[lupire]

This called "affine transformation" in linear algebra language. (Linear algebra is stretches and rotations. Affine (affinity?) adds translations)

https://people.computing.clemson.edu/~dhouse/courses/401/not...

In 2 dimensions:

Rotation = multiplying by an imaginary unit.

Stretches = multiplying by a real number

Translation = adding a complex number.

In higher dimensions, the analogy to complex numbers breaks down.

[xeonmc]

It is not affine transforms per se but rather the expansion into homogeneous coordinates that enables translation by treating it as if it's a shear that leaves the reciprocal dimension untouched.

> Rotation = multiplying by an imaginary unit.

This is also not quite right.

Rotation is multiplying by a complex number with a magnitude of 1 (or perhaps you meant to say "raising a number to the power of i"?)

[lupire]

Sorry I meant "complex unit".

"complex number with a magnitude of 1" is the definition of a "complex unit".

[xeonmc]

I like "roots of unity" (not necessarily rational) or "unit phasor" or "non-integer powers of -1"