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
Course Chapters
1. Introduction3
1. Introduction
3
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.
Concept Overviews
3 Lessons
2. Algebra of Complex Numbers34
2. Algebra of Complex Numbers
3
4
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.
Concept Overviews
3 Lessons
Problem Walkthroughs
4 Lessons
3. The Argand Diagram and Polar Form34
3. The Argand Diagram and Polar Form
3
4
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(cos??+isin??)) forms.
Concept Overviews
3 Lessons
Problem Walkthroughs
4 Lessons
4. De Moivre's Theorem24
4. De Moivre's Theorem
2
4
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.
Concept Overviews
2 Lessons
Problem Walkthroughs
4 Lessons
5. Roots of Unity24
5. Roots of Unity
2
4
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.
Concept Overviews
2 Lessons
Problem Walkthroughs
4 Lessons
6. Conclusion2
6. Conclusion
2
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.
Concept Overviews
2 Lessons