How the Other Half Thinks: Adventures in Mathematical Reasoning


Some topics in advanced mathematics require nothing more than arithmetic and common sense. How the Other Half Thinks makes use of this phenomenon to offer both the mathematically adept and mathematical beginner eight fascinating illustrations of the mathematical way.

Each chapter starts with a question about strings made up of nothing more than two letters. This question in turn suggests thought-provoking problems. After these problems are explored and solved, the author shows ...

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Some topics in advanced mathematics require nothing more than arithmetic and common sense. How the Other Half Thinks makes use of this phenomenon to offer both the mathematically adept and mathematical beginner eight fascinating illustrations of the mathematical way.

Each chapter starts with a question about strings made up of nothing more than two letters. This question in turn suggests thought-provoking problems. After these problems are explored and solved, the author shows how the related mathematics has been applied in areas as varied as computers, cellphones, measurement of astronomical distances, and cell growth.

An experienced educator, prize-winning expositor, and researcher, Stein engagingly presents each concept. The leisurely pace allows a reader to move slowly through each chapter, omitting no steps. This approach makes complex concepts like topology, set theory, and probability accessible and exciting. The book creates a bridge across the gulf between the two cultures: humanities and the sciences.

Stein shows how the mathematical style of thinking is one that everyone can use to understand the world. This charming book speaks to both those who employ the intuitive, creative right half of the brain, and to those who rely more on the analytical, numerical left half. How the Other Half Thinks is for the novice and the skilled, the poet and the scientist, the left-brained and the right-brained. When you read this book, you are immersed in the world of mathematics, not as a spectator, but as an involved participant.

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Product Details

  • ISBN-13: 9780071373395
  • Publisher: McGraw-Hill Companies, The
  • Publication date: 6/1/2001
  • Edition number: 1
  • Pages: 177
  • Product dimensions: 5.75 (w) x 8.59 (h) x 0.74 (d)

Table of Contents

1 The Needle and the Noodle 1
2 Win by Two 19
3 The Complete Triangle 43
4 Slumps and Streaks 61
5 Thrifty Strings 73
6 Counting Ballots 97
7 Infinity 121
8 Twins 135
Epilogue: A Backward Glance 151
App. A Triangles 157
App. B Twins: A Supplement 161
For Further Reading 167
Index 173
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Occasionally, in some difficult musical compositions there are beautiful, but easy parts-- so simple a beginner could play them. So it is with mathematics as well.

There are some discoveries in advanced mathematics that do not depend on specialized knowledge, not even on algebra, geometry, or trigonometry. Instead they may involve, at most, a little arithmetic, such as "the sum of two odd numbers is even," and common sense. Each of the eight chapters in this book illustrates this phenomenon. A layperson can understand every step in the reasoning.

One of my purposes in writing this little 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. I want to reveal not only some of the fascinating discoveries, but, more important, the reasoning behind them.

In that respect, this book differs from most books on mathematics written for the general public. Some present the lives of colorful mathematicians. Others describe important applications of mathematics. Yet others go into mathematical resoning, but assume that the reader is adept in using algebra.

The thinking in each chapter uses at most only elementary arithmetic, and sometimes not even that. 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.

Each chapter begins with a simple question about strings made up of two letters, usually a and b. The opening question may lead to others, which are answered in the course of the chapter. Strings of letters may arise in many ways, for instance, from the wins and losses of a baseball team, the heads and tails of a tossed penny, or the pulses and no-pulses of an electronic stream of data. Some originate in chance events, some in carefully planned arrangements.

Typical of this approach is Chapter 5, which raises the question, "How long a string of a's and b's can you make without repeating any triplet?" This question quickly leads to more general ones, which we then settle with the aid of maps of towns and one-way roads. The answer turns out to be of use in measuring by radar the distance to a planet, in transmitting confidential information and checking the reliability of a computer. Thus, while my primary goal is to illustrate the mathematical way of thinking, if a particular result has applications, so much the better.

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, holistic) and the left brain (analytical, numerical). As the chapters will illustrate, mathematics is not restricted to the analytical and numerical; intuition plays a significant role. 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. 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.

Other scientists have written books to explain their fields to outsiders, but have necessarily had to omit the mathematics, though it provides the foundation of their theories. 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 the expanding universe, subatomic particles, or chromosomes. 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.

Still, the non-mathematical reader can go far in understanding mathematical reasoning. This book presents the details that illustrate the mathematical style of thinking, which involves sustained, step-by-step analysis, experiments, and insights. You will turn these pages much more slowly than when reading a novel or a newspaper. It may help to have pencil and paper ready to check claims and carry out experiments.

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.

This book also serves readers who simply want to sharpen their analytical skills. Many careers, such as law and medicine, require extended, precise analysis. Each chapter offers practice in following a sustained and closely argued line of thought. That mathematics can help develop this skill is shown by these two testimonials:

A physician wrote, "The discipline of analytical thought processes [in mathematics] prepared me extremely well for medical school. In medicine one is faced with a problem which must be thoroughly analyzed before a solution can be found. The process is similar to doing mathematics."

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. I attribute much of my success there to having learned, through the study of mathematics, and, in particular, theorems, how to analyze complicated principles. Lawyers who have studied mathematics can master the legal principles in a way that most others cannot."

I hope you will share my delight in watching as simple, even naive, questions lead to remarkable solutions and purely theoretical discoveries find unanticipated applications.

Sherman Stein
Davis, California
November, 2000

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  • Anonymous

    Posted February 9, 2004


    A wonderful book, even for those with considerable mathematical training. Using no more than arithmetic, Sherman Stein carefully illustrates some very powerful results from many branches of mathematics in a completely intuitive, non-rigorous way (though they could be made rigorous - good exercise for the math professionals). The results often involve generalizations from the original problem and are, therefore, of considerable mathematical interest and even beauty. Though this book requires careful thought, all of the necessary steps are very well explained. It should have great appeal to High School Seniors who want to be entranced by the power of some elementary mathematical reasoning, as well as Undergraduates (the first chapter is a wonderful generalization of Buffon's Needle problem without calculus (the normal method of solution and for a less general result), and adults who have an interest in mathematical problems. I can't recommend this book too highly.

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