I have decided to chronicle my efforts to write and publish a book, the Illustrated Theory of Numbers. These efforts really began about seven or eight years ago, the first time I taught an introductory number theory class (Math 115 at UC Berkeley). Around the same time, I had started working with K-12 teachers, and I was inspired to take a visual and dynamic approach. Some unusual aspects of my class were: I introduced Gaussian integers and Eisenstein integers from the beginning, I used Conway’s topograph to study binary quadratic forms, and I followed Zolotarev’s proof of quadratic reciprocity.
Since then, I have taught similar material at UC Santa Cruz, and at the MMSS summer program at the University of Michigan. Each time I teach number theory, I find a few new approaches. Making number theory elementary might not be a virtuous goal in itself, but illustrating number theory is most rewarding.
When I first learned about number theory, I didn’t like it. I was given (or maybe I bought) a copy of Hardy and Wright’s classic text (An Introduction to the Theory of Numbers). From what I hear, this book has inspired generations of number theorists. But to me, early in my undergraduate career, it seemed like a large collection of unmotivated algebraic and analytic tricks. I didn’t get it. Instead, I enjoyed topology and complex analysis.
Later as an undergraduate, I had the great benefit of taking many (five, I think) courses with Goro Shimura. The first, I think, was a course on Riemann surfaces, meant as a second course in complex analysis. The second was a course in number theory. Incredibly, to me at the time, Shimura introduced aspects of number theory in parallel to what he had taught about Riemann surfaces — primes were analogous to points on a Riemann surface, and I was hooked. Around the same time, I took a “junior seminar” with John Conway, who taught us his beautiful approach to quadratic forms via the “topograph”. Since then, I have always taken a geometric approach to teaching number theory.
A few things have convinced me to write a number theory text — to enter a somewhat saturated market. First is that I cannot find any number theory texts which I find visually beautiful and suitable as an introductory text. This is the gap I hope to fill. A one-day course with Edward Tufte opened my eyes to the possibilities and importance of visual design. As Tufte explored visual design related to statistics in his first book (Visual Display of Quantitative Information), I hope to explore visual design related to introductory number theory in my first book. Hopefully I will not resort to taking out a second mortgage to self-publish!
Last year after teaching Math 110 at UCSC for the second time, I found myself with about 12 unpolished chapters of an illustrated number theory text. Now (with the security of tenure), I have begun polishing these chapters, adding more content and illustrations, with the goal of submitting parts of the book to a publisher by the end of the summer.
I have polished the first three chapters now, so this blog does not begin at the beginning. But with the next few posts I’ll try to catch up and share my experiences.