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1、New Words&Expressions:irrational number 无理数 rational 有理的the order axiom 序公理 rational number 有理数ordered 有序的 reasoning 推理product 积 scale 尺度，刻度quotient 商 sum 和第1页/共20页There exist certain subsets of R which are distinguished because they have special properties not shared by all real numbers.In this sec
2、tion we shall discuss such subsets,the integers and the rational numbers.4A Integers and rational numbers有有一一些些R的的子子集集很很著著名名，因因为为他他们们具具有有实实数数所所不不具具备备的的特特殊殊性性质质。在在本本节节我我们们将将讨讨论论这这样样的的子子集集，整整数数集集和有理数集。和有理数集。第2页/共20页To introduce the positive integers we begin with the number 1,whose existence is guaran
3、teed by Axiom 4.The number 1+1 is denoted by 2,the number 2+1 by 3,and so on.The numbers 1,2,3,obtained in this way by repeated addition of 1 are all positive,and they are called the positive integers.我我们们从从数数字字1开开始始介介绍绍正正整整数数，公公理理4保保证证了了1的的存存在在性性。1+1用用2表表示示，2+1用用3表表示示，以以此此类类推推，由由1重重复复累累加加的的方方式式得得到到
4、的的数数字字1,2,3，都都是是正正的的，它它们们被被叫叫做正整数。做正整数。第3页/共20页Strictly speaking,this description of the positive integers is not entirely complete because we have not explained in detail what we mean by the expressions“and so on”,or“repeated addition of 1”.严严格格地地说说，这这种种关关于于正正整整数数的的描描述述是是不不完完整整的的，因因为为我我们们没没有有详详细细解解
5、释释“等等等等”或或者者“1的的重重复复累累加加”的含义。的含义。第4页/共20页Although the intuitive meaning of expressions may seem clear,in careful treatment of the real-number system it is necessary to give a more precise definition of the positive integers.There are many ways to do this.One convenient method is to introduce first t
6、he notion of an inductive set.虽虽然然这这些些说说法法的的直直观观意意思思似似乎乎是是清清楚楚的的，但但是是在在认认真真处处理理实实数数系系统统时时有有必必要要给给出出一一个个更更准准确确的的关关于于正正整整数数的的定定义义。有有很很多多种种方方式式来来给给出出这这个个定定义义，一一个个简便的方法是先引进归纳集的概念。简便的方法是先引进归纳集的概念。第5页/共20页DEFINITION OF AN INDUCTIVE SET.A set of real numbers is called an inductive set if it has the followi
7、ng two properties:(a)The number 1 is in the set.(b)For every x in the set,the number x+1 is also in the set.For example,R is an inductive set.So is the set .Now we shall define the positive integers to be those real numbers which belong to every inductive set.现在我们来定义正整数，就是属于每一个归纳集的实数。现在我们来定义正整数，就是属于
8、每一个归纳集的实数。第6页/共20页Let P denote the set of all positive integers.Then P is itself an inductive set because(a)it contains 1,and(b)it contains x+1 whenever it contains x.Since the members of P belong to every inductive set,we refer to P as the smallest inductive set.用用P表表示示所所有有正正整整数数的的集集合合。那那么么P本本身身是是一
9、一个个归归纳纳集集，因因为为其其中中含含1，满满足足(a)；只只要要包包含含x就就包包含含x+1,满满足足(b)。由由于于P中中的的元元素素属属于于每每一一个个归归纳纳集集，因因此此P是最小的归纳集。是最小的归纳集。第7页/共20页This property of P forms the logical basis for a type of reasoning that mathematicians call proof by induction,a detailed discussion of which is given in Part 4 of this introduction.P的
10、的这这种种性性质质形形成成了了一一种种推推理理的的逻逻辑辑基基础础，数数学学家家称称之之为为归归纳纳证证明明，在在介介绍绍的的第第四四部部分分将将给给出出这这种种方方法法的详细论述。的详细论述。第8页/共20页The negatives of the positive integers are called the negative integers.The positive integers,together with the negative integers and 0(zero),form a set Z which we call simply the set of integers
11、.正正整整数数的的相相反反数数被被叫叫做做负负整整数数。正正整整数数，负负整整数数和和零构成了一个集合零构成了一个集合Z，简称为整数集。，简称为整数集。第9页/共20页In a thorough treatment of the real-number system,it would be necessary at this stage to prove certain theorems about integers.For example,the sum,difference,or product of two integers is an integer,but the quotient
12、of two integers need not be an integer.However,we shall not enter into the details of such proofs.在在实实数数系系统统中中，为为了了周周密密性性，此此时时有有必必要要证证明明一一些些整整数数的的定定理理。例例如如，两两个个整整数数的的和和、差差和和积积仍仍是是整整数数，但但是是商商不不一一定定是是整整数数。然然而而还还不不能能给给出出证证明明的的细节。细节。第10页/共20页Quotients of integers a/b(where b0)are called rational numbers
13、.The set of rational numbers,denoted by Q,contains Z as a subset.The reader should realize that all the field axioms and the order axioms are satisfied by Q.For this reason,we say that the set of rational numbers is an ordered field.Real numbers that are not in Q are called irrational.整整数数a与与b的的商商被被
14、叫叫做做有有理理数数，有有理理数数集集用用Q表表示示，Z是是Q的的子子集集。读读者者应应该该认认识识到到Q满满足足所所有有的的域域公公理理和和序序公公理理。因因此此说说有有理理数数集集是是一一个个有有序序的的域域。不不是是有有理数的实数被称为无理数。理数的实数被称为无理数。第11页/共20页The reader is undoubtedly familiar with the geometric representation of real numbers by means of points on a straight line.A point is selected to represen
15、t 0 and another,to the right of 0,to represent 1,as illustrated in Figure 2-4-1.This choice determines the scale.4B Geometric interpretation of real numbers as points on a line毫毫无无疑疑问问，读读者者都都熟熟悉悉通通过过在在直直线线上上描描点点的的方方式式表表示示实实数数的的几几何何意意义义。如如图图2-4-1所所示示，选选择择一一个个点点表表示示0，在在0右右边边的的另另一一个个点点表表示示1。这这种种做做法法决决定
16、定了了刻刻度。度。第12页/共20页If one adopts an appropriate set of axioms for Euclidean geometry,then each real number corresponds to exactly one point on this line and,conversely,each point on the line corresponds to one and only one real number.如果采用欧式几何公理中一个恰当的集合如果采用欧式几何公理中一个恰当的集合，那么，那么每一个实数刚好对应直线上的一个点，反之，直每一个
17、实数刚好对应直线上的一个点，反之，直线上的每一个点也对应且只对应一个实数。线上的每一个点也对应且只对应一个实数。第13页/共20页For this reason the line is often called the real line or the real axis,and it is customary to use the words real number and point interchangeably.Thus we often speak of the point x rather than the point corresponding to the real numbe
18、r.为此直线通常被叫做实直线或者实轴，习惯上使用为此直线通常被叫做实直线或者实轴，习惯上使用“实数实数”这个单词，而不是这个单词，而不是“点点”。因此我们经常说点。因此我们经常说点x不是指与实数对应的那个点。不是指与实数对应的那个点。第14页/共20页This device for representing real numbers geometrically is a very worthwhile aid that helps us to discover and understand better certain properties of real numbers.However,th
19、e reader should realize that all properties of real numbers that are to be accepted as theorems must be deducible from the axioms without any references to geometry.这种几何化的表示实数的方法是非常值得推崇的，它这种几何化的表示实数的方法是非常值得推崇的，它有助于帮助我们发现和理解实数的某些性质。然而，有助于帮助我们发现和理解实数的某些性质。然而，读者应该认识到，拟被采用作为定理的所有关于实数读者应该认识到，拟被采用作为定理的所有关
20、于实数的性质都必须不借助于几何就能从公理推出。的性质都必须不借助于几何就能从公理推出。第15页/共20页This does not mean that one should not make use of geometry in studying properties of real numbers.On the contrary,the geometry often suggests the method of proof of a particular theorem,and sometimes a geometric argument is more illuminating than
21、a purely analytic proof(one depending entirely on the axioms for the real numbers).这并不意味着研究实数的性质时不会应用到几何。相这并不意味着研究实数的性质时不会应用到几何。相反，几何经常会为证明一些定理提供思路，有时几何反，几何经常会为证明一些定理提供思路，有时几何讨论比纯分析式的证明更清楚。讨论比纯分析式的证明更清楚。第16页/共20页In this book,geometric arguments are used to a large extent to help motivate or clarity a particular discuss.Nevertheless,the proofs of all the important theorems are presented in analytic form.在本书中，几何在很大程度上被用于激发或者阐明一在本书中，几何在很大程度上被用于激发或者阐明一些特殊的讨论。不过，所有重要定理的证明必须以分些特殊的讨论。不过，所有重要定理的证明必须以分析的形式给出。析的形式给出。第17页/共20页作业：P 43 2.汉译英(2)3.英译汉(2)第18页/共20页谢 谢！第19页/共20页感谢您的观看。第20页/共20页