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
