Re: Defining "<" for the rationals
- From: Virgil <ITSnetNOTcom#virgil@xxxxxxxxxxx>
- Date: Sat, 26 Nov 2005 12:34:09 -0700
In article <200511261621.jAQGLl016502@xxxxxxxxxxxxxxxxxxxx>,
mstemper@xxxxxxxxxxxxxxxx (Michael Stemper) wrote:
> I've been looking at the rational numbers as an equivalence relation.
> I've been able to show that the definitions of multiplication and addition
> give the same answer, regardless of which member of an equivalance set
> is chosen. I wanted to also define the "<" relation, but hit a stumbling
> block. Most of the time, you can say (a,b)<(c,d) iff ad<bc. However, this
> falls apart if b or d is negative.
>
> I could say "chose a member of [(a,b)] with a positive second element,"
> or I could say "if b<0, change the signs of a and b." Neither of these
> seem particularly elegant. Is there a cleaner way to define the less
> than relation on rationals?
The "standard" representation of a rational requires that the numerator
and denominator share no positive integer factor larger than 1 AND that
the denominator be positive.
In comparing standard representations, a/b < c/d iff a*d < b*c works
fine.
Alternately, you can define your equivalence relation on the Cartesian
product of the integers with the (positive) naturals. Then your
"denominators" are automatically positive. This automatically avoids the
problem of zero denominators as well.
.
- References:
- Defining "<" for the rationals
- From: Michael Stemper
- Defining "<" for the rationals
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