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Using the basic properties of the base-b representation of rational numbers, we will give an elementary proof of Gauss's lemma: Every real root of a monic polynomial with integer coefficients is either an integer or irrational. The paper offers a new perspective in understanding the meaning of ‘irrational numbers’ from a deeper perspective, and a good occasion to improve the student's skills with point of view on number systems. [ABSTRACT FROM AUTHOR]

In this paper, we will show a characterization for pairs of commensurable diagonals of regular n-gons. This result gives a positive answer to a recent question. [ABSTRACT FROM AUTHOR]

In this paper, we propose an elementary proof of Niven's Theorem in which the tangent function will have a primary role. [ABSTRACT FROM AUTHOR]

In this article, we will give a geometric interpretation of certain finite arithmetic progressions. For this purpose, we will introduce the concept of the n-regular partition { P n } of a quadrilateral. [ABSTRACT FROM AUTHOR]

When dealing with elementary algorithms in the classroom, we almost always limit ourselves to considering the purely procedural aspects. In so doing, we run the risk of providing students with a distorted and reductive vision of mathematics, which could foster an attitude of disaffection towards the discipline among students. This work aims to show that we can overturn this situation by changing our approach and thus make the study of some elementary algorithms an opportunity to increase interest in mathematics and encourage the development of skills and competences in students. Our idea is to use the inquiry-based learning methodology in the study of algorithms in order to re-evaluate their role in school mathematics classes. Below we will describe an experiment on the teaching of elementary algorithms conducted between 2016 and 2018 as part of the 'Math High School' project of the University of Salerno and based on thorough historical and scientific research. The project involved a group of 120 first-year high school students and the skills that the students were able to develop suggest that the experiment should be extended to the use of IBL methodology in handling other algorithms. The inquiry-based activities proposed can be applied in the classroom by teachers and can offer practical strategies and tools for teaching algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we give possible suggestions for a classroom lesson about an application of probability using basic mathematical notions. We will approach to some combinatoric results without using ‘induction’, ‘polynomial identities’ nor ‘generating functions’, and will give a proof of the ‘Vandermonde Identity’ using elementary notions of probability. [ABSTRACT FROM AUTHOR]

In line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of theElementsof Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann inAusdehnungslehrein 1844. Assuming as axioms, the cognitive contents of the theorems of Pappus and Desargues, through their configurations, in an Euclidean plane a natural field structure can be identified that reveals the purely geometric nature of complex numbers. Reasoning based on figures is becoming a growing interdisciplinary field in logic, philosophy and cognitive sciences, and is also of considerable interest in the field of education, moreover, recently, it has been emphasized that the mutual assistance that geometry and complex numbers give is poorly pointed out in teaching and that a unitary vision of geometrical aspects and calculation can be clarifying. [ABSTRACT FROM PUBLISHER]

A well-known result of Feinberg and Shannon states that the tribonacci sequence can be detected by the so-called Pascal's pyramid. Here we will show that any tribonacci-like sequence can be obtained by the diagonals of the Feinberg's triangle associated to a suitable generalized Pascal's pyramid. The results also extend similar properties of Fibonacci-like sequences. [ABSTRACT FROM AUTHOR]