z = -2 + (2root3)i. Find the modulus and argument of z.

The first step to answering this question is to pinpoint exactly what it means. We remind ourselves that when we are given a number in the form of a + bi, we are dealing with a complex number. A complex number is a number with a real part (the 'a') and an imaginary part (the 'b'). So for z = -2 + (2root3)i, the real part is -2 and the imaginary part is 2root3. Next the complex number would be plotted on an Argand Diagram.
Finding the modulus of the complex number is the same as finding the length of the complex number when plotted on an Argand Diagram. This can most easily be explained using the whiteboard as a diagram. So we have a triangle where the side lengths are 2 and 2root3 and we have to find the hypotenuse. This is done by Pythagoras' Theorem. (2)^2 + (2root3)^2 = 4 + 4 x 3 = 16. So the modulus is root16 = 4. Note that the modulus is a length, so it is always positive.
Next we must find the argument of z. The argument is the angle taken from the x axis to the complex number. Because our complex number is in the second quadrant of the Argand Diagram, our argument will be positive. Note that because the argument is taken from the (positive) x-axis, it is always greater than or equal to -π and less than or equal to +π. Using our diagram we first have to find the angle inside the triangle at the centre. We do this using trigonometry and SohCahToa. We use tan because we have been given the opposite and adjacent, and so we know that tan(x) = 2root3/2 = root3. We then use inverse tan to find that x = pi/3. But we want the angle from the positive x-axis so we have to take this value away from pi. So arg z = 2pi/3.