数学专业英语第二版 课文翻译

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 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 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 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 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

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 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).

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 mentioned earlier, one of the applications of the integral is thecalculation

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 to arrive at a treatment of area that will enable us to deal with many different

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 geometric points by procedure for points in a plane

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 after the rise of modern analysis in the early 19th centurycould the concept

of function 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 of y, then y is a function of definition serves

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 mathematics 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。