[Disclosure of Material Connection: The AMS kindly provided me with a free review copy of this book. I was not required to write a positive review. The opinions expressed are my own.]

An Illustrated Theory of Numbers
Martin H. Weissman
American Mathematical Society, 2017

“Sorry… I couldn’t tell if you were napping, or praying, or reading…”

Thus apologized the barista at my favorite coffee shop for her hesitancy in telling me that my tea was ready. Or perhaps she was just apologizing for interrupting my nap/prayer/reading. In any case, what was it I was actually doing?

In fact, as you may have guessed from the subject of this post, I was reading Martin Weissman’s new book, An Illustrated Theory of Numbers. The reason the barista was so confused was that I was hunched over the book, deep in fascinated concentration over one of the many rich data visualizations. If I recall correctly it was actually a visualization of the distribution of prime numbers (and while staring at it I had a sudden epiphany that I have been implementing prime sieves inefficiently, but that is a story for another blog post!).

As any reader of this blog knows, I love number theory and I love visualizations, so this book is right up my alley. Weissman is a really great teacher, and has obviously spent a lot of time thinking carefully about the best way to structure and explain various topics—even without the illustrations I think the book would still make a valuable contribution to the state-of-the-art in number theory pedagogy. But with the visualizations and illustrations it is truly wonderful! They are plentiful (almost 500!), beautiful, and pedagogically powerful. I had never thought of number theory as a particularly visual subject before, but Weissman makes an impassioned—and, I think, successful—case that it is, or should be. I’d love to give an example but I’m not sure I could do it justice, and it would make this post too long. I will consider whether there is a cool example I can share in a future blog post.

Although the book makes it to “advanced” topics by the end (quadratic reciprocity and quadratic forms), it is fairly self-contained and doesn’t formally require much background beyond high school or basic undergraduate mathematics. All the necessary results are carefully explained and proved, each new result building upon previous ones. It also has a large collection of exercises after each chapter, making it useful for either self-study or for using as part of a class. To be honest I kind of wish I could teach a number theory class so I could use this as the textbook. And if I knew any mathematically inclined, self-motivated high school students I would get them a copy of this book in a heartbeat (well, modulo budgetary constraints).