# 一代数学大师 Rota 的经验与忠告

### Ten Lessons I wish I Had Been Taught

#### 讲座（Lecturing）

（a）每次讲座都应该只有一个重点。（Every lecture should make only one main point.）

Every lecture should state one main point and repeat it over and over, like a theme with variations. An audience is like a herd of cows, moving slowly in the direction they are being driven towards. If we make one point, we have a good chance that the audience will take the right direction; if we make several points, then the cows will scatter all over the field. The audience will lose interest and everyone will go back to the thoughts they interrupted in order to come to our lecture.

（b）不要超时。（Never run overtime.）

Running overtime is the one unforgivable error a lecturer can make. After fifty minutes (one micro-century as von Neumann used to say) everybody’s attention will turn elsewhere even if we are trying to prove the Riemann hypothesis. One minute overtime can destroy the best of lectures.

As you enter the lecture hall, try to spot someone in the audience with whose work you have some familiarity. Quickly rearrange your presentation so as to manage to mention some of that person’s work. In this way, you will guarantee that at least one person will follow with rapt attention, and you will make a friend to boot.

Everyone in the audience has come to listen to your lecture with the secret hope of hearing their work mentioned.

（d）给听众一些值得回忆的东西。（Give them something to take home.）

Most of the time they admit that they have forgotten the subject of the course and all the mathematics I thought I had taught them. However, they will gladly recall some joke, some anecdote, some quirk, some side remark, or some mistake I made.

#### 板书技巧（Blackboard Technique）

（a）开讲前保持黑板干净（Make sure the blackboard is spotless.）

By starting with a spotless blackboard you will subtly convey the impression that the lecture they are about to hear is equally spotless.

（b）从黑板的左上角开始书写（Start writing on the top left-hand corner.

What we write on the blackboard should correspond to what we want an attentive listener to take down in his notebook. It is preferable to write slowly and in a large handwriting, with no abbreviations.

When slides are used instead of the blackboard, the speaker should spend some time explaining each slide, preferably by adding sentences that are inessential, repetitive, or superfluous, so as to allow any member of the audience time to copy our slide. We all fall prey to the illusion that a listener will find the time to read the copy of the slides we hand them after the lecture. This is wishful thinking.

#### 多次公布同样的结果（Publish the Same Result Several Times）

The mathematical community is split into small groups, each one with its own customs, notation, and terminology. It may soon be indispensable to present the same result in several versions, each one accessible to a specific group; the price one might have to pay otherwise is to have our work rediscovered by someone who uses a different language and notation and who will rightly claim it as his own.

#### 说明性的工作反而更有可能被记得（You Are More Likely to Be Remembered by Your Expository Work）

When we think of Hilbert, we think of a few of his great theorems, like his basis theorem. But Hilbert’s name is more often remembered for his work in number theory, his Zahlbericht, his book Foundations of Geometry, and for his text on integral equations.

#### 每个数学家只有少数的招数（Every Mathematician Has Only a Few Tricks）

You admire Erdös’s contributions to mathematics as much as I do, and I felt annoyed when the older mathematician flatly and definitively stated that all of Erdös’s work could be “reduced” to a few tricks which Erdös repeatedly relied on in his proofs. What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over. But on reading the proofs of Hilbert’s striking and deep theorems in invariant theory, it was surprising to verify that Hilbert’s proofs relied on the same few tricks. Even Hilbert had only a few tricks!

There are two kinds of mistakes. There are fatal mistakes that destroy a theory, but there are also contingent ones, which are useful in testing the stability of a theory.

#### 使用费曼的方法（Use the Feynman Method）

You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say, “How did he do it? He must be a genius!”

#### 不要吝啬你的赞美（Give Lavish Acknowledgments）

I have always felt miffed after reading a paper in which I felt I was not being given proper credit, and it is safe to conjecture that the same happens to everyone else.

#### 写好摘要（Write Informative Introductions）

If we wish our paper to be read, we had better provide our prospective readers with strong motivation to do so. A lengthy introduction, summarizing the history of the subject, giving everybody his due, and perhaps enticingly outlining the content of the paper in a discursive manner, will go some of the way towards getting us a couple of readers.

#### 为老年做好心理准备（Be Prepared for Old Age）

You must realize that after reaching a certain age you are no longer viewed as a person. You become an institution, and you are treated the way institutions are treated. You are expected to behave like a piece of period furniture, an architectural landmark, or an incunabulum.

### Ten Lessons for the Survival of a Mathematics Department

#### 不要在其他系讲自己系同事的坏话（Never wash your dirty linen in public）

Departments of a university are like sovereign states: there is no such thing as charity towards one another.

Your letter will be viewed as evidence of disunity in the rank and file of mathematicians. Human nature being what it is, such a dean or provost is likely to remember an unsolicited letter at budget time, and not very kindly at that.

#### 不要进行领域评价（Never Compare Fields）

You are not alone in believing that your own field is better and more promising than those of your colleagues. We all believe the same about our own fields. But our beliefs cancel each other out. Better keep your mouth shut rather than make yourself obnoxious. And remember, when talking to outsiders, have nothing but praise for your colleagues in all fields, even for those in combinatorics. All public shows of disunity are ultimately harmful to the well-being of mathematics.

#### 别看不起别人使用的数学（Remember that the grocery bill is a piece of mathematics too）

The grocery bill, a computer program, and class field theory are three instances of mathematics. Your opinion that some instances may be better than others is most effectively verbalized when you are asked to vote on a tenure decision. At other times, a careless statement of relative values is more likely to turn potential friends of mathematics into enemies of our field. Believe me, we are going to need all the friends we can get.

#### 善待擅长教学的老师（Do not look down on good teachers）

Mathematics is the greatest undertaking of mankind. All mathematicians know this. Yet many people do not share this view. Consequently, mathematics is not as self-supporting a profession in our society as the exercise of poetry was in medieval Ireland. Most of our income will have to come from teaching, and the more students we teach, the more of our friends we can appoint to our department. Those few colleagues who are successful at teaching undergraduate courses should earn our thanks as well as our respect. It is counterproductive to turn up our noses at those who bring home the dough.

#### 学会推销自己的数学成果（Write expository papers）

When I was in graduate school, one of my teachers told me, “When you write a research paper, you are afraid that your result might already be known; but when you write an expository paper, you discover that nothing is known.”

It is not enough for you (or anyone) to have a good product to sell; you must package it right and advertise it properly. Otherwise you will go out of business.

When an engineer knocks at your door with a mathematical question, you should not try to get rid of him or her as quickly as possible.

#### 不要把提问者拒之门外（Do not show your questioners to the door）

What the engineer wants is to be treated with respect and consideration, like the human being he is, and most of all to be listened to with rapt attention. If you do this, he will be likely to hit upon a clever new idea as he explains the problem to you, and you will get some of the credit.

Listening to engineers and other scientists is our duty. You may even learn some interesting new mathematics while doing so.

#### 联合阵线（View the mathematical community as a United Front）

Grade school teachers, high school teachers, administrators and lobbyists are as much mathematicians as you or Hilbert. It is not up to us to make invidious distinctions. They contribute to the well-being of mathematics as much as or more than you or other mathematicians. They are right in feeling left out by snobbish research mathematicians who do not know on which side their bread is buttered. It is our best interest, as well as the interest of justice, to treat all who deal with mathematics in whatever way as equals. By being united we will increase the probability of our survival.

#### 把科学从不可靠中拯救出来（Attack Flakiness）

Flakiness is nowadays creeping into the sciences like a virus through a computer, and it may be the present threat to our civilization. Mathematics can save the world from the invasion of the flakes by unmasking them and by contributing some hard thinking. You and I know that mathematics is not and will never be flaky, by definition.

This is the biggest chance we have had in a long while to make a lasting contribution to the well-being of Science. Let us not botch it as we did with the few other chances we have had in the past.

#### 善待所有人（Learn when to withdraw）

Let me confess to you something I have told very few others (after all, this message will not get around much): I have written some of the papers I like the most while hiding in a closet. When the going gets rough, we have recourse to a way of salvation that is not available to ordinary mortals: we have that Mighty Fortress that is our Mathematics. This is what makes us mathematicians into very special people. The danger is envy from the rest of the world.

When you meet someone who does not know how to differentiate and integrate, be kind, gentle, understanding. Remember, there are lots of people like that out there, and if we are not careful, they will do away with us, as has happened many times before in history to other Very Special People.

### 参考资料：

1. Rota, Gian-Carlo. “Ten lessons I wish I had been taught.” Indiscrete thoughts. Birkhäuser, Boston, MA, 1997. 195-203.
2. Rota, Gian-Carlo. “Ten Lessons for the Survival of a Mathematics Department.” Indiscrete Thoughts. Birkhäuser, Boston, MA, 1997. 204-208.

# 从对数学的贡献上来讲，丘成桐有多厉害？

1.丘成桐教授不仅有数学才华，还很有商业天赋。他在Boston地区有三十多套房产。因为Harvard是个很有钱的学校，所以有很多闲置的房产，他们会用极低的价格把这些房产卖给教授。丘成桐教授以其杰出的商业眼光，前前后后一共买了三十多套，租给他的博士后，每年盈利不可胜计，真是令人钦佩！后来丘教授又看中了一处房子，但是学校却不愿意批准卖给他，所以他让当时是系主任的Ben Gross教授去询问缘由，后来Gross说，学校得知你在Boston地区有三十多套房产，实在太多了，所以不能卖给你。大家知道，在数学界，要想组织seminar和conference，经费是必不可少的。正因为丘教授有杰出的商业头脑和投资眼光，所以为中国数学的蓬勃发展输入了大量的物质财富，可谓是中国版的Simons。但是他的数学水平又远胜Simons，所以丘教授无愧为古往今来第一大师！

2.丘教授通过这些seminar和conference让大量的中国年轻数学家有了抛头露面和展示自己的机会。虽然这些年轻人的数学水平只可意会，但是相信通过丘教授的帮助会很快发展成为华人数学界的领军人物，继承他的资源和衣钵。近年来，丘教授在中国大陆，中国香港和台湾地区设立了大量的研究所。这些研究所的设立不但给不少人提供了很好的工作机会，也给不少想学数学的年轻人提供了优秀的平台。比如清华大学的丘成桐数学中心，可以说是亚洲第一数学中心，连日本京都的RIMS都是远远不如的，我想即使放到宇宙上也是名列前茅的。在这里我们应该特别欢迎广大二本和三本的数学系学生报考这些研究所，因为丘先生的理念就是要给普通高校热爱数学的学生以机会。

3.丘教授每年都到中国的各所高校讲学，尤其是他开设的几个数学中心，这些讲座传授给年轻人许多高深的数学知识和实用的数学技巧。他演讲的话题包括：数学之美、我的成功经验、Harvard数学系的历史和我的一个不听话的学生等等。内容丰富，发人深省，不但能从中学到数学知识，还能体会到许多做（中国）人的道理。可悲的是，一些反动派受到西方自由思想的荼毒，对这样高质量的讲座却视而不见，拒绝参加，其中包括一些数学界的同行。丘教授知悉此事后，给这些人发了一封邮件，明确要求他们：今后只要是我来你们学校做讲座，所有中国人就必须参加！丘先生的严厉做法很好地整肃了华人数学界的风气，提高了凝聚力。相信在丘先生的领导下，大家一定能鼓足干劲，力争上游，多快好省地建设中国数学！

4.丘教授亲自培养的许多学生都有极高的数学水准，在国际上获得广泛承认，多次荣获重大国际奖项，比如晨兴数学奖、新世界数学奖、陈省身奖之中国版等等。这些学生不仅自己水平惊人，对年轻人也提供了无微不至的关怀和细致周到的帮助。比如，丘教授的不少学生害怕学生没有自己的想法，经常亲自给学生提供idea，来帮助学生找到研究的思路。即使学生不需要也要苦口薄心，再三敦促。这样一来，不仅学生可以发paper，他们自己也因为贡献了一个“关键的”idea而顺便加到了名字，可谓是一举两得的做法。丘教授另一些学生因为害怕国际上一些著名杂志的编辑是势利眼，不让年轻学生单独发paper，所以不惜牺牲自己的名节，主动要求在paper上加名字。这样一来，学生发文章的时候就不会吃亏了。他们为学生的付出令人感动。可悲的是，一些年轻人不但不知道感恩，反而对此感到苦恼。对这样的人，我们就应该毫不犹豫地把他们踢出华人数学界，让他们去落后的西方世界吃点苦头！

5.丘教授掌握了国际上一本极为重要的数学杂志，即Journal of Differential Geometry。这本杂志现在成为许多年轻人展示自己只可意会的数学水平和找到教职的最佳平台。为了方便某些中国学生在杂志上发表论文，丘教授提供了一些非同寻常的便捷渠道。比如文章不用发给编辑，可以直接发给自己，再由他转发给编辑。这样一来，中国数学家的文章就经常出现在顶级杂志上，他们的研究水准得到了空前飞跃！丘教授控制的另一本杂志就是大名鼎鼎的Asian Journal。这本杂志上发表了人类在20世纪到21世纪一些最伟大的数学工作，比如朱熹平教授和曹怀东教授对Poincare猜想的最终证明，封顶了人类一百余年来悬而未决的难题。这篇文章长达300多页，但是经过Asian Journal的编辑不知疲倦的辛勤工作，该论文在极短的时间内就获得了发表。可以看到，丘教授在经营杂志以后，杂志审核文章的效率大大提高了。可以说，正是丘教授勤劳刻苦，生命不息，奋斗不止的精神感召了这些编辑，让他们不再玩忽职守和放松懈怠。

6.丘成桐教授对自己学生的关怀可以说是无微不至。有些学生一时糊涂涉嫌抄袭和剽窃，丘教授知道以后果断采取措施，息事宁人，避免了家丑外扬。中国数学界正是在丘先生的努力下才能铁板一块地团结在一起，大家毫无私心，全心全意为中国数学的发展添砖加瓦。但是有些人却不明白丘教授的苦心，经常在丘教授面前投诉，甚至还写匿名信把事情闹到别的学校。对此，丘教授态度坚决，铁面无私地无视了这些无理要求，可以说很好地体现了一位领袖的英明果决。而那些闹事的逆流虽然可能有一点点数学水平，但是今天也没办法站出来领导数学界了。就是因为某些人只知道做研究和思考数学问题，没有意识到帮助中国数学发展才是更有意义的事。思想境界比起丘教授差的太远了。可以说，丘先生高瞻远瞩，气盖环宇，数风流人物，还看今朝。

7.丘教授对中国学生的关心不仅仅局限在数学系，还遍及到各个非数学领域。从前，只要是中国、香港和台湾去Harvard读数学的学生，丘教授都要亲自过问，热情关怀，把他们一一纳入自己门下。比如某学生要跟Taubes，他会亲自找到Taubes，告诉他，这位学生就托付给你了。这样一来，这些西方数学家慑于丘先生的气魄和威望，就不敢再歧视中国学生了。到了后来，只要去Harvard的中国、香港和台湾学生，无论学什么专业，丘先生都要跟他们打交道。据说他还曾经举办过大型party，邀请Harvard商学院大中华地区的所有学生参加。这些活动使他亲民的形象更加突出，在各界广受好评。相信不久的将来，丘教授会吸引到亚洲其他地区的学生参与他的party。像他这的一代王者，相信任何人都会被他的魅力所感召。毕竟只有深入到人民群众中去，才能发现问题所在。丘教授真不愧为一代明君！

8.丘教授虽然已经接近70高龄，仍然老骥伏枥，近年来在数学研究上非常活跃。仅2015一年就在arxiv贴文23篇，以每个月两篇论文的速度进行高质量的数学研究，这是古往今来其他任何数学家都望尘莫及的！要知道，丘教授作为华人数学界的领袖，每天要处理几百封邮件。熟悉丘教授的朋友们都知道，即使是在seminar上他也要一边摁手机收发邮件，一边听talk。能在如此繁忙的情况下一个月写两篇论文，效率之高真是令人震惊！丘教授还特别注意与年轻人的合作，近年来每篇论文几乎都要提携一些年轻数学家，大度地和他们一起署名发表。由于他提携的年轻数学家太多，很多时候甚至会忘记自己的合作者。比如某韩国数学家之前跟他有合作，到了找教职的时候希望丘教授能帮自己写推荐信，但是丘教授却坦言自己并不认识对方。实际上，丘教授不认识自己的合作者正可以反映出他已经帮助了太多年轻人，以至于自己都想不起来自己干的那些好事！范仲淹说：云山苍苍，江水泱泱，先生之风，山高水长。丘先生年近七旬而笔耕不辍，真可谓吾辈典范！

9.丘成桐教授对于人才优劣的判断也是明察秋毫，一望即知。早先，北大一个学生仗着自己是那一届最优秀的就自不量力，想要去Harvard跟丘教授学数学，丘教授对他说：你水平不行。想跟我也可以，先去Boston待两年，经我考察合格了，再来跟我。这个学生不得已之下去了另一个inferior的学校跟了一个比丘教授差了十万八千里的数学家M。事实证明，这个学生现在虽然出了一点小名，在Yale做教授，但是确实不够资格在Harvard做丘教授的学生：因为他只拿到了晨兴数学银奖，而丘教授的学生一般都是拿金奖的。

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# [转载]痛批计算数学所

※ 来源:．南京大学小百合站 http://bbs.nju.edu.cn

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# Controversy over Yau-Tian-Donaldson

http://www.math.columbia.edu/~woit/wordpress/?p=6430

# Controversy over Yau-Tian-Donaldson

Posted on November 25, 2013 by woit

The last posting here was about an unusually collaborative effort among mathematicians, whereas this one is about the opposite, an unusually contentious situation surrounding important recent mathematical progress.

What’s at issue is the proof of what has become known as the “Yau-Tian-Donaldson” conjecture, which describes when compact Kähler manifolds with positive first Chern class have a Kähler-Einstein metric. This is analogous to the Calabi conjecture, which deals with the case of vanishing first Chern class. Progress by Donaldson on this was first mentioned on this blog here (based on his talk at Atiyah’s 80th birthday conference in 2009). Last fall a proof of the conjecture was announced by Chen-Donaldson-Sun, with an independent claim for a proof by Gang Tian, see here. I wrote a bit about this last winter here, after the details appeared of the Chen-Donaldson-Sun proof, and that posting gives some links to expository articles about the subject.

I had heard that there were complaints about Tian’s behavior in this story, including claims that he did not have a complete proof of the conjecture and was not acknowledging his use of ideas from Chen-Donaldson-Sun. Recently this controversy has become public, with Chen-Donaldson-Sun deciding to put out a document (linked to from Donaldson’s website) that challenges Tian’s claims to have an independent proof. The introduction includes:

Gang Tian has made claims to credit for these results. The purpose of this document is to rebut these claims on the grounds of originality, priority and correctness of the mathematical arguments. We acknowledge Tian’s many contributions to this field in the past and, partly for this reason, we have avoided raising our objections publicly over the last 15 months, but it seems now that this is the course we have to take in order to document the facts. In addition, this seems to us the responsible action to take and one we owe to our colleagues, especially those affected by these developments.

I should make it clear I’m no expert on this mathematics, so ill-equipped to judge many of the technical claims being made. The Chen-Donaldson-Sun document is giving one side of a complicated story, so it would be useful to have Tian’s side for comparison, but I have no idea if he intends to respond.

On a more positive note, perhaps this controversy will not interfere much with future progress in this area, as Donaldson and Tian are jointly organizing a Spring 2016 workshop on this topic at MSRI.

Update
: I hear from Tian that he has recently written a response to the Chen-Donaldson-Sun document, which is available here, and he may at some point write some more about this. Anyone who has read the CDS side of this should also take a look at what Tian has to say in response.

# 6月8日聚会 张益唐问答录

6月8日聚会  张益唐问答录

HL先生：我在网上看到，你早先的博士论文是做代数几何的（张：对）。你刚才说的经典的那些是指筛法吧，（张：对）我原来记得在陈景润做1+2时，魏伊说这达到了筛法的顶峰。但你的结果说明那还不是顶峰，你能把筛法加上后来的代数几何。

HL先生：也许你所做的意义更大，难度可能各有千秋。但你等于是证明有这回事，让大家不要做了，以前都没有这个具体的数值，现在有了七千万这个数，至于能不能压到2是另外一回事。

（众插话：还是好好去做数学。）

＊＊＊＊＊＊

is an honor for me, an old math major to be sitting here with Professor Zhang .He just does a remarkable thing, wonderful thing, in the field of mathematical where it is difficult to make much progress.

If you understand his paper, you will understand that he has shown that there is a number, which solved this problem, and the number is less than seventy million, actually we all believe that number is 2. So we have some distance to go, from seventy million down to 2. But that is not the worst situation in mathematics. There is another difficult problem in a different area of math, and we know there is a number that solves that problem. But the number is so big you can’t even express it, beyond millions, beyond trillions. It has a special name, Graham’s Number , a very very big number. But we actually believe that the true solution to this problem is 6. So this is not the worst case in mathematics. But it is marvelous achievement. I congratulate Professor Zhang, it is an honor to be here with him and it is very happy to make some old friends who I have known for twenty years or more, although we lost in touch.

It’s pleasure to be here. Thank you.

 作 者 :张益唐 等 出 处 :北京之春 整 理 :2013年7月8日20:26

# 素数并不孤独

http://songshuhui.net/archives/82114

——高斯

# 素数何时成双对

2、3、5、7、11、13……最初的几个素数，要找出来并不困难，但随着数字增大，如果一个一个数字按照定义去筛选是否素数，工作量会很快变得十分庞大。同为古希腊数学家的埃拉托色尼，给出了一个比较省力的算法，后人称之为埃拉托色尼筛法。

【埃拉托色尼筛法，图片出处：维基百科】

# 漫天星河难理清

【画在平面上的素数分布，图片出处：维基百科】

【霸气的哈代，图片出处：维基百科】

# 狂沙淘尽始得金

【出问题的那款芯片，图片出处：维基百科】

【陈景润的雕像，图片出处：维基百科】

# 梅花香自苦寒来

【张益唐，图片出处：新罕布什尔大学】

# 路漫漫其修远兮

6月5号，40万，连原来的百分之一都不到。

Bounded gaps between primes, Yitang Zhang, Annals of Mathematics

Open question: The parity problem in sieve theory, Terence Tao,http://terrytao.wordpress.com/2007/06/05/open-question-the-parity-problem-in-sieve-theory/

Are there infinitely many twin primes?, D. A. Goldston,http://www.math.sjsu.edu/~goldston/twinprimes.pdf

# 哈洛德•贺欧夫各特：彻底证明弱哥德巴赫猜想

http://songshuhui.net/archives/85342

“任一大于 2 的整数都可以写成三个质数之和。”271 年前，德国人哥德巴赫告诉欧拉这句话时，可能自己也没想到一下就在解析数论这个领域挖了一个东非大裂谷级别的“坑”。

1937 年，苏联数学家伊万•维诺格拉多夫更进一步，在无需广义黎曼猜想的情形下，直接证明了充分大的奇数可以表示为三个素数之和，被称为“三素数定理”。不过他无法给出“充分大”的界限。他的学生博罗兹金于 1939 年确定了一个“充分大”的下限：314348907。这个数字有 6846169 位，要验证比该数小的所有数完全不可行。

1995 年，法国数学家奥利维耶•拉马雷证明，不小于 4 的偶数都可以表示为最多六个素数之和。莱塞克•卡涅茨基证明了在黎曼猜想成立的前提下，奇数都可表示为最多五个素数之和。2012年，陶哲轩在无需黎曼猜想的情形下证明了这一结论。

2013年5月13日，法国国家科学研究院和巴黎高等师范学院的数论领域的研究员哈洛德•贺欧夫各特，在线发表两篇论文宣布彻底证明了弱哥德巴赫猜想。贺欧夫各特在文章“Minor arcs for Goldbach’s problem”中，给出了指数和形式的一个新界。在文章“Major arcs for Goldbach’s theorem”中，贺欧夫各特综合使用了哈迪-利特伍德-维诺格拉多夫圆法、筛法和指数和等传统方法，把下界降低到了1030左右，贺欧夫各特的同事 David Platt 用计算机验证在此之下的所有奇数都符合猜想，从而完成了弱哥德巴赫猜想的全部证明。

# 谈谈张益唐

【摄影：方弦】

Platt做的就是用计算机完成这样的计算，而且是以严格的方式。对于数学验证而言，严谨性很重要。我们知道，计算机只能表达有理数，它不能直接处理像圆周率这样的无理数。所以，实际上计算机不能处理实数，它只能处理一个区间[a,b]，其中a和b都是有理数。而你只能问你的计算机，能不能给出一个尽量短的区间[c,d]，使得区间[a,b]中的实数的正弦值（或者别的什么函数值）都落在区间[c,d]中。这就是所谓的区间算术。