Modulus and argument

The modulus is the distance from the origin. The argument is the angle of the position vector from the positive real axis, using the interval stated in the question. Check the quadrant before accepting an arctangent result.

For nonzero z, write z = r(cos θ + i sin θ). The argument of zero is undefined. Adding 2π to an angle describes the same point, but a principal-argument question demands an angle in its prescribed interval.

Worked example

Find the modulus and principal argument of −1 + i, using −π < θ ≤ π.

1. The modulus is √(1 + 1) = √2.
2. The point lies in quadrant two.
3. Its reference angle is π/4.

Answer: Modulus √2; argument 3π/4

Revise first: Complex arithmetic.