Imaginary Numbers and Properties: A Puzzling Case

[[C]alculus]

-1/1=1/-1
sqrt(-1/1)=sqrt(1/-1)
i/1=1/i
i*(i/1)=i*1/i
i^2/1=i/i
-1/1=1
-1=1 <-- Well my conclusion is that properties don't work with imaginary numbers or did i do something wrong?

 

[arildno]

Lines 2-to-3 wrong.
Don't bother to post similar riddles at these forums.

 

[kntsy]

Lines 2-to-3 wrong.
Don't bother to post similar riddles at these forums.

Why\sqrt\frac x{y}\not =\frac{\sqrt x}{\sqrt y} ??

 

[Gib Z]

If x and y are both positive real numbers that is true, but not in general for x and y being complex numbers.

 

[arildno]

Why\sqrt\frac x{y}\not =\frac{\sqrt x}{\sqrt y} ??

To expand on GibZ's comment.
In order to extend the square root operation to work on real&complex number, and not just on the positive reals, you will not be able to retain <i>all</i> properties of the square root operation if you wish it to be self-consistent, and not self-contradictory.