An Imaginary Tale: The Story of v-1

This book gets five stars because I had a major "wow, now I get it" moment reading it (the elegant explanation of De Moivre's theorem). I also really appreciated the focus on the geometry and coordinate mapping of i. Anyone that has previously worked with multiple integrals and elementary differential equations should have no trouble plodding through the math. It took me a lot of (rewarding) time to follow the harder parts, but I usually was able to unpack things enough to get the point. Yes, there are a couple of misprints (more distracting than critical) and, perhaps, a bit too much electrical engineering for my taste (although maybe I simply haven't got enough EE background to benefit). However, what matters most is that this author knows how to highlight the really important things. I cannot deduct any stars, the "wow" moment I had with De Moivre was just too good.

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.

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!

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

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!