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
