AN IMAGINARY TALE: THE STORY OF v-1

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.

Author
Paul J. Nahin is professor emeritus of electrical engineering at the University of New Hampshire and the author of many best-selling popular math books, including The Logician and the Engineer and Will You Be Alive 10 Years from Now? (both Princeton).

Contents
List of Illustrations xi
Preface to the Paperback Edition xiii
Preface xxi
Introduction 3
How Heron and Diophantus of Alexandria overlooked imaginary numbers
nearly 2,000 years ago.
CHAPTER ONE
The Puzzles of Imaginary Numbers 8
The early work of Scipione del Ferro in cubic equations, and of Niccolo
Tartaglia, Girolamo Cardano, and Rafael Bombelli on complex numbers as
the roots of cubic equations. Francoise Viète and noncomplex
trigonometric solutions to the irreducible cubic.
CHAPTER TWO
A First Try at Understanding the Geometry of 31
René Descartes’ interpretation of imaginary numbers as meaning physical
impossibility in geometric constructions, and John Wallis on physically
interpreting imaginary numbers.
CHAPTER THREE
The Puzzles Start to Clear 48
The long-lost work of Casper Wessel on the geometric interpretation of
complex numbers. as the rotation operator in the complex plane.
The easy derivation of trigonometric identities with De Moivre’s theorem.
Complex exponentials. Factoring the cyclotomic equation. The rediscovery
of Wessel’s ideas by the Abbé Adrien-Quentin Buée and Jean-Robert Argand.
Warren and Mourey rediscover Buée and Argand. William Rowan Hamilton
and complex numbers as couples of real numbers. Carl Friedrich Gauss.
CHAPTER FOUR
Using Complex Numbers 84
Complex numbers as vectors. Doing geometry with complex vector algebra.
The Gamow problem. Solving Leonardo’s recurrence. Imaginary time in
spacetime physics.
CHAPTER FIVE
More Uses of Complex Numbers 105
Taking a shortcut through hyperspace with complex functions. Maximum walks
in the complex plane. Kepler’s laws and satellite orbits. Complex numbers in
electrical engineering.
CHAPTER SIX
Wizard Mathematics 142
The mathematical gems of Leonhard Euler, John Bernoulli, Count Fagnano,
Roger Cotes, and Georg Riemann. Many-valued functions. The hyperbolic
functions. Karl Schellbach’s method of using to calculate
.
Euler again, using complex numbers to calculate real integrals, and the
gamma and zeta functions.
CHAPTER SEVEN
The Nineteenth Century, Cauchy, and the Beginning
of Complex Function Theory 187
Introduction. Augustin-Louis Cauchy. Analytic functions and the CauchyRiemann
equations. Cauchy’s first result. Cauchy’s second integral theorem.
Kepler’s third law: the final calculation. Epilog: what came next.
APPENDIXES 227
A. The Fundamental Theorem of Algebra 227
B. The Complex Roots of a Transcendental Equation 230
C. to 135 Decimal Places,
and How It Was Computed 235
D. Solving Clausen’s Puzzle 238
E. Deriving the Differential Equation for the
Phase-Shift Oscillator 240
F. The Value of the Gamma Function on the Critical Line 244
Notes 247
Name Index 261
Subject Index 265
Acknowledgments 269