At this point my mathematics teacher would hesitantly say "There are many different types of infinity," whilst scratching his head trying to think of a way to simplify his thoughts to something we could understand.

My other mathematics teacher would have slaughtered you (or me) before we even mentioned zero, let alone dividing by it.

Ralix, you say that: "Anything divided by zero is infinity. This is obvious when you start dividing a number by increasingly smaller numbers."

However this is an oversimplification, and quite incorrect. What you actually mean to say is that the limit of 1/x (as x approaches zero) would approach infinity.

What is zero divided by zero?
Well, it could be anything you want it to be, really... Keep in mind that anything divided by zero is undefined, so what you actually mean to say is:
What is the limit of x/y as x, y approach zero?

If you think about two numbers which are getting closer and closer to zero, divided by each other, the result could be anything, depending on how quickly they each approached zero.

Take this example:

2x/x = 2    (right?)
But surely the limit of 2x as x approaches zero would be zero.
Similarly the limit of x as x approaches zero is also zero!

So you see your quotient is entirely dependant on how quickly your divisor and dividend are converging to zero.

Another more complex example would be the following:

Evaluate the limit, as x approaches zero, of:

sin(x) - x
sin(x) - x.cos(x)

At first look you'd probably think "sin(0) - 0 == 0 - 0 == 0, and if the numerator is 0 then the whole thing is zero, right?" Wrong. As I said, it's all to do with the rate of convergence...

sin(x) can be approximated by the following infinite series:

sin(x) ~ x - x3/3! + x5/5! - ... + (-1)r ( x2r+1 / (2r+1)! ) + ...

Hence it is reasonable to assume that for x approaching 0,

sin(x) ~ x - x3/3!

The index of x increases by two with each term, and x is approaching 0, which means the significance of each subsequent term diminishes very quickly... so we can ignore anything with a coefficient of x4 or less.

Similarly we can say that:

cos(x) ~ 1 - x2/2!

If we put these together, by substituting into the original we get:
(x - x3/3!) - x
(x - x3/3!) - x.(1 - x2/2!)

==         - x3/6
( - x3/6) + (x3/2)

==      (-1/6)
(-1/6) + (1/2)

== (-1/6)
(1/3)

== -1/2

Not quite the zero you were hoping for, was it?

I wouldn't call myself a mathematician, so if there's anything wrong with the above don't hesitate to /msg me.