Re: Defining "<" for the rationals

On 26.11.2005 17:21, 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?
>

If you require the relation < to be compatible with addition in the
obvious way, you could reduce the situation by defining when a rational
is positive.

J.
.

Relevant Pages

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  • Re: Defining "<" for the rationals
    ... >>> I've been looking at the rational numbers as an equivalence relation. ... >>> than relation on rationals? ... you can define < by requiring ...
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  • Re: Defining "<" for the rationals
    ... >On 26.11.2005 17:21, Michael Stemper wrote: ... >> I've been looking at the rational numbers as an equivalence relation. ... Is there a cleaner way to define the less ... >> than relation on rationals? ...
    (sci.math)
  • Re: Defining "<" for the rationals
    ... >> I've been looking at the rational numbers as an equivalence relation. ... >> seem particularly elegant. ... Is there a cleaner way to define the less ... >> than relation on rationals? ...
    (sci.math)
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