real numberWord格式文档下载.docx
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real numberWord格式文档下载.docx
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decimalrepresentations"
togetherwithpreciseinterpretationsforthearithmeticoperationsandtheorderrelation.Thesedefinitionsareequivalentintherealmofclassicalmathematics.
Contents
[hide]
∙1Basicproperties
∙2Inphysics
∙3Incomputation
∙4Notation
∙5History
∙6Definition
o6.1Axiomaticapproach
o6.2Constructionfromtherationalnumbers
∙7Properties
o7.1Completeness
o7.2"
Thecompleteorderedfield"
o7.3Advancedproperties
∙8Generalizationsandextensions
∙9"
Reals"
insettheory
∙10Realnumbersandlogic
∙11Seealso
∙12Notes
∙13References
∙14Externallinks
[edit]Basicproperties
Arealnumbermaybeeitherrationalorirrational;
eitheralgebraicortranscendental;
andeitherpositive,negative,orzero.Realnumbersareusedtomeasurecontinuousquantities.Theymayintheorybeexpressedbydecimalrepresentationsthathaveaninfinitesequenceofdigitstotherightofthedecimalpoint;
theseareoftenrepresentedinthesameformas324.823122147…Theellipsis(threedots)indicatethattherewouldstillbemoredigitstocome.
Moreformally,realnumbershavethetwobasicpropertiesofbeinganorderedfield,andhavingtheleastupperboundproperty.Thefirstsaysthatrealnumberscompriseafield,withadditionandmultiplicationaswellasdivisionbynonzeronumbers,whichcanbetotallyorderedonanumberlineinawaycompatiblewithadditionandmultiplication.Thesecondsaysthatifanonemptysetofrealnumbershasanupperbound,thenithasaleastupperbound.Thesecondconditiondistinguishestherealnumbersfromtherationalnumbers:
forexample,thesetofrationalnumberswhosesquareislessthan2isasetwithanupperbound(e.g.1.5)butnoleastupperbound:
hencetherationalnumbersdonotsatisfytheleastupperboundproperty.
[edit]Inphysics
Inthephysicalsciences,mostphysicalconstantssuchastheuniversalgravitationalconstant,andphysicalvariables,suchasposition,mass,speed,andelectriccharge,aremodeledusingrealnumbers.Infact,thefundamentalphysicaltheoriessuchasclassicalmechanics,electromagnetism,quantummechanics,generalrelativityandthestandardmodelaredescribedusingmathematicalstructures,typicallysmoothmanifoldsorHilbertspaces,thatarebasedontherealnumbersalthoughactualmeasurementsofphysicalquantitiesareoffiniteaccuracyandprecision.
Insomerecentdevelopmentsoftheoreticalphysicsstemmingfromtheholographicprinciple,theUniverseisseenfundamentallyasaninformationstore,essentiallyzeroesandones,organizedinmuchlessgeometricalfashionandmanifestingitselfasspace-timeandparticlefieldsonlyonamoresuperficiallevel.ThisapproachremovestherealnumbersystemfromitsfoundationalroleinphysicsandevenprohibitstheexistenceofinfiniteprecisionrealnumbersinthephysicaluniversebyconsiderationsbasedontheBekensteinbound.[2]
[edit]Incomputation
Computerarithmeticcannotdirectlyoperateonrealnumbers,butonlyonafinitesubsetofrationalnumbers,limitedbythenumberofbitsusedtostorethem,whetherasfloating-pointnumbersorarbitraryprecisionnumbers.However,computeralgebrasystemscanoperateonirrationalquantitiesexactlybymanipulatingformulasforthem(suchas
or
)ratherthantheirrationalordecimalapproximation;
[3]however,itisnotingeneralpossibletodeterminewhethertwosuchexpressionsareequal(theConstantproblem).
Arealnumberissaidtobecomputableifthereexistsanalgorithmthatyieldsitsdigits.Becausethereareonlycountablymanyalgorithms,[4]butanuncountablenumberofreals,almostallrealnumbersfailtobecomputable.Someconstructivistsaccepttheexistenceofonlythoserealsthatarecomputable.Thesetofdefinablenumbersisbroader,butstillonlycountable.
[edit]Notation
MathematiciansusethesymbolR(oralternatively,
theletter"
R"
inblackboardbold,Unicodeℝ–U+211D)torepresentthesetofallrealnumbers(asthissetisnaturallyendowedwithastructureoffield,theexpressionfieldoftherealnumbersismorefrequentlyusedthansetofallrealnumbers).ThenotationRnreferstotheCartesianproductofncopiesofR,whichisann-dimensionalvectorspaceoverthefieldoftherealnumbers;
thisvectorspacemaybeidentifiedtothen-dimensionalspaceofEuclideangeometryassoonasacoordinatesystemhasbeenchoseninthelatter.Forexample,avaluefromR3consistsofthreerealnumbersandspecifiesthecoordinatesofapointin3-dimensionalspace.
Inmathematics,realisusedasanadjective,meaningthattheunderlyingfieldisthefieldoftherealnumbers(ortherealfield).Forexamplerealmatrix,realpolynomialandrealLiealgebra.Asasubstantive,thetermisusedalmoststrictlyinreferencetotherealnumbersthemselves(e.g.,The"
setofallreals"
).
[edit]History
VulgarfractionshadbeenusedbytheEgyptiansaround1000BC;
theVedic"
SulbaSutras"
("
Therulesofchords"
)in,ca.600BC,includewhatmaybethefirst'
use'
ofirrationalnumbers.TheconceptofirrationalitywasimplicitlyacceptedbyearlyIndianmathematicianssinceManava(c.750–690BC),whowereawarethatthesquarerootsofcertainnumberssuchas2and61couldnotbeexactlydetermined.[5][verificationneeded]Around500BC,theGreekmathematiciansledbyPythagorasrealizedtheneedforirrationalnumbers,inparticulartheirrationalityofthesquarerootof2.
TheMiddleAgessawtheacceptanceofzero,negative,integralandfractionalnumbers,firstbyIndianandChinesemathematicians,andthenbyArabicmathematicians,whowerealsothefirsttotreatirrationalnumbersasalgebraicobjects,[6]whichwasmadepossiblebythedevelopmentofalgebra.Arabicmathematiciansmergedtheconceptsof"
number"
and"
magnitude"
intoamoregeneralideaofrealnumbers.[7]TheEgyptianmathematicianAbūKāmilShujāibnAslam(c.850–930)wasthefirsttoacceptirrationalnumbersassolutionstoquadraticequationsorascoefficientsinanequation,oftenintheformofsquareroots,cuberootsandfourthroots.[8][verificationneeded]
Inthe16thcentury,SimonStevincreatedthebasisformoderndecimalnotation,andinsistedthatthereisnodifferencebetweenrationalandirrationalnumbersinthisregard.
Inthe17thcentury,Descartesintroducedtheterm"
real"
todescriberootsofapolynomial,distinguishingthemfrom"
imaginary"
ones.
Inthe18thand19thcenturiestherewasmuchworkonirrationalandtranscendentalnumbers.JohannHeinrichLambert(1761)gavethefirstflawedproofthatπcannotberational;
[citationneeded]Adrien-MarieLegendre(1794)completedtheproof,andshowedthatπisnotthesquarerootofarationalnumber.PaoloRuffini(1799)andNielsHenrikAbel(1842)bothconstructedproofsofAbel–Ruffinitheorem:
thatthegeneralquinticorhigherequationscannotbesolvedbyageneralformulainvolvingonlyarithmeticaloperationsandroots.
É
varisteGalois(1832)developedtechniquesfordeterminingwhetheragivenequationcouldbesolvedbyradicals,whichgaverisetothefieldofGaloistheory.JosephLiouville(1840)showedthatneitherenore2canbearootofanintegerquadraticequation,andthenestablishedexistenceoftranscendentalnumbers,theproofbeingsubsequentlydisplacedbyGeorgCantor(1873).CharlesHermite(1873)firstprovedthateistranscendental,andFerdinandvonLindemann(1882),showedthatπistranscendental.Lindemann'
sproofwasmuchsimplifiedbyWeierstrass(1885),stillfurtherbyDavidHilbert(1893),andhasfinallybeenmadeelementarybyAdolfHurwitzandPaulGordan.
Thedevelopmentofcalculusinthe18thcenturyusedtheentiresetofrealnumberswithouthavingdefinedthemcleanly.ThefirstrigorousdefinitionwasgivenbyGeorgCantorin1871.In1874heshowedthatthesetofallrealnumbersisuncountablyinfinitebutthesetofallalgebraicnumbersiscountablyinfinite.Contrarytowidelyheldbeliefs,hisfi
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