Complex Numbers

In this module of Maths Extension 2, you will learn about the 'imaginary' number $i = \sqrt{-1}$ and how we can define an entire number system that we will call the complex numbers $\mathbb{C}$ , which is an extension of the real numbers $\R$ that we have worked with so far.

You will learn how to express complex numbers and combine them in a range of ways, and how to prove special properties of the operations between them. You will also learn how complex numbers can be applied to find solutions to all quadratic equations, including quadratics where even the coefficients are complex numbers!

Introduction to Complex Numbers

In this section, you will be introduced the the complex number system and operations that can be performed on and between complex numbers. You will also learn how to represent complex numbers in algebraic and geometric forms, and about how complex numbers can be plotted as points within the complex plane.

Solving equations with complex numbers

In this section, you will learn about De Moivre's Theorem, and how this can be used to demonstrate trigonometric identities. You will also learn how to find complex solutions to quadratics which have only real coefficients, and for quadarativs which have complex coefficients.

Geometrical implications of complex numbers

In this section, you will learn how the representation of numbers on the complex plane can be used to relate complex numbers to problems in geometry. You will learn how to describe the effect of operations on and between complex numbers in the complex plane, and how we can define regions in the complex plane by using inequalities.

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