Question

For k=1,2,…9, if we define zk=cos(3kπ/10)+isin(2kπ/10), then is it true that for each zk, there exists zj satisfying zk× zj=1?

(a) True

(b) False

I have been asked this question at a job interview.

Origin of the question is Roots of Complex Numbers topic in section Complex Numbers of Engineering Mathematics

Answer 1

The correct option is (a) True

For explanation I would say: zk= e^i(2kπ/10)⇒ zk× zj=e^i(2π/10)(k+j),  zk is 10^th root of unity.

⇒\(\overline{z_k}\) is also 10^th root of unity. Taking zj as \(\overline{z_k}\), we have zk×zj=1.