Reform rational

[gifnksm]

std::ratio module contains BigRational type, but the type is not usable by following reasons.

• Ratio::new requires T: Copy + Num + Ord, but BigInt is not implicitly copyable, because it contains unique vector.
• BigInt is not implements Num

So, I rewrite Ratio as follows.

• Ratio requires T: Clone + Integer + Ord.
  • Copy -> Clone: to be able to use BigRational
  • Num -> Integer: It is incorrect that a rational number constructed by two non-integer numbers.
• BigInt implements Num and Orderable which are required by Integer bound

[huonw]

My reasoning for Num rather than Integer was a Ratio of two polynomials (for example) is sensible, although it might make sense for polynomials to implement Integer.

[gifnksm]

@huonw I think it makes sense that Rational of polynomials too.
I rebase to revert 41eaa97. (It may not break other commits)

[huonw]

Cool! (Nice clean-up.)

[huonw]

Actually thinking about it a bit more, it would be better to use the gcd implementation on the Integer trait than to have the one in the rational.rs. (Since the performance of the type-specific one will presumably be much better.)

So I think that the Integer bound is better than the Num bound.

[gifnksm]

@huonw It is correct in the performance sense, but it is strange in the naming convention sense (polynomials is Integer? I think not.)
I think it better that gcd method belongs to the other trait, such as CommonDivisor or Factor, if gcd supports polynomials.
(This may be the issue that should be discussed in #4819)

For now, in this pull request, using Integer instead of Num, and I stop working on reverting 41eaa97.
Please review.

[huonw]

I agree about the naming, but names are always hard, and I added a note to #4819.

(I don't have magical r+ powers, by the way.)

[gifnksm]

Oh, I didn't know you don't have the power, sorry.
I like the phrase "magical r+ powers" :)