# An Imaginary Tale: The Story of [the Square Root of Minus One]

## Synopsis

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 asi. 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 fori. In the first century, the mathematician-engineer Heron of Alexandria encountered 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 notoriousifinally 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.

## Excerpt

The hardcover edition of this book appeared in 1998, and for the eight long years since, I have had to go to bed every single night thinking about dumb typos, stupid missing minus signs, and embarrassingly awkward phrases. Each has been like a sliver of wood under a fingernail. None of them was life threatening, of course, but together they made my intellectual life less than perfect. For the first six months after the book first came out I’d wake up at night, muttering to myself much as the eccentric Victorian electrical physicist Oliver Heaviside had done when he was approaching his sixties: “I must be getting as stupid as an owl.” Those were interesting times—the life of the elderly mathematics writer can be a stressful one.

But no more! Over the years readers have generously taken the time to send me their discoveries of my oversights, and with their lists in one hand and a red pen in the other, I have joyfully pulled those irritants out of this paperback edition. Probably not all of them, alas, but still, I do feel better (a feeling that will no doubt vanish upon receiving the next e-mail or letter telling me of one or more errors I have missed). Two long, detailed letters I received in 1999 from Professors Robert Burckel (Dept. of Mathematics, Kansas State University) and David Wunsch (Dept. of Electrical Engineering, University of Massachusetts at Lowell) were particularly helpful. Princeton University Press published the sequel to this book, Doctor Euler’s Fabulous Formula, earlier this year, and so the appearance so soon after of a corrected and updated edition of An Imaginary Tale is particularly gratifying. I thank my editor at Princeton, Vickie Kearn, for providing me this opportunity to revisit the book. Now, on to what’s changed in this edition besides corrected misprints.

Actually, and perhaps somewhat inconsistently, I’ll start by mentioning some things that haven’t changed. While I carefully read everything that readers sent me, and while I admit that the vast majority of what was sent to me ended up having a significant impact on the corrections to the book, there were a few exceptions. Let me give you just two examples of this. To start, one reader took me to task rather severely for declaring (on p. 9) that del Ferro’s breakthrough idea that cracked the depressed cubic equation was “of the magician class.” No, no, no, said the reader (who claimed he was a professional mathematician), that was merely a “good idea” that lots of people could have had. Well, to that I can only say that nobody before del Ferro did have that idea. Whatever might be asserted about what mathematicians might have done, or . . .

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