Euler's Formula

$e^{θ} = 1 + θ + \frac{θ^{2}}{2!} + \dots$
which converges for any $θ \in R$
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$= (1 - \frac{(i θ)^{2}}{2!} + \dots) + i (θ - \frac{θ^{3}}{3!} + \frac{θ^{5}}{5!} + \dots)$
where $(1 - \frac{(i θ)^{2}}{2!} + \dots) = \cos θ$ and $(θ - \frac{θ^{3}}{3!} + \frac{θ^{5}}{5!} + \dots) = \sin θ$

$e^{i θ} = \cos θ + i \sin θ$
$e^{i θ} = 1 + i θ + \frac{(i θ)^{2}}{2!} + \dots$

$z = x + y i = r e^{i θ} = r (\cos θ + i \sin θ)$
Let $z_{1} = r_{1} e^{i θ_{1}}$ , $z_{2} = r_{2} e^{i θ_{2}}$
$\Rightarrow z_{1} \cdot z_{2} = r_{1} r_{2} e^{i (θ_{1} + θ_{2})}$
$\Rightarrow | z_{1} \cdot z_{2} | = | z_{1} | \cdot | z_{2} |$
and $a r g (z_{1} \cdot z_{2}) = a r g (z_{1}) + a r g (z_{2})$ which uses Set Equality.
$| \frac{z_{1}}{z_{2}} | = \frac{| z_{1} |}{| z_{2} |}$ , $a r g (\frac{z_{1}}{z_{2}}) = a r g (z_{1}) - a r g (z_{2})$
$\frac{z_{1}}{z_{2}} = \frac{r_{1} e^{i θ_{1}}}{r_{2} e^{i θ_{2}}} = \frac{r_{1}}{r_{2}} e^{i (θ_{1} - θ_{2})}$