Constructing a complex number geometrically, without putting it in standard form.

Question

Is there any way to construct geometrically a number like $1/(3+5i)$ without putting it in the standard form (i.e. like $a+bi$)? How to do it? I want the localization of the number, without changing the way it is expressed.

Answer 1

A complex multiplicative inversion ($z\mapsto\frac1z$) can be expressed as a reflection in the real axis ($z\mapsto\bar z$) combined with a circle inversion in the unit circle ($z\mapsto\frac1{\bar z}$). The order doesn't matter. See Wikipedia for instructions on how to construct an inversion.