Question

Find the modulus and the arguments of the complex number: $z=-1-i\sqrt{3}$.

Answer 1

Given the complex number $z=-1-i\sqrt{3}$.

Now, $|z|=\sqrt{(-1{)}^2+(-\sqrt{3}{)}^2}=2$.

The point $z$ lies in the third quadrant so the amplitude is $-\pi +{\tan }^{-1}\left|\frac{-\sqrt{3}}{-1}\right|=-\pi +\frac{\pi }{3}=-\frac{2\pi }{3}$.