数学专业英语第二版 课文翻译
2.4 整数、有理数与实数 4－A Integers and rational numbers 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. 有一些R的子集很著名，因为他们具有实数所不具备的特殊性质。在本节我们将讨论这样的子集，整数集和有理数集。 To introduce the positive integers we begin with the number 1, whose existence is guaranteed by Axiom 4. The number 1+1 is denoted by 2, the number 2+1 by 3, and so on. The numbers 1,2,3,…, obtained in this way by repeated addition of 1 are all positive, and they are called the positive integers. 我们从数字1开始介绍正整数，公理4保证了1的存在性。1+1用2表示，2+1用3表示，以此类推，由1重复累加的方式得到的数字1,2,3，…都是正的，它们被叫做正整数。 Strictly speaking, this description of the positive integers is not entirely complete because we have not explained in detail what we mean by the expressions “and so on”, or “repeated addition of 1”. 严格地说，这种关于正整数的描述是不完整的，因为我们没有详细解释“等等”或者“1的重复累加”的含义。 Although the intuitive meaning of expressions may seem clear, in careful treatment of the real-number system it is necessary to give a more precise definition of the positive integers. There are many ways to do this. One convenient method is to introduce first the notion of an inductive set. 虽然这些说法的直观意思似乎是清楚的，但是在认真处理实数系统时必须给出一个更准确的关于正整数的定义。 有很多种方式来给出这个定义，一个简便的方法是先引进归纳集的概念。 DEFINITION OF AN INDUCTIVE SET. A set of real numbers is called an inductive set if it has the following two properties: (a) The number 1 is in the set. (b) For every x in the set, the number x+1 is also in the set. For example, R is an inductive set. So is the set . Now we shall define the positive integers to be those real numbers which belong to every inductive set. 现在我们来定义正整数，就是属于每一个归纳集的实数。 Let P 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。 This property of P forms the logical basis for a type of reasoning that mathematicians call proof by induction, a detailed discussion of which is given in Part 4 of this introduction. P的这种性质形成了一种推理的逻辑基础，数学家称之为，在介绍的第四部分将给出这种方法的详细论述。归纳证明 The negatives of the positive integers are called the negative integers. The positive integers, together with the negative integers and 0 (zero), form a set Z which we call simply the set of integers. 正整数的相反数被叫做负整数。正整数，负整数和零构成了一个集合Z，简称为整数集。 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。 Quotients of integers a/b (where b≠0) are called rational numbers. The set of rational numbers, denoted by Q, contains Z as a subset. The reader should realize that all the field axioms and the order axioms are satisfied by Q. For this reason, we say that the set of rational numbers is an ordered field. Real numbers that are not in Q are called irrational. 整数a与b的商被叫做有理数，有理数集用Q表示，Z是Q的子集。读者应该认识到Q满足所有的域公理和序公理。因此说有理数集是一个有序的域。不是有理数的实数被称为无理数。 4－B Geometric interpretation of real numbers as points on a line 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. 毫无疑问，读者都熟悉通过在直线上描点的方式表示实数的几何意义。如图2-4-1所示，选择一个点表示0，在0右边的另一个点表示1。这种做法决定了刻度。 If one adopts an appropriate set of axioms for Euclidean geometry, then each real number corresponds to exactly one point on this line and, conversely, each point on the line corresponds to one and only one real number. 如果采用欧式几何公理中一个恰当的集合，那么每一个实数刚好对应直线上的一个点，反之，直线上的每一个点也对应且只对应一个实数。 For this reason the line is often called the real line or the real axis, and it is 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。 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。
全新版大学英语(第二版)综合教程2课文翻译
全新版大学英语(第二版)综合教程2课文翻译全新版大学英语(第二版)综合教程2课文翻译Lesson 1: A Trip to the Great Wall 乘夏空飞翱翔Today, I'd like to share with you the translation of the first lesson in the textbook "A Trip to the Great Wall" from the new edition of the Comprehensive English Course 2. This lesson introduces a lively and memorable trip to the Great Wall of China.As we begin our journey, our eyes are immediately met with breathtaking views. Stretching over 21,000 kilometers across China, the Great Wall is an emblem of ancient Chinese civilization. Its grandeur and historical significance attract millions of tourists each year.The Great Wall has a history of over 2,000 years. Construction first began during the Qin Dynasty and continued throughout various dynasties. It served as a key defense system, protecting the Chinese empire from invasions. In the present day, it stands as a prominent symbol of China's rich cultural heritage.During our trip, we had the opportunity to walk along the Wall and immerse ourselves in its history. The feeling of standing on the Wall, gazing at the vast landscape, was truly indescribable. It was as if we were transported back in time, experiencing the challenges and triumphs of the past.The Wall's architectural features are truly remarkable. Its solid structure is built from layers of brick, stone, and other materials, providing stability and strength. Along the Wall, there are watchtowers strategically located to ensure timely communication and defense.Not only is the Great Wall a historical treasure, but it also offers stunning natural scenery. The Wall winds through mountains, valleys, and even deserts, offering breathtaking views at every turn. We were captivated by the beauty of the surrounding landscape, enhancing our overall experience.Furthermore, the Great Wall has become a symbol of unity and perseverance. From the hands of countless laborers who dedicated their lives to its construction, the Wall stands as a testament to human determination and ambition. It serves as a reminder that through unity and hard work, great achievements can be made.In conclusion, our trip to the Great Wall was a truly unforgettable experience. It allowed us to appreciate the historical significance, magnificent architecture, and natural beauty that the Wall embodies. It taught us valuable lessons about the importance of preserving and cherishing our cultural heritage. It is an experience we will cherish for a lifetime.Through the translation of this lesson, we hope to inspire more people to explore and appreciate the rich history and cultural wonders of the Great Wall. Let us continue to celebrate and protect this iconic symbol of China's greatness.。
全新版大学英语(第二版)unit4课文翻译
Unit4爱因斯坦是外星人么艾伯特爱因斯坦被搞得筋疲力尽.连续第三个晚上，他的宝贝儿子汉斯，哭泣，让家人清醒直到黎明。
当艾伯特终于睡着了是时候起床去工作。
他不能跳过一天。
他需要工作来养活家人。
他轻快地走到专利局，在那里他是一个“技术专家，第三级，”艾伯特担心他的母亲。
她越来越虚弱，她不赞成他与米列娃结婚，关系紧张。
艾伯特看了一眼路过商店的橱窗。
他的头发是一个烂摊子；他忘了梳一遍。
工作。
家庭。
使收支平衡。
艾伯特感受到任何年轻的丈夫和父亲所有的压力和责任的。
放松，他彻底改变了物理学。
1905年，在年龄26时，四年前他找到了工作作为一个物理学教授，爱因斯坦出版了五个最重要的论文在科学史”——所有在他的空余时间写的。
”他证明了原子和分子的存在。
1905年之前，科学家们不清楚那些。
他认为光是小块（后来被称为“光子”），从而奠定了量子力的学基础，。
他描述了他的狭义相对论理论：空间和时间是同一个织物的线，他提出那是可弯曲，拉伸和扭曲的。
哦，顺便说一句，E = mc2。
在爱因斯坦之前，最后一个有这样突出创意的科学家，是艾萨克牛顿先生。
它发生在1666时，牛顿隔离自己母亲的农场去避免爆发在剑桥的瘟疫。
没有什么更好的事，他提出了他的万有引力。
几个世纪以来,历史学家称为1666牛顿的“奇迹年。
现在这些话有不同的意义:爱因斯坦和1905。
联合国已经宣布2005年“世界物理年“庆祝爱因斯坦“奇迹年的100周年。
现代流行文化吧爱因斯坦画一个bushy-haired superthinker。
我们被告之他的想法，是不可能远远领先于其他科学家。
他一定是从其他星球来的——也许是牛顿长大的同一个星球。
“爱因斯坦不是外星人，”哈佛大学物理学家和科学史家彼得笑到。
“他是他那个时代的人。
”他所有的1905年的文件揭开问题正在被其他科学家研究，成败参半，“如果爱因斯坦没有出生的，[文件]将最终由他人以某种形式写出来”Galison说。
1905年值得注意的是，一个人撰写的五个文件的全部，加上原有的，爱因斯坦以不敬的方式得到自己的结论。
(完整版)全新版大学英语(第二版)综合教程3课文原文及翻译Until1-8较完整版
8 The next year we grew even more food and managed to get through the winter on firewood that was mostly from our own trees and only 100 gallons of heating oil. At that point I began thinking seriously about quitting my job and starting to freelance. The timing was terrible。பைடு நூலகம்By then， Shawn and Amy， our oldest girls were attending expensive Ivy League schools and we had only a few thousand dollars in the bank。 Yet we kept coming back to the same question: Will there ever be a better time？ The answer， decidedly， was no， and so —— with my employer’s blessings and half a year’s pay in accumulated benefits in my pocket -- off I went.
全新版大学英语第二版综合教程2课文译文共11页文档
BOOK2课文译文UNIT 1TextA中国式的学习风格1987年春，我和妻子埃伦带着我们18个月的儿子本杰明在繁忙的中国东部城市南京住了一个月，同时考察中国幼儿园和小学的艺术教育情况。
然而，我和埃伦获得的有关中美教育观念差异的最难忘的体验并非来自课堂，而是来自我们在南京期间寓居的金陵饭店堂。
我们的房门钥匙系在一块标有房间号的大塑料板上。
酒店鼓励客人外出时留下钥匙，可以交给服务员，也可以从一个槽口塞入钥匙箱。
由于口子狭小，你得留神将钥匙放准位置才塞得进去。
本杰明爱拿着钥匙走来走去，边走边用力摇晃着。
他还喜欢试着把钥匙往槽口里塞。
由于他还年幼，不太明白得把钥匙放准位置才成，因此总塞不进去。
本杰明一点也不在意。
他从钥匙声响中得到的乐趣大概跟他偶尔把钥匙成功地塞进槽口而获得的乐趣一样多。
我和埃伦都满不在乎，任由本杰明拿着钥匙在钥匙箱槽口鼓捣。
他的探索行为似乎并无任何害处。
但我很快就观察到一个有趣的现象。
饭店里任何一个中国工作人员若在近旁，都会走过来看着本杰明，见他初试失败，便都会试图帮忙。
他们会轻轻握牢本杰明的手，直接将它引向钥匙槽口，进行必要的重新定位，并帮他把钥匙插入槽口。
然后那位“老师”会有所期待地对着我和埃伦微笑，似乎等着我们说声谢谢——偶尔他会微微皱眉，似乎觉得我俩没有尽到当父母的责任。
我很快意识到，这件小事与我们在中国要做的工作直接相关：考察儿童早期教育（尤其是艺术教育）的方式，揭示中国人对创造性活动的态度。
因此，不久我就在与中国教育工作者讨论时谈起了钥匙槽口一事。
两种不同的学习方式我的中国同行，除了少数几个人外，对此事的态度与金陵饭店工作人员一样。
既然大人知道怎么把钥匙塞进槽口——这是走近槽口的最终目的，既然孩子还很年幼，还没有灵巧到可以独自完成要做的动作，让他自己瞎折腾会有什么好处呢？他很有可能会灰心丧气发脾气——这当然不是所希望的结果。
为什么不教他怎么做呢？他会高兴，他还能早些学会做这件事，进而去学做更复杂的事，如开门，或索要钥匙——这两件事到时候同样可以（也应该）示范给他看。
全新版大学英语（第二版）综合教程2课文翻译[高效]
![全新版大学英语（第二版）综合教程2课文翻译[高效]](https://360docs-1393072327.cos.ap-beijing.myqcloud.com/935021076d175f0e7cd184254b35eefdc8d315a2.webp)
全新版大学英语(第二版)综合教程2课文翻译参考译文第一单元美国人的立场可以概括起来这么说，我们比中国人更重视创新和自立。
我们两种文化的差异也可以从我们各自所怀的忧虑中显示出来。
中国老师担心，如果年轻人不及早掌握技艺，就有可能一辈子掌握不了；另一方面，他们并不同样地急于促进创造力的发展。
美国教育工作者则担心，除非从一开始就发展创造力，不然创造力就有可能永不再现；而另一方面，技艺可于日后获得。
但我并不想夸大其辞。
无论在过去还是在当今，中国在科学、技术和艺术革新方面都展示了巨大的创造力。
而西方的创新突破则有被夸大的危险。
如果仔细审视任何一项创新，其对以往成就的依赖则都显而易见（“站在巨人肩膀之上”的现象）。
然而，假定我这里所说的反差是成立的，而培养技艺与创造力两者都是值得追求的目标，那么重要的问题就在于：我们能否从中美两个极端中寻求一种更好的教育方式，它或许能在创造力与基本技能这两极之间获得某种较好的平衡？第二单元可是我觉得只不过是昙花一现的念头更达到物质的东西很多其他人都有。
我在1999年车展的磨损10.5万英里。
但它仍是可靠的。
我的公寓是温和的，但宁静和放松。
我的衣服很适合我的工作，这主要是室外。
我最小的电脑需要能够得到满足图书馆看书。
在我没有什么，尽管我不觉得差。
为什么呢？我已经享受了53年来特别好的健康。
这不仅是因为我一直在生病，免费的，它的，我觉得充满活力和精神。
其实对我来说是锻炼的乐趣。
我期待着长，活力散步。
我还喜欢“我能”的态度如下。
我也珍惜创造力的礼物。
当我写一个美丽的诗行，或者编造一个笑话，发痒的人，我感到充实进去。
我不断惊讶的见解，通过我的写作过程中来。
而且有这么多有趣的作家朋友聊天，是享受我的主要来源之一。
第四单元以前我从未意识到日常的生活起居是多么重要，如穿戴整齐去上班，按时就寝。
以前我从未想过自己会那么依赖同事做伴。
我开始理解为什么长时间的失业会那么伤人，为什么一个人的生活缺少了外部支持的日常计划就会导致吸毒、犯罪、自杀率的增长。
全新版大学英语(第二版)综合教程1课文原文课文翻译
Unit 1 Growing UPPart Text A Writing for MyselfⅡWhen we are writing we are often told to keep our readers in mind, to shape what we say to fit their tastes and interests. But there is one reader in particular who should not be forgotten. Can you guess who? Russell Baker surprised himself and everyone else when he discovered the answer.我们写作时常常被告诫，脑子里要有读者，笔者所云一定要符合读者的口味和兴趣。
但有一位读者特别不该忘记。
你能猜出是谁吗？当拉塞尔·贝克找到这个问题的答案时，他自己和别人都感到大为惊讶。
Writing for MyselfRussell Baker1The idea of becoming a writer had come to me off and on since my childhood in Belleville, but it wasn't until my third year in high school that the possibility took hold. Until then I've been bored by everything associated with English courses. I found English grammar dull and difficult. I hated the assignments to turn out long, lifeless paragraphs that were agony for teachers to read and for me to write.为自己而写拉塞尔·贝克从孩提时代，我还住在贝尔维尔时，我的脑子里就断断续续地转着当作家的念头，但直等到我高中三年级，这一想法才有了实现的可能。
(2020年7月整理)新编大学英语第二版Book2 Unit1-4课文原文加翻译.doc
新编大学英语第二版Book2 Unit1-4课文原文加翻译Unit 1 A Good Heart to Lean OnMore than I realized, Dad has helped me keep my balance.[1] When I was growing up, I was embarrassed to be seen with my father. He was severely crippled and very short, and when we would walk together, his hand on my arm for balance , people would stare. I would be ashamed of the unwanted attention. If he ever noticed or was bothered, he never let on.[2] It was difficult to coordinate our steps—his halting, mine impatient —and because of that, we didn't say much as we went along. But as we started out, he always said, “You set the pace. I will try to adjust to you. ”[3] Our usual walk was to or from the subway, which was how he got to work. He went to work sick, and despite nasty weather. He almost never missed a day, and would make it to the office even if others could not. It was a matter of pride for him.[4] When snow or ice was on the ground, it was impossible for him to walk, even with help. At such times my sisters or I would pull him through the streets of Brooklyn , N.Y., on a child's sleigh to the subway entrance. Once there, he would cling to the handrail until he reached the lower steps that the warmer tunnel air kept ice-free.In Manhattan the subway station was the basement of his office building, and he would not have to go outside again until we met him in Brooklyn on his way home..[5] When I think of it now, I marvel at how much courage it must have taken for a grown man to subject himself to such indignity and stress. And I marvel at how he did it—without bitterness or complaint.[6] He never talked about himself as an object of pity, nor did he show any envy of the more fortunate or able . What he looked for in others was a “good heart”, and if he found one, the owner was good enough for him.[7] Now that I am older, I believe that is a proper standard by which to judge people , even though I still don't know precisely what a “good heart” is. But I know the times I don't have one myself.[8] Unable to engage in many activities, my father still tried to participate in some way. When a local baseball team found itself without a manager , he kept it going. He was a knowledgeable baseball fan and often took me to Ebbets Field to see the Brooklyn Dodgers play. He liked to go to dances and parties, where he could have a good time just sitting and watching.[9] On one memorable occasion a fight broke out at a beach party, with everyone punching and shoving .He wasn't content to sit and watch, but he couldn't stand unaided on the soft sand. In frustration he began to shout, “I'll fight anyone who will sit down with me! I'll fight anyone who will sit down with me! ”[10] Nobody did. But the next day people kidded him by saying it was the first time any fighter was urged to take a dive even before the bout began.[11] I now know he participated in some things vicariously through me, his only son. When I played ball (poorly), he “played” too. When I joined the Navy, he “joined” too. And when I came home on leave, he saw to it that I visited his office. Introducing me, he was really saying, “This is my son, but it is also me, and I could have done this, too, if things had been different. ” Those wordswere never said aloud.[12] He has been gone many years now, but I think of him often. I wonder if he sensed my reluctance to be seen with him during our walks. If he did, I am sorry I never told him how sorry I was, how unworthy I was, how I regretted it. I think of him when I complain about trifles, when I am envious of anoth er's good fortune, when I don't have a “good heart”.[13] At such times I put my hand on his arm to regain my balance, and say, “You set the pace. I will try to adjust to you.” ( 703 words)【译文】善良之心，久久相依1 随着我渐渐长大，当别人看见我和爸爸在一起，我会觉得很尴尬。
全新版大学英语第二版第二册Unit2课文译文及参考答案
一个疑惑不解的小孩提的一个问题促使卡尔·格林思考：虽然他没有贵重的财物，但他在其他许多方面却是富有的。
富足的一生卡尔·格林首次面对这个问题，是在2003年12月初，我第一次为救世军摇铃募捐的时候。
当时我就站在沃尔玛商场入口处门外，对每一位向我的红壶里投入捐款的人都报以一声“谢谢”和一个微笑。
一位穿着整洁的妇人牵着她的幼子向放壶的台子走过来。
她在钱包里摸着找钱时，孩子抬头看了我一眼，问我：“你穷吗？”当时他眼里充满疑惑和好奇，时至今日仍历历在目。
“嗯，”我结结巴巴，边想边回答，“我比有些人拥有的多，但比其他人拥有的少。
”母亲因为孩子问了一个在社交上不该问的问题，训斥了他一顿，他俩便匆匆地赶去购物。
但是孩子的问题却一直在我的心头挥之不去。
我从不认为自己“穷”，但有些事我不可否认。
每当我填1040税务申报表时，我都属于收入最低的档次之一。
在过去的三十五年中，我只出去度过一次假。
我的电视机是黑白的，还是八年前别人送给我的。
然而，想要得到其他那么多人都有的物质的东西，对我来说，只不过是转瞬即逝的念头而已。
我的汽车是1999年的产品，到现在开了十万五千英里，已经很破很旧了，但是它依然可靠。
我的住房不大，但是很安静，住着挺舒心。
我的衣服很适合于我的工作，主要都在户外。
我对计算机的很少的需求，可以在图书馆得到解决。
尽管有些东西我没有，我并不感到贫穷。
这是为什么？五十三年来我一直非常健康。
我不但不生病，而且精力充沛，情绪饱满。
锻炼对我而言是确确实实的快事，我乐意长距离步行，越走越有劲。
我喜爱步行后随之产生的一种“什么都干得了”的心态。
我还十分珍惜我的创作才能。
当我写出美丽的诗句或编造出能把人逗乐的笑话时，我内心感到很富有。
通过写作而获得的洞察力，不断地令我惊奇。
而与那么多写作朋友交谈，是我乐趣的主要源泉之一。
但是在我生活中，有一个重要方面我并不那么富有。
在一个对物资财富的追求投入如此之多心力的社会中，我觉得很不自在。
全新版大学英语第二版综合教程2课文译文
Unit 1One way of summarizing the American position is to state that we value originality and independence more than the Chinese do. The contrast between our two cultures can also be seen in terms of the fears we both harbor. Chinese teachers are fearful that if skills are not acquired early, they may never be acquired; there is, on the other hand, no comparable hurry to promote creativity. American educators fear that unless creativity has been acquired early, it may never emerge; on the other hand, skills can be picked up later.美国人的立场可以概括起来这么说，我们比中国人更重视创新和自立。
我们两种文化的差异也可以从我们各自所怀的忧虑中显示出来。
中国老师担心，如果年轻人不及早掌握技艺，就有可能一辈子掌握不了；另一方面，他们并不同样地急于促进创造力的发展。
美国教育工作者则担心，除非从一开始就发展创造力，不然创造力就有可能永不再现；而另一方面，技艺可于日后获得。
However, I do not want to overstate my case. There is enormous creativity to be found in Chinese scientific, technological and artistic innovations past and present. And there is a danger of exaggerating creative breakthroughs in the West. When any innovation is examined closely, its reliance on previous achievements is all too apparent (the "standing on the shoulders of giants" phenomenon).但我并不想夸大其辞。
全新版大学英语第二版综合教程2课文译文
Unit 1One way of summarizing the American position is to state that we value originality and independence more than the Chinese do. The contrast between our two cultures can also be seen in terms of the fears we both harbor. Chinese teachers are fearful that if skills are not acquired early, they may never be acquired; there is, on the other hand, no comparable hurry to promote creativity. American educators fear that unless creativity has been acquired early, it may never emerge; on the other hand, skills can be picked up later.美国人的立场可以归纳起来这么说，咱们比中国人更重视创新和自立。
咱们两种文化的不同也可以从咱们各自所怀的忧虑中显示出来。
中国老师担忧，若是年轻人不及早掌握技艺，就有可能一生掌握不了；另一方面，他们并非一样地急于增进创造力的发展。
美国教育工作者则担忧，除非从一开始就发展创造力，不然创造力就有可能永再也不现；而另一方面，技艺可于往后取得。
However, I do not want to overstate my case. There is enormous creativity to be found in Chinese scientific, technological and artistic innovations past and present. And there is a danger of exaggerating creative breakthroughs in the West. When any innovation is examined closely, its reliance on previous achievements is all too apparent (the "standing on the shoulders of giants" phenomenon).但我并非想夸大其辞。