An Imaginary Tale

by Paul J. Nahin

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Publisher: Princeton University
Copyright: 1998, 2007
Printing: 2010
ISBN: 0-691-14600-4
Format: Trade paperback
Pages: 259

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Subtitled The Story of √-1, An Imaginary Tale is a history of the use of the number mathematicians normally call i. Nahin starts in the 15th century with the discussion of the time on solving cubic equations, specifically del Ferro's solution to the depressed cubic. He walks through how to approach that solution with imaginary numbers, provides a brief introduction to the complex number plane, and then explains that del Ferro didn't follow the logic in those directions at all. Mathematicians at the time were dubious about negative numbers because they were not intuitive representations of real-world quantities. The square root of negative numbers was considered simply impossible.

Nahin continues on in this historical vein for three chapters, walking through the mathematical problems that arose from analysis of cubics and the constant appearance of imaginary numbers, the early attempts to find a geometric interpretation, and then the slow development of the modern conception of imaginary numbers in the 19th century. The emphasis throughout is on the specifics of the math, not on the personalities, although there are a few personal tidbits. Along the way, he takes frequent side journeys to highlight the various places complex numbers are useful in solving otherwise-intractable problems.

After that initial history come two chapters of applications of complex numbers: vector analysis, Kepler's laws, applications in electrical engineering, and more. He does win a special place in my heart by walking through the vector analysis problem that George Gamow uses to demonstrate complex numbers in One Two Three... Infinity: a treasure map whose directions depend on landmarks that no longer exist. Following that is a great chapter on deeper mathematical wizardry involving i, including Euler's identity, ii, and a pretty good explanation of hyperbolic functions. The final chapter is an introduction to complex function theory.

One's opinion of this book is going to vary a lot depending on what type of history of math you like to read. Unfortunately, it wasn't what I was hoping for. That doesn't make it a bad book — other reviewers have recommended it highly, and I think it would be a great book for someone with slightly different interests. But it's a very mathematical book. It's full of proofs, calculations, and analysis, and assumes that you remember a reasonable amount of algebra and calculus to follow along. It's been a long time since I studied math, and while I probably could have traced the links between steps of his proofs and derivations with some effort, I found myself skimming through large chunks of this book.

I love histories of mathematics, and even popularizations of bits of it (particularly number theory), but with the emphasis on the popularization. If you're like me and are expecting something more like The Music of the Primes, or even One Two Three... Infinity, be warned that this is less about the people or the concepts and more about the math itself. If you've read books like that and thought they needed more equations, more detail, and more of the actual calculations, this may be exactly what you're looking for.

Nahin is fun to read, at least when I wasn't getting lost in steps that are obvious to him. He's quite enthusiastic about the topic and clearly loves being able to show how to take apart a difficult mathematical equation using a novel technique. His joy reminds me of when I was most enjoying my techniques of integration class in college. Even when I started skimming past the details, I liked his excitement.

This wasn't what I was looking for, so I can't exactly recommend it, but hopefully this review will help you guess whether you would like it. It's much heavier on the mathematics and lighter on the popularization, and if that's the direction you would have preferred the other similar books I've reviewed to go, this may be worth your attention.

Rating: 6 out of 10

Reviewed: 2015-01-01

Last modified and spun 2015-10-22