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Question
Find the modulus and argument of the following complex number and hence express in the polar form:
\[\frac{1 - i}{1 + i}\]
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Solution
\[\frac{1 - i}{1 + i}\]
\[\text { Rationalising the denominator }: \]
\[\frac{1 - i}{1 + i} \times \frac{1 - i}{1 - i}\]
\[ \Rightarrow \frac{1 + i^2 - 2i}{1 - i^2} \]
\[ \Rightarrow \frac{- 2 i}{2} \left( \because i^2 = - 1 \right)\]
\[ \Rightarrow - i\]
\[r = \left| z \right|\]
\[ = \sqrt{0 + 1}\]
\[ = 1\]
\[\text { Since point } (0, - 1) \text { lies on the negative direction of the imaginary axis, the argument of z is given by } \frac{3\pi}{2} . \]
\[\text { Polar form } = r\left( \cos \theta + i\sin \theta \right)\]
\[ = \left( cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2} \right)\]
\[ = \left\{ cos\left( 2\pi - \frac{\pi}{2} \right) + i\sin\left( 2\pi - \frac{\pi}{2} \right) \right\}\]
\[ = \left( \cos\frac{\pi}{2} - i\sin\frac{\pi}{2} \right)\]
