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. An Imaginary Tale can be read as an engaging history, almost a biography, of one of the most evasive and pervasive "numbers" in all of mathematics.
List of Illustrations
Ch. 1 The Puzzles of Imaginary Numbers
Ch. 2 A First Try at Understanding the Geometry of [the square root of] -1
Ch. 3 The Puzzles Start to Clear
Ch. 4 Using Complex Numbers
Ch. 5 More Uses of Complex Numbers
Ch. 6 Wizard Mathematics
Ch. 7 The Nineteenth Century, Cauchy, and the Beginning of Complex Function Theory
App. A The Fundamental Theorem of Algebra
App. B The Complex Roots of a Transcendental Equation
App. C ([the square root of] -1)[superscript [square root of] -1] to 135 Decimal Places, and How It Was Computed
Paul J. Nahin is the author of many best-selling popular math books, including Digital Dice, Chases and Escapes, Dr. Euler's Fabulous Formula, When Least Is Best, Duelling Idiots and Other Probability Puzzlers, and Mrs. Perkins's Electric Quilt (all Princeton). He is professor emeritus of electrical engineering at the University of New Hampshire.
"A book-length hymn of praise to the square root of minus one."
- Brian Rotman, Times Literary Supplement
"An Imaginary Tale is marvelous reading and hard to put down. Readers will find that Nahin has cleared up many of the mysteries surrounding the use of complex numbers."
- Victor J. Katz, Science
"[An Imaginary Tale] can be read for fun and profit by anyone who has taken courses in introductory calculus, plane geometry and trigonometry."
- William Thompson, American Scientist
"Someone has finally delivered a definitive history of this 'imaginary' number [...] A must read for anyone interested in mathematics and its history."
- D. S. Larson, Choice
"Attempting to explain imaginary numbers to a non-mathematician can be a frustrating experience [...] On such occasions, it would be most useful to have a copy of Paul Nahin's excellent book at hand."
- A. Rice, Mathematical Gazette
"Imaginary numbers! Threeve! Ninety-fifteen! No, not those kind of imaginary numbers. If you have any interest in where the concept of imaginary numbers comes from, you will be drawn into the wonderful stories of how i was discovered."
- Rebecca Russ, Math Horizons
"There will be something of reward in this book for everyone."
- R.G. Keesing, Contemporary Physics
"Nahin has given us a fine addition to the family of books about particular numbers. It is interesting to speculate what the next member of the family will be about. Zero? The Euler constant? The square root of two? While we are waiting, we can enjoy An Imaginary Tale"
- Ed Sandifer, MAA Online
"Paul Nahin's book is a delightful romp through the development of imaginary numbers."
- Robin J. Wilson, London Mathematical Society Newsletter