1!=-1

(-1)^2=1
(1)^2=1

This does not mean that -1=1.

I think your first move

1=sqrt (1)

should be

1= THE POSITIVE VALUE OF (Or MODULUS OF) sqrt(1)

By glossing over that you have generated a false paradox since
MOD -1 = 1

1 = sqrt(1) (ok, sqrt is positive real sqrt)
sqrt(1) = sqrt((-1)(-1)) (ok, sqrt is still positive real sqrt)
sqrt((-1)(-1))) = sqrt(-1)sqrt(-1) (not ok, left sqrt is positive real sqrt, right sqrts are complex sqrts)
sqrt(-1)sqrt(-1) = i*i (ok, sqrt is complex sqrt)
i*i = -1 (ok)

I agree, drewdad, but where did I break the law?

hmmm fresco I don't understand your concern. Maybe you could dumb it down a little?

markr, I also think the prob is in that step. But I did not know that left sqrt being positive real and right sqrt being complex means they are not equal. Do you have anything to back that up, or is that something that is intuitive to you?

My point is that sqrt(1) is both 1 and -1. You cannot simply take the one definition and ignore the other.

I suspect you're missing an absolute value notation somewhere.

More like:

1=|sqrt(1)|=|sqrt[(-1)(-1)]|=|sqrt(-1)sqrt(-1)|=|ixi|=|-1|=1.

yes that's what I was looking for thx

Powers and Radicals
Fresco hit it on the head, but I would like to add the following:

The square root of any positive number necessarily has two values, one positive and the other negative. You must insure that your equation is a true statement.
Consider the equation x^2 = 1. The solution set is {-1,1}. That is, there are two possible but exclusive answers, x= -1 OR x= 1.
The same applies for numerical expressions (those without variables):
sqrt(1)= -1 or sqrt(1)= 1. You cannot use both in the same argument.

Simply, it is forbidden.