Making Sense of Division by Zero

Hello there! This isn’t going to be very long, but make yourself comfortable anyway.

When I was in school, we used to live in a tiny single-bedroom apartment. As you would expect in such a small apartment, it was quite full of things. One of the games I used to play was to find a way to go from one point to another without touching the ground. Some version of _The Floor is Lava_, but that sounds childish. I would rather go with _Crocodiles!_ It was fun. If it was too easy, I would add extra constraints – do it without touching a certain wall, or maybe within a fixed time, or whatever other idea came to mind. If it became too difficult, to the degree that it stopped being fun, I would similarly let myself break a rule here or there, or perhaps just change the destination to something more doable.

Turns out maths is fundamentally no different from this concept! We make a rules, like the rules that govern how addition takes place. Given numbers $a,b,c$, we specify that $a+b = b+a$ and $a+0 = a$, and $a+(b+c) = (a+b)+c$. Those are the rules of addition.
It’s arguable that Uno’s rules are more complicated than those of group theory. Try looking them up! If you do, you’ll notice that the rules of a group do not require that $a+b$ must be the same as $b+a$. In fact, the groups in which $a+b = b+a$ is always true have a special name for themselves: they are called Abelian groups.

But wait. Isn’t it a fact that $a+b=b+a$? Isn’t that just addition?
To that, I say, exactly. Addition is just a bunch of rules. In group theory, the plus sign ($+$) is just a symbol to denote some operation. It could be multiplication of numbers, the rotation of a square, or just plain actual addition.

Group theory doesn’t care what the operation is, as long as that operation follows the rules. And that’s what makes things interesting!

Look at any sport. It’s another bunch of rules. I’ll take lawn tennis for example, but you can think of any sport familiar to you. If tennis had too many restrictions for the size, shape, and material of the racket, then there would be no room left for exploring new ways to play. If there were too few rules, then it would lead to chaos: the computer-assisted, hydraulic rackets will end up competing against each other, leaving the players’ aside. Cool, but not interesting, nothing new, and not tennis.

Group theory, like tennis, has enough rules to be interesting, but not too much to be a dead end. These rules allow us to study a whole class of things, because they all happen to be following the rules of group theory.

Take for example, $\R$, the set of all real numbers. This, paired with the addition operation, is a group. It also happens to be a vector space! But vector spaces and groups are governed by different rules – different sports!

Here’s where the sport analogy ends. There’s no question of winning or losing here. The point is that we are the ones who make the rules of mathematics. Then we are the same ones that try to see what happens if we stick to them.

That brings me to the title. According to the rules of real numbers, division by zero is undefined. Reals are not the only numbers, by the way, there's other sets of numbers that have more, less, or different rules! $\R$ doesn't allow square roots of negative numbers, but $\mathbb C$ does I used to wonder why we didn’t just define it then. Unfortunately, there is no way to define division by zero without breaking our own rules. We can associate any physical intuition, or set it to something based on the context, but in the pure mathematical setting, there’s no answer that fits.

Dividing a real by a real must be a real. So which real number should $1\div0$ be? There is no such real number as infinity. Don’t get me wrong, there are ways to treat infinity like a number, but that will involve breaking rules too.
Can $1\div0=0$? That would then mean that $\frac10< \frac12$, and then following rules of real numbers, we would have $0>2$. This is a problem if we also want real numbers to follow our usual notions of numbers and counting.

So then the answer can’t be $0$.

Trying to set $1\div 0$ equal to any other real leads to roadblocks just like this one.

And so we leave it undefined, that is, $1\div0$ is not a real number.

What next? Real numbers have rules that make them useful to real world? Sometimes we need to make sense of situations when we reach a division by zero. What do we do then

Well, let’s see what happens when we divide by numbers that are very small, but not zero.

Before anybody has a chance to jump to conclusions, I want to show this:

We seem to have some pattern here! We can actually build some more rules to make sense of such patterns, which we shall do in the next chapter called limits!