Proof of de Moivre's Theorem

 The following post is based of how de Moivre's theorem can be shown through a matrix representation of complex multiplication and linear transformations. The 'proof' is not by any means rigorous, but instead is used to show an interesting connection. For those unfamiliar with complex numbers (with concepts such as modulus-argument form) and matrices, I suggest familiarising yourself first. De Moivre's Theorem is stated below:

Where theta is the argument and r is the modulus of a complex number. The expression in brackets is a complex number raised to the n power. For simplicity we will assume n is real. Note how instead of expanding the bracket using the binomial expansion, the theorem states that the n can just be multiplied by theta. This is quite unique and offers a route for exploration.

Complex Multiplication:

For multiplication of two complex numbers of the form a+ib and c+id, it can be seen as:

However, this can be represented in the form of a linear transformation of a vector by a matrix:

(note the top part of the vector is the real component and the bottom part is imaginary, also what complex number is a vector and what is the transformation is not important)

This can now be extended for a general complex number a+ib to the power of n:

The first step is used to make the following step more obvious, we will now replace the concept of repeated multiplication by a+ib with repeated linear transformations:

And therefore:

(note how the matrix is in the form of a rotation transformation)

Modulus-Argand Form

For a given complex number a+ib, it can be represented in modulus argument form as such:

r is the modulus and theta is the argument. This can be seen as like the polar form of complex numbers. Below is the number 2+i, in modulus argument form it is:

Rotation Transformation

Now back to the original theorem, this can be rewritten as:

r to the n can be cancelled out to give:

Now let:

Therefore, the equation above can be written as:

We now need to prove this statement. To do so we will look at this transformation geometrically, consider the point:

Under the rotation transformation the point is rotated by theta degrees and therefore after a single rotation the point becomes:

This can be proved by using matrix multiplication of the point and using the double angle identities. More generally by n-1 rotations the point is transformed to:

We can now see that:

If you are struggling with the last step check out rotation transformations. I hope you have enjoyed this 'proof' with thanks to desmos again.-naBla5040