Divison by Zero

Question

I have seen multiple answers on the web, but I can't get my mind around why division by zero outputs an error and not zero. Can anyone explain this in laymen's terms?

Answer 1

Well, let's try it. Let's divide $x$ by $0$ and say it's equal to some number $y$. Then

$$\frac x0=y.$$

By definition, we may multiply both sides by $0$, and see $x=y\cdot0=0$. This isn't true; we didn't assume $x$ to be zero. However, this tells us the only number we can divide by $0$ is $0$ itself; indeed, $\tfrac00=y$ doesn't result in a contradiction.

The problem with saying $\tfrac00$ is defined, is that it is everything at once. It doesn't have only one value. $\tfrac00=1$ wouldn't reach a contradiction (not by multiplying by $0$, that is) and neither would $\tfrac00=0$. In mathematics, we like things to only represent one thing and one thing only. If we would let $\tfrac00$ mean both $0$ and $1$ at the same time, then we wouldn't be able to use $=$ like we want; if we did, then

$$0=\frac00=1\implies 0=1$$

So, that would be a hassle for just letting this thing that is everything at once exist. Hence, we just agree to not divide by $0$, so we can keep all our rules, axioms, and theorems we know.