Monotonically Computable Real Numbers

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.

On countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show that with respect to computable equivalence there is one and only one equivalence class of representations of the real numbers which make the basic operations and the infinitary normed limit operator computable. This characterizes the real numbers in terms of the theory of effective algebras or computable structures, and is reflected by observations made in real number computer arithmetic. Demanding computability of the normed limit operator turns out to be essential: the basic operations without the normed limit operator can be made computable by more than one class of representations. We also give further evidence for the well‐known non‐appropriateness of the representation to some basebby proving that strictly less functions are computable with respect to these representations than with respect to a standard representation of the real numbers. Furthermore we consider basic constructions of representations and the countable substructure consisting of the computable elements of a represented, possibly uncountable structure. For countable structures we compare effectivity with respect to a numbering and effectivity with respect to a representation. Special attention is paid to the countable structure of the computable real numbers.

This paper presents a metod of inductive inference of realvalued functions from given pairs of observed data of (x,h(x)), where h is a target function to be inferred. Eac of suc observed data inevitably involves some ranges of errors, and hence it is usually represented by a pair of rational numbers sow te approximate value and te error bound, respectively. On the other hand, a real number called a recursive real number can be represented by a pair of two of rational numbers, which converges to the real number and converges to zero, respectively. These sequeces show an approximate value of te real number and an error bound at each point. Such a real number can also be represented by a sequence of closed intervals wit rational end points converges to a singleton interval with the real number as both end points.In this paper, we propose a notion of recursive real-valued functions that can enjoy the merits of the both representations of the recursive real numbers. Then we present an algorithm which approximately infers real-valued functions from numerical data with some error bounds, and show that there exists a rich set of real-valued functions which is approximately inferable in the limit from such numerical data. We also discuss the precision of the guesses from the machine when sufficient data have not yet given.

Real numbers are the basic objects in analysis. For most non-mathematicians a real number is an infinite decimal fraction, for example π = 3●14159 .... Mathematicians prefer to define the real numbers axiomatically as follows: (ℝ, +,·,0,1, <) is, up to isomorphism, the only Archimedean ordered field satisfying the axiom of continuity [Die60]. The set of real numbers can also be constructed in various ways, for example by means of Dedekind cuts or by completion of the (metric space of) rational numbers. We will neglect all foundational problems and assume that the real numbers form a well-defined set ℝ with all the properties which are proved in analysis. We will denote the real line topology, that is, the set of all open subsets of ℝ, by τℝ.KeywordsReal NumberRational NumberComputable FunctionComputable SequenceAdmissible RepresentationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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...

In computable analysis, sequences of rational numbers which effectively converge to a real number x are used as the (rho-) names of x. A real number x is computable if it has a computable name, and a real function f is computable if there is a Turing machine M which computes f in the sense that, M accepts any rho-name of x as input and outputs a rho-name of f(x) for any x in the domain of f. By weakening the effectiveness requirement of the convergence and classifying the converging speeds of rational sequences, several interesting classes of real numbers of weak computability have been introduced in literature, e.g., in addition to the class of computable real numbers (EC), we have the classes of semi-computable (SC), weakly computable (WC), divergence bounded computable (DBC) and computably approximable real numbers (CA). In this paper, we are interested in the weak computability of continuous real functions and try to introduce an analogous classification of weakly computable real functions. We present definitions of these functions by Turing machines as well as by sequences of rational polygons and prove these two definitions are not equivalent. Furthermore, we explore the properties of these functions, and among others, show their closure properties under arithmetic operations and composition.

A real valued function defined on a subset E of R, the set of real numbers, is lacunary statistically upward continuous if it preserves lacunary statistically upward half quasi-Cauchy sequences where a sequence (xn) of points in R is called lacunary statistically upward half quasi-Cauchy if limr→∞1hr|{k∈Ir:xk−xk+1≥ε}|=0 for every ε > 0, and θ = (kr) is an increasing sequence θ = (kr) of non-negative integers such that k0 = 1 and hr : kr−kr−1 → ∞. We investigate lacunary statistically upward continuity, and prove interesting theorems. It turns out that any lacunary statistically upward continuous function on a below bounded subset of R is uniformly continuous.

1 - Variables and Functions

Exact real number computation is a fast growing field with applications varying from debugging to specification of numerical to program. We present a specification of the real numbers represented as infinite lists of signed digits in Z. The expressive power and closeness to usual set theoretical mathematical notation gives us a clean and readable specification which is further directly implementable. A comparison with other formal methods is given together with a partial proof that the object being specified is actually the real numbers.

Reasoning about real number expressions in a proof assistant is challenging. Several problems in theorem proving can be solved by using exact real number computation. I have implemented a library for reasoning and computing with complete metric spaces in the Coq proof assistant and used this library to build a constructive real number implementation including elementary real number functions and proofs of correctness. Using this library, I have created a tactic that automatically proves strict inequalities over closed elementary real number expressions by computation.KeywordsReal NumberRational NumberTheorem ProveRegular FunctionStrict InequalityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

There are two well known constructions of the real numbers from the rationals—namely the Dedekind cuts method in which a real number is defined as a class of rationals, and the Cantor–Cauchy completion method in which a real number is defined as an equivalence class of Cauchy sequences of rational numbers. In this article we give a new construction of the reals from the rationals in which a real number is defined as an equivalence class of sets of natural numbers.

1. The Natural Numbers. The Basics. The Fundamental Theorem of Arithmetic. Searching for Primes. Number Fascinations. 2. The Integers. Diophantine Equations. Congruence Arithmetic. Pell and Pythagoras. Factoring Large Numbers. 3. The Rational Numbers. Rational Numbers as Decimals. Decimals as Rational Numbers. Continued Fractions. Solving Equations on the Rational Plane 4. The Real Numbers. Algebraic Representations. Geometric Representations. Analytic Representations. Searching for Transcendental Numbers. 5. Mathematical Projects. Rings of Factors. Sums of Consecutive Numbers. Measuring Abundance. Inside the Fibonacci Numbers. Pictures at an Iteration. Eenie Meenie Miney Mo. Factoring with the Pollard ...r Method. Charting the Integral Universe. Triangles on the Integral Lattice. The Gaussian Integers. Writing Fractions the Egyptian Way. Building Polygons with Dots. The Decimal Universe of Fractions, I. The Decimal Universe of Fractions, II. The Making of a Star. Making Your Own Real Numbers. Building 1 the Egyptian Way. Continued Fraction Expansions of x1/2 N. A Special Kind of Triangle. Polygon Numbers. Continued Fraction Expansions.

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.

I.4 - Rational Approximation