$Z^n = (Z + 1)^n$ roots in complex plane

Question

Find the roots of the equation $Z^n = (Z + 1)^n$ and show that the points which represent them are collinear on the complex plane.

Answer 1

Hints:

• $\dfrac{Z}{Z+1} = \omega_k\,$ where $\,\omega_k \mid k = 1,\ldots n-1\,$ are the non-unit $\,n^{th}\,$ roots of unity;

• the $\omega_k$ all lie on the unit circle, and $Z = \dfrac{\omega}{1-\omega}$ is a Möbius transformation which maps circles and lines to circles and lines.

Answer 2

For the collinear part:

$z^n = (z + 1)^n \implies |z|=|z+1| \implies x^2+y^2=(x+1)^2+y^2 \implies x= \dfrac 12$.