What is square root of i?

http://www.wolframalpha.com/input/?i=sqrt(i)

Hello Max,

you may find this interesting to read

https://www.math.toronto.edu/mathnet/questionCorner/rootofi.html

(-1)^(1/4)=

0.70710678118654752440084... +
0.70710678118654752440084... i

$\text{Why is }\sqrt{i}=\frac{1+i}{\sqrt2}? \text{I still don't understand how to obtain this answer?}$

Here is a direct algebraic answer:

Suppose that z=c+di, and we want to find √z=a+bi lying in the first two quadrants. So what are a and b?

Precisely we have:
a=Sqrt[c+sqrt(c^2+d^2)/2]

and

b=d/|d|Sqrt[-c+sqrt(c^2+d^2)/2]
(The factor of d/|d| is used so that b has the same sign as d) To find this, we can use brute force and the quadratic formula. Squaring, we would need to solve:
a^2−b^2+2abi=c+di.
This gives two equations and two unknowns (seperate into real and imaginary parts) which can then be solved by substitutions and the quadratic formula.

Hope that helps,