Modulus–Argument (Polar) Form

Modulus–argument (polar) form is a key part of the Complex numbers topic in NSW Year 12 Mathematics Extension 2 (NESA outcome ME2-12-03). Here a complex number is described by its distance from the origin and its angle: \(z = r(\cos\theta + i\sin\theta)\), with modulus \(r\), an argument \(\theta\), and the principal argument \(\operatorname{Arg}(z)\) chosen in \((-\pi, \pi]\).

Students and teachers will cover converting between Cartesian and polar form, why a product has the product of the moduli and the sum of the arguments, \(\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)\), with matching rules for quotients and powers, and what this means on the Argand diagram — used throughout the HSC Extension 2 course.