At 12:50 AM 5/16/01 +0000, you wrote:
>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
>
>Alberto Monteiro
So Alberto has shown us how to get from Z to Q. Who recalls how to get
from Q to R?
Or did everyone cut that day?
-- Ronn! :)
