数学专业英语(吴炯圻)

New Words & Expressions:

algebra 代数学geometrical 几何的

algebraic 代数的identity 恒等式

arithmetic 算术, 算术的measure 测量，测度

axiom 公理numerical 数值的, 数字的

conception 概念，观点operation 运算

constant 常数postulate 公设

logical deduction 逻辑推理proposition 命题

division 除，除法subtraction 减，减法formula 公式

term 项，术语

trigonometry 三角学variable 变化的，变量

2.1 数学、方程与比例

Mathematics, Equation and Ratio

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Mathematics c omes from man’s social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches.

1－A What is mathematics

数学来源于人类的社会实践，比如工农业生产，商业活动，军事行动和科学技术研究。And in turn, mathematics serves the practice and plays a great role in all fields. No modern scientific and technological branches could be regularly developed without the application of mathematics.

反过来，数学服务于实践，并在各个领域中起着非常重要的作用。没有应用数学，任何一个现在的科技的分支都不能正常发展。

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From the early need of man came the concepts of numbers and forms. Then, geometry developed out of problems of measuring land , and trigonometry came from problems of surveying. To deal with some more complex practical problems, man established and then solved equation with unknown numbers , thus algebra occurred.

很早的时候，人类的需要产生了数和形的概念。接着，测量土地问题形成了几何学，测量问题产生了三角学。为了处理更复杂的实际问题，人类建立和解决了带未知数的方程，从而产生了代数学。

Before 17th century, man confined himself to the elementary mathematics, i.e. , geometry, trigonometry and algebra, in which only the constants are considered.

17世纪前，人类局限于只考虑常数的初等数学，即几何学，三角学和代数学。

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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。

Now there are many branches in higher mathematics, among which are mathematical analysis, higher algebra, differential equations, function theory and so on.

现在高等数学里面有很多分支，其中有数学分析，高等代数，微分方程，函数论等。

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Mathematicians study conceptions 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.

数学家研究的是概念和命题，公理，公设，定义和定理都是命题。符号是数学中一个特殊而有用的工具，常用于表达概念和命题。

Formulas ,figures and charts are full of different symbols. Some of the best known symbols of mathematics are the Arabic numerals 1,2,3,4,5,6,7,8,9,0 and the signs of addition “+”, subtraction “-” , multiplication “×”, division “÷” and equality “=”.

公式，图形和图表都是不同的符号……..

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The conclusions in mathematics are obtained mainly by logical deductions and computation. For a long period of the history of mathematics, the centric place of mathematics methods was occupied by the logical deductions.

数学结论主要由逻辑推理和计算得到。在数学发展历史的很长时间内，逻辑推理一直占据着数学方法的中心地位。

Now , since electronic computers are developed promptly and used widely, the role of computation becomes more and more important. In our times, computation is not only used to deal with a lot of information and data, but also to carry out some work that merely could be done earlier by logical deductions, for example, the proof of most of geometrical theorems.

现在，由于电子计算机的迅速发展和广泛使用，计算机的地位越来越重要。现在计算机不仅用于处理大量的信息和数据，还可以完成一些之前只能由逻辑推理来做的工作，例如，证明大多数的几何定理。

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回顾：

1. 如果没有运用数学，任何一个科学技术分支都不可能正常的发展。

2. 符号在数学中起着非常重要的作用，它常用于表示概念和命题。

1－A What is mathematics

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An equation is a statement of the equality between two equal numbers or number symbols.

1－B Equation

等式是关于两个数或者数的符号相等的一种描述。

Equation are of two kinds---- identities and equations of condition.

An arithmetic or an algebraic identity is an equation. In such an equation either the two members are alike, or become alike on the performance of the indicated operation.

等式有两种－恒等式和条件等式。算术或者代数恒等式都是等式。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。

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An identity involving letters is true for any set of numerical values of the letters in it.

含有字母的恒等式对其中字母的任一组数值都成立。

An equation which is true only for certain values of a letter in it, or for certain sets of related values of two or more of its letters, is an equation of condition, or simply an equation. Thus 3x-5=7 is true for x=4 only; and 2x-y=10 is true for x=6 and y=2 and for many other pairs of values for x and y.

一个等式若仅仅对其中一个字母的某些值成立，或对其中两个或者多个字母的若干组相关的值成立，则它是一个条件等式，简称方程。因此3x-5=7仅当x=4 时成立，而2x-y=0，当x=6,y=2时成立，且对x, y的其他许多对值也成立。

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A root of an equation is any number or number symbol which satisfies the equation.

To obtain the root or roots of an equation is called solving an equation.

方程的根是满足方程的任意数或者数的符号。求方程根的过程被称为解方程。

There are various kinds of equations. They are linear equation, quadratic equation, etc.

方程有很多种，例如：线性方程，二次方程等。

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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.

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。

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.

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。

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Equations are of very great use. We can use equations 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 to.

方程作用很大，可以用方程解决很多数学问题。注意到几乎每一个问题都给出一个或多个关于一个事情与另一个事情相等的陈述，这就给出了方程，利用该方程，如果我们需要的话，可以解方程。

New Words & Expressions:

numerical 数值的，数的position 位置，状态

cube n. 立方体sphere n. 球

cylinder n. 柱体cone 圆锥

geometrical 几何的triangle 三角形

surface 面，曲面pyramid 菱形

plane 平面solid 立体，立体的

straight line 直线line segment 直线段

broken line 折线ray 射线

equidistant 等距离的curve 曲线，弯曲

2.2 几何与三角

Geometry and Trigonology

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New Words & Expressions:

side 边angle 角

radius（radii）半径diameter 直径

endpoint 端点circle 圆周，圆

semicircle 半圆arc 弧

minor arc 劣弧major arc 优弧

acute angle 锐角right angle 直角

hypotenuse 斜边adjacent side 邻边

chord 弦circumference 周长

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Many leading institutions of higher learning have recognized that positive benefits can be gained by all who study this branch of mathematics.

2－A Why study geometry?

许多居于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确实的受益。This is evident from the fact that they require study of geometry as a prerequisite to matriculation in those schools.

许多学校把几何的学习作为入学考试的先决条件，从这一点上可以证明。

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Geometry had its origin long ago in the measurement by the Babylonians and Egyptians of their lands inundated by the floods of the Nile River.

几何学起源于很久以前巴比伦人和埃及人测量他们被尼罗河洪水淹没的土地。

The greek word geometry is derived from geo, meaning “earth” and metron, meaning “measure” .

希腊语几何来源于geo ，意思是”土地“，和metron意思是”测量“。

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As early as 2000 B.C. we find the land surveyors of these people re-establishing vanishing landmarks and boundaries by utilizing the truths of geometry .

公元前2000年之前，我们发现这些民族的土地测量者利用几何知识重新确定消失了的土地标志和边界。

One of the most important objectives derived from a study of geometry is making the student be more critical in his listening, reading and thinking. In studying geometry he is led away from the practice of blind acceptance of statements and ideas and is taught to think clearly and critically before forming conclusions.

几何的学习使学生在思考问题时更周密、审慎，他们将不会盲目接受任何结论.

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A solid is a three-dimensional figure. Common examples of solids are cube, sphere, cylinder, cone and pyramid.

2－B Some geometrical terms

立体是一个三维图形，立体常见的例子是立方体，球体，柱体，圆锥和棱锥。

A cube has six faces which are smooth and flat. These faces are called plane surfaces or simply planes.

立方体有6个面，都是光滑的和平的，这些面被称为平面曲面或者简称为平面。

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A plane surface has two dimensions, length and width. The surface of a blackboard or of a tabletop is an example of a plane surface.

平面曲面是二维的，有长度和宽度，黑板和桌子上面的面都是平面曲面的例子。

A circle is a closed curve lying in one plane, all points of which are equidistant from a fixed point called the center.

平面上的闭曲线当其中每点到一个固定点的距离均相当时叫做圆。固定点称为圆心。

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A line segment drawn from the center of the circle to a point on the circle is a radius of the circle. The circumference is the length of a circle.

经过圆心且其两个端点在圆周上的线段称为这个园的直径，这条曲线的长度叫做周长。One of the most important applications of trigonometry is the solution of triangles. Let us now take up the solution to right triangles.

三角形最重要的应用之一是解三角形，现在我们来解直角三角形。

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A triangle is composed of six parts three sides and three angles. To solve a triangle is to find the parts not given.

一个三角形由6个部分组成，三条边和三只角。解一个三角形就是要求出未知的部分。

A triangle may be solved if three parts (at least one of these is a side ) are given. A right triangle has one angle, the right angle, always given. Thus a right triangle can be solved when two sides, or one side and an acute angle, are given.

如果三角形的三个部分（其中至少有一个为边）为已知，则此三角形就可以解出。直角三角形的一只角，即直角，总是已知的。因此，如果它的两边，或一边和一锐角为已知，则此直角三角形可解。

New Words & Expressions:

brace 大括号roster 名册

consequence 结论，推论roster notation 枚举法

designate 标记，指定rule out 排除，否决

diagram 图形，图解subset 子集

distinct 互不相同的the underlying set 基础集

distinguish 区别，辨别universal set 全集

divisible 可被除尽的validity 有效性

dummy 哑的，哑变量visual 可视的

even integer 偶数visualize 可视化

irrelevant 无关紧要的void set(empty set) 空集

2.3 集合论的基本概念

Basic Concepts of the Theory of Sets

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The concept of a set has been utilized so extensively throughout modern mathematics that an understanding of it is necessary for all college students. Sets are a means by which mathematicians talk of collections of things in an abstract way.

3－A Notations for denoting sets

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。

Sets usually are denoted by capital letters; elements are designated by lower-case letters.

集合通常用大写字母表示，元素用小写字母表示。

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We use the special notation to mean that “x is an element of S” or “x belongs to S”. If x does not belong to S, we write .

我们用专用记号来表示x是S的元素或者x属于S。如果x不属于S，我们记为。

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.”

如果方便，我们可以用在大括号中列出元素的方式来表示集合。例如，小于10的正偶数的集合表示为{2,4,6,8}，而所有正偶数的集合表示为{2,4,6,…}, 三个圆点表示“等等”。

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The dots are used only when the meaning of “and so on” is clear. The method of listing the members of a set within braces is sometimes referred to as the roster notation.

只有当省略的内容清楚时才能使用圆点。在大括号中列出集合元素的方法有时被归结为枚举法。

The first basic concept that relates one set to another is equality of sets:

联系一个集合与另一个集合的第一个基本概念是集合相等。

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DEFINITION OF SET EQUALITY Two sets A and B are said to be equal (or identical) if they consist of exactly the same elements, in which case we write A=B. If one of the sets contains an element not in the other, we say the sets unequal 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。

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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.

根据这个定义，两个集合{2,4,6,8}和{2,8,6,4}是相等的，因为他们都包含了四个整数2,4,6,8。因此，当我们用枚举法来描述集合的时候，元素出现的次序是无关紧要的。

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EXAMPLE 2. The sets {2,4,6,8} and {2,2,4,4,6,8} are equal even though, in the second set, each of the elements 2 and 4 is listed twice. Both sets contain the four elements 2,4,6,8 and no others;

therefore, the definition requires that we call these sets equal.

例2. 集合{2,4,6,8} 和{2,2,4,4,6,8}也是相等的，虽然在第二个集合中，2和4都出现两次。两个集合都包含了四个元素2,4,6,8，没有其他元素，因此，依据定义这两个集合相等。This example shows that we do not insist that the objects listed in the roster notation be distinct.

A similar example is the set of letters in the word Mississippi, which is equal to the set {M,i,s,p}, consisting of the four distinct letters M,i,s, and p.

这个例子表明我们没有强调在枚举法中所列出的元素要互不相同。一个相似的例子是，在单词Mississippi中字母的集合等价于集合{M,i,s,p}, 其中包含了四个互不相同的字母M,i,s,和p. 7

From a given set S we may form new sets, called subsets of S. For example, the set consisting of those positive integers less than 10 which are divisible by 4 (the set {4,8}) is a subset of the set of all even integers less than 10. In general, we have the following definition.

3－B Subsets

一个给定的集合S可以产生新的集合，这些集合叫做S的子集。例如，由可被4除尽的并且小于10的正整数所组成的集合是小于10的所有偶数所组成集合的子集。一般来说，我们有如下定义。

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In all our applications of set theory, we have a fixed set S given in advance, and we are concerned only with subsets of this given set. The underlying set S may vary from one application to another; it will be referred to as the universal set of each particular discourse. （35页第二段）

当我们应用集合论时，总是事先给定一个固定的集合S，而我们只关心这个给定集合的子集。基础集可以随意改变，可以在每一段特定的论述中表示全集。

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It is possible for a set to contain no elements whatever. This set is called the empty set or the void set, and will be denoted by the symbol . We will consider to be a subset of every set.（35页第三段）

一个集合中不包含任何元素，这种情况是有可能的。这个集合被叫做空集，用符号表示。空集是任何集合的子集。

Some people find it helpful to think of a set as analogous to a container (such as a bag or a box) containing certain objects, its elements. The empty set is then analogous to an empty container. 一些人认为这样的比喻是有益的，集合类似于容器（如背包和盒子）装有某些东西那样，包含它的元素。

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To avoid logical difficulties, we must distinguish between the elements x and the set {x} whose only element is x. In particular, the empty set is not the same as the set . （35页第四段）

为了避免遇到逻辑困难，我们必须区分元素x和集合{x}，集合{x}中的元素是x。特别要注意的是空集和集合是不同的。

In fact, the empty set contains no elements, whereas the set has one element. Sets consisting of exactly one element are sometimes called one-element sets.

事实上，空集不含有任何元素，而有一个元素。由一个元素构成的集合有时被称为单元素集。

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Diagrams often help us visualize relations between sets. For example, we may think of a set S as a

region in the plane and each of its elements as a point. Subsets of S may then be thought of the collections of points within S. For example, in Figure 2-3-1 the shaded portion is a subset of A and also a subset of B. （35页第五段）

图解有助于我们将集合之间的关系形象化。例如，可以把集合S看作平面内的一个区域，其中的每一个元素即是一个点。那么S的子集就是S内某些点的全体。例如，在图2-3-1中阴影部分是A的子集，同时也是B的子集。

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Visual aids of this type, called Venn diagrams, are useful for testing the validity of theorems in set theory or for suggesting methods to prove them. Of course, the proofs themselves must rely only on the definitions of the concepts and not on the diagrams.

这种图解方法，叫做文氏图，在集合论中常用于检验定理的有效性或者为证明定理提供一些潜在的方法。当然证明本身必须依赖于概念的定义而不是图解。

New Words & Expressions:

conversely 反之geometric interpretation 几何意义

correspond 对应induction 归纳法

deducible 可推导的proof by induction 归纳证明

difference 差inductive set 归纳集

distinguished 著名的inequality 不等式

entirely complete 完整的integer 整数

Euclid 欧几里得interchangeably 可互相交换的

Euclidean 欧式的intuitive直观的

the field axiom 域公理irrational 无理的

2.4 整数、有理数与实数

Integers, Rational Numbers and Real Numbers

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New Words & Expressions:

irrational number 无理数rational 有理的

the order axiom 序公理rational number 有理数

ordered 有序的reasoning 推理

product 积scale 尺度，刻度

quotient 商sum 和

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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.

4－A Integers and 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。

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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。

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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。

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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。6

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 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。

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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。

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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，数学家称之为归纳证明，在介绍的第四部分将给出这种方法的详细论述。

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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。10

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。

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

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 re present geometric points by numbers. The 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：

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 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 a pair 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-A the coordinate system of Cartesian geometry

7

A geometric figure, such as a curve in the plane , is a collection of points 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 in question , for example, consider a circle of radius r with its center at the origin, as show in figure 2-5-2. let P be an arbitrary point on this circle, and suppose P has 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).

5-B Geometric figure

9

New Words & Expressions:

prime 素数displacement 位移

edge 棱，边domain 定义域，区域

real variable 实变量schematic representation 图解表示

tabulation 作表，表mass 质量，许多，群众

absolute-value function 绝对值函数

2.6 function concept and function idea

10

Seldom has a single concept played so important a role in mathematics as has the concept of function. It is desirable to know 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。

6-C The concept of function

11

This concept, like many others ,originates in physics. The physical quantities were the forerunners of mathematical variables. And relation among them was called a function relation 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。

6-C The concept of function

12

For example , the formula s=16t2 for 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 study of such relations led people in the 18th century to think of a function 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。

6-C The concept of function

13

Only after the rise of modern analysis in the early 19th century could the concept of function be extended. In the extended sense , 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 x. this 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-C The concept of function

14

Not specified by this definition is the manner of setting up the correspondence. 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.

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。

6-C The concept of function

15

A typical example is the room temperature, which obviously is a function of time. But this function admits of no formula representation, 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。

6-C The concept of function

16

The modern definition of a function y of x is simply a mapping from a space X to another space Y.

a mapping is defined when every point x of X has a definition image y, 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 space concept is incorporated in this modern definition, its generality contributes 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。

New Words & Expressions:

alphabet 字母表prime 素数，质数

displacement 位移proportional 成比例的

domain 定义域the real-valued function实值函数

edge 棱，边spring constant 弹性系数graph 图，图形limit 极限stretch 拉伸volume 体积，容积，卷

2.6 函数的概念与函数思想

Function concept and function idea

1

New Words & Expressions(二)P59:

admit 准许mapping 映射

forerunner 先行者presume 假定

incorporate 并入，结合trace 追踪

2

Various fields of human have to do with relationships that exist between one collection of objects and another.

6-A Informal description of functions

各行各业的人们都必须处理一类事物与另一类事物之间存在的关系。

Graphs, charts, curves, tables, formulas, and Gallup polls are familiar to everyone who reads the newspapers.

几乎每个人都熟悉图形，图表，曲线，公式和盖洛普民意测验。

3

These are merely devices for describing special relations in a quantitative fashion. Mathematicians refer to certain types of these relations as functions.

这些只是以定量的方式描述特定关系的方法。数学家将这些关系中的某些类型视作函数。

In this section, we give an informal description of the function concept. A formal definition is given in Section 3.

在本节中，我们给出一个非正式的描述函数的概念。在第3节给出一个正式的定义。

4

EXAMPLE 1. The force F necessary to stretch a steel spring a distance x beyond its natural length is proportional to x. That is, F=cx, where c is a number independent of x called the spring constant.

把一个钢制的弹簧拉伸到超过其自然长度的距离为x时所需要的力F与x成正比。即，F=cx,这里c是不依赖与x的数，叫做弹性系数。

This formula, discovered by Robert Hooke in the mid-17th century, is called Hooke’s law, and it is said to express the force as a function of the displacement.

这个公式是在17世纪中叶被胡克发现的，叫做胡克定律，它用来表示力关于位移的函数。5

EXAMPLE 2. The volume of a cube is a function of its edge-length. If the edges have length x, the volume V is given by the formula V=x3.

立方体的体积是它棱长的函数。如果棱长为x，那么体积的公式为：V=x3。

6

EXAMPLE 3. A prime is any integer n>1 that cannot be expressed in the form n=ab, where a and b are positive integers, both less than n. The first few primes are 2,3,5,7,11,13,17,19.

素数是大于1且不能表示成n=ab形式的整数，这里a和b都是小于n的正整数。开始的几个素数是2,3,5,7,11,13,17,19.

7

For a given real number x>0, it is possible to count the number of primes less than or equal to x. This number is said to be a function of x even though no simple algebraic formula is known for computing it (without counting) when x is known.

对于一个给定的实数x>0，数出小于或者等于x的素数的个数是有可能的。这个数称为x的函数，尽管还没有一个简单代数式可以由已知的x计算(不通过计数求)出它的值。

8

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推广。

9

Today, the meaning of function is essentially this: Given two sets, say X and Y, a function is a correspondence which associates with each element of X one and one only element of Y.

如今，从本质上讲，函数的定义如下：给定两个集合X 和Y，函数是X中元素与Y中元素的一一对应。

10

The set X is called the domain of the function. Those elements of Y associated with the elements in X form a set called the range of the function. (This may be all of Y, but it need not be)

集合X叫做函数的定义域，与X中的元素相对应的Y中的元素的集合叫做函数的值域。(值域可能是整个集合Y,也可能不是。)

11

Letters of the English and Greek alphabets are often used to denote functions. The particular letters f,g,h,F,G,H, and are frequently used for this purpose.

英语字母和希腊字母表通常用于表示函数。为此，一些特定的字母如：f,g,h,…频繁使用。

12

If f is a given function and if x is an object of its domain, the notation f(x) is used to designate that object in the range which is associated to x by the function f; and it is called the value of f at x or the image of x under f. The symbol f(x) is read as “f of x.”

如果f是一个给定的函数，x是它定义域中的一个点，符号f(x)表示值域中按照f对应于x的点，它叫做f在x点的值或者x在f下的像。符号f(x)读作“f of x.”

13

Seldom has a single concept played so important a role in mathematics as has the concept of function. It is desirable to know how the concept has developed.

6-C The concept of function

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.

这个概念像许多其他概念一样，起源于物理学。

14

The physical quantities were the forerunners of mathematical variables, and relation among them was called a function relation in the later 16th century.

物理量是数学变量的先驱，他们之间的关系在16世纪后期称为函数关系。

For example, the formula s=16t2 for 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.

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。

15

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。

Not specified by this definition is the manner of setting up the correspondence.

至于如何建立这种对应关系，这个定义并未详细规定。

16

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.

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。

17

A typical example is the room temperature, which obviously is a function of time. But this function admits of no formula representation, although it can be recorded in a tabular form or traced out 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表示，不过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 function y of x is simply a mapping from a space X to another space Y.

A mapping is defined when every point x of X has a definite image y, a point of Y.

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。

19

The mapping concept is close to intuition, and therefore desirable to serve as a basis of the function concept. Moreover, as the space concept is incorporated in this modern definition, its generality contributes much to the generality of the function concept.

这个映射概念接近于直观，因此，值得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，它的普及性对函数概念的普及性有很大的贡献。

New Words & Expressions:

assume 假定sequence 序列，数列

converge 收敛series 级数，序列

diverge 发散subscript 下标

imaginary part 虚部succession 连贯性imply 蕴含，推出successor 后继

recursion formula 递推公式

2.7 序列及其极限

Sequences and Their Limits

1

Key points:

the definition of sequences

Difficult points:

some relevant terms

2

In everyday usage of the English language, the words “sequence” and “series” are synonyms, and they are used to suggest a succession of things or events arranged in some order.

7-A The definition of sequences

在日常英语中，单词“sequence”和“series”是同义词，用以表示按某种次序排列的一串东西或事件。

In mathematics these words have special technical meanings.

在数学中这些单词有特殊的专业含义。

3

The word “sequence” is employed as in the common use of the term to convey the idea of a set of things arranged in order, but the word “series” is used in a somewhat different sense.

像通常用法一样，术语“sequence”用以表达按次序排列的一串东西的意思，但是“series”一词则用于表示别的意思（级数）。

The concept of a sequence will be discussed in this section, and series will be defined in section 11.

序列的概念在本节讨论，级数的概念将在第11节定义。

4

If for every positive integer n there is associated a real or complex number an, then the ordered set a1 , a2, …, an ,… is said to define an infinite sequence.

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

5

Each term has a successor a nd hence there is no “last ” term.

每一项都有下面的一项，因此没有最后一项。

The most common examples of sequences can be constructed if we give some rule or formula for describing the nth term.

如果我们给出描述第n项的规则或者公式，就可以构造出序列的常见例子。

6

Thus, for example, the formula an=1/n defines a sequence whose first five terms are 1, 1/2, 1/3, 1/4, 1/5.

例如，公式an=1/n定义了一个序列，它的前五项是1, 1/2, 1/3, 1/4, 1/5.

Sometimes two or more formulas may be employed as, for example, a2n-1=1, a2n=2n2, the first few terms in this case being 1,2,1,8,1,18,1,32,1.

有时可以使用两个或者多个公式(来定义一个序列)，例如，a2n-1=1, a2n=2n2，此时，前几项是……

7

Another common way to define a sequence is by a set of instructions which explains how to carry on after a given start. Thus we may have a1=a2=1, an+1=an+an-1 for n ≥2 .

另一种常用的定义序列的方法是，通过一串指令说明在给定首项后如何给出后面的各项。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…...

8

※In any sequence the essential thing is that there be some function f defined on the positive integers such that f(n) is the nth term of the sequence for each n=1,2,3,…

对任意序列，最本质的事就是存在某一个定义在正整数集上的函数，使得对n=1,2,3,…f(n) 为序列的第n项。

In fact, this is probably the most convenient way to state a technical definition of sequence.

事实上，这可能是给出序列专业定义的一种最方便的方法。

9

DEFINITION. A function f whose domain is the set of all positive integers 1,2,3,… is called an infinite sequence. The function value f(n) is called the nth term of the sequence.

定义域为所有正整数集合的函数f 被称为无穷序列。函数值f(n) 称为序列的第n项。

10

The range of the function (that is, the set of function values) is usually displayed by writing the terms in order.

函数的值域（即是函数值的集合）通常以按顺序书写各项的方式来表示。

Very often the dependence on n is denoted by using subscripts, and we write an, xn…, or something similar instead of f(n). Unless otherwise specified, all sequences in this chapter are assumed to have real or complex terms.

序列各项对n 的依赖性常利用下标来表示，记为an 或者其他类似的记号，而不记作f(n) 。除非另有说明，本章研究的序列都是假定具有实的项或复的项。

11

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.

7-B The limit of a sequence

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。

To treat this problem, we must extend the limit concept to sequences. This is done as follows.

为了解决这个问题，必须把极限的概念推广到序列中。做法如下：

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DIFINITION. A sequence {f(n)} is said to have a limit L if, for every positive number e, there is another positive number N ( which may depend on e ) such that

| an－L | < e for all n ≥N .

In this case, we say the sequence {f(n)} converges to L. A sequence which does not converge is called divergent.

……不收敛的序列被称为发散序列。

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In this definition the function values f(n) and the limit L may be real or complex numbers. If f and L are complex, we may decompose them into their real and imaginary parts, say f=u+iv and L=a+ib.

在这个定义中，函数值f(n) 和极限L都可能是实数或者复数。如果f和L是复数, 可以将其分解为实部和虚部。

In other words, a complex-valued sequence f converges if and only if both the real part u and the imaginary part v converge separately. (P61 第三段最后一句)

换句话说，复值序列f收敛当且仅当实部和虚部(序列)分别收敛。

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※It is clear that any function defined for all positive real x may be used to construct a sequence by restricting x to take only integer values.

显然, 每一个对所有正实数x 有定义的函数都可以用来构造一个序列，其办法是限制x 只取整数值。

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The phrase “convergent sequence” is used only for a sequence whose limit is finite. A sequence with an infinite limit is said to diverge. There are, of course, divergent sequences that do not have infinite limits.

仅当一个序列的极限有限时，才使用短语“收敛序列”。一个具有无穷极限的序列称为发散

序列。当然，一个发散序列未必存在一个无穷极限。

[函数的导数和它的几何意义]

8-A 函数的导数

前一节中描述的例子给出了引进导数概念的方法。我们从至少定义在x-轴上的某个开区间（a,b）内的函数f(x)开始，然后我们在这个区间内选择一点x，引进差商

这里，数h (可以是正的或者负的但不能是0）要使得x+h还在（a, b）内。这个商的分子测量了当x从x变到x+h时函数的变化。称这个商为f在连接x与x+h的区间内的平均变化率。现在让h→0，看看这个商会发生什么。如果商趋于某个确定的值作为极限（这就推得无论h 是从正的方向还是负的方向趋于0，这个极限是一样的），成这个极限为f在x点的导数，记为f / (x)（读作“f一撇x”）。因此，f / (x)的正规定义可以陈述如下：

导数定义。如果

存在极限，导数f / (x)由等式（8.2）定义。数f / (x)也称为f在x点的变化率。

对比（8.2）与前一节的（7.3），我们看到瞬时速度仅仅是导数概念的一个例子。速度v(t)等于f / (t)，这里f是位移函数，这就是常常被描述为速度是位移关于时间的变化率。在7.2节算出的例子中，位移函数由等式f (t)=144t-32t2表示，而它的导数f / 是由 f / (t) =144-32t 给出的新的函数（速度）。

一般地，从f(x)产生f / (x)的极限过程给我们从一个给定函数f获得一个新函数f / 的方法。这个过程称为微分法，f / 称为f的一阶导数。依次地，如果f / 定义在开区间上，我们可以设法求出它的一阶导数，记为f // 并称其为f的二阶导数。类似地，由f (n-1)定义的一阶导数是f的n阶导数记为f (n)，我们规定f (0)= f，即零阶导数是函数本身。

对于直线运动，速度的一阶导数（位移的二阶导数）称为加速度。例如，要计算7.2节中的例子的加速度，我们可以用等式（7.2）形成差商

因为这个差商对每一个h≠0都是常数值-32，因此当h→0时它的极限也是-32.于是在这个问题中，加速度是常数且等于-32. 这个结论告诉我们速度是以每秒32尺/秒的速率递减的。9秒内，速度总共减少了9?32=288尺/秒。这与运动9秒期间，速度从v(0)=144变到v(9)=-144是一致的。

8-B 导数作为斜率的几何意义

通常定义导数的过程给出了一个几何意义，就是以自然的方式导出关于曲线的切线的思想。图2-8-1是一个函数的部分图像。两个坐标(x,f(x)) 和(x+h,f(x+h))分别表示P, Q两个点坐标，考虑斜边为PQ的直角三角形，它的高度：f(x+h)- f(x)，表示P, Q两个点纵坐标的差，因此差商

表示PQ与水平线的夹角α的正切，实数tanα称为通过P, Q两点直线的斜率，而它提供了一种测量这条直线“陡度”的方法。例如，如果f是线性函数，记为f=mx+b，则（8.4）的差商是m, 所以m是这条直线的斜率。图2-8-2表示的是一些各种斜率的直线的例子。对于水平线而言，α=0，因而tanα也是0. 如果α位于0与π/2之间, 直线是从左到右上升的，斜率是正的。如果α位于π/2与π之间，直线是从左到右下降的，斜率是负的。对于α=π/4的直线，斜率是1. 当α从0增加到π/2时，tanα递增且无界，斜率为tanα相应的直线趋于垂直的位置，因为tanπ/2没有定义，所以我们说垂直的直线没有斜率。

假设f在x点有导数，这就意味着，当h→0时，P点保持不动，Q沿曲线向P移动，通过