Question

Find the modulus and argument of the following complex number and hence express in the polar form:

\[\frac{1}{1 + i}\]

Answer 1

\[ \frac{1}{1 + i}\]

\[\text { Rationalising the denominator }: \]

\[\frac{1}{1 + i} \times \frac{1 - i}{1 - i}\]

\[ \Rightarrow \frac{1 - i}{1 - i^2} \]

\[ \Rightarrow \frac{1 - i}{2} \left( \because i^2 = - 1 \right)\]

\[ \Rightarrow \frac{1}{2} - \frac{i}{2}\]

\[r = \left| z \right|\]

\[ = \sqrt{\frac{1}{4} + \frac{1}{4}}\]

\[ = \frac{1}{\sqrt{2}}\]

\[\text { Let } \tan \alpha = \left| \frac{Im(z)}{Re(z)} \right|\]

\[ \therefore \tan \alpha = \left| \frac{\frac{1}{2}}{\frac{- 1}{2}} \right|\]

\[ = 1 \]

\[ \Rightarrow \alpha = \frac{\pi}{4}\]

\[\text { Since point } \left( \frac{1}{2}, - \frac{1}{2} \right) \text { lies in the fourth quadrant, the argument is given by }\]

\[\theta = - \alpha = \frac{- \pi}{4}\]

\[\text{ Polar form} = r\left( \cos \theta + i\sin \theta \right) \]

\[ = \frac{1}{\sqrt{2}}\left\{ cos\left( \frac{- \pi}{4} \right) + i\sin\left( \frac{- \pi}{4} \right) \right\}\]

\[ = \frac{1}{\sqrt{2}}\left( cos\frac{\pi}{4} - i\sin\frac{\pi}{4} \right)\]

\[\]