Consider 1 + i and - 1 - i

In both cases arctan(y/x) = pi/4

since 1 + i is in the first quadr arg(z) = pi/4

hoewever for - 1 -i we know this is in the 3d quadrant so we add pi for the arg

Similarly consider 1 - i and -1 + i

arctan(y/x) = -pi/4 for -1 + i is in the 2 quadrant so add pi to find argz

for 1- i is in the fourth quadrant so add 2pi to find argz

In general locate the quadrant in which z lies to determine arg z