Algebra • Topic 9

Complex Numbers

Complex numbers are not just a tool for solving equations but a powerful language for geometry and algebra.

Definition and Operations

Definition (Complex Number)

A complex number is a number of the form

z = a + bi

, where

a, b \in R

and

i^{2} = - 1

Complex numbers can be viewed as points

(a, b)

or vectors in the plane.

Theorem (Polar Form)

Any complex number

z = x + i y

can be written as:

z = r (cos θ + i sin θ)

where

r = ∣ z ∣

and

θ = arg (z)

Theorem (De Moivre's Theorem)

For any integer

n

(cos θ + i sin θ)^{n} = cos (n θ) + i sin (n θ)

This allows for easy powering of complex numbers. $z^{n} = r^{n} (cos (n θ) + i sin (n θ))$ .

The equation

z^{n} = 1

has

n

distinct solutions, called the $n$ -th roots of unity.

z_{k} = cos (\frac{2 πk}{n}) + i sin (\frac{2 πk}{n}), k = 0, 1, \dots, n - 1

These roots form a regular $n$ -gon inscribed in the unit circle.

Exercise (Problem 1)

Compute

(1 + i)^{10}

Exercise (Problem 2)

Let

ω

be a complex cube root of unity (

ω \neq = 1

). Calculate

(1 - ω + ω^{2}) (1 + ω - ω^{2})

Exercise (Problem 3)

Solve

z^{3} = 8 i

for all complex

z