The reason that the result of a division by zero is undefined is the fact that any attempt at a definition lead to a contradiction.

To begin with, how do we define division? The ratio r of two numbers a and b, "r=a/b" is that number "r" that satisfies "a=r*b."

if we are trying to divide by zero, the condition above would be that "b=0", thus we have to find a number r such that "r*0=a", But "r*0=0" for all numbers "r", and so, unless "a=0", there is no possible solution.

Now you could say that "r=infinity" satisfies would satisfy the condition. That's a common way of putting things, however "Infinity" is not a number but a concept, because if we treated "infinity" like a number, more contradictions would arise. 
For example, what do we obtain when adding a number to infinity?

The common perception is that infinity plus any number is still infinity. If that's so, then "infinity = infinity+1 = infinity + 2" which would imply that "1 equals 2" if "infinity" was a number. That in turn would imply that all integers are equal, for example, and our whole number system would collapse.

I hope this answers your question.
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