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![\bf \cfrac{1+i}{1-i}\implies \cfrac{1+i}{1-i}\cdot \cfrac{1+i}{1+i}\implies \cfrac{(1+i)^2}{\underset{\textit{difference of squares}}{(1-i)(1+i)}}\implies \cfrac{\stackrel{FOIL}{1^2+2i+i^2}}{1^2-i^2} \\\\\\ \stackrel{\textit{recalling that }i^2=-1}{\cfrac{1+2i+(-1)}{1-(-1)}}\implies \cfrac{2i}{1+1}\implies \cfrac{~~\begin{matrix} 2 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~i}{~~\begin{matrix} 2 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}\implies i\implies \stackrel{\textit{a+bi form}}{0 + 1i}](/tpl/images/0189/3777/31360.png)
![\bf \begin{cases} r = \sqrt{0^2+1^2}\\ \qquad 1\\ \theta =\frac{\pi }{2} \end{cases} \qquad \qquad 1\left[cos\left( \frac{\pi }{2} \right) +i~sin\left( \frac{\pi }{2} \right) \right]](/tpl/images/0189/3777/eed0f.png)
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