数学专业英语(吴炯圻-第2版)2-4市公开课一等奖省赛课获奖课件

NewWords&Expressions:conversely反之geometricinterpretation几何意义correspond对应induction归纳法deducible可推导proofbyinduction归纳证实difference差inductiveset归纳集distinguished著名inequality不等式entirelycomplete完整integer整数Euclid欧几里得interchangeably可相互交换Euclidean欧式intuitive直观thefieldaxiom域公理irrational无理2.4整数、有理数与实数Integers,RationalNumbersandRealNumbers第1页

NewWords&Expressions:irrationalnumber无理数rational有理theorderaxiom序公理rationalnumber有理数ordered有序reasoning推理product积scale尺度，刻度quotient商sum和第2页ThereexistcertainsubsetsofRwhicharedistinguishedbecausetheyhavespecialpropertiesnotsharedbyallrealnumbers.Inthissectionweshalldiscusssuchsubsets,theintegersandtherationalnumbers.4－AIntegersandrationalnumbers有一些R子集很著名，因为他们含有实数所不具备特殊性质。在本节我们将讨论这么子集，整数集和有理数集。第3页Tointroducethepositiveintegerswebeginwiththenumber1,whoseexistenceisguaranteedbyAxiom4.Thenumber1+1isdenotedby2,thenumber2+1by3,andsoon.Thenumbers1,2,3,…,obtainedinthiswaybyrepeatedadditionof1areallpositive,andtheyarecalledthepositiveintegers.我们从数字1开始介绍正整数，公理4确保了1存在性。1+1用2表示，2+1用3表示，以这类推，由1重复累加方式得到数字1,2,3，…都是正，它们被叫做正整数。第4页Strictlyspeaking,thisdescriptionofthepositiveintegersisnotentirelycompletebecausewehavenotexplainedindetailwhatwemeanbytheexpressions“andsoon”,or“repeatedadditionof1”.严格地说，这种关于正整数描述是不完整，因为我们没有详细解释“等等”或者“1重复累加”含义。第5页Althoughtheintuitivemeaningofexpressionsmayseemclear,incarefultreatmentofthereal-numbersystemitisnecessarytogiveamoreprecisedefinitionofthepositiveintegers.Therearemanywaystodothis.Oneconvenientmethodistointroducefirstthenotionofaninductiveset.即使这些说法直观意思似乎是清楚，不过在认真处理实数系统时有必要给出一个更准确关于正整数定义。有很各种方式来给出这个定义，一个简便方法是先引进归纳集概念。第6页DEFINITIONOFANINDUCTIVESET.Asetofrealnumbersiscalledaninductivesetifithasthefollowingtwoproperties:Thenumber1isintheset.Foreveryxintheset,thenumberx+1isalsointheset.Forexample,Risaninductiveset.Soistheset.Nowweshalldefinethepositiveintegerstobethoserealnumberswhichbelongtoeveryinductiveset.现在我们来定义正整数，就是属于每一个归纳集实数。第7页LetPdenotethesetofallpositiveintegers.ThenPisitselfaninductivesetbecause(a)itcontains1,and(b)itcontainsx+1wheneveritcontainsx.SincethemembersofPbelongtoeveryinductiveset,werefertoPasthesmallestinductiveset.用P表示全部正整数集合。那么P本身是一个归纳集，因为其中含1，满足(a)；只要包含x就包含x+1,满足(b)。因为P中元素属于每一个归纳集，所以P是最小归纳集。第8页ThispropertyofPformsthelogicalbasisforatypeofreasoningthatmathematicianscallproofbyinduction,adetaileddiscussionofwhichisgiveninPart4ofthisintroduction.P这种性质形成了一个推理逻辑基础，数学家称之为归纳证实，在介绍第四部分将给出这种方法详细叙述。第9页Thenegativesofthepositiveintegersarecalledthenegativeintegers.Thepositiveintegers,togetherwiththenegativeintegersand0(zero),formasetZwhichwecallsimplythesetofintegers.正整数相反数被叫做负整数。正整数，负整数和零组成了一个集合Z，简称为整数集。第10页Inathoroughtreatmentofthereal-numbersystem,itwouldbenecessaryatthisstagetoprovecertaintheoremsaboutintegers.Forexample,thesum,difference,orproductoftwointegersisaninteger,butthequotientoftwointegersneednotbeaninteger.However,weshallnotenterintothedetailsofsuchproofs.在实数系统中，为了周密性，此时有必要证实一些整数定理。比如，两个整数和、差和积仍是整数，不过商不一定是整数。然而还不能给出证实细节。第11页Quotientsofintegersa/b(whereb≠0)arecalledrationalnumbers.Thesetofrationalnumbers,denotedbyQ,containsZasasubset.ThereadershouldrealizethatallthefieldaxiomsandtheorderaxiomsaresatisfiedbyQ.Forthisreason,wesaythatthesetofrationalnumbersisanorderedfield.RealnumbersthatarenotinQarecalledirrational.整数a与b商被叫做有理数，有理数集用Q表示，Z是Q子集。读者应该认识到Q满足全部域公理和序公理。所以说有理数集是一个有序域。不是有理数实数被称为无理数。第12页Thereaderisundoubtedlyfamiliarwiththegeometricrepresentationofrealnumbersbymeansofpointsonastraightline.Apointisselectedtorepresent0andanother,totherightof0,torepresent1,asillustratedinFigure2-4-1.Thischoicedeterminesthescale.4－BGeometricinterpretationofrealnumbersaspointsonaline毫无疑问，读者都熟悉经过在直线上描点方式表示实数几何意义。如图2-4-1所表示，选择一个点表示0，在0右边另一个点表示1。这种做法决定了刻度。第13页IfoneadoptsanappropriatesetofaxiomsforEuclideangeometry,theneachrealnumbercorrespondstoexactlyonepointonthislineand,conversely,eachpointonthelinecorrespondstooneandonlyonerealnumber.假如采取欧式几何公理中一个恰当集合，那么每一个实数刚好对应直线上一个点，反之，直线上每一个点也对应且只对应一个实数。第14页Forthisreasonthelineisoftencalledthereallineortherealaxis,anditiscustomarytousethewordsrealnumberandpointinterchangeably.Thusweoftenspeakofthepointxratherthanthepointcorrespondingtotherealnumber.为此直线通常被叫做实直线或者实轴，习惯上使用“实数”这个单词，而不是“点”。所以我们经常说点x不是指与实数对应那个点。第15页Thisdeviceforrepresentingrealnumbersgeometricallyisaveryworthwhileaidthathelpsustodiscoverandunderstandbettercertainpropertiesofrealnumbers.However,thereadershouldrealizethatallpropertiesofrealnumbersthataretobeacceptedastheoremsmustbededuciblefromtheaxiomswithoutanyreferencestogeometry.这种几何化表示实数方法是非常值得推崇，它有利于帮助我们发觉和了解实数一些性质。然而，读者应该认识到，拟被采取作为定理所相关于实数性质都必须不借助于几何就能从公理推出。第16页This

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