An Imaginary Tale: The Story of i [the square root of minus one] (Hardcover)

by Paul J. Nahin (Author) "At the end of his 1494 book Summa de Arithmetical, Geometria, Proportioni et Proportionalita, summarizing all the knowledge of that time on arithmetic, algebra (including..." (more)
Key Phrases: first integral theorem, second integral theorem, cyclotomic equation, John Bernoulli, John Wallis, Lord Kelvin (more...)

Editorial Reviews

Amazon.com Review
At the very beginning of his book on i, the square root of minus one, Paul Nahin warns his readers: "An Imaginary Tale has a very strong historical component to it, but that does not mean it is a mathematical lightweight. But don't read too much into that either. It is *not* a scholarly tome meant to be read only by some mythical, elite group.... Large chunks of this book can, in fact, be read and understood by a high school senior who has paid attention to his or her teachers in the standard fare of pre-college courses. Still, it will be most accessible to the million or so who each year complete a college course in freshman calculus.... But when I need to do an integral, let me assure you I have not fallen to my knees in dumbstruck horror. And neither should you."

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

Review
There will be something of reward in this book for everyone. -- Review

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Product Details

• Hardcover: 257 pages
• Publisher: Princeton University Press (August 24, 1998)
• Language: English
• ISBN-10: 0691027951
• ISBN-13: 978-0691027951
• Product Dimensions: 9.3 x 6 x 1.1 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 3.9 out of 5 stars See all reviews (50 customer reviews)
• Amazon.com Sales Rank: #208,400 in Books (See Bestsellers in Books)

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Customer Reviews

Most Helpful Customer Reviews

136 of 140 people found the following review helpful:

4.0 out of 5 stars Excellent introductory treatment of complex analysis, but..., November 30, 1999

By	Jon McAuliffe (Philadelphia, PA United States) - See all my reviews (REAL NAME)

Despite its billing as a history of science book, I would really categorize "An Imaginary Tale" as a supplemental math text with occasional historical color, much as you'll find, for example, in offset boxes of "friendly" freshman calculus treatments. The author largely concedes this in the preface. Granted, the first couple of chapters have a more historical emphasis, but by the end of chapter 3 we've largely left behind the etiology of complex analysis.

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 [e-mail], 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.

76 of 79 people found the following review helpful:

5.0 out of 5 stars A great book, chock full of equations, September 22, 1999

By	Duwayne Anderson (Saint Helens, Oregon) - See all my reviews (REAL NAME)

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

35 of 35 people found the following review helpful:

5.0 out of 5 stars Excellent, if you have the background, July 21, 2000

By	Kevin P. Costello (Atlanta, GA) - See all my reviews

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.