數(shù)學(xué)專業(yè)英語(第2版)24課件

1、New Words & Expressions:conversely 反之 geometric interpretation 幾何意義correspond 對(duì)應(yīng) induction 歸納法deducible 可推導(dǎo)的 proof by induction 歸納證明difference 差 inductive set 歸納集distinguished 著名的 inequality 不等式entirely complete 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the field axio

2、m 域公理 irrational 無理的2.4 整數(shù)、有理數(shù)與實(shí)數(shù)Integers, Rational Numbers and Real NumbersNew Words & Expressions:irrational number 無理數(shù) rational 有理的the order axiom 序公理 rational number 有理數(shù)ordered 有序的 reasoning 推理product 積 scale 尺度，刻度quotient 商 sum 和There exist certain subsets of R which are distinguished because t

3、hey 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。To introduce the positive integers we begin with the number 1, whose ex

4、istence 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 positive, and they are called the positive integers.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

5、做正整數(shù)。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”. 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。 Although the intuitive meaning of expressions

6、 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 the notion of an inductive set.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

7、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。DEFINITION OF AN INDUCTIVE SET. A set of real numbers is called an inductive set if it has the following two properties:The number 1 is in the set.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

8、define the positive integers to be those real numbers which belong to every inductive set.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)。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

9、 to every inductive set, we refer to P as the smallest inductive set.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。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 g

10、iven in Part 4 of this introduction. 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。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.正整數(shù)的相反數(shù)被叫做負(fù)整

11、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。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 of two integers need not be an integer. However, we

12、 shall not enter into the details of such proofs.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)。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 axio

13、ms 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)。The reader is undoubtedly familiar with the

14、geometric representation of real numbers 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毫無疑問，讀者都熟

15、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。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. 如果采用歐式幾何公理中一個(gè)恰當(dāng)?shù)?/p>

16、集合，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)。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 number.為此直線通常被叫做實(shí)直線或者實(shí)軸

17、，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)。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, the reader should realize that all properties of real numbers that are to be accep

18、ted as theorems must be deducible from the axioms without any references to geometry.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。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 a purely analytic proof (one depending entirely on the axioms for the real numbers). 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