The negative square root of $-1$ as the value of $i$ I have a small point to be clarified.We all know $ i^2 = -1 $ when we define complex numbers, and we usually take the positive square root of $-1$ as the value of "$i$" , i.e, $i =  (-1)^{1/2} $. 
I guess it's just a convention that has been accepted in maths and the value $i =  -[(-1)^{1/2}] $ is neglected as I have never seen this value of "$i$" being used. What I wanted to know is, if we will use $i =  -[(-1)^{1/2}] $ instead of $ (-1)^{1/2} $, would we be doing anything wrong? My guess is there is nothing wrong in it as far as the fundamentals of maths goes. Just wanted to clarify it with you guys. Thanks.

The square roots with the properties we know are used only for Positive Real numbers. 
We say that $i$ is the square root of $-1$ but this is a convention. You cannot perform operations with the usual properties of radicals if you are dealing with complex numbers, rather than positive reals. 
It is a fact that, $-1$ has two complex square roots. We just define one of them to be $i$.
You have to regard the expression $i=\sqrt {-1}$ just as a symbol, and not do operations. 
Counterexample: 
$$\dfrac{-1}{1}=\dfrac{1}{-1}\Rightarrow$$
$$\sqrt{\dfrac{-1}{1}}=\sqrt{\dfrac{1}{-1}}\Rightarrow $$
$$\dfrac{\sqrt{-1}}{\sqrt{1}}=\dfrac{\sqrt{1}}{\sqrt{-1}}\Rightarrow$$
$$\dfrac{i}{1}=\dfrac{1}{i}\Rightarrow$$
$$i^2=1\text { ,a contradiction}$$
