Approximation of quadratic irrationals and their pierce expansions

In this article two aims are pursued: on the one hand, to present a rapidly converging algorithm for the approximation of square roots; on the other hand and based on the previous algorithm, to find the Pierce expansions of a certain class of quadratic irrationals as an alternative way to the method presented in 1984 by J.O. Shallit; we extend the method to find also the Pierce expansions of quadratic irrationals of the form $2 (p-1) (p - \sqrt{p^2 - 1})$ which are not covered in Shallit's work.Quadratic irrationals, Pierce series

A total order in [0,1] defined through a 'next' operator

A `next' operator, s, is built on the set R1=(0,1]-{ 1-1/e} defining a partial order that, with the help of the axiom of choice, can be extended to a total order in R1. Besides, the orbits {sn(a)}n are all dense in R1 and are constituted by elements of the same arithmetical character: if a is an algebraic irrational of degree k all the elements in a's orbit are algebraic of degree k; if a is transcendental, all are transcendental. Moreover, the asymptotic distribution function of the sequence formed by the elements in any of the half-orbits is a continuous, strictly increasing, singular function very similar to the well-known Minkowski's ?(×) function.Total orders, pierce series, singular functions

A singular function and its relation with the number systems involved in its definition

Minkowski's ?(x) function can be seen as the confrontation of two number systems: regular continued fractions and the alternated dyadic system. This way of looking at it permits us to prove that its derivative, as it also happens for many other non-decreasing singular functions from [0,1] to [0,1], when it exists can only attain two values: zero and infinity. It is also proved that if the average of the partial quotients in the continued fraction expansion of x is greater than k* =5.31972, and ?'(x) exists then ?'(x)=0. In the same way, if the same average is less than k**=2 log2(F), where F is the golden ratio, then ?'(x)=infinity. Finally some results are presented concerning metric properties of continued fraction and alternated dyadic expansions.Singular function, number systems, metric number theory

Fermat's treatise on quadrature: A new reading

The Treatise on Quadrature of Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, R x+m/n dx, or under a higher hyperbola, R x-m/n dx— with the appropriate limits of integration in each case—, has a second part which was not understood by Fermat’s contemporaries. This second part of the Treatise is obscure and difficult to read and even the great Huygens described it as 'published with many mistakes and it is so obscure (with proofs redolent of error) that I have been unable to make any sense of it'. Far from the confusion that Huygens attributes to it, in this paper we try to prove that Fermat, in writing the Treatise, had a very clear goal in mind and he managed to attain it by means of a simple and original method. Fermat reduced the quadrature of a great number of algebraic curves to the quadrature of known curves: the higher parabolas and hyperbolas of the first part of the paper. Others, he reduced to the quadrature of the circle. We shall see how the clever use of two procedures, quite novel at the time: the change of variables and a particular case of the formula of integration by parts, provide Fermat with the necessary tools to square very easily curves as well-known as the folium of Descartes, the cissoid of Diocles or the witch of Agnesi.History of mathematics, quadratures, integration methods

On the concept of optimality interval

The approximants to regular continued fractions constitute `best approximations' to the numbers they converge to in two ways known as of the first and the second kind. This property of continued fractions provides a solution to Gosper's problem of the batting average: if the batting average of a baseball player is 0.334, what is the minimum number of times he has been at bat? In this paper, we tackle somehow the inverse question: given a rational number P/Q, what is the set of all numbers for which P/Q is a `best approximation' of one or the other kind? We prove that in both cases these `Optimality Sets' are intervals and we give a precise description of their endpoints.Diofantine approximations, continued fractions, metric theory

A mathematical excursion: From the three-door problem to a Cantor-type perfect set

We start with a generalization of the well-known three-door problem: the n-door problem. The solution of this new problem leads us to a beautiful representation system for real numbers in (0,1] as alternated series, known in the literature as Pierce expansions. A closer look to Pierce expansions will take us to some metrical properties of sets defined through the Pierce expansions of its elements. Finally, these metrical properties will enable us to present 'strange' sets, similar to the classical Cantor set.Pierce expansions, cantor-type sets

Sobre una sèrie de Goldbach i Euler

El teorema 1 de l'article d'Euler «Variae observationes circa series infinitas», publicat el 1737, enuncia un resultat sorprenent: la sèrie dels recíprocs de les potències enteres menys la unitat té suma 1. Euler atribueix el teorema a Goldbach. La demostració que ofereix és un dels exemples —tan freqüents als segles xvii i xviii— de mal ús d'una sèrie divergent que acaba produint un resultat correcte. Examinem amb detall la demostració d'Euler i, amb l'ajut de les intuïcions que ens proporciona una demostració moderna (i totalment diferent), presentem una reconstrucció racional en termes que es podrien considerar rigorosos per als estàndards weierstrassians moderns. Al mateix temps, amb l'ajut d'algunes idees de l'anàlisi no estàndard, veiem com la mateixa reconstrucció també es pot considerar correcta per als estàndards robinsonians moderns. Aquest últim enfocament, però, s'adiu completament amb la demostració d'Euler i de Goldbach. Esperem, doncs, convèncer el lector de com unes poques idees d'anàlisi no estàndard són suficients per reivindicar el treball d'Euler.Theorem 1 of Euler’s paper of 1737 «Variae observationes circa series unfinitas », states the astonishing result that the series of all unit fractions whose denominators are perfect powers of integers minus unity has sum 1. Euler attributes the theorem to Goldbach. The proof is one of those examples of misuse of divergent series to obtain correct results so frequent during the seventeenth and eighteenth centuries. We examine this proof closely and, with the help of some insight provided by a modern (and completely different) proof of the Goldbach-Euler Theorem, we present a rational reconstruction in terms which could be considered rigorous by modern weierstrassian standards. At the same time, with a few ideas borrowed from nonstandard analysis we see how the same reconstruction can be also be considered rigorous by modern robinsonian standards. This last approach, though, is completely in tune with Goldbach and Euler’s proof. We hope to convince the reader then how a few simple ideas from nonstandard analysis vindicate Euler’s work

El mètode de quadratures de Fermat

El Tractat de quadratures de Fermat (c. 1659) és conegut perquè conté la primera demostració de la qual hom té constància del còmput de l'àrea sota una paràbola superior, $\int x^{+m/n} dx$, o una hipèrbola superior, $\int x^{-m/n} dx$ —amb els límits d'integració adequats a cada cas. Però també conté una segona part que va ser gairebé ignorada pels seus contemporanis. Aquesta part és força obscura i difícil de llegir. En aquesta part, Fermat redueix la quadratura d'un gran nombre de corbes algebraiques a la quadratura de corbes conegudes: les paràboles i hipèrboles de la primera part. En altres casos, aconsegueix la reducció a la quadratura del cercle. En aquest article s'examina el mètode de quadratures de Fermat, que combina de manera molt intel·ligent dos procediments innovadors a l' època: el canvi de variables i un cas particular de la integració per parts. Amb el seu mètode, Fermat aconsegueix quadrar corbes tan conegudes com el foli de Descartes, la cissoide de Diocles o la bruixa d'Agnesi.Fermat’s Method of Quadrature. The Treatise on Quadrature of Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, $\int x^{+m/n} dx$ , or under a higher hyperbola, $\int x^{-m/n} dx$ — with the appropriate limits of integration in each case — has a second part which was mostly unnoticed by Fermat’s contemporaries. This second part of the Treatise is obscure and difficult to read. In it Fermat reduced the quadrature of a great number of algebraic curves to the quadrature of known curves: the higher parabolas and hyperbolas of the first part of the paper. Others, he reduced to the quadrature of the circle. We shall see how the clever use of two procedures, quite novel at the time: the change of variables and a particular case of the formula of integration by parts, provide Fermat with the necessary tools to square — quite easily — as well-known curves as the folium of Descartes, the cissoid of Diocles or the witch of Agnesi