## Review: An Imaginary Tale

An Imaginary Tale is an enjoyable historical and hands-on exploration of imaginary numbers. One Amazon reviewer, K. Jazayeri, characterized the book perfectly:

You may think of it as an "appetizer sampler" for the topic - "A Splash From The Complex Plane" - to quote the title of a diagram in the final pages of the book.

And a great sampler it is. Seven chapters, six appendices, and numerous illustrations provide a nice and deep introduction to the subject.

An Imaginary Tale begins with an exploration of how del Ferro, a mathematician living in the 1400's, solved a special case of the cubic equation, and how his solution relates to complex numbers. I'm always excited to discover connections between different seemingly unrelated branches of mathematics so I was excited with this cubic discussion coming early.

The book nicely explains the relationship of complex numbers to roots of equations, geometry, trigonometry and trigonometric identities, vector algebra, physics, hyperspace, electrical engineering and more. I'm the kind of person who learns much better if I can understand how things are related. This book did that for me, tying together ideas I already was exposed to but hadn't connected the dots between.

Another Amazon reviewer, David J. Lewis, exposed one of the great gems of the book:

Here, for example, is one extremely elementary application that I did not know about. Prove: the product of two sums of squares is itself the sum of two squares in two different ways. Symbolically, given any integers a, b, c, d, there are integers p, q, r, s with...

(a^2 + b^2)(c^2 + d^2) = p^2 + q^2 = r^2 + s^2

This was demonstrated by mathematicians a long time ago, but not particularly easily. Using complex numbers, it's almost trivial to see, however, certainly within reach of a student of Algebra I. (There's an even simpler version of the proof that Nahin presents, but it's a bit messy to write without properly typeset mathematics.) This also makes the important point that complex numbers are very useful to help understand non-complex mathematical phenomena, a point Nahin makes throughout the book.

I'm always excited when one branch of mathematics provides a result in a new and simple way.

I should warn you that this is not a spectator book. In order to get the Aha's you'll need to work at it. But, most of this book is quite accessible to bright high school students who have had some calculus. And, fortunately, this is not a dry complex analysis book.

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