Talk:Gaussian rational

Puzzling statement
The section Ford spheres contains this statement:

"For a Gaussian rational represented in lowest terms as $$p/q$$, the radius of this sphere should be $$1/q\bar q$$ where $$\bar q$$ represents the complex conjugate of $$q$$."

I'm not sure how "lowest terms" is defined here (nothing is mentioned about lowest terms in the article about Gaussian rationals), but every Gaussian rational is of the form a/b + c/d i for integers a, b, c, d, so it can be expressed as (ad + bc i)/bd. This can then be expressed in "lowest terms" by canceling any common integer factor of the numerator and denominator.

Then the denominator is just an integer, so there is no need to refer to its "complex conjugate".

I hope someone knowledgeable about this subject can fix this.

— Preceding unsigned comment added by 2601:204:f181:9410:6d65:1246:9fe5:9202 (talk) 14:45, 26 June 2024 (UTC)


 * I'm not sure, but I would assume that 'lowest terms' means the numerator and denominator are relatively prime. To get there you could (1) put the number into the form $(a + bi) \big/ (c + di)$, (2) take the prime factorization of numerator and denominator (in terms of Gaussian primes), then (3) eliminate any common factors. (Or for faster computation, find the GCD of the numerator and denominator then divide by it.) –jacobolus (t) 17:57, 26 June 2024 (UTC)