Hans Georg Schaathun
September 2016
$$z = a+bi$$
$$|z| = \sqrt{a^2+b^2}$$
$$|z| = \sqrt{z\bar z}$$
$$\bar z = a-bi$$
$$\arg z = \theta$$
$$\tan \theta = \frac{b}{a}$$
$$\textbf{Arg}\;z = \tan^{-1}\frac{b}{a}$$
$$z = a+bi$$
$$a=r\cdot \cos \theta$$
$$b=r\cdot \sin \theta$$
$$z=r(\cos\theta+i\sin\theta)$$
$$z\cdot w = \big[r\cdot(\cos\theta+i\sin\theta) \big] \cdot\big[s\cdot(\cos\phi+i\sin\phi)\big]$$
$$z\cdot w = (rs) \big[(\cos\theta)(\cos\phi) + i(\sin\theta)(\cos\phi) + i(\cos\theta)(\sin\phi)+i^2(\sin\theta)(\sin\phi)\big]$$
$$z\cdot w = (rs) \bigg( \big[(\cos\theta)(\cos\phi) - (\sin\theta)(\sin\phi)\big] + i\big[(\sin\theta)(\cos\phi) + (\cos\theta)(\sin\phi) \big] \bigg) $$
$$z\cdot w = (rs) \big[ \cos(\theta+\phi) + i(\sin(\theta+\phi)\big] $$
$$|zw| = |z|\cdot|w|$$
$$\arg (zw) = \arg z + \arg w$$