This works perfectly fine with complex numbers. Consider Gaussian integers or Gaussian rationals (complex numbers of the form a + b i, where a and b are integers or rationals). Like complex numbers in general, these amount to the combination of scaling by some magnitude and rotating by some angle. If we only care about the angle, we can consider two Gaussian rationals to be equivalent if their ratio is a positive real number.
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.
I don't see how that would help: e^(i*âs) is only further analyzable when s is the square of a rational number, but even then it won't help you go further: e^i*k is not rational (kâ 0). In something like DFT or FIR analysis, it wouldn't help to work with spread instead of angle, I think. Integrating over s^2 only adds more steps, and I also don't see how it could help numerical approximation.
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.