TITLE: What is the meaning of $\frac{0}{0}$?
QUESTION [7 upvotes]: I asked my teacher what is the real meaning of $\cfrac{0}{0}$, and the answer I got was "nobody knows".
I can't leave this subject "as is". I need a decent explanation, at least an explanation to why "nobody knows". I'm sure you'll come up with a few good answers.

REPLY [2 votes]: Well, if $0/0$ is well-defined then it should play nicely with our definition of division. Specifically, $a/b = c$ if $a = cb$. So there's gotta be some c such that $0c = 0$. Let's try 1. $0\cdot 1 = 0$, so $0/0 = 1$. 
Not too bad. Now let's try two. $0\cdot 2 = 0$, so $0/0 = 2$. Since equality is transitive, $1=2$.
Aaaaaand we just broke math. We resolve this by saying that $0/0$ is not defined, which keeps everything running nicely with a minimum of fuss.