HELP: restore my faith in Python

[Neel Krishnaswami]

Mikael Olofsson <mikael at isy.liu.se> wrote:
> 
> On 05-Mar-00 Moshe Zadka wrote:
>  >  On Sat, 4 Mar 2000, Tim Peters wrote:
>  > > Your fear of the binary point is much deeper <wink>.
>  >  
>  >  Actually, no: both scare me equally. For what I do, 2 and 10 are just as
>  >  bad -- it's the finite numbers of bits I'm fighting against. And I can't
>  >  even take the rational way out: I have to deal with square roots and
>  >  trigonometry.
> 
> Ahh, it's time to include extensions of the rational field. Let's get
> more abstract. Abstraction is beautiful! And we need to represent 
> transcendental numbers consistently aswell.

Is this possible? I've forgotten most of my analysis, but I was under
the impression that you could divide the continuum as follows:

  o Rationals -- eg 2/3. Countably infinite.
  o Algebraic irrationals -- eg sqrt(2). Countably infinite.
  o Transcendental numbers -- everything else. Uncountably infinite.

So presumably you could write numeric classes that properly (if you
don't care about speed or memory usage) handle rationals and algebraic
irrationals, but the full transcendentals are just plain impossible. 
Maybe some interesting subset of the transcendentals are possible; 
if you throw in e and pi most people will be happy most of the time? 
Though on reflection this is getting pretty close to a full symbolic 
computation package.

I guess Mathematica is that expensive for a reason -- arithmetic is
harder than I thought. :)


Neel