JAMIA 2022 — Mathematics PYQ

Let $z$ be a complex number represented in polar form as:

$z=r(\cos \theta +i\sin \theta )=r{e}^{i\theta }$

Where $r=|z|$ and $\theta =\arg (z)$.

Step 2: Argument of the Conjugate

The conjugate of $z$, denoted as $\bar{z}$, is obtained by reflecting $z$ across the real axis. In polar form:

$\bar{z}=r(\cos \theta -i\sin \theta )=r(\cos (-\theta )+i\sin (-\theta ))=r{e}^{-i\theta }$

Therefore:

$\arg (\bar{z})=-\theta =-\arg (z)$

Step 3: Sum of the Arguments

Now, substitute these values into the given expression:

$\arg (z)+\arg (\bar{z})=\theta +(-\theta )$

$\arg (z)+\arg (\bar{z})=0$

Note on Principal Argument

While $\arg (z)+\arg (\bar{z})$ is generally $0$, if we consider the standard principal argument range $(-\pi ,\pi ]$, there is one special case:

If $z$ lies on the negative real axis, then $\arg (z)=\pi$ and $\arg (\bar{z})=\pi$ (since the conjugate of a real number is itself). In this specific case, the sum could be $2\pi$, which is equivalent to $0$ modulo $2\pi$. However, in most general mathematical contexts:

$\arg (z)+\arg (\bar{z})=0$

Final Answer:

$0$