An introduction to mathematical writing

In chapter 2 of Suugaku Girl, we’re introduced to the third component of the little love triangle that’s forming. Tetra is the underclassman that the main character is tutoring and she’s one of the many people who think they might like math but school eventually beats that silly thought out of them.

Anyone who’s taken a real math class will know that by the end of it, your notes are essentially a giant list of definitions, theorems, and proofs along with an example thrown in once in a while. By a real math class, I mean a math class that’s actually concerned about reasoning about how we get our theorems instead of focusing on how to use them for practical applications. I did my undergrad at a school where math has its own faculty and where there’s a lot of rivalry between the various faculties. Everyone in math often jokes about how an example suffices as a proof for engineers and their ilk.

I checked your textbook for a proof and it said that we’ve done enough examples for it to be plausible. Must have been written by engineers.
— Vanderburg, PMATH 340 (F09)

Here, Tetra falls into the same trap when the main character asks her to define what prime numbers are. So MC-kun has to explain how definitions work. First of all, definitions have to be precise, just like theorems. It’s pretty easy to lose marks for misstating theorems and definitions by leaving off an edge case for 0 or 1 or something silly like that. Of course, that leads Tetra to question why these things need to be so precise and arbitrary.

One of the things I realized about math is that it’s all about trying to do things with the definitions you start out with. You can kind of see that in how the number system is built up. We can start with plain old numbers that we use to count things. And then we can add things to it. Like, what happens if we have less of a thing, how would we represent that? Tada, we’ve got negative numbers. Okay, now what if we have a part of a thing? Now we’ve got fractions. And so on and so forth. We realize that not every number can be written as a fraction and suddenly we’ve defined real numbers.

I’m sure we’ve all wondered why complex numbers work out the way they do. It’s because we’ve defined everything to work out like that. Some guy tried to take the square root of a negative number and found that it didn’t work out very well, so we defined the square root of -1 to be $i$. Now we have this $i$ thing, what can we do with it? Well, we can just start writing crazy things like $3 + i + 4i^2 + 5i^3 + 9i^4 + 2i^5 + 6i^6$, but then we realize that it just becomes $2-2i$. Well, okay that’s kind of neat.

But now we know that all complex numbers can be written as $a+bi$ with $a,b\in\mathbb{R}$. So someone along the line must’ve been like, what would happen if we tried to graph these things? So we treat $a$ as the $x$ component and $b$ as the $y$ component and it turns out we can think of complex numbers as other structures like a vector or just a 2-tuple or something. And suddenly, this gives us a way to compare complex numbers, by taking the length of the vector that they define. And now that we have vectors, we can do some weird geometry stuff with them. We can think of these things as the length of the vector and the angle that they form. And then you can go crazy and talk about roots of unity or what multiplication of complex numbers might mean.

The point is that all of mathematics is built up like this. You start off with some definitions or premises and you go nuts. Of course, you don’t have to use the same definitions as everyone else. The problem with doing that is you might not end up with anything interesting. Like, what if we did something nuts and defined 1 to be a prime number like so many people do when they leave off that condition? Well, not much, we’d probably just rephrase all of our theorems to exclude 1.

On the other hand, sometimes when we play around with definitions, we do end up with something interesting. For instance, we can define something called the extended complex numbers, which is just the set $\mathbb{C}\cup\{\infty\}$. Yep, we just say okay, infinity is a number now, deal with it. So what can we do now?

Well, we can divide by 0 now.

I imagine there might be a few people who might flip out at this notion, but yes, since $\infty$ is included in our number system, we can define $\frac{1}{0}=\infty$. Of course, we can’t do everything — $0\times\infty$ and $\infty-\infty$ still don’t mean anything. But if you’ve been paying attention, you might be going, okay well, we can divide by 0, but what else can we do? Dividing by 0 is kinda meaningless if there’s nothing new we can do.

As it turns out, the extended complex numbers defines a very different geometric object. If we remember from the example of the complex numbers, we can basically define every complex number as a vector over a two-dimensional plane. Here, with the extended complex numbers, we can define every number as a line passing through something called the Riemann sphere. And like the complex plane, this sphere lets us create weird relationships between numbers and angles and stuff. This turns out to have interesting properties in complex analysis and quantum mechanics.

So yes, definitions are often arbitrary. Why? Because it’s just useful and interesting that way. You could argue that it’s because nature forces us to define things a certain way. Kind of like, of course you can’t take a square root of a negative number, you just can’t do it! What happens, though, is that we always seem to end up finding useful things that line up with our mathematics rather than inventing our mathematics to do useful things with. After all, mathematicians have been playing around with imaginary numbers for at least a few decades before electromagnetism was even discovered.