The function y = 1/x. As xapproaches 0 from the right, yapproaches infinity. As x approaches 0 from the left, y approaches negative infinity.
The one thing 0000 is not is “undefined”. “Undefined” means that there are no valid choices, which is precisely the opposite of the conundrum 0000 faces: everything is a valid choice, and there's no way to pick any one possible value as being a better choice than any of the others. It could be 0, 1, π, e, i, -5, ¾, negative one million, or any number of other possibilities: all work equally well as possible solutions.
(Proof: if 𝑥=00x=00, then 0𝑥=00x=0; what value, when multiplied by zero, gives you a result of zero? All of them.)
So we say that 0000 is indeterminate. “Indeterminate” means that you cannot determine what the value is: there's an infinitude of equally valid possibilities and no way to choose one over the others.
This is similar to being undefined in that both bring your calculations to a screeching halt; but the reasons why are diametrically opposed. If your calculations ever become undefined, you have nowhere to go; if they ever become indeterminate, you have no way to know where to go.
Now, calculus provides a way to do an end-run around both of these issues: you can look at a similar problem that is very near to the point that's undefined or indeterminate, and often that problem has a distinct answer that can be used as a reasonable substitute for the impossible answer. This is done through a process known as limits (where you add an arbitrarily small amount, often called 𝜖ϵ, to one or more parts of the equation), and it's a common mistake to think that a limit lets you solve the otherwise unsolvable problem. To be clear, it doesn't; 0000 remains indeterminate, and 1010 remains undefined, even if 0+2𝜖0+𝜖=20+2ϵ0+ϵ=2 and 1+2𝜖0+𝜖→∞1+2ϵ0+ϵ→∞.
EDIT: since writing this answer, I've come to a clearer understanding of the situation. Rather than saying that one case is undefined and the other is indeterminate, it's better to say that both are undefined but for opposite reasons: 1010 is undefined because nothing is a valid solution, while 0000 is undefined because everything is a valid solution. It's still fair to call the latter “indeterminate”; but the more precise understanding is that “indeterminate” is a type of “undefined”, not an alternative to it. If I had to peg a word for the opposed type of “undefined”, I'd probably go with something like impossible.
File:Division by zero on android 2.2.1 calculator.png
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