Question

What is the square root of i (the imaginary unit)?

Answer 1

hhmm - so embarasing - i knew how to do this once....

Answer 2

lol - ty

Answer 3

The sqrt{i} must equal a complex number of the form a+bi such that (a+bi)^2=i. We just have to find a and b.

Square the LHS: a^2+2abi-b^2=i
Or in standard form: (a^2-b^2)+2abi=i
where
a^2-b^2=0 (real part) and
2abi=i (imaginary part) which simplifies to 2ab=1.

Consider the real part: a^2-b^2=0
so it follows that a^2=b^2
and so a=+/-b.

But since 2ab=+1 we must have a=b or -a=-b, which simplifies to a=b.

If 2ab=1 and a=b then 2aa=1 or a^2=1/2,
and
a=b=+/-sqrt{1/2}.

So there are two square roots of i:

sqrt{1/2}+sqrt{1/2}i and -sqrt{1/2}-sqrt{1/2}i

Answer 4

which is why it isn't easy to define the complex square root if you want a function.

Answer 5

there is an easier geometric way to do it but I can't remember how! some easier!