Sqrt (i) much faster answer

When dealing with nth roots of a complex number z:

There are always n of them, each of which of which is just as equally valid as all the rest. As far as square roots are concerned there are always 2 of them. An important property of square roots of complex numbers is that they will always negatives of each other.

An very common misconception is in the definition of i itself. You see many people (and some text books for that matter) define i as the sqrt(-1). While this is not technically incorrect it is, however, INCOMPLETE. i is just one of them.

The sqrt(-1) is also equal to -i, and -i is just as valid a sqrt(-1) as i is.

A complete definition of i :

i is the symbol that has the property that i^2 -= -1 which leads to (-1)^1/2 = +/- i

Remember, there is no such thing as the "principal" square root of a complex number. The term has no meaning here as you cannot single one of them as "principal". If I had my way we would not use the radical sign at all when dealing with complex numbers as it is a symbol which leads to misunderstanding. Use ^1/2 instead.

The original post is correct. The photo is incomplete and therefore misleading.

Faster answer:

I have a faster answer. (1+i)^2=1+2i-1=2i. So i=(1+i)^2/2. Therefore sqrt(i)=(1+i)/sqrt(2)=1/sqrt(2)+1/sqrt(2)i. I went off of your answer though, so thanks.