Further Complex Numbers

In Pure 1, the modulus-argument form of a complex number was introduced, with multiplication of two complex numbers involving multiplying their moduli and adding their arguments. Repeating this process give's De Moivre's Theorem:

This is true for any integer power , and can be shown by proof by induction.

Euler's formula relates complex exponentiation to trigonometric functions:

While the textbook claims "it makes no sense to ask why it is true or how to prove it", we leave a proof in the footnotes using Maclaurin series.

Euler's formula also gives a new way of writing complex numbers:

The complex conjugate of is :

For , this gives .

De Moivre's theorem can also be used to find roots of complex numbers, to solve equations of the form :

The roots of form a regular -gon with a circumcircle centred on the origin.

The roots of unity are such that . There are th roots of unity, each with modulus 1 and differing by an argument of . The roots of unity are:

The roots of unity can also be written as:

There are symmetries of the roots of unity that can be applied, e.g. many are complex conjugates. Additionally, as the roots of unity are spaced equally around the circle, the sum must be zero (this can also be shown by considering roots of polynomials).

This can be used to factorise expressions of the form .

Multiplication by corresponds to a rotation about the origin through angle and an enlargement by scale factor .

Division by corresponds to a rotation about the origin through angle and an enlargement by scale factor .

Proof of Euler's Formula
This is taken from non-RDB private notes.
The Maclaurin expansion of is given by:

For example, considering = :

Considering :

Considering :

By considering , we can observe that:

Proving Euler's formula.