Real, Complex or Imaginary?

Algebra Level 3

Suppose $z$ is a complex number and $n$ is a natural number. If | $z$ | = 1, and $z^{2n}$ does not equal -1, then $\dfrac{z^{n}}{1+z^{2n}}$ is

Neither purely real nor purely imaginary Purely Real Purely Imaginary

1 solution

Aareyan Manzoor
Dec 9, 2015

if it is purely real/imaginary, so is its reciprocal: $\dfrac{z^{2n}+1}{z^n}=z^n+z^{-n}$ now since |z|=1, it is a $e^{xi}=\cos(x)+i\sin(x)$ . put this to find $\cos(x)+i\sin(x)+\cos(-x)+i\sin(-x)=\cos(x)+i\sin(x)+\cos(x)-i\sin(x)=2\cos(x)$ which is always real.

-QED, hence proved.

Real, Complex or Imaginary?

1 solution

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