NewWords&Expressions:conversely反之geometricinterpretation幾何意義correspond對應induction歸納法deducible可推導的proofbyinduction歸納證明difference差inductiveset歸納集distinguished著名的inequality不等式entirelycomplete完整的integer整數Euclid歐幾里得interchangeably可互相交換的Euclidean歐式的intuitive直觀的thefieldaxiom域公理irrational無理的,2.4整數、有理數與實數Integers,RationalNumbersandRealNumbers,NewWords&Expressions:irrationalnumber無理數rational有理的theorderaxiom序公理rationalnumber有理數ordered有序的reasoning推理product積scale尺度，刻度quotient商sum和,ThereexistcertainsubsetsofRwhicharedistinguishedbecausetheyhavespecialpropertiesnotsharedbyallrealnumbers.Inthissectionweshalldiscusssuchsubsets,theintegersandtherationalnumbers.,4AIntegersandrationalnumbers,有一些R的子集很著名，因為他們具有實數所不具備的特殊性質。在本節我們將討論這樣的子集，整數集和有理數集。,Tointroducethepositiveintegerswebeginwiththenumber1,whoseexistenceisguaranteedbyAxiom4.Thenumber1+1isdenotedby2,thenumber2+1by3,andsoon.Thenumbers1,2,3,obtainedinthiswaybyrepeatedadditionof1areallpositive,andtheyarecalledthepositiveintegers.,我們從數字1開始介紹正整數，公理4保證了1的存在性。1+1用2表示，2+1用3表示，以此類推，由1重復累加的方式得到的數字1,2,3，都是正的，它們被叫做正整數。,Strictlyspeaking,thisdescriptionofthepositiveintegersisnotentirelycompletebecausewehavenotexplainedindetailwhatwemeanbytheexpressions“andsoon”,or“repeatedadditionof1”.,嚴格地說，這種關于正整數的描述是不完整的，因為我們沒有詳細解釋“等等”或者“1的重復累加”的含義。,Althoughtheintuitivemeaningofexpressionsmayseemclear,incarefultreatmentofthereal-numbersystemitisnecessarytogiveamoreprecisedefinitionofthepositiveintegers.Therearemanywaystodothis.Oneconvenientmethodistointroducefirstthenotionofaninductiveset.,雖然這些說法的直觀意思似乎是清楚的，但是在認真處理實數系統時有必要給出一個更準確的關于正整數的定義。有很多種方式來給出這個定義，一個簡便的方法是先引進歸納集的概念。,DEFINITIONOFANINDUCTIVESET.Asetofrealnumbersiscalledaninductivesetifithasthefollowingtwoproperties:Thenumber1isintheset.Foreveryxintheset,thenumberx+1isalsointheset.Forexample,Risaninductiveset.Soistheset.Nowweshalldefinethepositiveintegerstobethoserealnumberswhichbelongtoeveryinductiveset.,現在我們來定義正整數，就是屬于每一個歸納集的實數。,LetPdenotethesetofallpositiveintegers.ThenPisitselfaninductivesetbecause(a)itcontains1,and(b)itcontainsx+1wheneveritcontainsx.SincethemembersofPbelongtoeveryinductiveset,werefertoPasthesmallestinductiveset.,用P表示所有正整數的集合。那么P本身是一個歸納集，因為其中含1，滿足(a)；只要包含x就包含x+1,滿足(b)。由于P中的元素屬于每一個歸納集，因此P是最小的歸納集。,ThispropertyofPformsthelogicalbasisforatypeofreasoningthatmathematicianscallproofbyinduction,adetaileddiscussionofwhichisgiveninPart4ofthisintroduction.,P的這種性質形成了一種推理的邏輯基礎，數學家稱之為歸納證明，在介紹的第四部分將給出這種方法的詳細論述。,Thenegativesofthepositiveintegersarecalledthenegativeintegers.Thepositiveintegers,togetherwiththenegativeintegersand0(zero),formasetZwhichwecallsimplythesetofintegers.,正整數的相反數被叫做負整數。正整數，負整數和零構成了一個集合Z，簡稱為整數集。,Inathoroughtreatmentofthereal-numbersystem,itwouldbenecessaryatthisstagetoprovecertaintheoremsaboutintegers.Forexample,thesum,difference,orproductoftwointegersisaninteger,butthequotientoftwointegersneednotbeaninteger.However,weshallnotenterintothedetailsofsuchproofs.,在實數系統中，為了周密性，此時有必要證明一些整數的定理。例如，兩個整數的和、差和積仍是整數，但是商不一定是整數。然而還不能給出證明的細節。,Quotientsofintegersa/b(whereb0)arecalledrationalnumbers.Thesetofrationalnumbers,denotedbyQ,containsZasasubset.ThereadershouldrealizethatallthefieldaxiomsandtheorderaxiomsaresatisfiedbyQ.Forthisreason,wesaythatthesetofrationalnumbersisanorderedfield.RealnumbersthatarenotinQarecalledirrational.,整數a與b的商被叫做有理數，有理數集用Q表示，Z是Q的子集。讀者應該認識到Q滿足所有的域公理和序公理。因此說有理數集是一個有序的域。不是有理數的實數被稱為無理數。,Thereaderisundoubtedlyfamiliarwiththegeometricrepresentationofrealnumbersbymeansofpointsonastraightline.Apointisselectedtorepresent0andanother,totherightof0,torepresent1,asillustratedinFigure2-4-1.Thischoicedeterminesthescale.,4BGeometricinterpretationofrealnumbersaspointsonaline,毫無疑問，讀者都熟悉通過在直線上描點的方式表示實數的幾何意義。如圖2-4-1所示，選擇一個點表示0，在0右邊的另一個點表示1。這種做法決定了刻度。,IfoneadoptsanappropriatesetofaxiomsforEuclideangeometry,theneachrealnumbercorrespondstoexactlyonepointonthislineand,conversely,eachpointonthelinecorrespondstooneandonlyonerealnumber.,如果采用歐式幾何公理中一個恰當的集合，那么每一個實數剛好對應直線上的一個點，反之，直線上的每一個點也對應且只對應一個實數。,Forthisreasonthelineisoftencalledthereallineortherealaxis,anditiscustomarytousethewordsrealnumberandpointinterchangeably.Thusweoftenspeakofthepointxratherthanthepointcorrespondingtotherealnumber.,為此直線通常被叫做實直線或者實軸，習慣上使用“實數”這個單詞，而不是“點”。因此我們經常說點x不是指與實數對應的那個點。,Thisdeviceforrepresentingrealnumbersgeometricallyisaveryworthwhileaidthathelpsustodiscoverandunderstandbettercertainpropertiesofrealnumbers.However,thereadershouldrealizethatallpropertiesofrealnumbersthataretobeacceptedastheoremsmustbededuciblefromtheaxiomswithoutanyreferencestogeometry.,這種幾何化的表示實數的方法是非常值得推崇的，它有助于幫助我們發現和理解實數的某些性質。然而，讀者應該認識到，擬被采用作為定理的所有關于實數的性質都必須不借助于幾何就能從公理推出。,Thisdoesnotmeanthatoneshouldnotmakeuseofgeometryinstudyingpropertiesofrealnumbers.Onthecontrary,thegeometryoftensuggeststhemethodofproofofaparticulartheorem,andsometimesageometricargumentismoreilluminatingthanapurelyanalyticproof(onedependingentirelyontheaxiomsfortherealnumbers).,這并不意味著研究實數的性質時不會應用到幾何。相反，幾何經常會為證明一些定理提供思路，有時幾何討論比純分