An Imaginary Tale: The Story of i [the square root of minus one] (Princeton Library Science Edition)

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. Some images inside the book are unavailable due to digital copyright restrictions. A Note to the Reader Contents Illustrations Preface to the Paperback Edition Preface Introduction CHAPTER ONE The Puzzles of Imaginary Numbers CHAPTER TWO. A First Try at Understanding the Geometry of √-1 CHAPTER THREE. The Puzzles Start to Clear CHAPTER FOUR. Using Complex Numbers CHAPTER FIVE. More Uses of Complex Numbers CHAPTER SIX. Wizard Mathematics CHAPTER SEVEN. The Nineteenth Century, Cauchy, and the Beginning of Complex Function Theory APPENDIX A. The Fundamental Theorem of Algebra APPENDIX B. The Complex Roots of a Transcendental Equation APPENDIX C. (√-1) (√-1) to 135 Decimal Places, and How It Was Computed APPENDIX D. Solving Clausen’s Puzzle APPENDIX E. Deriving the Differential Equation for the Phase-Shift Oscillator APPENDIX F. The Value of the Gamma Function on the Critical Line Notes Name Index Subject Index Acknowledgments 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. 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. This book tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. Annotation Complex numbers have such widespread practical use. This title tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. It offers an account of mathematical discussions, and the application of complex numbers and functions to important problems 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.