Rational Trigonometry

Just because I can do a Google Scholar query using "rational trigonometry" and find papers saying things like:

"Explicit tough Ramsey graphs" at https://arxiv.org/pdf/0807.2692.pdf . "We present here two new explicit constructions ... The first construction is based on a remarkable new approach of Wildberger to trigonometry and Euclidean geometry."

does not mean I am qualified to judge what is "interesting", or figure out which use non-real fields.

Any angle with a rational tangent or cotangent has some representation as a Gaussian rational (usually requiring a non-unit magnitude, though), and these angles are closed under addition and subtraction (corresponding to multiplying and dividing these Gaussian rationals). And given a Gaussian integer or rational, we can readily take its squared magnitude (a^2 + b^2) to get an integer or rational, and we can in the same way take the squared cosine or squared sine of its angle (a^2/(a^2 + b^2) or b^2/(a^2 + b^2)) to get a rational.

RT boils down to doing trigonometry with dot products and without square roots. There's a temptation to replace dot products with ordinary trigonometry, but RT says don't do it because it's a step back.

> 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

He highlights in some of his Youtube videos that in respected Math textbooks the definition of real numbers is left vauge.

In his opinion set theory has the same kind of holes the we are expected to accept that we can add an infinite quantity of things to a Set by describing a function or simply having a desciption of the elements of the Set.

    |P Q R|
    |1 1 1| / 2