what is real number reallyWord文档下载推荐.docx
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tasteandtooanalytictopleasethealgebraists.
Inthiswebpage,I'
lldiscussthemathematicalmeaningof"
realnumber."
Beforethat,Iwanttodiscussthismoreelementaryquestion:
wheredidthename"
real"
comefrom?
(Itturnsouttohavelittletodowiththedeeperpropertiesofrealnumbers.)Toanswerthatquestion,Ifirstneedtotalkaboutcomplexnumbers.
Treatingpointsintheplaneasnumbers
Thereisanaturalwayto"
add"
or"
multiply"
twopointsintheEuclideanplane.By"
natural"
Imeanthatthedefinitionshaveturnedouttobeusefulformanyapplications,andthatthedefinitionsarefairlysimple.Unfortunately,thedefinitionstaketheirsimplestformsifweusedifferentcoordinatesystemsfortheadditionandmultiplicationoperations.
∙
The"
addition"
ofpointsisdescribedmostsimplyasvectoraddition.Avectorcanberepresentedbyadirectedline-segment;
twovectorsareconsideredequaliftheypointinthesamedirectionandhavethesamelength.(Seediagram.)Wecanchangetherepresentationofavectorbymovingit(i.e.,"
translating"
it)toanewpositionparalleltotheoriginalposition.
ToaddtwovectorsV1andV2,representthemwithdirectedline-segmentssothattheinitialendofV2islocatedattheterminalendofV1.Thusthearrowsinthediagramformapath:
startattheinitialendofV1,proceedtoitsterminalend,thenturnacornerandfollowV2fromitsinitialendtoitsterminalend.Thesum,orresultant,V1+V2,isthejourneygoingfromtheinitialendofV1totheterminalendofV2.Thatsumisrepresentedbyasingledirectedline-segment,thedashedthirdsideofthetriangle.
TorepresentvectorswiththeCartesiancoordinatesystem,drawavectorVsothatitsinitialendisattheorigin(0,0).Thenthecoordinatesofthelocationofitsterminalendareusedasthecoordinatesofthevector.(Seediagram.)
Ifweusethatcoordinatesystem,thentheformulaforvectoradditionisverysimple:
ThefirstcoordinateofV1+V2isthesumofthefirstcoordinatesofV1andV2,andthesecondcoordinateofV1+V2isthesumofthesecondcoordinatesofV1andV2.Thatis,
(a,b)+(c,d)=(a+c,b+d)
multiplication"
thatwewanttousecanalsobedescribedinCartesiancoordinates:
(a,b)⋅(c,d)=(ac−bd,ad+bc).Butthat'
sabitcomplicatedandnonintuitive;
itlookssomewhatarbitraryandcontrived.Wegetamuchsimpler,moregeometricallyappealingdefinitionifweswitchtopolarcoordinates.Letapointberepresentedby<
r,θ>
ifithasradiusrandangleθ--i.e.,ifitislocatedrunitsawayfromtheorigin,andonaraythatisθradianscounterclockwisefromtheraythatpointstowardtheright.ThatpointhasCartesiancoordinates(r
cosθ,r
sinθ).IfyousubstitutethosevaluesintoourCartesianformulaformultiplication,andthensimplifyusingsometrigonometricidentities,you'
llendupwiththismuchsimplerdefinitionofmultiplication:
IfP1haspolarcoordinates<
r1,θ1>
andP2haspolarcoordinates<
r2,θ2>
then
theproductP1P2isdefinedtobethepointwithpolarcoordinates<
r1r2,θ1+θ2>
.
∙Inotherwords,multiplytheradiiandaddtheangles.TheeffectofmultiplyingpointsintheplanebyP2istorotatetheplanethroughanangleofθ2andstretch(orshrink)theplanebyamagnificationfactorofr2.Thisconceptisverysimple,andit'
squiteusefulinengineering,whichisoftenconcernedwithdescribingrotations(e.g.,ofengines).
Whenadditionandmultiplicationaredefinedasabove,thenthepointsintheplanearecalledcomplexnumbers,forreasonsthatwillbediscussedafewparagraphsfromnow.
Since(a,0)+(c,0)=(a+c,0)and(a,0)×
(c,0)=(ac,0),thepointsalongthehorizontalaxishaveanarithmeticjustlike"
ordinary"
numbers;
wewillwrite(a,0)morebrieflyasa.Forinstance,(5,0)willbewrittenas5.Thepointsalongtheverticalaxisalsohaveashorternotation:
thepoint(0,b)willbewrittenmorebrieflyasbi;
forinstance,(0,5)willbewrittenas5i.Theistandsfor"
imaginary"
forreasonsexplainedbelow.
Importantexercises.Usingeithertheformula(a,b)×
(c,d)=(ac−bd,ad+bc)orthedefinitionintermsofpolarcoordinates,thebeginnershouldnowverifythati2
=
−1.Thatwillbeimportantinthediscussionbelow.
What'
s"
abouttherealnumbers?
Probablythesimplestwaytounderstand"
complexnumbers"
istostartwithpointsintheplane,asIhavedoneintheprecedingparagraphs.However,byahistoricalaccident,thesimplestexplanationwasnotthefirstexplanationdiscovered.Indeed,thegeometric,points-in-the-planeviewpointwasn'
tdiscovereduntilthe19thcentury,longafterthealgebraiccomputationshadbeeninvestigated.Asearlyasthe16thcentury,mathematiciansweredevisingnew"
numbers"
asawayofsolvingpolynomialequations;
theywerethinkingintermsofalgebraicformulasratherthanpictures.Theywereparticularlyinterestedinthethirdandfourthdegreeequationsatthattime,buttheyevenhadnewinsightsintothequadraticequation.Theattitudethattheytookwassomethinglikethis:
Weallknowthatthereisn'
treallyany"
number"
pthatcansatisfytheequationp2
−1.Sucha"
canonlyexistinourimagination.Butifitsomehowdidexist,whatkindofarithmeticruleswouldithavetofollow?
Youhavetoadmirethegeniusofthe16thcenturymathematicians:
Theycorrectlyworkedoutthearithmeticrulesofthecomplexnumbersdespitetheirlackofthesimplegeometricmodel;
theycalculatedwith"
whoseexistencetheydidn'
tevenbelievein!
Theirterminologywasunfortunate,however.Thereisnothingfictitiousordreamlikeaboutrotationsofengines,butthenamestuck.Thepointsontheverticalaxisarenowcalledimaginarynumbers,despitethefactthattheyhaveverytangibleapplications.Thepointsonthehorizontalaxisare(bycontrast)calledrealnumbers.Allthepointsintheplanearecalledcomplexnumbers,becausetheyaremorecomplicated--theyhavebotharealpartandanimaginarypart.
Thusendsourtaleaboutwherethename"
realnumber"
comesfrom.Butwehavebarelybeguninvestigatingthemathematicalpropertiesassociatedwiththatname.
Gettingridofthepictures
pointonaline"
answerisnotafullysatisfactoryanswer,becauseitisnotaxiomaticoralgebraic.Itreliesonpicturesthatwedon'
treallyunderstand.Forinstance,thesetofrealnumbersandthesetofrationalnumbershaveessentiallythesamepicture,buttheiralgebraicpropertiesdifferinwaysthatareveryimportantforanalysts.
Imaginestudyingthatpictureofalineunderasupermicroscope.Ifyoucouldmagnifythelineataveryhighpower--sayatamagnificationofagoogolplex,orbetteryetamagnificationofinfinity--woulditstilllookthesame?
Orwouldyouseearowofdotsseparatedbyspaces,likethedotsinapictureinanewspaper?
(Itturnsoutthat,insomesense,therealnumberswouldstilllooklikealineunderinfinitemagnification,buttherationalnumberswouldbedotsseparatedbyspaces.Butthatisonlyavagueandintuitivestatement,notanythingprecisethatwecanuseinproofs.)
Theonlywaytogetrigorousanswerstothesequestionsistosetupaverycarefulsystemofaxiomsaboutgeometry...butthatamountstothesamethingassettingupacarefulsetofaxiomsaboutthealgebraicpropertiesoftherealnumbers.Itturnsoutthatthelatterisalittleeasier,sowemayaswellconcentrateonthealgebraicaspectsofthesituation.Toanswerquestionslikethis,ultimatelywehavetogetawayfromthepictures;
wehavetounderstandtherealnumbersentirelyintermsofformulas.
Asapreview,hereisthedefinitionthatwe'
regoingtoendupwith:
thereallineisaDedekind-completeorderedfield.That'
scomplicated,sowe'
llworkourwayuptoitinstages.We'
lldiscuss:
∙Whatisafield?
∙Whatisanorderedfield?
∙WhatisaDedekind-completeorderedfield?
∙WhydoIsaythatthereallineisaDedekind-completeorderedfield?
Howcanthatbeadefinition?
Groupsandfields
Firstofall,agroupisamathematicalobject;
itisatriple(X,e,*)withtheseproperties:
∙Xisanonemptyset.
∙eisaspeciallychosenmemberofthesetX.Itiscalledtheidentityofthegroup.
∙*isabinaryoperationonX,whichwemaycallthegroupoperation.ThismeansthatwheneverpandqaremembersofX,thenp*qisalsoamemberofX.
∙(p*q)*r=p*(q*r)forallp,q,rinX.
∙p*e=e*p=pforeverypinX.
∙ForeachpinX,thereexistsatleastonecorrespondingqinXthatsatisfiesp*q=q*p=e.(Itcanbeshownthatthereisatmostonesuchq,andthusqisuniquelydeterminedbyp;
wecallqtheinverseofp.)
Exercises:
∙Theidentityisuniquelydetermined---i.e.,ifp*e1=e1*p=pandp*e2=e2*p=pforallpinX,thene1=e2.
∙Inversesareuniquelydetermined---i.e.,if
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