Search

This paper uses generalized continued fractions to approximate er, tan (r), tanh (r), r ∈ Q*, and shows that they are irrational numbers, and their irrationality measures are 2. [ABSTRACT FROM AUTHOR]

Let (t (m)) m ≥ 0 be Thue–Morse sequence and β > 2 be a Pisot or Salem number. In this paper, we prove that the real number ∑ m = 0 ∞ t (m 2) β m + 1 does not belong to Q (β). [ABSTRACT FROM AUTHOR]

Previous studies have developed pseudo-random number generators, where a pseudo-random number is not perfectly random but is practically useful. In this paper, we propose a new system for pseudo-random number generation. Recurrent neural networks with long short-term memory units are used to mimic the appearance of a given sequence of irrational number (e.g., pi), and these are intended to generate pseudo-random numbers in an iterative manner. We design algorithms to ensure that the output sequence contains no repetition or pattern. Through experimental results, we can observe the potential of the proposed system in terms of its randomness and stability. As this system can be used for parameter approximation in machine learning techniques, we believe that it will contribute to various industrial fields such as traffic management and frameworks for sensor networks. [ABSTRACT FROM AUTHOR]

The subject of this paper is the current state of art in theory of continued fractions, intermediate fractions and their relation to the best rational approximations of the first and second kind. The paper provides an overview of the some well known and even some new properties of continued fractions, and the various terms associated with them. In addition to intermediate fractions, paper considers the fine intermediate fractions and gave some statements to position these fractions in the continued fraction representation of numbers. [ABSTRACT FROM AUTHOR]

In this paper we study the b-ary expansions of the square roots of the function defined by the recurrence ƒb(n) = b ƒb(n−1) + n with initial value ƒ(0) = 0 taken at odd positive integers n, of which the special case b = 10 is often referred to as the ''schizophrenic'' or ''mock-rational'' numbers. Defined by Darling in 2004 and studied in more detail by Brown in 2009, these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases b ≥ 2 by formally defining the schizophrenic pattern present in the b-ary expansion of these numbers and the study of the lengths of the non-repeating and repeating digit sequences that appear within. [ABSTRACT FROM AUTHOR]

We locate gaps in the spectrum of a Hamiltonian on a periodic cuboidal (and generally hyperrectangular) lattice graph with δ couplings in the vertices. We formulate sufficient conditions under which the number of gaps is finite. As the main result, we find a connection between the arrangement of the gaps and the coefficients in a continued fraction associated with the ratio of edge lengths of the lattice. This knowledge enables a straightforward construction of a periodic quantum graph with any required number of spectral gaps and—to some degree—to control their positions; i.e. to partially solve the inverse spectral problem. [ABSTRACT FROM AUTHOR]

Abstract: Mathematical cognition research has largely emphasized concepts that can be directly perceived or grounded in visuospatial referents. These include concrete number systems like natural numbers, integers, and rational numbers. Here, we investigate how a more abstract number system, the irrationals denoted by radical expressions like 2, is understood across three tasks. Performance on a magnitude comparison task suggests that people interpret irrational numbers (specifically, the radicands of radical expressions) as natural numbers. Strategy self‐reports during a number line estimation task reveal that the spatial locations of irrationals are determined by referencing neighboring perfect squares. Finally, perfect squares facilitate the evaluation of arithmetic expressions. These converging results align with a constellation of related phenomena spanning tasks and number systems of varying complexity. Accordingly, we propose that the task‐specific recruitment of more concrete representations to make sense of more abstract concepts (referential processing) is an important mechanism for teaching and learning mathematics. [ABSTRACT FROM AUTHOR]

Let α and β be two irrational real numbers satisfying α ± β ∉ Z . We prove several inequalities between min k ∈ { 1 , … , n } ⁡ ‖ k α ‖ and min k ∈ S ⁡ ‖ k β ‖ , where S is a set of positive integers, e.g., S = { n } , S = { 1 , … , n − 1 } or S = { 1 , … , n } and ‖ x ‖ stands for the distance between x ∈ R and the nearest integer. We also give some constructions of α and β which show that the result of Kan and Moshchevitin (asserting that the difference between min k ∈ { 1 , … , n } ⁡ ‖ k α ‖ and min k ∈ { 1 , … , n } ⁡ ‖ k β ‖ changes its sign infinitely often) and its variations are best possible. Some of the results are given in terms of the sequence d ( n ) = d α , β ( n ) defined as the difference between reciprocals of these two quantities. In particular, we prove that the sequence d ( n ) is unbounded for any irrational α , β satisfying α ± β ∉ Z . [ABSTRACT FROM AUTHOR]

The aim of this study is to investigate students' difficulties about the relation between irrational number set, rational number set and real number set. For this purpose, 'Irrational Number Concept Test' which was composed of open-ended questions has been developed. The Data collection instrument was applied to 58 students in grade 8 and 50 students in grade 9. Semi-structured interviews with ten students who were selected from different levels with the maximum diversity sampling were conducted. In each grade, it was found that students had difficulties in understanding the relationship between real number set and other number sets. There have been some thoughts such as 'all of the irrational numbers are not real numbers' and 'a number can be both rational and irrational'. It is found that the rate of students that have these wrong thoughts in 8th grades is more than those in 9th grades. [ABSTRACT FROM AUTHOR]

The Güting algorithm for constructing multidimensional continued fractions is considered. It is proved that, in the case of dimension 2, this algorithm can be used to find the coefficients of the linear dependence of numbers; a criterion is given for verifying that the partial quotients furnished by the algorithmare, indeed, elements of the continued fraction for the expanded (generally irrational) numbers. [ABSTRACT FROM AUTHOR]

We establish asymptotically sharp estimates for the sums of the inverses (and more general sums) of the fractional parts { iθ} of irrational numbers θ, depending on the arithmetical characteristics of the numbers θ. [ABSTRACT FROM AUTHOR]

In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet's proof (1842) of existence of the solution [12], [1]. [ABSTRACT FROM AUTHOR]

We propose another method of deriving the Marcovecchio estimate for the irrationality measure of the number ln 2 following, for the most part, the method of proof of the irrationality of the number ζ(3) proposed by the author in 1996. The proof uses single complex integrals, the so-called Meyer G-functions, and is much simpler than that of Marcovecchio. [ABSTRACT FROM AUTHOR]