# Equation of Circle in Complex Plane/Formulation 1/Exterior

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## 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 exterior of $C$ can be defined by:

- $\left\lvert{z - \alpha}\right\rvert > r$

where $\left\lvert{\, \cdot \,}\right\rvert$ denotes complex modulus.

## Proof

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

- $\left\lvert{z - \alpha}\right\rvert = r$

## Sources

- 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 2$. Geometrical Representations: Example $1$.