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Summary
Summary
More stimulating mathematics puzzles from bestselling author Paul Nahin
How do technicians repair broken communications cables at the bottom of the ocean without actually seeing them? What's the likelihood of plucking a needle out of a haystack the size of the Earth? And is it possible to use computers to create a universal library of everything ever written or every photo ever taken? These are just some of the intriguing questions that best-selling popular math writer Paul Nahin tackles in Number-Crunching . Through brilliant math ideas and entertaining stories, Nahin demonstrates how odd and unusual math problems can be solved by bringing together basic physics ideas and today's powerful computers. Some of the outcomes discussed are so counterintuitive they will leave readers astonished.
Nahin looks at how the art of number-crunching has changed since the advent of computers, and how high-speed technology helps to solve fascinating conundrums such as the three-body, Monte Carlo, leapfrog, and gambler's ruin problems. Along the way, Nahin traverses topics that include algebra, trigonometry, geometry, calculus, number theory, differential equations, Fourier series, electronics, and computers in science fiction. He gives historical background for the problems presented, offers many examples and numerous challenges, supplies MATLAB codes for all the theories discussed, and includes detailed and complete solutions.
Exploring the intimate relationship between mathematics, physics, and the tremendous power of modern computers, Number-Crunching will appeal to anyone interested in understanding how these three important fields join forces to solve today's thorniest puzzles.
Author Notes
Paul J. Nahin is the author of many best-selling popular math books, including Mrs. Perkins's Electric Quilt , Digital Dice , Chases and Escapes , Dr. Euler's Fabulous Formula , When Least Is Best , and An Imaginary Tale (all Princeton). He is professor emeritus of electrical engineering at the University of New Hampshire.
Reviews (1)
Choice Review
Nahin (emer., electrical engineering, Univ. of New Hampshire), author of numerous mathematics-related books, has assembled a challenging set of topics and problems that overlap physics, mathematics, and the use of computers. This is not a book for the uninitiated. The mathematics is at the level of an upper-division math or physics major, and some of the challenge problems are quite difficult. The book is impressive for several reasons. First, Nahin has found the right level--not too easy and not too hard. Second, the problem selections and topics are interesting and in several cases give surprising results. Finally, the book is just plain fun. This reviewer did not work through all of the chapters and challenge problems, but just after doing a few he was hooked. This book is for those that love working math/physics problems with clever or unusual solutions. The work could also be considered a survey of mathematical and computational techniques. Though, again, a fun book, it might have a place in an upper-division undergraduate course. This reviewer plans on using a few of the computational examples from Newtonian gravity in his advanced mechanics course. Summing Up: Highly recommended. Upper-division undergraduates, graduate students, and researchers/faculty. E. Kincanon Gonzaga University
Table of Contents
| Introduction | p. x |
| Chapter 1 Feynman Meets Fermat | p. 1 |
| 1.1 The Physicist as Mathematician | p. 1 |
| 1.2 Fermat's Last Theorem | p. 2 |
| 1.3 "Proof" by Probability | p. 3 |
| 1.4 Feynman's Double Integral | p. 6 |
| 1.5 Things to come | p. 10 |
| 1.6 Challenge Problems | p. 11 |
| 1.7 Notes and References | p. 13 |
| Chapter 2 Just for Fun: Two Quick Number-Crunching Problems | p. 16 |
| 2.1 Number-Crunching in the Past | p. 16 |
| 2.2 A Modern Number-Cruncher | p. 20 |
| 2.3 Challenge Problem | p. 25 |
| 2.4 Notes and References | p. 25 |
| Chapter 3 Computers and Mathematical Physics | p. 27 |
| 3.1 When Theory Isn't Available | p. 27 |
| 3.2 The Monte Carlo Technique | p. 28 |
| 3.3 The Hot Plate Problem | p. 34 |
| 3.4 Solving the Hot Plate Problem with Analysis | p. 38 |
| 3.5 Solving the Hot Plate Problem by Iteration | p. 44 |
| 3.6 Solving the Hot Plate Problem with the Monte Carlo Technique | p. 50 |
| 3.7 ENIAC and MANIAC-I: the Electronic Computer Arrives | p. 55 |
| 3.8 The Fermi-Pasta-Ulam Computer Experiment | p. 58 |
| 3.9 Challenge Problems | p. 73 |
| 3.10 Notes and References | p. 74 |
| Chapter 4 The Astonishing Problem of the Hanging Masses | p. 82 |
| 4.1 Springs and Harmonic Motion | p. 82 |
| 4.2 A Curious Oscillator | p. 87 |
| 4.3 Phase-Plane Portraits | p. 96 |
| 4.4 Another (Even More?) Curious Oscillator | p. 99 |
| 4.5 Hanging Masses | p. 104 |
| 4.6 Two Hanging Masses and the Laplace Transform | p. 108 |
| 4.7 Hanging Masses and MATLAB | p. 113 |
| 4.8 Challenge Problems | p. 124 |
| 4.9 Notes and References | p. 124 |
| Chapter 5 The Three-Body Problem and Computers | p. 131 |
| 5.1 Newton's Theory of Gravity | p. 131 |
| 5.2 Newton's Two-Body Solution | p. 139 |
| 5.3 Euler's Restricted Three-Body Problem | p. 147 |
| 5.4 Binary Stars | p. 155 |
| 5.5 Euler's Problem in Rotating Coordinates | p. 166 |
| 5.6 Poincaré and the King Oscar II Competition | p. 177 |
| 5.7 Computers and the Pythagorean Three-Body Problem | p. 184 |
| 5.8 Two Very Weird Three-Body Orbits | p. 195 |
| 5.9 Challenge Problems | p. 205 |
| 5.10 Notes and References | p. 207 |
| Chapter 6 Electrical Circuit Analysis and Computers | p. 218 |
| 6.1 Electronics Captures a Teenage Mind | p. 218 |
| 6.2 My First Project | p. 220 |
| 6.3 "Building" Circuits on a Computer | p. 230 |
| 6.4 Frequency Response by Computer Analysis | p. 234 |
| 6.5 Differential Amplifiers and Electronic Circuit Magic | p. 249 |
| 6.6 More Circuit Magic: The Inductor Problem | p. 260 |
| 6.7 Closing the Loop: Sinusoidal and Relaxation Oscillators by Computer | p. 272 |
| 6.8 Challenge Problems | p. 278 |
| 6.9 Notes and References | p. 281 |
| Chapter 7 The Leapfrog Problem | p. 288 |
| 7.1 The Origin of the Leapfrog Problem | p. 288 |
| 7.2 Simulating the Leapfrog Problem | p. 290 |
| 7.3 Challenge Problems | p. 296 |
| 7.4 Notes and References | p. 296 |
| Chapter 8 Science Fiction: When Computers Become Like Us | p. 297 |
| 8.1 The Literature of the Imagination | p. 297 |
| 8.2 Science Fiction "Spoofs" | p. 300 |
| 8.3 What If Newton Had Owned a Calculator? | p. 305 |
| 8.4 A Final Tale: the Artificially Intelligent Computer | p. 314 |
| 8.5 Notes and References | p. 324 |
| Chapter 9 A Cautionary Epilogue | p. 328 |
| 9.1 The Limits of Computation | p. 328 |
| 9.2 The Halting Problem | p. 330 |
| 9.3 Notes and References | p. 333 |
| Appendix (FPU Computer Experiment MATLAB Code) | p. 335 |
| Solutions to the Challenge Problems | p. 337 |
| Acknowledgments | p. 371 |
| Index | p. 373 |
| Also | p. 377 |