Consider the basic algebraic principles: they state that if one number or expression is equal to another, then you may apply basic arithmetic operations to one side as long as you mirror them on the other. For example, the expression 7=14/2 is mathematically true. Multiplying both sides by 2 will give you 7x2=14 which of course is also true according to our number system. More generally, if a=b then ac=bc as well.
We can also apply this principle to the expression 0/0. Say, we are assuming that we do not know the value of 0/0, so we will set the expression equal to the letter X. So now X=0/0. Applying the principle that multiplying zero by both sides will result in an equal expression, we do so, giving us Xx0=0. But wait a minute; the value of any number multiplied by 0 is equal to 0, so can't X take on an infiinte set of values, not just 0? Strange as it may seem this is mathematically true, however we can't simply leave the definition of 0/0 as "all real or imaginary numbers", since this goes against the rules for forming legitimate funtions.
If we define the function f(X)=X/0, we should be able to set Y=X/0. However, this means that again Yx0=X, which as we have seen means Y is equal to any possible number. This cannot happen however, since the function isn't valid. It is not a many-to-one or one-to-one function. It is in fact a one-to-many function meaning that in our system of mathematics it is undefined due to the fact that for the value X=0, we get more than 1 equally likely result. So, in actual fact 0/0 is not 0, it is undefined.
There are a few strange things to note however, about multiplying and dividing by 0. The first of these things is that if Y=X/0, then if X takes on any value other than 0, the resulting value for Y is infinity.
So for example, if X=3, then Y=3/0, which of course is infinity. This becomes stranger however when we take Y as infinity in terms of the algebra.
if ∞=3/0
∞x0=3
and
0=3/∞
Now, that last statement makes sense: if 3 is divided infinitely, then it will approach 0. This at least seems feasible in terms of convergence to a limit. However, what about the statment ∞x0=3. Haven't we already assumed that any number multiplied by 0 is equal to 0? Infinity however multiplied by 0 seems to be equal to an infinite number of differnent numbers, as with the value of 0/0. So does this not mean that since both expressions are equal to the same vast range of different values, we can set ∞x0=X, where X comes from the expression Xx0=0, so X=0/0 and has a range of different values, just as ∞x0 has. However, since both expressions may take on a range of different values, then at the same time as ∞x0 is equal to 3 in one case, X may be equal to any number, for example 4, in the other. We then come out with the strange result 3=4. Now, this proves at least that many to one functions cannot exist at the same time as our normal number system, since strange results occur. The only solution to the problem is to set 0=∞. Now everything begins to seem more rational. By saying this we can now say that ∞=3/0 is not valid in the first place, since any number is 1 divided by itself, and that value only. So the only acceptable value for ∞=X/0 is X=0 (or X=∞ since we are now assuming 0=∞). This now becomes a much simpler 1 to 1 function, which is acceptable in mathematics.
Our other problem of Xx0=0 now becomes one which is much easier to solve. If 0=∞ then Xx∞=0 and the only value that X can now take is 0. This also means however that strangely ∞^2=0.
In summary, the strangeness of the whole dividing by zero thing, seems to have a logical solution: that 0 and infinity are one and the same. If you have read Godel, Escher, Bach by Douglas Hofstadter, you will be familiar with the term strange loop - and this is one. A strange loop is essentially a figure which comes back to its original value or state, as you get further and further away from the original: a strange sort of paradox. It is a weird concept, but one which perhaps could be used to explain why the universe seems to curve in a strange way I have realised: since it has been said that the universe's conept of space is undefinable in physical terms since if you travel infinitely far away from the place you are at then after an infinite distance you will come back to position 0 (or back to where you started). But of course it would be impossible for us to ever reach that infinite distance, since we would need to be able to travel at an infinite speed, and one thing is for certain: the speed of light is the limit.
Drawing Hands by M.C.Escher - a drawing which embodies strange loops
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