One of the classic theorems concerning the real numbers states that every open cover of a closed and bounded subset of the real line contains a finite subcover. Compactness is an abstraction of that notion, and there are several ideas concerning it which are equivalent and many which are similar. The purpose of this paper is to synthesize the more important of these ideas. This synthesis is accomplished by demonstrating either situations in which two ordinarily different conditions are equivalent or combinations of two or more properties which will guarantee a third.

The purpose of this paper is to review and summarize the topics involved in the study of real analysis. Real analysis is a branch of mathematics that studies the field of real numbers including the calculus of real numbers, analytical properties of real functions and sequences. This includes limits of sequences of real numbers, continuity, completeness, and related properties of real functions. While all topics in the course were important and vital to understanding analysis, the goal of this paper is to review, research, and report on a few of the more interesting topics covered in the class. / text

The purpose of this paper is the analysis of functions of real numbers which have a special additive property, namely, f(x+y) = f(x)+f(y).

The purpose of this paper is to construct the real number system. The foundation upon which the real number system will be constructed will be the system of counting numbers.

The purpose of this paper is to examine certain questions concerning infinite series. The first chapter introduces several basic definitions and theorems from calculus. In particular, this chapter contains the proofs for various convergence tests for series of real numbers. The second chapter deals primarily with the equivalence of absolute convergence, unconditional convergence, bounded multiplier convergence, and c0 multiplier convergence for series of real numbers. Also included in this chapter is a proof that an unconditionally convergent series may be rearranged so that it converges to any real number desired. The third chapter contains a proof of the Silverman-Toeplitz Theorem together with several applications.

Submitted by ALINE DE FREITAS MORONI null (ali-moroni@hotmail.com) on 2016-05-17T15:39:59Z No. of bitstreams: 1 Rotações no espaço tridimensional por meio de produtos quaterniônicos..pdf: 761793 bytes, checksum: 0d00cba6c4297a34f603a3b61904efc4 (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-05-19T13:44:08Z (GMT) No. of bitstreams: 1 moroni_af_me_rcla.pdf: 761793 bytes, checksum: 0d00cba6c4297a34f603a3b61904efc4 (MD5) / Made available in DSpace on 2016-05-19T13:44:08Z (GMT). No. of bitstreams: 1 moroni_af_me_rcla.pdf: 761793 bytes, checksum: 0d00cba6c4297a34f603a3b61904efc4 (MD5) Previous issue date: 2016-04-20 / Neste trabalho pretendemos descrever o processo de construção da álgebra dos quatérnios, e a interpretação da multiplicação desses objetos via rotações no espaço. Para isto, vimos a necessidade de iniciar com conceitos que formam a base da álgebra, listando axiomas para o sistema de números reais e complexos. / The aim of this work is to describe the construction of the quaternion algebra and to interpret the multiplication operation via tridimensional rotations. For that we begin with basic algebraic concepts, and we list the axioms for the real and complex number systems.

The principal focus of this thesis is the study of the real numbers regarded as a structure endowed with its usual addition and multiplication and the operations of raising to real powers. For our first main result we prove that any statement in the language of this structure is equivalent to an existential statement, and furthermore that this existential statement can be chosen independently of the concrete interpretations of the real power functions in the statement; i.e. one existential statement will work for any choice of real power functions. This result we call uniform model completeness. For the second main result we introduce the first order theory of raising to an infinite power, which can be seen as the theory of a class of real closed fields, each expanded by a power function with infinite exponent. We note that it follows from the first main theorem that this theory is model-complete, furthermore we prove that it is decidable if and only if the theory of the real field with the exponential function is decidable. For the final main theorem we consider the problem of expanding an arbitrary o-minimal expansion of a field by a non-trivial exponential function whilst preserving o-minimality. We show that this can be done under the assumption that the structure already defines exponentiation on a bounded interval, and a further assumption about the prime model of the structure.

Nesta dissertação são apresentadas algumas propriedades de números reais. Descrevemos de maneira breve os conjuntos numéricos N, Z, Q e R e apresentamos demonstrações detalhadas da irracionalidade dos números \'pi\' e e. Também, apresentamos um texto sobre o número e, menos técnico e mais intuitivo, na tentativa de auxiliar o professor no preparo de aulas sobre o número e para alunos do Ensino Médio, bem como, alunos de cursos de Licenciatura em Matemática / In this thesis we present some properties of real numbers. We describe briefly the numerical sets N, Z, Q and R, and we present detailed proofs of irrationality of numbers \'pi\' and e. We also present a text about the number e less technical and more intuitive in an attempt to assist the teacher in preparing lessons about number e for High School students as well as for Teaching degree in Mathematics students

Submitted by Tassia Roberta Zangiacomo null (tassia_zangiacomo@hotmail.com) on 2017-03-23T22:04:31Z No. of bitstreams: 1 TASSIA ROBERTA ZANGIACOMO - MESTRADO.pdf: 1004175 bytes, checksum: 12925ba240f8d9a89e295b32b2efb13e (MD5) / Approved for entry into archive by Luiz Galeffi (luizgaleffi@gmail.com) on 2017-03-24T17:23:14Z (GMT) No. of bitstreams: 1 zangiacomo_tr_me_rcla.pdf: 1004175 bytes, checksum: 12925ba240f8d9a89e295b32b2efb13e (MD5) / Made available in DSpace on 2017-03-24T17:23:15Z (GMT). No. of bitstreams: 1 zangiacomo_tr_me_rcla.pdf: 1004175 bytes, checksum: 12925ba240f8d9a89e295b32b2efb13e (MD5) Previous issue date: 2017-02-20 / Este trabalho tem como objetivo construir os sistemas numéricos usuais, a saber, o conjunto dos números naturais N, o conjunto dos números inteiros Z, o conjunto dos números racionais Q e o conjunto dos números reais R. Iniciamos o trabalho tratando de noções sobre conjuntos e relações binárias. Em seguida, apresentamos o conjunto dos números naturais, definido através dos axiomas de Peano; o conjunto dos números inteiros via uma relação de equivalência com o conjunto dos números naturais; o conjunto dos números racionais, que são obtidos também via relação de equivalência, mas dessa vez com o conjunto dos números inteiros; a construção do conjunto dos números reais, feita via cortes no conjunto dos números racionais; e, para todos esses casos, mostramos a imersão do conjunto anterior no conjunto que surge na sequência. Por fim, observamos alguns materiais do ensino fundamental e médio com o intuito de investigar de que forma esses temas estão sendo apresentados para os alunos. / This work aims to construct the usual numerical systems, namely the set of natural numbers N, the set of integers Z, the set of rational numbers Q and the set of real numbers R. We begin the work dealing with notions about sets and binary relations. Next, we present the set of natural numbers, defined by Peano's axioms; the set of integers via an equivalence relation with the set of natural numbers; the set of rational numbers, which are also obtained via equivalence relation, but this time with the set of integers; the construction of the set of real numbers, made through cuts in the set of rational numbers; end for all these cases we show the immersion of the previous set in the ensemble that appears in the sequence. Finally, we observed some materials in elementary school and high school in order to investigate how these themes are being presented to the students.

Orientador: Rosa Lúcia Sverzut Baroni / Banca: Doherty Andrade / Banca: Sandra M. Semensato Godoy / Banca: Ubiratan D'Ambrosio / Banca: Vanderlei M. Nascimento / Resumo: Entendendo o Material Um tratamento, via medição para os números reais como uma alternativa para abordar os números reais em cursos de formação de professores, buscou-se as possibilidades que o mesmo poderia apresentar ao introduzir os números reais a partir de um processo de medição de segmentos. Esta investigação foi realizada a partir do acompanhamento da utilização do Material numa sala de aula, cujos alunos eram professores de Matemática. A análise foi conduzida em dois momentos. Um deles denominado Apresentação/Comentário, que situou os dados do Diário de Campo ao centro, e, o outro, que traz essas possibilidades da análise de Entrevistas. A literatura mostra que a forma como o conjunto dos números reais tem sido apresentada aos futuros professores furta a oportunidade de que noções e conceitos inerentes ao mesmo sejam discutidos. Introduzir os números reais via medição, oportuniza que noções e conceitos intrínsecos ao número real possam ser explorados, em particular, conceitos básicos da Análise, como convergência, continuidade, completude, etc., e mesmo que indiretamente àqueles relacionados a outros campos da Matemática como a Álgebra, a Geometria e a História da Matemática. / Abstract: Understanding the Material A treatment, via measurement for the real numbers as an alternative to deal with the real numbers in teacher education courses, we looked for the possibilities the Material presents when introducing the real numbers from a process of measuring segments. This investigation was done through the observation of using the Material in a classroom, whose students were Mathematics teachers. The analysis was conducted in two moments. One of them denominated Presentation/Comment, sited the data of the Camp Diary to the center, and, the other one that brought these possibilities of the analysis of the Interviews. The literature shows that the way how the set of the real numbers has been presented to the future teachers steals the opportunity for notions and concepts inherent to this set to be discussed. Introducing the real numbers via measurement, gives the opportunity that notions and intrinsic concepts to the real number can be explored, in particular, basic concepts of Analysis as convergence, continuity, completion, etc; and even indirectly those related to other branches of Mathematics as Algebra, Geometry and History of Mathematics. / Doutor