数学专业英语(吴炯圻)(4)

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。

11

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。

12

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.

4－B Geometric interpretation of real numbers as points on a line

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。

13

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。

14

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。

15

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。

16

第二版 课文翻译及课后习题

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。

17

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。

3

New Words & Expressions:

polygonal 多边形的 circular regions 圆域

parabolic 抛物线的 coordinate axis 坐标轴

the unit distance单位长度 the origin 坐标原点

horizontal 水平的 coordinate system 坐标系

perpendicular 互相垂直的，垂线 vertical 竖直的

an ordered pair 一个有序对 abscissa 横坐标

quadrant 象限 ordinate 纵坐标

intersect 相交 the theorem of Pythagoras 勾股

定理

2.5 basic concepts of Cartesian geometry

4

As mentioned earlier, one of the applications of the integral is the calculation 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 first find 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。

5-A the coordinate system of Cartesian geometry

5

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 numbers. The procedure for points in a plane is this ：

第二版 课文翻译及课后习题

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：

5-A the coordinate system of Cartesian geometry

6

Two perpendicular reference lines (called coordinate axes) are chosen, one horizontal (called the

“x-axis”), the other vertical (the “y-axis”). Their point of intersection denoted by O, is …… 此处隐藏：4186字，全部文档内容请下载后查看。喜欢就下载吧 ……