Help with complex numbers?

I was doing some complex numbers problems today and something has me really confused. I had to find the arument of -2-2root3i so I plotted the point on an argand diagram and used tan to find the reference angle. Then I added pi to it because thats what the book said to do for the 3rd quadrant in the previous questions and because I thought the argument was the angle between the positive x axis and the line joining the complex number to the origin. I always that it was in an anticlockwise direction. But this time the book said to add -pi and continued to do this for the rest of the questions. Now i am confused as to how i find the argument. Can someone please help.

Thanks

Re: Can someone help me with complex numbers?

The "standard" argument for complex numbers lies between 0 and $\displaystyle 2\pi$ while "arctan" always gives a value between $\displaystyle -\pi/2$ and $\displaystyle \pi/2$. In the case of $\displaystyle -2- 2\sqrt{3}$ we are looking at $\displaystyle arctan(\sqrt{3})= \pi/3$ which is ih the first quadrant so we add $\displaystyle \pi$ to get $\displaystyle \frac{4\pi}{3}$.

**However**, the "argument" of a complex number is not "uniquely" defined. Adding or subtracting $\displaystyle 2\pi$ to the argument will give another that is also a valid argument for that same complex number. Subtracting $\displaystyle 2\pi$ from $\displaystyle \frac{4\pi}{3}$, of course, gives the same as subtracting $\displaystyle \pi$ from $\displaystyle \frac{\pi}{3}$. As I said before the "standard" is between 0 and $\displaystyle 2\pi$ but there might be good reason to use another representation.

Re: Can someone help me with complex numbers?

Thanks so much for your answer! It definitely makes more sense now