0 divided by 0...

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I think the question was already correctly answered above. Just this:

Dividing "0 / 0" is "worse" than dividing, say "1 / 0".
In algebra, they say that dividing by 0 never makes sense in a field. It's just not defined.

You sometimes here people saying "well, if you divide by zero, you get infinity", which is not that wrong...
Even mathematicians sometimes define "1 / 0 = ∞". This is OK, if you know what you're doing.
The idea behind it is that you consider limits. The limit for "1 / x" with "x → 0" is ∞ if you require "x > 0" (otherwise, the limit doesn't exist).
You can do this, but you shouldn't be surprised that "1 / 0" and "2 / 0" are the same thing then (namely ∞). That's why you have to be careful when solving equations.

But "0 / 0" is worse. You can still try limits and consider "x / y" for "x → 0" and "y → 0" (with positive x,y). But then you still don't know... If you apply "x → 0" first, then the answer should be 0 (because any fraction with 0 in its nominator is 0). If you apply "y → 0" first, then the answer should be ∞.

tl;dr: Don't divide by zero. If you insist, don't be surprised about the implications.

Here's my non-mathematical answer. It's completely incorrect (mathematically), but I like it.

If you think of zero as nothing:

"4 / 0 = ?" is like saying "how many nothings are there in 4?". There can't be any nothings, because there is something. If there is something there cannot be nothing. So the answer is zero.

"0 / 0 = ?" is like saying "how many nothings are there in nothing?". Well, there's one. One whole nothing. There aren't any more or less than that, because you are not asking "how many nothings are there in two nothings?" or "how many nothings are there in an infinite number of nothings?" or "how many nothings are there in half a nothing?". And there aren't no nothings, because you've just referred to a nothing.

Therefore, 0 / 0 = 1.

I know, it's pathetic. And this is a maths forum so I'll be leaving now

When you divide something by any number less than zero it is the same as multiplying by that number's reciprical . Thus dividing by one third is the same as multiplying by three.

The smaller the sub-one number your are using as a divisor the closer you get to having the effect of multiplying by infinity.

Thus if you divide by zero you in effect "multiplying by infinity."

Three over zero is infinity.

But when you divide zero into zero- you are in effect muliplying zero by infinity.

Infinite amount of nothing is....still nothing.

So dividing any non zero number equals infinity (because you are in effect multiplying by infinity). But dividing zero by zero is zero ( not one- but zero!).

It was creative at least. I'll give you that.

I can easily say, though, that 4 / 0 is 4 using your argument.

If there are no nothings, then there's a something. That something is 4.

There's another way to look at the problem, which is to talk about the limits of infinite sequences. The problem is that there are good limit arguments for all of the possibilities in the original post. So it's wise to just not define it, unless you have some particular goal in mind. Here are limit arguments for all four possibilities:

Consider the infinite sequence
(0/2, 0/1, 0/0.5, 0/0.25, ...)
where the top of the fraction is always 0, and the bottom of the fraction keeps getting halved.
All of the things in this sequence are actually just 0, so the sequence is approaching 0.
When we look at the pattern of the sequence, we see that it is approaching "0/0".
So we want 0/0 = 0.

Consider the infinite sequence
(2/2, 1/1, 0.5/0.5, 0.25/0.25, ...)
where the top of the fraction and the bottom of the fraction keep getting halved.
All of the things in this sequence are actually just 1, so the sequence is approaching 1.
When we look at the pattern of the sequence, we see that it is approaching "0/0".
So we want 0/0 = 1.

Consider the infinite sequence
(2/32, 1/8, 0.5/2, 0.25/0.5, ...)
where the top of the fraction keeps getting halved and the bottom of the fraction keeps getting quartered.
This sequence is actually (1/16, 1/8, 1/4, 1/2, 1, 2, 4, 8,...), so the sequence is approaching infinity.
When we look at the pattern of the sequence, we see that it is approaching "0/0".
So we want 0/0 = infinity.

Pick any number x.
Consider the infinite sequence
(2x/2, x/1, 0.5x/0.5, 0.25x/0.25, ...)
where the top of the fraction and the bottom of the fraction keep getting halved.
All of the things in this sequence are actually just x, so the sequence is approaching x.
When we look at the pattern of the sequence, we see that it is approaching "0/0".
So we want 0/0 = x.

I had left, but I can't resist responding.

It was creative at least. I'll give you that.

I can easily say, though, that 4 / 0 is 4 using your argument.

If there are no nothings, then there's a something. That something is 4.

I guess Australian math is different, kind of like how water drains differently.