# An Imaginary Tale

The Story of √-1

Publication Year: 2010

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.

Published by: Princeton University Press

Series: Princeton Science Library

#### Comic Strip, Title Page, Copyright, Dedication, Note to Reader

#### Preface to the Paperback Edition

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. ...

#### Preface

Long ago, in a year so far in the past (1954) that my life then as a high school
freshman now seems like a dream, my father gave me the gift of a subscription
to a new magazine called *Popular Electronics*. He did this because he was a
scientist, and his oldest son seemed to have talents in science and mathematics
that were in danger of being subverted by the evil of science fiction. ...

#### Introduction

In 1878 a pair of brothers, the soon-to-become-infamous thieves Ahmed and Mohammed Abd er-Rassul, stumbled upon the ancient Egyptian burial site in the Valley of Kings, at Deir el-Bahri. They quickly had a thriving business going selling stolen relics, one of which was a mathematical papyrus; one of the brothers sold it to the Russian Egyptologist V. S. Golenishchev in 1893, ...

#### Chapter One: The Puzzles of Imaginary Numbers

At the end of his 1494 book *Summa de Arithmetica, 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. He declared that the
solution of the cubic equation is “as impossible at the present state of science
as the quadrature of the circle.” ...

#### Chapter Two: A First Try at Understanding the Geometry of √–1

Despite the success of Bombelli in giving formal meaning to √–1 when it appeared in the answers given by Cardan’s formula, there still lacked a physical interpretation. Mathematicians of the sixteenth century were very much tied to the Greek tradition of geometry, and they felt uncomfortable with concepts to which they could not give a geometric meaning. ...

#### Chapter Three: The Puzzles Start to Clear

More than a hundred years after Wallis’ valiant but flawed attempt to tame
complex numbers geometrically, the problem was suddenly and quite undramatically
solved by the Norwegian^{1} Caspar Wessel (1745–1818). This is
both remarkable and, ironically, understandable, when you consider that
Wessel was not a professional mathematician but a surveyor. ...

#### Chapter Four: Using Complex Numbers

In this chapter, and in the next, I will show you some specific examples or case studies of the application of complex numbers to the solution of interesting problems in mathematics and applied science. Most of the underlying theory in this chapter will be based on the elementary idea that complex numbers can represent vectors, i.e., quantities with magnitude and direction, in the complex plane. ...

#### Chapter Five More Uses of Complex Numbers

I ended the last chapter in spacetime, and this next example of using complex
numbers remains at least a little bit connected with that part of mathematical
physics. Watchers of science fiction movies are well acquainted with the idea
of *hyperspace wormholes* as spacetime shortcuts from one point to another,
paths that are traversable in less time than it takes light to make the straight
line trip; ...

#### Chapter Six: Wizard Mathematics

While it is general practice today to date the beginning of the modern theory of complex numbers from the appearance of Wessel’s paper, it is a fact that many of the particular properties of √–1 were understood long before Wessel. The Swiss genius Leonhard Euler (1707–83), for example, knew of the exponential connection to complex numbers. ...

#### Chapter Seven: The Nineteenth Century, Cauchy, and the Beginning of Complex Function Theory

With the completion of the previous chapter we have really, I think, done pretty much everything we can do with just the imaginary √–1 itself, and its extension to complex numbers. To continue on, the next logical step is to consider functions of variables that are complex valued, i.e., functions f (z) where z = x iy. ...

#### Appendix E. Deriving the Differential Equation for the Phase-Shift Oscillator

#### Acknowledgments

It is always a pleasure for me to thank those who have helped me create a new book. At the University of New Hampshire the physicist Roy Torbert enthusiastically endorsed the idea of an electrical engineer writing a math-history book; since he is my dean, that support was most important! ...

E-ISBN-13: 9781400833894

E-ISBN-10: 1400833892

Print-ISBN-13: 9780691146003

Print-ISBN-10: 0691146004

Page Count: 296

Publication Year: 2010

Edition: Princeton Library Science Edition

Series Title: Princeton Science Library

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