数学专业英语

数学专业英语－The Real Number System

Th

Real-Number System

The real-number system is collection of mathematical objects, called real number, which acquire mathematical life by virtue fundamental principles, or rules, that we adopt. The situation is somewhat similar to a game, like chess, for example. The chess system, or game, is a collection of objects, called chess pieces, which acquire life by virtue of the rules of the game, that is, the principles that are adopted to define allowable moves for the pieces and the way in which they may interact.

Our working experience with numbers has provided us all with some familiarity with the principles that govern the real-number system. However, to establish a common ground of understanding and avoid certain errors that have become very common, we shall explicitly(明确的，清楚地) state and illustrate many of these principles.

The real-number system includes such numbers as –27,-2,2/3,… It is worthy of note that positive numbers（正数）, 1/2, 1, for examples, are sometimes expressed as +(1/2), +1. The plus sign, “+”, used here does not express the operation of addition, but is rather part of the symbolism for the numbers themselves. Similarly, the minus sign, “-“, used in expressing such numbers as -(1/2), -1, is part of the symbolism for these numbers.

Within the real number system, numbers of various kinds are identified and named. The numbers 1, 2, 3, 4,… which are used in the counting process, are called natural numbers. The natural numbers（自然数）, together with–1,-2,-3,-4,…and zero, are called integers（整数）. Since

1,2,3,4,…are greater than 0, they are also called positive integers; -1,-2,-3,-4,…are less than 0, and for this reason are called negative integers. A real number is said to be a rational number（有理数） if it can be expressed as the ratio（比率，比例） of two integers, where the denominator （分母）is not zero. The integers are included among the rational numbers since any integer can be expressed as the ratio of the integer itself and one. A real number that cannot be expressed as the ratio of two integers is said to be an irrational number（无理数）.

One of the basic properties of the real-number system is that any two real numbers can be compared for size. If a and b are real numbers, we write

aa, which is read “b is greater than a “.

Geometrically, real numbers are identified with points on a straight line. We choose a straight line, and an initial（开始的，最初的） point f reference called the origin（原点）. To the origin we assign the number zero. By marking off（划出，划下分割） the unit of length in both directions from the origin, we assign positive integers to marked-off points in one direction (by convention（惯例）, to the right of the origin ) and negative integers to marked-off point in the other direction. By following through in terms of the chosen unit of length, a real number is attached to one point on the number line, and each point on the number line has attached to it one number.

Geometrically, in terms of our number line, to say that aa means that b is to the right of a.

Properties of Addition and Multiplication（乘法）

Addition and multiplication are primary operations on real numbers. Most, if not all, of the basic properties of these operations are familiar to us from experience.

(a)Closure property （闭包特性）of addition and multiplication.

Whenever two real numbers are added or multiplied, we obtain a real number as the result. That is, performing the operations of addition and multiplication leaves us within the real-number system.

(b)Commutative（互相的，交替的，交换的） property（交换性） of addition and multiplication.

The order in which two real numbers are added or multiplied does not affect the result obtained. That is, if a and b are any two real numbers, then we have (i) a+ b=b+ a and (ii) ab = ba. Such a property is called a commutative property. Thus（这样，因此，从而）, addition and multiplication of real numbers are commutative operations.

(c)Associative property of addition and multiplication.

Parentheses（parenthesis的复数，圆括号）, brackets, and the like, we recall, are used in algebra to group together whatever terms are within them. Thus 2+(3+4) means that 2 is to be added to the sum of 3 and 4 yielding 2+7 =9 whereas (2+3)+4 means the sum of 2 and 3 is to be added to 4 yielding also 9. Similarly, 2?(3?4) yields（生成）2?(12)=24 whereas (2?3) ?4 yields

the same end result（最终结果）by the route 6?4=24 . That such is the case in general is the content of the associative property of addition and multiplication of real numbers.

(d)Distributive property of multiplication over addition.

We know that 2?(3?4)=2?7=14 and that 2?3+ 2?4=14 ,thus 2?(3+4)=2?3+ 2?4. That such is the case in general for all real numbers is the content of the distributive property of multiplication over addition, more simply called the distributive property.

Substraction and Division

The numbers zero and one. The following are the basic properties of the numbers zero and one.

(a)There is a unique real number, called zero and denoted by 0, with the property that a+0=0+a, where a is any real number.

There is a unique real number, different from zero, called one and denoted by 1, with the property that a?1=1?a=a, where a is any real number.

(b)If a is any real number, then there is a unique real number x, called the additive inverse of a , or negative of a, with the property that a+ x = x+ a .If a is any nonzero real number, then there is a unique real number y, called the multiplicative inverse of a, or reciprocal of a, with the property that ay = ya = 1

The concept of the negative of a number should not be confused with the concept of a negative number; they are not the same. ”Negative of“ means additive inverse of “. On the other hand, a “negative number” is a number that is less than zero.

The multiplicative inverse of a is often represented by the symbol 1/a or a-1. Note that since the product of any number y and 0 is 0, 0 cannot have a multiplicative inverse. Thus 1/0 does not exist.

Now substraction is defined in terms of addition in the following way.

If a and b are any two real numbers, then the difference a-b is defined by a- b= c where c is such that b+ c=a or c= a+(-b). That is, to substract b from a means to add the negative of b (additive inverse of b) to a.

Division is defined in terms of multiplication in the following way.

If a and b are any real numbers, where b≠0, then a+ b is defined by a +b= a?(1/b) =a?b-1. That is, to divide a by b means to multiply a by the multiplicative inverse ( reciprocal)of b. The quotient a +b is also expressed by the fraction symbol a/b.

Exercise

I．Translate the following sentences into Chinese ( pay attention to the phrases underlined:

1. Note that a+ib=c+id means a=c and b=d

2. We recall that log z: C－{0} C is an inverse for when is restricted to a strip

3. Notice that if ,angles need not be preserved.

4. To show that the test fails when ,observe that, by elementary analysis, and but diverges while converges.

5. To prove the results of this section, we shall use the techniques developed in the last section.

6. We can deduce, in a way similar to the way we deduced theorem A, the following theorem.

7. We are now in a position to draw important consequences from Cauchy’s theorem.

8. We are now in a position to prove easily an otherwise difficult theorem stating that any polynomial of degree n has a root.

9. Unless otherwise specified (stated), curves will always be assumed to be continuous and piecewise differentiable.

10. We shall prove a theorem that appears to be elementary and that the student has, in the past, taken for granted.

11. The solution to this differential equation is unique up to the addition of a constant.

12. The function that maps the simply connected domain onto the unit disc is unique up to a Mobius transformation.

II．Translate the following passages into Chinese:

1．If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standing point, not only is this problem frequently more accessible to our investigation ,but at the same time we come into possession of a method which is applicable also to related problems.

2．In dealing with mathematical problems, specialization plays, as I believe, a still more important part than generalization. Perhaps in most cases where we seek in vain the answer to a question, the cause of the failure lies in the

fact that problems simpler and easier than the one in hand have been either not at all or incompletely solved. All depends then, on finding out these easier problems, and on solving them by means of methods as perfect as possible.

How to Write Mathematics?------ Honesty is the Best Policy

The purpose of using good mathematical language is, of course, to make the understanding of the subject easy for the reader, and perhaps even pleasant. The style should be good not in the sense of flashy brilliance, but good in the sense of perfect unobtrusiveness. The purpose is to smooth the reader’s wanted, not pedantry; understanding, not fuss.

The emphasis in the preceding paragraph, while perhaps necessary, might seem to point in an undesirable direction, and I hasten to correct a possible misinterpretation. While avoiding pedantry and fuss, I do not want to avoid rigor and precision; I believe that these aims are reconcilable. I do not mean to advise a young author to be very so slightly but very very cleverly dishonest and to gloss over difficulties. Sometimes, for instance, there may be no better way to get a result than a cumbersome computation. In that case it is the author’s duty to carry it out, in public; the he can do to alleviate it is to extend his sympathy to the reader by some phrase such as “unfortunately the only known proof is the following cumbersome computation.”

Here is the sort of the thing I mean by less than complete honesty. At a certain point, having proudly proved a proposition P, you feel moved to say: “Note, however, that p does not imply q”, and then, thinking that you’ve done a good expository job, go happily on to other things. Your motives may be perfectly pure, but the reader may feel cheated just the same. If he knew a ll about the subject, he wouldn’t be reading you; for him the nonimplication is, quite likely, unsupported. Is it obvious? (Say so.) Will a counterexample be supplied later? (Promise it now.) Is it a standard present purposes irrelevant part of the literature? (Give a reference.) Or, horrible dictum, do you merely mean that you have tried to derive q from p, you failed, and you don’t in fact know whether p implies q? (Confess immediately.) any event: take the reader into your confidence.

There is nothing wrong with often derided “obvious” and “easy to see”, but there are certain minimal rules to their use. Surely when you wrote that something was obvious, you thought it was. When, a month, or two months, or six months later, you picked up the manuscript and re-read it, did you still think that something was obvious? (A few months’ ripening always improves manuscripts.) When you explained it to a friend, or to

a seminar, was the something at issue accepted as obvious? (Or did someone question it and subside, muttering, when you reassured him? Did your assurance demonstration or intimidation?) the obvious answers to these rhetorical questions are among the rules that should control the use of “ obvious”. There is the most frequent so urce of mathematical error: make that the “ obvious” is true.

It should go without saying that you are not setting out to hide facts from the reader: you are writing to uncover them. What I am saying now is that you should not hide the status of your statements and your attitude toward them either. Whenever you tell him something, tell him where it stands: this has been proved, that hasn’t, this will be proved, that won’t. Emphasize the important and minimize the trivial. Th e reason saying that they are obvious is to put them in proper perspecti e for the uninitiated. Even if your saying so makes an occasional reader angry at you, a good purpose is served by your telling him how you view the matter. But, of course, you must o bey the rules. Don’t let the reader down; he wants to believe in you. Pretentiousness, bluff, and concealment may not get caught out immediately, but most readers will soon sense that there is something wrong, and they will blame neither the facts nor themselves, but quite properly, the author. Complete honesty makes for greatest clarity.

---------Paul R.Haqlmos

vocabulary

flashy 一闪的counter-example 反例

unobtrusiveness 谦虚dictum 断言；格言

forestall 阻止，先下手deride嘲弄

anticipate 预见subside沉静

pedantry 迂腐；卖弄学问mutter出怨言，喃喃自语

fuss 小题大做intimidation威下

reconcilable 使一致的rhetorical合符修辞学的

gloss 掩饰pretentiousness自命不凡

alleviate 减轻，缓和bluff 欺骗

implication 包含，含意concealment隐匿

notes

1. 本课文选自美国数学学会出版的小册子How to write mathematics 中Paul R.Halmos. 的文章第9节

2. The purpose is smooth the reader’ way, to anticipates his difficulties and to forestall them. Clarity is what’s wanted, not pedantry; understanding, not fuss.

意思是：目的是为读者扫清阅读上的障碍，即预先设想读者会遇到什么困难，并力求避免出现这类困难。我们需要的是清晰明了，而不是故弄玄虚。这里fuss的意思是“小题大做”。Understanding后面省去is what’s wanted 以避免重复。

3. I do not mean to advise a young author to be ever so slightly but very very cleverly dishonest and to gloss over difficulties.

意思是：我的意思是青年的作者绝不可有哪怕只是些少，当却是掩饰得非常巧妙得虚伪，我也劝告他们不要去掩饰困难。

4．Here is the sort of thing I mean by than complete honesty.

意思是：这就是我所认为的不够完全诚实的那类事情（东西）。注意：Here is 的意思是：“这里就是---”，然后把要说的事情在随后给出，若用This is the sort of thing---一般是当你把要说的事情已经说了然后用指示代词This 来概括所说的事，注意这一区分。

5．In any event: take the reader into your confidence.

意思是：在任何情况，要敢于对读者讲出真相。这里take---into one’s confidence意思是：“对---吐露秘密；把---当成心腹朋友”。

6．Don’t let the reader down..

意思是：不要使读者丧气。这里down是形容词。

7．Complete honesty makes for greatest clarity.

意思是：彻底的诚实就是最大的明嘹。Make for 是“有助于”的意思。这样简洁而又充满哲理的句子还有Emphasize the important minimize the trivial.

Exercise

(Miscellaneous Exercises)

Ⅰ.Fill in each blank with a suitable word.

1. is called the ________ ________of .

2.p (x)= is a_________of_________n.

3. called a __________ _________.

4. is called the________of .

5.The graph of (p>0) is a___________.

6. then is called a________ _________sequence.

7. is a_________of two equations with three_______.

8.Numbers such as and πare called________numbers.

9.The relation between the celements of a set of real numbers denoted

by<(or<;>;>) is called an_________relation.

10.The relation between sets, denoted by is called an_________relation.

Ⅱ.Each ofthe following sentences is grammatically wrong. Correct these sentences.

1. Let is a continuous function defined on[a,b].

2. Differentiating both sides of with respect to x, the equation becomes y’=—x/y

3. Take the derivatives of both sides of the equation ,we get x+yy’=0.

4. The primtive of here C is a constant.

5. We say that has a limit A at if approaches to A when X tends to .

Ⅲ.Translate the following sentences into Chinese (pay attention to the phrases underlined):

1. We are now in a position to prove the main theorem.

2. An analogous argument gives a proof of the corresponding theorem for decreasing functions.

3. An immediate consequence of Bolzano’s theorem is the intermediate-value theorem for continuous functions.

4. We claim that has no real solution, In fact if is a real solution, then we have which is impossible.

5. It is clear that the method described above also applies to the general case.

6. It is easy to show that has derivatives up to order n at the point x=0,where n>1.

Ⅳ.Translate the following passage into Chinese:

1. It is helpful to introduce the words‖local‖and“global”to contrast two types of situations that frequently arise. If we are considering a given set D, then we say that any specific property holds“locally”at of D if it is tre at if and at all points near ；thus there will be an open ball B apout and the property will hold for all .On the other hand, a property that holds at all points in D is said to hold ―globally‖in D.

2. The study of sequences is concerned primarily with the following type of question:if each term of a sequence has a certain property, such as continuity, differentiability or integrability, to what extend is this property transferred to the

limit function?

数学专业英语[19]－How to Organize a paper (For Beginers)?

数学专业英语－How to Organize a paper (For Beginers)?

The usual journal article is aimed at experts and near-experts, who are the people most likely to read it. Your purpose should be say quickly what you have done is good, and why it works. Avoid lengthy summaries of known results, and minimize the preliminaries to the statements of your main results. There are many good ways of organizing a paper which can be learned by studying papers of the better expositors. The following suggestions describe a standard acceptable style.

Choose a title which helps the reader place in the body of mathematics.

A useless title: Concerning some applications of a theorem of J. Doe. A. good title contains several well-known key words, e. g. Algebraic solutions of linear partial differential equations. Make the title as informative as possible; but avoid redundancy, and eschew the medieval practice of letting the title serve as an inflated advertisement. A title of more than ten or twelve words is likely to be miscopied, misquoted, distorted, and cursed.

The first paragraph of the introduction should be comprehensible to any mathematician, and it should pinpoint the location of the subject matter. The main purpose of the introduction is to present a rough statement of the principal results; include this statement as soon as it is feasible to do so, although it is sometimes well to set the stage with a preliminary paragraph. The remainder of the introduction can discuss the connections with other results.

It is sometimes useful to follow the introduction with a brief section that establishes notation and refers to standard sources for basic concepts and results. Normally this section should be less than a page in length. Some authors weave this information unobtrusively into their introductions, avoiding thereby a dull section.

The section following the introduction should contain the statement of

one or more principal results. The rule that the statement of a theorem should precede its proof a triviality. A reader wants to know the objective of the paper, as well as the relevance of each section, as it is being read. In the case of a major theorem whose proof is long, its statement can be followed by an outline of proof with references to subsequent sections for proofs of the various parts.

Strive for proofs that are conceptual rather than computational. For an example of the difference, see A Mathematician’s Miscellany by

J.E.Littlewood, in which the contrast between barbaric and civilized proofs is beautifully and amusingly portrayed. To achieve conceptual proofs, it is often helpful for an author to adopt an initial attitude such as one would take when communicating mathematics orally (as when walking with a friend). Decide how to state results with a minimum of symbols and how to express the ideas of the proof without computations. Then add to this framework the details needed to clinch the results.

Omit any computation which is routine (i.e. does not depend on unexpected tricks). Merely indicate the starting point, describe the procedure, and state the outcome.

It is good research practice to analyze an argument by breaking it into a succession of lemmas, each stated with maximum generality. It is usually bad practice to try to publish such an analysis, since it is likely to be long and uninteresting. The reader wants to see the path-not examine it with a microscope. A part of the argument is worth isolating as a lemma if it is used at least twice later on.

The rudiments of grammar are important. The few lines written on the b lackboard during an hour’s lecture are augmented by spoken commentary, and aat the end of the day they are washed away by a merciful janitor. Since the published paper will forever speak for its author without benefit of the cleansing sponge, careful attention to sentence structure is worthwhile. Each author must develop a suitable individual style; a few general suggestions are nevertheless appropriate.

The barbarism called the dangling participle has recently become more prevalent, bu t not less loathsome. “Differentiating both sides with respect to x, the equation becomes---”is wrong, because “the equation” cannot be the subject that does the differentiation. Write instead “differentiating both sides with respect to x, we get the equat ion---,” or “Differentiation of both sides with respect to x leads to the equation---”

Although the notion has gained some currency, it is absurd to claim that informal “we” has no proper place in mathematical exposition. Stri ct formality is appropriate in the statement of a theorem, and casual chatting should indeed be banished from those parts of a paper which will be printed in italics. But fifteen consecutive pages of formality are altogether foreign to the spirit of the twentieth century, and nearly all authors who try to sustain an impersonal dignified text of such length succeed merely in erecting elaborate monuments to slumsiness.

A sentence of the form ―if P,Q‖ can be understood. However ―if P,Q,R,S,T‖ is not so good, even if it can be deduced from the context that the third comma is the one that serves the role of ―then.‖ The reader is looking at the paper to learn something, not with a desire for mental calisthenics

Vocabulary

preliminary 序,小引(名)开端的,最初的(形)

eschew 避免

medieval 中古的，中世纪的

inflated 夸张的

comprehensible 可领悟的，可了解的

pinpoint 准确指出（位置）

weave 插入，嵌入

unobtrusivcly 无妨碍地

triviality 平凡琐事

barbarism 野蛮，未开化

portray 写真，描写

clinch 使终结

rudiment 初步，基础

commentary 注解，说明

janitor 看守房屋者

sponge 海绵

dangling participle 不连结分词

prevalent 流行的，盛行

loathsome 可恶地

absurd 荒谬的

banish 排除

sustain 维持，继续

slumsiness 粗俗，笨拙

monument 纪念碑

calisthenics 柔软体操，健美体操

notes

1. 本课文选自美国数学会出版的小册子A mamual for authors of mathematical paper的一节，本文对准备投寄英文稿件的读者值得一读。

2. Choose a title which helps the reader place in the body of mathematics.

意思是：选择一个可帮助读者进入数学核心的标题。

3．For an example of the difference, ……in which the contrast between barbaric and civilized proofs is beautifully and amusingly portrayed.

意思是：作为这种差别的一个例子，可参看J.E.L ittlewood A mathematician’s Miscellany一文，在那里，他把野蛮的（令人讨厌的）证明与文明的证明这两者之间的对比很漂亮地和有趣地给予描绘出来，这里“差别”是指conceptual proof 与computation proof 差别。Portray的意思是：“人像”，这里作动词用，作“描绘”解。

4．The reader wants to see the path--------not to examine it with a microscope.

意思是：读者想知道的是有关论证的途径——而不想使用显微镜去观察。这里作者所要表达的意思是：写文章的人只需把论证的要点写出即可，无需把论证的整个分析过程写得过于冗长。

5．The barbarism called the dangling participles had recently become prevalent, but not less loathsome.

意思是：一种称为“不连结分词”的句子，最近变得盛行起来，但这类句子毕竟是令人讨厌的。关于“不连结分词”，请参看ⅡA第三课注2。

6．The reader is looking at the paper to learn something, not with a desire for mental calisthenics.

意思是：读者阅读文章是为了学到一点东西，而不是抱着一种做智力体操的愿望去阅读的。这里作者是在批评有些写文章的人使用了一些令读者摸不着头脑，而要读者去猜其真实意思的句子（例如用‖if P,Q,R,S,T‖表达‖if P,Q,R then S,T这样的句子。）

Exercise

(Miscellaneous exercises (continued))

Ⅰ.Translate the following sentensces into English:

1. 若行列式中有两行成比例，则行列式为零。

2. 矩阵和矩阵是两种常用的矩阵

3. 两个次多项式上的值相等，则

4. 对任意两个多项式一定存在有多项式或者相等于零或者它的次数小于的次数。

5. 如果取双曲线的渐近线做为坐标轴，则双曲线方程将得到特别简单的形式。

6. 抛物线与椭圆和双曲线不同，它没有中心，它的另一个特殊性是它仅有一个焦点。

7. 通过平面上任何5个不同的点，其中没有4点同在一直线上（共线），有一条仅有一条二级曲线。

8. 当椭圆的长轴等于它的短轴时，它化为一圆。

9. 显然，无界序列不收敛。

10. 设在[a,b]上连续，若存在两个点，则必有

11. 若我们能证明连续函数级数在紧集D上一致收敛，则在此集上可对级数进行逐项积分。

12. 此定理给出了用n次泰勒多项式来近似代替时余项太小的一种估计。

13. 用同样的方法，我们还可以证明定理A。

14. 定理中的条件是缺一不可的。

15. 最后，我们再举出两个能说明问题的例子来结束文章。

Ⅱ.Translate the following sentences into Chinese(Pay attention to the words underlined):

1. The compact and Fredholm operators lately have been receiving renewed attention because of the applications to integral operators and partial differential elliptic operators.

2. We dcnote by [x] the greatest integral part which is less than or equal to x.

3. Global analysis on manifolds has come into its owm, both in its integral and differential aspects. It is therefore desirable to integrate manifolds in analysis courses.

Ⅲ.Translate the following passages into Chinese:

1. The concept of ordering was abstracted form various relations, such as the inequality relation between real numbers and the inclusion relation between sets. Suppose that we are given a set X={x, y, z ,…}，the relation between the elements of X, denoted by

2. Suppose we are given a relation R (usually denoted by the symbol ) between elements of a set X such that for any elements x and y of X, either xRy orits negation holds The relation is called an equivalence relation (on X)if it satisfies the following three conditions:

(i) xRx

(ii) xRy implies yRx,

(iii) xRy and yRz imply xRz.

Conditions (i),(ii),and(iii)are called the reflexive, symmetric and transitive laws respectively. Together they are called equivalence properties. The relations of congruence and similarity between figures are equivalence relations.

数学专业英语[20]－Mathematical Discovery

数学专业英语－Mathematical Discovery

To give the flavor of Polya’s thinking and writing in a very beautiful but subtle case , a case that involve a change in the conceptual mode , I shall quote at length from his Mathematical Discovery (vol.II , pp.54 ff):

EXAMPLE I take the liberty a little experiment with the reader , I shall state a simple but not too commonplace theorem of geometry , and then I shall try to reconstruct the sequence of idoas that led to its proof .

I shall proceed slowly , very slowly , revealing one clue after the other , and revealing each gradually . I think that before I have finished the whole story , the reader will seize the main idea (unless there is some special hampering circumstance ) . But this main idea is rather

unexpected , and so the reader may experience the pleasure of a little discovery .

A.If three circles having the same radius pass through a point , the circle through their other three points of intersection also has the same radius .Fig.1 Three circles through one point.

This is the theorem that we have to prove . The statement is short and clear , but does not show the details distinctly enough . If we draw a figure (Fig .1) and introduce suitable notation , we arrive at the following more explicit restatement :

B . Three circles k , l , m have the same radius r and pass through the same point O . Moreover , l and m intersect in the point A , m and k in B , k and l in

C . Then the circle e through A , B , C has also the radius Fig .2 too crowded .

Fig .1 exhibits the four circles k , l , m , and e and their four points of intersection A, B , C , and O . The figure apt to be unsatisfactory , however , for it is not simple , and it is still incomplete ; something seems to be missing ; we failed to take into account something essential , it seems .

We are dialing with circles . What is a circle ? A circle is determined by center and radius ; all its points have the same distance , measured by the length of the radius , from the center . We failed to introduce the common radius r , and so we failed to take into account an essential part of the hypothesis . Let us , therefore , introduce the centers , K of k , L of l , and M of m . Where should we exhibit the radius r ? there seems to be no reason to treat any one of the three given circles k ; l , and m or any one of the three points of intersection A , B , and C better than the others . We are prompted to connect all three centers with all the points of intersection of the respective circle ; K with B , C , and O , and so forth .

The resulting figure (Fig . 2) is disconcertingly crowded . There are so many lines , straight and circular , that we have much trouble old－fashioned magazines . The drawing is ambiguous on purpose ; it presents a certain figure if you look t it in the usual way , but if you turn it to a certain position and look at it in a certain peculiar way , suddenly another figure flashes on you , suggesting some more or less witty comment on the first . Can you recognize in our puzzling figure , overladen with straight and circles , a second figure that makes sense ?

We may hit in a flash on the right figure hidden in our overladen drawing , or we may recognize it gradually . We may be led to it by the effort to solve the proposed problem , or by some secondary , unessential circumstance . For instance , when we are about to redraw our unsatisfactory figure , we may observe that the whole figure is determined by its rectilinear part (Fig . 3) .

This observation seems to be significant . It certainly simplifies the geometric picture , and it possibly improves the logical situation . It leads us to restate our theorem in the following form .

C . If the nine segments

KO , KC , KB ,

LC , LO , LA ,

MB , MA , MO ,

are all equal to r , there exists a point E such that the three segments

EA , EB , EC ,

are also equal to r .

Fig . 3 It reminds you －of what ?

This statement directs our attention to Fig . 3 . This figure is attractive ; it reminds us of something familiar . (Of what ?)

Of course , certain quadrilaterals in Fig .3 . such as OLAM have , by hypothesis , four equal sided , they are rhombi , A rhombus I a familiar object ; having recognized it , we can “see “ the figure better . (Of what does the whole figure remind us ?)

Oppositc sides of a rhombus are parallel . Insisting on this remark , we realize that the 9 segments of Fig . 3 . are of three kinds ; segments of the same kind , such as AL , MO , and BK , are parallel to each other . (Of what does the figure remind us now ?)

We should not forget the conclusion that we are required to attain . Let us assume that the conclusion is true . Introducing into the figure the center E or the circle e , and its three radii ending in A , B , and C , we obtain (supposedly ) still more rhombi , still more parallel segments ; see Fig . 4 . (Of what does the whole figure remind us now ?)

Of course , Fig . 4 . is the projection of the 12 edges of a parallele piped having the particularity that the projection of all edges are of equal length .

Fig . 4 of course !

Fig . 3 . is the projection of a “nontransparent “ parallelepiped ; we see only 3 faces , 7 vertices , and 9 edges ; 3 faces , 1 vertex , and

3 edges are invisible in this figure . Fig . 3 is just a part of Fig .

4 . but this part defines the whole figure . If the parallelepiped and the direction of projection are so chosen that the projections of the 9 edges represented in Fig . 3 are all equal to r (as they should be , by hypothesis ) , the projections of the 3 remaining edges must be equal to r . These 3 lines of length r are issued from the projection of the 8th , the invisible vertex , and this projection E is the center of a circle passing through the points A , B , and C , the radius of which is r .

Our theorem is proved , and proved by a surprising , artistic conception of a plane figure as the projection of a solid . (The proof uses notions of solid geometry . I hope that this is not a treat wrong , but if so it is easily redressed . Now that we can characterize the situation of the center E so simply , it is easy to examine the lengths EA , EB , and EC independently of any solid geometry . Yet we shall not insist on this point here .)

This is very beautiful , but one wonders . Is this the “ light that breaks forth like the morning . “ the flash in which desire is fulfilled ? Or is it merely the wisdom of the Monday morning quarterback ? Do these ideas work out in the classroom ? Followups of attempts to redu ce Polya’s program to practical pedagogics are difficult to interpret . There is more to teaching , apparently , than a good idea from a master .

——From Mathematical Experience

Vocabulary

subtle 巧妙的，精细的

clue 线索，端倪

hamper 束缚，妨碍

disconcert 使混乱，使狼狈

ambiguous 含糊的，双关的

witty 多智的，有启发的

rhombi 菱形（复数）

rhombus 菱形

parallelepiped 平行六面体

projection 射影

solid geometry 立体几何

pedagogics 教育学，教授法

commonplace 老生常谈；平凡的

数学专业英语[21]－Notations and Abbreviations (I) Learn to understand

N set of natural numbers

Z set of integers

R set of real numbers

C set of complex numbers

+ plus; positive

－minus; negative

× multiplied by; times

÷ divided by

＝equals; is equal to

≡ identically equal to

≈,≌ approximately equal to

＞greater than

≥ greater than or equal to

＜less than

≤ less than or equal to

》much greater than

《much less than

square root

cube root

nth root

│a│ absolute value of a

n! n factorial

a to the power n ; the nth power of a

[a] the greatest integer≤a

the reciprocal of a

Let A, B be sets

∈belongs to ; be a member of

not belongs to

x∈A x is a member of A

∪ union

A∪B A union B

∩ intersection

A∩B A intersection B

A符合B A is a subset of B;A is contained in B（子集）A B A contains B

complement of A A的补集

the closure of A A 的闭集

empty set 空集

( ) i=1,2,…,r j=1,2,…,s r-by-s（r×s）matrix 矩阵

││I,j=1,2,…,n determinant of order n n 阶的行列式det( ) the determinant of the matrix ( ) 矩阵的行列式

vector F 矢量

x=( , ,…, ) x is an n-tuple of

‖‖the nor m of …

‖ parallel to

┴ perpendicular to

the exponential function of x

lin x the logarithmic function of x sie sine

cos cosine

tan tangent

sinh hyperbolic sine

cosh hyperbolic cosine