- I understand that $\displaystyle |z| = \sqrt{x^2+y^2}$ (from the Argand Diagram). Is there an algebraic way to show that $\displaystyle |z| = \sqrt{x^2+y^2}$?

- Why is the principal value of the argument taken to be between $\displaystyle -\pi$ and $\displaystyle \pi$? I was asked to find the modulus and argument of $\displaystyle -1-i$. Here is what I did:

$\displaystyle |z| = \sqrt{\left(-1\right)^2+\left(-1\right)^2} = \sqrt{1+1} = \sqrt{2}$. Right.

$\displaystyle \arg{z} = \arctan\left(\frac{-1}{-1}\right)+\frac{\pi}{2} = \arctan(1)+\frac{\pi}{2} = \frac{\pi}{4}+2\pi = \frac{2\pi+\pi}{4} = \frac{3\pi}{4}$. Obviously that isn't between $\displaystyle -\pi$ & $\displaystyle \pi$. What am I supposed to do then?

PS. For the last part (as noticed by Archie Meade), I meant: $\displaystyle \arg{z} = \arctan\left(\frac{-1}{-1}\right)+\pi = \arctan(1)+\pi = \frac{\pi}{4}+\pi = \frac{4\pi+\pi}{4} = \frac{5\pi}{4}$.