Equation of Unit Circle in Complex Plane/Corollary 1
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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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 3$. Roots of Unity