# Division by fictional number

I faced a trouble separating by fictional numbers lately. I was attempting to streamline :

$2 \over i$

I thought of 2 approaches, which generated various outcomes :

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 recognize from making use of the formula from this Wikipedia article that method 1 generates the proper outcome. My inquiry is : **why does method 2 offer the wrong outcome **? What is the void action?

The wrong action is claiming:

$\sqrt{4}/\sqrt{-1} = \sqrt{4/-1}$

The identification:

$\sqrt{a}/\sqrt{b} = \sqrt{a/b}$

is just warranted when $a$ and also $b$ declare.

This is specifically the very same concern as in this question. Each non - absolutely no intricate number has 2 numbers that make even to offer it, with the very same size, yet with contrary indicator. When we specify the square origin function, we need to determine which of the origins we desire. For favorable numbers, it is noticeable to pick the favorable origin. For adverse number, we pick to have the favorable fictional values, although as a result of proportion the selection does not suggest much anyhow.

So, to see if the typical reproduction and also department regulations use, after that we need to take into consideration domain name the numbers remain in. We currently recognize they look for non - adverse actual numbers. It is very easy sufficient to validate that for adverse numbers sqrt (a) *sqrt (b) = - sqrt (abdominal muscle) and also sqrt (a)/ sqrt (b) = sqrt (a/b). We additionally see that, if a declares and also b adverse, after that sqrt (abdominal muscle) = sqrt (a) sqrt (b), sqrt (a/b) = - sqrt (a)/ sqrt (b) and also sqrt (b/a) = sqrt (b)/ sqrt (a).

The only soundproof means to ensure to locate the appropriate outcome while separating 2 intricate numbers

(a+bi)/ (c+di)

is lowering it to a reproduction. The solution is of the kind x+yi ; consequently

(c+di) (x+yi) = a+bi

and also you will certainly wind up with 2 straight formulas, one for the actual coefficient and also an additional for the fictional one. As Simon and also Casebash currently created, taking a square origin brings about troubles, given that you can not make certain which value has to be picked.

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