Argument of complex numbers Questions in English

(C) Let $\arg(z) = \theta$,where $0 < \theta < \pi$.
Since $-z$ is the reflection of $z$ through the origin,its argument is given by $\arg(-z) = \theta - \pi$ (if $0 < \theta < \pi$).
Therefore,$\arg(z) - \arg(-z) = \theta - (\theta - \pi) = \theta - \theta + \pi = \pi$.

(A) Let $\arg (z) = \theta$,where $-\pi < \theta < -\frac{\pi}{2}$.
Since $\bar{z}$ is the reflection of $z$ across the real axis,$\arg (\bar{z}) = -\arg (z) = -\theta$.
Since $-\pi < \theta < -\frac{\pi}{2}$,we have $\frac{\pi}{2} < -\theta < \pi$.
Now,$-\bar{z} = -1 \cdot \bar{z} = e^{i\pi} \cdot \bar{z}$.
Therefore,$\arg (-\bar{z}) = \arg (e^{i\pi}) + \arg (\bar{z}) = \pi + (-\theta) = \pi - \theta$.
We need to calculate $\arg (\bar{z}) - \arg (-\bar{z}) = -\theta - (\pi - \theta) = -\theta - \pi + \theta = -\pi$.
Since the argument is typically defined in the range $(-\pi, \pi]$,we can add $2\pi$ to get the principal value: $-\pi + 2\pi = \pi$.