A Certain Ambiguity: A Mathematical Novel - Softcover

Gaurav Suri & Hartosh Singh Bal

Softcover

ISBN 10: 0691146012 ISBN 13: 9780691146010

Verlag: Princeton Univers. Press, 2010

Inhaltsangabe

While taking a class on infinity at Stanford in the late 1980s, Ravi Kapoor discovers that he is confronting the same mathematical and philosophical dilemmas that his mathematician grandfather had faced many decades earlier--and that had landed him in jail. Charged under an obscure blasphemy law in a small New Jersey town in 1919, Vijay Sahni is challenged by a skeptical judge to defend his belief that the certainty of mathematics can be extended to all human knowledge--including religion. Together, the two men discover the power--and the fallibility--of what has long been considered the pinnacle of human certainty, Euclidean geometry. As grandfather and grandson struggle with the question of whether there can ever be absolute certainty in mathematics or life, they are forced to reconsider their fundamental beliefs and choices. Their stories hinge on their explorations of parallel developments in the study of geometry and infinity--and the mathematics throughout is as rigorous and fascinating as the narrative and characters are compelling and complex. Moving and enlightening, A Certain Ambiguity is a story about what it means to face the extent--and the limits--of human knowledge.

Über diesen Titel

Über die Autorin bzw. den Autor

Gaurav Suri, a partner at a global management consulting firm in San Francisco, holds a master's degree in mathematics from Stanford. Hartosh Singh Bal, a leading independent journalist in New Delhi, holds a master's degree in mathematics from New York University.

Von der hinteren Coverseite

"A Certain Ambiguity is an amazing narrative that glows with a vivid sense of the beauty and wonder of mathematics."--Martin Gardner

"This is a truly captivating thriller that will take you on a whirlwind tour to infinity--and beyond. But be warned: once you start reading, you won't be able to put it aside until finished! A masterly-told story that weaves together criminal law, ancient and modern history, a young man's quest to know his deceased grandfather-and some highly intriguing mathematics."--Eli Maor, author of e: the Story of a Number and The Pythagorean Theorem: A 4,000-Year History

"This rich and engaging novel follows the path that leads one young person to become a professional mathematician. By deftly blending the young man's story with mathematical ideas and historical developments in the subject, the authors succeed brilliantly in taking the reader on a tour of some of the major highlights in the philosophy of mathematics. If that were not enough, the book also examines, through the minds of its characters, the natures of faith (religious and other) and truth. I am strongly thinking of building a university non-majors math course around this novel."--Keith Devlin, Stanford University, author of The Math Gene

"A Certain Ambiguity is a remarkably good effort to work through some fundamental issues in the philosophy of mathematics in the context of a novel. Crucial to the success of such a venture is creating characters and a plot that are strong enough to hold a reader's interest. Suri and Bal succeed particularly well in the story of Vijay Sahni and Judge Taylor. This well-written book will, I believe, find readers not only among mathematicians, but in a wider audience that is intrigued by mathematical meaning."--Joan Richards, Brown University

"Suri and Bal convey the beauty and elegance--as well as the fascination--of basic mathematical concepts."--Alexander Paseau, University of Oxford

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

A Certain Ambiguity

A Mathematical NovelBy Gaurav Suri Hartosh Singh Bal

PRINCETON UNIVERSITY PRESS

Copyright © 2007 Princeton University Press
All right reserved.
ISBN: 978-0-691-14601-0

Chapter One

Yesterday I found the calculator my grandfather gave me on my 12th birthday. It had fallen behind the bookcase and I saw it when I was rearranging the study. I had not thought about it for years, yet when I held it, it seemed as familiar as ever. The "I" in Texas Instruments was missing, as it had been for all but two days of its life; the buttons still made a confirming clicking sound when pressed; and when I put in some new batteries, the numbers on the LCD shone through with a blazing greenness, more extravagant than the dull grey of the modern calculator. My grandfather had intended this calculator to mark a change in my life—a new direction. As it turned out, it did mark a change, though not the one he had in mind.

I punched in the number 342 without thinking about it. It was the same number I had entered 25 years ago when the calculator was brand new.

"Want to see some number magic?" my grandfather had asked as he watched me push the buttons more or less randomly. I was sitting in his room completely taken by his birthday present, if not quite sure what to do with it. He put his notebook down, temporarily giving up on the math problem that had resisted solution since morning.

"Yes, Bauji!" I had rushed over to him.

"Enter any three-digit number in your calculator and do not let me see it." That is when I had first entered 342, the same three digits I entered now. "OK. Now enter the same number again, so you have a six-digit number," he had said. I punched in 342 again, so now I had 342342 entered in my calculator. "Now, I do not know the number you have in there, Ravi, but I do know that it is evenly divisible by 13."

By "evenly divisible" he meant that there would be no remainder. For example, 9 is evenly divisible by 3 but not by 4.

Bauji's claim seemed fantastic to me. How could he know that my number, randomly chosen and completely unknown to him, would be evenly divisible by 13? But it was! I divided 342342 by 13 and I got 26334 exactly, with no remainder.

"You're right," I said, amazed.

He wasn't finished, though. "Now, Ravi, I also know that whatever number you got after you divided by 13 is further divisible by 11." He was right once again. 26334 divided by 11 was 2394. Why was this working? "Take the number you got and divide by 7. Not only will it divide evenly, but you will be surprised at what you get." He had begun his pacing and I knew that he was as excited as I was.

I divided 2394 by 7 and I got 342! "Oh! Oh! It's the number I started with! Bauji, how did this happen?"

My grandfather just sat there, grinning at the completeness of my astonishment. "You will just have to figure that one out Ravi," he said, walking out to check the state of his tomato plants, the newest additions to his vegetable garden in the backyard. He seemed to be the only person who could grow tomatoes in New Delhi's dry summer heat.

The first thing I did was to check the divisions by hand. My hypothesis was that Bauji had rigged the new calculator somehow. But no, the numbers worked out exactly the same way when I did the long divisions by hand. Next, I decided to try this with some other three-digit numbers. The same thing worked every time. Whatever the repeated number, I could divide it evenly by 13, 11, and 7, and each time I got back to the number I started with. A few minutes of checking and rechecking convinced me that this property was true of any three-digit number. I tried doing the same thing with four-digit numbers, and it did not work any more. Neither did it work for two-digit numbers. What was going on?

I tried reversing the order. Instead of dividing first by 13, then 11, and then 7, I divided the six-digit number first by 7, then 11, and then 13. It made no difference at all. After dividing by each of those three numbers I would get back my original three-digit number. Why was this happening?

I wasn't getting anywhere and it was getting to be dinnertime. Ma had already called me twice. I knew that risking a third "I'll be right there" would be unwise, and so I put my notebook away and headed to the kitchen. I stopped thinking about the problem. And then mysteriously, out of nowhere, just when I was wondering if Ma would let me have ice cream, a new idea occurred to me. Even now, with more experience in such things, I cannot quite explain the inception of the moment of insight—the Aha moment—when out of nowhere a new idea comes, and chaos is replaced by understanding.

My first real Aha moment was at the dinner table, two days after my 12th birthday. The idea that loosened the knot was the realization that division was the reverse of multiplication, a fact I had long known, but never applied in quite the fashion this problem demanded. Instead of dividing by 13, 11, and 7 one at a time, why not multiply them together and then divide by the product all at once? Would this approach even lead to the same answer? I thought it would, and a confirming example showed this to be the case. In my head I divided the number 24 first by 2 and then by 3. I got 4 as the answer. Next I divided 24 by 6 (which is 2 x 3) and got 4 as well. So, it should work to divide the six-digit number by the product 13 x 11 x 7. I did some more examples on a paper towel just to be sure. It appeared that I might be onto something.

Ma noticed that I was completely distracted. "Ravi, what's going on? Why aren't you eating?" But I hardly heard her. I had to find out what 13 x 11 x 7 was; perhaps that would lead me to understand why Bauji's magic worked.

"I'll be right back, Ma," I said, getting up quickly before she could react.

"No, sir, you won't. Sit here and finish your dinner." She looked like she meant it.

But my grandfather must have known what I was going through.

"Its okay Anita. Let him go." He must have been convincing enough, for I saw unwilling permission in my mother's eyes.

I ran up the stairs two at a time and fired up the calculator. 13 x 11 x 7 was ... 1001.

I knew this was terribly significant, though as yet I was not quite sure why. I tried dividing 342342 by 1001. As I expected, I got 342. But wait a minute. That must mean that the reverse is true as well. So if I multiply 342 by 1001 I should get 342342. Of course! 342 x (1001) = 342 x (1000 + 1) = 342,000 + 342 = 342342. So, taking a three-digit number and repeating it was just like multiplying it by 1001. And if you multiplied it by 1001, you could divide the six-digit number by 1001 to get the original three-digit number. What had confused me was dividing by 13, 11, and 7—but by dividing by those numbers I was in effect dividing by 1001. How simple! How could I have not seen it?

"Bauji! Bauji! I've got it!"

And he too bounded up the stairs two at a time, just as I had a few minutes before. "Tell me," he gasped. He was out of breath, but not overly so—not bad for 85.

"When you asked me to repeat the three-digit number, you were actually having me multiply it by 1001. And then you made me divide it by 1001, except you did it in three stages. So of course I ended up with the same number I started with!"

He looked at me and smiled. "Good work," he said ruffling my hair, his most characteristic gesture of affection. "I'll give you another one to think about tomorrow." When I told him I wanted another one right then,...

Details zur Ausgabe

• Verlag: Princeton Univers. Press
• Erscheinungsdatum: 2010
• Sprache: Englisch
• ISBN 10: 0691146012
• ISBN 13: 9780691146010
• Einband: Taschenbuch
• Anzahl der Seiten: 296
• Kontakt zum Hersteller: Princeton University Press; https://press.princeton.edu/books; ; 7th Floor, New South Building; PRINCETON; NJ; 8544; USA
• Verantwortliche Person: GPSR Kontakt; gpsr@libri.de; ; Europaallee 1; Bad Hersfeld; 36244; Deutschland