Equation of Circle in Complex Plane/Formulation 1/Interior

Theorem
Let $\C$ be the complex plane.

Let $C$ be a circle in $\C$ whose radius is $r \in \R_{>0}$ and whose center is $\alpha \in \C$.

The points in $\C$ which correspond to the interior of $C$ can be defined by:
 * $\cmod {z - \alpha} < r$

where $\cmod {\, \cdot \,}$ denotes complex modulus.

Proof
From Equation of Circle in Complex Plane, the circle $C$ itself is given by:


 * $\cmod {z - \alpha} = r$