Course Summary — Complex Numbers

Complex numbers extend the real number system and allow equations with no real solutions to be solved. This course builds your understanding from the basics through to advanced applications, developing both algebraic skill and geometric insight.

You will learn how to add, subtract, multiply and divide complex numbers, including using conjugates and identifying real and imaginary parts. You will then represent complex numbers on the complex plane and convert between Cartesian and polar form by finding modulus and argument.

The course also covers solving and factorising equations with complex solutions, including higher-degree polynomials and the use of De Moivre’s Theorem. You will sketch subsets of the complex plane by interpreting modulus and real/imaginary conditions as geometric regions.

Finally, you will study roots of unity — finding nth roots of complex numbers, locating them on the unit circle, and understanding the symmetry they form.

By the end of this course, you will confidently perform complex number operations, solve polynomial equations, interpret geometric representations, and apply advanced techniques to unfamiliar problems.