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Question
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
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Solution
Let z be a complex number with argument θ.
Then,
\[z = r e^{i\theta} \]
\[ \Rightarrow \bar{z} = \bar{{r e^{i\theta}}} = r e^{- i\theta}\]
⇒ argument of \[\bar{z}\] is −θ.
Thus,
\[\arg\left( z \right) + \arg\left( \bar{z} \right) = 0\].
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