Complex Modulus
Complex Modulus, also known as the "Norm" of a complex number, is represented as $ |z| $.
$ |x + iy| = \sqrt{x^2 + y^2} $
In exponential form for $ |z| $
$ |re^{i\phi}| = r $
(This format is used when dealing with Phasors)
Basics
- $ |z|^2 $ of $ |z| $ is known as the Absolute Square.
- $ \frac{|Ae^{i\phi_{1}}|}{|Be^{i\phi_{2}}|} = \frac{A}{B}\frac{|e^{i\phi_{1}}|}{|e^{i\phi_{2}}|} = \frac{A}{B} $
- $ |\frac{Ae^{i\phi_{1}}}{Be^{i\phi_{2}}}| = \frac{A}{B}|e^{i(\phi_{1}-\phi_{2})}| = \frac{A}{B} $
- $ |\frac{Ae^{i\phi_{1}}}{Be^{i\phi_{2}}}| = \frac{|Ae^{i\phi_{1}}|}{|Be^{i\phi_{2}}|} $
- $ |Ae^{i\phi_{1}}||Be^{i\phi_{2}}| = {A}{B}|e^{i\phi_{1}}||e^{i\phi_{2}}| = {A}{B} $
- $ |(Ae^{i\phi_{1}})(Be^{i\phi_{2}})| = {A}{B}|e^{i\phi_{1}+i\phi_{2}}| = {A}{B} $
- $ |Ae^{i\phi_{1}}||Be^{i\phi_{2}}| = |(Ae^{i\phi_{1}})(Be^{i\phi_{2}})| $
- $ |z^n|=|z|^n $
- $ zz=|z|^2 $
Where $ z $ is the complex conjugate $ z=x-iy $
