Re: int2interval, frac2interval, rat2interval

[Vincent Lefevre <vincent@xxxxxxxxxx>]

On 2012-04-20 18:18:31 +0100, N.M. Maclaren wrote:
> On Apr 20 2012, Vincent Lefevre wrote:
> >>Er, no.  Firstly, in mathematics, 1/0 is undefined in the integers,
> >>reals, rationals and complex numbers.  It can be added to any of them,
> >>but only at the cost of breaking some of their important properties.
> >
> >It can be added, but AFAIK, there are well-known no conventions
> >about it (contrary to Rbar).
> 
> Eh?  The usual completion of the real line by adding infinities leaves
> 1/0 as undefined.

Yes, that's precisely my point. Though Rbar (signed infinities)
is a well-known convention, everything else, like defining 1/0 is
just an IEEE 754 extension (well, this also became a well-known
convention but with weak mathematical grounds).

> While there are conventions that define extend the reals to include
> 1/0, none are widely used, unlike the usual extension.

You don't extend the reals by including 1/0. You include either a
single (unsigned) infinity or you include two (signed) infinities.
These are the usual conventions. Then IEEE 754 has its own rules
to define 1/+0, 1/-0 and so on.

Though this could also be done, I've never seen a similar convention
on the integers (probably because it isn't much useful).

> IEEE 754 is mathematically inconsistent, in treating a single value
> both as true zero and a positive infinitesimal, and that leads to no
> end of trouble.

I agree, and I don't think that the same thing should be done with
integers (with more trouble, because I don't see how an infinitesimal
would make sense on integers). Anyway, even though IEEE 754 defines
1/0, it isn't a valid math expression in the context of an exact
arithmetic.

-- 
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
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