Consider a complex number
\(z=a+bi\)
We can write this as:
\(z=r\cos(\theta ) + ir\sin(\theta )\)
Because the functions loop:
\(ae^{i\theta }=a(\cos(\theta )+i\sin(\theta ))\)
\(ae^{i\theta }=a(\cos(\theta +n\tau )+i\sin(\theta +n\tau ))\)
\(ae^{i\theta }= ae^{i\theta +n\tau}\)
Additionally:
\(ae^{i\theta }=a(\cos(\theta )+i\sin(\theta ))\)
\(ae^{i\theta }=a(\cos(\theta )+i\sin(\theta ))\)
\(ae^{i\theta }=-a(\cos(\theta )-i\sin(\theta ))\)
\(ae^{i\theta }=-a(\cos(\theta +\dfrac{\pi }{2})+i\sin(\theta +\dfrac{\pi }{2}))\)
We can extract the real and imaginary parts of this number.
\(Re(z):=r\cos (\theta )\)
\(Im(z):=r\sin (\theta )\)
Alternatively:
\(Re(z)=r\dfrac{e^{i\theta }+e^{-i\theta }}{2}\)
\(Im(z)=r\dfrac{e^{i\theta }-e^{-i\theta }}{2i}\)
All polar numbers can be shown as Cartesian
\(ae^{i\theta }=a(\cos(\theta )+i\sin(\theta ))\)
\(ae^{i\theta }=a\cos(\theta )+ia\sin(\theta )\)
\(z=a+bi\)
\(e^{i\theta }=\)
\(e^x=\sum^{\infty }_{i=0} \dfrac{x^i}{i!}\)
Addition
\(z_3=z_1+z_2\)
\(z_3=a_1e^{i\theta_1}+a_2e^{i\theta_2}\)
\(z_3=a_1[\cos(\theta_1)+i\sin(\theta_1)]+a_2[\cos(\theta_2)+i\sin(\theta_2)]\)
\(z_3=[a_1\cos(\theta_1)+a_2\cos(\theta_2)]+i[a_1\sin(\theta_1)+a_2 \sin(\theta_2)]\)
Multiplication
\(z_3=z_1.z_2\)
\(z_3=a_1e^{i\theta_1}a_2e^{i\theta_2}\)
\(z_3=a_1a_2e^{i(\theta_1+\theta_2)}\)
\(a_3=a_1a_2\)
\(\theta_3=\theta_1+\theta_2\)