*Bits, ink, particles, and words.*

There are plenty of ways to enjoy mathematics. You can attend a classroom lecture, you can read a textbook, you can look at a news article, you can watch a video, or you could just play with some concepts yourself. There’s not one way in particular that is better than any other. Rather, each one has its own advantages and disadvantages. It depends what you’re looking to get out of your session.

As a student in mathematics and physics, I’m part of two different worlds. On the one hand, proofs and abstraction come from the side of mathematics. On the other hand, physics is where concrete examples and applications are the norm. In physics (at least, within the scope of undergraduate education), we only care about the mathematical tools that we can apply to a given problem.

Learning mathematics in school and doing mathematics in general are *not* the same thing.

In mathematics, the terms “necessary” and “sufficient” have technical meanings. These terms come about when looking at two statements *P* and *Q*. If we say that *P* is sufficient for *Q*, then that means if *P* is true, *Q* automatically has to be true (*P* implies *Q*). On the other hand, if *P* is only necessary for *Q*, having *P* be true doesn’t mean *Q* has to be true (but the other way works, so *Q* implies *P*). If we have the *P* is both necessary and sufficient for *Q*, that means having one gives us the other for free. They are tied together and are inseparable.