This is just a delightful book, a prequel to the equally delightful [book: Dr. Euler’s Fabulous Formula]. It’s got the same approach, and much of the same subject matter, as its sequel. The method is to show you a bunch of cool math, to experiment, to avoid complete rigor in proofs as a way to drive the reader forward. It’s an *exciting* book.

[book: An Imaginary Tale] tells the story of √-1 historically. At many points throughout history, people could have made great strides in mathematics if only they had been willing to treat √-1 as a real thing — as something no more or less real than, say, π or [math: e]. Recall that the existence of irrational numbers led to the dissolution of the Pythagorean cult: an isosceles triangle with two legs of unit length has a hypotenuse of length √2, and the proof that √2 is irrational is trivial. When your worldview is centered around the belief that integers and their ratios are fundamental to the structure of the universe, and yet you can easily demonstrate that some basic operations lead to irrational numbers, you’re going to have problems.

Yet we acknowledge that irrational numbers “exist”, even though in some sense they’re fictional. (This gets to a whole discussion about mathematical existence, which I’m going to bypass for now.) In part that’s because they’re useful. And it turns out that √-1 is incredibly useful as well; many theorems about *real* numbers (which is to say, rational or irrational numbers) turn out to have short, elegant proofs by way of the number [math: i]. Indeed, throughout Nahin’s book I wondered why I didn’t learn about complex numbers as a natural part of undergraduate or even high-school calculus.

The quickest way to appreciate √-1 as a useful tool is to treat it as a rotation operator. Multiplying a vector by [math: i] is, geometrically, equivalent to rotating the vector counterclockwise by 90 degrees. But to appreciate this, it’s helpful to first picture the complex numbers as a plane, with the real part on the [math: x]-axis and the “imaginary” part on the [math: y]-axis; we now call this the “Argand diagram”, though Nahin shows that any number of other mathematicians were very close to making the same discovery.

Having made the discovery, mathematicians were off and running, and so is Nahin. Just like [book: Dr. Euler’s Fabulous Formula], [book: An Imaginary Tale] is filled with fun little discoveries of both a mathematical and an engineering sort (Nahin is an electrical engineer by profession). He gives you just enough of an introduction to, say, the Euler product formula to not be scared of it, and to say, “Oh, that’s all that was?” And you feel always like you’re in the presence of childlike glee; Nahin is genuinely excited about what he’s teaching you.

There’s something of a thread connecting a few of the books I’ve read recently: the two Nahin books, [book: The Mathematical Experience], and the biography of the geometer Donald Coxeter. Mathematics took a turn for the formal in the 20th century, under the influence of ([foreign: inter alia]) Bourbaki. All the formality probably purged mathematics of all doubt, but it might have lost something — intuition or geometric understanding or even “a sense of play” — along the way. This thread of popular math writing tries to restore that sense of play; Paul Nahin is one of its greatest practitioners.