Re:[Axiom-developer] Re: [sage-devel] Re: sage thoughts

[Raymond Rogers]

I haven't been paying close attention but I think the following might work:
define the gcd() implicitly: i.e. minimize over [m,n integer,G>0]( m(a/b)+n(c/d))=G
This seems to make sense in Euclidean domains.
This leads to
G=gcd(da,bc)/bd
let's see how this works
gcd(1/4,1/6) would yield 2/24=1/12
gcd(3/12,9/54) would yield gcd(3*54,12*9)=gcd(3*9*6,3*4*9)=3*9*2
3*9*2/(3*4*9*6)=1/12

So it seems consistent.
Sorry if this is off-topic or I have overlooked something obvious.  Of course the actual reasonableness and verification needs proof.
I think I have developed a formalism that makes sense over Principal Ideal Rings, extended  to include inverses.  Bur the ideas are not  mathematically well defined.

Ray


On 02/11/2011 03:55 AM, daly wrote:

On Fri, 2011-02-11 at 01:49 -0800, Simon King wrote:

Hi,

On 11 Feb., 09:56, Simon King<address@hidden>  wrote:

Well, I had the impression that a couple of people are in favour of
the following:
  gcd(a/b,c/d) := gcd(a,c)/lcm(b,d)
  lcm(a/b,c/d) := lcm(a,c)/gcd(b,d)

It just occurs to me that I am incredibly stupid.

The definition above wouldn't work at all, it isn't even well-defined.
Just replace gcd(1/4,1/6) by gcd(3/12,9/54). You obtain gcd(1,1)/
lcm(4,6) = 1/12,  but gcd(3,9)/lcm(12,54) = 1/36.

Does anyone have a better idea? Would it be a correct definition if
one insisted on reduced fractions?

Cheers,
Simon

That's why I was asking for an algorithm for gcd and lcm
in the subdomain. I'm not sure what answer is expected.
The unit (1) is correct but not by your definition, and
apparently not helpful for the original poster.

Tim Daly



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