11–12Mathematics Extension 2 11–12 Syllabus

Define and perform complex number addition, subtraction and multiplication, with and without digital tools

Use multiplication, division and powers of complex numbers in polar form and interpret these geometrically

Convert between complex numbers in Cartesian form and polar form and use complex numbers in Cartesian form and polar form to solve problems

Recognise that solutions to quadratic equations with real coefficients are complex conjugates of each other and use this to solve problems

Solve problems involving complex conjugate roots of polynomial equations with real coefficients

Use de Moivre’s theorem to find any integer power of a given complex number

Use de Moivre’s theorem to derive trigonometric identities

Recognise that a complex number can be represented as a vector, where the magnitude and direction of the vector are determined by the modulus and argument of the complex number respectively

Examine and use addition and subtraction of complex numbers as vectors on the complex plane

Examine and use the geometric interpretation of multiplying complex numbers, including rotation and dilation on the complex plane, with and without graphing applications

Prove geometric results using complex numbers as vectors

Graph regions associated with lines, rays and circles defined using complex numbers, giving a geometrical description of any such curves or regions, and using circle geometry theorems where necessary