数学专业英语第二版-课文翻译-converted

2.4 整数、有理数与实数

4－A Integers and rational numbers

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.

有一些R 的子集很著名，因为他们具有实数所不具备的特殊性质。在本节我们将讨论这样的子集，整数集和有理数集。

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 positive, and they are called the positive integers.

我们从数字 1 开始介绍正整数，公理 4 保证了 1 的存在性。1+1 用2 表示，2+1 用3 表示，以此类推，由 1 重复累加的方式得到的数字 1,2,3，…都是正的，它们被叫做正整数。

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”.

严格地说，这种关于正整数的描述是不完整的，因为我们没有详细解释“等等”或者“1的重复累加”的含义。

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 the notion of an inductive set.

虽然这些说法的直观意思似乎是清楚的，但是在认真处理实数系统时必须给出一个更准确的关于正整数的定义。有很多种方式来给出这个定义，一个简便的方法是先引进归纳集的概念。

DEFINITION OF AN INDUCTIVE SET. A set of real number s is cal led an i n ductiv e 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 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。

Let P d enote t he s et o f a ll p ositive i ntegers. T hen P i s i tself a n i nductive set b ecause (a) i t contains 1, a nd (b) i t c ontains x+1 w henever i t c ontains x. Since the m embers o f P b elong t o e very inductive s et, w e r efer t o P a s t he s mallest i nductive set.

用 P 表示所有正整数的集合。那么 P 本身是一个归纳集，因为其中含 1，满足(a)；只要包含x 就包含x+1, 满足(b)。由于 P 中的元素属于每一个归纳集，因此 P 是最小的归纳集。

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 given in Part 4 of this introduction.

P 的这种性质形成了一种推理的逻辑基础，数学家称之为，在介绍的第四部分将给出这种方法的详细论述。归纳证明

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.

正整数的相反数被叫做负整数。正整数，负整数和零构成了一个集合 Z，简称为整数集。 In

a t horough t reatment o f t he r eal-number s ystem, i t w ould

b e n ecessary a t t his 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 to ne an integer. However, we shall not enter into the details of such proofs. 在实数系统中，为了周密性，此时有必要证明一些整数的定理。例如，两个整数的和、

差和积仍是整数，但是商不一定是整数。然而还不能给出证明的细节。

Quotients of integers a/b (where b≠0) 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 ordered field. Real numbers that are not in Q are called irrational.

整数 a 与 b 的商被叫做有理数，有理数集用 Q 表示，Z 是 Q 的子集。读者应该认识到 Q 满

足所有的域公理和序公理。因此说有理数集是一个有序的域。不是有理数的实数被称为无理数。4－B Geometric interpretation of real numbers as points on a line

The reader is undoubtedly familiar with the geometric interpretation 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.

毫无疑问，读者都熟悉通过在直线上描点的方式表示实数的几何意义。如图 2-4-1 所示，选

择一个点表示 0，在 0 右边的另一个点表示 1。这种做法决定了刻度。

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.

如果采用欧式几何公理中一个恰当的集合，那么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。

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.

为此直线通常被叫做实直线或者实轴，习惯上使用“实数”这个单词，而不是“点”。因

此我们经常说点x 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 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的性质都必须不借助于几何

就能从公理推出。

This d oes n ot m ean t hat o ne s hould n ot m ake u se o f g eometry i n s tudying p roperties 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).

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。

2-5 理解笛卡儿几何学的基本概念 basic

concepts of Cartesian geometry

5-A the coordinate system of Cartesian geometry

As m entioned e arlier, o ne of t he a pplications o f t he i ntegral i s t hecalculation of area. Ordinarily , we do not talk about area by itself ,instead,we talk about the area of something .

This means that we have certain objects (polygonal regions, circular regions, parabolic segments etc.) whose areas we wish to measure. If we hope t o a rrive a t a t reatment o f a rea that w ill e nable u s t o d eal w ith m any d ifferent kinds of objects, we must firstfind an effective way to describe these objects.就像前面提到的积分的一个应用就是面积的计算,通

常我们不讨论面积本身，相反，是讨论某事物的面积。这意味着我们有些想测量的面积的对象（多边形区域，圆域，抛物线弓形等），如果我们希望获得面积的计算方法以便能够用它来处理各种不同类型的图形我们就必须首先找出表述这些对象的有效方法。

The most primitive way of doing this is by drawing figures, as was done by the ancient Greeks. A much better way was suggested by Rene Descartes, who introduced the subject of analytic geometry (also known as Cartesian geometry). Descartes’ idea was to represent g eometric p oints b y n umbers.The p rocedure f or p oints i n a p lane is this：描述对象最基本的方法是画图，就像古希腊人做的那样。R 笛卡儿提出了一种比较好的方法，并建立了解析几何（也称为笛卡儿几何）这门学科。笛卡儿的思想就是用数来表示几何点，在平面上找点的过程如下

Two perpendicular reference lines (called coordinate axes) are chosen, onehorizontal (called the“x-axis”),the other vertical (the“y-axis”). Their point ofintersection denoted by O, is called the origin. On the x-axis a convenient point is chosen to the right of O and its distance from O is called the unit distance. Vertical distances along the Y-axis are usually measured with the same unit distance ,although sometimes it is convenient to use a different scale on the y-axis. Now each point in the plane (sometimes called the xy-plane) is assigned apair of numbers, called its coordinates. These numbers tell us how to locate the

points.

选两条互相垂直的参考线（称为坐标轴）一条水平（称为x 轴）另一条竖直（称为y 轴）。他们的交点记为 O,称为原点。在 x 轴上，原点的右侧选择一个合适的点该点与原点之间的距离称为单位长度，沿着 y 轴的垂直距离通常用同样的单位长度来测量虽然有时候采用不同的尺度比较方便。现在平面上的每一个点都分配了一对数，称为坐标。这些数告诉我们如何定义一个点。

5-B

A geometric figure, such as a curve in the plane , is a collection ofpoints

satisfying one or more special conditions. By translating these conditions into expressions,, involving the coordinates x and y, we obtain one or more equations which characterize the figure inquestion , for example, consider a circle of radius r with its center atthe origin, as show in figure 2-5-2. let P be an arbitrary point on this circle, and suppose Phas coordinates (x, y).

一个几何图形是满足一个或多个特殊条件的点集，比如平面上的曲线。通过把这些条件转化成含有坐标 x 和 y 的表达式，我们就得到了一个或多个能刻画该图形特征的方程。例如如图 2-5-2 所示的中心在原点，半径为 r 的圆，令P是原上任意一点，假设 P 的坐标为(x, y).

2.6 function concept and function idea 6-C

The concept of function

Seldom has a single concept played so important a role inmathematics as has the concept of function. It is desirable toknow how the concept has developed.

在数学中，很少有个概念象函数的概念那样，起那么重要的作用的。因此需要知道这个概念是如何发展起来的。

This concept, like many others ,originates in physics. The physical quantities were the forerunners of mathematical variables. And relation among them was called a functionrelation in the later 16th century.

这个概念像许多其他概念一样起源于物理学。物理的量是数学的变量的先驱他们之间的关系在 16 世纪后期称为函数关系。

For example , the formula s=16t2for the number of feet s a body falls in any number of seconds t is a function relation between s and t. it describes the way s varies with t. the studyof such relations led people in the 18th century to think of afunction relation as nothing but a formula.

例如代表一物体在若干秒 t 中下落若干英尺s 的公式 s=16t2 就是 s 和t 之间的函数关系。它描述了 s 随 t 变化的公式对这种关系的研究导致了 18 世纪的人们认为函数关系只不过是一个公式罢了。

Only a fter t he r ise o f m odern a nalysis i n t he e arly 19th c enturycould t he c oncept of f unction be extended. In the extendedsense , a function may be defined as follows: if a variable y depends on another variable x in such a way that to each value of x corresponds a definite value o f y, t hen y i s a f unction o f x.this d efinition s erves many a practical purpose even today.

只有在19 世纪初期现代分析出现以后函数的概念才得以扩大。在扩大的意义上讲函数可定义如下如果一变量y 随着另一个变量x 而变换即x 的每一个值都和y 的一定值相对应那么 y 就是x 的函数。这个定义甚至在今天还适用于许多实际的用途。6

Not specified by this definition is the manner of setting up the correspondence. It may be done by a formula as the 18thcentury m athematics presumed but it can equally well be doneby a tabulation such as a statistical chart, or by some other form of description.

至于如何建立这种对应关系这个定义并未详细规定。可以如18 世纪的数学所假定的那样用公式来建立但同样也可以用统计表那样的表格或用某种其他的描述方式来建立。

A typical example is the room temperature, which obviously isa function of time. But this function admits of no formularepresentation, although it can be recorded in a tabular form or traced but graphically by an automatic device.

典型的例子是室温这显然是一个时间的函数。但是这个函数不能用公式来代表但可以用表格的形式来记录或者用一种自动装置以图标形式来追踪

The modern definition of a functionyofxis simply a mapping from a space X to another space Y. a mapping is defined whenevery pointxof X has a definition imagey, a point of Y. the mapping concept is close to intuition, and therefore desirable to serve as a basis of the function concept, Moreover, as the spaceconcept is incorporated in this modern definition, its generalitycontributes much to the generality of the function concept.

现代给 x 的一个函数y 所下的定义只是从一个空间 X 到另一个空间 Y 的映射。当 X 空间的每一个点 x 有一个确定的像点 y 即 Y 空间的一点那么映射就确定了。这个映射概念接近于直观因此很可能作为函数概念的一个基础。此外由于这个现代的定义中体现了空间的概念所以它的普遍性对函数概念的普遍性有很大的贡献。