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Regiões Circulares E O Número Pi

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Regiões Circulares E O Número Pi Universidade Federal de Goiás Instituto de Matemática e Estatística Programa de Mestrado Profissional em Matemática em Rede Nacional Regiões Circulares e o número Pi THIAGO VERÍSSIMO PEREIRA Goiânia 2013 THIAGO VERÍSSIMO PEREIRA Regiões Circulares e o número Pi Trabalho de Conclusão de Curso apresentado ao Programa de Pós–Graduação do Instituto de Matemática e Estatística da Universidade Federal de Goiás, como requisito parcial para obtenção do título de Mestre em matemática Área de concentração: Matemática do ensino básico. Orientador: Prof. Dr. José Yunier Bello Cruz Goiânia 2013 Dados Internacionais de Catalogação na Publicação (CIP) GPT/BC/UFG Pereira, Thiago Veríssimo. P436r Regiões circulares e o número pi [manuscrito] / Thiago Veríssimo Pereira. – 2013. 40 f. : il., figs., tabs. Orientador: Prof. Dr. José Yunier Bello Cruz. Dissertação (Mestrado) – Universidade Federal de Goiás, Instituto de Matemática e Estatística, 2013. Bibliografia. 1. Matemática, História da. 2. Geometria. 3. Pi. I. Título. CDU: 51:930.1 Todos os direitos reservados. É proibida a reprodução total ou parcial do trabalho sem autorização da universidade, do autor e do orientador(a). Thiago Veríssimo Pereira Licenciado em Matemática pela UnB. Professor da Secretaria de Educação do Distrito Federal e da rede particular desde 2009. A Deus e a minha família, em especial minha esposa, que me deram apoio e força para chegar até aqui, e a meus professores que me inspiraram. Agradecimentos A Deus, minha família, em especial a minha esposa, aos professores, tutores e coordenadores do IME-UFG pelo empenho e dedicação mostrados ao longo do curso, em especial ao Prof. Dr. José Yunier Bello Cruz e aos colegas de turma pelo apoio e compreensão nos momentos difíceis. E ainda a meus antigos professores que me inspiraram. Resumo Pereira,Thiago Veríssimo . Regiões Circulares e o número Pi . Goiânia , 2013. 40p. Trabalho de conclusão de curso. Instituto de Matemática e Estatística, Universidade Federal de Goiás. Ao longo da história em várias situações foi necessário determinar a área de um círculo e nesses momentos matemáticos foram levados a testar sua criatividade e suas habilidades. Várias foram as formas de determinar a área de uma região circular desenvolvidas até hoje. Muita teoria foi criada, direta ou indiretamente, ligada a esse problema como o problema da quadratura do círculo e o número p. Palavras–chave <História da Matemática, Geometria, p.> Abstract Pereira,Thiago Veríssimo . Circular regions and the number p . Goiânia , 2013. 40p. Completion of course work. Instituto de Matemática e Estatística, Universidade Federal de Goiás. Throughout history in various situations was necessary to determine the area of a circle and those moments mathematicians were led to test their creativity and skills. There were several ways to determine the area of a circular region developed until today. Some theory was created, directly or indirectly, related to this issue as the problem of squaring the circle and the number p. Keywords <Geometry, History, p.> Sumário 1 Introdução 10 2 O cálculo da área do circulo no decorrer da história 12 2.1 Egito 12 2.2 Babilônios e Chineses 14 2.3 Quadratura do Círculo 16 2.4 A impossibilidade da Quadratura do Círculo 19 3 Histórias e Curiosidades do número p 23 3.1 A origem do símbolo p 23 3.2 Aproximações do número p 23 3.3 Maneiras que p foi calculado 24 3.4 Uma breve cronologia do número p 25 3.5 Calculando p 26 4 Aplicações no ensino básico 31 4.1 Área de uma região circular pelo método egípcio 32 4.2 Método de Arquimedes para determinar o valor de p 32 4.3 Pode-se quadrar as Lunas de Hipócrates! 34 5 Considerações finais 36 Referências Bibliográficas 38 A Valores aproximados de p durante a história 39 CAPÍTULO 1 Introdução Esse trabalho tem por objetivo apresentar, a professores do ensino básico, o desenvolvimento histórico de conceitos que envolvem a área de uma região circular, que muitas vezes aparece relacionada ao quadrado e ao problema da quadratura do círculo, problemática que pode ser utilizada como uma proposta didática. O texto apresentará algumas resoluções aproximadas, outras exatas, de problemas que envolvem o cálculo de áreas de regiões circulares no decorrer da história, problemas e soluções que podem ser colocados como motivação para alunos do ensino básico. Registros mostram que os povos antigos sabiam como calcular as áreas de figuras poligonais simples. São abundantes nos textos históricos exemplos de usos das fórmulas 1 A = b · h e A = · b · h, ou ideias equivalentes, para o cálculo das áreas de retângulos R T 2 e triângulos, respectivamente, onde b representa medida da base e h medida da altura da figura. Os babilônios apresentaram muitas fórmulas para cálculo de áreas nas quais 7 apresentavam o que são hoje chamados coeficientes. Por exemplo, o número = 0;875 8 era utilizado como o coeficiente para a altura de um triângulo equilátero, isso significa 7 que a altura de um triângulo equilátero era calculada como da base, aqui, naturalmente, 8 p 3 o resultado é apenas aproximação como ilustra a Figura 1.1. O múltiplo correto seria , 2 cuja aproximação, com três casas decimais, é 0,866. ℎ = (푐표푒푓푐푒푛푡푒) ∙ 푙푎푑표 Figura 1.1 Muitas vezes os antigos escribas encontravam aproximações razoáveis para com- primentos e áreas. Serão discutidos alguns métodos desenvolvidos por certas civilizações 11 para cálculo da áreas de regiões circulares. O Capítulo 2 apresenta alguns problemas históricos, dentre eles o método egípcio e o babilônico para o cálculo da áreas de regiões circulares e a Quadratura do Círculo. O Capítulo 3 apresenta uma breve história do número p, a origem do símbolo utilizado para representá-lo, algumas aproximações, em especial o método de Arquime- des. O Capítulo 4 trata de como apresentar esse tipo de conhecimento para alunos do ensino básico, comparar com o método utilizado hoje em dia. Sugere-se uma forma de apresentar ou consolidar esses conceitos para alunos do ensino básico. Nos apêndices há tabelas com aproximações utilizadas para o número p no decorrer da história. CAPÍTULO 2 O cálculo da área do circulo no decorrer da história 2.1 Egito Toda matemática conhecida do Antigo Egito está baseada em papiros que sobre- viveram ao tempo e chegaram até nós. O papiro Rhind é o nome de um desses papiros e será citado a seguir. Os problemas 48 e 50 do papiro Rhind; ver Figura 2.1, são muito interessantes do ponto de vista matemático e eles podem dar uma pista de como os egípcios chegaram a um método que podemos traduzir como uma fórmula para o cálculo da área do círculo. Figura 2.1: papiro Rhind No problema 48 do Papiro Rhind propõe-se comparar a área do círculo com a área do quadrado circunscrito. E no problema 50 do mesmo papiro, calcula-se a área de uma região circular cuja solução apresentada pelo escriba é equivalente ao seguinte: "Subtrair do diâmetro sua nona parte e elevar o restante ao quadrado.". 2.1 Egito 13 Não sabemos como os egípcios chegaram a esse método, uma possível explica- ção, ver mais em [2], é a que segue. Justificativa Tomando um círculo inscrito, de área Ac, em um quadrado como na Figura 2.2, temos que ∼ Ac = Aoctogono: (2-1) Figura 2.2 A partir de um quadrado idêntico ao da figura anterior, podemor retirar uma linha e uma coluna, como ilustra a Figura 2.3 a seguir. Figura 2.3 2.2 Babilônios e Chineses 14 Dessa forma obtém-se um quadrado menor, cuja área Aq, é tal que ∼ Aq = Aoctogono; (2-2) logo, por (2-1) e (2-2), ∼ Ac = Aq: (2-3) 8 Tem-se que o lado do quadrado menor da Figura 2.3 é do diâmetro do círculo 9 construído na Figura 2.2, assim, 8 2 A = D ; (2-4) q 9 onde D é o diâmetro do círculo. Por (2-3) e (2-4) tem-se 8 2 A =∼ D : (2-5) c 9 Sendo r o raio da circunferência da Figura 2.2. A partir de (2-5) tem-se 16 2 162 A =∼ D = (r)2 : c 9 9 Fazendo as contas tem-se 162 =∼ 3;1604: 9 como aproximação, de quatro casas decimais, para p. 2.2 Babilônios e Chineses Os resultados dessa seção são comuns a essas duas civilizações. Os babilônios, e também os Chineses, estavam familiarizados com os conceitos de diâmetro, comprimento e área de um círculo e calculavam a área do círculo pela fórmula CD A = ; (2-6) 2 2 em que C representa o comprimento da circunferência que limita o círculo em questão e D seu diâmetro. Note que essa fórmula é equivalente a atual. Babilônios e Chineses usavam 2.2 Babilônios e Chineses 15 também a fórmula C2 A = ; 12 C facilmente deduzida de (2-6) substituindo D por e que comparada com a fórmula atual, 3 conduz à aproximação de p = 3, encontrada na tábua babilônica YBC7302.1 Como esses povos encontraram o resultado (2-6) CD A = ; 2 2 não sabe-se exatamente. Katz, em [2], apresenta duas possíveis justificativas: A primeira justificativa é que eles tenham considerado o corte de um círculo em setores e sua reorganização em um retângulo aproximado, como sugere a Figura (2-4) a seguir. Figura 2.4 Como já conheciam o cálculo da área de um retângulo, AR, fizeram AR = b · h C D com b = e h = , daí segue o resultado 2 2 CD A = : 2 2 Essa justificativa pode gerar atividades para serem aplicadas em sala de aula. "Essa explicação é conhecida dos professores de ensino fundamental e médio que a fazem com a utilização de material concreto. Eles tomam um círculo de 1 Os babilônicos faziam seus registros em tábuas de argila, que os historiadores colocaram nomes como esse.
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