Calculating the modulus

Question

I have a function $z=|\frac{-4i\pm2i\sqrt3}{2}|$

Now I would simplify this by doing:

$|\frac{-4i+2i\sqrt3}{2}|=|-2i+i\sqrt3|=|-2+\sqrt3|=2+\sqrt3$

However the book I am using gives: $|\frac{-4i+2i\sqrt3}{2}|=|2-\sqrt3|=2-\sqrt3$

and then $|\frac{-4i-2i\sqrt3}{2}|=|2+\sqrt3|=2+\sqrt3$

Could someone explain my mistake and how to work out the correct answer?

Answer 1

You're line $|-2+\sqrt3| = 2 + \sqrt3$ is incorrect. Remember taking the modulus multiplies by +1 or -1 to make the number inside positive. Therefore $|-2+\sqrt3| = -2 + \sqrt3$ or $|-2+\sqrt3| = 2-\sqrt3$, but since $\sqrt3<2$ we have that $$|-2+\sqrt3| = 2-\sqrt3$$ like the book says.

Hope this helps