Domingo Gómez Morín

This mathematician has studied the rational approximation of solutions to polynomial equations.
I find it very interesting, and I here post something from his Web Page:

"The Cartesian-decimal-system fundamentals have imposed the replacement of Number by its corresponding decimal value, i.e.: 3/2=6/4=1.5, that is, it has restricted 3/2 to the decimal result of dividing 3 by 2, and this constitutes a crude transfiguration and degradation of Number.

In the same way flowers bring us their multiple natural properties: beauty, color and scent, also Number brings out much more than just the decimal result of a division, it bears a relative value (The form of the ratio) and a very specific location within a set of ratios which plays an important role in the development of root-solving algorithms. Cartesian system has depersonalized Number confining it to just an absolute value.

Indeed, it is really striking to realize that ancient mathematicians (Babylonians, Greeks, etc.) certainly had at hand the most elemental arithmetic tool (The rational mean) for achieving all those "advanced" algorithms that were consecrated as the most outstanding successes brought to light by the Cartesian system and decimal fractions. Believe It or Not!, based on all the evidence at hand, it seems that the extremely simple arithmetical methods shown in the book “The Fifth Arithmetic Operation” have no precedents at all, all through the very long history of roots solving.

These new trivial methods and the comments shown above are certainly very sound arguments to cogitate on the following important issues:

• The Cartesian system should not be considered as a fundamental system of Natural Philosophy but just as an artificial creation which apart from being extrinsic to the natural properties of Number also contributes to distort and vitiate the genuine image of Quantity.

• The results of the arithmetical operations of irrational numbers can be easily established by agency of the Rational Process (based on the Rational Mean in accordance with Number itself), rather than by using Dedekind's and Cantor's judgments.

• The traditional continued fractions expressions are just a particular case (Second Order Continued Fractions) of a more general conception called: Generalized Continued Fractions (Fractal Fractions), which yield periodic representations for algebraic numbers of higher degrees. You will realize that any representation of irrational numbers of higher degree get distorted when using the traditional continued fractions, that is, the “Second Order Continued Fractions” . It remains so much to be investigated on this matter mainly the issue on generating best approximants for roots of higher degrees, but these new generalized continued fractions are a step ahead on the comprehension of periodicity and best convergents.

• There are so few precedents on the analysis of the rational mean.

• The rational mean is the most intelligible arithmetical operation to understand irrational numbers, and the arithmonic mean is a particular case and an essential operation for root-solving.

Based on all the very important and simple arithmetical methods brought to light by the Rational Mean, and considering all the wonderful arithmetical properties that have been passed over during so long time, one can say now that it is certainly a pathetic arrogance to think that any result (i.e.: imaginary numbers, higher dimensions, cartesian system, relativity theory, etc.) coming out from artifices and opinions extrinsic to Number could ever supersede the Natural Order determined “in accordance with number by the forethought and the mind of Him that created all things, for the pattern was fixed, like a preliminary sketch,...” (quoted text: Nicomachus, chap.VI, [1])."