Talk:Irreducible fraction

Refering to the last sentence of the article. The "fastness" of simplification of fractions by each method depends on the ingenuity of who is doing it (for example by finding common factors faster). It would be interesting to compare the methods in practical use. But I am not sure how would be the best way in doing that, any suggestions? Ricardo sandoval 14:18, 17 April 2007 (UTC)

Is there an equivalent term for ratios of more than two numbers? 60:80:100 can be reduced to 3:4:5, for instance. "Irreducible ratio"? —Preceding unsigned comment added by 71.167.66.173 (talk) 04:39, 31 May 2009 (UTC)

Case of Gaussian integers

These definitions do not account for Gaussian integers in the numerator and denominator. The second definition would also not work if a∈~{ℝ∪iℝ} (any Gaussian integer that doesn't have a zero real or imaginary part). TheExceptionCloaker (talk) 16:45, 16 September 2018 (UTC)

I guess that you are talking of the definitions given in the lead. These definitions are given only for integers. A definition for Gaussian integers is given in section "Generalization". In fact, Gaussian integers form a unique factorization domain. Every fraction of integers is equal to two irreducible fractions, which can be deduced each from the other by multiplying the numerator and the denominator by –1. Similarly, every fraction of Gaussian integers has exactly 4 irreducible forms, obtained by multiplying the numerator and the denominator of one of them by 1, –1, i, or –i. However, another reduced form is generally preferred, which is ${\frac {a+ib}{d}},$ where d is a positive integer, and a, b, d are setwise coprime integers. Every Gaussian fraction has a unique such form. Note that this form is not necessarily a reduced fraction. For example,

${\frac {2+i}{1+i}}={\frac {(2+i)(1-i)}{(1+i)(1-i)}}={\frac {3-i}{2}}.$

The first fraction is irreducible, while the last one is in normal form, but not irreducible. D.Lazard (talk) 17:24, 16 September 2018 (UTC)

Every positive rational number can be represented as an irreducible fraction in exactly one way.

I think this is false. The number a/b, a and b integers with gcd = 1, can be represented as a/b or -a/-b, which are both irreducible. -a/-b is irreducible because the gcd of -a and -b is 1. For example, 2/3 can be represented as 2/3 or -2/-3, which are both irreducible. TheGoatOfSparta (talk) 17:48, 31 January 2024 (UTC)

Good remark. Also, there is no reason to limit the result to positive rational numbers. I have edited the article for fixing this point. D.Lazard (talk) 18:17, 31 January 2024 (UTC)

Thank you for fixing that. TheGoatOfSparta (talk) 12:46, 2 February 2024 (UTC)