# Complex numbers ( Finding principal argument ).

We're given a complex number , $Z=-1- \sqrt{3}i$ , we need to find its principal argument $\phi$.

Clearly it lies in the third quadrant , so its , argument $\theta$ will be :

$\theta = \pi + \dfrac{\pi}{3} = \dfrac{4 \pi}{3}$ ,

But , the principal arguments lies between $[- \pi , \pi]$ , so ,

$\phi = 2 \pi - \dfrac{4 \pi}{3} = \dfrac{2 \pi}{3}$ ,

But the solution says , $\phi = - \dfrac{2 \pi}{3}$ , could anyone tell , what am I doing wrong ?

• Your $\phi$ should be $\phi = \dfrac{4 \pi}{3} - 2 \pi = -\dfrac{2 \pi}{3}$. – Raskolnikov Sep 9 '15 at 6:30
• Can you elaborate , please ? @Raskolnikov – User9523 Sep 9 '15 at 6:36
• I don't know what more I should say. Angles are determined modulo $2\pi$. So, your angle can be $\phi = \dfrac{4 \pi}{3} + 2 k \pi$ for any integer $k$. Just choose the $k$ that gives you an angle between $-\pi$ and $\pi$. Why you write the opposite of the angle is a mystery to me. – Raskolnikov Sep 9 '15 at 6:41