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Complex Numbers


1. Irrational numbers

Irrational numbers​ - surds

Construct root 2

Using a compass and straight edge

Construct root 3

Using a compass and straight edge

2. Complex numbers introduction

Equation with imaginary roots

Quadratic equation with imaginary roots

Using the quadratic formula

Addition and multiplication of complex numbers 


3. Division and equity of complex numbers

Dividing by a complex number - conjugate

Expressing a fraction with a real denominator

Complex number equations

"Real = real, imaginary = imaginary"

4. Argand diagram - modulus

Plot on an Argand diagram
Calculate modulus


5. Transformations of complex numbers

Understanding multiplication

Multiplication involves scaling and rotating
​Addition involves shifting

Transformations involving multiplication


​6. Conjugate roots theorem

Quadratic equation - conjugate roots

Showing that a conjugate is a root

Solving a cubic equation with two imaginary roots


7. Polar form of a complex number

Converting from polar to rectangular form

Express in polar form

Finding the modulus and argument

8. Products and quotients in polar form

Multiplying and dividing in polar form

Multiplication: Multiply the module and add the arguments
Division: Divide the module and subtract the arguments

9. De Moivre's Theorem

Raising a number to a power - expanding

Express in polar form and expand


10. Applications of de Moivre's Theorem

Raising a number to a power

Proving trigonometric identities using de Moivre's Theorem

Finding roots



12 Revision Questions

Complex Number Revision Questions
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1. Solution
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2. Solution
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3. Solution
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4. Solution
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5. Solution
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6. Solution
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12. Solution

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Revision Notes

Complex Number Video
Powers of i
KA Powers of i
KA Complex Numbers and i
Introduction to Complex Numbers: Addition, subtraction, multiplication, powers of i.
KA Adding & Subtracting Complex Numbers
KA Multiplying Complex Numbers
Complex division in rectangular form
KA Dividing Complex Numbers
Solving Complex Equations:
 
Real parts = Real parts
Imaginary parts = Imaginary parts
Graphing complex numbers on an Argand diagram and finding the modulus of a complex number.
KA Argand Diagram (Complex Plane)
KA Modulus (Absolute Value) of a Complex Number
Transformations in the Complex Plane
Complex conjugate roots
Solving quadratic and cubic equations with imaginary roots.
Writing a complex number in Polar Form
Multiplying and dividing in Polar Form and using deMoivre's Theorem to expand a complex number to a power.

Using deMoivre's Theorem to find roots of a Complex Equation. 
Using deMoivre's Theorem to prove Trigonometric identities. This question also requires use of the Binomial Theorem.

The Syllabus

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3.1 Ordinary & Higher Levels
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4.1 Higher Levels
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