At 08:18 AM 5/17/01 +0100, you wrote:
>From: "Alberto Monteiro"
>
>
>
> >
> > Robert Shaw wrote:
> > >
> > >Using hyperbolic functions, since cosh^2-sinh^2=1
> > >you get x++= square(cosh(asinh(sqrt(x))))
> > >
> > >If you don't use hyperbolic functions it's slightly more complicated.
> > >You have to use 1+tan^2=sec^2 to find
> > >x++= 1/(square(cos(atan(sqrt(x)))))
> > >which takes one extra button press.
> > >
> > >Of course, repeatedly incrementing is generally not the faster
> > >way to reach an integer, but it is a proof of principle.
> > >
> > Spoilsport :-/
> >
> > Then it's trivial to prove that every rational positive number
> > p / q can be constructed by induction over (p + q);
> > because 1 / 1 can be constructed, and, in general:
> >
> > if p = 1 then p / q can be constructed because q
> > can be constructed and 1/x over q generates p/q
> >
> > if p > q then p / q can be constructed because
> > p / q = 1 + (p - q) / q and (p - q) / q can be constructed
> > [induction argument]
> >
> > if 1 < p < q, then p / q = 1 / (q / p), and q / p can
> > be constructed because (q - p) / p can be constructed
> > [induction argument], etc
> >
>We never got that far.
>
>Actually, this is basically the continued fraction expansion.
>Any rational can be written as a finite continued fraction
>(a+1/(b+1/(c+...
>and any real can be written as an infinite continued fraction.
>
>That means any real can be produced by some infinite
>sequence of button presses, which wasn't self-evident.
>
>Which numbers can be reached with a finite sequence is
>a harder question, but I don't think it's a well known set.
>
>The largest standard countable set is the algebraic numbers,
>the other infinite countable sets being subsets of that.
>For the set of constructable numbers to be that set you'd
>have to be able to take nth roots of any number with these
>operations (and solve more general polynomials) but thats
>equivalent to deviding or multiplying by an arbitary integer.
>I don't see any way of doing that so I suspect the constructable
>set is not the algebraic numbers, but I don't see any proof.
>
>The constructable set does include all quadratic and cubic
>irrationals, e, pi, and the log of every integer, so it includes
>most of the best known transcendentals.
>
>--
>Robert
How about sum (10^-n! | n = 1, 2, 3, ...) =
0.110001000000000000000001000..., the first number to be proven transcendental?
-- Ronn! :)
