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.

However, as long as you are aware of this going in, you'll be treated to an absolutely first-rate trip through the motivation, development and application of complex function theory, including several thoroughly worked out real-world examples. I was delighted by Nahin's painstaking efforts to build intuition about the meaning of complex algebra. If nothing else, drilling in the idea that i is a pi/2 rotation operator in the complex plane would give a conceptual toehold to thousands of high school students who never learn anything about complex algebra beyond formal symbol manipulation. One can easily imagine "An Imaginary Tale" as recommended reading for interested high school seniors, or for undergraduates looking for some background and motivation of ideas they are required to understand.

Make no mistake, when the author says he will not "fall to his knees in dumbstruck horror" at the sight of an integral, you should take him at his word --- this book is packed with integral calculus equations, and you're not going to get much out of it if you're not prepared to follow along with them. But I think Nahin has achieved the right blend of explaining each step versus leaving algebra to the reader (here I disagree somewhat with smlauer@mindspring.com, though I am sympathetic to his point).

I have deducted a star for exactly the reasons Duwayne Anderson and others complained about: (1) we need to have *many* of the results numbered, but unfortunately we only get a box or two in the entire text, and (2) who proofed this thing? I mean, honestly, stating Green's theorem correctly twice and then misprinting it in the section where it's proved? Randomly leaving the circle off of contour integrals? (sqrt(15)i)^2 = 15? It's fun to work the algebra that's left to the reader, but it's *tedious* to work out which results are misprinted and which aren't.

Despite these typographical problems, I can enthusiastically recommend "An Imaginary Tale" to all readers at a moderate level of mathematical sophistication who are curious about the origins, theory and application of complex analysis.

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A great book, chock full of equations  Sep 22, 1999 By Duwayne Anderson
When I first took a copy of Nahin's book off the shelf, I expected a history book operating under the usual rules that seem to dominate easy reading books on science today - no equations. What I found instead was an unexpected surprise that immediately cemented my decision to purchase the book - it is chuck full of equations. But then, how do you write a book about mathematics without using equations? I'm glad that for this one, at least, the publishers listened to reason.

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

43 of 43 found the following review helpful:


Excellent, if you have the background  Jul 22, 2000 By Kevin P. Costello
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.

29 of 29 found the following review helpful:


This gives you what's usually left out of textbooks  May 25, 2000 By Stan Vernooy
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!

17 of 17 found the following review helpful:


How the imaginary became real  Mar 11, 2003 By V. N. Dvornychenko
This marvelous book fulfills a long-standing need for a history of how "i" (the square-root-of-minus-one) went from a disreputable construct, to an indispensable tool in the mathematician's toolbox. The author, Paul J. Nahin, is an electrical engineer with an unmistakable flair for mathematics. He is also a good writer who has done his homework. The result is an outstanding book covering an important chapter of mathematical history.

The book has something to offer to a broad cross-section of readers: from bright high-school students, to professional mathematicians, to historians. For the professional mathematician, Nahin offers many arcane tidbits, such as how Euler first summed the reciprocals of the integers-squared. (Such information is usually not found in text books.)

The book is a case study of how important mathematical concepts arrive at maturity. The history of "i" may be divided into six phases: 1) initial recognition of the "impossibility" of taking the square-root of minus one; 2) need to reconsider "i" in connection with the equations for the solution of the cubic (the delFerro-Tartaglia-Cardano equations); 3) Euler introduces the notation "i", and publishes his celebrated formula connecting the circular and exponential functions; 4) Wessel, Argand, and Gauss independently discover the correct geometric interpretation of complex numbers, 5) Cauchy introduces the theory of complex functions, 6) complex numbers are recognized as special instances of abstract fields. The author correctly points out that - contrary to what is taught in introductory courses - the deciding impetus to take "imaginary" numbers seriously came not from quadratic equations, but from cubics.

On a larger scale the book raises a fascinating question: why do some concepts (such as the zero, or "i") produce boundless fruit, while others (e.g., "perfect numbers"), upon final analysis, appear sterile.

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