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:

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

This theorem requires a proof. In particular: This needs to be put into the rigorous context of Jordan curves, so as to define what is actually meant by "interior". At the moment, the understanding is intuitive. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

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