Over the past four years I’ve been writing a book, A Programmer’s Introduction to Mathematics. It teaches someone with programming knowledge and experience how to engage with mathematics. I can achieve this goal largely because of the implicit overlap in the content and ways of thinking between math and programming. Despite that, there is no smooth bridge from one world to the other—a bridge that explains the unfamiliar parts and focuses on the important concepts, while leveraging programming to demonstrate applications.
Until now. If you’re a programmer who wants to really grok math, this book is for you.
In this article I’ll explain how I wrote and self-published the book.
Having written technical blog articles for eight years, the writing process felt natural. I wrote about a third of the book within a few months after starting, and I was 100% happy with the content—though not the style, cf. “Editing.”
I could have kept writing, but I knew I had to constrain myself to something that could fit in a reasonable-sized book. I aimed for 300 pages (final book is about 375 pages of content), but by the time I realized that I needed to cut some content, I had already written about 100 pages.
This first 100 pages covered the three chapters “Polynomials,” “Sets,” and “Graphs.” I wanted chapters on calculus, linear algebra, optimization, probability, group theory, complex numbers, and differential equations. But polynomials, sets, and graphs are the easiest chapters content-wise. These other chapters would be necessarily longer. I wanted to keep each chapter to ~50 pages. Including all the chapters would push the book above 500 pages. I also didn’t want it to be too dense (I could cram linear algebra into 50 pages, if I leave out the exposition that actually makes my book worth reading; screw that!).
This process of cutting out entire chapters, and whittling each remaining chapter to its core while still providing useful high-level insights, proved to be the most challenging part of writing.
This resulted in glaring omissions. The book has a chapter on calculus, but omits integration. Instead, the calculus chapter focuses primarily on derivatives and Taylor series. This fits largely with my personal experience — Taylor series are more applicable to math (and the kind of math that shows up in programming) than computing convoluted integrals. Taylor series also had use in later chapters for optimization, big-O notation, and physics.
Cutting integration was a deliberate choice. The general ethos of approximation is what I want the reader to internalize, and once that’s done learning why integration makes sense is much easier. I felt this way about most topics I omitted. Once you have a strong understanding of sets and the basic terminology around combinatorics, the average book on discrete probability is more digestible. Perhaps that is wishful thinking, but I had to make compromises somewhere or I’d never finish the book.
Instead of trying to cram every subject in the book, or guess the best topics, I chose topics that crystallize essential skills and patterns, which I could also demonstrate with code. Then, with ample exercises to go out and explore math beyond the book, the reader will build the cognitive skills to learn whatever math topic they desire.
I had to do some extra research for the book. While graphs and eigenvalues are readily in my brain, other topics I had learned and long forgotten. Hyperbolic geometry was one (specifically how to compute in the hyperbolic plane), I had to refer to a few physics books about the typical way they present mathematical models for waves, and I admit I did gain a newer and more complete understanding of the core definitions of multivariable calculus.
As I repeat in the book, every time you re-learn a subject, you gain new and useful insights.
Typesetting (LaTeX in vim with latexmk)
The entire book was typeset in LaTeX.
I was able to jump into the writing without thinking too much about formatting because I started with a copy/pasted book template from Overleaf. I ended up changing that template quite a bit (for some reason, it mixed serif and sans-serif text?), mostly simplifying it, configuring code samples, exercise lists (using the exercise package), and theorem environments. If you ask me later, I will happily upload these structural TeX files so they can be reused, but be aware I’m not a TeX expert, so it’s probably better to start from a template.
I stored all my files on Dropbox. In 2014, that made sense for me, as I was just starting to use git in earnest. If I were to do it again today, I’d do it in git on principle, but there were no major drawbacks to storing on Dropbox.
Each chapter is in a different file, and no TeX file is longer than about 3k lines. At 80 chars per line, the entire book is almost exactly 20k lines (including all TeX annotations in the chapter files, like index markers and figures).
I wrote the book in vim, and used latexmk to automate compilation (more details).
At the end of the book, I realized I wanted an index, and I went through the entire book just hunting for places to put index markers. LaTeX will build an index for you automatically, but you have to tell it precisely what location to index, and what word refers to that location. This worked out quite nicely for some things; for example, I indexed the location where I define each bit of new notation. It was also a slog.
Finally, at the very end I did another pass over the book using chktex with some slight configuration to tell it to ignore code listings, texttt, etc. I learned a few new things about TeX when doing this, but it probably wouldn’t have killed me to skip that step.
The cover art is an early 1900’s painting by Piet Mondrian. I did the cover organization and layout using Sketch. One hiccup is that Sketch doesn’t export pdfs in a way that you can choose a resolution (it used 72ppi by default), so I had to use a different tool (ImageMagick convert CLI) to adjust the resolution.
During writing, I read each chapter draft roughly three times. After writing the whole book, I read it cover to cover twice.
I made many embarrassing mistakes, and I fixed the ones I found, but I bet there are still mistakes in the book. If you find any, submit an erratum at pimbook.org.
During early edits I rewrote large portions of each chapter. I struggled to achieve the right the pace and density for the first few chapters. I wanted the book to have a ramp. The earlier chapters should be accessible to anyone without—as I put it mildly insensitively—a phobia of fractions. The goal of those chapters is to introduce notation, to wean readers onto cultural expectations like writing down examples, to lay out basic definitions and proof techniques, and to get past that inevitable part of reading math where your eyes glaze over looking at a formula.
In addition to proofing the book myself, I sent it out to basically anyone who would ask for it via mailing list. I trolled Hacker News posts where programmers griped about math and slipped in an advertisement, and many of them contacted me via email and some provided useful feedback. I also gave early rough drafts to any of my friends who wanted to take a look, and many provided good feedback.
However, one glaring omission was a thorough, paid technical review of the math and programs. I struggled to find someone who would commit to reviewing more than one chapter. I think in part it was because friends and colleagues didn’t want to start a business relationship and be on the hook for a friendship or something (I would never end a friendship over this!). Finally, I got in touch with an old classmate from my time in the Budapest Semesters in Mathematics who agreed to review two chapters that I was particularly concerned about. If I did it again, I’d work harder to find multiple technical reviewers and get better coverage in that regard.
A few days after publication, it was already clear that I should have spent more time editing. I had somehow skipped my final spell check on the Sets chapter, and an early reader submitted a number of typo corrections. Rats.
But around a month before my ultimate publication date, I was just exhausted by all the editing. I felt that I was ultimately OK publishing it with some minor imperfections, and that I was proud of it in its current state. And that was how I decided I was done editing.
Self-publish vs. publisher
Over the years I worked on the book, I had discussed a book deal with a few publishers, including O’Reilly. What I took away from those conversations was that the added work of working with a publisher, and the added deadlines, wasn’t worth the benefit.
The offer was something like “$5k advance paid now, then 10–15% royalties on sales after those royalties pay back the $5k advance.” What I’d get in return was a copy editor, the brand, the printing and distribution channels, and the advertising.
There were a few catches. Most non-academic publishers won’t let the author write in LaTeX (and for mathematical typesetting there is simply no other way), so I was told multiple times I’d have to convert my existing 150 page draft into Word or OpenOffice format.
If that alone wasn’t a deal breaker, some publishers told me that I was on my own for finding technical reviewers. Couple this with my difficulty meeting a deadline (see “Timing”), and I felt the publisher would add more stress than support. I also knew in the back of my head that they’d force me to change the style of my book in a number of ways, and that added some uncertainty.
I had built up an audience from my blog, with a mailing list of 1,000 people (closer to 2,000 by publication), so I felt I had a brand and decent distribution. I was confident I could typeset the book professionally enough. The editing wouldn’t be perfect, but I could live with that.
Most importantly, I knew about on-demand publishers like CreateSpace, since (disclaimer) I had worked at CreateSpace part time as an undergraduate. CreateSpace (owned by Amazon) allows you to upload a book interior as a pdf and a cover, and “publish” instantly. Then whenever someone orders the book on Amazon, they print it and ship it that day. That covers distribution. I can sign up for Google Ads and schedule promotional talks and interviews. I can’t print, store, and ship books to customers, full stop. So CreateSpace was essential in my choice.
CreateSpace also gives much better royalties, around 40% of the list price for a paperback. Self-publishing also gave me the flexibility of choosing a venue to sell the ebook. A friend of mine published a book using Gumroad, and they seemed pretty good, providing about 95% royalties on sales. The ebook was simply a pdf download.
Code repository, website, and referenced material
Along with the book I had associated programs, some of which were very long, with the important bits showing up in the book itself. So I needed a website and a GitHub repository to allow readers to see the full code.
The book code was written in Python 3.x, and naturally it has a full test suite with Travis CI and Coveralls running against the code repository to ensure it stays healthy. I have already received one pull request for a version incompatibility I overlooked.
I wrote the early chapters at a time when I had been competent in Python, but I hadn’t used it professionally. By the time I finished the first draft, I decided I should go back and rewrite all the early Python code so it conformed to better engineering practices docstrings, more modular organization, standardized test framework, etc. I would have liked to use more modern Python features like dataclasses and type hints, but I felt they were too new and might have thrown off some readers who are either used to Python 2 or new to Python entirely. Type hints in particular add a lot of noise.
I worked on this book for four years. During that time, I finished a PhD thesis, got married, moved across the US, and switched jobs twice.
Periods of time during which I completely lost steam. I got bored, or hit a wall trying to decide what topics to cover or what applications to use. I went out and made new friends, picked up new hobbies like rock climbing and craft cocktails, played video games, went back to blogging, and worked on random side projects.
I like to tell myself I could have finished the book in a year if I worked on it full time. But to be honest, I think the book turned out better with the wait. I write better software now, and I have a stronger understanding of the industry since I’ve been out of graduate school. And my writing style continually improves.
Pricing, customers, sales
I listed the book at $35 for the paperback and $20+ for the ebook. (i.e., $20 or pay what you want above that) The royalties worked out to about $15-$18 per sale. I originally thought I should list the book for cheaper, but I undersell myself and so I looked around at what other books were selling for, imagined to myself what I thought was a fair price, and added $10.
Initial sales have been great. I set a baseline for myself that if I couldn’t sell a thousand copies of the book, then I would consider it an abject failure. At the time of this writing I’ve sold more than 2,500 copies, with about 60% of that being paperback, and the majority of the sales coming from the US. More importantly, initial reception has been heartwarmingly positive. I can’t wait to see how people feel after they’ve read the whole book.
Many folks have been asking for a format of the ebook that is compatible with Kindles (epub or mobi format). They want to resize text and such. Sadly, I don’t know of any good way to do this that is cost effective for me and still produces a book with the quality I expect. In particular, every website I looked that that showed how to put math formulas in epub/mobi produced results that looked awful. And you have to manually rewrite all the offset formulas so that they fit within the expected form factor. Six inches minus margins is quite small for equations! I’m sure there is a way to engineer it, but I don’t have the time to do it myself. I’d much rather be writing another book.
Two customers have asked for “refunds” so far related to the ebook format, and I’ve gotten a handful of odds-and-ends requests for other bits of customer service and accessibility. A few people have asked if it will be available in other languages (Spanish and Chinese, so far). I’m more than happy to address customer concerns it if it’s not too much work, but a major undertaking like a translation will probably not happen for a little while.
Next up: book tour
Finally, I’m starting to organize a series of talks and interviews to promote the book. I can talk about learning math as a software engineer, about cool math stuff on its own, or even about the process of self-publishing with all the nitty-gritty typesetting.
If you want to coordinate, please email meat firstname.lastname@example.org.