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Abstract(s)
Todas as sociedades humanas, desdes as mais rudimentares às mais sofisticadas
necessitam do conceito de número e de alguma forma de contagem. De acordo com
muitos estudiosos (ver [6]), todas as sociedade humanas tem uma designação para
os primeiros números naturais, embora em tribos mais primitivas essa nomenclatura
não ultrapasse 2 ou 3.
Relativamente aos processos de contagem, parece que os seres humanos sempre
usaram os dedos como forma mais conveniente de fazer a contagem de números
naturais. No entanto, embora os dedos das mãos permitam fazer cálculos simples, a
necessidade de contar um vasto número de objetos, sejam cabeças de gado, amigos,
dias ou anos, levou a uma sistematização do processo de contagem. Um primeiro
passo neste sentido consistiu na criação de grupos de números, a partir dos quais
os restantes seriam construídos. O leitor estará provavelmente mais familiarizado
com os 10 primeiros números inteiros para esse grupo de números, a partir dos quais
são construídos os restantes números naturais e depois todos os números reais. A
preponderância da base 10 deve-se ao facto da contagem ser habitualmente feita
com os dedos das mãos. No entanto, outras bases foram também bastante usadas
ao longo da história por outras civilizações, como a base 5, onde a contagem é
feitas usando os dedos de uma única mão, ou a base 20, esta bastante comum, e
que corresponde ao uso dos dedos das mãos e dos pés. Este sistema de base 20 foi
bastante usado nas civilizações pré-Colombianas da América como os Maias, mas
o seu uso parece ter-se entendido bem par a além da América Central. Na Europa
é possível encontrar vestígios da utilização desse antigo sistema de base 20. Por
exemplo em francês, oitenta diz-se " quatre-vingf', um resquício desse antigo sistema
de contagem.
Este trabalho debruça-se sobre diferentes formas de representar números reais. Iremos
apresentar e estudar dois modos distintos de representar números reais. Assim,
iniciamos este trabalho com um capítulo onde estudaremos sucessões e séries. Estes
assuntos são necessários aos capítulos seguintes uma vez que a resolução de
vários problemas que nos a parecem dependem do estudo da convergência de certas
sucessões e séries.
No Capítulo II , a representação de um número real numa base g, sendo 9 um inteiro
maior do que 1. Iremos provar várias propriedades dos números reais, propriedades
essas bem conhecidas, estando os números representados numa base g.
No Capítulo III iremos estudar outra forma de representar números reais, através das
frações contínuas. A principal questão que vamos abordar é saber se qualquer fração
contínua representa sempre um número real. Se bem que para frações contínuas
finitas este problema é facilmente resolúvel, quando se trata de frações contínuas
infinitas o problema torna-se bastante mais complexo. Iremos resolvê-lo para uma
classe particular de frações contínuas infinitas, chamadas simples.
All human societies, from the most rudimentary to the most sophisticated, need the concept of number and some form of counting. According to many scholars (sec cite Ore), all human societies have a designation for the first natural numbers, although in more primitive tribes this nomenclature does not exceed 2 or 3. With regard to counting processes, it seems that humans have always used fingers as the most convenient way of counting natural numbers. However, although the fingers allow simple calculations to be made, the need to count a vast number of objects, be they cattle, friends, days or years, has led to a systematization of the counting process. A first step in this direction was the creation of groups of numbers, from which the rest would be built. The reader will probably be more familiar with the first 10 integer numbers for that group, from which the remaining natural numbers and all real numbers are constructed. The preponderance of the base 10 is due to the fact that the count is usually done with the fingers. However, other bases have also been used throughout history by other civilizations, such as the 5 base, where the counting is done using the fingers of a single hand, or the 20 base, which is quite common, and which corresponds to the use fingers and toes. This 20 base system was widely used in pre-Colombian civilizations in America such as the Mayans, but its use seems to have been well understood beyond Central America. ln Europe it is possible to find traces of the use of this old base system 20. For example in French, eighty is called " it quatre-vingt “, a remnant of that old counting system. This work focuses on different ways of representing real numbers. \~re will present and study two different ways of representing real numbers. Thus, we begin this work with an introductory chapter where we will study successions and series. These issues are necessary for the following chapters since the resolution of several problems that appear to us depends on the study of the convergence of certain successions and series. ln Chapter II, the representation of a real number on a g basis, g being an integer greater than 1. \~re will prove several properties of real numbers, properties that are well known, the numbers being represented on a g basis. ln Chapter III we will study another way of representing real numbers, using continuous fractions. The main question we are going to address is to know if any fraction continued fraction always represent a real number. Although for finite continuous fractions this problem is easily solved, when it comes to infinite continuous fractions the problem becomes much more complex. We will solve it for a particular class of infinite continuous fractions, called simple.
All human societies, from the most rudimentary to the most sophisticated, need the concept of number and some form of counting. According to many scholars (sec cite Ore), all human societies have a designation for the first natural numbers, although in more primitive tribes this nomenclature does not exceed 2 or 3. With regard to counting processes, it seems that humans have always used fingers as the most convenient way of counting natural numbers. However, although the fingers allow simple calculations to be made, the need to count a vast number of objects, be they cattle, friends, days or years, has led to a systematization of the counting process. A first step in this direction was the creation of groups of numbers, from which the rest would be built. The reader will probably be more familiar with the first 10 integer numbers for that group, from which the remaining natural numbers and all real numbers are constructed. The preponderance of the base 10 is due to the fact that the count is usually done with the fingers. However, other bases have also been used throughout history by other civilizations, such as the 5 base, where the counting is done using the fingers of a single hand, or the 20 base, which is quite common, and which corresponds to the use fingers and toes. This 20 base system was widely used in pre-Colombian civilizations in America such as the Mayans, but its use seems to have been well understood beyond Central America. ln Europe it is possible to find traces of the use of this old base system 20. For example in French, eighty is called " it quatre-vingt “, a remnant of that old counting system. This work focuses on different ways of representing real numbers. \~re will present and study two different ways of representing real numbers. Thus, we begin this work with an introductory chapter where we will study successions and series. These issues are necessary for the following chapters since the resolution of several problems that appear to us depends on the study of the convergence of certain successions and series. ln Chapter II, the representation of a real number on a g basis, g being an integer greater than 1. \~re will prove several properties of real numbers, properties that are well known, the numbers being represented on a g basis. ln Chapter III we will study another way of representing real numbers, using continuous fractions. The main question we are going to address is to know if any fraction continued fraction always represent a real number. Although for finite continuous fractions this problem is easily solved, when it comes to infinite continuous fractions the problem becomes much more complex. We will solve it for a particular class of infinite continuous fractions, called simple.
Description
Keywords
Base Fração Contínua Números Reais Representação