*Bits, ink, particles, and words.*

Throughout secondary school, we learn about solving simple systems of equations. We learn that there are several methods (which are more or less the same thing): comparison, substitution, and elimination are the big three. We then go through tons of practice questions which all focus on doing this kind of solving. In particular, we solve systems of *two* equations, and I don’t recall ever doing more than that. The sense I get from students is that solving these kinds of equations is sometimes confusing. I personally think it’s because there’s a lack of equivalence between the methods, so they all seem like pulling magic tricks out of a hat. That’s a pedagogical/time problem, but I don’t want to focus on that today. Instead, I want to focus on something that makes solving these systems of two equations super quick. It’s called Cramer’s rule, and this method makes it possible to forego using comparison, substitution, or elimination to solve for variables. Instead, you apply this procedure, and you can simply read off the answer.

The distance formula is one of the most frequently used relations in physics, allowing us to decompose a variety of vectors into different components. It’s something that every physics student uses, and so it becomes second-nature for most of us. However, I’ve come across the sad fact that many secondary schools don’t seem to teach how the distance formula comes about and its connections with earlier work. As such, the link between algebra and geometry is lost and the distance formula gets lost in calculating differences between $x$ and $y$. Here, the goal is for us to look at the distance formula and see how it relates to other concepts that are of much use.

When learning a new topic, there’s always a certain tension between two approaches: going straight to abstraction, or starting off easier with examples. I see this more and more as I learn about more complex and detailed physics and mathematics, and it has always made me wonder which way I should go about trying to learn. Just like anyone else, I want to get to a place where I feel fully comfortable with the concept in abstraction, but I don’t want to subject myself to a painful learning process by hitting myself against the brick wall of abstraction.

This is going to be a very quick post, but it’s something I wanted to share since I think it could give some insight into a concept that is used to make proofs in mathematics. When writing proofs, it is often difficult to show that your proof holds for *every* case (say, if you’re trying to show that the square root of two is not capable of being represented by a ratio of integers). Checking every single combination of integer ratios would take an infinite amount of time, so we want to come up with a strategy that is better. To do this, we try to prove our statement by *contradiction*. The idea is that we negate our conclusion, and from that, we need to show that there is a logical contradiction. Therefore, when we come to a contradiction, we know that our assumption of negating the conclusion was false, so that means our conclusion is true.