An Imaginary Tale: The Story of "i" [the square root of minus one]

Nahin is a professor of electrical engineering at the University of New Hampshire; he has also written a number of science fiction short stories. His style is far more lively and humane than a mathematics textbook while covering much of the same ground. Readers will end up with a good sense for the mathematics of i and for its applications in physics and engineering. --Mary Ellen Curtin

Product Description

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.

Of course, the book isn't all equations. There is some downright interesting history in it as well. For the most part, however, this is a book that illustrates the equations (or at least their modern counter parts) that led mathematicians to develop the concept of the square root of a negative number, eventually leading to the branch of mathematics we call today complex analysis. Having said that, I should point out that this is not a mathematics book on complex analysis [for that, a better choice is "Complex Variables," by Mark J. Ablowitz and Athanassios S. Fokas, Cambridge University Press, 1997]. The author does not develop theorems or proofs, and many of the demonstrations stretch the notion of mathematical proofs - but they are not intended to be mathematical proofs at all, but just that - demonstrations. Think of this book as a mathematicians leisurely romp through the mathematical history of root negative one, with an average of at least two or three equations on every page. The mathematics isn't advanced by any means. If you are reasonably grounded in algebra, geometry, trigonometry (and lots of it), and a little calculus (including a few differential equations) you should have no trouble at all. Plan on working through the equations, though, step by step. You won't want to miss a single "aaaahhh."

I really have only two complaints about Nahin's book, both of which are really pretty minor. The first complaint is that none of the equations are numbered. This means the author is constantly saying things like "now go back to the first equation in the last section and notice ...." I found this sometimes hard to follow, and would have appreciated a few key equations having numbers (and a box) associated with them. Another complaint is that the book has some typographical errors in some of the equations that can sometimes interfere with following the derivations.

Don't misunderstand, though. This is one of the best leisure books on mathematics I've read in a long time. The author writes clearly, has an incredible breadth of knowledge, and presents some really beautiful mathematics. It was a real let down when I finally finished, and realized how tough it was going to be finding another book to which I would look with such yearning at the end of the day for a relaxing evening of intellectual entertainment.

The book begins with the story of cubics, and how their solutions involved the square root of negative numbers. From there the book moves toward early work, or the "first try" at understanding complex numbers. There is some interesting history about Rene Descartes and John Wallis, as well as stories about Casper Wessel, Gauss, Argand, Warren, Mourey, and, of course, De Moivre.

The books first three chapters have the most history. The last four chapters offer more examples of how complex analysis has played a pivotal role in science and technology. The author offers some interesting uses of complex analysis in the solving of integrals, trigonometric identities, Kepler's laws of satellite orbits, and, of course, circuit analysis in electrical engineering.

My favorite chapter by far is chapter six, titled "wizard mathematics." It seems there was a "aaaahhh" on at least every other page. This chapter is devoted to illuminating some of the mathematical prowess of wizards such as Euler, Bernoulli, Fagnano, Cotes, Riemann, and Schellback. Plan on using up at least one highlighter on this chapter alone.

Nahin ends with a chapter on complex analysis in the nineteenth century, and Cauchy's integral formulas (there is also a brief discussion and derivation of Green's theorem). Then, as with the other chapters, Nahin gives lots of examples of what you can do with these mathematical tools, and where they can take you.

Easily one of the best books I've ever read. If you love mathematics, your library really cannot be considered complete unless this book, tattered and worn with lots of dog-eared pages and scribbles all over the margins, is on the shelf.

Duwayne Anderson September 22, 1999


Excellent, if you have the background   July 22, 2000
 32 out of 32 found this review helpful

As a few of the other reviewers have noted, this book is not for those people whose only mathematical knowledge comes from the science pages of the New York Times. For many of the chapters and proofs shown, a background consisting of at least the basics of Freshman Calculus (through power series or so) is assumed and indeed is necessary to know what is going on. If you don't have this knowledge, you'll probably become lost quite frequently. However, the fact that Nahin is writing for a more knowledgable audience is indeed quite refreshing. Because he IS willing to include the mathematics, the historical information becomes that much more interesting. Instead of just telling how imaginary numbers came about, he works through the steps of many of the exact problems that first led people to consider (and ignore) imaginary numbers. The chapter on "Wizard Mathematics" is worth the price of the book all by itself. Some of the proofs shown there are so beautiful to make one want to cry out in the joy of discovery. In addition, he includes a chapter on the applications of Complex Numbers which is also quite enlightening.


This gives you what's usually left out of textbooks   May 25, 2000
 26 out of 26 found this review helpful

If all math textbooks included the kind of material and discussions in this book, students would learn better and be more interested in math. The standard math book is a continuous list of definitions and theorems, interspersed with examples of how to do certain kinds of problems. Never does anyone explain how and why people came up with the ideas in the first place, or why such and such a theorem is important, or what kinds of problems triggered the research and investigations which have been done. "Shut up and learn it!" seems to be the universal slogan. Nahin's book can't really be used as a textbook, but it provides an all-important context for the material found in various courses all the way from Intermediate Algebra to Complex Analysis. In fact, I think the primary beneficiaries of a book like this are math teachers (like me!). The material in this book will enable me to flesh out and personalize some ideas which are found in a variety of courses which I teach. When someone asks me why anyone ever thought of having a square root of negative one, or what kinds of problems it's good for, this book will enable me to give some interesting answers. And, of course, I'll pretend that I came up with those answers all by myself!


Fantastic! Thorough, scholarly, interesting!   March 6, 1999
 14 out of 14 found this review helpful

This is an excellent, beautiful book! Just the section on Kepler's laws is worth the price of the book (hardcover to boot!)

If you like math, if you are willing to spend a bit of time understanding the wonderful results -- get it! Some calculus background needed -- nothing beyond high school.

The book goes well beyond providing a narrative on the history of "square root of -1". It actually shows in complete detail how to use "i" to do wonderful things. Along the way the author provides the important historical events and plenty of notes and references for anyone interested in getting some more. It is clear the author took his time to research and study the subject. He has presented it well, thouroghly, and in an interesting way -- without sacrificing detail!


the complex clarified   March 29, 2006
 11 out of 11 found this review helpful

I really enjoyed this book even though it was quite a bit of work for me. I found for a book of this scope, this one takes three to four times longer to read. This is no fault of the author. The text is clear, interesting and very informative. Equations are typeset in a format suited to algebraic equations in contrast to some similar books where equations are embedded in sentences. The reason for the long read time is the amount of material presented in a condensed format. Literature teachers would appreciate the economy of it all. The intermediate steps left out of some proofs are to be either trusted or calculated by the reader. To truly experience this history and gain an appreciation of the math skills, one should work through these steps.

The author gives the reader an appreciation of many key mathematicians. Complex problems are solved. One of my favorite solutions is Gamow's problem of finding the treasure without the gallows for reference. I found the problems on spacetime physics, hyperspace, and Kepler's laws especially keen. But, the total scope of material is diverse. The author covers the zeta function, the gamma function, and the relationships between pi and i. There is so much more. The book feels deceptively light in your hands, it's content dense.

The last chapter is a real reward; and I really appreciate the author's approach on complex function theory which I would have had no hope of understanding on my own. The reader is guided in integration through the complex plane with all the required steps shown for some elementary functions. I have never read a better introduction to these fundamentals. Obviously, Nahin's goal is to educate.

This is the first book by Nahin that I have read. I expect the next one to be challenging and rewarding as well.