Equation of Unit Circle in Complex Plane/Proof 1

Theorem
Consider the unit circle $C$ whose center is at $\left({0, 0}\right)$ on the complex plane.

Its equation is given by:
 * $\left\vert{z}\right\vert = 1$

where $\left\vert{z}\right\vert$ denotes the complex modulus of $z$.

Proof
From Equation of Unit Circle, the unit circle whose center is at the origin of the Cartesian $xy$ coordinate plane has the equation:
 * $x^2 + y^2 = 1$

Identifying the Cartesian $xy$ coordinate plane with the complex plane: