数学专业英语第二版-课文翻译-converted
课文2—AB数学专业英语翻译(第二版)吴炯圻
2-A Why study geometry?Why do we study geometry? The student beginning the study of this text may well ask, "What is geometry? What can I expect to gain from this study?2-A为什么研究几何学？为什么我们研究几何学？刚开始学习这篇文章的学生会疑问，“几何是什么？研究几何我们能学到什么呢？Many leading institutions of higher learning have recognized that positive benefits can be gained by all who study this branch of mathematics. This is evident from the fact that they require study of geometry as a prerequisite to matriculation in those schools.许多居领导地位的学术机构承认，所有学习这个数学分支的人都将得到很好的收益。
事实是，他们需要学习几何作为学校入学考试的先决条件。
Geometry had its origin long ago in the measurements by the Babylonians and Egyptians of their lands inundated by the floods of the Nile River. The greek word geometry is derived from geo, meaning "earth," and metron, meaning "measure." As early as 2000 B. C. we find the land surveyors of these people reestablishing vanishing landmarks and boundaries by utilizing the truths of geometry.很早以前，几何学源于测量被尼罗河的洪水淹没了的巴比伦人和埃及人的土地。
数学专业英语翻译2-7
7-B The limit of a sequence
The main question we are concerned with here is to decide whether or not the terms f(n) tend to a finite limit as n increases infinitely. 这里我们关心的主要问题是当n无限增加时，项 f(n) 是否会趋于一个有限的极限。
如果对每一个正整数 n都有一个实数或复数an与之对 应, 则有序集a1 , a2, …, an ,… 称为一个无穷序列.
The important thing here is that each member of the set has been labeled with an integer so that we may speak of the first term a1, the second term a2, and, in general, the nth term an. 这里重要的是集合中的每一个元素都由一个整数标 记，因此我们可以说第一项 , 第二项, 一般的，第n项
This particular rule is known as a recursion formula and it defines a famous sequence whose terms are called Fibonacci numbers. The first few terms are 1,1,2,3,5,8,13,21,34. 这个特殊的规则就是常见的递推公式，它定义了一个 著名的序列，其中的项称为菲波那契数。前几项是…...
数学专业英语2-10翻译
Although dependence and independence are properties of sets of elements, we also apply these terms to the elements themselves. For example, the elements in an independent set are called independent elements.虽然相关和无关是元素集的属性，我们也适用于这些元素本身。
例如，在一个独立设定的元素被称为独立元素。
If s is finite set, the foregoing definition agrees with that given in Chapter 8 for the space n V . However, the present definition is not restricted to finite sets.如果S 是有限集，同意上述定义与第8章中给出的空间n V ，然而，目前的定义不局限于有限集。
If a subset T of a set S is dependent, then S itself is dependent. This is logically equivalent to the statement that every subset of an independent set is independent.如果集合S 的子集T 是相关的，然后S 本身是相关的，这在逻辑上相当于每一个独立设置的子集是独立的语句。
If one element in S is a scalar multiple of another, then S is dependent. 如果S 中的一个元素是另一个集中的多个标量的，则S 是相关的。
If S ∈0,then S is dependent. 若S ∈0，则 S 是相关的。
数学专业英语 【第二版】(吴炯圻)
数学专业英语【第二版】(吴炯圻)数学专业英语【第二版】1- A 什么是数学数学来自于人的社会实践，例如，工业和农业生产、商业活动、军事行动和科研工作。
与数学反过来，为实践服务和所有字段中的伟大作用。
没有现代的科学和技术分支机构可以定期制定中的数学，应用无早有需要的人来了数字和形式的概念。
然后，开发出的几何土地和三角测量的问题来自测量的问题。
若要对付一些更复杂的实际问题，男子成立，然后解决方程未知号码，因此代数发生。
17 世纪前, 男子向自己限于小学数学，即几何、三角和代数，只有常量被认为在其中。
17 世纪产业的快速发展促进了经济和技术的进展和所需变量的数量、处理从常量到带来两个分支的数学-解析几何和微积分，属于高等数学，现在有很多分支机构，其中有数学分析、高等代数、微分方程的高等数学中的可变数量的飞跃函数理论等。
数学家研究理念和主张。
所有命题公理、假设、定义和定理都。
符号是一种特殊和功能强大的数学工具，用于表示很多时候的理念和主张。
公式、数字和图表是阿拉伯数字 1,2,3,4,5,6,7,8,9,0 与另外的符号"+"、减法"-"，乘"*"，除"\"和平等"="。
数学中的结论得到主要由逻辑推理和计算。
长期的数学史上，以中心地点的数学方法被占领逻辑扣除。
现在，由于电子计算机是迅速发展和广泛应用，计算的作用变得越来越多重要。
在我们这个时代计算不只用于处理大量的信息和数据，而且还进行一些只是可以做的工作较早前的逻辑推理，例如，大部分的几何定理的证明。
1--B 方程方程是平等的语句的两个相等的数字或数字符号之间。
因此 (a-5） = 一 5a 和 x 3 = 5 是方程。
方程的两种― ― 身份和方程的条件。
方程的算术或代数的身份。
这种方程中两名成员是相似的或成为相似的指示操作的性能。
因此 12-2=2+8,(m+n)(m-n) = m n 是身份。
数学专业英语课文翻译
1-A：什么是数学数学来源于人类的社会实践，包括工农业的劳动，商业、军事和科学技术研究等活动。
反过来，数学服务于实践并在所有领域扮演一个重要的角色。
没有数学的应用，现代化科学和技术的分支都不能有规律的发展。
从早期人类的需求引出了数和形状。
然后，几何学因测量陆续的发展出来，三角学来自于勘探问题。
为了处理一些更复杂的实践问题，人们建立了方程，通过求解方程的未知数，从而代数学出现了。
17世纪之前，人们局限于初等数学，例如几何、三角和代数，那些只考虑常数。
17世纪工业的迅速发展促进了经济学和科技的发展，并且我们需要处理变量。
从常数到变量的跳跃带来了两个属于高等数学的新的数学分支，解析几何和微积分学。
现在，高等数学中有了许多分支，数学分析、高等代数、微分方程、函数论等。
数学家们研究概念和命题。
公理、公社、定义和定理都是命题。
符号是一种特别并且很重要的数学工具，它常用于表示概念和命题。
公式、图形和表格充满着不同的符号。
阿拉伯数字1,2,3,4,5,6,7,8,9,0和加”+”减”-”乘”*”除”/”等号”=”使我们最熟悉的数学符号。
主要通过逻辑推导和计算来获得数学结论。
在数学史的很长的时期内，逻辑推论一直占据数学方法的中心地位。
现在，自从电子计算机迅速发展和广泛应用，计算的角色越来越重要。
现在，计算不仅用来处理信息与数据，而且用来完成一些在以前只能靠逻辑推理来做的工作，例如证明大多数的几何定理。
1-B：等式等式是关于两个数或数的符号相等的一种陈述。
因此a(a-5)=a^2-5a和x-3=5是等式。
等式有两种，恒等式和条件等式。
算术和代数恒等式是等式。
这种等式的两端要么一样，要么经过执行指定的运算后变成一样。
因此12-2=2+8，(m-n)(m+n)=m^2-n^2是恒等式。
含有字母的恒等式对其中字母的任何一组数值都成立。
因此恒等式x(a+2)=ax+2x变成3(7+2)=21+6或27=27，比如当x=3和a=7。
大学英语第二版第二册原文翻译
新视野大学英语第二版第二册课文原文Unit1A foreigner's first impression of the US is likely to be that everyone is in a rush—often under pressure. City people always appear o be hurrying to get where they are going, restlessly seeking attention in a store, or elbowing others as they try to complete their shopping. Racing through daytime meals is part of the pace of life in this country. Working time is considered precious. Others in public eating-places are waiting for you to finish so they, too, can be served and get back to work within the time allowed. You also find drivers will be abrupt and people will push past you. You will miss smiles, brief conversations, and small exchanges with strangers. Don't take it personally. This is because people value time highly, and they resent someone else "wasting" it beyond a certain appropriate point.外国人对美国的第一印象很可能是：每个人都匆匆忙忙──常常处于压力之下。
数学专业英语(吴炯圻-第2版)2-6
It may be done by a formula as the 18th century mathematics presumed but it can equally well be done by a tabulation such as a statistical chart, or by some other form of description.
The study of such relations led people in the 18th century to think of a function relation as nothing but a formula. 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。
The word “function” was introduced into mathematics by Leibniz, who used the term primarily to refer to certain kinds of mathematical formulas. “函数”这个词是由莱布尼茨引入到数学中的，他主 要使用这个术语来指代某种数学公式。 It was later realized that Leibniz’s idea of function was much too limited in its scope, and the meaning of the word has since undergone many stages of generalization. 后来人们才认识到，莱布尼茨的函数思想适用的范围 太过局限了，这个术语的含义从那时起已经过了多次
数学专业英语(吴炯圻-第2版)2-1
The rapid development of industry in 17th century promoted the progress of economics and technology and required dealing with variable quantities. The leap from constants to variable quantities brought about two new branches of mathematics----analytic geometry and calculus, which belong to the higher mathematics.
17世纪工业的快速发展推动了经济技术的进步， 从而遇到需 要处理变量的问题。从常量到变量的跳跃产生了两个新的数 学分支-----解析几何和微积分，他们都属于高等数学。
现在高等数学里面有很多分支，其中有数学分析，高等代数，ceptions and propositions, Axioms, postulates, definitions and theorems are all propositions. Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often.
2.1 数学、方程与比例 Mathematics, Equation and Ratio
New Words & Expressions:
algebra 代数学
geometrical 几何的
algebraic 代数的
数学专业英语课文翻译2(吴炯圻)
7A在日常使用的英文单词"序列"和' '系列"是同义词，和他们用来建议一系列的事情或按某种顺序排列的事件。
在数学中，这句话有特别技术的意义。
"序列"一词被受雇如在共同使用这一术语，传达的理念的一套东西排列顺序，但"系列"一词用于稍有不同的意义。
概念在本节中，将讨论序列和系列将定义第11 节。
如果为每个正整数n 有关联的真实或复数a，那时有序的集据说是定义一个无限的序列。
这里最重要的是每个成员集的已标记的整数，使我们可以发言的第一届、第二个任期，以及，一般的第n 个词。
每个学期了继任者，因此，没有任何"最后"一词。
如果我们给一些规则或第n 个词描述的公式，可以构造序列的最常见的例子。
因此，例如，公式= 1 定义的序列的第五个任期是1.有时两个或多个公式可受雇作为，例如，a=1.the 第一次在这种情况下被1 的一些术语。
另一种常见方法定义一系列是一套的说明解释了如何在一个给定的开始后进行的。
因此，我们可能= 1。
此特定的规则被称为递归公式，它定义了著名的序列，其条款被称为斐波那契数。
第一次的几个术语are1。
最重要的事情是序列的序列的这样f(n) 的每个n=1.In 事实的第n 个燕鸥是序列的序列的正整数上定义一些函数 f 的任何序列，这可能是序列的序列的最方便的方法，国家技术定义。
定义。
其域是所有积极integers1 的一组函数f 称为一个无限的序列。
函数值f(n) 调用序列的第n 个词。
通过按顺序，因此编写条款通常显示的功能（即，函数值的集合）的范围：f (2)。
为简便起见，{f(n)} 符号用于指示第n 个任期是f(n) 的序列。
由使用下标，很多时候表示，n 的依赖和我们写，或类似的而不是f (n0。
除非另外指定，否则所有的序列，在这一章中假定有真实的或复杂的条款。
7B我们担心在这里主要的问题在于决定是否条款f(n) 倾向于有限的n 无限增加。
专业学位研究生英语通用教程_第二版,课文翻译
In the United States most people are simultaneously consumers and producers; they are also voters who help inf luence the decisions of the government. The mixture among consumers, producers, and government changes constantly, making a dynamic rather than a static economy. In the last decade consumers have made their concern known and government has responded by creating agenci es to protect consumer interests and promote the general public welfare. In another development, the population and the labor f orce have moved dramatically f rom farms to cities, f rom the f ields to the f actori es, and above all to service industries, thus providing more personal and public servi ces. In today’s economy these providers of services f ar outnumber producers of agricultural and manuf actured goods.﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏Every year 80000 Chinese die f rom antibiotics abuse, making China one of the worst offenders in the world. Statistics show that among the country’s 15 best-selling medicines, 10 are antibiotics. More than 50 per cent of the medical expenses f or Chinese in-patients goes to cover di ff erent kinds of antibiotics, while the f igure in Western countries is just 15 to 30per cent.The side eff ects of antibiotics use can damage organs, cause disorders in the body’s normal bacteria and increase the resistance of disease causing germs. Many Chinese believe the myth that antibiotics can diminish inf lammation, so they use them to treat everything f rom toothache to f ever. A story from the Xinhua News Agency said that more than 80 per cent of Chinese families have antibiotics stored on shelves at home in case of different kinds of infections.Children have becom e the biggest victims of antibiotic abuse where the drug kills normal bacteria leading to damage of children’s still growing organs. Statistics in Guangzhou showed the some 17 per cent of local residents now suffer from antibiotic-resisitant diseases while the f igure was 50 per cent for children, according to data f rom the local children’s hospital. The marked increase of asthma in children recently has a lot to do with antibiotics abuse.﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏Television news reports and programs bring about both positive and negative inf luence. From the news reports, people will happen in the world, such as wars in the Middle East and famine in Af rica. As a result they broaden their mind. By watching television programs people also get to know other peoples and their cultures. However, television news reports and programs may mislead people. V iolence and drug problems are harm f ul to young people. Political scandals, illegal campaign contributions an illegal f inancial manipulation greatly damage the nation’s image. Things t o worse especially when international audiences lack context f ram e of reference.﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏The last century saw a steady gradual increase in literacy, and thus in the number of readers. As readers increased, so the number of potential listeners declined, and thus there was some reduction in the need of read aloud. As reading f or the benef it of listeners grew less common, so came the f lourishing of reading as a private activity in such public places as libraries, railway carriages and off i ces, where reading aloud would cause distraction to other readers.Towards the end of the century there was still considerable argument over whether books should be used f or inf ormation or treated respect f ully, and over whether the reading of materi al such as newspapers was n some way mentally weakening. Indeed this argument remains with us still in education. However, whatever its virtues, the old shared literacy culture had gone and was replaced by the printed mass media on the one hand by books and periodicals f or specialized readership on the other.﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏﹏什么叫抠门？你是否会在快用完的洗发水瓶里加点水可以再多洗一次？或者在垃圾筐里陈一个塑料袋？是否会把大孩子穿过的一条牛仔裤或一条裙子给小孩子穿？或者用前一天的剩菜再做一顿新的吃？这些事情是否表明你是一个守财奴或吝啬鬼呢？尽管“抠门主义者”会说一分钱扮成两半用并不意味着抠门的生活方式，但是如果我们有意节省，我们会更能掌控手中的钱财。
数学专业英语(吴炯圻-第2版)2-4
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.
(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 R .
Now we shall define the positive integers to be those real numbers which belong to every inductive set.
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 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.
数学专业英语第二版的课文翻译
线性方程，二次方程等。
To solve an equation means to find the value of the unknown term. To do this , we must, of course,change the terms about until the unknown term stands alone on one side of the equation, thus making it equal to something on the other side. We then obtain the value of the unknown and the answer to the question. To solve the equation,therefore,means to move and change the terms about without making the equation untrue,until only the unknown quantity is left on one side ,no matter which side.解方程意味着求未知项的值，为了求未知项的值，当然必须移项，直到未知项单独在方程的一边，令其等于方程的另一边，从而求得未知项的值，解决了问题。
因此解方程意味着进行一系列的移项和同解变形，直到未知量被单独留在方程的一边，无论那一边。
Equation are of very great use. We can use equation in many mathematical problems. We may notice that almost every problem gives us one or more statements that something is equal to something, this gives us equations, with which we may work if we need it.方程作用很大，可以用方程解决很多数学问题。
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 numbersThere 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。
数学专业英语
数学专业英语（1）、Given ε>0,there exists a number N >0 such that ε<-a a n for all N n ≥ 译文：对给定的ε>0，存在一个数N >0使得不等式ε<-a a n 对所有的N n ≥都成立。
（2）、The function )(x f approaches infinity as x tends to zero.译文：当x 趋于0时，函数)(x f 趋于∞。
（3）、Suppose D is an open set with its closure in G .译文：假定D 是一个开区间，且其闭包在G 中。
（4）、Suppose )(x f is a function on domain D such that M x f <)( for all x ∈D ,where M is a constant .译文：假定)(x f 是区域D 上的一个函数，使得对所有x ∈D ，不等式M x f <)(成立，其中M 是一个常数。
（可用“satisfying ”代替上述“such that ”。
）（5）、表示推理的根据常用“by 短语”，有时也用“according to ”。
By Lemma 2 we have x y ≥.译文：根据引理2可推出x y ≥。
（6）、有时用现在分词表示“经过……而得到……”（推论）。
Integrating the above inequality twice,we see that . log )(0t t c t y ≥译文：将上一不等式两次积分得到. log )(0t t c t y ≥。
（7）、表示充分必要条件The sufficient and necessary condition for the equality isα>0 and ≥p 3. 译文：该等式成立的充分必要条件是α>0 且≥p 3。
数学专业英语翻译2-3
We use the special notation x S to mean that “x is an element of S” or “x belongs to S”. If x does not belong to S, we write x S .
When convenient, we shall designate sets by displaying the elements in braces; for example, the set of positive even integers less than 10 is displayed as {2,4,6,8} whereas the set of all positive even integers is displayed as {2,4,6,…}, the three dots taking the place of “and so on.”
Sets usually are denoted by capital letters; elements are designated by lower-case letters.
集合论的概念已经被广泛使用，遍及现代数学，因此对大学生 来说，理解它的概念是必要的。集合是数学家们用抽象的方式 来表述一些事物的集体的工具。 集合通常用大写字母表示，元素用小写字母表示。
and we write A≠B.
集合相等的定义 如果两个集合A和B确切包含同样的元素,则 称二者相等，此时记为A=B。如果一个集合包含了另一个集 合以外的元素，则称二者不等，记为A≠B。
EXAMPLE 1. According to this definition, the two sets {2,4,6,8} and {2,8,6,4} are equal since they both consist of the four integers 2,4,6 and 8. Thus, when we use the roster notation to describe a set, the order in which the elements appear is irrelevant.
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2.4 整数、有理数与实数4－A Integers and rational numbersThere 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。
Ina t horough t reatment o f t he r eal-number s ystem, i t w ouldb e n ecessary a t t his stage to prove certain theorems about integers. For example, the sum, difference, or product of two integersis 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 lineThe 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 customaryto 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”。