Complex Numbers

Complex numbers

Complex Numbers

• Real and imaginary numbers

• Addition, subtraction and multiplying complex numbers and simplifying powers of i

• Complex conjugates

• Division of a complex number by a complex number

• Argand diagrams

• Modulus and argument of a complex number

• Solving problems with complex numbers

• Square roots of a complex number

• Solving quadratic equations with complex roots

• Solving cubic equations

• Solving quartic equations

• Exam Questions - Complex numbers

• Geometrical effects of conjugating a complex number

• Exam Questions - Finding roots

• Modulus-argument form of a complex number

• Exponential Form (Euler's relation)

• Multiplication and division rules for mod and argument of two complex numbers

• Exam Questions - Further complex numbers

De Moivre’s Theorem - Not needed in A level but i think it makes things easier in roots

• De Moivre's theorem

• Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ

• Exam Questions - Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ

• Expressing sinnθ and cosnθ in terms of sin(kθ) and cos(kθ)

• nth roots of a complex number

• Exam Questions - nth roots of a complex number

Epansion using De Moivre. This was in the scholarship exam 2022

Tutorial 1

Tutorial 2

Exam question from FM ( same as schol qu from 2022)

Part b looks hard!

Loci in the Complex Plane

• The locus of a point moving in a circle

• The locus of a point moving along a perpendicular bisector

• The locus of a point moving along a half-line

• Using complex numbers to represent regions on an Argand diagram

• Loci of ellipse

• Exam Questions - Loci in the complex plane

Common exam qs

https://www.youtube.com/watch?v=oKvK-Bg15dI

Common exam question of finding cartesian equation of a line / perpendicular bisector (algebraic method)

Complex numbers by Jack 

A history for those interested

a. Introducing Complex Numbers

b. Working with Complex Numbers

c. Complex Conjugates

Complex numbers in modulus and argument form, and argand diagrams

a. Introducing the Argand Diagram

b. Introducing Modulus-Argument Form

c. Multiply and Divide in Modulus-Argument Form

d. Loci with Argand Diagrams

De Moivre's Theorem and Polar form

a. De Moivre's Theorem

b. z = re^(iθ)