How do I find the square roots of ii?

1 Answer
Oct 26, 2014

Let z=re^{i theta}z=reiθ be the square-roots of ii.

z^2=i Rightarrow r^2e^{i(2theta)}=e^{i(pi/2+2npi)}z2=ir2ei(2θ)=ei(π2+2nπ)

Rightarrow {(r^2=1 Rightarrow r=1),(2theta=pi/2+2npi Rightarrow theta=pi/4+npi):}

z={e^{i pi/4}, e^{i {5pi}/4}}

={cos(pi/4)+isin(pi/4), cos({5pi}/4)+isin({5pi}/4)}

={1/sqrt{2}+1/sqrt{2}i, -1/sqrt{2}-1/sqrt{2}i}


I hope that this was helpful.