This is Chapter 4 – Complex Numbers for the Year 12 Specialist Mathematics course.

This chapter extends the complex-number foundations developed in Year 11 and introduces several powerful new ways of representing and manipulating them. Students move beyond calculations in Cartesian form and begin using geometry, trigonometry and exponential notation to describe complex quantities.

We begin with the complex plane, where complex numbers are represented as points and vectors. The modulus describes the distance from the origin, while the argument describes direction. Together, these ideas allow a complex number to be expressed in polar form, connecting its algebraic components with its geometric position.

Euler’s form provides a compact exponential representation of a complex number and creates a direct connection between exponential and trigonometric functions. Students learn to move confidently between Cartesian, polar and Euler forms, choosing the representation that makes a particular calculation most efficient.

De Moivre’s theorem is then used to raise complex numbers to integer powers and simplify expressions involving repeated multiplication. The theorem also provides a systematic method for finding all roots of a complex number. These roots appear as equally spaced points around a circle in the complex plane, creating a strong visual connection between algebra and geometry.

This chapter relies on accurate work with exact trigonometric values, arguments and angle conventions. It also connects closely with the Fundamental Theorem of Algebra, since polynomial equations may have non-real roots.

Complex numbers are widely used in electrical engineering, signal processing, control systems, fluid dynamics and quantum mechanics, where they simplify the representation of rotation, oscillation and wave behaviour.

Subchapters:

• 4A: The complex plane

• 4B: Modulus and argument
• 4C: Polar form
• 4D: Euler’s form
• 4E: De Moivre’s theorem
• 4F: Roots of complex numbers

By the end of this chapter, students will be able to move between complex-number representations and use geometric and algebraic methods to calculate powers and roots.