Continued Fractions & Rational Approximations

“Continued fractions can be thought of as an alternative to digit sequences for representing numbers, based on division rather than multiplication by a base. Studied occasionally for at least half a millennium, continued fractions have become increasingly important through their applications to dynamical systems theory and number theoretic algorithms. Mathematica has highly efficient original algorithms for finding large numbers of terms in continued fractions, as well as for handling exact continued fractions for quadratic irrationals.”

The above comes from Wolfram´s Research. Stephen Wolfram himself has written at least one program to calculate the continued fraction expansion of roots of integers. One can download this as a demo. For instance one can learn from this that:

4^1/3 = [1;1,1,2,2,1,3,2,3,1,3,1,30,1,4,1,2,9,6,4,1,1,2,7,2,....]

This is equivalent to:

4^1/3 = 1.587401052...

The second is based on multiplication, and the first on division.

The question of which is more natural seems more and more artificial to me.

Both can have periods, and maybe quasiperiods. There are more irrational numbers than rationals, maybe there are more quasiperiodic numbers than periodic. By the same token, there may be more without periods, than with quasiperiods.

What is interesting to me is persistence. Phenomena described by periodic mathematics seem persistent, somehow in presence of quasiperiodic mathematical objects I get surprised when after a long period of chaos, suddenly some order appears. With complete periodic praedicere, information is zero, with quasiperidicity, information arrives.

We live to get surprised. As Freman Dyson wrote, “The Universe is as interesting as it can be”. Lately Lee Smolin has looked into an evolving Universe, according to him, the great amount of black holes is a clue to understand the World we live in.

If Smolin is right, there must be tests, maybe finding Kaon Condensates inside stars. In 1973 Ray Sawyer from UCSB told me that maybe there were condensed pions in astrophysical objects, it will be really surprising to find out that not only there are condensed pion, but also kaons. Strange stars really.