An Imaginary Tale: The Story of "i" [the square root of minus one]

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.Paul J. Nahin is professor emeritus of electrical engineering at the Universit Cover Page 1 Title: An Imaginary Tale 2 Copyright 6 ISBN 0691127980 6 Contents 11 List of Illustrations 13 Preface to the Paperback Edition 15 Preface 23 CHAPTER ONE: The Puzzles of Imaginary Numbers 34 CHAPTER TWO: A First Try at Understanding the Geometry of Sqrt(-1) 57 CHAPTER THREE: The Puzzles Start to Clear 74 CHAPTER FOUR: Using Complex Numbers 110 Complex numbers as vectors 110 Doing geometry with complex vector algebra 111 The Gamow problem 118 Solving Leonardo’s recurrence 120 Imaginary time in spacetime physics 123 CHAPTER FIVE More Uses of Complex Numbers 131 Taking a shortcut through hyperspace with complex functions 131 Maximum walks in the complex plane 133 Kepler’s laws and satellite orbits 135 Complex numbers in electrical engineering 151 CHAPTER SIX Wizard Mathematics 168 The hyperbolic functions 193 CHAPTER SEVEN: The Nineteenth Century, Cauchy, and the Beginning of Complex Function Theory 213 Introduction. 213 Augustin-Louis Cauchy 214 Analytic functions and the Cauchy-Riemann equations 216 Cauchy’s first result 221 Cauchy’s second integral theorem 234 Kepler’s third law: the final calculation 244 Epilog: what came next 247 APPENDIXES A. The Fundamental Theorem of Algebra 253 APPENDIXES B. The Complex Roots of a Transcendental Equation 256 APPENDIXES C. Sqrt(-1)^(Sqrt(-1) to 135 Decimal Places, and How It Was Computed 261 APPENDIXES D. Solving Clausen’s Puzzle 264 APPENDIXES E. Deriving the Differential Equation for the Phase-Shift Oscillator 266 APPENDIXES F. The Value of the Gamma Function on the Critical Line 270 Notes 273 Name Index 287 A 287 B 287 C 287 D 287 E 287 F 287 G 288 H 288 J 288 K 288 L 288 M 288 N 288 O 288 P 288 R 289 S 289 T 289 U 289 V 289 W 289 Subject Index 291 A 291 B 291 C 291 D 291 E 291 F 292 G 292 H 292 I 292 J 292 K 292 L 292 M 292 N 292 O 293 P 293 Q 293 R 293 S 293 T 293 V 293 W 293 X 293 Z 293 Acknowledgments 295 Book Flaps 296 Back Page 297 9780691127989

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.

Cover Page......Page 1 Title: An Imaginary Tale......Page 2 ISBN 0691127980......Page 6 Contents......Page 11 List of Illustrations......Page 13 Preface to the Paperback Edition......Page 15 Preface......Page 23 CHAPTER ONE: The Puzzles of Imaginary Numbers......Page 34 CHAPTER TWO: A First Try at Understanding the Geometry of Sqrt(-1)......Page 57 CHAPTER THREE: The Puzzles Start to Clear......Page 74 Complex numbers as vectors......Page 110 Doing geometry with complex vector algebra......Page 111 The Gamow problem......Page 118 Solving Leonardo’s recurrence......Page 120 Imaginary time in spacetime physics......Page 123 Taking a shortcut through hyperspace with complex functions......Page 131 Maximum walks in the complex plane......Page 133 Kepler’s laws and satellite orbits......Page 135 Complex numbers in electrical engineering......Page 151 CHAPTER SIX Wizard Mathematics......Page 168 The hyperbolic functions......Page 193 Introduction.......Page 213 Augustin-Louis Cauchy......Page 214 Analytic functions and the Cauchy-Riemann equations......Page 216 Cauchy’s first result......Page 221 Cauchy’s second integral theorem......Page 234 Kepler’s third law: the final calculation......Page 244 Epilog: what came next......Page 247 APPENDIXES A. The Fundamental Theorem of Algebra......Page 253 APPENDIXES B. The Complex Roots of a Transcendental Equation......Page 256 APPENDIXES C. Sqrt(-1)^(Sqrt(-1) to 135 Decimal Places, and How It Was Computed......Page 261 APPENDIXES D. Solving Clausen’s Puzzle......Page 264 APPENDIXES E. Deriving the Differential Equation for the Phase-Shift Oscillator......Page 266 APPENDIXES F. The Value of the Gamma Function on the Critical Line......Page 270 Notes......Page 273 F......Page 287 P......Page 288 W......Page 289 E......Page 291 N......Page 292 Z......Page 293 Acknowledgments......Page 295 Book Flaps......Page 296 Back Page......Page 297 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 this book, the author tells the 2000 year old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i, re-creating 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, the author weaves into this narrative entertaining historical facts, mathematical discussions, and 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 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, re-creating the baffling mathematical problems that conjured it up and the colorful characters who tried to solve them. Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts, mathematical discussions, and the application of complex numbers and functions to important problems. ch. 1. ch. 1. -- ch. 2. ch. 1. -- ch. 2. -- ch. 3. ch. 1. -- ch. 2. -- ch. 3. -- ch. 4. ch. 1. -- ch. 2. -- ch. 3. -- ch. 4. -- ch. 5. ch. 1. -- ch. 2. -- ch. 3. -- ch. 4. -- ch. 5. -- ch. 6. ch. 1. -- ch. 2. -- ch. 3. -- ch. 4. -- ch. 5. -- ch. 6. -- ch. 7. ch. 1.-- ch. 2.-- ch. 3.-- ch. 4.-- ch. 5.-- ch. 6.-- ch. 7. At the end of his 1494 book Summa de Arithmetical, Geometria, Proportioni et Proportionalita, summarizing all the knowledge of that time on arithmetic, algebra (including quadratic equations), and trigonometry, the Franciscan friar Luca Pacioli (circa 1445-1514) made a bold assertion.