Number and Representation

In his Web page Domingo Gómez Morín writes:

"So, in the fourth step we got 479 exact digits for the square root of 2, just by means of the most simple arithmetic. We did not require any derivatives, nor the Cartesian system, nor any cumbersome-&-mysterious concept coming out from Infinitesimal Calculus."

Gómez Morín accomplished an impressive feat with simple arithmetic. In a few algebraic steps with a new algorithm, he goes - it seems - to the heart of the matter. The order he maintains is worth studying.

This is an example of how different algorithms can be used to represent the same number. Maybe we could define number, as the set of all algorithms that produce it. Some clearly defined processes, or algorithms, will be better than others depending on the context; but in any case I am looking for a periodic representation of irrational numbers, just like Minkowski's question mark function, ?(x), seems to bring some order to the square root of natural numbers, so Gómez Morín's algorithms, which he calls, Fractal Fractions, seem to bring even more order at a lower complexity prize. Not only can he get 479 exact digits in four steps for quadratics, he can improve the solution precision to general polynomial equations.