Teacher's manual for complex numbers

Operations with complex numbers: Introduction to complex numbers

Teacher's manual for complex numbers

Teacher's manual for Complex numbers

Introduction

Welcome to the Complex numbers module. This module provides a self-contained introduction to complex numbers that can also be incorporated into a linear algebra course or added to complement a high school curriculum.

Course description

This module is suitable for high school students, as well as for early college or applied university students. It can be employed in a flipped classroom setting for both and also in a remedial setting for early college and applied university levels. The minimal needed theory is described in the boxes on the pages, while additional and more advanced theory, like proofs, is found in the tabs.

Below, find the key learning objectives of this module, which outline the essential skills and knowledge students are expected to gain.

Learning objectives

The student...

...understands complex numbers as an extension of real numbers, evaluates square roots of negative numbers, and recognises that complex numbers arise as solutions of equations of the form #x^2 + c^2=0#;
...distinguishes between their imaginary and real components, represents them in both cartesian and polar form, as well as performs basic operations between them, such as addition, subtraction, multiplication, complex conjugation, and division;
...works with complex-valued functions, such as exponentials and logarithms, and reproduces and applies Euler’s representation of complex numbers and de Moivre’s theorem;
...understands complex polynomial functions, how to factorise them and obtain their zeros, as well as reproduces and applies the fundamental theorem of algebra.

The course assumes an understanding of basic algebra, functions, polynomials, trigonometry, exponential, and logarithmic functions, which can be obtained with our Basic Mathematics course.

Description of the course

This module contains two chapters. The first chapter introduces the imaginary unit and operations with complex numbers, both in cartesian and polar form. The second continues with complex functions and polynomials, covering Euler's formula, de Moivre's theorem, and roots. Both chapters contain a package with applications in science and engineering.

Module contents

This module consists of two chapters with two subchapters each, excluding the introduction and conclusion.

Chapters	Subchapters	Number of theory pages	Number of exercises
Operations with complex numbers	Introduction to complex numbers	1	0
	Calculating with complex numbers	3	24
	More operations with complex numbers	4	49
	Conclusion	1	10
Complex functions and polynomials	Introduction to complex functions and polynomials	1	0
	Complex functions	5	54
	Complex polynomials	3	33
	End of complex functions and polynomials	1	12

Study load: 1 ECT.

How to organise your course

Below, you can find a selection of suggested exercises for classroom demonstration and for student practice (exercises suitable for demonstration are also suitable for practice). Later in this document, you can also find a sample diagnostic test. Assuming a three- to four-hour per week class load, this module can be covered in three weeks.

Sample course plan 1

Chapter	Subchapter	Recommended for demonstration (solve together)	Recommended for practice (student)
Operations with complex numbers	Calculating with complex numbers Imaginary numbers Complex numbers Basic operations with complex numbers	Exercise 2 (107208) and 4 (107219) on Imaginary numbers Exercise 1 (107252) on Complex numbers Exercise 1 (107264), 4 (107272), and 6 (107269) on Basic operations with complex numbers	Exercise 1 (107209) and 5 (107221) on Imaginary numbers Exercise 3 (107261) on Complex numbers Exercise 2 (107266), 3 (107271), and 5 (107267) on Basic operations with complex numbers
	More operations with complex numbers Complex conjugation and norm The quotient of two complex numbers Polar coordinates Operations on complex numbers in polar coordinates	Exercise 3 (141039) and 8 (140657 + 140658) on Complex conjugation and norm Exercise 3 (141076) and 6 (140934) on The quotient of two complex numbers Exercise 1 (141931) and 4 (142872) on Polar coordinates Exercise 3 (142090) and 4 (141812 + 141813) on Operations on complex numbers in polar coordinates	Exercise 1 (140829) and 9 (140772) on Complex conjugation and norm Exercise 2 (140576), 5 (140600), and 7 (140930) on The quotient of two complex numbers Exercise 2 (141381) and 3 (141516) on Polar coordinates Exercise 1 (146820), 2 (146821), and 5 (141843 + 141844) on Operations on complex numbers in polar coordinates
	End of operations with complex numbers Applications
Complex functions and polynomials	Complex functions The complex exponential function Euler's formula and trigonometric functions The complex logarithm De Moivre's theorem Roots of unity and complex numbers	Exercise 3 (141725) and 6 (142631) on The complex exponential function Exercise 6 (142615 + 142827 + 142828) on Euler's formula and trigonometric functions Exercise 1 (143117) and 4 (143489) on The complex logarithm Exercise 2 (143662) and 6 (143753 + 143858) on De Moivre's theorem Exercise 1 (143826) and 4 (143308) on Roots of unity and complex numbers	Exercise 4 (141521) and 7 (142399) on The complex exponential function Exercise 7 (142732) and 8 (142846) on Euler's formula and trigonometric functions Exercise 2 (143078) and 3 (143243) on The complex logarithm Exercise 3 (143733) on De Moivre's theorem Exercise 3 (143771) and 5 (143951) on Roots of unity and complex numbers
	Complex polynomials Complex polynomials Quadratic formula and the fundamental theorem of algebra Real coefficients and conjugate roots	Exercise 1 (143948) and 5 (144236) on Complex polynomials Exercise 2 (144543) and 7 (144945) on The complex quadratic formula and the fundamental theorem of algebra Exercise 2 (144637) and 8 (144683) on Real coefficients and conjugate roots	Exercise 4 (144541) and 7 (144298) on Complex polynomials Exercise 1 (144359 + 144364) and 9 (144114) on The complex quadratic formula and the fundamental theorem of algebra Exercise 3 (144561) and 9 (144692) on Real coefficients and conjugate roots
	End of complex functions and polynomials Applications

Assessments

Below, you can find a selection of exercises that are suitable for diagnosing a student's level in the contents of this module.

Sample diagnostic test

Chapter	Subchapter	Recommended for diagnostic testing
Operations with complex numbers	Calculating with complex numbers Imaginary numbers Complex numbers Basic operations with complex numbers	Exercise 3 (107218) and 6 (107220) on Imaginary numbers Exercise 5 (140771) on Complex numbers Exercise 9 (145700) on Basic operations with complex numbers
	More operations with complex numbers Complex conjugation and norm The quotient of two complex numbers Polar coordinates Operations on complex numbers in polar coordinates	Exercise 10 (140887) and 11 (140889) on Complex conjugation and norm Exercise 8 (140487) and 10 (141025) on The quotient of two complex numbers Exercise 8 (141906 + 141907) on Polar coordinates Exercise 7 (142099) on Operations on complex numbers in polar coordinates
	End of operations with complex numbers Applications
Complex functions and polynomials	Complex functions The complex exponential function Euler's formula and trigonometric functions The complex logarithm De Moivre's theorem Roots of unity and complex numbers	Exercise 10 (142317) on The complex exponential function Exercise 9 (142832) on Euler's formula and trigonometric functions Exercise 8 (143536) on The complex logarithm Exercise 7 (143811) on De Moivre's theorem Exercise 7 (143872) on Roots of unity and complex numbers
	Complex polynomials Complex polynomials Quadratic formula and the fundamental theorem of algebra Real coefficients and conjugate roots	Exercise 10 (144529) on Complex polynomials Exercise 3 (143991) and 10 (144237) on The complex quadratic formula and the fundamental theorem of algebra Exercise 7 (144194) on Real coefficients and conjugate roots
	End of complex functions and polynomials Applications

Teacher's tools and resources

Best practices and tips

In the first chapter, students are introduced to complex numbers. They learn to represent them both analytically and in an Argand diagram.

Visual demonstrations of complex numbers and their operations in an Argand diagram can help build intuition and ease the students' work in the second chapter.
The norm of a complex number and complex conjugation are introduced in the first chapter and become crucial to understand the content of the second chapter. Using visualisation to illustrate complex conjugation and to connect the norm to the Pythagorean theorem can help students internalise these concepts quickly.
Mastering the switch between cartesian and polar representations of complex numbers is crucial to follow the introduction of Euler's form in the second chapter. Repeated practice of these transformations is quite helpful, in particular with an emphasis on the difference between the principal value and general polar coordinates.

Additional resources

The theory pages contain video explanations of the most important concepts. These and additional video materials are available at the Sowiso Explains' YouTube channel.