Os artigos de Euler sobre os números amigáveis

Among the many methodological resources that the mathematics teacher can use in the classroom, we can cite the History of Mathematics which has contributed to the development of activities that promotes students curiosity about mathematics and its history. In this regard, the present dissertation aims to translate and analyze, mathematically and historically, the three works of Euler about amicable numbers that were writed during the Eighteenth century with the same title: De numeris amicabilibus. These works, despite being written in 1747 when Euler lived in Berlin, were published in different times and places. The first, published in 1747 in Nova Acta Eruditorum and which received the number E100 in the Eneström index, summarizes the historical context of amicable numbers, mentions the formula 2nxy & 2nz used by his precursors and presents a table containing thirty pairs of amicable numbers. The second work, E152, was published in 1750 in Opuscula varii argument. It is the result of a comprehensive review of Euler s research on amicable numbers which resulted in a catalog containing 61 pairs, a quantity which had never been achieved by any mathematician before Euler. Finally, the third work, E798, which was published in 1849 at the Opera postuma, was probably the first among the three works, to be written by Euler

Retratos, imagens, letras e numeros colados nas paredes:representaçãoes sociais de escola para ribeirinhos dos rios Môa e Azul - Acre

Remote Communities. Absence of artifacts and minimization of the exacerbated consumption of modernity. The desire which spread beyond what reality can provide. Expressions like this are present in this paper which focus in the social representations of school built by residents who live at the riversides of Môa and Azul Rivers, in Mâncio Lima, Acre State. To do so, we used the methodological contribution of the semi-structured interview, observation of the place while a natural inhabitant of the region, and also photos analyses of local reality. A key feature of the riverside homes is the glued paper on the walls of houses forming a panel set of portraits, pictures, letters and numbers for all appreciated. Regardless of whether or not read, there is admiration for the color of the images, the layout of the letters, and the things of the city awakening the desire to obtain school knowledge. The resident of this Amazon region maintains a close relationship between thinking, acting and feeling living harmonically with nature that connects them to the ideal landscape which is revisited by the graphic material that attracts wondering what exists beyond the shores of the river, beyond the horizon of green forests. It is a life entirely accomplished by the imaginary where exist a framed landscape merged and confused by the real and the supernatural, in which men and gods walk together by the forest, sailing by the rivers and seek a possible aesthetic between the real and ideal. The Theory of Social Representations spread by Serge Moscovici (2005) and Jodelet (2001) guided our gaze on the understanding what the school is and its representation to the riversides, as well to reveal the relation they practice with the knowledge that is spread by the mystification and the knowledge that is practice daily. Based in Bardin s thematic analysis (2004) we tried to raise such contents combining them in five analysis categories

Aspectos linguístico-ergonômicos em rótulos: avaliação da linguagem verbo-visual de rótulos de embalagens para alimentos achocolatados

We have been living in a world of packed products. The package and the labels support the companies to communicate with the customers in addition to give protection, storage and convenience in proportion to the products that move in the price list. The labels mainly add up a value which helps the companies differ their products and increase the value of the brands among the final customers. However, the information given in the label are not clear sometimes. It displays a verbal-visual defective language resulted from a poor visibility, legibleness and comprehensibleness of the verbal and visual marks. The aim of this research is to verify, according to the costumers‟ view, the level of the clarity in the informative texts, harmony and ergonomic conformity of the package labels in the chocolate powder of the Claralate brand, considering the linguistic aspects presented on the labels. The criteria to evaluate the chocolate package selected were based on the linguistic field: the organization and the structure of the text derided from the classification of the textual genre; the clarity and the comprehension of the language utilized on those labels. From the ergonomic view, the informative and ergonomic conformity, based on the following requirements: legibility, symbols, characters, reading fields and intermission of the written lines. Therefore, the research done july 2007 and added july 2011 had a structured questionnaire in the interview put to the 118 customers of the chocolate package that go shopping in one of the two supermarkets in Floriano, Piauí São Jorge and/or Super Quaresma. The main results of the investigation show that the linguistic aspects in the informative texts of the labels provide the customers‟ expectancy partially, while the consideration of the informative ergonomic analyzed can contribute to the improvement of the information and consequent visual progress of those, on the labels of chocolate package investigated. As recommendation towards the maker of the product, the outcome of the research indicates: harmonize the proportion of the letters and numbers; enlarge the letters size; make the visual information more comprehensive determined by the reading field; put the expiry date in a better visual place

Núcleos de acessibilidade e o atendimento a alunos com necessidades educacionais especiais nas universidades federais do nordeste brasileiro

In the current context of policies for inclusiveness in the Brazilian higher education, the accessibility centers are responsible for the organization of actions toward the fulfillment of the legal requirements regarding accessibility and the elimination of barriers that interfere with the participation and learning of persons with special learning needs (SLN), providing conditions for the full inclusion of these students in the activities of learning, research and community outreach in this learning level. This research had the goal to analyse the work developed by the accessibility centers in the federal universities in the northeastern region of Brazil towards the care of students with SLN. It is a descriptive work, with quantitative and qualitative approaches. Twelve federal universities participated through their accessibility center’s coordinators. The data - gathered by means of an electronic survey filled out in 2014 - was organized and analyzed through descriptive statistics and content analysis; their discussion was made around four topics: organization of accessibility centers for the care of students with SLN, kinds and numbers of students with SLN cared for, actions developed, and suggestions for the improvement of the accessibility centers. It was found that in the sphere of the researched universities, the accessibility centers have been taking actions involving many parts of the academic community towards the improvement of the conditions and permanence of students with SLN. However, in some institutional realities, these action need to be expanded and/or consolidated. The coordinators suggest further actions towards the betterment of these centers, regarding expansion of financial and human resources, professional qualification, awareness of the academic community, institutionalization and the formation of a collaborative network among the accessibility centers. Thus, although there are still challenges to overcome, the presence of the accessibility centers is shown to advance the realization of policies for the inclusion of students with SLN in the post-secondary education, towards the democratization of learning starting from the universal right to education.

Núcleos de acessibilidade e o atendimento a alunos com necessidades educacionais especiais nas universidades federais do nordeste brasileiro

In the current context of policies for inclusiveness in the Brazilian higher education, the accessibility centers are responsible for the organization of actions toward the fulfillment of the legal requirements regarding accessibility and the elimination of barriers that interfere with the participation and learning of persons with special learning needs (SLN), providing conditions for the full inclusion of these students in the activities of learning, research and community outreach in this learning level. This research had the goal to analyse the work developed by the accessibility centers in the federal universities in the northeastern region of Brazil towards the care of students with SLN. It is a descriptive work, with quantitative and qualitative approaches. Twelve federal universities participated through their accessibility center’s coordinators. The data - gathered by means of an electronic survey filled out in 2014 - was organized and analyzed through descriptive statistics and content analysis; their discussion was made around four topics: organization of accessibility centers for the care of students with SLN, kinds and numbers of students with SLN cared for, actions developed, and suggestions for the improvement of the accessibility centers. It was found that in the sphere of the researched universities, the accessibility centers have been taking actions involving many parts of the academic community towards the improvement of the conditions and permanence of students with SLN. However, in some institutional realities, these action need to be expanded and/or consolidated. The coordinators suggest further actions towards the betterment of these centers, regarding expansion of financial and human resources, professional qualification, awareness of the academic community, institutionalization and the formation of a collaborative network among the accessibility centers. Thus, although there are still challenges to overcome, the presence of the accessibility centers is shown to advance the realization of policies for the inclusion of students with SLN in the post-secondary education, towards the democratization of learning starting from the universal right to education.

De solutione problematum diophanteorum per números integros : o primeiro trabalho de Euler sobre equações diofantinas

The present dissertation analyses Leonhard Euler´s early mathematical work as Diophantine Equations, De solutione problematum diophanteorum per números íntegros (On the solution of Diophantine problems in integers). It was published in 1738, although it had been presented to the St Petersburg Academy of Science five years earlier. Euler solves the problem of making the general second degree expression a perfect square, i.e., he seeks the whole number solutions to the equation ax2+bx+c = y2. For this purpose, he shows how to generate new solutions from those already obtained. Accordingly, he makes a succession of substitutions equating terms and eliminating variables until the problem reduces to finding the solution of the Pell Equation. Euler erroneously assigns this type of equation to Pell. He also makes a number of restrictions to the equation ax2+bx+c = y and works on several subthemes, from incomplete equations to polygonal numbers

Euler e os números pentagonais

The present investigation includes a study of Leonhard Euler and the pentagonal numbers is his article Mirabilibus Proprietatibus Numerorum Pentagonalium - E524. After a brief review of the life and work of Euler, we analyze the mathematical concepts covered in that article as well as its historical context. For this purpose, we explain the concept of figurate numbers, showing its mode of generation, as well as its geometric and algebraic representations. Then, we present a brief history of the search for the Eulerian pentagonal number theorem, based on his correspondence on the subject with Daniel Bernoulli, Nikolaus Bernoulli, Christian Goldbach and Jean Le Rond d'Alembert. At first, Euler states the theorem, but admits that he doesn t know to prove it. Finally, in a letter to Goldbach in 1750, he presents a demonstration, which is published in E541, along with an alternative proof. The expansion of the concept of pentagonal number is then explained and justified by compare the geometric and algebraic representations of the new pentagonal numbers pentagonal numbers with those of traditional pentagonal numbers. Then we explain to the pentagonal number theorem, that is, the fact that the infinite product(1 x)(1 xx)(1 x3)(1 x4)(1 x5)(1 x6)(1 x7)... is equal to the infinite series 1 x1 x2+x5+x7 x12 x15+x22+x26 ..., where the exponents are given by the pentagonal numbers (expanded) and the sign is determined by whether as more or less as the exponent is pentagonal number (traditional or expanded). We also mention that Euler relates the pentagonal number theorem to other parts of mathematics, such as the concept of partitions, generating functions, the theory of infinite products and the sum of divisors. We end with an explanation of Euler s demonstration pentagonal number theorem

A relação de Euler para poliedros

In this paper we analyze the Euler Relation generally using as a means to visualize the fundamental idea presented manipulation of concrete materials, so that there is greater ease of understanding of the content, expanding learning for secondary students and even fundamental. The study is an introduction to the topic and leads the reader to understand that the notorious Euler Relation if inadequately presented, is not sufficient to establish the existence of a polyhedron. For analyzing some examples, the text inserts the idea of doubt, showing cases where it is not fit enough numbers to validate the Euler Relation. The research also highlights a theorem certainly unfamiliar to many students and teachers to research the polyhedra, presenting some very simple inequalities relating the amounts of edges, vertices and faces of any convex polyhedron, which clearly specifies the conditions and sufficient necessary for us to see, without the need of viewing the existence of the solid screen. And so we can see various polyhedra and facilitate understanding of what we are exposed, we will use Geogebra, dynamic application that combines mathematical concepts of algebra and geometry and can be found through the link http://www.geogebra.org