TITLE: Division by imaginary number
QUESTION [14 upvotes]: 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?

REPLY [5 votes]: The only soundproof way to be sure to find the right result while dividing two complex numbers
$$\frac{a+bi}{c+di}$$
is reducing it to a multiplication. The answer is of the form $x+yi$; therefore 
$$(c+di)(x+yi) = a+bi$$
and you will end up with two linear equations, one for the real coefficient and another for the imaginary one. As Simon and Casebash already wrote, taking a square root leads to problems, since you cannot be sure which value must be chosen.