(A Level) Complex numbers: modulus–argument form and loci

Learning objectives

Represent complex numbers as points or vectors on an Argand diagram, and calculate the modulus |z| and the argument arg z in the interval −π < arg z ≤ π, taking care over the quadrant

Express a complex number in the form r(cos θ + i sin θ) or re^(iθ), and multiply or divide in this form using |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg z₁ + arg z₂

Interpret conjugation as a reflection in the real axis, z₁ ± z₂ as vector addition and subtraction, and multiplication by re^(iθ) as an enlargement by factor r combined with a rotation through θ

Sketch loci and regions such as |z − 2 + i| ≤ 3, |z − 1| = |z + 3i| and arg(z + 1) = π/4 on an Argand diagram

Use the geometry of a sketched locus or region to find the greatest or least possible value of |z| or arg z