Forms of Complex Numbers
Rectangular:
The basic form of a complex number in rectangular form is $ \ z = a + jb $ where $ j = \sqrt{-1} $. $ i $ is sometimes used instead of $ j $
Polar:
The polar representation of a complex number is given by $ \ \rho cos(\Theta)+j\rho sin(\Theta) = \rho e^{j\Theta} $. This is also known as Euler's Identity. Using the coefficients of the rectangular form $ \rho = \sqrt{a^2 + b^2} $ and $ \ \Theta = tan^{-1}(b/a) $.
Operations of Complex Numbers
Product:
In rectangular form: $ \ z_{1}=(a + bj) , z_{2} =(c + jd) $ so $ \ z_{1}(z_{2}) = (a + jb)(c + jd) = ac - bd + j(bc + ad) $
In polar form $ z_{1}(z_{2}) = \rho_{1}e^{j\Theta_{1}}\rho_{2}e^{j\Theta{2}} = \rho_{1}\rho_{2}e^{j(\Theta_{1}+\Theta{2})} $
Division:
In rectangular form: $ \frac{z_{1}}{z_{2}} = \frac{a + jb}{c + jd} = \frac {(a + jb)(c - jd)}{c^2 + d^2} $
In polar form: $ \frac{z_{1}}{z_{2}} = \frac{\rho_{1}e^{j\Theta_{1}}}{\rho_{2}e^{j\Theta{2}}} = \frac {\rho_{1}}{\rho_{2}}e^{j(\Theta_{1}-\Theta{2})} $
