Question

Can someone explain why (sqrt3+i)^2 is 4e^i(pi/3) instead of 4e^i(2pi/3) when I try to solve it using De Moivres formula?

Answer 1

\[\sqrt3+i=\sqrt{(\sqrt3)^2+1^2}\bigg(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\bigg)=2\bigg(\cos\frac{\pi}{6}+i\sin\frac{\pi}{6}\bigg)\]
and squaring gives
\[(\sqrt3+i)^2=4\bigg(\cos\frac{2\pi}{6}+i\sin\frac{2\pi}{6}\bigg)\]
by DeMoivre's theorem. Then \(\dfrac{2\pi}{6}=\dfrac{\pi}{3}\).

Answer 2

In the complex plane:

Answer 3

Thanks a lot! I had the argument wrong...

Answer 4

You're welcome