Resources for Math Learning (Intro to Math)

Michael Hartl • Oct 4, 2023

Resources for the kind of math learning project I’ve recently undertaken have never been more readily available. This post offers summaries of some of the resources I’m using for the initial phase of the project, which I’m calling “intro to math” as a shorthand for a breadth-first overview and introduction to proof. I make specific reference to resources that support my personal goals and strategies for math learning, including academic math, contest math, computational math, and special topics. I plan to make more detailed standalone posts on several of these introductory resources, and I also expect to share information about more specialized and advanced resources as I progress in my program. Stay tuned for more!

1 Finding good resources

An extraordinary variety of resources on math learning have become available in recent years. This includes books, which are an old format but are more accessible now than ever before. Newer formats include videos and online apps. In order to create a sort of “math immersion” environment, I’m using a bunch of different such resources, but my primary sources are textbooks.

One great thing about learning math (or really any subject) nowadays is that it’s easier than ever to find first-rate resources. Instead of using some random textbook selected by an apathetic faculty committee, you can use one that hundreds or thousands of real students have studied and enjoyed. My general strategy for finding such books, especially ones that align with my math learning goals, involves googling the general subject and then reading suggestions at places like Math Stack Exchange and Quora, as well as consulting reviews at Amazon. Examining the table of contents, reading the preface, and taking a look at the first couple of chapters is usually enough for me to reach an informed conclusion about whether the book meets my needs.

As previously noted, I have a strong preference for PDFs, which are often difficult to find for purchase but are clearly the best format for math ebooks.1 In addition to using highly recommended books, I also prefer to use books in their second edition or later. Among other reasons, using a later edition means that many of the inevitable first-edition typos will have been fixed. (Typos in math textbooks are the worst—with natural language, I can generally error-correct, but with math it’s hard to know if the problem is me or the text.) Finally, because I’m mostly learning alone, I generally give precedence to books that have full solutions manuals available.2 This is a key part of my strategy for using books to learn math without having to rely too much on outside help. And I find that it greatly lowers the psychological barrier to attempting to solve problems myself—an essential part of math learning—without worrying that I might get permanently stuck.

For other resource types, I typically use a similar process. For videos, I pay especially close attention to what the YouTube algorithm surfaces. For example, I found Start Learning Mathematics (Section 2.2) by googling for videos on constructing the real numbers, and YouTube quickly started recommending other useful videos as well.

2 Resources for intro to pure math

To support this goal, I’m using mainly books, with some videos. The techniques above led to a large number of possibilities, and there’s no one right answer that works for everyone. After an extensive winnowing process, these are the main sources I decided to use for an introduction to pure mathematics, with a focus on breadth and proof-writing.

Books for intro to pure math

These are just brief summaries. I’m planning standalone posts on at least two of the following titles.

Videos for intro to pure math

There are literally jillions of math videos out there. Here are just a couple of ones I’ve found particularly useful as beginner-level introductions to rigorous math.

  • Start Learning Mathematics from The Bright Side of Mathematics. This remarkable course builds up the various number systems of mathematics from scratch, starting with logic and set theory, moving through the definition of the natural numbers and constructions of the integers, rationals, and reals, and ending with the extension to complex numbers. I’ll have a lot more to say about this amazing YouTube channel in the future.

  • Introduction to Higher Mathematics by Bill Shillito. The production values of Introduction to Higher Mathematics are a little dated—as Bill Shillito himself notes, the course is “ages old at this point”—but the videos provide a great overview of the subject and the material is hard to find all in one place like this.

Math quals

I mentioned before that I’m using Ph.D. qualifying exams to provide general guidance about which direction to go. There are actually a lot of options to choose from; I’m currently broadly aiming toward the Ph.D. quals used by the Harvard University math department. In addition to coming from one of the top math graduate programs, Harvard’s quals have significantly more breadth compared to most of the other quals I’ve found online. Having taken several undergraduate courses in the Harvard math department back in the day,5 there’s some sentimental value here as well.

3 Resources for contest math

In the spirit of “as easy as possible (but no easier)”, I’m initially pursuing this goal using books aimed at high-school students, namely The Art of Problem Solving (a.k.a. AoPS). These books are not to be trifled with, though; even as a trained theoretical physicist, I still find many of the problems contained in these books quite challenging. They also have remarkable breadth, including algebra, Euclidean geometry, analytic geometry, combinatorics, probability and statistics, number theory, and graph theory.

  • The Art of Problem Solving, Volume 1: the Basics by Sandor Lehoczky and Richard Rusczyk (solutions). Great coverage of both standard high-school topics (logarithms, algebra, geometry, trigonometry) and more advanced topics (probability and statistics, number theory).

  • The Art of Problem Solving, Volume 2: and Beyond by Richard Rusczyk and Sandor Lehoczky (solutions). Even more advanced topics, including vectors and matrices, analytic geometry, combinatorics, graph theory, and number theory (linear and quadratic congruences, Euler’s \( \phi \) function, Diophantine equations, and more—I told you these “high school” books weren’t to be trifled with).

These are the only contest resources I’m using for now (I’m currently about halfway through Volume 1 of AoPS), but many other resources are available. There are books for Math Olympiad (extremely challenging “high school” math) and the Putnam exam (brutal college-level math), and the Art of Problem Solving website has tons of great content (including an archive of problems from the AMC 12).

4 Resources for computational math

Pursuing this goal is fairly straightforward. I’m taking a lightweight approach, reading just enough to get started (one might say, learning enough to be dangerous) and focusing on applying the techniques to resources for the other goals.

5 Resources for special topics

As outlined above, my hands are quite full learning the basics, but I have taken some tentative steps toward one of my preferred special topics: geometric (Clifford) algebra.

6 Current slots

Because of the huge variety of great resources available, it’s important to have strategies for maintaining focus. One technique I’m using involves thinking of myself as having some small number of “slots” (corresponding roughly to a typical undergraduate or graduate course load) and moving back and forth between them. For this early intro-to-math phase, I’m using four slots, corresponding to four primary resources:

  1. Pure Mathematics for Beginners by Steve Warner

  2. How to Prove It by Daniel Velleman

  3. The Art of Problem Solving, Volume 1 by Lehoczky and Rusczyk

  4. Sage for Undergraduates by Gregory Bard

This list is up to date as of this writing; as I progress in my program, I’m planning to update my /now page with the changes, so see that page for more on what resources I’m using now.

1. My preferred reading tools for PDFs are GoodReader for iOS and Acrobat Reader for macOS. Among other things, with this setup it’s possible to configure iCloud to sync PDF annotations automatically across different devices.
2. In this context, Library Genesis often proves indispensable.
3. The shorter version of Pure Mathematics for Beginners better matches my current learning goals, but I have purchased the longer version from Amazon both for reference and to reward Warner for his excellent work. Among other things, I can confirm that the accelerated and extended edition includes free access to a full solution guide.
4. If you do use Epp as a primary text, you might consider the third edition, which (unlike the fourth or fifth editions) currently has an instructor’s manual available.
5. Math 22ab (advanced calculus and linear algebra, including differential forms and an introduction to algebraic topology), Math 112 (real analysis), and Math 115 (Complex variables and applications, Fourier series and boundary-value problems, and calculus of variations). Links are to the books we actually used and do not necessarily represent endorsements (though all of the Math 115 books are excellent).