Question

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

1 + i

Answer 1

\[ z = 1 + i \]

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

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

\[ = \sqrt{2}\]

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

\[ \Rightarrow \tan \alpha = \left( \frac{1}{1} \right)\]

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

\[\text { Since point (1, 1) lies in the first quadrant, the argument of z is given by } \]

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

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

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