Re: [boost] Constructible numbers? (was: Re: Interest in an arbitrary precision library?)

From: Guillaume Melquiond (guillaume.melquiond_at_[hidden])
Date: 2005-09-14 02:10:15

Le mercredi 14 septembre 2005 à 02:12 -0400, Daryle Walker a écrit :
> On 9/13/05 8:17 AM, "Guillaume Melquiond" wrote:
> 
> [SNIP]
> > [1] Rational numbers are constructible. Irrational numbers are
> > constructible. Pi, E, and other common mathematical constants are
> > constructible. Numbers derived from them are constructible. Omega (the
> > halting problem encoding) is not constructible.
> 
> I couldn't find this mathematical definition of "constructible;" is it known
> by another term?  (The only definition I know of is for those ancient Greek
> compass & [unmarked] straight-edge puzzles.  But all the associated numbers
> for those are a subset of algebraic numbers.  By that definition, pi and e
> are not constructible, since they're transcendental.)  I looked in the
> Wikipedia, BTW.  But maybe you mistyped; rational and irrational numbers
> cover _every_ real number, so you didn't have to specify pi, e, common
> constants, or derived values.

Sorry, "algebraic" is missing in my sentence; I wanted to express a
progression in the complexity of the numbers. As for what I called
constructible reals, you can look at the "computable numbers" page in
the Wikipedia. I should have checked beforehand that the word I use in
French was the one used in English :-)

Anyway, my point was: "by definition", computable numbers are the ones a
computer can handle in arbitrary precision (emphasis on arbitrary), and
they cover a wider range than just rational numbers.

Best regards,

Guillaume