Master of Science / Department of Mathematics / Todd Cochrane / Polygonal numbers are nonnegative integers constructed and represented by geometrical arrangements of equally spaced points that form regular polygons. These numbers were originally studied by Pythagoras, with their long history dating from 570 B.C, and are often referred to by the Greek mathematicians. During the ancient period, polygonal numbers were described by units which were expressed by dots or pebbles arranged to form geometrical polygons. In his "Introductio Arithmetica", Nicomachus of Gerasa (c. 100 A.D), thoroughly discussed polygonal numbers. Other Greek authors who did remarkable work on the numbers include Theon of Smyrna (c. 130 A.D), and Diophantus of Alexandria (c. 250 A.D). Polygonal numbers are widely applied and related to various mathematical concepts. The primary purpose of this report is to define and discuss polygonal numbers in application and relation to some of these concepts. For instance, among other topics, the report describes what triangle numbers are and provides many interesting properties and identities that they satisfy. Sums of squares, including Lagrange's Four Squares Theorem, and Legendre's Three Squares Theorem are included in the paper as well. Finally, the report introduces and proves its main theorems, Gauss' Eureka Theorem and Cauchy's Polygonal Number Theorem.

This report explores the effect of interactive visualizations on the advancement of mathematics understanding. Not only do interactive visualizations aid mathematicians to expand the body of knowledge of mathematics but it also allows students an efficient way to process the information taught in schools. There are many concepts in mathematics that utilize interactive visualizations and examples of such concepts are illustrated within this report. / text

Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2015-10-27T14:48:51Z No. of bitstreams: 2 Dissertação - Ronaldo Pires da Silva - 2015.pdf: 1653286 bytes, checksum: 63a72d8fbcc7390f80fb41dbadaaa9fe (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2015-10-27T14:53:07Z (GMT) No. of bitstreams: 2 Dissertação - Ronaldo Pires da Silva - 2015.pdf: 1653286 bytes, checksum: 63a72d8fbcc7390f80fb41dbadaaa9fe (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-10-27T14:53:07Z (GMT). No. of bitstreams: 2 Dissertação - Ronaldo Pires da Silva - 2015.pdf: 1653286 bytes, checksum: 63a72d8fbcc7390f80fb41dbadaaa9fe (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2015-07-06 / Our work had as main objective to study the intersection of integer sequences, denominated polygonal numbers, through Pell's equation. In this context, the solution of two equations will be treated: x2 􀀀 Dy2 = 1 and x2 􀀀 Dy2 = N, jNj > 1. For the rst one we have used results from the theory of continued fractions. For the last one, we have used the method of solution delineated in literature. Besides, propositions referring to the intersection of polygonal numbers for some particular cases are presented and demonstrated. Also, the proposition of the general case is presented and demonstrated. Finally, we have performed the solution of some of Pell's equations in order to determine the intersection of some polygonal numbers. / Nosso trabalho teve como objetivo central estudar a interseção de sequências de inteiros, denominadas números geométricos, através da equação de Pell. Neste contexto, a resolução de duas equações serão tratadas: x2 􀀀 Dy2 = 1 e x2 􀀀 Dy2 = N com jNj > 1. Para a primeira utilizamos importantes resultados presentes na teoria das frações contínuas. Para última, utilizamos o método de resolução delineado na literatura. Além disso, proposições referentes a interseção de números geométricos para alguns casos particulares são apresentadas e demonstradas. Também a proposição do caso geral é apresentada e demonstrada. Por m, realizamos a resolução de algumas equações de Pell para determinarmos a interseção de alguns números geométricos.

Made available in DSpace on 2016-06-02T20:29:21Z (GMT). No. of bitstreams: 1 5050.pdf: 4225946 bytes, checksum: eef403ca886d7a47c3c8886fee5bc0f4 (MD5) Previous issue date: 2013-03-11 / Financiadora de Estudos e Projetos / The shortage of activities regarding recursive defined sequences, allied to the importance of this theme to young students has been contested in at least twenty years of experience teaching Mathematics. The elaboration of a learning product, in a form of sheets of activities which gradually lead the student to the understanding of the concept of recursion, the pattern recognition, conjectures tests and formulas acquisition, could be inferred by the application of these sheets of activities in two high school classrooms in a technical state school. The data attained from these activities applications were analyzed and compared to the previously formed hypothesis (previous analysis) which were formed during the elaboration of the sheets of activities. The investigation method used was the Didactical Engineering. The students did the activities in groups of two or three students .They were participative and felt challenged in doing every step (lessons) proposed in the sheets. It was verified that the teaching material developed works, as it reached its main goals, the biggest one, the students learning. It is believed that the material developed may be useful to other teachers who may wish to develop the theme proposed in their classrooms, even adapting them to their students needs. This work brought to this author a great profession evolution which began with the theme choice, passed through the development of the material, the application in the classes and finally in the reflection of everything that was done and that is recorded in these notes. / A escassez de atividades envolvendo sequências definidas recursivamente, aliada à importância desse tema para jovens estudantes, foi constatada em pelo menos vinte anos de experiência dando aulas de matemática. A elaboração de um produto de ensino, na forma de folhas de atividades que gradativamente levam o estudante à compreensão do conceito de recursividade, reconhecimento de padrões, testes de conjecturas e obtenção de fórmulas, pôde ser aferida através da aplicação dessas folhas de atividades em duas salas do ensino médio de uma escola técnica estadual. Os dados obtidos dessas aplicações foram analisados e comparados com as hipóteses levantadas (análises prévias), sendo estas feitas durante a elaboração das folhas de atividades. A metodologia de investigação usada foi a Engenharia Didática. Os alunos fizeram as atividades em grupos de dois ou três, foram participativos e se sentiram bastante estimulados em realizar todas as etapas (lições) propostas nas folhas. Constatou-se que o material de ensino produzido funciona, pois atingiu seus objetivos principais, sendo o maior deles o aprendizado do aluno. Acredita-se que o material elaborado possa ser útil a outros professores que desejarem desenvolver o tema proposto em suas aulas, podendo mesmo adaptá-los à realidade de suas turmas. Este trabalho trouxe, para esse autor, uma grande evolução profissional que se iniciou na escolha do tema, passou pela elaboração do material construído, pela aplicação nas turmas e finalmente pela reflexão de tudo que foi feito e que se encontra registrado nessas notas.