数学专业英语4文档格式.docx
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数学专业英语4文档格式.docx
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2.4Integers,RationalNumbersandRealnumbers
TEXTAIntegersandrationalnumbers
ThereexistcertainsubsetsofRwhicharedistinguishedbecausetheyhavespecialpropertiesnotsharedbyallrealnumbers.Inthissectionweshalldiscusstwosuchsubsets,theintegersandtherationalnumbers.
Tointroducethepositiveintegerswebeginwiththenumber1,whoseexistenceisguaranteedbyAxiom4.Thenumber1+1isdenotedby2,thenumber2+1by3,andsoon.Thenumbers1,2,3,…,obtainedinthiswaybyrepeatedadditionof1areallpositive,andtheyarecalledthepositiveintegers.Strictlyspeaking,thisdescriptionofthepositiveintegersisnotentirelycompletebecausewehavenotexplainedindetailswhatwemeanbytheexpressions“andsoon”,or“repeatedadditionof1”.Althoughtheintuitivemeaningofexpressionsmayseemclear,inacarefultreatmentofthereal-numbersystemitisnecessarytogiveamoreprecisedefinitionofthepositiveintegers.Therearemanywaystodothis.Oneconvenientmethodistointroducefirstthenotionofaninductiveset.
DEFINITIONOFANINDUCTIVESETAsetofrealnumbersiscalledaninductivesetifithasthefollowingtwoproperties:
(a)Thenumber1isintheset.
(b)Foreveryxintheset,thenumberx+1isalsointheset.
Forexample,Risaninductiveset.SoisthesetR+.Nowweshalldefinethepositiveintegerstobethoserealnumberswhichbelongtoeveryinductiveset.
DEFINITIONOFPOSITIVEINTEGERSArealnumberiscalledapositiveintegerifitbelongstoeveryinductiveset.
LetPdenotethesetofallpositiveintegers.ThenPisitselfaninductivesetbecause(a)itcontains1,and(b)itcontainsx+1wheneveritcontainsx.SincethemembersofPbelongtoeveryinductiveset,werefertoPasthesmallestinductiveset.ThispropertyofthesetPformsthelogicalbasisforatypeofreasoningthatmathematicianscallproofbyinduction,adetaileddiscussionofwhichisgiveninPart4ofthisIntroduction.
Thenegativesofthepositiveintegersarecalledthenegativeintegers.Thepositiveintegers,togetherwiththenegativeintegersand0(zero),formasetZwhichwecallsimplythesetofintegers.
Inathoroughtreatmentofthereal-numbersystem,itwouldbenecessaryatthisstagetoprovecertaintheoremsaboutintegers.Forexample,thesum,difference,orproductoftwointegersisaninteger,butthequotientoftwointegersneednotbeaninteger.However,weshallnotenterintothedetailsofsuchproofs.
Quotientsofintegersa/b(whereb0)arecalledrationalnumber.Thesetofrationalnumbers,denotedbyQ,containsZasasubset.ThereadershouldrealizethatallthefieldaxiomsandtheorderaxiomsaresatisfiedbyQ.Forthisreason,wesaythatthesetofrationalnumbersisanorderedfield.RealnumbersthatarenotinQarecalledirrational.
Notations
FieldaxiomsAfieldisanysetofelementsthatsatisfiesthefieldaxiomsforbothadditionandmultiplicationandisacommutativedivisionalgebra,wheredivisionalgebra,alsocalleda"
divisionring"
or"
skewfield,"
meansaringinwhicheverynonzeroelementhasamultiplicativeinverse,butmultiplicationisnotnecessarilycommutative.
OrderaxiomsAtotalorder(or"
totallyorderedset,"
linearlyorderedset"
)isasetplusarelationontheset(calledatotalorder)thatsatisfiestheconditionsforapartialorderplusanadditionalconditionknownasthecomparabilitycondition.ArelationisatotalorderonasetS("
totallyordersS"
)ifthefollowingpropertieshold.
1.Reflexivity:
aaforallaS.
2.Antisymmetry:
abandbaimpliesa=b.
3.Transitivity:
abandbcimpliesac.
4.Comparability(trichotomylaw):
Foranya,bS,eitheraborba.
Thefirstthreearetheaxiomsofapartialorder,whileadditionofthetrichotomylawdefinesatotalorder.
TEXTBGeometricinterpretationofrealnumbersaspointsonaline
Thereaderisundoubtedlyfamiliarwiththegeometricrepresentationofrealnumbersbymeansofpointsonastraightline.Apointisselectedtorepresent0andanother,totherightof0,torepresent1,asillustratedinFigure2-4-1.Thischoicedeterminesthescale.IfoneadoptsanappropriatesetofaxiomsforEuclideangeometry,theneachrealnumbercorrespondstoexactlyonepointonthislineand,conversely,eachpointonthelinecorrespondstooneandonlyonerealnumber.Forthisreasonthelineisoftencalledthereallineortherealaxis,anditiscustomarytousethewordsrealnumberandpointinterchangeably.Thusweoftenspeakofthepointxratherthanthepointcorrespondingtotherealnumbers.
Theorderingrelationamongtherealnumbershasasimplegeometricinterpretation.Ifx<
y,thepointxliestotheleftofthepointyasshowninFigure2-4-1.Positivenumberslietotherightof0andnegativenumberstotheleftof0.Ifa<
b,apointxsatisfiestheinequalitiesa<
x<
bifandonlyifxisbetweenaandb.
Thisdeviceforrepresentingrealnumbersgeometricallyisaveryworthwhileaidthathelpsustodiscoverandunderstandbettercertainpropertiesofrealnumbers.However,thereadershouldrealizethatallpropertiesofrealnumbersthataretobeacceptedastheoremsmustbededuciblefromtheaxiomswithoutanyreferencetogeometry.Thisdoesnotmeanthatoneshouldnotmakeuseofgeometryinstudyingpropertiesofrealnumbers.Onthecontrary,thegeometryoftensuggeststhemethodofproofofaparticulartheorem,andsometimesageometricargumentismoreilluminatingthanapurelyanalyticproof(onedependingentirelyontheaxiomsfortherealnumbers).Inthisbook,geometricargumentsareusedtoalargeextenttohelpmotivateorclarifyaparticulardiscuss.Nevertheless,theproofsofalltheimportanttheoremsarepresentedisanalyticform.
SUPPLEMENTPrimeNumber
Aprimenumber(orprimeinteger,oftensimplycalleda"
prime"
forshort)isapositiveintegerp>
1thathasnopositiveintegerdivisorsotherthan1andpitself.Forexample,theonlydivisorsof13are1and13,making13aprimenumber,whilethenumber24hasdivisors1,2,3,4,6,8,12,and24(correspondingtothefactorization24=233),making24notaprimenumber.Positiveintegersotherthan1whicharenotprimesarecalledcompositenumbers.
Primenumbersarethereforenumbersthatcannotbefactoredor,moreprecisely,arenumbersnwhosedivisorsaretrivialandgivenbyexactly1andn.
Thenumber1isaspecialcasewhichisconsideredneitherprimenorcomposite.With1excluded,thesmallestprimeistherefore2andsince2istheonlyevenprime,itisalsosomewhatspecial.Notealsothatwhile2isconsideredaprimetoday,atonetimeitwasnot.
Thenthprimenumberiscommonlydenotedpn,sop1=2,p2=3,andsoon,andmaybecomputedinMathematicaasPrime[n].ThesetofprimesissometimesdenotedP,representedinMathematicaasPrimes.
Eulercommented"
Mathematicianshavetriedinvaintothisdaytodiscoversomeorderinthesequenceofprimenumbers,andwehavereasontobelievethatitisamysteryintowhichthemindwillneverpenetrate"
.Ina1975lecture,D.Zagiercommented"
TherearetwofactsaboutthedistributionofprimenumbersofwhichIhopetoconvinceyousooverwhelminglythattheywillbepermanentlyengravedinyourhearts.Thefirstisthat,despitetheirsimpledefinitionandroleasthebuildingblocksofthenaturalnumbers,theprimenumbersgrowlikeweedsamongthenaturalnumbers,seemingtoobeynootherlawthanthatofchance,andnobodycanpredictwherethenextonewillsprout.Thesecondfactisevenmoreastonishing,foritstatesjusttheopposite:
thattheprimenumbersexhibitstunningregularity,thattherearelawsgoverningtheirbehavior,andthattheyobeytheselawswithalmostmilitaryprecision"
.
LargeprimesincludethelargeMersenneprimes,Ferrier'
sprime.ThelargestknownprimeasofFeb.2005istheMersenneprime225964951-1(Weisstein2005).
Primenumberscanbegeneratedbysievingprocesses(suchasthesieveofEratosthenes),andluckynumbers,whicharealsogeneratedbysieving,appeartosharesomeinterestingasymptoticpropertieswiththeprimes.Primenumberssatisfymanystrangeandwonderfulproperties.
Thefunctionthatgivesthenumberofprimeslessthanorequaltoanumbernisdenoted(n)andiscalledtheprimecountingfunction.Thetheoremgivinganasymptoticformfor(n)iscalledtheprimenumbertheorem.
Thefundamentaltheoremofarithmeticstatesthatanypositiveintegercanberepresentedinexactlyonewayasaproductofprimes.Euclid'
ssecondtheoremdemonstratedthatthereareaninfinitenumberofprimes.However,itisnotknownifthereareaninfinitenumberofprimesoftheformn2+1,whetherthereareaninfinitenumberoftwinprimes(thetwinprimeconjecture),orifaprimecanalwaysbefoundbetweenn2and(n+1)2.ThelattertwoofthesearetwoofLandau'
sproblems.
Primesconsistingofconsecutivedigits(counting0ascomingafter9)include2,3,5,7,23,67,89,4567,78901,....Primesconsistingofdigitsthatarethemselvesprimesinclude23,37,53,73,223,227,233,257,277,337,353,373,523,557,...,whichisoneoftheSmarandachesequences.
Becauseaprimenumberphasthetrivialfactors1andp,BillGatesaccidentallyreferredtoatrivialoperationwhenhestated"
Becauseboththesystem'
sprivacyandthesecurityofdigitalmoneydependonencryption,abreakthroughinmathematicsorcomputersciencethatdefeatsthecryptographicsystemcouldbeadisaster.Theobviousmathematicalbreakthroughwouldbethedevelopmentofaneasywaytofactorlargeprimenumbers[emphasisadded]"
(Ga
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