3.9.7—Complex numbers

|z−a|=r is a circle centred at a with radius r; arg(z−a)=θ is a ray from a. Inequalities give interiors, exteriors or angular regions.

Sketch the reference point a first, then apply the distance or angle condition. Intersections of loci must satisfy every condition simultaneously.

|z−(2+i)|=3 is a circle centred at (2,1); |z|<2 is its interior.

An argument condition describes a direction, not a distance, and arg is defined modulo 2π unless a principal range is stated.