數(shù)學(xué)專業(yè)英語(吳炯圻-第2版)2-4

NewWords&Expressions:conversely反之geometricinterpretation幾何意義correspond對(duì)應(yīng)induction歸納法deducible可推導(dǎo)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(shù)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(shù)、有理數(shù)與實(shí)數(shù)Integers,RationalNumbersandRealNumbers,NewWords&Expressions:irrationalnumber無理數(shù)rational有理的theorderaxiom序公理rationalnumber有理數(shù)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?yàn)樗麄兙哂袑?shí)數(shù)所不具備的特殊性質(zhì)。在本節(jié)我們將討論這樣的子集，整數(shù)集和有理數(shù)集。,Tointroducethepositiveintegerswebeginwiththenumber1,whoseexistenceisguaranteedbyAxiom4.Thenumber1+1isdenotedby2,thenumber2+1by3,andsoon.Thenumbers1,2,3,obtainedinthiswaybyrepeatedadditionof1areallpositive,andtheyarecalledthepositiveintegers.,我們從數(shù)字1開始介紹正整數(shù)，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(fù)累加的方式得到的數(shù)字1,2,3，都是正的，它們被叫做正整數(shù)。,Strictlyspeaking,thisdescriptionofthepositiveintegersisnotentirelycompletebecausewehavenotexplainedindetailwhatwemeanbytheexpressions“andsoon”,or“repeatedadditionof1”.,嚴(yán)格地說，這種關(guān)于正整數(shù)的描述是不完整的，因?yàn)槲覀儧]有詳細(xì)解釋“等等”或者“1的重復(fù)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(rèn)真處理實(shí)數(shù)系統(tǒng)時(shí)有必要給出一個(gè)更準(zhǔn)確的關(guān)于正整數(shù)的定義。有很多種方式來給出這個(gè)定義，一個(gè)簡便的方法是先引進(jìn)歸納集的概念。,DEFINITIONOFANINDUCTIVESET.Asetofrealnumbersiscalledaninductivesetifithasthefollowingtwoproperties:Thenumber1isintheset.Foreveryxintheset,thenumberx+1isalsointheset.Forexample,Risaninductiveset.Soistheset.Nowweshalldefinethepositiveintegerstobethoserealnumberswhichbelongtoeveryinductiveset.,現(xiàn)在我們來定義正整數(shù)，就是屬于每一個(gè)歸納集的實(shí)數(shù)。,LetPdenotethesetofallpositiveintegers.ThenPisitselfaninductivesetbecause(a)itcontains1,and(b)itcontainsx+1wheneveritcontainsx.SincethemembersofPbelongtoeveryinductiveset,werefertoPasthesmallestinductiveset.,用P表示所有正整數(shù)的集合。那么P本身是一個(gè)歸納集，因?yàn)槠渲泻?，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(gè)歸納集，因此P是最小的歸納集。,ThispropertyofPformsthelogicalbasisforatypeofreasoningthatmathematicianscallproofbyinduction,adetaileddiscussionofwhichisgiveninPart4ofthisintroduction.,P的這種性質(zhì)形成了一種推理的邏輯基礎(chǔ)，數(shù)學(xué)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(xì)論述。,Thenegativesofthepositiveintegersarecalledthenegativeintegers.Thepositiveintegers,togetherwiththenegativeintegersand0(zero),formasetZwhichwecallsimplythesetofintegers.,正整數(shù)的相反數(shù)被叫做負(fù)整數(shù)。正整數(shù)，負(fù)整數(shù)和零構(gòu)成了一個(gè)集合Z，簡稱為整數(shù)集。,Inathoroughtreatmentofthereal-numbersystem,itwouldbenecessaryatthisstagetoprovecertaintheoremsaboutintegers.Forexample,thesum,difference,orproductoftwointegersisaninteger,butthequotientoftwointegersneednotbeaninteger.However,weshallnotenterintothedetailsofsuchproofs.,在實(shí)數(shù)系統(tǒng)中，為了周密性，此時(shí)有必要證明一些整數(shù)的定理。例如，兩個(gè)整數(shù)的和、差和積仍是整數(shù)，但是商不一定是整數(shù)。然而還不能給出證明的細(xì)節(jié)。,Quotientsofintegersa/b(whereb0)arecalledrationalnumbers.Thesetofrationalnumbers,denotedbyQ,containsZasasubset.ThereadershouldrealizethatallthefieldaxiomsandtheorderaxiomsaresatisfiedbyQ.Forthisreason,wesaythatthesetofrationalnumbersisanorderedfield.RealnumbersthatarenotinQarecalledirrational.,整數(shù)a與b的商被叫做有理數(shù)，有理數(shù)集用Q表示，Z是Q的子集。讀者應(yīng)該認(rèn)識(shí)到Q滿足所有的域公理和序公理。因此說有理數(shù)集是一個(gè)有序的域。不是有理數(shù)的實(shí)數(shù)被稱為無理數(shù)。,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(diǎn)的方式表示實(shí)數(shù)的幾何意義。如圖2-4-1所示，選擇一個(gè)點(diǎn)表示0，在0右邊的另一個(gè)點(diǎn)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(gè)恰當(dāng)?shù)募?，那么每一個(gè)實(shí)數(shù)剛好對(duì)應(yīng)直線上的一個(gè)點(diǎn)，反之，直線上的每一個(gè)點(diǎn)也對(duì)應(yīng)且只對(duì)應(yīng)一個(gè)實(shí)數(shù)。,Forthisreasonthelineisoftencalledthereallineortherealaxis,anditiscustomarytousethewordsrealnumberandpointinterchangeably.Thusweoftenspeakofthepointxratherthanthepointcorrespondingtotherealnumber.,為此直線通常被叫做實(shí)直線或者實(shí)軸，習(xí)慣上使用“實(shí)數(shù)”這個(gè)單詞，而不是“點(diǎn)”。因此我們經(jīng)常說點(diǎn)x不是指與實(shí)數(shù)對(duì)應(yīng)的那個(gè)點(diǎn)。,Thisdeviceforrepresentingrealnumbersgeometricallyisaveryworthwhileaidthathelpsustodiscoverandunderstandbettercertainpropertiesofrealnumbers.However,thereadershouldrealizethatallpropertiesofrealnumbersthataretobeacceptedastheoremsmustbededuciblefromtheaxiomswithoutanyreferencestogeometry.,這種幾何化的表示實(shí)數(shù)的方法是非常值得推崇的，它有助于幫助我們發(fā)現(xiàn)和理解實(shí)數(shù)的某些性質(zhì)。然而，讀者應(yīng)該認(rèn)識(shí)到，擬被采用作為定理的所有關(guān)于實(shí)數(shù)的性質(zhì)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(shí)數(shù)的性質(zhì)時(shí)不會(huì)應(yīng)用到幾何。相反，幾何經(jīng)常會(huì)為證明一些定理提供思路，有時(shí)幾何討論比純分