The maths
We'll start off with a simple proof:
Lets say a=b
then a²=ab
and a²-b²=ab-b²
and since a²-b²=(a-b)(a+b)
it must be true that (a-b)(a+b)=b(a-b)
and if we divide out (a-b) we get a+b=b
and since we said that a=b it must be true that 2a=a
and if we divide out a we've proven that 2=1
Now I'm pretty sure everyone here knows that 2 does not equal 1. So where did we go wrong?
Our proof is true up to a²-b²=(a-b)(a+b), but after that it goes horribly wrong. Because what does a²-b²=(a-b)(a+b) say? Since we've said that a=b, then a-b=0. So in our proof we divided by 0 and we ended up proving that 2=1.
I'll try to make it a bit more clear how this problem arises. (Whenever I type "a/b" I mean a divided by b, if only the ning supported LateX input...)
So lets say we can divide by 0. Then we can say for sure that 1/0=1/0.
But we also know that 500*0=0 and the same for 2*0=0.
So we can also say that 1/(500*0)=1/(2*0).
We can rewrite the identity above as (1/0)*(1/500)=(1/0)*(1/2).
But if we then divide out 1/0 we end up with 1/500=1/2, which is clearly not true.
The problem with division by zero lies in the fact that every number multiplied by 0, is again 0. So if you allow division by zero all numbers are equal to each other and thus lose their meaning.
A slightly different perspective
We can also look at it from a slightly different (less purely mathematical) perspective.
Yesterday in the comments Book Chic said the following:
"I've always wondered why you can't do it. Because, in my mind, if you're dividing by nothing, then nothing should happen to the number. So if you're dividing 6 by 0, the answer should be six because you're not doing anything to it."
The nature of mathematical division is that a number is broken into parts, where all the parts equals the number you divide by and the answer (of the equation) equals the number of parts. But how would you divide 6 into parts of nothing? If you were able to do that you would be able to make the 6 disappear into nothing (because 0+0+0...+0=0). Lets say you have an apple pie (this reminds me of elementary school), in how many equal parts would you have to cut it before every part equals nothing? You can throw the pie into a wood chipper for all I care, but you're still not going to end up with parts that equal 0.
Infinity
Now you may have read or heard somewhere that if you divide something by zero it equals infinity (the symbol for infinity is ∞). This is NOT true. Infinity is not a number, it is a concept. What is true is that the limit of 1/x goes to infinitely high values as x goes to 0. You can see that if we plot the graph of y=1/x:
As you can see, if x approaches 0, the value of 1/x skyrockets, but never reaches 0 (because it can't). This is what we mean with the limit of 1/x goes to infinity as x goes to 0 (or in mathematical notation lim(x->0) 1/x=∞), But that doesn't mean that 1/0=∞!
So if you ever hear anyone say that you can divide by 0, laugh at them and please tell them that they're wrong.
Sources
http://home.ubalt.edu/ntsbarsh/zero/ZERO.HTM
http://en.wikipedia.org/wiki/Invalid_proof
PS: if you want me to blog about other mathematical subjects you can still leave your requests in the comments blow. Thus far I have Google, Chaos Theory and the Axiom of Choice.
