An Introduction to Complex Numbers

Explains why we need it and how to work with it.
What you'll learn
Understand the need for complex numbers.
Differentiate between imaginary numbers and complex numbers.
Identify when to use complex numbers.
Perform basic operations with complex numbers in Cartesian form.
Calculate the powers of Imaginary numbers in Cartesian form
Calculate the square root of complex number in Cartesian form
Solve quadratic equations with complex coefficients.
Create quadratic equations form the cartesian form of the complex roots
Solve for unknowns when two complex numbers in cartesian form are equivalent
Learn to draw Argand diagrams
Show that complex numbers represented on an Argand diagram can be treated as vectors.
Convert a complex number in Cartesian form to Polar form.
Show the difference between polar form and polar coordinates
Perform basic operations with complex numbers in Polar form
Define De Moivre's theorem for the polar form of a complex number
Use De Moivre's theorem to solve problems
Represent the locus of a variable complex point on an Argand diagram
Solve problems related to intersecting loci
Use Argand diagrams to solve problems involving inequalities
Convert a complex number in Polar form to Exponential form
Define Euler's formula for the exponential form of a complex number
Perform basic operations with the exponential form of a complex number
Express sin(x) and cos(x) as complex numbers in exponential form
Requirements
You should know how to solve quadratic equations
You should know how to perform operations with algebra terms
You should know how to perform operations with surds
You should know how to find the quadrant an angle lies in based on its coordinates
You should know what the locus of a variable point is
You should know the cartesian equation of a circle
You should know how to apply circle theorems
You should know how to use the Binomial theorem
* You should know the Maclaurin expansion
You should know how to solve trigonometric equations
Description
This course is an introduction to complex numbers. The content provided in the course will be below the university level. The course will take the point of view of the student asking questions and I will answer these questions. I tried to make each successive question build on the information you received from earlier lectures and it should answer any question you had from an earlier lecture. You will start from the foundation of complex numbers (i.e. the imaginary unit) and why complex numbers are needed. You will learn when to use complex numbers and how to use the knowledge you already have(such as algebra and surds) to work with them. You will also learn about the different forms of a complex number, the benefits of the different forms and how to use each of the forms to perform operations (such as addition, subtraction, multiplication, division and evaluating powers). You will also learn how to represent complex numbers on an Argand diagram and use Argand diagrams to solve problems of intersecting loci for variable complex points and inequalities. I will also show you how to apply De Moivre's theorem for trigonometric expansion and how to use Euler's formula to convert the polar form of a complex number to the exponential form.  At the end of the course you will be given a mix problem sheet with challenging problems.

Who this course is for:
Students unfamiliar with complex numbers
CAPE Unit 2 Pure Mathematics students
Anyone who needs a basic understanding of how to work with complex numbers

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