0 divided by any number always give you 0 but
any number divided by 0 is imposible...
Think of it this way, Anon.
Pretend you had four quarters of one pizza, or 1/4 for each quarter. You could divide the pizza so that it mysteriously 'vanishes', or so that when the pizza is served, well, there is no pizza to give out. Simply put, this means that there are zero quarters of pizza to give to three other people and yourself. This is why 0/4 is perfectly normal, and equals zero. Sadly, taking all four quarters for yourself will probably leave you with three angry friends and one satisfied (and probably stuffed) self.
However, if you have four slices of pizza and three friends plus yourself, you can't just not give the pizza away, even to yourself. This option is not feasible, and thus is invalid. I like to think of it as the pizza goes nasty, and thus nobody wants it, and it isn't good. This is why 4/0 is undefined, and this is simply because you can't just have something that is givable without giving it to something.
It's my weird way of thinking about it.