Complex Numbers - Mathematical Methods (Undergraduate Advanced)

Standard numbers cannot solve every square root. This course teaches you to use complex numbers for such problems. You will learn number systems, algebra, and how to use the Argand plane. Lessons cover polar forms, De Moivre's theorem, and finding roots. You will also study sine, cosine, and log functions through many worked examples. These methods are vital for electrical engineering, physics, and signal processing. You need these skills to calculate electricity flow and wave behaviour. Engineers use these tools to build machines and structures. Mastering this allows you to solve the difficult maths used in modern technology and research. It provides the logic needed for all advanced technical work. You will learn to add, multiply, and divide complex numbers. You will switch between rectangular, polar, and exponential forms using Taylor series. You will gain the skill to find Nth roots of unity and solve hyperbolic equations. You will also learn to draw complex equations on a graph accurately. By the end, you will handle advanced mathematical expressions without errors. This course is for university students in engineering or maths who must pass advanced modules. It also serves technical workers who need to refresh their skills. The step-by-step examples make the topics easy to follow for anyone starting out. This knowledge is a requirement for anyone pursuing a career in high-level science or technology.

Enrolment valid for 12 months

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

MTH 202: Mathematical Methods II

Comprehensive treatise of advanced mathematics covering vector calculus, complex numbers, linear vector spaces, linear maps, matrices, eigenvalues and eigenvectors. Curated for second-year students of engineering and physical sciences at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Master the foundational mathematical structures essential for success in quantitative undergraduate degrees and professional technical roles. This comprehensive learning track delivers a rigorous treatment of algebra and trigonometry, moving rapidly from fundamental set theory and real number operations to advanced topics including matrix algebra, complex numbers, and analytical trigonometry. You will establish the critical problem-solving framework required for advanced study in calculus, engineering mechanics, and data science. This programme is primarily designed for first-year university students in STEM disciplines requiring strong analytical bases, particularly engineering, physics, computer science, and economics. It also serves as an intensive, high-level refresher for professionals returning to academia or shifting into data-driven roles demanding precise numerical literacy and logical structuring. Prior competence in standard secondary school mathematics is assumed; focus is placed strictly on mastery and application of core definitions. Upon completion, you will possess the skills to construct rigorous logical arguments using set theory and mathematical induction, model complex relationships with functions and matrices, and analyze periodic systems using advanced trigonometry. You will demonstrate competence in solving diverse equation types, from quadratics to linear systems, and manipulating complex numbers in engineering applications. This track prepares you directly for the mathematical demands of second-year university studies and technical professional certification exams.

Course Chapters

1. Introduction

Standard maths cannot solve square roots of negative numbers. This chapter reviews number systems and introduces the imaginary unit. It provides the foundation for every advanced method covered in this course. You will define natural, integer, rational, and real number sets; explain why complex numbers exist; calculate increasing powers of i; and solve fundamental introductory problems.

Concept Overviews

4 Lessons

1:23:49

Problem Walkthroughs

1 Lesson

14:45

2. Algebra

Arithmetic for complex numbers starts here. You need addition, multiplication, and division to solve engineering formulae. These rules form the foundation for every future calculation. You will master adding real and imaginary parts; multiplying scalars and complex terms; using conjugates for division; and applying identities to solve for unknowns.

Concept Overviews

6 Lessons

1:44:23

Problem Walkthroughs

3 Lessons

2:08:34

3. Use of the Argand Plane

This chapter teaches you to plot complex numbers as points on a 2D graph. Visualising numbers this way is essential for understanding phase and magnitude in engineering and physics. You will learn to represent numbers on the Argand plane; calculate the modulus and general argument; determine the unique principal argument; and apply modulus properties to solve complex geometric problems.

Concept Overviews

4 Lessons

1:44:48

Problem Walkthroughs

2 Lessons

1:29:15

4. Polar Form

This chapter introduces polar coordinates to simplify complex number calculations. Using angles and magnitudes makes multiplying, dividing, and raising numbers to powers much faster than using standard algebra. You will learn to convert numbers into polar form; multiply and divide using angles; apply De Moivre's theorem to find powers; and prove mathematical identities. This section provides the speed needed for advanced engineering problems.

Concept Overviews

5 Lessons

1:15:37

Problem Walkthroughs

2 Lessons

1:30:03

5. Roots

This chapter teaches you to find all roots and rational powers of complex numbers. These techniques are essential for solving polynomial equations and understanding symmetric patterns on the Argand plane. You will master comparing polar numbers for equality; calculating all roots of any complex value; evaluating rational powers; and identifying the properties of the Nth roots of unity through detailed worked examples.

Concept Overviews

4 Lessons

2:03:40

Problem Walkthroughs

4 Lessons

4:01:13

6. Exponential Form

Exponential form uses Euler's formula to simplify calculus and engineering maths. It links trigonometry with exponents for the most efficient way to solve complex problems. You will master Taylor series expansions; represent numbers in Eulerian form; and solve multiplication, division, and power operations using exponent rules.

Concept Overviews

3 Lessons

55:21

Problem Walkthroughs

3 Lessons

1:05:01

7. Trigonometric Functions

This chapter links complex numbers to trigonometry to simplify calculus and wave analysis. You will use polar and exponential forms to derive trigonometric identities more efficiently than using standard geometry. You will learn to express sine and cosine using Euler's formula; expand powers of trigonometric functions into multiple angles; and solve complex identities using polar representations and worked algebraic examples.

Concept Overviews

2 Lessons

38:38

Problem Walkthroughs

4 Lessons

4:06:38

8. Hyperbolic Functions

This chapter explores the relationship between complex numbers and hyperbolic functions. You need these expressions to solve advanced problems in structural engineering and heat transfer. You will learn to derive hyperbolic sine and cosine using exponential forms; convert between circular and hyperbolic functions; and solve complex algebraic identities through step-by-step worked examples.

Concept Overviews

1 Lesson

21:29

Problem Walkthroughs

1 Lesson

21:09

9. Logarithmic Functions

This chapter explains how to find the logarithm of a complex number using its exponential form. This tool is vital for solving equations where the unknown is an exponent in complex space. You will learn to derive the general logarithmic expression; identify the principal value of a complex log; and solve practical problems involving natural logarithms of imaginary and complex values.

Concept Overviews

1 Lesson

22:29

Problem Walkthroughs

1 Lesson

53:19

10. Graphing on the Complex Plane

Convert complex algebraic equations into geometric shapes on the Argand plane. This skill is critical for mapping electromagnetic fields and fluid flow boundaries in engineering design. You will master identifying equations for circles and ellipses; converting modulus expressions into coordinates; sketching loci like perpendicular bisectors and rays; and shading regions while excluding undefined points.

Concept Overviews

5 Lessons

19:11

Problem Walkthroughs

7 Lessons

1:01:16