A-Level Mathematics: Complex Numbers: Argand Diagram & Loci

Argand basics

Every complex number z = x + iy can be represented as a point (x, y) on the Argand diagram. The modulus is the distance from the origin: |z| = √(x² + y²). The argument θ is the angle with the positive real axis.

Polar form

Write z = r(cosθ + i sinθ). Handy for multiplication/division: multiply moduli, add arguments. This is the gateway to De Moivre’s Theorem.

Exam habits

• Always draw a quick sketch of the quadrant before quoting the argument.
• Check if the question wants radians or degrees.
• Use conjugates to rationalise denominators: z × z̄ = |z|².