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173 of 177 people found the following review helpful
4.0 out of 5 stars Excellent introductory treatment of complex analysis, but...
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...
Published on November 30, 1999 by Jon McAuliffe

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41 of 44 people found the following review helpful
3.0 out of 5 stars packed with math and worth the effort
There is a lot to be learned from An Imaginary Tale. However, it will take some effort on part of the reader. Overall, I think it is a worthwhile read, but I do have the following criticisms.

1. In my opinion, the book is a bit deceptive about the level of mathematics required by the reader. It states that a freshman calculus student could follow the math,...
Published on April 30, 2006 by book reviewer


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173 of 177 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
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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 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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106 of 110 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
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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
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45 of 45 people found the following review helpful
5.0 out of 5 stars Excellent, if you have the background, July 21, 2000
By 
Kevin C. (Riverside, CA) - 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.
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31 of 31 people found the following review helpful
5.0 out of 5 stars This gives you what's usually left out of textbooks, May 24, 2000
By 
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!
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41 of 44 people found the following review helpful
3.0 out of 5 stars packed with math and worth the effort, April 30, 2006
There is a lot to be learned from An Imaginary Tale. However, it will take some effort on part of the reader. Overall, I think it is a worthwhile read, but I do have the following criticisms.

1. In my opinion, the book is a bit deceptive about the level of mathematics required by the reader. It states that a freshman calculus student could follow the math, but even those students will probably get stuck on the problems the author presents that uses differential equations and multi-variable calculus. One section tackling a topic from electrical engineering was far too technical. Either you are already familiar with this material or you would do best to just skip it as I did.

2. I caught a few typos in the math equations that could confuse less astute readers and some (but not most) of the diagrams are crude, like they were drawn hastily. I found that a little insulting, considering the mathematical sophistication expected of the reader.

3. In the section entitled Wizard Mathematics, I became a little tired of seeing different ways to produce expressions with pi using complex numbers. The author gave at least one too many of these, in my opinion.

4. There are ALOT of equations and it would have been easier to follow everything if they had been numbered. In one derivation, he plugs an expression from about three pages back into an equation presently used, with no reference or warning, leaving the reader to wonder for a minute, where did that come from?

I doubt most people could passively read this book and fully appreciate it. You should be prepared to scribble calculations at some parts. I would not have appreciated this book nearly as much if I didn't have pencil and paper ready to make sure I understood the intermediate steps he left out in his various derivations. That's how I caught some typos. Fortunately, I was able fill in most of the blanks so the material was meaningful. For the few places where I couldn't figure out how he got from step A to step B, I was able to keep going without loss of continuity. Reading this book definitely sharpened my math skills and I learned some genuinely interesting uses for complex numbers so it was worth the time and effort.
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16 of 16 people found the following review helpful
5.0 out of 5 stars Clears up mysteries, June 21, 2005
By 
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.
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16 of 16 people found the following review helpful
5.0 out of 5 stars Fantastic! Thorough, scholarly, interesting!, March 5, 1999
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!
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14 of 14 people found the following review helpful
5.0 out of 5 stars the complex clarified, March 28, 2006
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.
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14 of 14 people found the following review helpful
5.0 out of 5 stars thumbs-up from an EE/physicist - not meant as a textbook, September 21, 2003
By 
This was an incredible book. I'm an electrical engineer by degree and a physicist by hobby, so I'm pretty familiar with imaginary numbers. While a lot of the concepts were a review to me, the book also introduced me to a lot of new and fascinating territory. But besides the pure math, it also introduced me to a lot of the history and personalities behind it all. Putting it in perspective and historical context helps breathe new life into it.
I must strongly disagree with the reviewers who said that the math was not rigorous enough, and that the presentation was lacking in personality (two opposite viewpoints).
The style had way more personality than any textbook on mathematics. And anyone with a high-school math background can get through most of the book (not all of it - they may need to skip the bits involving calculus). And whoever says the presentation lacks rigor is missing the point entirely, because this is NOT a textbook and was never meant to be. The author never intended to scare away the casual reader with lenghty proofs - he wants to explain in accessible terms, not alienate.
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17 of 18 people found the following review helpful
4.0 out of 5 stars How the imaginary became real, March 11, 2003
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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An Imaginary Tale: The Story of [the Square Root of Minus One] (Princeton Science Library)
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