My theory

[Banana fanna fo fanna]

for all n > 0, n / 0 = infinity
for all n < 0, n / 0 = - infinity
for n = 0, n / 0 = 1

Thoughts?

[Rule]

for all n > 0, n / 0 = infinity
for all n < 0, n / 0 = - infinity
for n = 0, n / 0 = 1

Thoughts?

Well, I think you're assuming n is a real number.  If you are, then I already think about the first two (magnitude wise) the same as you do, although the "+" and "-" before your infinities don't seem natural to me.  For example,  let z = n/epsilon,
epsilon --> 0+,   then  z --> infinity,  but if epsilon --> 0-,  z --> -infinity.

Also, your idea of "0/0 =1," on the other hand, while may be a little intuitive, is dangerous.   Let's say we have the equation x + 0*y = z.  You couldn't say then 1+1..+1....+x*infinity + y = z*infinity + 1 + 1 + 1 + 1 + 1 + ....     
Well, perhaps I shouldn't say "couldn't," because you might argue that you can subtract the infinite number of 1s from both sides (after all, they are "equal" infinities).  Aside from things getting a little messy, I think there are too many ways to approach 0/0 that lead to different results to let 0/0=1.

Generally I think it's good to think of these things intuitively (if you're careful), rather than being blinded by the superfluous rigour that often underscores statements like "does not exist."

[Yoni]

What Rule said is precise.

For the first 2 statements, you could define an additional symbol, "+0" - for positive zero (i.e., infinitesimal approaching zero from the positive side).
Then you can say n/+0 = infinity for positive n.
You can define a "-0" similarly. I've seen some books in calculus actually use this notation, though only briefly - it's not popular.

For 0/0, here are some examples why you can't say it's 1. All of these can be written in formal notation but I will skip that.

1) If 0/0 is 1, then should (0+0)/0 be 2? Should (0-0)/0 be 0?
2) If 0/0 is 1, then should (0*0)/0 = 0*(0/0) be 0?
3) If 0/0 is 1, and 1/0 is infinity, then should 0/(0*0) = (0/0)/0 be infinity?

In fact, I have shown that expressions of the form "0/0" can take any real value or +/- infinity.
The same is true for expressions of the form "infinity/infinity". Study calculus for more information.

[rabbit]

I've always had the pie system in my brain, where dividing n by 0 gives n, since you're not splitting it at all (equivolent to not cutting any slices).  Though, with this system, negatives don't really work, at least when you think of the entire thing in terms of pie.

[Yoni]

That brings up a question of mathematical logic.

If you divide n pieces of pie to 1 person, he gets n pieces of pie.
If you divide n pieces of pie to 0 people, each person can get any amount of pie and it doesn't even have to add up because no people get any pie.

And in formal logic:
Every element of an empty set satisfies any proposition.
i.e., every student in this empty classroom is from Mars. This formally is true (even if you assert that students exist and no student is from Mars).
Or: Every person who was given some of the pie, got infinite pieces of pie. This is true since no people got any pie. But it is also true that every person who was given some of the pie got 3 pieces of pie. So n/0 is undefined by the pie analogy as well.