On the basis of its title and cover art, you might believe that [book: Dr. Euler’s Fabulous Formula] is a work of popular mathematics on the level of, say, [book: How To Lie With Statistics] or [book: Innumeracy]. Don’t get me wrong: both of those books are spectacular — must-reads, in fact. But you don’t read them in the expectation that they’ll contain interesting mathematical content. If you have a background in mathematics or statistics, you in fact don’t read them; you assume that you already know everything that’s in them. ([book: Naked Statistics] is like that for me, though the author’s recent appearance on Planet Money makes me think again about reading it.)

[book: Dr. Euler’s Fabulous Formula] is not like that. It is accessible to anyone with a basic college calculus education, and its rewards are astonishing. Starting from the premise that the Euler formula “e^(i theta) = i sin theta + cos theta” is amazing — which is a correct premise — Nahin is off and running. He runs through Fourier series, Fourier transforms, proof that pi is irrational, how to design radio circuitry, whether a tailwind helps or harms a runner on a circular track, and a hundred other things besides.

But not only is the content remarkable; Nahin pulls off the trick — which is incredibly rare among mathematical writers — of being completely, 100%, crystal clear in his proofs. His book is filled with full-frontal integrals, but every step is spelled out so clearly, and so conversationally, that I never missed a single step in the argument. I love mathematics, but I’ve long wished that I were better at it. Nahin makes me wonder if the mathematicians are the problem, rather than me. (Though it doesn’t matter what the answer to that question is: if I want to learn more math, I need to learn to read mathematics as she is written [by people other than Nahin]. Sad but true.)

So Nahin’s book is both filled to the brim with extremely interesting mathematics, and written clearly enough that any college sophomore could understand it. It’s a trick that I’m not sure I’ve ever seen before. On this basis, I’m strongly inclined to read Nahin’s other work, starting with [book: An Imaginary Tale: The Story of the Square Root of Minus One]; apparently Nahin’s Euler book is best viewed as the second half of [book: An Imaginary Tale].

Many thanks to Chris Young, of the long-defunct Explananda blog, for the pointer to this fabulous book.