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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]

The purpose of this study was to explore the possibility of interconnection between non-fractional representation and non-circular decimal representation in irrational teaching and learning of school mathematics and to examine the possibility of field application. First, through the analysis of previous studies, it was found that the core of understanding irrational concepts is the intuitive perception of convergence and the inference of acyclicity. And through mathematical analysis, we confirmed that finite decimal sequences and approximate fractional sequences should be learning contents. In this regard, we applied engineering tools in teaching and learning activity of irrational numbers and analyzed the process by which engineering tools become learners' mathematical tools and then learning occurs, that is, instrument genesis. As a result, in the activity of finding finite fractions of square roots, students gained a meaningful understanding by focusing on the meaning of square root symbols and square roots as objects from inputting functions and operations and using drag. And in the activity to solve the task 'Can a graph cross a grid point?', Students could infer the acyclicity of the decimal representation of irrational numbers by observing the approximate fraction as the slope. Thus, this study is expected to be a basic reference of teaching and learning of irrational numbers to construct a mental image of real numbers completeness axiom.

Copyright of International Journal of Educational Studies in Mathematics (IJESIM) is the property of International Journal of Educational Studies in Mathematics (IJESIM) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Calculations typically performed on a calculator or computer show the result as a decimal fraction if it is not an integer. It would be easier to interpret the result if a value could be expressed with integers and operations, such as the root subtraction operation. This article shows how this can be done with a developed algorithm in Microsoft Excel, which recognizes the most famous irrational numbers and displays them in text form together with the character of the operation sign. For example, “5√3/2” is given for 4.330127019. It is also useful to display irrational numbers with integers because only an infinite number of decimal places in a decimal fraction could show the exact value, and that is not possible. So, the developed algorithm can display a more interpretable and accurate form of the irrational number. In addition to the results that can be written as square roots, the algorithm is capable of displaying irrational numbers that can be expressed as the number Pi, using the π character. The Excel algorithm which was implemented in Visual Basic for Applications shows all rational numbers as the quotient of two integers that are relative primes. [ABSTRACT FROM AUTHOR]

We consider here the 'meta-problem' of creating 'nice' problems in elementary mathematics, 'nice' being defined as a problem in which the input data, as well as the answers, are rational numbers. The classical example of this is that of generating Pythagorean triples. Many examples of this kind arise when we study Euclidean geometry and the theory of equations. We consider a few problems of this genre. [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]

Copyright of Current Social & Economic Processes / Jelenkori Társadalmi és Gazdasági Folyamatok is the property of University of Szeged, Faculty of Engineering and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

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]