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

Princeton University Press, Mar 20, 2010 - Mathematics - 296 pages

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.

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This book was a lot of fun to read and is quite well written. By the end I really felt a lot more comfortable with the concept and use of imaginary numbers. The history is fascinating, of course dominated by Euler and Gauss but many other names that are known from different math classes. There is a lot of detailed algebra which requires being half-awake to follow.

### Contents

 The Puzzles of Imaginary Numbers 8 A First Try at Understanding the Geometry of 87301 31 The Puzzles Start to Clear 48 Using Complex Numbers 84 CHAPTER FIVE More Uses of Complex Numbers 105 CHAPTER SIX Wizard Mathematics 142 The Nineteenth Century Cauchy and the Beginning of Complex Function Theory 187 APPENDIXES A The Fundamental Theorem of Algebra 227
 APPENDIXES C 87301sup87301to 135 Decimal Places and How It Was Computed 235 APPENDIXES D Solving Clausens Puzzle 238 APPENDIXES E Deriving the Differential Equation for the PhaseShift Oscillator 240 APPENDIXES F The Value of the Gamma Function on the Critical Line 244 Notes 247 Name Index 261 Subject Index 265 Acknowledgments 269

 APPENDIXES B The Complex Roots of a Transcendental Equation 230