数学专业英语.docx
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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
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