Inversion 
Inversion in the complex plane
By remembering some elementary properties of complex numbers, we can easily give an alternative description of geometric inversion defined previously and deduce some of the listed properties.
A complex is determined by its module (distance to the origin) and its argument (angle of the positive real half-line with the half-line emitted from the origin and determined by the complex).
And it is not difficult to conclude that the product of two complex numbers is equal to the product of the modules and that the argument of the product is the sum of the arguments.
From these properties immediately follows that, given a nonzero complex z:
- the module of its inverse (in the sense of complex multiplication, i.e., the number that multiplied by z gives 1) is equal to the inverse of its module: the product of the distances to the origin of the corresponding points is 1.
- the sum of the arguments of the complex number and its inverse (in the same sense) is 0 (Argument 1).
If we compare this inversion for the multiplication with the geometric inversion defined above in the plane, we see they do not match. But the inverses of a same point, in both senses, are symmetrical to each other in relation to the real line in the complex plane. This symmetry corresponds to the function that changes the signal to the imaginary part of a complex, that is, it corresponds to the passage to the conjugate.
Therefore, the geometric inversion previously considered can be described in terms of complex numbers by the function \(z\rightarrow\frac{1}{\overline{z}}\) ou \(z\rightarrow\overline{\left(\frac{1}{z}\right)}\).
Obviously the passage to the conjugate, translated geometrically by a reflection, preserves the angles (in addition to changing the orientation). And it results from general properties of the analytic functions, that the function \(z\rightarrow\frac{1}{z}\) also preserves angles (in addition to conserve orientation). Therefore, the geometric inversion, as a composition of the two, also preserves angles (in addition to changing the orientation).
