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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]

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]

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.

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]

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 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]

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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]