Made available in DSpace on 2016-04-27T16:58:52Z (GMT). No. of bitstreams: 1 Aloisio Daniel Vendeniatti.pdf: 1272014 bytes, checksum: 1262d89ac2880970c73eca396d22ca43 (MD5) Previous issue date: 2009-04-24 / Secretaria da Educação do Estado de São Paulo / The goal of this essay is to show aspects of genesis of number π, inherent to the question of squaring the circle, which consists in constructing a square which has the same area as a given circle. This problem does not refer to a practical application of mathematics, but to the theoretic question that involves the distinction between a valid approach and thinking accuracy. The first attempt to squaring the circle dates back in the fifth century before Christ. After that, it was established that this construction should be carried through using a finite number of times, also the non-graduated ruler and the drawing compass itself. In the constructions with ruler and drawing compass we are referring to the first three postulates of Euclides Elements: 1) It´s possible to join two points by a straight line, 2) to expand a straight line until the necessary point, and 3) to draw a circumference around any point and with any radius. These postulates are the base of these constructions, sometimes called euclidean´s constructions. A real number α is constructible, if feasible building a segment of legth α with ruler and drawing compass, since a segment is taken as a unity. We show the idea of translating the geometrical problem of constructions made with ruler and drawing compass to the algebraic language and this allowed us to solve the problem of squaring the circle. We exposed that all constructible numbers are algebraic, over the rational numbers, establishing the impossibility of squaring the circle, with Lindemann´s demonstration, in 1882, of the number π transcendence. This problem has been fascinating people for more than twenty centuries. We tried to supply all mathematical tools needed for this demonstration. Demonstrations play a fundamental role in the development of this essay, which purpose is not only to contribute to the math teacher formation, but also to detail the resolution of the problem of squaring the circle / O objetivo deste trabalho é apresentar aspectos da gênese do número π, inerentes à questão da quadratura do círculo, a qual consiste em construir um quadrado de área igual à área de um círculo de raio r dado. Esse problema não diz respeito a uma aplicação prática da matemática, mas sim a uma questão teórica que envolve uma distinção entre uma boa aproximação e a exatidão do pensamento. O registro da primeira tentativa de se quadrar o círculo remonta a Anaxágoras, no século V a.C. Posteriormente, ficou estabelecido que essa construção deveria ser realizada utilizando-se, um número finito de vezes, a régua não graduada e o compasso. Nas construções com régua e compasso, estamos nos referindo aos três primeiros postulados dos Elementos de Euclides: 1) é possível unir dois pontos por uma reta, 2) prolongar uma linha reta até onde seja necessário e 3) traçar uma circunferência em torno de qualquer ponto e com qualquer raio. Esses postulados são a base dessas construções, muitas vezes chamadas de construções euclidianas. Um número real α é construtível, se for possível "construir com régua e compasso um segmento de comprimento igual a α, a partir de um segmento tomado como unidade". Apresentamos a idéia de traduzir o problema geométrico das construções com régua e compasso para a linguagem algébrica, e isso permitiu solucionar o problema da quadratura do círculo. Expomos que todo número construtível é algébrico sobre os racionais, estabelecendo a impossibilidade de quadrar o círculo com a demonstração de Lindemann, em 1882, da transcendência do número π. Vemos que esse problema fascinou estudiosos por mais de 20 séculos. Procuramos fornecer todas as ferramentas matemáticas necessárias para essa demonstração. As demonstrações desempenham um papel fundamental no desenvolvimento deste trabalho, que tem por finalidade não só contribuir para a formação do professor de matemática, mas também detalhar a resolução do problema da quadratura do círculo

In its theoretical part, this thesis sums up the basic knowledge concerning permutations. Besides the representation of permutations and determination of their fundamental characteristics, the theoretical part is, first of all, aimed at results concerning the decomposition of permutations into disjoint cycles and at finding the number of permutations with a certain characteristic. We introduce the fundamental bijection that is useful for solving many problems concerning the permutations. Further on, we focus on the number of permutations without a fixed point, Eulerian numbers expressing the number of permutations with a given number of descents, and the number of permutations with a given number of excedances, Stirling numbers of the first kind expressing the number of permutations with a given number of cycles, and Catalan numbers representing the number of permutations avoiding a chosen pattern of length three. Attention is also paid to the Gilbreath permutations and their characteristics. The practical part consists of 14 solved problems. The solutions rely on the results presented in the theoretical part, and there are deduced some further interesting results concerning random permutations.

Neste trabalho é apresentado um breve levantamento da história dos números primos e de que maneira o assunto acerca desses números aparecem no novo cenário trazido pela BNCC. Provamos o Teorema Fundamental da Aritmética e apresentamos duas ferramentas importantes de cálculo, que são as Congruências e o Pequeno Teorema de Fermat. Apresentamos ainda uma proposta didática e um material diferenciado para ser utilizado em sala de aula. / In the present work is presented a brief data collection about the history of prime numbers and how this subject is shown in the new scenario brought by BNCC (Common Curricular National Base) . It was proved the Fundamental Arithmetic Theorem and it was presented two important ways to calculate that are the Congruence and the Fermet Theorem. It is given a teaching method and a differentiated material to be used in class.

We study bounds on the distances of roots of integer polynomials and applications of such results. The separation of complex roots for reducible monic integer polynomials of fourth degree is thoroughly explained. Lemmas on roots of polynomials in the p-adic setting are proved. Explicit families of polynomials of general degree as well as families in some classes of quadratic and cubic polynomials with very good separation of roots in the same setting are exhibited. The second part of the thesis is concerned with results on p-adic versions of Mahler's and Koksma's functions wn and w*n and the related classifications of transcendental numbers in Cp. The main result is a construction of numbers such that the two functions wn and w*n differ on them for every n and later on expanding the interval of possible values for wn-w*n. The inequalities linking values of Koksma's functions for algebraically dependent numbers are proved.

After reviewing some properties of the two dimensional hyperbolic toral automorphism called Arnold's discrete cat map, including its generalizations with matrices having positive unit determinant, this thesis contains a definition of a novel cat map where the elements of the matrix are found in the sequence of Pell numbers. This mapping is therefore denoted as Pell's cat map. The main result of this thesis is a theorem determining the upper bound for the minimal period of Pell's cat map. From numerical results four conjectures regarding properties of Pell's cat map are also stated. A brief exposition of some applications of Arnold's discrete cat map is found in the last part of the thesis.

Submitted by Raniere Barreto (raniere.barros@ufvjm.edu.br) on 2018-04-12T16:51:05Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) raphael_neves_matos.pdf: 4286914 bytes, checksum: 4faddab9001b8b035017adfd9a2d6d75 (MD5) / Approved for entry into archive by Rodrigo Martins Cruz (rodrigo.cruz@ufvjm.edu.br) on 2018-04-20T14:12:28Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) raphael_neves_matos.pdf: 4286914 bytes, checksum: 4faddab9001b8b035017adfd9a2d6d75 (MD5) / Made available in DSpace on 2018-04-20T14:12:28Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) raphael_neves_matos.pdf: 4286914 bytes, checksum: 4faddab9001b8b035017adfd9a2d6d75 (MD5) Previous issue date: 2017 / Este trabalho teve como objetivo principal apresentar uma contribui??o para o ensino aprendizagem dos n?meros racionais, destacando principalmente a rela??o entre d?zimas peri?dicas e progress?es geom?tricas. A metodologia utilizada permitiu a an?lise da abordagem e sequ?ncia did?tica dos t?picos D?zima peri?dica e Progress?o Geom?trica Infinita, contemplada nos livros did?ticos aprovados pelo Programa Nacional do Livro Did?tico. Nesta abordagem as fra??es e os n?meros decimais, especialmente os decimais infinitos e peri?dicos, e por consequ?ncia o c?lculo de sua fra??o geratriz, foram objetos de estudo centrais e instigadores dessa pesquisa. Realizou-se um estudo mais detalhado sobre a representa??o decimal dos n?meros racionais e analisando a compreens?o destes n?meros em n?vel fundamental e m?dio. Foi ainda proposto uma abordagem das maneiras mais usuais do c?lculo da fra??o geratriz, bem como, explorado a rela??o entre os decimais infinitos e peri?dicos e as progress?es geom?tricas. Durante o desenvolvimento deste trabalho, foi poss?vel perceber que h? mais de uma abordagem did?tica dos t?picos de ensino inerentes ao tema central analisado. O reconhecimento de que a parte decimal das d?zimas peri?dicas pode ser expressa como uma soma infinita de parcelas que, a partir de certo ponto, descreve uma progress?o geom?trica infinita de raz?o compreendida entre zero e um, ? um ponto chave na proposta de interven??o apresentada para a sala de aula. Diante desse quadro, foi verificado a ordem atualmente seguida pelos professores do 1? Ano do Ensino M?dio, o que permitiu constatar que os conte?dos D?zimas Peri?dicas e Progress?es Geom?tricas Infinitas s?o tratados sem liga??o significativa e, diante disso, foi proposta uma altera??o na ordem de abordagem desses conte?dos no Ensino M?dio. Ao final foram propostas algumas sugest?es de atividades resolvidas e outras para serem desenvolvidas em sala de aula. / Disserta??o (Mestrado Profissional) ? Programa de P?s-Gradua??o Matem?tica, Universidade Federal dos Vales do Jequitinhonha e Mucuri, 2017. / The aim of this work was to present a contribution to the teaching of rational numbers, emphasizing mainly the relation between periodic tithe and geometric progression. The methodology used allowed the analysis of the approach and didactic sequence of the topics Periodic Dizima and Infinite Geometric Progression, contemplated in textbooks approved by the National Textbook Program. In this approach fractions and decimal numbers, especially the infinite and periodic decimals, and consequently the calculation of their generative fraction, were central objects and instigators of this research. A more detailed study on the decimal representation of rational numbers was carried out and the understanding of these numbers at the fundamental and medium levels was analyzed. It was also proposed an approach of the most usual ways of calculating the generative fraction, as well as exploring the relationship between infinite and periodic decimals and geometric progressions. During the development of this work, it was possible to perceive that there is more of a didactic approach of the teaching topics inherent to the central theme analyzed. The recognition that the decimal part of the periodic tithe can be expressed as an infinite sum of plots which, from a certain point, describes an infinite geometric progression of ratio between zero and one, is a key point in the proposal of intervention presented for the classroom. In view of this situation, we verified the order currently being followed by teachers of the 1? Year of High School, which allowed to verify that the Periodic Dictionaries and Infinite Geometric Progressions are treated without significant connection and, accordingly, a change was proposed in order to approach these contents in High School. At the end, some suggestions for solved activities and others to be developed in the classroom were proposed.

Nous étudions les bornes sur les distances des racines des polynômes entiers et les applications de ces résultats. La séparation des racines complexes pour les polynômes réductibles normalisés de quatrième degré à coefficients entiers est examinée plus à fond. Différents lemmes sur les racines des polynômes en nombres p-adiques sont prouvés. Sont fournies les familles explicites de polynômes de degré général, ainsi que les familles dans certaines classes de polynômes quadratiques et cubiques avec une très bon separation des racins dans le cadre p-adique. Le reste de la thèse est dédié aux résultats liés aux versions p-adiques des fonctions de Mahler et de Koksma wn et w*n , ainsi qu'aux classifications correspondantes des nombres transcendants dans Cp. Le résultat principal est une construction des nombres pour lesquelles les deux fonctions wn et w*n sont différentes pour tous les n et puis l'intervalle de valeurs possibles pour wn-w*n est élargi. Les inégalités reliant les valeurs des fonctions de Koksma en nombres algébriquement dépendants sont prouvées. / We study bounds on the distances of roots of integer polynomials and applications of such results. The separation of complex roots for reducible monic integer polynomials of fourth degree is thoroughly explained. Lemmas on roots of polynomials in the p-adic setting are proved. Explicit families of polynomials of general degree as well as families in some classes of quadratic and cubic polynomials with very good separation of roots in the same setting are exhibited. The second part of the thesis is concerned with results on p-adic versions of Mahler's and Koksma's functions wn and w*n and the related classifications of transcendental numbers in Cp. The main result is a construction of numbers such that the two functions wn and w*n differ on them for every n and later on expanding the interval of possible values for wn-w*n. The inequalities linking values of Koksma's functions for algebraically dependent numbers are proved.

Submitted by Emanuel Varela Cardoso (emanuel.varela@ufcg.edu.br) on 2018-11-09T18:09:57Z No. of bitstreams: 1 ANA CLÁUDIA GUEDES DOS SANTOS – DISSERTAÇÃO (PPGMat) 2013.pdf: 24981615 bytes, checksum: d442e8df3b32727e30684e3cbd516a9b (MD5) / Made available in DSpace on 2018-11-09T18:09:57Z (GMT). No. of bitstreams: 1 ANA CLÁUDIA GUEDES DOS SANTOS – DISSERTAÇÃO (PPGMat) 2013.pdf: 24981615 bytes, checksum: d442e8df3b32727e30684e3cbd516a9b (MD5) Previous issue date: 2013-08 / Capes / Este trabalho tem como proposta pedagógica apresentar aos alunos o conceito de segmentos comensuráveis e de segmentos incomensuráveis, mostrando a importância desses conceitos para o estudo dos números racionais e irracionais. Veremos um processo de verificação da comensurabilidade de dois segmentos, doravante P.V.C.D.S, que é um processo geométrico de verificação de comensurabilidade de dois segmentos. A partir do P.V.C.D.S, apresentamos a demonstração clássica de que p2 é irracional, com uma abordagem geométrica, mostrando que o segmento do lado de um quadrado de medida 1 e o segmento de sua diagonal são incomensuráveis. Ainda apresentamos um estudo sobre expressões decimais, no qual será apresentado um teorema que nos permite verificar se uma fração irredutível possui representação decimal finita ou infinita e periódica. Também apresentamos outro teorema que nos permite transformar expressões decimais finitas e infinitas e periódicas na sua forma de fração. Por fim, apresentaremos algumas sugestões de atividades, que englobam todo conteúdo do presente TCC. Essas atividades foram aplicadas a uma turma de 1 ano do Ensino Médio de uma escola pública, e as respostas dos alunos estão anexadas ao trabalho. / This work have pedagogical proposed to introduce the concept of commensurable segments and incommensurable segments, showing the importance of these concepts for the study of rational and irrational numbers. We will stabelish a verification process to detect the mensurability of two segments, which is a geometric process. We present the classic demonstration that root of 2 is irrational with a geometric approach, showing that the segment of the side of a square measuring its diagonal are immeasurable. We still will present a study on decimal expressions, and prove a theorem that allows to check that an irreducible fraction has decimal representation finite or infinite and periodic. We also present another theorem that allows us to turn decimal expressions finite or infinite and periodic on its fraction form. Finally we present some suggestions for activities that include all content of the TCC. These activities have been applied to a class of 1st year of high school at a public school, and the students’ answers are attached to the work.

Made available in DSpace on 2016-06-02T20:02:58Z (GMT). No. of bitstreams: 1 6514.pdf: 4306078 bytes, checksum: 049474ab016a0a90e3ef1b9a2412ff69 (MD5) Previous issue date: 2014-12-19 / This work aimed to facilitate the study of the content of complex numbers for students in the 3rd year of High School, within a focus covering a geometric context, with the use of Geogebra software, combined with notebook student (SÃO PAULO, 2014). The main motivation of this study was to search for an innovative practice, aiming to improve the quality of the teaching-learning process, in relation to the content, since the numerous difficulties demonstrated by students. The theoretical research was based mainly on the theory of representation semiotics registers of Raymond Duval and analysis of curriculum documents for Middle School. Already the methodology was built based on the steps of the Didactics Engineering idealized by Michele Artigue. The production of information involved mathematical activities performed by 31 students of Higher Education of a Civil Engineering course and between 10 to 12 students (there was variation) of a 3rd year of High School; providing the basis for answering the following research issues: I What knowledge about complex numbers, students of Higher Education bring the baggage of high School? II - That prospect of construction of knowledge the notebook of the Student gives the learning of complex numbers? III - The Geogebra can add the construction of knowledge when articulated to the notebook of the Student, made available by the Department of the São Paulo State of Education? The instruments used to collect the amount of information produced by participants, in empirical step of our research were: questionnaire, written records of tasks (protocols) and images of mathematical activities carried out with the help of Geogebra. The results obtained in the analysis of the questionnaires showed that the knowledge of the students in higher education, as well as the students in the middle school, are negligible in relation to the need of those who follow their studies in the field of exact area. With respect to the results obtained from the implementation of tasks for high school students, using the software proved to be efficient for a geometrized viewing the content in relation to construction of its concepts, which allowed students a much better understanding across the studied content. / Este trabalho teve como objetivo viabilizar o estudo do conteúdo de números complexos para alunos do 3° ano do Ensino Médio, dentro de um enfoque abrangendo uma contextualização geométrica com a utilização do software Geogebra, combinado com o caderno do aluno (SÃO PAULO, 2014). Observa-se no entanto que este trabalho não se trata de uma sequência didática. A motivação principal deste estudo foi a busca por uma prática inovadora, visando melhorar a qualidade do processo ensinoaprendizagem, em relação a esse conteúdo, frente as inúmeras dificuldades demonstradas pelos estudantes. O percurso teórico da pesquisa fundamentou-se principalmente na teoria dos registros de representação semiótica de Raymond Duval e análise de documentos curriculares para o Ensino Médio. Já o percurso metodológico foi construído com base nas etapas da Engenharia Didática idealizada por Michèle Artigue. A produção de informações envolveu atividades matemáticas realizadas por 31 estudantes de Ensino Superior de um curso de Engenharia Civil e entre 10 a 12 alunos (houve variação) de uma 3ª série do Ensino Médio; servindo de base para responder as seguintes questões de investigação: I - Que saberes sobre números complexos, alunos do Ensino Superior trazem como bagagem do Ensino Médio? II - Que perspectiva de construção de saberes o Caderno do aluno e do professor proporciona ao aprendizado de números complexos? III - O Geogebra pode agregar a construção de saberes quando articulado ao Caderno do aluno, disponibilizado pela Secretaria da Educação do Estado de São Paulo? Os instrumentos utilizados para reunir o montante de informações produzidas pelos sujeitos participantes, na etapa empírica da nossa investigação foram: questionário, registros escritos das tarefas (protocolos) e imagens das atividades matemáticas realizadas com o auxílio do Geogebra. Os resultados obtidos na análise dos questionários apontaram que os conhecimentos dos alunos do ensino superior, bem como dos alunos do ensino médio, são ínfimos em relação a necessidade daqueles que seguirão seus estudos, no campo das exatas. No que diz respeito aos resultados obtidos a partir da aplicação das tarefas para os alunos do Ensino Médio, a utilização do software revelou-se eficiente para uma visualização geometrizada do conteúdo em relação a construção de seus conceitos, o que permitiu aos estudantes uma compreensão muito maior frente ao conteúdo estudado.

Submitted by Susiquine Silva (susi.bibliotecaufpb@hotmail.com) on 2015-10-20T13:41:36Z No. of bitstreams: 2 Arquivototal.pdf: 1918900 bytes, checksum: faffe2a0b39bb0b14b7718f5a5b1b326 (MD5) license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) / Rejected by Susiquine Silva (susi.bibliotecaufpb@hotmail.com), reason: Corrigir título. on 2015-10-20T14:44:19Z (GMT) / Submitted by Susiquine Silva (susi.bibliotecaufpb@hotmail.com) on 2015-10-20T14:46:20Z No. of bitstreams: 2 Arquivototal.pdf: 1918900 bytes, checksum: faffe2a0b39bb0b14b7718f5a5b1b326 (MD5) license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) / Approved for entry into archive by Susiquine Silva (susi.bibliotecaufpb@hotmail.com) on 2015-10-20T14:46:58Z (GMT) No. of bitstreams: 2 Arquivototal.pdf: 1918900 bytes, checksum: faffe2a0b39bb0b14b7718f5a5b1b326 (MD5) license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) / Made available in DSpace on 2015-10-20T14:46:58Z (GMT). No. of bitstreams: 2 Arquivototal.pdf: 1918900 bytes, checksum: faffe2a0b39bb0b14b7718f5a5b1b326 (MD5) license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) Previous issue date: 2013-09-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The purpose of this work is to present a special category of integers: Prime numbers. It will be presented a historical retrospective, quoting the most important and interesting results achieved by great mathematicians over the years. Then, most of these results will be formally announced with propositions or theorems and their respective demonstrations, starting with the basic properties of divisibility and cul- minating in some primality tests. / O propósito deste trabalho é apresentar uma categoria especial de números inteiros: Os Números Primos. Será apresentada uma retrospectiva histórica,citando os resultados mai s importantes e interessantes obtidos por grandes matemáticos ao longodos anos. Em seguida, a maioria destes resultados serão formalmente enunciados com proposições ou teoremas e suas respectivas demonstrações,começando com as propriedades básicas da divisibilidade e culminando em alguns testes de primalidade.