complex numbers: express (1+i)/(1-i) in re^(i theta) form

[cheffy]

Express
\(\displaystyle \
\frac{{1 + i}}{{1 - i}}
\\)

in the form \(\displaystyle \
re^{i\vartheta }
\\)

where \(\displaystyle \
r \ge 0
\

and

\
- \pi < \vartheta \le \pi
\\)

and draw an argand diagram. I don't know how to make this into a complex number. =\

Thanks.

 

[skeeter]

\(\displaystyle \L \frac{1+i}{1-i} \cdot \frac{1+i}{1+i} = 0+i\)

sketch (0 + i) on the complex plane ...

pretty straight-forward from the diagram, r = 1 and \(\displaystyle \L \theta = \frac{\pi}{2}\)

 

[pka]

Here is another way:
\(\displaystyle \L\begin{array}{l}
1 + i = \sqrt 2 e^{\frac{\pi }{4}i} \quad \quad 1 - i = \sqrt 2 e^{ - \frac{\pi }{4}i} \\
\frac{{1 + i}}{{1 - i}} = \frac{{\sqrt 2 e^{\frac{\pi }{4}i} }}{{\sqrt 2 e^{ - \frac{\pi }{4}i} }} = e^{\frac{\pi }{2}i} \\
\end{array}.\)