How do I find the square roots of #i#?

Question

3300 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2014-10-26T09:59:54

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

#z^2=i Rightarrow r^2e^{i(2theta)}=e^{i(pi/2+2npi)}#

#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.