Here's also a fun read about complex numbers [1]. It shows how rotation is inherently embedded in it.

[1] https://acko.net/blog/how-to-fold-a-julia-fractal/

This is really wonderful! Thanks.

Is anyone aware of a resource that collects examples of what complex numbers “encode” in various settings in which they naturally appear?

For example, complex numbers in the eigendecomposition of a real-valued square matrix encode permutations among dimensions in a particular eigenspace via the roots of unity — which amount to rotations. So they’re a bookkeeping device that allows the eigendecomposition of an automorphism of a real vector space to actually work in all cases.

They're also used for analytic continuation. For instance, the zeta function can be analytically continued to negative values of its argument using complex numbers. I'm wondering why these analytic continuations "really" work.

Quaternion is a 4D complex number representing rotation in 3D space. It has the form of: w + xi + yj + zk, where w is the real scalar value and x, y, z are the values of the i, j, k imaginary axes.

Complex number is a natural fit for rotation because operations of unit complex numbers end up on the same complex unit circle/sphere. e.g. Multiplying two quaternions performs two rotations one after another.

Of course, there's the octonion complex number with 8 axes for representing rotation in 4D hyperspace.

Complex roots of unity show up in discrete Fourier transforms (DFTs) [1]. DFTs allow you to move between the time domain and frequency domain for a given signal. They’re used all over the place in signal processing and image processing.

[1] https://en.wikipedia.org/wiki/Discrete_Fourier_transform

That'd be a very useful resource. Another example is the use in impedances [1]

[1]http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.h...

In Yaglom's Complex Numbers in Geometry, the complex numbers can be interpreted as either:

- points on a plane

- points on a sphere (via one-point compactification)

- points on the hyperbolic plane

The last is linked to the Poincare half-plane model and the Poincare disk models of hyperbolic geometry.

In communication they are used for representing the sine and cosine components of a modulation at a carrier frequency at baseband. The general concept behind this is called phasors which has many applications in physics and engineering.

They show up in roots of polynomials with real coefficients.

Do you have a reference or example of how this encoding works? Curious about this interpretation of complex numbers. Thanks for sharing!