Introduction to Complex Numbers - Mathematics (Undergraduate Foundation)

This course provides a complete introduction to the complex number system, from the definition of the imaginary unit to the Argand diagram and polar form. It then covers De Moivre’s theorem and its application to finding the nth roots of unity. This material extends algebra into two dimensions and provides essential tools for advanced mathematics. Complex numbers are an essential tool in engineering and the physical sciences. They are used extensively in electrical engineering to analyse AC circuits, in signal processing for Fourier transforms, in quantum mechanics to describe wave functions, and in fluid dynamics. This course provides the practical foundation for these applications. By the end of this course, you will be able to perform all arithmetic operations on complex numbers, represent them on an Argand diagram, convert between Cartesian and polar forms, apply De Moivre’s theorem to find powers of complex numbers, and calculate the nth roots of unity. This course is designed for first-year university students of mathematics, physics, and engineering. It is a critical prerequisite for the study of electrical circuits, signal processing, quantum mechanics, and any field that uses wave analysis or advanced vector mathematics.

Enrolment valid for 12 months

This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.

[NUC Core] MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Master the mathematics every scientist, engineer, and computer scientist must know. This track covers the NUC Core syllabus for MTH 101: Elementary Mathematics I. From set theory and sequences to trigonometry and complex numbers, you will gain the exact tools used in higher mathematics, algorithm design, data analysis, physics, and engineering. This programme is for first-year university students in mathematics, computer science, engineering, or physics, as well as serious learners preparing for undergraduate study. It also serves professionals who need a rigorous mathematical foundation for technical fields. By the end, you will handle formal set notation, progressions, quadratic models, combinatorics, induction proofs, binomial expansions, trigonometric analysis, and complex arithmetic. You will be equipped to solve real problems in science, engineering, computing, and finance, and ready for advanced study in calculus, statistics, and applied mathematics.

Course Chapters

1. Introduction

This chapter introduces the complex number system. It establishes the need for a new type of number by defining the imaginary unit, i, and explains the standard form of a complex number, laying the groundwork for the entire course. Key topics include an overview of the course structure, the formal definition of a complex number, and the properties of the imaginary unit, i.

Chapter lessons

1-1. Welcome

A direct statement of the course's purpose and structure. It outlines the progression from basic definitions to De Moivre's theorem and its applications.

1-2. What are complex numbers?

Formally defines a complex number as a number of the form a + bi. This lesson explains the real and imaginary parts of a complex number.

1-3. The imaginary unit, i

Defines the imaginary unit i as the solution to the equation x² = -1. This lesson covers the fundamental properties and the cyclical nature of the powers of i.

2. Algebra of Complex Numbers

This chapter covers the fundamental arithmetic of complex numbers. It details the procedures for addition, subtraction, multiplication, and division, which form the basis for all algebraic manipulation in the complex plane. Key topics include the addition and subtraction of complex numbers, multiplication, and the use of the complex conjugate for division. Worked examples demonstrate each operation.

Chapter lessons

2-1. Addition and subtraction

Details the process for adding and subtracting complex numbers by combining their respective real and imaginary parts. The process is analogous to combining like terms in algebra.

2-2. Multiplication

Covers the multiplication of complex numbers using standard algebraic distribution, including the substitution of i² with -1. This forms the basis for more complex operations.

2-3. Division and the conjugate

Introduces the complex conjugate as the tool for dividing complex numbers. The lesson explains how to rationalise the denominator by multiplying by its conjugate.

3. The Argand Diagram and Polar Form

This chapter introduces the geometric representation of complex numbers. It covers the Argand diagram for visualising numbers in a two-dimensional plane and the conversion to polar form, a prerequisite for De Moivre's theorem. Key topics include plotting complex numbers, defining the modulus and argument, and converting between Cartesian (a+bi) and polar (r(cosθ+isinθ)) forms.

Chapter lessons

3-1. The complex plane

Introduces the Argand diagram as a two-dimensional plane for plotting complex numbers. It establishes the horizontal axis for the real part and the vertical axis for the imaginary part.

3-2. Modulus and argument

Defines the modulus of a complex number as its distance from the origin and the argument as its angle from the positive real axis. These are the core components of the polar form.

3-3. Polar form of complex numbers

Explains how to express a complex number in polar form using its modulus and argument. This form is essential for efficient multiplication, division, and finding roots.

4. De Moivre's Theorem

This chapter focuses entirely on De Moivre's theorem, a powerful tool for finding powers of complex numbers. The theorem provides an efficient method for calculation when a complex number is expressed in polar form. Key topics include the formal statement of the theorem, its proof by mathematical induction, and its direct application to finding integer powers of complex numbers.

Chapter lessons

4-1. Statement of the theorem

Formally states De Moivre's theorem for finding an integer power of a complex number in polar form. It provides the direct formula for the resulting modulus and argument.

4-2. Proving the theorem by induction

Provides a brief outline of the proof of De Moivre's theorem for positive integer exponents using the principle of mathematical induction.

5. Roots of Unity

This chapter covers the primary application of De Moivre's theorem: finding the nth roots of complex numbers. This is a critical technique for solving polynomial equations in the complex plane. Key topics include the general formula for the nth roots of unity and its application to finding the roots of any complex number. The geometric interpretation of the roots is also covered.

Chapter lessons

5-1. The nth roots of unity

Introduces the concept of the nth roots of unity. It provides the general formula for finding all n distinct roots of 1 in the complex plane.

5-2. Finding roots of complex numbers

Extends the concept of roots of unity to finding the nth roots of any complex number. The general formula and procedure are detailed.

6. Conclusion

This chapter consolidates the core concepts of complex numbers. It provides a structured summary of their algebra, geometric representation, and the application of De Moivre's theorem for finding powers and roots. The conclusion summarises key definitions and formulas. It also provides a forward look to how complex numbers are applied in engineering and physics.

Chapter lessons

6-1. Summary of complex numbers

A concise review of the key methods and formulas for complex numbers. This lesson ensures all procedural and theoretical material has been consolidated.

6-2. Applications in science and engineering

Provides a brief overview of how complex numbers serve as an essential tool in electrical engineering, signal processing, and quantum mechanics.