Problem
Find two distinct square roots of i.
Solution
First write i in complex exponential form, then take it’s square root and then use euler’s formula, as such: $$\sqrt{i} = \sqrt{e^{\frac{\pi}{2} i}} = \pm e^{\frac{\pi}{4}i} = \pm \cos\left(\frac{\pi}{4}\right) \pm i\sin\left(\frac{\pi}{4}\right) = \pm \frac{\sqrt{2}}{2}(1 + i)$$ Another approach is to let the square root of i equal a + bi and then solve for a and b as follows: $$ \sqrt{i} = a + bi \to i = (a + bi)^2 = a^2 - b^2 + 2abi$$ From which it must follow that: $$a^2 - b^2 = 0$$ $$2ab = 1$$ Therefore $a = b$ and $ab = 1/2$. Implying that: $$ a^2 = b^2 = \frac{1}{2} \to a = b = \pm \frac{\sqrt{2}}{2} $$ And we’re done.