the argument of a complex number
If the modulus |z| tells you how far the point z is from the origin, the argument tells you which way it points. The argument of z, written arg z, is the angle (measured counterclockwise from the positive real axis) that the arrow from 0 to z makes. Together, the distance |z| and the direction arg z pin down z completely — they are the polar coordinates of the point in the complex plane.
You compute it from z = x + i y by recognizing that x = |z| cos(arg z) and y = |z| sin(arg z); the angle is essentially arctan(y/x), but you MUST look at which quadrant (x, y) lies in to get the right answer, because arctan alone cannot tell apart, say, the first and third quadrants. For z = -1 - i, both x and y are negative, so the point is in the third quadrant and arg z is about 225 degrees (or 5 pi / 4), NOT the 45 degrees a naive arctan(1) would give.
Here is the crucial honesty: the argument is not a single number. Adding a full turn (2 pi radians) brings you back to the same point, so if theta is an argument of z, so is theta + 2 pi, theta - 2 pi, and so on. So arg z is really a whole family of angles differing by multiples of 2 pi — it is 'multivalued'. This is not a defect to be patched away cheaply; it is the seed of branch cuts, the multivalued logarithm, and Riemann surfaces later on.
For z = -1 - i: |z| = the square root of 2, and the point is in the third quadrant, so arg z = 5 pi / 4 (plus any multiple of 2 pi). Naively writing arctan((-1)/(-1)) = arctan(1) = pi/4 gives the WRONG quadrant.
Always use the quadrant of (x, y) to fix the argument — arctan alone is ambiguous.
arg 0 is undefined: the point at the origin has no direction. And arg z is only determined up to adding multiples of 2 pi — choosing one specific value is what the principal argument does.