Today complex numbers have such widespread practical usefrom electrical engineering to aeronauticsthat 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 rootsnow 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.
About the 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).
Table of Contents
List of Illustrations xi
Preface to the Paperback Edition xiii
Introduction: How Heron and Diophantus of Alexandria overlooked imaginary numbers nearly 2,000 years ago 3
Chapter 1 The Puzzles of Imaginary Numbers 8
Chapter 2 A First Try at Understanding the Geometry of √-1 31
Chapter 3 The Puzzles Start to Clear 48
Chapter 4 Using Complex Numbers 84
Chapter 5 More Uses of Complex Numbers 105
Chapter 6 Wizard Mathematics 142
Chapter 7 The Nineteenth Century, Cauchy, and the Beginning of Complex Function Theory 187
A The Fundamental Theorem of Algebra 227
B The Complex Roots of a Transcendental Equation 230
C (√-1)√-1 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
Name Index 261
Subject Index 265
What People are Saying About This
"Dispelling many common myths about the origin of the mystic 'imaginary' unit, Nahin tells the story of i from a historic as well as human perspective. His enthusiasm and informal style easily catch on to the reader. An Imaginary Tale is a must for anyone curious about the evolution of our number concept."Eli Maor, author of Trigonometric Delights, e: The Story of a Number, and To Infinity and Beyond