Complex Numbers

Do you want to learn how to work with numbers that go beyond the real line? Do you want to understand the concepts of imaginary unit, conjugate, modulus, argument, and polar and exponential forms of complex numbers? Do you want to master the skills of performing algebraic and geometric operations on complex numbers using different methods and tools? If you answered yes to any of these questions, then this course is for you! In this course, you will learn how to: - Define and classify complex numbers and their real and imaginary parts - Perform addition, subtraction, multiplication, and division of complex numbers using the standard form a + bi - Find the conjugate, modulus, and argument of a complex number and use them to compare and simplify complex numbers - Represent complex numbers on the Argand plane and visualize their geometric properties and transformations - Convert complex numbers from rectangular to polar and exponential forms and vice versa - Use De-Moivre's theorem and Euler's formula to find the powers and roots of complex numbers in polar and exponential forms - Use complex numbers to define and manipulate trigonometric and hyperbolic functions and their inverses - Use complex numbers to define and manipulate logarithmic functions and their properties - Use complex numbers to graph and solve equations of circles, lines, and other curves on the complex plane This course is suitable for anyone who wants to learn or review the basics of complex numbers and their applications. It is especially useful for students and professionals in engineering, physics, computer science, cryptography, and other related fields. By the end of this course, you will have a solid understanding of complex numbers and their operations. You will also be able to apply the knowledge and skills you gain to real-world problems and challenges that involve complex numbers. Once enrolled, you have access to dynamic video lessons, interactive quizzes, and live chat support for an immersive learning experience. You engage with clear video explanations, test your understanding with instant-feedback quizzes and interact with our expert instructor and peers in the chat room. Join a supportive learning community to exchange ideas, ask questions, and collaborate with peers as you master the material, by enrolling right away.

34 hrs

$ 10.00

Payment required for enrolment

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.

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.

Course Chapters

Introduction

Natural numbers, integers, rational numbers, real numbers, and an introduction to complex numbers and their descriptions.

Chapter lessons

1.Welcome13:11

Welcome to the course and an overview of the course outline.

2.Number systems25:22

What are natural numbers, integers, rational numbers, irrational numbers and real numbers?

3.Complex numbers21:57

Why are complex numbers necessary?

4.Powers of i23:19

Powers of i.

5.Worked examples (1)14:45

Worked examples on powers of i.

Algebra of Complex Numbers

Operations on complex numbers; conjugates of complex numbers and their properties; equality of complex numbers.

Chapter lessons

1.Addition7:28

Operation of addition (and subtraction) of complex numbers.

2.Multiplication19:56

Multiplication of a complex number by a scalar (real number) and by another complex number.

3.Conjugates11:44

Conjugates of complex numbers.

4.Division25:10

Division of a complex number by a constant (real number) and by another complex number.

5.Equality8:21

Equality (and inequality) of complex numbers.

6.Properties31:44

Properties and identities of algebra of complex numbers and their conjugates.

7.Worked examples (1)33:21

Worked examples on algebra of complex numbers.

8.Worked examples (2)26:50

More worked examples on algebra of complex numbers.

9.Worked examples (3)1:08:23

More worked examples on algebra of complex numbers.

Complex Numbers on the Argand Plane

Geometric representation of complex numbers on the Argand plane; modulus of a complex number; general and principal arguments of a complex number.

Chapter lessons

1.Representation21:04

Representation of a complex number on the Argand plane.

2.Modulus and argument8:08

Modulus (magnitude) and general argument of a complex number.

3.Principal argument36:35

Principal argument of a complex number.

4.Properties of the modulus39:01

Properties of the modulus of a complex number.

5.Worked examples (1)43:20

Worked examples on the modulus and argument of a complex number.

6.Worked examples (2)45:55

More worked examples on the modulus and argument of a complex number.

Polar Form

Polar representation of complex numbers; multiplication and division of complex numbers in polar form; powers of complex numbers in polar form (De-Moivre's theorem).

Chapter lessons

1.Representation7:18

Representation of a complex number in polar coordinates.

2.Multiplication14:39

Multiplication of complex numbers in polar form.

3.Division13:26

Division of complex numbers in polar form.

4.Powers18:00

Powers of complex numbers in polar form and an introduction to De-Moivre's theorem.

5.De-Moivre's theorem22:14

Statement and proof of De-Moivre's theorem.

6.Worked examples (1)28:15

Worked examples on representation, multiplication, division and powers (De-Moivre's theorem) of complex numbers in polar forms.

7.Worked examples (2)1:01:48

More worked examples on representation, multiplication, division and powers (De-Moivre's theorem) of complex numbers in polar forms.

Roots

Equality and roots of complex numbers in polar form.

Chapter lessons

1.Equality15:04

Equality of complex numbers in polar form.

2.Roots17:58

How to find all roots of complex numbers.

3.Rational powers5:33

How to find rational powers of complex numbers.

4.Worked examples (1)1:23:42

Worked examples on rational powers and roots of complex numbers.

5.nth roots of unity1:25:05

Properties of the nth roots of unity.

6.Worked examples (2)44:55

More worked examples on rational powers and roots of complex numbers.

7.Worked examples (3)1:02:35

More worked examples on rational powers and roots of complex numbers.

8.Worked examples (4)50:01

More worked examples on rational powers and roots of complex numbers.

Exponential Form

Exponential (Euler's) representation of complex numbers; powers of complex numbers in exponential form.

Chapter lessons

1.Taylor's series30:10

Review of Taylor's series expansion of sine, cosine and exponential functions.

2.Representation13:50

The Eulerian representation of a complex number.

3.Multiplication, division and powers11:21

Multiplication, division and powers of complex numbers in exponential form.

4.Worked examples (1)30:27

Worked examples on representation, multiplication, division and powers of complex numbers in exponential form.

5.Worked examples (2)15:52

More worked examples on representation, multiplication, division and powers of complex numbers in exponential form.

6.Worked examples (3)18:42

More worked examples on representation, multiplication, division and powers of complex numbers in exponential form.

Trigonometric Functions

Manipulating sines and cosines with complex numbers.

Chapter lessons

1.Expressions (1)27:14

Expressions for Sine and Cosine functions and their powers using complex numbers in polar and exponential forms.

2.Expressions (2)11:24

Expressions for Sine and Cosine functions and their powers using complex numbers in polar and exponential forms.

3.Worked examples (1)44:41

Worked examples on manipulating Sines and Cosines using complex numbers.

4.Worked examples (2)1:07:14

More worked examples on manipulating Sines and Cosines using complex numbers.

5.Worked examples (3)1:03:05

More worked examples on manipulating Sines and Cosines using complex numbers.

6.Worked examples (4)1:11:38

More worked examples on manipulating Sines and Cosines using complex numbers.

Hyperbolic Functions

Manipulating hyperbolic functions with complex numbers.

Chapter lessons

1.Expressions21:29

Expressions for hyperbolic Sine and Cosine using complex numbers.

2.Worked examples (1)21:09

Worked examples on manipulating hyperbolic functions using complex numbers in exponential and polar forms.

Logarithmic Functions

Manipulating logarithms with complex numbers.

Chapter lessons

1.Expression22:29

General expression for logarithms of complex numbers.

2.Worked examples (1)53:19

Worked examples on logarithms of complex numbers.

Graphing on the Complex Plane

Equations in two-dimensional coordinate geometry using complex numbers.

Chapter lessons

1.Some general equations19:11

General equations of circles and ellipses on the complex plane.

2.Worked examples (1)29:06

Worked examples on graphing on the complex plane.

3.Worked examples (2)15:57

More worked examples on graphing on the complex plane.

4.Worked examples (3)16:13

More worked examples on graphing on the complex plane.