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

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Contents

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pp. ix-x

List of Illustrations

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pp. xi-xii

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Preface to the Paperback Edition

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pp. xiii-xx

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. ...

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Preface

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pp. xxi-xiv

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. ...

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Introduction

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pp. 3-7

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, ...

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Chapter One: The Puzzles of Imaginary Numbers

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pp. 8-30

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.” ...

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

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pp. 31-47

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. ...

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Chapter Three: The Puzzles Start to Clear

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pp. 48-83

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 Norwegian1 Caspar Wessel (1745–1818). This is both remarkable and, ironically, understandable, when you consider that Wessel was not a professional mathematician but a surveyor. ...

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Chapter Four: Using Complex Numbers

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pp. 84-104

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. ...

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Chapter Five More Uses of Complex Numbers

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pp. 105-141

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; ...

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Chapter Six: Wizard Mathematics

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pp. 142-186

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. ...

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

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pp. 187-226

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 A. The Fundamental Theorem of Algebra

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pp. 227-229

Appendix B. The Complex Roots of a Transcendental Equation

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pp. 230-234

Appendix C. (√–1)(√–1) to 135 Decimal Places, and How It Was Computed

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pp. 235-237

Appendix D. Solving Clausen’s Puzzle

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pp. 238-239

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

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pp. 240-243

Appendix F. The Value of the Gamma Function on the Critical Line

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pp. 244-246

Notes

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pp. 247-260

Name Index

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pp. 261-264

Subject Index

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pp. 265-268

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Acknowledgments

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p. 269

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! ...