Hermite-Padé Rational Approximation to Irrational Numbers

I.4 - Rational Approximation

For a continuous random variable in real number field, there must be a distribution and also a probability density function of this random variable. If there is a known function with this random variable as independent variable, its image is a smooth or piecewise smooth line, there must be at least one function that takes this random variable as its independent variable, these functions are bounded on the image of the first function. Any one of these functions conduct line integral operation to the line segment or arc length of the certain image of the first known function is the cumulative probability of this continuous random variable interval corresponding to the section of the image for line integral operation. A general designation for these functions are linear probability density function of continuous random variables. Conduct line integral operation to the linear probability density function and conduct integral operation to the probability density function have same results of the cumulative probability of continuous random variable. By the way, Line integration including curve integration. According to the uniqueness of the probability, the existence and the number of linear probability density function can be proved and calculated.

This article reports on a study of high-school and of technologist students (prospective engineers and economists) understanding of real numbers. Our study was based on written response to a properly designed questionnaire and on interviews taken from students. The quantitative results of our experiment showed an almost complete failure of the technologist students to deal with processes connected to geometric constructions of incommensurable magnitudes. However, this didn’t prevent them in answering satisfactorily the other questions. The superiority of their correct answers with respect to those of high-school students, although the majority of them correspond to mediocre graduates of the secondary education, was evident in most cases. This is a strong indication that the age and the width of mathematical knowledge of the individual play an important role for the better understanding of the real numbers. The results of our experiment suggest also that the ability to transfer in comfort among several representations of real numbers helps students in obtaining a better understanding of them. A theoretical explanation about this is obtained through the adoption of the conceptual framework of dimensions of knowledge, introduced by Tirosh et al. (1998) for studying the comprehension of rational numbers. Following in part the idea of generic decomposition of the APOS analysis (Weller et al. 2009) we suggest a possible order for development of understanding the real numbers by students when teaching them at school. Some questions open to further research are also mentioned at the end of the paper. Keywords: Real, rational, irrational, algebraic and transcendental numbers, fractions, decimals, representations of real numbers.

Any real number is made accessible through its rational approximations, for example, cutting off the decimals starting with \((n+1)\)th one

1 - THE FOUNDATIONS OF ALGEBRA

The Thomist 64 (2000): 101-25 CANTOR'S TRANSFINITE NUMBERS AND TRADITIONAL OBJECTIONS TO ACTUAL INFINITY JEANW. RIOUX Benedictine College Atchison, Kansas Georg Cantor was one of the most prominent mathematicians of the late nineteenth and early twentieth centuries. His development of a theory of transfinite numbers resurrected philosophical questions about infinity and led to a division of mathematics into schools of thought such as formalism and intuitionism. Cantor's published attempts to justify his mathematical theories were directed not only toward the mathematicians of his day but also toward philosophers, both ancient and contemporary. His efforts on the latter front were rooted in his desire to deal with objections to the very idea of the actual infinite in quantity, and he attached great importance to those objections that came from traditional philosophy.1 I intend to review the basic philosophical issues: Cantor's claims to a workable mathematics of real and actually infinite quantities, his response to Aristotelian objections to those claims, and my reflections on whether Cantor finally settled the matter, as he had hoped. 1 I shall focus upon Cantor's philosophical defense of transfinite numbers against this fundamental Aristotelian position as it is found in part 5 of his "Ueber unendliche, lineare Punktmannigfaltigkeiten," Mathematische Annalen 21 :545-86 (published separately as Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ei11 mathematisch-philosophischer Versuch in der Lehre des Unend.lichen [Leipzig: B. G. Teubner, 1883], hereafter Grumilage11). Cantor saw Aristotle as the source of the Scholastic position on infinity, and in the Gnmdlagen he addressed the basic error involved in all 'finitist' reasoning, as he saw it. A few years after writing the Grundlagen, Cantor published papers dealing with specific Scholastic arguments against the actual infinite in detail. For example, see Robin Small, "Cantor and the Scholastics," American Catholic Philosophical Quarterly 66 (1992): 407-28, where the question concerns the eternity of the world. 101 102 JEANW. RIOUX Cantor was well aware of the prevailing mathematical and philosophical climates of his day.2 As he drew nearer to a completed theory of transfinite numbers, he became increasingly interested in justifying it, not only as a consistent and practical3 exercise of mathematical thought, but also as one having a basis in the real world,4 as somehow providing as real a view of the natural world as did the relatively unproblematic theory of finite integers. While other mathematicians seemed unconcerned with such metaphysical questions, Cantor devoted much of his time and effort to addressing them, especially later in his life.5 Ironically, Cantor'sfascination with these metaphysical aspects of the theory, so foreign to his contemporaries in mathematics, turned out to be somewhat prophetic, since the dubious ontological character of transfinite numbers was to figure prominently in later developments in mathematics.6 I will begin by giving an overview of Cantor's transfinite number theory, focusing in particular upon his claim that transfinite numbers possess an objective, or real, infinity which is actual, not merely potential.7 I. SUMMARY OF CANTOR'S TRANSFINITE NUMBER THEORY Cantor discovered thatmerely 'potential' infinity, the sortwith which mathematicians were comfortable to that day, was not the 2 See Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton, N.J.: Princeton University Press, 1979), 120-22. 3 As he notes in Grundlagen, § 8, the success in application of a mathematical concept is a major factor in whether it is accepted as legitimate or abandoned. 4 Cantor meant atleastthree thingsby 'real number': 'real' as distinguished from complex, rational, or irrational numbers; 'real' as existing in the understanding, as consistent and definite ideas in the mind; and 'real' as existing in the extramental world. Cantor held that all numbers that are real in the second sense are images of those that are real in the third sense. For more on the distinction between the last two types of reality, see Grundlagen, § 8. See also Dauben, Georg Cantor, 125-26. 5 Though many of the mathematicians of his day were able to look past Cantor's concern with the philosophical and theological implications of his transfinite theory (and in this way many came to defend the theory), to Cantor they were an inseparable part of his...

The problem of radio channel assignments with multiple levels of interference can be modeled using graph theory. Given a graph G, possibly infinite, and real numbers k/sub 1/, k/sub 2/, ..., k/sub p/ /spl ges/ 0, a L(k/sub 1/, k/sub 2/, ..., k/sub p/)-labeling of G assigns real numbers f(x) /spl ges/ 0 to the vertices x, such that the labels of vertices u and v differ by at least k/sub i/ if u and v are at distance i apart. We denote by /spl lambda/(G; k/sub 1/, k/sub 2/, ..., k/sub p/) the infimum span over such labelings f. We survey this new theory of real number labelings. When p - 2 it is enough to determine /spl lambda/(G; k, 1) for reals k /spl ges/ 0; which will be a piecewise linear function. We present the function for the square lattice (grid) and for the hexagonal lattice. For the triangular lattice, we have also solved it except for the range 1/2 /spl les/ k /spl les/ 4/5.

Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs. The extracted programs construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching non-wellfounded trees describing when the algorithm writes and reads digits. We discuss several examples including the extraction of programs for polynomials up to degree two and the definite integral of continuous maps.

A uniform approach to computing with infinite objects like real numbers, tuples of these, compacts sets and uniformly continuous maps is presented. In the work of Berger, it was shown how to extract certified algorithms working with the signed digit representation from constructive proofs. Berger and the present author generalised this approach to complete metric spaces and showed how to deal with compact sets. Here, we unify this work and lay the foundations for doing a similar thing for the much more comprehensive class of compact Hausdorff spaces occurring in applications. The approach is of the same computational power as Weihrauch’s Type-Two Theory of Effectivity.

On the computational complexity of the languages of general symbolic dynamical systems and beta-shifts

1 - Variables and Functions

Let ξ \xi be a real irrational number. We are interested in sequences of linear forms in 1 and ξ \xi , with integer coefficients, which tend to 0. Does such a sequence exist such that the linear forms are small (with given rate of decrease) and the coefficients have some given rate of growth? If these rates are essentially geometric, a necessary condition for such a sequence to exist is that the linear forms are not too small, a condition which can be expressed precisely using the irrationality exponent of ξ \xi . We prove that this condition is actually sufficient, even for arbitrary rates of growth and decrease. We also make some remarks and ask some questions about multivariate generalizations connected to Fischler-Zudilin’s new proof of Nesterenko’s linear independence criterion.

In the general case a real irrational number cannot be approximated by infinitely many rationalsp/q involving error terms less than q-2 when the denominatorsq are taken from a given thin set of positive integers. The distribution of irrationals which are situated in close neighborhoods of infinitely many fractionsp/q, whereq is restricted to the elements of a thin set, depends on the asymptotic behaviour of theq’s and on their arithmetic properties.

Throughout the present Chapter, we are essentially concerned with the following problem: for which functions Ψ : ℝ ≥1 → : ℝ ≥0 is it true that, for a given real number ξ, or for all real numbers ξ in a given class, the equation |ξ – p/q | q ) has infinitely many solutions in rational numbers p/q ? We begin by stating the results on rational approximation obtained by Dirichlet and Liouville in the middle of the nineteenth century. In Section 1.2, we define the continued fraction algorithm and recall the main properties of continued fractions expansions. These are used in Section 1.3 to give a full proof of a metric theorem of Khintchine. The next two Sections are devoted to the Duffin–Schaeffer Conjecture and to some complementary results on continued fractions. Dirichlet and Liouville Every real number ξ can be expressed in infinitely many ways as the limit of a sequence of rational numbers. Furthermore, for any positive integer b , there exists an integer a with |ξ – a/b | ≤ 1/(2 b ), and one may hope that there are infinitely many integers b for which |ξ – a/b | is in fact much smaller than 1/(2 b ). For instance, this is true when ξ is irrational, as follows from the theory of continued fractions.

This paper illustrates a TI N-Spire .tns file created by the author for generating continued fraction representations of real numbers and doing arithmetic with them. The continued fraction representation provides an alternative to the decimal representation. The .tns file can be used as tool for studying continued fractions and their properties as well as introducing the ideas of a best rational approximation to a real number and exploring some elementary ideas in error analysis. While the paper is a review of some of the classical works on continued factions, it does not present the material in a theorem/proof style. It motivates the results by citing examples and noting some interesting properties that will need to be verified. Notations are given to the references where exact and complete proofs of these facts can be found. The type of investigations done in this paper are motivational and the style of presentation is designed to pique students and teachers curiosity about the subject. Another purpose of this paper is to show that by looking at alternative representations of something that seems to be well known, such as the real numbers, can lead to exciting new and useful outcomes and applications.