Question

why do we say that -1/1=1/-1
if we take square root on both sides it becomes i/1=1/i _______1
and i=1
but actually left and right hand sides of equation 1 are not equal and i is never equal to -1

Answer 1

....

Answer 2

you are making the assumption that

\[\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\]

holds for all a,b (b not zero). This is not the case

Answer 3

There is no theorem says a/b = b/a
this is equal just because it is specified as 1.