AncientEgyptiansusedtheirEgyptianfractionnotationforrationalnumbersin

mathematicaltextssuchastheRhindMathematicalPapyrusandtheKahunPapyrus.

ClassicalGreekandIndianmathematiciansmadestudiesofthetheoryofrational

numbers,aspartofthegeneralstudyofnumbertheory.Thebestknownoftheseis

Euclid'sElements,datingtoroughly300BC.OftheIndiantexts,themostrelevantis

theSthanangaSutra,whichalsocoversnumbertheoryaspartofageneralstudyof

mathematics.

Theconceptofdecimalfractionsiscloselylinkedwithdecimalplace-valuenotation;

thetwoseemtohavedevelopedintandem.Forexample,itiscommonfortheJain

mathsutratoincludecalculationsofdecimal-fractionapproximationstopiorthe

squarerootof2.Similarly,Babylonianmathtextshadalwaysusedsexagesimal

(base60)fractionswithgreatfrequency.

Irrationalnumbers[edit]

Furtherinformation:Historyofirrationalnumbers

TheearliestknownuseofirrationalnumberswasintheIndianSulbaSutras

composedbetween800and500BC.[18]Thefirstexistenceproofsofirrational

numbersisusuallyattributedtoPythagoras,morespecificallytothePythagorean

HippasusofMetapontum,whoproduceda(mostlikelygeometrical)proofofthe

irrationalityofthesquarerootof2.ThestorygoesthatHippasusdiscovered

irrationalnumberswhentryingtorepresentthesquarerootof2asafraction.

However,Pythagorasbelievedintheabsolutenessofnumbers,andcouldnotaccept

theexistenceofirrationalnumbers.Hecouldnotdisprovetheirexistencethrough

logic,buthecouldnotacceptirrationalnumbers,andso,allegedlyandfrequently

reported,hesentencedHippasustodeathbydrowning,toimpedespreadingofthis

disconcertingnews.[19]

The16thcenturybroughtfinalEuropeanacceptanceofnegativeintegraland

fractionalnumbers.Bythe17thcentury,mathematiciansgenerallyuseddecimal

fractionswithmodernnotation.Itwasnot,however,untilthe19thcenturythat

mathematiciansseparatedirrationalsintoalgebraicandtranscendentalparts,and

oncemoreundertookthescientificstudyofirrationals.Ithadremainedalmost

dormantsinceEuclid.In1872,thepublicationofthetheoriesofKarlWeierstrass

(byhispupilE.Kossak),EduardHeine(Crelle,74),GeorgCantor(Annalen,5),and

RichardDedekindwasbroughtabout.In1869,CharlesMerayhadtakenthesame

pointofdepartureasHeine,butthetheoryisgenerallyreferredtotheyear1872.

Weierstrass'smethodwascompletelysetforthbySalvatorePincherle(1880),and

Dedekind'shasreceivedadditionalprominencethroughtheauthor'slaterwork

(1888)andendorsementbyPaulTannery(1894).Weierstrass,Cantor,andHeine

basetheirtheoriesoninfiniteseries,whileDedekindfoundshisontheideaofacut

(Schnitt)inthesystemofrealnumbers,separatingallrationalnumbersintotwo

groupshavingcertaincharacteristicproperties.Thesubjecthasreceivedlater

contributionsatthehandsofWeierstrass,Kronecker(Crelle,101],andMeray.

Thesearchforrootsofquinticandhigherdegreeequationswasanimportant

development,theAbel-Ruffinitheorem(Ruffini1799,Abel1824]showedthatthey

couldnotbesolvedbyradicals(formulasinvolvingonlyarithmeticaloperationsand

roots].Henceitwasnecessarytoconsiderthewidersetofalgebraicnumbers(all

solutionstopolynomialequations).Galois(1832)linkedpolynomialequationsto

grouptheorygivingrisetothefieldofGaloistheory.

Continuedfractions,closelyrelatedtoirrationalnumbers(andduetoCataldi,1613),

receivedattentionatthehandsofEuler,andattheopeningofthe19thcenturywere

broughtintoprominencethroughthewritingsofJosephLouisLagrange.Other

noteworthycontributionshavebeenmadebyDruckenmiiller(1837),Kunze(1857),

Lemke(1870),andGunther(1872).Ramus(1855)firstconnectedthesubjectwith

determinants,resulting,withthesubsequentcontributionsofHeine,Mobius,and

Gunther,inthetheoryofKettenbruchdeterminanten.

Transcendentalnumbersandreals[edit]

Furtherinformation:HistoryofIT

Theexistenceoftranscendentalnumbers[20]wasfirstestablishedbyLiouville

(1844,1851).Hermiteprovedin1873thateistranscendentalandLindemann

provedin1882thatnistranscendental.Finally,Cantorshowedthatthesetofall

realnumbersisuncountablyinfinitebutthesetofallalgebraicnumbersis

countablyinfinite,sothereisanuncountablyinfinitenumberoftranscendental

numbers.

Infinityandinfinitesimals[edit]

Furtherinformation:Historyofinfinity

TheearliestknownconceptionofmathematicalinfinityappearsintheYajurVeda,

anancientIndianscript,whichatonepointstates,"Ifyouremoveapartfrom

infinityoraddaparttoinfinity,stillwhatremainsisinfinity."Infinitywasapopular

topicofphilosophicalstudyamongtheJainmathematiciansc.400BC.They

distinguishedbetweenfivetypesofinfinity:infiniteinoneandtwodirections,

infiniteinarea,infiniteeverywhere,andinfiniteperpetually.

AristotledefinedthetraditionalWesternnotionofmathematicalinfinity.He

distinguishedbetweenactualinfinityandpotentialinfinity—thegeneralconsensus

beingthatonlythelatterhadtruevalue.GalileoGalilei'sTwoNewSciences

discussedtheideaofone-to-onecorrespondencesbetweeninfinitesets.Butthe

nextmajoradvanceinthetheorywasmadebyGeorgCantor;in1895hepublisheda

bookabouthisnewsettheory,introducing,amongotherthings,transfinitenumbers

andformulatingthecontinuumhypothesis.

Inthe1960s,AbrahamRobinsonshowedhowinfinitelylargeandinfinitesimal

numberscanberigorouslydefinedandusedtodevelopthefieldofnonstandard

analysis.Thesystemofhyperrealnumbersrepresentsarigorousmethodoftreating

theideasaboutinfiniteandinfinitesimalnumbersthathadbeenusedcasuallyby

mathematicians,scientists,andengineerseversincetheinventionofinfinitesimal

calculusbyNewtonandLeibniz.

Amoderngeometricalversionofinfinityisgivenbyprojectivegeometry,which

introduces"idealpointsatinfinity",oneforeachspatialdirection.Eachfamilyof

parallellinesinagivendirectionispostulatedtoconvergetothecorresponding

idealpoint.Thisiscloselyrelatedtotheideaofvanishingpointsinperspective

drawing.

Complexnumbers[edit]

Furtherinformation:Historyofcomplexnumbers

Theearliestfleetingreferencetosquarerootsofnegativenumbersoccurredinthe

workofthemathematicianandinventorHeronofAlexandriainthe1stcenturyAD,

whenheconsideredthevolumeofanimpossiblefrustumofapyramid.They

becamemoreprominentwheninthe16thcenturyclosedformulasfortherootsof

thirdandfourthdegreepolynomialswerediscoveredbyItalianmathematicians

suchasNiccoldFontanaTartagliaandGerolamoCardano.Itwassoonrealizedthat

theseformulas,evenifonewasonlyinterestedinrealsolutions,sometimes

requiredthemanipulationofsquarerootsofnegativenumbers.

Thiswasdoublyunsettlingsincetheydidnotevenconsidernegativenumberstobe

onfirmgroundatthetime.WhenReneDescartescoinedtheterm"imaginary"for

thesequantitiesin1637,heintendeditasderogatory.(Seeimaginarynumberfora

discussionofthe"reality"ofcomplexnumbers.]Afurthersourceofconfusionwas

thattheequation

[-1]2=-1-1=-1{\displaystyle\left({\sqrt{-l}}\right]A{2}={\sqrt{-l}}{\sqrt

{-1}}=-!}

seemedcapriciouslyinconsistentwiththealgebraicidentity

ab=ab,{\displaystyle{\sqrt{a}}{\sqrt}={\sqrt{ab}},}

whichisvalidforpositiverealnumbersaandb,andwasalsousedincomplex

numbercalculationswithoneofa,bpositiveandtheothernegative.Theincorrect

useofthisidentity,andtherelatedidentity

1a=1a{\displaystyle{\frac{l}{\sqrt{a}}}={\sqrt{\frac{l}{a}}}}

inthecasewhenbothaandbarenegativeevenbedeviledEuler.Thisdifficulty

eventuallyledhimtotheconventionofusingthespecialsymboliinplaceof

-1{\displaystyle{\sqrt{-1}}}

toguardagainstthismistake.

The18thcenturysawtheworkofAbrahamdeMoivreandLeonhardEuler.De

Moivre'sformula(1730)states:

(cos?0+isin?0)n=cos@n0+isin回n0{\displaystyle(\cos\theta+i\sin

\theta)A{n}=\cosn\theta+i\sinn\theta}

whileEuler'sformulaofcomplexanalysis(1748)gaveus:

cos?0+isin?0=ei0.{\displaystyle\cos\theta+i\sin\theta=eA{i\theta}.}

TheexistenceofcomplexnumberswasnotcompletelyaccepteduntilCasparWessel

describedthegeometricalinterpretationin1799.CarlFriedrichGaussrediscovered

andpopularizeditseveralyearslater,andasaresultthetheoryofcomplex

numbersreceivedanotableexpansion.Theideaofthegraphicrepresentationof

complexnumbershadappeared,however,asearlyas1685,inWallis'sDeAlgebra

tractatus.

Alsoin1799,Gaussprovidedthefirstgenerallyacceptedproofofthefundamental

theoremofalgebra,showingthateverypolynomialoverthecomplexnumbershasa

fullsetofsolutionsinthatrealm.Thegeneralacceptanceofthetheoryofcomplex

numbersisduetothelaborsofAugustinLouisCauchyandNielsHenrikAbel,and

especiallythelatter,whowasthefirsttoboldlyusecomplexnumberswithasuccess

thatiswellknown.

Gaussstudiedcomplexnumbersoftheforma+bi,whereaandbareintegral,or

rational(andiisoneofthetworootsofx2+1=0).Hisstudent,GottholdEisenstein,

studiedthetypea+whereisacomplexrootofx3-1=0.Othersuchclasses

(calledcyclotomicfields)ofcomplexnumbersderivefromtherootsofunityxk-1=

0forhighervaluesofk.ThisgeneralizationislargelyduetoErnstKummer,who

alsoinventedidealnumbers,whichwereexpressedasgeometricalentitiesbyFelix

Kleinin1893.

In1850VictorAlexandrePuiseuxtookthekeystepofdistinguishingbetweenpoles

andbranchpoints,andintroducedtheconceptofessentialsingularpoints.This

eventuallyledtotheconceptoftheextendedcomplexplane.

Primenumbers[edit]

Primenumbershavebeenstudiedthroughoutrecordedhistory.Eucliddevotedone

bookoftheElementstothetheoryofprimes;initheprovedtheinfinitudeofthe

primesandthefundamentaltheoremofarithmetic,andpresentedtheEuclidean

algorithmforfindingthegreatestcommondivisoroftwonumbers.

In240BC,EratosthenesusedtheSieveofEratosthenestoquicklyisolateprime

numbers.ButmostfurtherdevelopmentofthetheoryofprimesinEuropedatesto

theRenaissanceandlatereras.

In1796,Adrien-MarieLegendreconjecturedtheprimenumbertheorem,describing

theasymptoticdistributionofprimes.Otherresultsconcerningthedistributionof

theprimesincludeEuler'sproofthatthesumofthereciprocalsoftheprimes

diverges,andtheGoldbachconjecture,whichclaimsthatanysufficientlylargeeven

numberisthesumoftwoprimes.Yetanotherconjecturerelatedtothedistribution

ofprimenumbersistheRiemannhypothesis,formulatedbyBernhardRiemannin

1859.TheprimenumbertheoremwasfinallyprovedbyJacquesHadamardand

CharlesdelaVallee-Poussinin1896.GoldbachandRiemann'sconjecturesremain

unprovenandunrefuted.

Mainclassification[edit]

"Numbersystem"redirectshere.Forsystemsforexpressingnumbers,seeNumeral

system.

Seealso:Listoftypesofnumbers

Numberscanbeclassifiedintosets,callednumbersystems,suchasthenatural

numbersandtherealnumbers,[21]Themajorcategoriesofnumbersareasfollows:

Mainnumbersystems

N{\displaystyle\mathbb{N}}

Natural

0,1,2,3,4,5,...or1,2,3,4,5,...

N0{\displaystyle\mathbb{N}_{0}}

or

N1{\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

/

b

whereaandbareintegersandbisnot0

R{\displaystyle\mathbb{R}}

Real

Thelimitofaconvergentsequenceofrationalnumbers

C{\displaystyle\mathbb{C}}

Complex

a+biwhereaandbarerealnumbersandiisaformalsquarerootof-1

Thereisgenerallynoprobleminidentifyingeachnumbersystemwithaproper

subsetofthenextone(byabuseofnotation),becauseeachofthesenumbersystems

iscanonicallyisomorphictoapropersubsetofthenextone,[citationneeded]The

resultinghierarchyallows,forexample,totalk,formallycorrectly,aboutreal

numbersthatarerationalnumbers,andisexpressedsymbolicallybywriting

NuZuQuRuC{\displaystyle\mathbb{N}\subset\mathbb{Z}\subset

\mathbb{Q}\subset\mathbb{R}\subset\mathbb{C}}

Naturalnumbers[edit]

Mainarticle:Naturalnumber

Thenaturalnumbers,startingwith1

Themostfamiliarnumbersarethenaturalnumbers(sometimescalledwhole

numbersorcountingnumbers):1,2,3,andsoon.Traditionally,thesequenceof

naturalnumbersstartedwith1(0wasnotevenconsideredanumberforthe

AncientGreeks.]However,inthe19thcentury,settheoristsandother

mathematiciansstartedincluding0(cardinalityoftheemptyset,i.e,0elements,

where0isthusthesmallestcardinalnumber]inthesetofnaturalnumbers.[22][23]

Today,differentmathematiciansusethetermtodescribebothsets,including0or

not.ThemathematicalsymbolforthesetofallnaturalnumbersisN,alsowritten

N{\displaystyle\mathbb{N}}

,andsometimes

N0{\displaystyle\mathbb{N}_{0}}

or

N1{\displaystyle\mathbb{N}_{1}}

whenitisnecessarytoindicatewhetherthesetshouldstartwith0or1,

respectively.

Inthebase10numeralsystem,inalmostuniversalusetodayformathematical

operations,thesymbolsfornaturalnumbersarewrittenusingtendigits:0,1,2,3,4,

5,6,7,8,and9.Theradixorbaseisthenumberofuniquenumericaldigits,

includingzero,thatanumeralsystemusestorepresentnumbers(forthedecimal

system,theradixis10).Inthisbase10system,therightmostdigitofanatural

numberhasaplacevalueof1,andeveryotherdigithasaplacevaluetentimesthat

oftheplacevalueofthedigittoitsright.

Insettheory,whichiscapableofactingasanaxiomaticfoundationformodern

mathematics,[24]naturalnumberscanberepresentedbyclassesofequivalentsets.

Forinstance,thenumber3canberepresentedastheclassofallsetsthathave

exactlythreeelements.Alternatively,inPeanoArithmetic,thenumber3is

representedassssO,wheresisthe"successor"function(i.e.,3isthethirdsuccessor

of0).Manydifferentrepresentationsarepossible;allthatisneededtoformally

represent3istoinscribeacertainsymbolorpatternofsymbolsthreetimes.

Integers[edit]

Mainarticle:Integer

Thenegativeofapositiveintegerisdefinedasanumberthatproduces0whenitis

addedtothecorrespondingpositiveinteger.Negativenumbersareusuallywritten

withanegativesign(aminussign),Asanexample,thenegativeof7iswritten-7,

and7+(-7)=0.Whenthesetofnegativenumbersiscombinedwiththesetof

naturalnumbers(including0),theresultisdefinedasthesetofintegers,Zalso

written

Z{\displaystyle\mathbb{Z}}

.HeretheletterZcomesfromGermanZahtmeaning'number'.Thesetofintegers

formsaringwiththeoperationsadditionandmultiplication.[25]

Thenaturalnumbersformasubsetoftheintegers.Asthereisnocommonstandard

fortheinclusionornotofzerointhenaturalnumbers,thenaturalnumberswithout

zeroarecommonlyreferredtoaspositiveintegers,andthenaturalnumberswith

zeroarereferredtoasnon-negativeintegers.

Rationalnumbers[edit]

Mainarticle:Rationalnumber

Arationalnumberisanumberthatcanbeexpressedasafractionwithaninteger

numeratorandapositiveintegerdenominator.Negativedenominatorsareallowed,

butarecommonlyavoided,aseveryrationalnumberisequaltoafractionwith

positivedenominator.Fractionsarewrittenastwointegers,thenumeratorandthe

denominator,withadividingbarbetweenthem.Thefraction

m

/

n

representsmpartsofawholedividedintonequalparts.Twodifferentfractions

maycorrespondtothesamerationalnumber;forexamp