数学专业英语.docx

数学专业英语.docx

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数学专业英语

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