A Occasionally, in some difficult musical compositions, there are beautiful, but easy parts - parts so simple a beginner could play them. A有时候，在一些较难的音乐作品中，存在一些很美但是也很简单的部分，这些部分简单到初学者们都可以演奏。 So it is with mathematics as well. 数学中也是如此。 There are some discoveries in advanced mathematics that do not depend on specialized knowledge, not even on algebra, geometry, or trigonometry. 在高等数学中有些发现也不需要专业知识，甚至不需要代数，几何或者三角函数知识。 Instead they may involve, at most, a little arithmetic, such as 'the sum of two odd numbers is even', and common sense. 相反，它们可能最多涉及到一些算术方法，例如，两个奇数相加的和是偶数，以及常识。 Each of the eight chapters in this book illustrates this phenomenon. 这本书中的八个章节分别论述了这种现象。 Anyone can understand every step in the reasoning. 任何人都可以理解推理中的每个步骤。

The thinking in each chapter uses at most only elementary arithmetic, and sometimes not even that. 每个章节中的推理用得最多的就是基本算术，甚至有些时候，算术都不用。 Thus all readers will have the chance to participate in a mathematical experience, to appreciate the beauty of mathematics, and to become familiar with its logical, yet intuitive, style of thinking. 因此，所有的读者都有机会参与到一场数学体验中，去领略数学之美，去熟悉有逻辑却也直观的思考方式。

B One of my purposes in writing this book is to give readers who haven't had the opportunity to see and enjoy real mathematics the chance to appreciate the mathematical way of thinking. B 我写这本书的目的之一就是给那些没有机会看到，享受到真正数学的读者们一个机会去理解数学思维方式。 I want to reveal not only some of the fascinating discoveries, but, more importantly, the reasoning behind them. 我不仅要展示一些让人着迷的发现，但更重要的是它们之后的推理。

In that respect, this book differs from most books on mathematics written for the general public. 在这个方面上，这本书和其它写给大众的数学书是不一样的。 Some present the lives of colorful mathematicians. 它们中一些书呈现的是数学家的多彩生活。 Others describe important applications of mathematics. 还有一些描述的是数学的运用。 Yet others go into mathematical procedures, but assume that the reader is adept in using algebra. 但是也有一些是关于数学程式的，不过这些书都是假定读者们很熟练的运用代数为基础的。

C I hope this book will help bridge that notorious gap that separates the two cultures: the humanities and the sciences, or should I say the right brain (intuitive) and the left brain (analytical, numerical). C我希望这本书有助于架起一座桥梁于缺口之上，这缺口分割了人文科学和社会科学，或者应该说分割了右脑（直觉方面）和左脑（分析和数字方面）。 As the chapters will illustrate, mathematics is not restricted to the analytical and numerical; intuition plays a significant role. 正如这些章节将阐释的，数学不是局限于分析和数字领域的；直觉也起着重要作用。 The alleged gap can be narrowed or completely overcome by anyone, in part because each of us is far from using the full capacity of either side of the brain. 这条所谓的缺口可以被任何人缩窄，或者完全跨越，部分原因在于我们中的任何人远没有使用我们大脑任何一半的全部能力。 To illustrate our human potential, I cite a structural engineer who is an artist, an electrical engineer who is an opera singer, an opera singer who published mathematical research, and a mathematician who publishes short stories. 为了说明我们人类的潜力，我列举了作为艺术家的结构工程师，作为歌剧演员的电子工程师，出版了数学研究专著的歌剧演员，以及一个发表了短篇故事的数学家。

D Other scientists have written books to explain their fields to non-scientists, but have necessarily had to omit the mathematics, although it provides the foundation of their theories. D其他科学家写过书像那些非科学界人士解释他们自己的研究领域，但是却必须略去数学，尽管数学给他们的理论提供了基础。 The reader must remain a tantalized spectator rather than an involved participant, since the appropriate language for describing the details in much of science is mathematics, whether the subject is expanding universe, subatomic particles, or chromosomes. 读者依然是个饱受引诱的旁观者，而不是一个涉入其中的参与者，因为在科学中很多的细节描述所用的适当的语言是数学，不管这个学科是宇宙膨胀，亚原子粒子还是染色体。 Though the broad outline of a scientific theory can be sketched intuitively, when a part of the physical universe is finally understood, its description often looks like a page in a mathematics text. 尽管科学理论的大致框架能被直观地勾勒出来，当物质世界的一部分最终被理解时，它的描述通常都看起来像数学文本中的一页。

E Still, the non-mathematical reader can go far in understanding mathematical reasoning. E然而，非数学领域的读者也能深入理解数学推理。 This book presents the details that illustrate the mathematical style of thinking, which involves sustained, step-by-step analysis, experiments, and insights. 这本书呈现细节，用来描述数学的思维风格，这涉及到持续的，一步步的分析，实验以及洞察力。 You will turn these pages much more slowly than when reading a novel or a newspaper. 你翻阅这些章节的速度会比读一个小说或者一份报纸的时侯缓慢。 It may help to have a pencil and paper ready to check claims and carry out experiments. 有支铅笔和一张纸可以帮你随时验证论断，做些实验。

F As I wrote, I kept in mind two types of readers: those who enjoyed mathematics until they were turned off by an unpleasant episode, usually around fifth grade, and mathematics aficionados, who will find much that is new throughout the book. F当我在写作的时候，我的脑海中有两种读者：那些喜欢数学的学生，一般到了五年级才会碰上有不愉快的经历让他们丧失兴趣；还有那些酷爱数学迷的人，他们也将在整本书中发现很多新东西。

This book also serves readers who simply want to sharpen their analytical skills. 这本书也是写给那些仅仅想改善他们的分析能力的读者。 Many careers, such as law and medicine, require extended, precise analysis. 很多专业领域，如法学，医学，都需要延伸的，精确的分析。 Each chapter offers practice in following a sustained and closely argued line of thought. 每个章节都提供了持续的，严谨辩证的思维。 That mathematics can develop this skill is shown by these two testimonials. 数学能培养这种技能的力证在以下两封信中有所体现。

G A physician wrote, 'The discipline of analytical thought processes [in mathematics] prepared me extremely well for medical school. G一个物理学家写过：“在数学中的分析性的思维过程训练让我以极好的状态进入了医学院。 In medicine one is faced with a problem which must be thoroughly analyzed before a solution can be found. 在医学中，一个人面临的问题在找到解决方案前是必须要彻底分析的。 The process is similar to doing mathematics.' 这个过程和做数学题很像。”

A lawyer made the same point, 'Although I had no background in law - not even one political science course - I did well at one of the best law schools. 一个律师也有同样的言论：“尽管我没有法学背景，甚至没有选修过一门政治科学课程，我也在最好的一所法学院中表现很出色。 I attribute much of my success there to having learned, through the study of mathematics, and, in particular, theorems, how to analyze complicated principles. 我把我成功的大部分原因归结于我已经通过学习数学获得的东西，尤其是怎样分析复杂原则的那些定理。 Lawyers who have studied mathematics can master the legal principles in a way that most others cannot.' 学习过数学的律师能够用某种大多数人不能做到的方式掌握法律原则。”

I hope you will share my delight in watching as simple, even naïve, questions lead to remarkable solutions and purely theoretical discoveries find unanticipated applications. 我希望你们能同我一样在看到简单甚至单纯的问题最后得到非凡的解决方案时，以及从纯粹的理论发现中获得出乎意料的应用时，会感到喜悦。