complex number power I have  question related to  power of i,which is   determined by equality  $i=\sqrt{-1}$
  actually from complex number  book  I know  that  $i^2=-1$, as much as  i know   if  we compare  physical meaning of  operations on real and  complex number, we get different pictures, like  while  operation on real numbers represents  like  compression or extension,for  example  $2*2=4$  extend  2  by  2 unit right,  the  power of  $i$  represent rotation by $90$ degree, but on the other hand  power of $i$  we can write into two different form, first like this                            $i^2=(({-1})^{0.5}))^2$   
or  $i^2=\sqrt{-1}*\sqrt{-1}$
while first we  can multiply  powers and  get  that  $i^2=-1$, second we can write
   $i^2=\sqrt{-1*-1}$    or  $i^2=\sqrt{1}$  which is equal to $1$ and   -$1$, so  why we are taking -$1$? Because of different physical application? Thanks
 A: Well, first thing your first claim that: $i^2 =((\sqrt{-1}^{0.5}))^2$ is wrong, it equals $i$.
The definition of $i$ is the solution of the equation $x^2+1=0$.
It's not meaningful what you wrote as $\sqrt{-1}\cdot\sqrt{-1} = \sqrt{(-1)\cdot(-1)}$ cause you use here a multiplication on $\mathbb{R}$, but $\sqrt{-1}$ is not a real number.
A: $$\sqrt{z}\times\sqrt{z} \ne \sqrt{(z)\cdot(z)}$$ for $z\in \mathbb{C}$


Because of the discontinuous nature of the square root function in the complex plane, the law $\sqrt{zw}$ = $\sqrt{z}\sqrt{w}$ is in general not true. (Equivalently, the problem occurs because of the freedom in the choice of branch. The chosen branch may or may not yield the equality; in fact, the choice of branch for the square root need not contain the value of √z√w at all, leading to the equality's failure. (A similar problem appears with the complex logarithm and the relation $\log z + \log w = \log(zw).$) Wrongly assuming this law underlies several faulty "proofs", for instance the following one showing that $–1 = 1$

~Wikipedia
