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Pell's equation, continued fractions and Diophantine approximations of irrational numbers
Kodýtek, Jakub ; Beran, Filip (advisor) ; Jančařík, Antonín (referee)
This bachelor's thesis deals with Pell's equation, while clearly presenting structured information from studied domestic and foreign books, articles, and other sources. The goal of this thesis is to create study material primarily for university students but also for inquisitive high school students, and thus explain as intuitively as possible what Pell's equation is, how to find its solutions, and how it is related, for example, to continued fractions, approximations of irrational numbers, and invertible elements in Z[√n ]. The main motivation for solving Pell's equation throughout the work is specifically that its solutions give best approximations of irrational square roots. Pell's equation is presented in a brief historical context. Further, it is proved that there is a non-trivial integer solution for every Pell equation, and the theory of continued fractions is used to find it. To make the creation of continued fractions easier, the so-called Tenner's algorithm is introduced. Specifically, the search for a solution to Pell's equation is derived using convergents and the periodicity of continued fractions of irrational roots. Subsequently, the structure of the solution is described: it is proved that there is a so-called minimal solution that generates all positive solutions, and a set of...
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Angles, areas, volumes: dot product and determinant
Ondič, Milan ; Beran, Filip (advisor) ; Zamboj, Michal (referee)
This bachelor thesis deals with the introduction of scalar product and determinant, which are important tools of analytic geometry. The purpose of the thesis is to provide a parallel interpretation of these two key concepts of advanced algebra - the dot product and the determinant - primarily from a geometric, not an algebraic, point of view. The aim of the thesis is to show how both representations can be derived just by solving geometric problems in two-dimensional space and then how to transfer them to three-dimensional space. The first part of the work is devoted to finding the angle between two vectors in the plane and to calculating the area of a triangle. Both of problems are solved in several ways and then the scalar product and determinant are derived. The second part of the work is devoted to three-dimensional space, in particular the angle between two vectors, lines and planes and the volume of a tetrahedron and parallelogram. This is then supplemented by the introduction of some notions of linear algebra, an investigation of the algebraic properties of the dot product and determinant, and a generalization of the notions to the n-dimensional space. The last part of the thesis is devoted to the analysis of selected czech high school mathematics textbooks in terms of the occurrence and...
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Transfer between mathematics and music in teaching mathematics
Kovačová, Petra ; Zamboj, Michal (advisor) ; Beran, Filip (referee)
Transfer between Mathematics and Music in Teaching Mathematics ABSTRACT The thesis targets the concept of learning transfer between mathematics and music and its usage in teaching mathematics in primary school. The aim of this thesis is to create interdisciplinary tasks for initiation of the learning transfer between mathematics and music, through an educational experiment identify whether the transfer occurs while solving them and determine its conditions. Another aim is to define the role of a teacher in this process. It focuses on the transfer between the area of grids, symmetry and melodies created by two tones. Moreover, it aims at fractions and creating tones by shortening a string. Educational areas Mathematics and its application and Music are described according to the Framework Education Programme for Elementary Education alongside with the possibilities of their connecting through interdisciplinary relations, cross-sectional and integrated learning. The concept of learning transfer and its kinds is elaborated in the final chapter of the theoretical part. Mathematical and musical terms used in tasks of the second part of the thesis are explained there too. Within the practical part there were tasks for initiating and testing transfer created. They were used in the educational experiment in the 9th...
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Linear Diophantine equations and congruences
Kaňáková, Natálie ; Beran, Filip (advisor) ; Jančařík, Antonín (referee)
This bachelor's thesis summarizes and systematizes knowledge about congruences and linear Diophantine equations. This work is divided into two parts. The first part is dedicated to congruences. At first, it shows where we can find congruences in real life, congruence as a relation, its properties, and applications in calculating the last ciphers of large numbers, proofs of divisibility rules, or calculating the date of Easter. Afterward, we look into congruences containing unknowns - linear congruence equations. It looks into methods of solving linear congruences and illustrates them in exercises. The last topic of the first part is oriented on systems of linear congruences and the Chinese remainder theorem, both for non-coprime and coprime moduli, the algebraic version, applications in various types of problems, and modular representation of numbers. The second part of this thesis is dedicated to linear Diophantine equations - equations with integer solutions. It shows various methods of solving linear Diophantine equations with two, three, or more unknowns - the extended Euclidean algorithm, reduction method, substitution method, and others. This part also describes the relationship between linear congruences and linear Diophantine equations and the use of this relationship in solving both linear...
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