Expert: Ahmed Salami - 6/24/2004

Question
Someone gave me this proof that one is equal to negative one. Can you tell me what's wrong with it? There must be something wrong with it. I'm going to use [ ] to mean square root of, as it makes it much easier to type.

(-1) = (-1)
(-1)/1 = 1/(-1)
[(-1)/1] = [1/(-1)]
[(-1)]/[1] = [1]/[(-1)]
[1]x[1] = [(-1)]x[(-1)]
[1]^2 = [(-1)]^2
1^2 = i^2
1 = (-1)

I can't find anything wrong with it, other than the fact that one does NOT equal negative one.


Answer
Hi Alia,
I really do understand your concern.
This is not the first time i have come across something like this. I think the problem has to do with the nature of roots generally.
The proof you provided is only one of a lot that do question the definition of the complex number i.
Let me show you another example,
i^5 = i
but i^5 = (i^4)^(5/4) = 1^(5/4) = 1
Now, how do we explain this one? The definition of i is still our main problem.
I think this remains one of the unresolved problems in mathematics.
Anyway, i would say you shouldn't worry too much about it for now. Just be sure that 1 is not -1.
I hope this helps you.
Regards.