I have read the Wikipedia articles (http://en.wikipedia.org/wiki/NaN and http://en.wikipedia.org/wiki/Division_by_zero) and I have come to a few indefinite conclusions.
1) 0/0 is not equal to 1
2) Infinity/Infinity is not equal to one
3) x/x = 1
So I have decided to redefine zero and infinity, and x.
Zero: The absence of any number, real or imaginary.
Infinity: The limitless expanse of numbers in all dimensions.
x: Any fathomable number, excepting when x does not equal 0.
If we take 0 x 1 = 0 x 2, it seems to make sense. However, when 0/0 x 1 = 0/0 x 2 is equated, it would seem that the equation would simplify to 1 = 2, obviously not correct. This may prove that 0/0, unlike x/x, does not equal 1.
If we had zero objects and wished to divide them amongst zero people, we would be handing out nothing to no one for eternity. (Obviously we would not bother in real life, but let us assume this just for the purpose of this example.) So I draw the conclusion that 0/0 = infinity.
I could now ambitiously define 0 as being the opposite of infinity (referring back to my definitions 1) and 2).
So if 0/0 = the opposite of zero = infinity, does this solve the paradox infinity/infinity? For if infinity is the opposite of zero, then surely infinity/infinity = the opposite of infinity = 0.
But then I am totally bamboozled by this:
If x/0 = infinity, then logically, x must be equal to 0 x infinity, right?
Does that mean that 0 x infinity can be expressed as any fathomable number?