Question

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

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

Answer 1

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

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

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

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

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

\[ \Rightarrow - i\]

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

\[ = \sqrt{0 + 1}\]

\[ = 1\]

\[\text { Since point } (0, - 1) \text { lies on the negative direction of the imaginary axis, the argument of z is given by } \frac{3\pi}{2} . \]

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

\[ = \left( cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2} \right)\]

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

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