## Book Review: An Imaginary Tale

An Imaginary Tale: The Story of √-1 by Paul Nahin

I’ve had this book for years, and have read the first few chapters several times but never actually finished it for some reason. I think I got bogged down in the geometry of Chapter 3, even though it’s not that difficult. I even read and reviewed its direct sequel a few years ago. Better late than never I suppose.

This book is about complex numbers, and especially the unit imaginary number i. The first few chapters are about its history, and how mathematicians were suspicious of such quantities for a long time (hell, they were suspicious of negative numbers for centuries, too). Interestingly, their origin traces more to solutions to cubic equations rather than quadratic equations, where they seem, to a modern eye, to more naturally arise.

Then the book jumps around to an assortment of different topics out of chronological order. For instance, there’s a section dealing with solutions to a particular type of electronic circuit, which comes before Euler’s foundational work. At times Nahin will prove some theorem, and at other times he’ll say something to the effect of “To prove this claim isn’t that hard, but this isn’t a math textbook!” Sometimes I wish he did just write a more thorough textbook with his style. So treat it more like “Here’s a variety of interesting things about complex numbers, as selected by the author.”

Which isn’t to say there’s not good stuff being proven. We get for instance Cauchy’s integral formula, which is proved though not to full generality (he only considers a rectangular contour; “not a textbook!”). I remember being blown away by this theorem when I took complex analysis in university, and Nahin expresses similar amazement.

I found some errors in the appendices in this 1998 copy: in Appendix B on p. 133 there are some denominators that are squared which I don’t think should be (it’s a simple matter of FOILing out a quadratic). In the original paper the squares are not present. Also in Appendix E when discussing the Laplace transform, he uses the pound sterling symbol £ rather than the script capital ℒ  which I found somewhat amusing.