Equation of Unit Circle in Complex Plane/Corollary 1

From ProofWiki
Jump to navigation Jump to search

Corollary to Equation of Unit Circle in Complex Plane

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


The equation of $C$ can be given by:

$z \overline z = 1$

where $\overline z$ denotes the complex conjugate of $z$.


Proof

From Equation of Unit Circle in Complex Plane, the equation of $C$ can also be given by:

$\cmod z = 1$

where $\cmod z$ denotes the complex modulus of $z$.

Thus:

\(\ds \cmod z\) \(=\) \(\ds 1\)
\(\ds \leadsto \ \ \) \(\ds \cmod z^2\) \(=\) \(\ds 1\)
\(\ds \leadsto \ \ \) \(\ds z \overline z\) \(=\) \(\ds 1\) Modulus in Terms of Conjugate

$\blacksquare$


Sources