A question that frequently comes up in the insalubrious sort of place a mathematician might hang around is, what is that value of $0^0$. We generally sigh and answer that the same way every time.
It was nice, then, to see someone ask a more fundamental one: what is $0 \div 0$?
The short answer is, it’s not defined, even though $0 \div a = 0$ pretty much everywhere. But why?
There are probably dozens of explanations for this. My favourite is to look at what division means.
$a \div b$ asks the question “what do you multiply by $b$ to get $a$?” So, $6 \div 2 = 3$ because $3 \times 2 = 6$.
In particular, $0 \div 0$ asks “what do you multiply by $0$ to get $0$?” The answer to that is “anything at all” – which is a Problem.
Basic arithmetic relies on operations giving unique answers to questions – if $2+2$ was both 4 and 5, we’d be in a terrible state. $0 \div 0$ can’t possibly be 0 and 7 and $\pi$ and Graham’s number and every other value you can multiply by 0 to get 0, for exactly the same reason.
In short, it’s undefined because the answer could be anything, and that’s not allowed.