Advice for Learning Math

Introduction

The best way to learn math is to pursue a degree. Obviously, not everyone has the opportunity to do that. The next best thing is a solid path for self-study, which tends to be quite book-heavy. Of course, if for some reason burning the midnight oil reading math textbooks isn’t your idea of a fun time, then I will jingle some shiny keys in the pop-math section at the end of the article that you will hopefully enjoy. Joking aside, there’s a lot of fantastic (and accurate) pop math out there for a general audience: this article has plenty of links to 3blue1brown (my favorite video if his is the hardest problem on the hardest test), who I highly recommend.

The Essentials

Discrete math and an introduction to proofs is a much better gateway to “real” math than calculus. I recommend Discrete Mathematics: An Open Introduction by Levin, Book of Proof by Hammack, and playing the natural number game.

The goal here is a strong foundation in basic set theory, boolean algebra, and proof techniques, as well as a broad familiarization with the many different kinds of objects and ideas in math.

Highly Recommended

After you have a solid foundation, I heavily recommend studying linear algebra, which is an essential backbone in virtually every area of math. Linear Algebra Done Right by Sheldon Axler teaches linear algebra from the “abstract” point of view, and lives up to its name. For computational linear algebra, I used Gilbert Strang’s Linear Algebra. Both should be supplemented with 3blue1brown’s linear algebra series, especially the episode on span, which is hard to understand without a visual reference. You’ll then be equipped to understand his videos on machine learning.

The final essential piece of math I recommend is group theory, which is the study of symmetry. Group theory provides the foundation for virtually all modern mathematics and has frequent applications in physics, chemistry, cryptography, and many other areas. Here’s an excellent video to motivate the subject. I’d recommend Abstract Algebra by John A. Beachy and William D. Blair, which also has a substantial introductory chapter on other basic subjects. I’d recommend reading through chapter TODO You can supplement your reading with with Socratica’s abstract algebra playlist.

Number Theory

At this point, there are several avenues open to you. If you liked group theory and some of the early topics from Beachy and Blair, I recommend pursuing number theory, which is the enormous field studying patterns in the integers. A Friendly Introduction to Number Theory by Joseph H. Silverman is a wide introduction and an excellent place to start; if you want a more advanced introduction to the subject (or a second course), A Classical Introduction to Modern Number Theory by Ireland and Rosen is a good choice. If you decide to pursue number theory, I recommend studying ring theory (a very closely related field of abstract algebra) in Beachy and Blair chapters TODO, and watching this playlist on number theory by the Fields medalist Richard E. Borcherds.

If you’re interested in elliptic curves, I recommend Rational Points on Elliptic Curves by Joseph H. Silverman and John T. Tate, which my undergraduate capstone project drew on heavily.

Theoretical Computer Science

With a crash course in proofs, set theory, and graph theory, you can jump into Introduction to the Theory of Computation by Michael Sipser, which is an excellent read. I recommend that you do exercise 28 in chapter 5, which covers Rice’s Theorem, a very useful theorem in theoretical computer science. (I’m honestly surprised Sipser put such little emphasis on it!)

I also recommend reading Scott Aaronson’s excellent blog Shtetl-Optimized, particularly Five Worlds of AI, the 8000th Busy Beaver number eludes ZF set theory, and Logicians on safari.

Graph Theory

Graph theory (no, not that kind of graph!) is another bit of math that is uniquely useful and applicable to everyday life and in virtually every area of computer science. Maybe you want to schedule tasks efficiently, find the fastest route from point A to point B, or solve puzzles like “who is the liar” and an 18th century bridge crossing riddle. Trudeau’s Graph Theory book is one of the standard texts, and reads quite nicely. Deistle’s Graph Theory is a much faster but more general introduction.

Modern Algebra

Keep reading modern algebra in Beachy and Blair. I think the crowning theorem of an introductory abstract algebra sequence is Abel-Ruffini and Galois theory. After seeing the isomorphism theorems proven multiple times and seeing the fundamental theorem of Galois theory, I recommend reading Category Theory In Context by Emily Riehl to get “a bird’s eye view of math”. If you’ve get that far and still want more, look at the most recent version of The Rising Sea by Vakil and consider pursuing a PhD in algebraic geometry.

Calculus

Calculus is the star of every ambitious highschooler’s senior year, and is the reason that I became a mathematician. It is the study of change, specifically “instantaneous change”. It has applications virtually everywhere: business, engineering, physics, computer science (particularly machine learning), biology, statistics, and everywhere else. You’re best served by picking up a copy of Calculus: Early Transcendentals by Stewart and supporting it 3blue1brown’s calculus series; if you’re a bit ambitious and want to see some of the nitty-gritty details that often get swept under the rug, consider adding on Calculus by Spivak. When you get to multivariable calculus, add on Khan Academy’s lecture series. (You might recognize the narrator.)

Probability

All knowledge degenerates into probability.
— David Hume

Now that you’ve learned calculus, it’s a good time to jump into probability and statistics, which are used literally everywhere. My textbook was An Introduction to Mathematical Statistics and its Applications by Larsen and Marx.

Real Analysis

Calculus is a treat, but eventually the time comes for every mathematician to discover out how the sausage is made. You may have noticed that the calculus textbooks were very light on proofs. (This is because most calculus students are physicists or engineers; the most important part of being a mathematician is learning to hate those “people”.) Unfortunately, the subject can’t meaningfully advance until you build up a real, rigorous foundation for the calculus on the real numbers: this field is called real analysis, and my two preferred books on the subject are Analysis I by Terence Tao and Understanding Analysis by Stephen Abbott. After you’re done with those and want to study calculus in a more general setting, the standard text is Calculus on Manifolds by Spivak.

Topology

After the beauty of calculus has been thoroughly shredded by real analysis, you can either learning some machinery to abstract over and simplify the heinously ugly proofs of important theorems (such as the intermediate value theorem). Abbott’s book points towards the subject of topology, which is the abstract study of space and continuity, which has applications all over physics and advanced math. Introduction to Topological Manifolds by John Lee is a nice read; Topology by Munkres is the standard text in the field; it’s a bit terse but worth referring to as a secondary text.

Complex Analysis

If you find yourself pining for the days before real analysis where you could be forgiven for believing fairy tales like “every continuous function is differentiable” and “every differentiable function is infinitely differentiable and converges to its Taylor series”, consider studying complex analysis — the extension of calculus to the complex numbers. Complex analysis is often considered the most beautiful and extraordinary field of math and has applications in physics, engineering, and elsewhere. You can get a visual understanding from Visual Complex Analysis by Needham; for a more rigorous standard introduction, I’ve heard good things about Complex Analysis by Stein and Shakarchi.

Differential Equations

A natural next step after learning calculus is solving differential equations; I recommend learning about separable ordinary differential equations, the Laplace transform, the eigenvalue method, and the Fourier transform. The theory of differential equations is actually something I know very little about, so I can’t make any solid recommendations there.

Pop Math

• For high-quality and accessible videos about math, I recommend 3blue1brown (especially the hardest problem on the hardest test) and some of Veritasium’s videos. Numberphile has some good videos too, but their -1/12 video is a real stinker.

• The natural number game is quite fun and a way to do short, guided, and rigorous mathematical proofs online using Lean, a popular proof assistant.

• If you’re interested in one of the most dramatic stories in recent math history, I highly recommend Manifold Destiny, an incredible account of how a mathematician solved a $1,000,000 math problem and the reasons he declined the prize money and quit math.

• In a similar vein, I recommend this phenomenal article about Alexander Grothendieck — the luminary who revolutionized the now-booming field of algebraic geometry and perhaps the greatest mathematician of the 20th century — and his abrupt disappearance to eat dandelion soup in the Pyrenees.

• If you like books but not textbooks, I loved Fermat’s Enigma by Simon Singh, about a 350 year-old math problem and Andrew Wiles’s proof in 1994. The book is famous for its quotation of Andrew Wiles’s moment of triumph:

  “Suddenly I had this incredible revelation. … It was so indescribably beautiful; it was so simple and so elegant. I couldn’t understand how I’d missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I’d keep coming back to my desk looking to see if it was still there. It was still there. I couldn’t contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much.”

Fun Facts

• You can’t comb all the hair on a coconut without making a cowlick
• Any Rubik’s cube can be solved in at most 20 moves
• There are just as many even whole numbers as there are whole numbers
• You can’t design a regular expression that will parse only valid HTML, Python, or regular expressions
• Godel’s incompleteness theorem implies that our system of mathematics (Zermelo-Fraenkal set theory, or ZF for short) can express mathematical statements that are true but that it cannot prove, and that ZFC is either contradictory or that it cannot prove its own consistency
• Subgroups of finitely generated groups may not be finitely generated
• See also: many other theorems that disappointed mathematicians

Fun Calculus Facts

• There are functions that are continuous everywhere but differentiable nowhere. In some sense, most continuous functions are like this
• We can at least approximate any continuous function with polynomials
• However, every continuous function is integrable. (This means that there are functions which are only differentiable a finite number of times)
• Even functions that are infinitely differentiable might not converge to their Taylor series
• However, complex-differentiable functions are much nicer: if a complex function is differentiable on a disk, then it is infinitely differentiable and converges to its Taylor series. This follows from the fact that we can integrate a complex-differentiable function to get its derivative
• In stark contrast to the single-variable case, partial derivatives can exist at points where the function is not continuous
• If an infinite series doesn’t converge absolutely, you can rearrange the terms to get the series to diverge or converge to any value you want

Random Lore

• Harvard’s Math 55

The Numberphile Video

• If you want an explanation of that abominable -1/12 Numberphile video, see here for an overview and here for an explanation of the Riemann zeta function and analytic continuation.