Re: More on trits & tetrits... (long)
On 2010-04-28 05:10:25 +0100, John Pryce wrote:
> P1788
>
> On 23 Apr 2010, at 15:24, Michel Hack wrote:
> > Vincent Lefèvre wrote:
> >> In floating-point arithmetic, if you have an exact zero and signed
> >> inexact zeros, you need a 4th zero (inexact zero with unknown sign)
> >> and a 3rd infinity (infinity with unknown sign).
> >
> > Right -- except it would be a 5th infinity...
> >
> > ...Now we just need a (paper/dream) arithmetic design for this!
> >
> > Interval Arithmetic is supposed to get us out of this mess.
>
> Hear, hear. I cited Siegfried already. In fact if any such system
> exists it would have been invented already by Gauss, or Cauchy, or
> someone of that time.
They weren't using floating-point arithmetic.
> If we go on in this way we might produce the Nonstandard Reals with
> infinitely many infinities and infinitesimals...
I think this would go too far. Ditto for the 5th infinity: only 3 are
necessary for floating-point (1 / exact 0 should be NaN, as this isn't
defined over the real numbers). Only interval arithmetic could need
additional infinities to do the difference between bounded intervals
with overflow and unbounded intervals (if one really wants to have a
difference).
Also note that if we had a special value +huge meaning that the value
comes from a positive overflow, so that [1,+huge] means that there
exists a real number X such that the real interval is included in
[1,X], then such a level-2 interval wouldn't have a well-determined
level-1 counterpart.
--
Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.net/>
100% accessible validated (X)HTML - Blog: <http://www.vinc17.net/blog/>
Work: CR INRIA - computer arithmetic / Arénaire project (LIP, ENS-Lyon)