Question. Find a square root of the complex number $\displaystyle i $.

Answer.

Let $\displaystyle i^2=w$. By inspection (of i), the modulus of w is 1; we need to find $\displaystyle \theta$.

$\displaystyle (\cos(\frac{\theta}{n}+\frac{2kn}{\pi})+i\sin(\fra c{\theta}{n}+\frac{2kn}{\pi}) $ where n=2 and k is 0 and 1 (from 0 to n-1).

Therefore there are two numbers w that, when squared, give $\displaystyle i$:

$\displaystyle \theta_1=\frac{\pi}{4}$ and $\displaystyle \theta_2=\frac{5\pi}{4}$

$\displaystyle w_1=\frac{1}{\sqrt{2}} + i \frac{1}{\sqrt{2}} $

$\displaystyle w_2=-\frac{1}{\sqrt{2}} - i \frac{1}{\sqrt{2}} $

The textbook answer, however, only gives one value of w ($\displaystyle w_1$). Am I wrong in thinking that there are two values of w? Or, the question has only asked for one of them ('a square root'), rather than 'find all square roots'?

PS I understand that there are always two numbers (one positive and one negative) that, when squared, give the same number. I only wanted to check whether the same principle applies to the questions about complex numbers.