I thought I was just drawing pretty lines, but my high school doodles turned out to be Linear Algebra over a finite field. When I was in high school and college, I would always try to read ahead in my classes so I’d be prepared when the topics came up…

I managed to submit a peek into the future of my articles to 3blue1brown’s #SoME2 competition. I also updated my profile picture. As a brief summary, I’ve been producing fewer articles since the start of the summer to focus on making a video for 3blue1brown’s #SoME2 contest. I learned a…

Lagrangian Mechanics is more than a shortcut — it’s the basis of modern Physics. Let’s see why that is. — If the Lagrangian were simply a shortcut to find the equations of motion, we wouldn’t care about it too much. It would be a nice calculation trick and it might have found use in some numerical algorithms, but it would not be the bedrock of modern Physics. Previous Article Next…

Finite fields are a finite set of objects you can add, subtract, multiply, and divide. We can use them for error detection and cryptography. — Given that I named this series “How to Discover Finite Fields While Bored in Class,” it’s time to talk about finite fields. In the previous article, we introduced the rationals, constructible, algebraic, real, and complex fields. These fields are all infinite, but our game only has five colors. We need…

We’re going to introduce fields in preparation for finite fields and show how they answer questions that went unanswered for millennia — In the last article, we got to the point where we could define polynomials using rings and we could have solutions to those polynomials with division rings. These solutions, however, were difficult to work with. Previous Article Next Article All Articles To solve this problem, we require that × be…

Lagrangian Mechanics makes finding the equations of motion almost trivial and will serve as the basis for modern Physics. As I’ve said before, there are many tasks that can be quite difficult in Newtonian Mechanics. These tasks include finding conserved quantities, dealing with constraint forces, converting the equations of motion…

Groups may allow for solvable equations, but we need more structure to even talk about polynomials. We call these structures rings. In the previous article, we established groups as the simplest algebraic structures with solvable equations and we showed that the five color group was isomorphic to the finite Abelian…

Differential Geometry is the study of Calculus on smooth manifolds, and it allows you to make statements that are true in any coordinate system. At this point on the Road to Quantum Mechanics, we know how to set up and solve some important partial differential equations in different coordinate systems…