Today complex numbers have such widespread practical use--from electrical engineering to aeronautics--that few people would expect the story behind their derivation to be filled with adventure and enigma. In An Imaginary Tale, Paul Nahin tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them. In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number. The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for i. In the first century, the mathematician-engineer Heron of Alexandria encountered I in a separate project, but fudged the arithmetic; medieval mathematicians stumbled upon the concept while grappling with the meaning of negative numbers, but dismissed their square roots as nonsense.
By the time of Descartes, a theoretical use for these elusive square roots--now called "imaginary numbers"--was suspected, but efforts to solve them led to intense, bitter debates. The notorious i finally won acceptance and was put to use in complex analysis and theoretical physics in Napoleonic times. Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts and mathematical discussions, including the application of complex numbers and functions to important problems, such as Kepler's laws of planetary motion and ac electrical circuits. This book can be read as an engaging history, almost a biography, of one of the most evasive and pervasive "numbers" in all of mathematics.
Rezensionen / Stimmen
One of Choice's Outstanding Academic Titles for 1999 Honorable Mention for the 1998 Award for Best Professional/Scholarly Book in Mathematics, Association of American Publishers "A book-length hymn of praise to the square root of minus one."--Brian Rotman, Times Literary Supplement "An Imaginary Tale is marvelous reading and hard to put down. Readers will find that Nahin has cleared up many of the mysteries surrounding the use of complex numbers."--Victor J. Katz, Science "[An Imaginary Tale] can be read for fun and profit by anyone who has taken courses in introductory calculus, plane geometry and trigonometry."--William Thompson, American Scientist "Someone has finally delivered a definitive history of this 'imaginary' number... A must read for anyone interested in mathematics and its history."--D. S. Larson, Choice "Attempting to explain imaginary numbers to a non-mathematician can be a frustrating experience... On such occasions, it would be most useful to have a copy of Paul Nahin's excellent book at hand."--A. Rice, Mathematical Gazette "Imaginary numbers! Threeve! Ninety-fifteen! No, not those kind of imaginary numbers. If you have any interest in where the concept of imaginary numbers comes from, you will be drawn into the wonderful stories of how i was discovered."--Rebecca Russ, Math Horizons "There will be something of reward in this book for everyone."--R.G. Keesing, Contemporary Physics "Nahin has given us a fine addition to the family of books about particular numbers. It is interesting to speculate what the next member of the family will be about. Zero? The Euler constant? The square root of two? While we are waiting, we can enjoy An Imaginary Tale."--Ed Sandifer, MAA Online "Paul Nahin's book is a delightful romp through the development of imaginary numbers."--Robin J. Wilson, London Mathematical Society Newsletter "You will definitely enjoy it. In fact it clearly reflects the the joy and delight that the author experienced when he was confronted with complex analysis during his engineering studies."--Adhemar Bultheel, European Mathematical Society
Reihe
Auflage
Sprache
Verlagsort
Editions-Typ
Produkt-Hinweis
Illustrationen
1 halftone. 47 line illus.
Maße
Höhe: 216 mm
Breite: 136 mm
Dicke: 20 mm
Gewicht
ISBN-13
978-0-691-16924-8 (9780691169248)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Klassifikation
Paul J. Nahin is professor emeritus of electrical engineering at the University of New Hampshire and the author of many best-selling popular math books, including The Logician and the Engineer and Will You Be Alive 10 Years from Now? (both Princeton).
*FrontMatter, pg. i*A Note to the Reader, pg. vii*Contents, pg. ix*Illustrations, pg. xi*Preface to the Paperback Edition, pg. xiii*Preface, pg. xxi*Introduction, pg. 1*CHAPTER ONE The Puzzles of Imaginary Numbers, pg. 8*CHAPTER TWO. A First Try at Understanding the Geometry of -1, pg. 31*CHAPTER THREE. The Puzzles Start to Clear, pg. 48*CHAPTER FOUR. Using Complex Numbers, pg. 84*CHAPTER FIVE. More Uses of Complex Numbers, pg. 105*CHAPTER SIX. Wizard Mathematics, pg. 142*CHAPTER SEVEN. The Nineteenth Century, Cauchy, and the Beginning of Complex Function Theory, pg. 187*APPENDIX A. The Fundamental Theorem of Algebra, pg. 227*APPENDIX B. The Complex Roots of a Transcendental Equation, pg. 230*APPENDIX C. ( -1)( -1) to 135 Decimal Places, and How It Was Computed, pg. 235*APPENDIX D. Solving Clausen's Puzzle, pg. 238*APPENDIX E. Deriving the Differential Equation for the Phase-Shift Oscillator, pg. 240*APPENDIX F. The Value of the Gamma Function on the Critical Line, pg. 244*Notes, pg. 247*Name Index, pg. 261*Subject Index, pg. 265*Acknowledgments, pg. 269
