Where to learn more math

If there’s one thing I hope 3blue1brown videos do, it’s to inspire people to spend more time playing with and learning about math. Below are a few selected resources which I’d recommend should you find yourself so inspired.

(Side note: Several of the companies listed below have sponsored 3b1b videos in the past. This page, however, includes no paid advertisements, just creators and organizations that comes to mind when I think of what this audience might enjoy.)


Practice

Passive consumption, say by watching videos or reading books, is certainly the lowest friction way to expose yourself to new math. But if you care about long-term retention, or gaining a deeper knowledge of the topics you come across, there really is no substitute for practice. So before getting to other YouTube channels you might enjoy, here are a few places you can go to be more active about the learning.

The first is brilliant.org. I know a few of the people there, and I can vouch for the fact that they are extremely thoughtful about what makes for good learning. It shows in the product. Rather than putting lectures and videos front and center, problems and puzzles take precedence, all presented in a structured progression.

I particularly like their principles, which are good to keep in mind wherever you do your practice. The site also benefits from an active and helpful community of users.

 

Another good reservoir or problems with a helpful community surrounding them is the Art of Problem Solving. In particular, they have an extensive archive of contest math problems, ranging from more approachable ones like the AMC8 to extremely hard ones like the IMO, all with excellent surrounding discussion from the community.

 

For building up your fundamentals, say starting with algebra, take a look at Khan Academy. They are probably most famous for the videos Sal does, but arguably the site is most valuable as a source of problems you can drill on, starting from the very basics if necessary. I may be biased, though, given that I used to work there!


 

Other YouTube channels

Mathologer, by Burkard Polster. Half of the appeal here comes from the delightful bits of math this channel covers, often with novel ways of presenting otherwise extremely complicated topics to make them easier. For example, he has a video showing why pi is transcendental, which is a famously difficult topic, but which he skillfully boils down into an approachable step-by-step presentation. The other half of the appeal is watching Burkard delightedly laugh as his own jokes every 2 minutes or so.

 

Numberphile, by Brady Haran. I’m guessing if you’re here, you already know about Numberphile. If somehow you don’t, it’s a great channel full of interviews with mathematicians. As a result, you see a wide array of the types of topics which different mathematicians find exciting to showcase or teach, and a little more of the human element behind this subject. There is also an associated podcast, less about the math, and more about those human stories.

 

Looking Glass Universe, by Mithuna Yoganathan. She covers ideas from physics and math explained simply and unusually honestly. Start with the one on spin, it’s great. She also places an emphasis on doing practice, often assigning “homework” at the end of each video.


 

Welch Labs, by Stephen Welch. Perhaps best known for his series on imaginary numbers, this channel is full of high production quality and well-written videos. There are also several series on machine learning, the topics underlying self-driving cars, so this is certainly one of the more applied math channels out there.

 

Black pen red pen, by Steve Chow. If you like seeing how to work out integrals in all sorts of crazy ways, boy is this the channel for you. Steven is a math teacher, and the videos have a feeling of being in office hours, honestly working through problems with a very skillful and enthusiastic teacher. Many of the problems covered are the kind you might come across in school, and many or just for fun.

 

Standup Maths, by Matt Parker. This is channel is more whimsical and comedic, but that’s not to say there isn’t a lot of serious math it covers.

 

Patrick JMT, by…well… Patrick. If you want good, honest worked examples of practice problems, especially for calculus, it’s hard to find a deeper well of examples than what this channel provides.


Books

For those in college, here's a selection of books I found particularly well-written.

  • "Vector Calculus, Linear Algebra, and Differential Forms" by Hubbard and Hubbard

  • "The Cauchy-Schwarz Master Class" by J. Michael Steele

  • "Visual Complex Analysis" by Tristan Needham

  • "Ordinary Differential Equations" by Vladimir Arnold

  • "Chaos and Nonlinear Dynamics" by Steven Strogatz

  • "Linear algebra done right" by Sheldon Axler

  • The expository papers written by Keith Conrad

  • "An Epsilon of Room" by Terry Tao

  • "Topology" by Munkres

  • "Proofs from the Book" by Aigner and Ziegler

  • "Primes of the form x^2 + ny^2" by David Cox

In reading, really try to predict what proofs will look like, and be willing to meditate on what the right way to think about a given object is.  Ask yourself if each new construct feels motivated, or if it's out of the blue.  If it is out of the blue, it's okay to move forward anyway, just keep note of the fact that there is a lurking question mark.  Read with a pencil and paper by your side so you can sketch things out and scribble solutions to exercises as you come across them.

I hope that helps.