> You can work over the rational numbers, but you don’t need to. You could work over real or complex numbers, or even finite fields. Because you don’t take square roots, you can work over fields that don’t necessarily have square roots. If you’re working with integers modulo a prime, half of your numbers have no square root and the other half have two square roots. ...
> Why would you want to do geometry over finite fields? Finite fields are important in applications: error-correcting codes, cryptography, signal processing, combinatorics, etc. And by thinking of problems involving finite fields as geometry problems, you can carry your highly developed intuition for plane geometry into a less familiar setting.
"Orientation Modeling Using Quaternions and Rational Trigonometry" at https://www.mdpi.com/2075-1702/10/9/749 has an example of using it with complex numbers:
We can define the quadrance of z by:
Q(z) = z · z̅ = a² + b².
Spread: The spread can be defined in several ways ...
in a more general way, the spread can be specified
for any complex number by the next expression:
s(w) ≡ b² / a²
The paper includes the section "Rotations on the Plane" which may address your e^(i*alpha).
> You can work over the rational numbers, but you don’t need to. You could work over real or complex numbers, or even finite fields. Because you don’t take square roots, you can work over fields that don’t necessarily have square roots. If you’re working with integers modulo a prime, half of your numbers have no square root and the other half have two square roots. ...
> Why would you want to do geometry over finite fields? Finite fields are important in applications: error-correcting codes, cryptography, signal processing, combinatorics, etc. And by thinking of problems involving finite fields as geometry problems, you can carry your highly developed intuition for plane geometry into a less familiar setting.
"Orientation Modeling Using Quaternions and Rational Trigonometry" at https://www.mdpi.com/2075-1702/10/9/749 has an example of using it with complex numbers:
The paper includes the section "Rotations on the Plane" which may address your e^(i*alpha).