I ran into a problem dividing by imaginary numbers recently. I was trying to simplify:
$2 \over i$
I came up with two methods, which produced different results:
Method 1: ${2 \over i} = {2i \over i^2} = {2i \over -1} = -2i$
Method 2: ${2 \over i} = {2 \over \sqrt{-1}} = {\sqrt{4} \over \sqrt{-1}} = \sqrt{4 \over -1} = \sqrt{-4} = 2i$
I know from using the formula from this Wikipedia article that method 1 produces the correct result. My question is: why does method 2 give the incorrect result? What is the invalid step?