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Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving PDF

153 Pages·2010·1.51 MB·English
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STREET-FIGHTING M ATH E MATICS THE ART OF EDUCATED GUESSING AND OPPORTUNISTIC PROBLEM SOLVING SANJOY MAHAJAN F OREWORD BY CARVER A. MEAD Street-Fighting Mathematics Street-Fighting Mathematics The Art of Educated Guessing and Opportunistic Problem Solving Sanjoy Mahajan Foreword by Carver A. Mead The MIT Press Cambridge, Massachusetts London, England (cid:2)C 2010 by Sanjoy Mahajan Foreword (cid:2)C 2010 by Carver A. Mead Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving by Sanjoy Mahajan (author), Carver A. Mead (foreword), and MIT Press (publisher) is licensed under the Creative Commons Attribution–Noncommercial–Share Alike 3.0 United States License. A copy of the license is available at http://creativecommons.org/licenses/by-nc-sa/3.0/us/ For information about special quantity discounts, please email [email protected] Typeset in Palatino and Euler by the author using ConTEXt and PDFTEX Library of Congress Cataloging-in-Publication Data Mahajan, Sanjoy, 1969– Street-fighting mathematics : the art of educated guessing and opportunistic problem solving / Sanjoy Mahajan ; foreword by Carver A. Mead. p. cm. Includes bibliographical references and index. ISBN 978-0-262-51429-3 (pbk. : alk. paper) 1. Problem solving. 2. Hypothesis. 3. Estimation theory. I. Title. QA63.M34 2010 510—dc22 2009028867 Printed and bound in the United States of America 10 9 8 7 6 5 4 3 2 1 For Juliet Brief contents Foreword xi Preface xiii 1 Dimensions 1 2 Easy cases 13 3 Lumping 31 4 Pictorial proofs 57 5 Taking out the big part 77 6 Analogy 99 Bibliography 123 Index 127 Contents Foreword xi Preface xiii 1 Dimensions 1 1.1 Economics: The power of multinational corporations 1 1.2 Newtonian mechanics: Free fall 3 1.3 Guessing integrals 7 1.4 Summary and further problems 11 2 Easy cases 13 2.1 Gaussian integral revisited 13 2.2 Plane geometry: The area of an ellipse 16 2.3 Solid geometry: The volume of a truncated pyramid 17 2.4 Fluid mechanics: Drag 21 2.5 Summary and further problems 29 3 Lumping 31 3.1 Estimating populations: How many babies? 32 3.2 Estimating integrals 33 3.3 Estimating derivatives 37 3.4 Analyzing differential equations: The spring–mass system 42 3.5 Predicting the period of a pendulum 46 3.6 Summary and further problems 54 4 Pictorial proofs 57 4.1 Adding odd numbers 58 4.2 Arithmetic and geometric means 60 4.3 Approximating the logarithm 66 4.4 Bisecting a triangle 70 4.5 Summing series 73 4.6 Summary and further problems 75

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In problem solving, as in street fighting, rules are for fools: do whatever works—don't just stand there! Yet we often fear an unjustified leap even though it may land us on a correct result. Traditional mathematics teaching is largely about solving exactly stated problems exactly, yet life often
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