数学专业英语4.docx

数学专业英语4.docx

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数学专业英语4

MathematicalEnglish

Dr.XiaominZhang

Email:

zhangxiaomin@

§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,"or"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

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[emph