数学专业英语第2版24课件

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1、New Words&Expressions:conversely 反之反之 geometric interpretation 几何意义几何意义correspond 对应对应 induction 归纳法归纳法deducible 可推导的可推导的 proof by induction 归纳证明归纳证明difference 差差 inductive set 归纳集归纳集distinguished 著名的著名的 inequality 不等式不等式entirely complete 完整的完整的 integer 整数整数Euclid 欧几里得欧几里得 interchangeably 可互相交换的可互相交

2、换的Euclidean 欧式的欧式的 intuitive直观的直观的the field axiom 域公理域公理 irrational 无理的无理的2.4 整数、有理数与实数整数、有理数与实数Integers,Rational Numbers and Real NumbersNew Words&Expressions:irrational number 无理数无理数 rational 有理的有理的the order axiom 序公理序公理 rational number 有理数有理数ordered 有序的有序的 reasoning 推理推理product 积积 scale 尺度，刻度尺度，刻

3、度quotient 商商 sum 和和There exist certain subsets of R which are distinguished because they have special properties not shared by all real numbers.In this section 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

4、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。To introduce the positive integers we begin with the number 1,whose existence is guaranteed 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 pos

5、itive,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。Strictly speaking,this description of the positive integers is not entirely complete because we have not explained

6、 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。Although the intuitive meaning of expressions may seem clear,in careful treatment of the real-number system it is necessary to g

7、ive a more precise definition of the positive integers.There are many ways to do this.One convenient method is to introduce first the 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

8、纳集的概念。简便的方法是先引进归纳集的概念。DEFINITION OF AN INDUCTIVE SET.A set of real numbers is called an inductive set if it has the following 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

9、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，就是属于每一个归纳集的实数。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

10、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。This property of P forms the logical basis for a type of reasoning that mathe

11、maticians call proof by induction,a detailed discussion of which is given in Part 4 of this introduction.P的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。The negatives of the positive integers are called the negative integers.The positive integers,to

12、gether with the negative integers and 0(zero),form a set Z which we call simply the set of integers.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。In a thorough treatment of the real-number system,it would be necessary at this stage to prove certain theorems about integer

13、s.For example,the sum,difference,or product of two integers is an integer,but the quotient 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

14、不不能能给给出出证证明明的的细节。细节。Quotients of integers a/b(where b0)are called rational numbers.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

15、 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。The reader is undoubtedly familiar with the geometric representation of real numb

16、ers by means of points on a straight line.A point is selected to represent 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

17、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定了了刻刻度。度。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

18、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。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

19、x rather than the point corresponding to the real number.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。This device for representing real numbers geometrically is a very worthwhile aid that helps us to discover and understand better

20、 certain properties of real numbers.However,the 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。

21、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。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 geometri

22、c argument is more illuminating than 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。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。作业：P 43 2.汉译英(2)3.英译汉(2)谢 谢！