tag:blogger.com,1999:blog-76491330332935980802024-03-13T08:51:24.470-03:00MatematicandoFrancisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.comBlogger151125tag:blogger.com,1999:blog-7649133033293598080.post-89807899672077042432019-04-01T19:29:00.002-03:002019-04-01T22:52:23.448-03:00Demonstração do volume da PirâmideNa figura abaixo, feita no software GeoGebra, vemos um prisma de base triangular, cujo volume é dado pela fórmula:<br />
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<img border="1" heigth="20" src="https://i.imgur.com/WQunWFs.png" width="100" /><br />
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Se movimentarmos o cursor da figura, veremos que o prisma se dividirá em três pirâmides de igual volume, consequentemente, cada uma delas com volume igual a 1/3 do volume do prisma que lhes deu origem, logo o volume da pirâmide vale:<br />
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<img heigth="30" src="https://i.imgur.com/MbnmyZv.png" width="150" /></div>
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<iframe height="350px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/yxjqctve/width/640/height/350/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/true/rc/false/ld/false/sdz/true/ctl/false" style="border: 0px;" title="Demonstração do volume da pirâmide" width="640px"> </iframe>
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Francisco Ismael Reis<br />
<img src="https://i.imgur.com/68WtLUu.jpg" /><br />
01/04/2019Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-79302624746412156392019-03-30T11:48:00.001-03:002019-03-30T15:56:22.665-03:00O Triângulo de Reuleaux<p><img width="114" height="179" align="right" style="margin: 0px 0px 0px 5px; float: right; display: inline;" src="https://upload.wikimedia.org/wikipedia/commons/thumb/e/e6/Franz_Reuleaux.jpg/200px-Franz_Reuleaux.jpg"><span style="font-family: "arial" , "helvetica" , sans-serif;">O <strong>Triângulo de Reuleaux</strong>, também conhecido como <strong>Triângulo esférico</strong>, recebeu esse nome em homenagem a Franz Reuleaux (1829-1905), engenheiro, cientista e professor de origem alemã, o primeiro a registrar as propriedades desse triângulo.</span><br>
<span style="font-family: "arial" , "helvetica" , sans-serif;"></span></p><p><span style="font-family: "arial" , "helvetica" , sans-serif;">Apesar do nome, o Triângulo de Reuleuax, não é propriamente um triângulo. Trata-se de uma forma geométrica plana, não circular, construída a partir de um triângulo equilátero. </span></p><p><span style="font-family: "arial" , "helvetica" , sans-serif;">Para a obtenção do Triângulo de Reuleaux, constróiem-se sobre os três lados de um triângulo equilátero, três arcos de circunferência com centro nos vértices do triângulo e raio de medida igual à medida do lado do triângulo.</span><br>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Ao se rotacionar o Triângulo de Reuleaux, seus vértices percorrem uma figura que se assemelha a um quadrado de “cantos arredondados”. </span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><br></span></div><div style="text-align: left;"><span style="font-family: "arial" , "helvetica" , sans-serif;">Na figura a seguir, construída no software Geogebra, vemos o Triângulo de Reuleaux. Movimente o cursor (ponto verde) para observar o movimento do Triângulo.</span></div>
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<iframe width="516" height="450" src="https://tube.geogebra.org/material/iframe/id/uHddRsKm/width/516/height/450/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/true/ld/false/sri/true/at/auto" scrolling="no" style="border: 0px currentcolor; border-image: none;"> </iframe><br></div>
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<span style="color: rgb(77, 77, 77); font-family: "uoltext" , "arial" , sans-serif; font-size: 18px;"><font size="3">Uma aplicação bastante curiosa do Triângulo de Reuleaux foi desenvolvida pelo engenheiro inglês Harry James Watt, que, em 1914, aproveitando as propriedades da curva, concebeu uma broca de furadeira com eixo flexível para fazer furos na forma aproximada de um quadrado.</font> </span><br>
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<img style="border: 2px solid rgb(255, 204, 102); border-image: none;" src="https://hypescience.com/wp-content/uploads/2014/08/conceitos-de-ciencia-gifs-2.gif"></div>
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Francisco Ismael Reis</div>
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30/03/2019Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-35129838654915274032019-03-29T19:30:00.000-03:002019-03-29T21:24:32.941-03:00A Cardioide<div style="text-align: justify;">
<span style="font-family: "arial" , "helvetica" , sans-serif;">A <b><i>Cardióde</i></b> é uma curva matemática que se assemelha a um coração </span><span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="background-color: white; color: #333333; font-size: 14px;">(o seu nome deriva da palavra grega </span><em style="background-color: white; box-sizing: border-box; color: #333333; font-size: 14px;">kardia</em><span style="background-color: white; color: #333333; font-size: 14px;"> + </span><em style="background-color: white; box-sizing: border-box; color: #333333; font-size: 14px;">eidos</em><span style="background-color: white; color: #333333; font-size: 14px;"> que significa coração + forma)</span></span><span style="font-family: "arial" , "helvetica" , sans-serif;">. Foi apresentada pela primeira vez por Johann Castillon (1708-1791), matemático e astrônomo italiano, em 1741.</span><br />
<span style="font-family: "arial" , "helvetica" , sans-serif;">Sua equação cartesiana é dada por:</span></div>
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<a href="https://imgur.com/g8CT2dO"><img src="https://i.imgur.com/g8CT2dO.png" title="source: imgur.com" /></a>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">A figura abaixo, construída no software GeoGebra, mostra o gráfico da equação acima. Movimente o cursor para fazer a Cardioide mudar de tamanho e de posição. </span><br />
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<iframe height="350px" scrolling="no" src="https://www.geogebra.org/material/iframe/id/ywpf2sz9/width/650/height/350/border/888888/sfsb/true/smb/false/stb/false/stbh/false/ai/false/asb/false/sri/false/rc/false/ld/false/sdz/true/ctl/false" style="border: 0px;" title="Cardióide" width="650px"> </iframe><br />
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Francisco Ismael Reis</span><br />
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29/03/2019Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com1tag:blogger.com,1999:blog-7649133033293598080.post-81015152700137065142019-03-27T18:05:00.001-03:002019-03-28T19:27:26.200-03:00A Catenária<p><img style="margin: 0px 0px 0px 1px; float: right; display: inline;" alt="Resultado de imagem para a catenária" src="data:image/jpeg;base64,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" align="right" />Uma corda, ou uma corrente, suspensa por suas extremidades, presa a dois pontos fixos e sob influência exclusiva da gravidade, gera uma curva parecida com uma parábola, como conjecturou Galileu Galilei (1564-1642). </p> <p>Christiaan Huygens (1629-1695), físico, matemático e astrônomo holandês, mostrou em 1646, aos 17 anos de idade, que a conjectura feita por Galileu era falsa. <br /> A essa curva, que se parece com uma parábola,mas que não é uma parábola, foi dado o nome de<em><strong> catenária</strong></em>, termo que deriva da palavra latina catena, que significa “cadeia” ou “corrente”. <br />A equação matemática que descreve a catenária é dada pela seguinte função hiperbólica: <br /></p> <p><img style="margin-right: auto; margin-left: auto; float: none; display: block;" src="https://i.imgur.com/bLlH8sC.png" /> <br /> que tem como equivalente exponencial a função: <br /><img style="margin-right: auto; margin-left: auto; float: none; display: block;" src="https://i.imgur.com/6y4kqs7.png" /> <br />que define uma família de curvas, entre as quais vemos algumas representadas no gráfico abaixo. <br /><img style="margin-right: auto; margin-left: auto; float: none; display: block;" alt="Catenarias para diferentes valores do parametro" src="https://upload.wikimedia.org/wikipedia/commons/thumb/6/6b/Catenary-pm.png/400px-Catenary-pm.png" /> <br /><strong>A catenária no nosso cotidiano</strong></p> <p>A catenária está muito presente em nosso cotidiano. Ela aparece com certa frequência nas construções civis dando forma a arcos e a certas casas construidas de pedra, na construção de pontes pênsis e até mesmo no formato adquirido pela barriga de uma mulher grávida.</p> <p><img style="margin-right: auto; margin-left: auto; float: none; display: block;" alt="Resultado de imagem para barriga de mulher grávida" src="data:image/jpeg;base64,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" width="412" height="274" /></p> <p>Veja os exemplos a seguir:</p> <p><img style="margin-right: auto; margin-left: auto; float: none; display: block;" alt="Resultado de imagem para Gateway Arch" src="data:image/jpeg;base64,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" width="411" height="308" /></p> <p> <br /> O Gateway Arch, também conhecido como Portal para o Oeste ou Portal em Arco, está situado na cidade de Saint Louis, Missouri, Estados Unidos.</p> <hr /> <br /> <p><img style="margin-right: auto; margin-left: auto; float: none; display: block;" alt="Resultado de imagem para clochan" 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" width="410" height="307" /> <br /> O Clochán Irlandês, uma construção típica irlandesa feita de pedras.</p> <hr /> <br /> <p><img style="margin-right: auto; margin-left: auto; float: none; display: block;" alt="Resultado de imagem para a catenária na natureza" 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" width="406" height="319" /> <br /> Muitos tuneis apresentam o formato de catenária.</p> <hr /> <br /> <p><img style="margin-right: auto; margin-left: auto; float: none; display: block;" src="http://hotsite.tvescola.org.br/matematica-em-toda-parte-2/fasciculos/img/transporte/fasciculo_transporte_imagem-3.png" width="412" height="251" /> <br /> Assim como diversas pontes pênseis.</p> <hr /> <br /> <p><img style="margin-right: auto; margin-left: auto; float: none; display: block;" alt="Resultado de imagem para redes de alta tensão e catenária" 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" width="414" height="243" /> <br /> E redes de transmissão.</p> <hr /> <br /> Francisco Ismael Reis <br /><img src="https://i.imgur.com/68WtLUu.jpg" /> <br /> 27/03/2019 Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-24079346433589222372019-03-26T17:30:00.001-03:002019-03-29T17:34:11.048-03:00Pitágoras<blockquote> <p><img align="right" style="margin: 0px 0px 0px 9px; float: right; display: inline;" alt="Resultado de imagem para pitágoras" 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">“Pitágoras nasceu em 582 a.C., em Samos. Era filho de Mnesarcos que contratou grandes homens da época, como Ferecides e Hermodamas para ensiná-lo. Pitágoras aprendeu rapidamente e, em pouco tempo, deixou para trás seus mestres matemáticos e filósofos; com dezoito anos participou dos jogos olímpicos como pugilista e ganhou todas as competições que disputou. Com mais ou menos dezenove anos, iniciou uma viagem em direção ao oriente; dirigiu-se primeiro à Babilônia onde esteve em contato com os sábios da terra, homens que pertenciam a uma raça já antiga em cultura; depois dirigiu-se à Índia onde encontrou Ciência e também a Filosofia budista, que o influenciou pelo resto de sua vida. No Egito, esteve em contato com sacerdotes do Nilo. Quando regressou à Grécia já estava com 53 anos e tornou-se líder de um movimento religioso por toda a Grécia, movimento este que tinha como base o misticismo, ritos de abstinência, pureza e meditações. Seus adeptos usavam roupas que os distinguiam das pessoas comuns. A escola pitagórica durou cerca de 150 anos, e a vida mística dos pitagóricos não impediu de fundarem a <em>Aritmética</em>, ou a <em>Ciência dos Números</em>.</p> </blockquote> <blockquote> <p>Como astrônomo, foi o primeiro a afirmar que a Terra era redonda e estava suspensa no espaço, assim como os planetas giravam em torno de uma chama central à qual dava o nome de <em>Héstia</em>, que não era o Sol; este por sua vez recebia a luz de <em>Héstia</em> e a refletia. O movimento dos planetas produzia sons que nós não ouvíamos, porque estamos acostumados com eles. Com essa forma de pensar o seu sistema foi o melhor que se conheceu até Copérnico.</p> </blockquote> <blockquote> <p>Obcecado por números, nos deixou a célebre frase: todas as coisas são números. Achava que o Universo era uma escala musical e ao número 1 (um) atribuía a própria essência da vida, pois tudo vem de um deus que é onipotente, onipresente e onisciente e é simplesmente um. Com sua teoria dos números, desenvolveu o estudo das notas musicais que o levou a descobrir que cordas em vibração emitem sons que dependem de seus comprimentos. <img align="left" style="margin: 0px 5px 0px 0px; float: left; display: inline;" alt="cordas-Pitágoras" src="https://escutecomigo.files.wordpress.com/2013/03/cordas-pitc3a1goras.jpg?w=529">Pitágoras construiu um instrumento de cordas para ensaiar e demonstrar essa teoria, acredita-se ter sido esse o primeiro instrumento de Física que existiu, em outras palavras, talvez tenha sido o primeiro experimento na história, onde a Matemática foi relacionada com a Natureza, assim como a primeira tentativa de equacionar um fenômeno natural. Ao tomar conhecimento de que as notas são obtidas por divisões da corda por números inteiros, atribuíram ao fato o significado de uma força mística. A relação entre os números e a Natureza era tão perfeita que não só os sons naturais, mas todos os eventos deveriam ser números representados por harmonias; até as órbitas dos corpos celestes estariam relacionadas aos intervalos musicais, assim os movimentos celestes representavam a Música das esferas.</p> </blockquote> <blockquote> <p><img width="284" height="154" align="left" style="margin: 0px 5px 0px 0px; float: left; display: inline;" alt="Monedas griegas con el Pentagrama pitagórico halladas en Metaponto (450 a.C)." src="https://virtual.uptc.edu.co/ova/estadistica/docs/autores/pag/mat/PitagDivProp3.gif">Em sua sociedade secreta, os pitagóricos tinham o pentagrama, ou a estrela de cinco pontas, como emblema; não era permitida a entrada de mulheres e eles passavam os dias buscando relações matemáticas para explicar a harmonia do mundo. Também toda e qualquer descoberta feita deveria ser atribuída ao mestre e jamais revelada a uma pessoa estranha; diz-se que foi Pitágoras (ou sua sociedade) o responsável pela introdução do sistema de pesos e medidas na Grécia.</p> </blockquote> <blockquote> <p>A cidade de Milos, palco principal dos pitagóricos, quando de uma revolução mais ou menos em 501 a.C., teve o governo derrubado, o palácio incendiado e vários pitagóricos foram assassinados; o próprio Pitágoras fugiu para Tarento, de onde se dirigiu para Metaponto e aí perdeu a vida numa revolta em 501 a.C. Estudos sobre Aritmética, Geometria, Música e Astronomia eram considerados fundamentais para os pitagóricos. Foi Pitágoras quem classificou a Matemática em Aritmética ou números absolutos, Música em números aplicados, Geometria em grandezas em repouso e Astronomia em grandezas em movimento,</p> </blockquote> <blockquote> <p>Assim como Cristo e Sócrates, Pitágoras não escreveu qualquer trabalho que tenha chegado até nós, logo o que sabemos tem origem em três fontes:</p> <ol> <li>Os escritos de seus discípulos e seguidores como Nicomano, autor de <em>Introdução à aritmética</em>, onde retrata os ensinamentos dos pitagóricos.</li> <li>Nas obras de Platão que, por sua vez, foi muito influenciado pelos seguidores de Pitágoras.</li> <li>Nas obras de Aristóteles que cita e resume o pensamento pitagórico, embora muitas vezes discorde do mesmo.</li> </ol> <p>A biografia de Pitágoras só foi escrita por volta de ano 300 d.C. por Diógenes Laércio, portanto muita coisa é desconhecida ou lendária.</p> <p>O homem Pitágoras morreu, mas seus ensinamentos continuaram e seu trabalho foi descrito por Filolau de Tebas cerca de cem anos depois de sua morte”</p> </blockquote> <p>Texto extraído do livro <strong><em>A Matemática na arte e na vida</em></strong> de <strong><em>Paulo Roberto Martins Contador</em></strong> – Editora Livraria da Física – 2ª edição – 2011 – Pág. 51 a 53.</p> <p>Uma leitura mais que recomendada e obrigatória para todos os amantes da Matemática e para os que buscam desvendar o fascínio provocado pela <em>Divina Proporção</em> ou <em>Proporção Áurea,</em> aqui explicada com muita clareza pelo autor.</p> <p><img style="margin-right: auto; margin-left: auto; float: none; display: block;" src="https://lh5.googleusercontent.com/-jPcHSThAGVg/TYOPd_jupqI/AAAAAAAAAC4/CX07ojF6U4w/s640/CAPA+2+edi%25C3%25A7%25C3%25A3o.jpg"></p>Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-22594749675092824152019-03-25T17:18:00.001-03:002019-03-25T22:37:35.680-03:00A espiral áureaA <strong>Sequência de Fibonacci</strong> é uma sucessão de números inteiros proposta pelo matemático italiano Leonardo Pisa (1170-1250), mais conhecido como Fibonacci: Essa sequência tem seus dois primeiros termos definidos por 1 e 1 sendo cada um dos demais obtidos por meio da soma dos seus dois antecessores. Assim sendo temos:<br />
<center>
<center>
</center>
<table align="center" bgcolor="#cef6ce" border="1" cellpadding="2" cellspacing="0" style="height: 270px; width: 204px;" table="table"><tbody>
<tr> <td align="center" valign="top">1º termo</td> <td align="center" valign="top"></td> <td align="center" valign="top">1</td> </tr>
<tr> <td align="center" valign="top">2º termo</td> <td align="center" valign="top"></td> <td align="center" valign="top">1</td> </tr>
<tr> <td align="center" valign="top">3º termo</td> <td align="center" valign="top">1+1</td> <td align="center" valign="top">2</td> </tr>
<tr> <td align="center" valign="top">4º termo</td> <td align="center" valign="top">2+1</td> <td align="center" valign="top">3</td> </tr>
<tr> <td align="center" valign="top">5º termo</td> <td align="center" valign="top">3+2</td> <td align="center" valign="top">5</td> </tr>
<tr> <td align="center" valign="top">6º termo</td> <td align="center" valign="top">5+3</td> <td align="center" valign="top">8</td> </tr>
<tr> <td align="center" valign="top">7º termo</td> <td align="center" valign="top">8+5</td> <td align="center" valign="top">13</td> </tr>
<tr> <td align="center" valign="top">8º termo</td> <td align="center" valign="top">13+8</td> <td align="center" valign="top">21</td> </tr>
<tr> <td align="center" valign="top">9º termo</td> <td align="center" valign="top">21+13</td> <td align="center" valign="top">34</td> </tr>
<tr> <td align="center" valign="top">10º termo </td> <td align="center" valign="top">34+21</td> <td align="center" valign="top">55</td> </tr>
<tr> <td align="center" valign="top">11º termo </td> <td align="center" valign="top">55+34</td> <td align="center" valign="top">89</td> </tr>
<tr> <td align="center" valign="top">12º termo <br />
<strong>…</strong> </td> <td align="center" valign="top">89+55 <br />
<strong>…</strong></td> <td align="center" valign="top">144 <br />
<strong>…</strong></td> </tr>
</tbody></table>
</center>
Ou:<br />
<blockquote>
<span style="color: red; font-size: small;"><strong><em>1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …</em></strong></span></blockquote>
A partir dos valores dessa sequência, podemos construir vários retângulos, chamados de <strong>retângulos áureos.</strong><br />
Começamos com um quadrado de lado 1. Ao seu lado colocamos outro quadrado de lado 1, ao lado desses dois quadrados colocamos outro, agora de lado 2. Ao lado do retângulo obtido com a construção anterior, colocamos um quadrado de lado 3, obtendo um novo retângulo, ao lado do qual colocamos outro quadrado, agora de lado 5, e assim indefinidamente.<br />
Veja essa construção na animação apresentada abaixo, feita para os primeiros sete elementos da sequência de Fibonacci. <br />
<center>
<table border="1" cellpadding="2" cellspacing="0"><tbody>
<tr> <td valign="top"><img alt="espiral2" src="http://clubes.obmep.org.br/blog/wp-content/uploads/2015/05/espiral2.gif" height="433" style="border-image: none;" width="606" /><br />
<span style="font-size: xx-small;">Fonte:</span> <a href="http://clubes.obmep.org.br/blog/atividade-a-razao-aurea-uma-construcao-da-espiral-aurea/"><span style="font-size: xx-small;">http://clubes.obmep.org.br/blog/atividade-a-razao-aurea-uma-construcao-da-espiral-aurea/</span></a> </td> </tr>
</tbody></table>
</center>
Se dentro de cada quadrado, desenharmos arcos de circunferência de raio igual ao lado do respectivo quadrado e unirmos os arcos assim obtidos, formaremos uma curva, uma espiral chamada de espiral áurea, também conhecida como espiral logarítmica.<br />
<img height="313" src="https://i.imgur.com/VZCxP0F.png" style="display: block; float: none; margin-left: auto; margin-right: auto;" width="497" /><br />
A espiral logarítmica aparece em diversas situações da natureza. Veja algumas delas.<br />
<img alt="Resultado de imagem para espiral logarÃtmica na natureza" height="384" 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" 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Francisco Ismael Reis<br />
<img src="https://i.imgur.com/68WtLUu.jpg" /><br />
25/03/2019Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-32603664404278028842019-03-19T18:10:00.001-03:002019-03-25T14:49:39.825-03:00O Pentagrama e a quantidade de triângulos<p>O <em><strong>Pentagrama</strong></em> ou <strong><em>Estrela de Cinco Pontas</em></strong>, para a Matemática, é uma figura geométrica de cinco pontas que tem a sua origem com Pitágoras, sendo usado por seus seguidores como um talismã. As razões dos segmentos lineares num pentagrama são todas iguais a PHI (<b>φ=1,618…)</b>, o que faz deste símbolo a expressão perfeita da <em><strong>Proporção Divina</strong></em>. O Pentagrama pode ser construído fazendo-se coincidir os seus vértices com os de um pentágono regular, conforme vemos na figura abaixo:</p> <p><img style="border: 0px currentcolor; border-image: none; margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" border="0" src="https://i.imgur.com/b6Ja1y7.png" />O Pentagrama possui inúmeras representações e vários significados, entre eles, é para muitos um sinal de força e energia. Já, para os exotéricos, representa os cinco elementos: a terra, o ar, a água, o fogo e o espírito. </p> <p>A questão que se propõe é a seguinte:</p> <blockquote> <p><em><font color="#0000ff">Olhando para a figura acima, você seria capaz de dizer quantos triângulos existem nela?</font></em></p> </blockquote> <p>Dica: São mais de  vinte, mas menos de cinquenta triângulos.</p> <p>Francisco Ismael Reis</p> <p><img src="https://i.imgur.com/68WtLUu.jpg" /></p> <p>19/03/2019</p>Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com12tag:blogger.com,1999:blog-7649133033293598080.post-58235547878499908152019-03-17T14:05:00.002-03:002019-03-22T20:43:12.389-03:00Fluxograma<span style="background-color: white; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; font-size: 16px; text-align: justify;">Um </span><b style="background-color: white; font-family: "Lucida Grande", Verdana, Arial, sans-serif; font-size: 16px; text-align: justify;">fluxograma</b><span style="background-color: white; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; font-size: 16px; text-align: justify;"> é um diagrama que tem como finalidade representar processos ou algoritmos (uma sequência finita de regras, raciocínios ou operações que, aplicada a um número finito de dados, permite solucionar classes semelhantes de problemas). No fluxograma é criado, com o auxílio de formas geométricas pré-estabelecidas, o mapa visual onde podemos mostrar as diversas etapas de um sistema, tornando o conjunto das etapas um trabalho padronizado, organizado, que possa ser entendido e utilizado por qualquer pessoa. </span><br />
<span style="background-color: white; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; font-size: 16px; text-align: justify;">Veja, a seguir, o fluxograma correspondente ao desenvolvimento dos conteúdos da Probabilidade trabalhados no ensino médio.</span><br />
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<span style="background-color: white; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; font-size: 16px; text-align: justify;">São inúmeras as vantagens em se utilizar um fluxograma no processo de aprendizagem de um determinado conteúdo. Podemos destacar, entre muitas que existem, as seguintes:</span><br />
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<ul>
<li><span style="background-color: white; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; text-align: justify;">Melhora e facilita a compreensão do conteúdo que se está estudando;</span></li>
<li><span style="background-color: white; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; text-align: justify;">Mostra as etapas necessárias para a realização estudo;</span></li>
<li><span style="background-color: white; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; text-align: justify;">Facilita a elaboração das atividades e tarefas.</span></li>
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<span style="background-color: white; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; font-size: 16px; text-align: justify;"><br /></span>
<span style="background-color: white; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; font-size: 16px; text-align: justify;">Francisco Ismael Reis</span><br />
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<a href="https://3.bp.blogspot.com/-B-OCBLeF6Ok/XI594c1BsXI/AAAAAAAADl8/85Gmr3ufyG0gD3g5RQcVkD1PG4G8rWmcwCLcBGAs/s1600/Assinatura.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="57" data-original-width="150" src="https://3.bp.blogspot.com/-B-OCBLeF6Ok/XI594c1BsXI/AAAAAAAADl8/85Gmr3ufyG0gD3g5RQcVkD1PG4G8rWmcwCLcBGAs/s1600/Assinatura.jpg" /></a><span style="clear: left; color: black; float: left; font-family: "arial" , "helvetica" , sans-serif; margin-bottom: 1em; margin-right: 1em;">17/03/2018</span></div>
<span style="background-color: white; font-family: "lucida grande" , "verdana" , "arial" , sans-serif; font-size: 16px; text-align: justify;"><br /></span>Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-44863906420239999612019-03-16T12:22:00.001-03:002019-03-23T18:27:56.702-03:00O problema dos nove copos<span style="font-family: "arial" , "helvetica" , sans-serif;">Sobre uma mesa estão colocados enfileirados nove copos, conforme mostra a figura:</span><br>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://3.bp.blogspot.com/-pRvSOiII6UQ/XI194ab9JdI/AAAAAAAADe4/Kg6srquStDATAp_rHWMzUZvyFb1lal6tgCLcBGAs/s1600/Copos1.jpg" imageanchor="1"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img src="https://3.bp.blogspot.com/-pRvSOiII6UQ/XI194ab9JdI/AAAAAAAADe4/Kg6srquStDATAp_rHWMzUZvyFb1lal6tgCLcBGAs/s1600/Copos1.jpg" border="0" data-original-width="270" data-original-height="72"></span></a></div>
<span style="font-family: "arial" , "helvetica" , sans-serif;">Inverte-se, sucessivamente e um de cada vez, a posição de seis desses copos. Um copo invertido ficará ao contrário se antes estava direito, e vice-versa.</span><br>
<span style="font-family: "arial" , "helvetica" , sans-serif;">Num primeiro movimento, inverte-se a posição do seis últimos copos que ficam assim distribuídos:</span><br>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://2.bp.blogspot.com/-IrkNWjLB3rA/XI1-suYQi3I/AAAAAAAADfE/L12rDikZVlsKyUN0TmLMrCNmSMl8ykNnwCLcBGAs/s1600/Copos2.jpg" imageanchor="1"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img src="https://2.bp.blogspot.com/-IrkNWjLB3rA/XI1-suYQi3I/AAAAAAAADfE/L12rDikZVlsKyUN0TmLMrCNmSMl8ykNnwCLcBGAs/s1600/Copos2.jpg" border="0" data-original-width="266" data-original-height="71"></span></a></div>
<span style="font-family: "arial" , "helvetica" , sans-serif;">Num segundo movimento, inverte-se a posição dos seis primeiros:</span><br>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://4.bp.blogspot.com/-6AvXMdf18v4/XI1_D9ddTHI/AAAAAAAADfM/uFn2DCHSrOw8jKxQy-5Ia8cY6gnjh0AMwCLcBGAs/s1600/Copos3.jpg" imageanchor="1"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img src="https://4.bp.blogspot.com/-6AvXMdf18v4/XI1_D9ddTHI/AAAAAAAADfM/uFn2DCHSrOw8jKxQy-5Ia8cY6gnjh0AMwCLcBGAs/s1600/Copos3.jpg" border="0" data-original-width="266" data-original-height="70"></span></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Num terceiro movimento, invertem-se seis copos a partir do segundo:</span><br>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://4.bp.blogspot.com/-L3RrzI5M4pk/XI1_uVA-cAI/AAAAAAAADfY/8U11ZbLZ5-ovABpHwz3qhcpMirwHb2VXACLcBGAs/s1600/Copos4.jpg" imageanchor="1"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img src="https://4.bp.blogspot.com/-L3RrzI5M4pk/XI1_uVA-cAI/AAAAAAAADfY/8U11ZbLZ5-ovABpHwz3qhcpMirwHb2VXACLcBGAs/s1600/Copos4.jpg" border="0" data-original-width="268" data-original-height="70"></span></a></div>
<span style="font-family: "arial" , "helvetica" , sans-serif;">Num quarto movimento, vamos inverter seis copos, a partir do terceiro:</span><br>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://3.bp.blogspot.com/-nvF3hO1iulw/XI2AFmdev5I/AAAAAAAADfg/dJobbjO0xXQ-BJWJEZ0C1X356NYMdovsACLcBGAs/s1600/Copos5.jpg" imageanchor="1"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img src="https://3.bp.blogspot.com/-nvF3hO1iulw/XI2AFmdev5I/AAAAAAAADfg/dJobbjO0xXQ-BJWJEZ0C1X356NYMdovsACLcBGAs/s1600/Copos5.jpg" border="0" data-original-width="266" data-original-height="72"></span></a></div><p>
<span style="font-family: "arial" , "helvetica" , sans-serif;">E assim, sucessivamente.</span><br>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">A questão é
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><font color="#0000ff"><em>Quantos movimentos sucessivos são necessários para que todos os nove copos fiquem na posição contrária àquela em que se encontravam originalmente?</em></font></span></blockquote>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Solução:</span><br>
<span style="font-family: "arial" , "helvetica" , sans-serif;">Vamos substituir por +1 os copos que estão com a boca voltada para cima (direitos) e por –1 os que estão com a boca voltada para baixo (invertidos) e multiplicar os valores assim obtidos.</span><br>
<span style="font-family: "arial" , "helvetica" , sans-serif;">De inicio temos:</span><br>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://1.bp.blogspot.com/-0H2icgEyHQg/XI2AzV63YMI/AAAAAAAADfs/Gm5M13t-c7gdciRx54NMyyaJ2xvVImjAwCLcBGAs/s1600/Produto1.jpg" imageanchor="1"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img src="https://1.bp.blogspot.com/-0H2icgEyHQg/XI2AzV63YMI/AAAAAAAADfs/Gm5M13t-c7gdciRx54NMyyaJ2xvVImjAwCLcBGAs/s1600/Produto1.jpg" border="0" data-original-width="414" data-original-height="32"></span></a></div>
<span style="font-family: "arial" , "helvetica" , sans-serif;">Após o primeiro movimento:</span><br>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://4.bp.blogspot.com/-0cOhAR4D7cI/XI2CYpBCokI/AAAAAAAADf4/t4mpnfLrB6gEMtzDPeiYWehg8Tm-5PfVgCLcBGAs/s1600/Produto2.jpg" imageanchor="1"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img src="https://4.bp.blogspot.com/-0cOhAR4D7cI/XI2CYpBCokI/AAAAAAAADf4/t4mpnfLrB6gEMtzDPeiYWehg8Tm-5PfVgCLcBGAs/s1600/Produto2.jpg" border="0" data-original-width="412" data-original-height="31"></span></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Após o segundo movimento teremos:</span></div>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://1.bp.blogspot.com/-s27Od_tEAYk/XI2C_rcPvUI/AAAAAAAADgA/SO1yQzHAKzw45UBaGlbpZfpElG9y4ZeGACLcBGAs/s1600/Produto3.jpg" imageanchor="1"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img src="https://1.bp.blogspot.com/-s27Od_tEAYk/XI2C_rcPvUI/AAAAAAAADgA/SO1yQzHAKzw45UBaGlbpZfpElG9y4ZeGACLcBGAs/s1600/Produto3.jpg" border="0" data-original-width="415" data-original-height="32"></span></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Após o terceiro:</span></div>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://3.bp.blogspot.com/-TaF8b-2AW3E/XI2DXKurFqI/AAAAAAAADgM/AKAilSSHv_QGI--RJ5FP_8ShvM4DtQFAgCLcBGAs/s1600/Produto4.jpg" imageanchor="1"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img src="https://3.bp.blogspot.com/-TaF8b-2AW3E/XI2DXKurFqI/AAAAAAAADgM/AKAilSSHv_QGI--RJ5FP_8ShvM4DtQFAgCLcBGAs/s1600/Produto4.jpg" border="0" data-original-width="420" data-original-height="30"></span></a></div>
<span style="font-family: "arial" , "helvetica" , sans-serif;">E após o quarto:</span><br>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://1.bp.blogspot.com/-BKq3EHAYCz0/XI2DtSxw3EI/AAAAAAAADgU/uM4t2ohsNrAwZuifnydxlzh6OtBv3_-XgCLcBGAs/s1600/Produto5.jpg" imageanchor="1"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img src="https://1.bp.blogspot.com/-BKq3EHAYCz0/XI2DtSxw3EI/AAAAAAAADgU/uM4t2ohsNrAwZuifnydxlzh6OtBv3_-XgCLcBGAs/s1600/Produto5.jpg" border="0" data-original-width="415" data-original-height="28"></span></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;"><span style="text-align: center;">Observe que, após a inversão de seis dos nove copos, o produto, em qualquer situação é sempre +1, isto porque </span><span style="text-align: center;">em cada movimento mudamos os sinais de uma quantidade par de números.</span></span></div>
<span style="font-family: "arial" , "helvetica" , sans-serif;">Supondo que todos os copos estivessem invertidos, teríamos a seguinte situação:</span><br>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://4.bp.blogspot.com/-QpO_vWDPORU/XI2EhRCvATI/AAAAAAAADgg/ELbxZbI6ZX4yi0_5pUsdI6mnIX4t-eq3gCLcBGAs/s1600/Produto6.jpg" imageanchor="1"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img src="https://4.bp.blogspot.com/-QpO_vWDPORU/XI2EhRCvATI/AAAAAAAADgg/ELbxZbI6ZX4yi0_5pUsdI6mnIX4t-eq3gCLcBGAs/s1600/Produto6.jpg" border="0" data-original-width="410" data-original-height="29"></span></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Algo que, como mostramos anteriormente, nunca ocorrerá, uma vez que, após a inversão dos seis primeiros copos, o produto é sempre igual a +1. Dessa maneira, o problema não tem solução.</span></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">Francisco Ismael Reis</span></div>
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<a style="clear: left; margin-right: 1em; margin-bottom: 1em; float: left;" href="https://3.bp.blogspot.com/-MTHwtqwLuHA/XI19kL0CcNI/AAAAAAAADew/MKRrgjeqEw4Y06hY9E-d3n9hON0RyOrsgCLcBGAs/s1600/Assinatura.jpg" imageanchor="1"><span style="font-family: "arial" , "helvetica" , sans-serif;"><img src="https://3.bp.blogspot.com/-MTHwtqwLuHA/XI19kL0CcNI/AAAAAAAADew/MKRrgjeqEw4Y06hY9E-d3n9hON0RyOrsgCLcBGAs/s1600/Assinatura.jpg" border="0" data-original-width="150" data-original-height="57"></span></a></div>
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<span style="font-family: "arial" , "helvetica" , sans-serif;">16/03/2019</span></div>Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-21362678136145258312019-03-12T10:06:00.001-03:002019-03-12T10:06:51.714-03:00Como a Índia revolucionou a matemática séculos antes do Ocidente<p align="right"><a href="https://www.bbc.com/portuguese/geral-47487130" target="_blank">Marcus du Sautoy BBC,</a> </p><p align="right">"A História da Matemática"</p><h4>À frente do tempo, os matemáticos indianos inventaram o número zero, mediram a circunferência da Terra e criaram o infinito matemático</h4><p><a href="https://lh3.googleusercontent.com/-QHNyGiemcgk/XIeuQLO_agI/AAAAAAAADcQ/ALxIEyEFvLUQKAMzNMxiItlyhysTGAIFQCHMYCw/s1600-h/image%255B5%255D"><img width="613" height="396" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-lzmXT3SKBEs/XIeuST1rLpI/AAAAAAAADcU/_sFhfSl9XCgQ6n1Y2P2hTECapLCBSWOIACHMYCw/image_thumb%255B3%255D?imgmax=800" border="0"></a></p><p>Assim como a China, a Índia descobriu os benefícios de um sistema decimal há muito tempo.<p>Eles, inclusive, já o utilizavam no século 3.<p>É o mesmo sistema que usamos atualmente, em que a posição numérica indica unidades, dezenas, centenas, milhares e assim por diante.<p>Nós não sabemos direito como chegaram ao sistema decimal, mas os indianos certamente o aperfeiçoaram, estabelecendo as bases para os números de um a nove que são usados em todo o mundo.<p>Eles também inventaram um novo número: zero.<h4>Nada além de um espaço vazio</h4><p>Seu primeiro uso conhecido data do século 9, mas é provável que tenha sido usado centenas de anos antes disso.<p>Este estranho número está gravado nas paredes de um pequeno templo dentro da fortaleza de Gwalior, na Índia Central.<p><a href="https://lh3.googleusercontent.com/-3tlZcOzb8xY/XIeuUNWTQ1I/AAAAAAAADcY/htIZ2dt20-oV5b3D5NI3txuaAnD999hxgCHMYCw/s1600-h/image%255B10%255D"><img width="615" height="411" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-zwW_OC3OELc/XIeuV6ztaBI/AAAAAAAADcc/gMXHf3Hy_NcmO89m5eCmWb1Br_vCCiaOwCHMYCw/image_thumb%255B6%255D?imgmax=800" border="0"></a><p>Tornou-se um lugar cultuado por matemáticos, porque ele hospeda o "início" do número zero.<p><a href="https://lh3.googleusercontent.com/-1HcUfBK1saU/XIeuXwIJ_iI/AAAAAAAADcg/oakso6ywrtcgw9zf6CgJFLW-0XDf6HSrACHMYCw/s1600-h/image%255B15%255D"><img width="615" height="397" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-8fWDZtKKZLU/XIeuZ4cC2PI/AAAAAAAADck/G7YX7WGo6k0uYZS2dKREUHNhfFizS5H_QCHMYCw/image_thumb%255B9%255D?imgmax=800" border="0"></a><p>No antigo Egito, na Mesopotâmia e na China, havia um zero, mas como marcador, um espaço vazio.<p>Foram os indianos que o transformaram em número de fato, um salto conceitual que revolucionou a matemática.<p>Desde então, tornou-se possível formar números de tamanhos astronômicos e de uma maneira muito eficiente.<p><a href="https://lh3.googleusercontent.com/-9IMpyViSE7Y/XIeubqE8PiI/AAAAAAAADco/w28tuw3vSTs-Kl81Mg8b-yNKid7TwSEKQCHMYCw/s1600-h/image%255B20%255D"><img width="595" height="383" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-gdxQE7JKito/XIeuelz2NUI/AAAAAAAADcs/o4CVM-YtZo4OpkfOItWXrLhFYqta60cdgCHMYCw/image_thumb%255B12%255D?imgmax=800" border="0"></a><h4>Como se chegou ao zero?</h4><p>Nós nunca saberemos com certeza, mas é possível que a ideia e o símbolo usados para o zero tenham se originado de cálculos feitos no chão com pedras.<p>Quando as pedras foram retiradas, um recuo redondo permaneceu no lugar, representando o movimento de algo para nada.<p>Mas também pode ter havido uma razão cultural para a invenção desse número.<p>Os conceitos de nada e de eternidade faziam parte do sistema de crenças dos antigos índios.<p>Tanto a religião budista quanto a hindu adotam o conceito de nada como parte de seus ensinamentos.<p>Logo, não deve ter sido algo tão surpreendente que uma cultura que abraçava entusiasticamente o nada pudesse acomodar a noção de um zero.<p><a href="https://lh3.googleusercontent.com/-XCInYO4nfiQ/XIeufvGI2dI/AAAAAAAADcw/PZCRK8ixCt8_fKCse3tsAknpZxf8QohEACHMYCw/s1600-h/image%255B25%255D"><img width="608" height="407" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-F-VZHsPiGIw/XIeuhHq8eCI/AAAAAAAADc0/SzLmbSuh2WIY-ZZbALfJcrs4O74uPCNQwCHMYCw/image_thumb%255B15%255D?imgmax=800" border="0"></a><p>Os indianos até usaram a palavra <i>shunya</i>, que representa a ideia filosófica do vazio, para representar o novo termo matemático.<h4>Do zero ao infinito</h4><p>Brahmagupta, famoso matemático indiano do século 7, demonstrou algumas das propriedades essenciais do zero.<p>Suas regras básicas para cálculos envolvendo zero ainda são ensinadas em escolas de todo o mundo.<p><strong>1 + 0 = 1</strong><p><strong>1 - 0 = 1</strong><p><strong>1 x 0 = 0</strong><p>Mas Brahmagupta encontrou mais quando tentou dividir 1 por 0.<p>Qual número multiplicado por zero é igual a um?<p>A solução exigiu um novo conceito matemático: o infinito.<p><a href="https://lh3.googleusercontent.com/-E92A3TP8x6o/XIeuibnn_jI/AAAAAAAADc8/QBJF6xVfuBIz57FaIiXKo80Ye9p8sVRBACHMYCw/s1600-h/image%255B30%255D"><img width="585" height="387" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-BiJDTrNbs6w/XIeuj3hDH2I/AAAAAAAADdA/GUTvct9gbn83rxqJYPtTSNE1ZJDbkfx4wCHMYCw/image_thumb%255B18%255D?imgmax=800" border="0"></a><p>Só isso daria sentido a divisões por zero.<p>E esse avanço também foi obra de um matemático indiano - Bhaskara, que o inventou no século 12.<h4>Como assim?</h4><p>Se você pegar uma fruta e cortá-la ao meio, você terá dois pedaços.<p>Se você cortar em três pedaços, terá três.<p>Outras divisões resultarão em frações cada vez menores, com mais e mais peças.<p>Eventualmente, você teria infinitas partes.<p>Bhaskara, então, raciocinou que um dividido por zero é igual ao infinito.<p><a href="https://lh3.googleusercontent.com/-ZILBTcJVvrs/XIeulV7_aSI/AAAAAAAADdE/546qgHW4txUCNILnXTOyw_nPbetuDh5nwCHMYCw/s1600-h/image%255B35%255D"><img width="601" height="392" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-YRC1Oj9wzmA/XIeum6bPG3I/AAAAAAAADdI/YrV90XEye5Ew0gWgwt3wssS72yZbNUPVACHMYCw/image_thumb%255B21%255D?imgmax=800" border="0"></a><p>Mas os cálculos usando zero foram além.<p>Embora fosse aceito que 3 menos 3 equivalia a zero, qual seria o resultado de 3 menos 4?<p>Parece que você não tem nada, mas os indianos perceberam que isso era um novo tipo de nada: os números negativos.<p>Os indianos conseguiram chegar aos números negativos e ao zero porque os conceberam como entidades abstratas.<p><a href="https://lh3.googleusercontent.com/-oJuGLUiUUGE/XIeupXwQzhI/AAAAAAAADdM/y8v639ICOHIn27o0vhs5-KP_BEuUN8GsgCHMYCw/s1600-h/image%255B40%255D"><img width="603" height="501" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-jD280lomXW8/XIeur2v8GEI/AAAAAAAADdU/p9UMtcb04eoaYjE1t8gBhu9vAFjvLKZUgCHMYCw/image_thumb%255B24%255D?imgmax=800" border="0"></a><p>Os números não eram apenas algo usado para contar ou medir; eles tinham vida, flutuavam sem amarras ao mundo real.<p>Essa linha de pensamento gerou uma explosão de ideias matemáticas.<h4>X e Y</h4><p>A abordagem abstrata indiana para a matemática revelou novas maneiras de resolver equações quadráticas - aquelas que incluem números quadrados.<p>A compreensão de Brahmagupta, nascido em 598, dos números negativos permitiu-lhe ver que as equações quadráticas teriam sempre duas soluções. E que uma delas poderia ser negativa.<p>Ele foi ainda mais longe resolvendo equações com duas variáveis (X e Y).<p>Esse movimento só aconteceria no Ocidente em 1657, quando o matemático francês Pierre de Fermat apresentou sua solução, alheio ao que seu colega indiano havia encontrado mil anos antes.<p><a href="https://lh3.googleusercontent.com/-qTMhWDqDRlY/XIeutkNOzLI/AAAAAAAADdc/qrLxQ0r5zggGF-4x-OcFwgQfqH41C1KVQCHMYCw/s1600-h/image%255B45%255D"><img width="597" height="384" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-VHGrsVtWbR0/XIeuvStnDhI/AAAAAAAADdk/tvjXSV5A4xgFu5rSlj_3RjieyD41NfiNgCHMYCw/image_thumb%255B27%255D?imgmax=800" border="0"></a><p>Brahmagupta também desenvolveu uma nova linguagem para expressar a solução dessas equações.<p>Enquanto experimentava maneiras de apresentar seus cálculos, ele usou as letras iniciais de dois nomes de cores para representar variáveis.<p>A decisão resultou na adoção de X e Y, que ainda usamos até hoje.<h4>Não terminou aí</h4><p>Os matemáticos indianos também foram responsáveis por mais descobertas em trigonometria.<p><a href="https://lh3.googleusercontent.com/-YLdU0P4StPc/XIeuwcFfLNI/AAAAAAAADdo/2n8yf01KjxUY8rsMTP566LdbOlL-qKdNgCHMYCw/s1600-h/image%255B50%255D"><img width="594" height="395" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-bJKLB-7YF4E/XIeuxboHWCI/AAAAAAAADds/cnwtF5jWtUMDc-4mSRyPhzWDhlyyCosCgCHMYCw/image_thumb%255B30%255D?imgmax=800" border="0"></a><p>É verdade que os gregos foram os primeiros a desenvolver o que podemos chamar de "dicionário" que traduzia a geometria em números e vice-versa.<p>Mas os indianos foram mais longe.<p>Eles usaram trigonometria para estudar o mundo ao seu redor, o que inclui navegar pelos mares e calcular distâncias espaciais.<p>Matemáticos indianos, por exemplo, conseguiram calcular a distância entre a Terra e a Lua e entre a Terra e o Sol.<p><a href="https://lh3.googleusercontent.com/-nquYi9YLCYY/XIeuzM6NMmI/AAAAAAAADdw/zqGQsXDZYkQ1A3WZ6D_07w1Vk73pzjebwCHMYCw/s1600-h/image%255B56%255D"><img width="592" height="380" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-92DlMYRKGXo/XIeu1UvorpI/AAAAAAAADd0/FkVU7P4Ukog_6icQqEbbNXbe4oD0zxv-gCHMYCw/image_thumb%255B34%255D?imgmax=800" border="0"></a><p>Os matemáticos da Índia também solucionaram o mistério de um dos números mais importantes da matemática: Pi.<p>Pi é o valor numérico da razão entre a circunferência de um círculo e seu diâmetro.<p>É um número que aparece em todos os tipos de cálculos, mas que é especialmente útil para engenheiros e arquitetos, já que toda medida envolvendo curvas requer Pi.<p>Durante séculos, os matemáticos procuraram o valor preciso de Pi.<p>Mas foi no século 6 que o matemático indiano Aryabhata deu uma aproximação bastante precisa: 3,1416.<p>Ele também usou o Pi para medir a circunferência da Terra, chegando ao valor de 39.968 km - um número muito próximo daquele que conhecemos hoje (40.075 km).<p><a href="https://lh3.googleusercontent.com/-co5_OAGW4F4/XIeu3cn2GwI/AAAAAAAADd4/5fIwExofWFgNOKP3cygraRUJdZ1zIqgwQCHMYCw/s1600-h/image%255B61%255D"><img width="599" height="384" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-cL3yEbo39UE/XIeu5aALO2I/AAAAAAAADeA/xFJp3u6srvglW2s3UWj8S3FupS4MYKv1wCHMYCw/image_thumb%255B37%255D?imgmax=800" border="0"></a><p>Madhava percebeu que, adicionando e subtraindo diferentes frações, era possível determinar uma fórmula exata para Pi.<p>Esta fórmula ainda é ensinada em muitas universidades ao redor do mundo como se tivesse sido descoberta pelo alemão Gottfried Wilhelm Leibniz no século 17.<p>Fonte:<p><a href="https://www.bbc.com/portuguese/geral-47487130" target="_blank"><img width="94" height="32" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-a21KvHSl3Us/XIeu5-wPOCI/AAAAAAAADeE/WBxM9PsaQDg0LOK0i_UzTtGhZEzYa0w8gCHMYCw/image%255B67%255D?imgmax=800" border="0"></a><a title="https://www.bbc.com/portuguese/geral-47487130" href="https://www.bbc.com/portuguese/geral-47487130" target="_blank">https://www.bbc.com/portuguese/geral-47487130</a>Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-27069341517492647262019-03-09T21:33:00.001-03:002019-03-16T12:16:50.702-03:00Citações de Pitágoras<p><br>
<a href="https://lh3.googleusercontent.com/-DrtsB5I_RpM/XIT3buH2PcI/AAAAAAAADbo/gjILf8juOn8Aao_dsj4wLq92CNA16pHPwCHMYCw/s1600-h/image%255B1%255D"><img width="120" height="159" title="image" align="right" style="margin: 0px 0px 0px 10px; float: right; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-hJLN3Wkb5IE/XIT3cZBF5CI/AAAAAAAADbs/_QOKCsZynSMgbfPYDQ38zZJKqgUqymtWwCHMYCw/image_thumb?imgmax=800" border="0"></a><font size="3">Pitágoras de Samos (c. 570 – c. 495 a.C.) foi um filósofo e matemático grego jônico creditado como o fundador do movimento chamado Pitagorismo. A maioria das informações sobre Pitágoras foram escritas séculos depois que ele viveu, de modo que há pouca informação confiável sobre ele. Nasceu na ilha de Samos e viajou o Egito e Grécia e talvez a Índia, em 520 a.C. voltou a Samos.Cerca de 530 a.C., se mudou para Crotona, na Magna Grécia.</font></p><font size="3">
</font><p><font size="3">Um problema não solucionado na época de Pitágoras era determinar as relações entre os lados de um triângulo retângulo. Pitágoras provou que a soma dos quadrados dos catetos é igual ao quadrado da hipotenusa, conhecido até os dias de hoje como Teorema de Pitágoras.</font></p><font size="2">
</font><p><a href="https://lh3.googleusercontent.com/-7z_5BeY_Q1w/XIRbT24k8lI/AAAAAAAADag/3XTZGgYQHasjOKgmnuXSDruQKVuJMVWggCHMYCw/s1600-h/image%255B12%255D"><img width="192" height="168" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-GFLlc3YfCHg/XIRbUs5orGI/AAAAAAAADak/lPD8_q8khpUwwoCFCr1qlFIi5phjBQ4qQCHMYCw/image_thumb%255B6%255D?imgmax=800" border="0"></a></p><font size="2">
</font><p><font size="3">Como filósofo, Pitágoras nos deixou muitas citações. Veja a seguir algumas delas:</font></p><font size="2">
</font><ol><li><font size="3">“Eduquem as crianças e não será necessário castigar os homens.”</font></li><li><font size="3">“Escuta e serás sábio. O começo da sabedoria é o silêncio.”</font></li><li>“Quem fala, semeia. Quem escuta, colhe.”</li><li>“O homem é mortal por seus temores e imortal por seus desejos.” </li><li>“Pensem o que quiserem de ti; faz aquilo que te parece justo.” </li><li>“Antes de fazer alguma coisa, pense. Quando achar que já pode faze-la, pense novamente.”</li><li>“Deus é uno. Ele não está jamais, como pesam alguns, fora do mundo, mas sim totalmente no mundo inteiro. Deus está no Universo e o Universo está em Deus. O Mundo e Deus não são mais que uma unidade.”</li><li>“A verdadeira amizade significa unir muitos corações e corpos num coração e num espírito.” </li><li>“Despreza as estradas largas, segue os carreiros.” </li><li>“Todas as coisas são números.” </li><li>“A prudência é o olho de todas as virtudes.” </li><li>“A felicidade consiste em poder unir o princípio com o fim.” </li><li>“Não peças nada nas tuas preces, porque tu não sabes o que é útil e só Deus conhece as tuas necessidades.”</li><li>“Não peças nada nas tuas preces, porque tu não sabes o que é útil e só Deus conhece as tuas necessidades.”</li><li>“Em três partes se divide o alma humana: em mente, em sabedoria e em ira.” </li><li>“Cala-te ou dê algo melhor do que o silêncio.” </li><li>“Não é livre quem não obteve domínio sobre si.”</li><li>“Os afetos podem, às vezes, somar-se. Subtrair-se, nunca.”</li><li>“Não te gabes de ser adorado por uma mulher que se adora muito.”</li><li>“O universo é uma harmonia de contrários.”</li><li>“Quando mais fazemos, mais podemos fazer; quando estamos mais ocupados é quando temos mais tempo para divertir-nos.”</li><li>“Os homens que sempre falam verdade são os que mais se aproximam a Deus.” </li><li>“Necessário é encontrar o infinitamente grande no infinitamente pequeno, para sentir a presença de Deus.” </li><li>“Observa o teu culto, a família e, cumpre teus deveres para com teu pai, tua mãe e todos os teus parentes. Educa as crianças e não precisarás castigar os homens.” <font size="2"><br></font></li><li>“As palavras são os suspiros da alma.”</li><li>“Nada perece no Universo; tudo o que acontece nele não passa de meras transformações.”</li><li>“Mais lhe vale a um homem ter a boca fechada, e que os demais lhe achem tonto, que a abrir e que os demais se convençam de que o é.” </li><li>“Não sejas ambicioso e mesquinho; a justa medida é excelente em tais casos.” </li><li>“O riso afasta o medo e quem não tem medo não tem fé.”</li><li>“Em matéria de injustiça, pior não é sofre-la, é comete-la.”<font size="2"><br><br></font></li></ol>
<p>Francisco Ismael Reis </p><p><a href="https://lh3.googleusercontent.com/-XlUUobCDtQ8/XIRbVKfVl2I/AAAAAAAADao/IoyXfPlZ2vUFEGMtAqwVVnuNVYbl3_KGACHMYCw/s1600-h/image%255B10%255D"><img width="141" height="41" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-7RlYBlDT5OI/XIRbV_n-imI/AAAAAAAADas/dg2I9eWcPIYzdKfOZsP-smCvhzGAZpx4QCHMYCw/image_thumb%255B4%255D?imgmax=800" border="0"></a></p>
<p>09/03/2019</p>Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-11020401914896374892019-03-08T10:47:00.001-03:002019-03-08T10:51:09.633-03:00O Problema de Monty Hall<p>Esse problema foi criado pelo estatístico Steve Selvin inspirado em um programa da TV dos Estados Unidos – “<em>Let’s Make a Deal</em> “– dos anos 70, comandado pelo apresentador Monty Hall. O problema, publicado numa carta à revista The American Statistician em 1975 é reproduzido num trecho do filme <a href="https://youtu.be/DSbtIa8NM5E" target="_blank">Quebrando a Banca</a>, lançado em 2008.</p><p>Eis o problema:</p><table border="1" cellspacing="0" cellpadding="2">
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<td valign="top"><p><font color="#0000ff"><em>Imagine que você é um participante de um programa de auditório, e o apresentador mostra a você três portas fechadas. Ele diz que atrás de uma das portas está um carro, e atrás das outras duas há apenas bodes. Se você escolher a porta certa, ganha o carro – caso contrário levará apenas um bode.</em></font><p><a href="https://lh3.googleusercontent.com/-srDak2w4d8w/XIJyZGaML9I/AAAAAAAADYY/-DO8RhW7xiQOwpxqKgCP48VcOnfnxvVfQCHMYCw/s1600-h/image%255B3%255D"><img width="240" height="155" title="image" style="border: 0px currentcolor; border-image: none; margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-YLj9lLAwWII/XIJyZr-6A5I/AAAAAAAADYc/Qb3Bcq79q9Yy0hQItuTBp3lISV40bD_BwCHMYCw/image_thumb%255B1%255D?imgmax=800" border="0"></a><p><font color="#0000ff"><em>Você escolhe uma das portas. O apresentador, então, sem abrir a porta que você escolheu, dirige-se para uma das outras duas portas que sobraram. Como ele sabe em qual das três portas está o carro, ele então abre uma das duas portas que você não tinha escolhido – exatamente uma porta que escondia um bode. Restaram então apenas duas portas fechadas: aquela que você tinha escolhido, e uma outra que não foi aberta pelo apresentador. Atrás de uma delas está o carro.</em></font><p><font color="#0000ff"><em>O apresentador então pergunta se você quer manter a escolha original ou se quer, agora, trocar de porta, escolhendo a outra que ele não abriu e que pode conter o carro.</em></font><p><font color="#0000ff"><em>O que você deve fazer:</em></font><p><font color="#0000ff"><em>1 – manter a escolha original, ou</em></font><p><font color="#0000ff"><em>2 – trocar de porta ?</em></font></p></td>
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<p>Diante do que foi apresentado, as pessoas, de maneira geral, acham que as chances são de 50%, afinal, ou o prêmio está na porta que você escolheu, ou está na outra porta que sobrou. Entretanto essa resposta não é correta<em>.</em> Vejamos:</p><p>As tabelas a seguir nos ajudarão a entender a solução do problema.<p>Admitindo que você tenha escolhido a porta 1 e lembrando que o apresentador nunca abrirá a porta com o carro, temos duas situações a considerar: <p><u>1ª. situação:</u> <em><strong>Manter a escolha</strong></em><p><a href="https://lh3.googleusercontent.com/-6L-J0yinY3s/XIJyaQm9wXI/AAAAAAAADYg/2NsWwMA3FoMJ-Btm-Jdhuhg-_1_hmK9_gCHMYCw/s1600-h/image%255B6%255D"><img width="244" height="142" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-7ziEnjkxQTk/XIJybJL_iwI/AAAAAAAADYk/SFTvz_CvEw0YUv03Mc5wwa3gGmmVg91kwCHMYCw/image_thumb%255B2%255D?imgmax=800" border="0"></a><p>Neste caso, manter a escolha, dá a você 1 chance de ganhar num total de 3, ou seja, a sua probabilidade de ganhar o carro é de 33,3%.</p><p><u>2ª. situação:</u><em> <strong>Trocar de porta</strong></em></p><p><a href="https://lh3.googleusercontent.com/-sKPZdWZfTXI/XIJybqK-q9I/AAAAAAAADYo/p9oqWJ7Qej8EsK6kXsi4qN7_Ky-XgXk8gCHMYCw/s1600-h/image%255B9%255D"><img width="244" height="142" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-zgmdKlGvX3g/XIJycBqSpjI/AAAAAAAADYs/vmC4KVXoRR8bzGolFgUPzdgDz-5oePSCgCHMYCw/image_thumb%255B3%255D?imgmax=800" border="0"></a></p><p>Neste caso, trocar de porta, dá a você 2 chances de ganhar num total de 3, ou seja, a sua probabilidade de ganhar o carro é de 66,6%.</p><p>Portanto, TROCAR dá a você duas vezes mais chance de ganhar e esta é a escolha a ser feita.<p><br><p>Francisco Ismael Reis<p><a href="https://lh3.googleusercontent.com/-czHH9I4he60/XIJycpANAHI/AAAAAAAADYw/Rsq1MtevqWQOf2IyawdwWbW1Lc0869kQQCHMYCw/s1600-h/Assinatura%255B3%255D"><img width="141" height="41" title="Assinatura" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="Assinatura" src="https://lh3.googleusercontent.com/-gKl26pGj7tw/XIJydCW-LoI/AAAAAAAADY0/S32yjD6jVAYbJbuhaZ7sRCXTbm13-7aYACHMYCw/Assinatura_thumb%255B1%255D?imgmax=800" border="0"></a><p>08/03/2019Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-75926436699712495932019-03-07T20:47:00.001-03:002019-03-07T21:04:19.928-03:00Símbolos matemáticos<p><a href="https://lh3.googleusercontent.com/-THwhH-rmXLE/XIGw5-3RqtI/AAAAAAAADYE/s0HxY04P-Csb3Hy5RUUO5Em2Gpp2Y549gCHMYCw/s1600-h/image%255B3%255D"><img width="56" height="53" title="image" align="left" style="margin: 0px 10px 0px 0px; float: left; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-hAzDPtOMZqo/XIGw6SqPZfI/AAAAAAAADYI/Xd4V68PospAtm60BKyliFutvavKP114BwCHMYCw/image_thumb%255B1%255D?imgmax=800" border="0"></a>Na tabela a seguir são mostrados os principais símbolos utilizados em Matemática e os seus respectivos significados. </p><table border="1" cellspacing="0" cellpadding="2">
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<td width="65" valign="top"><p align="center"><strong>Símbolo</strong></p></td>
<td width="40%" valign="top"><p align="center"><strong>Significado</strong></p></td>
<td width="65" valign="top"><p align="center"><strong>Símbolo</strong></p></td>
<td width="40%" valign="top"><p align="center">Significado</p></td>
</tr>
<tr>
<td width="65" valign="top">+</td>
<td width="40%" valign="top">mais, positivo</td>
<td width="65" valign="top">-</td>
<td width="40%" valign="top">menos, negativo</td>
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<tr>
<td width="65" valign="top">x, * ou <strong>.</strong></td>
<td width="40%" valign="top">multiplicado por; vezes</td>
<td width="65" valign="top">/<i>,</i> ÷ <i>ou</i> :</td>
<td width="40%" valign="top">dividido por</td>
</tr>
<tr>
<td width="65" valign="top">±</td>
<td width="40%" valign="top">mais ou menos</td>
<td width="65" valign="top">=</td>
<td width="40%" valign="top">igual a</td>
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<tr>
<td width="65" valign="top">≅</td>
<td width="40%" valign="top">aproximadamente igual</td>
<td width="65" valign="top">≡</td>
<td width="40%" valign="top">equivalente a; congruente</td>
</tr>
<tr>
<td width="65" valign="top">≠</td>
<td width="40%" valign="top">diferente de</td>
<td width="65" valign="top"><</td>
<td width="40%" valign="top">menor que</td>
</tr>
<tr>
<td width="65" valign="top">></td>
<td width="40%" valign="top">maior que</td>
<td width="65" valign="top">≤</td>
<td width="40%" valign="top">menor ou igual</td>
</tr>
<tr>
<td width="65" valign="top">≥</td>
<td width="40%" valign="top">maior ou igual</td>
<td width="65" valign="top">≪</td>
<td width="40%" valign="top">muito menor que</td>
</tr>
<tr>
<td width="65" valign="top">≫</td>
<td width="40%" valign="top">muito maior que</td>
<td width="65" valign="top">⊀</td>
<td width="40%" valign="top">não é menor que</td>
</tr>
<tr>
<td width="65" valign="top">⊁</td>
<td width="40%" valign="top">não é maior que</td>
<td width="65" valign="top">∝</td>
<td width="40%" valign="top">proporcional a</td>
</tr>
<tr>
<td width="65" valign="top">∞</td>
<td width="40%" valign="top">infinito</td>
<td width="65" valign="top">%</td>
<td width="40%" valign="top">por cento</td>
</tr>
<tr>
<td width="65" valign="top">‰</td>
<td width="40%" valign="top">por mil</td>
<td width="65" valign="top">∃</td>
<td width="40%" valign="top">existe</td>
</tr>
<tr>
<td width="65" valign="top">∄</td>
<td width="40%" valign="top">não existe</td>
<td width="65" valign="top">∉</td>
<td width="40%" valign="top">não pertence a</td>
</tr>
<tr>
<td width="65" valign="top">∈</td>
<td width="40%" valign="top">pertence a</td>
<td width="65" valign="top">∀</td>
<td width="40%" valign="top">qualquer que seja</td>
</tr>
<tr>
<td width="65" valign="top">⇒ <i>ou</i> →</td>
<td width="40%" valign="top">implica</td>
<td width="65" valign="top">⇔ <i>ou</i> ↔</td>
<td width="40%" valign="top">se e somente se</td>
</tr>
<tr>
<td width="65" valign="top">| |</td>
<td width="40%" valign="top">valor absoluto; módulo</td>
<td width="65" valign="top">∅ <i>ou</i> { }</td>
<td width="40%" valign="top">conjunto vazio</td>
</tr>
<tr>
<td width="65" valign="top">#</td>
<td width="40%" valign="top">cardinalidade</td>
<td width="65" valign="top">∩</td>
<td width="40%" valign="top">intersecção</td>
</tr>
<tr>
<td width="65" valign="top">∪</td>
<td width="40%" valign="top">união</td>
<td width="65" valign="top">⊃</td>
<td width="40%" valign="top">contém</td>
</tr>
<tr>
<td width="65" valign="top">⊅</td>
<td width="40%" valign="top">não contém</td>
<td width="65" valign="top">⊂</td>
<td width="40%" valign="top">está contido</td>
</tr>
<tr>
<td width="65" valign="top">⊄</td>
<td width="40%" valign="top">não está contido</td>
<td width="65" valign="top">[ ]</td>
<td width="40%" valign="top">colchetes</td>
</tr>
<tr>
<td width="65" valign="top">( )</td>
<td width="40%" valign="top">parênteses</td>
<td width="65" valign="top">{ }</td>
<td width="40%" valign="top">chaves</td>
</tr>
<tr>
<td width="65" valign="top">√</td>
<td width="40%" valign="top">raiz quadrada</td>
<td width="65" valign="top">∛</td>
<td width="40%" valign="top">raiz cúbica</td>
</tr>
<tr>
<td width="65" valign="top">log</td>
<td width="40%" valign="top">logaritmo de base 10</td>
<td width="65" valign="top">ln</td>
<td width="40%" valign="top">logaritmo natural</td>
</tr>
<tr>
<td width="65" valign="top">^</td>
<td width="40%" valign="top">potência, exponenciação</td>
<td width="65" valign="top">n!</td>
<td width="40%" valign="top">fatorial</td>
</tr>
<tr>
<td width="65" valign="top">dy/dx</td>
<td width="40%" valign="top">derivada</td>
<td width="65" valign="top">∫</td>
<td width="40%" valign="top">integral</td>
</tr>
<tr>
<td width="65" valign="top">∇</td>
<td width="40%" valign="top">gradiente</td>
<td width="65" valign="top">∏</td>
<td width="40%" valign="top">produtório</td>
</tr>
<tr>
<td width="65" valign="top">∑</td>
<td width="40%" valign="top">somatório</td>
<td width="65" valign="top">sen</td>
<td width="40%" valign="top">seno</td>
</tr>
<tr>
<td width="65" valign="top">cos</td>
<td width="40%" valign="top">cosseno</td>
<td width="65" valign="top">tg</td>
<td width="40%" valign="top">tangente</td>
</tr>
<tr>
<td width="65" valign="top">cotg</td>
<td width="40%" valign="top">cotangente</td>
<td width="65" valign="top">sec</td>
<td width="40%" valign="top">secante</td>
</tr>
<tr>
<td width="65" valign="top">cossec</td>
<td width="40%" valign="top">cossecante</td>
<td width="65" valign="top">°</td>
<td width="40%" valign="top">grau</td>
</tr>
<tr>
<td width="65" valign="top">‘</td>
<td width="40%" valign="top">minuto</td>
<td width="65" valign="top">“</td>
<td width="40%" valign="top">segundo</td>
</tr><tr><td width="65" valign="top">π</td><td width="40%" valign="top">pi (3,1415…)</td><td width="65" valign="top">φ</td><td width="40%" valign="top">fi, o número de ouro (1,618…)</td></tr><tr><td width="65" valign="top">e</td><td width="40%" valign="top">número de Euler (2,718…)</td><td width="65" valign="top">∟</td><td width="40%" valign="top">ângulo reto</td></tr><tr><td width="65" valign="top">⊥</td><td width="40%" valign="top">perpendicular a</td><td width="65" valign="top">//</td><td width="40%" valign="top">paralelo a</td></tr><tr><td width="65" valign="top">¬ ou ~</td><td width="40%" valign="top">negação lógica</td><td width="65" valign="top">∴</td><td width="40%" valign="top">portanto</td></tr><tr><td width="65" valign="top">∵</td><td width="40%" valign="top">porque</td><td width="65" valign="top">∨</td><td width="40%" valign="top">ou</td></tr><tr><td width="65" valign="top">∧</td><td width="40%" valign="top">e</td><td width="65" valign="top">ℕ</td><td width="40%" valign="top">conjunto dos números naturais</td></tr><tr><td width="65" valign="top">ℤ</td><td width="40%" valign="top">conjunto dos números inteiros</td><td width="65" valign="top">ℚ</td><td width="40%" valign="top">conjunto dos números racionais</td></tr><tr><td width="65" valign="top">I<sub>r</sub></td><td width="40%" valign="top">conjunto dos números irracionais</td><td width="65" valign="top"><font color="#000000">ℝ</font></td><td width="40%" valign="top">conjunto dos números reais</td></tr><tr><td width="65" valign="top">ℂ</td><td width="40%" valign="top">conjunto dos números complexos</td><td width="65" valign="top">i</td><td width="40%" valign="top">unidade imaginária (i<sup>2</sup> = –1)</td></tr>
</tbody>
</table>
<p><br></p><p>Francisco Ismael Reis</p><p><a href="https://lh3.googleusercontent.com/-eP3YIKZ6XV8/XIGtqBVtDpI/AAAAAAAADXw/WmlqPk3oTnw2bEKdxV82xKj58w1gJs3cwCHMYCw/s1600-h/Assinatura%255B3%255D"><img width="141" height="41" title="Assinatura" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="Assinatura" src="https://lh3.googleusercontent.com/-NFCDPLGZ7Ig/XIGtqmCY2WI/AAAAAAAADX0/nM6I10Zv3_IdvP1UuLZowhnuHnIXotu2QCHMYCw/Assinatura_thumb%255B1%255D?imgmax=800" border="0"></a></p><p>07/03/2019</p>Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-83745811021915570662019-03-07T16:38:00.001-03:002019-03-17T18:11:09.303-03:00Probabilidade num dado viciadoOs dados são figuras geométricas, mais precisamente poliedros, e como tal podem se apresentar sob diferentes formatos, como mostrado na figura abaixo.<br />
<a href="https://lh3.googleusercontent.com/-_GqCmg9ZHJ8/XIFzO5Pjm8I/AAAAAAAADW4/HVqUBmGoY-AI8NfsWnooPb4HTlzgFrBNQCHMYCw/s1600-h/image%255B5%255D"><img alt="image" border="0" height="195" src="https://lh3.googleusercontent.com/-13TZjFeId8I/XIFzPlzcnTI/AAAAAAAADW8/qaC1gwUSJxcqlAs4bMeIftpVkDbiMfQ1wCHMYCw/image_thumb%255B3%255D?imgmax=800" style="background-image: none; display: block; float: none; margin-left: auto; margin-right: auto;" title="image" width="244" /></a><br />
Entretanto, o mais comum é o de seis faces, que são numeradas de 1 a 6. Em um desses dados a frequência com que cada uma dessas faces ocorre, após o lançamento do mesmo é a mesma e igual a 1/6 o seja 0,1666 … ≅ 16,7%. Nessas condições, o dado é dito <strong><em>dado honesto</em></strong>. <br />
Existem, porém, os <strong><em>dados viciados</em></strong>, aqueles em que as faces não ocorrem com a mesma frequência.<br />
O problema a seguir, refere-se a um desses dados viciados.<br />
<a href="https://lh3.googleusercontent.com/-UpdqyyEnzC8/XIFzQUzEj8I/AAAAAAAADXA/v_0WHQsivQApVAqrO7W_PN4IuTf6X48NwCHMYCw/s1600-h/image%255B9%255D"><img alt="image" border="0" height="104" src="https://lh3.googleusercontent.com/-8U1rFgu1HL4/XIFzQzkQgpI/AAAAAAAADXE/mKjdnDowFwcLRePz9qC6vgE2WAFP7_McgCHMYCw/image_thumb%255B5%255D?imgmax=800" style="background-image: none; display: block; float: none; margin-left: auto; margin-right: auto;" title="image" width="464" /></a><br />
<br />
Resolução:<br />
Representemos por P(6) = p a probabilidade de ocorrência da face 6 desse dado.<br />
De acordo com o enunciado do problema temos então que:<br />
<blockquote>
<a href="https://lh3.googleusercontent.com/-eJKARcypNI8/XIFzRVft8HI/AAAAAAAADXI/TLJoIvBrN2U0DE4DEYTGBBZ3nrgahKLmwCHMYCw/s1600-h/image%255B19%255D"><img align="right" alt="image" border="0" height="200" src="https://lh3.googleusercontent.com/-ypqaZIl9ZWE/XIFzSMD1i8I/AAAAAAAADXM/uAbDv-l2xLw0JU-fveQiqpmemcNKDtHCwCHMYCw/image_thumb%255B13%255D?imgmax=800" style="background-image: none; display: inline; float: right;" title="image" width="191" /></a>P(1) = 2p</blockquote>
<blockquote>
P(2) = 2p</blockquote>
<blockquote>
P(3) = 2p</blockquote>
<blockquote>
P(4) = 2p</blockquote>
<blockquote>
P(5) = 2p</blockquote>
<blockquote>
P(6) = p</blockquote>
Ocorre que P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 1<br />
Ou seja:<br />
<blockquote>
2p + 2p + 2p + 2p + 2p + p = 1</blockquote>
<blockquote>
11p = 1</blockquote>
<blockquote>
p = 1/11 ≅ 0,090909 …</blockquote>
Logo:<br />
A probabilidade de ocorrência da face 6 é P(6) ≅ 9,1%<br />
Francisco Ismael Reis<br />
<a href="https://lh3.googleusercontent.com/-m8MG-4o3vDI/XIFzSvvVnqI/AAAAAAAADXQ/EBQ2aToQ-XgkdpaVqupGkpThQhE449Y0QCHMYCw/s1600-h/Assinatura%255B3%255D"><img alt="Assinatura" border="0" height="41" src="https://lh3.googleusercontent.com/-B5-Dg-LAkB8/XIFzTNi4rvI/AAAAAAAADXU/KsXuaoMp0kIEKGRTrjydAtzrWJqZiufbQCHMYCw/Assinatura_thumb%255B1%255D?imgmax=800" style="background-image: none; border: 0px currentcolor; display: inline;" title="Assinatura" width="141" /></a><br />
07/03/2019Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-76366017973563591972019-03-06T22:36:00.001-03:002019-03-07T08:42:16.929-03:00Uma breve história dos números<p><strong>O conceito</strong></p><p><a href="https://lh3.googleusercontent.com/-RJbn3uYnAbA/XIB1BAYj4xI/AAAAAAAADNQ/I6-2vI27PsYEacJknv68Drak78iMW6PAwCHMYCw/s1600-h/image%255B2%255D"><img width="244" height="134" title="image" align="right" style="margin-right: 0px; margin-left: 0px; float: right; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-RBruGFHGsIs/XIB1B0ksYII/AAAAAAAADNU/JCIwrKzxqTwNvv2enkpd1w_3FMxUEKsrgCHMYCw/image_thumb?imgmax=800" border="0"></a>A construção e o desenvolvimento dos conjuntos numéricos aparece como resultado das necessidades que o ser humano teve, em algum período da história, de criar um símbolo que desse significado matemático a certas operações que até então não o tinham.<p><br><p><strong>O conjunto dos números naturais</strong><p><a href="https://lh3.googleusercontent.com/-hR9qzdFbFuA/XIB1CjXX_pI/AAAAAAAADNY/xENCuKOnVc0x_lCFlyMORR3u7yDkCTHggCHMYCw/s1600-h/image%255B5%255D"><img width="244" height="167" title="image" align="right" style="float: right; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-YCuPgkG692g/XIB1DehGvfI/AAAAAAAADNc/1CqXUbf-BNA4pcb0Qga5fWaVMhcyVR_HACHMYCw/image_thumb%255B1%255D?imgmax=800" border="0"></a>É o primeiro conjunto de que se tem notícia. Foi criado a partir da simples necessidade que o homem tem de contar. É representado pelo símbolo N e formado pelos seguintes elementos:<p>N = {0, 1, 2, 3, 4, 5, ⋯}<p><br><p><br><p><strong>O conjunto dos números inteiros</strong><p><a href="https://lh3.googleusercontent.com/-x2sweRS6cu4/XIB1Dx3Xp9I/AAAAAAAADNg/ZGKKROAIkAcVcxoqnQuLjKDNqCMRkK-nACHMYCw/s1600-h/image%255B9%255D"><img width="226" height="217" title="image" align="right" style="float: right; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-gfRh9XOBIv0/XIB1EtkZsyI/AAAAAAAADNk/4DJ-a-Z_1G4T9fyUeGrvvOhXTrKn6e3vACHMYCw/image_thumb%255B3%255D?imgmax=800" border="0"></a>É uma ampliação do conjunto dos números naturais. Nasceu da necessidade de se criar um símbolo que representasse, na operação de subtração, o resultado que se obtém quando de uma quantidade menor se pretende tirar uma maior. É representado pelo símbolo Z e formado pelos seguintes elementos:<p>Z = { ⋯, -3, -2, -1, 0, +1, +2, +3, ⋯}<p><br><p><br><p><strong>O conjunto dos números racionais</strong><p><a href="https://lh3.googleusercontent.com/-XN32qkwXytE/XIB1FE8UjgI/AAAAAAAADNo/RXYMJdWGX4wCplbglDSTxRhDyOlKZ2HsQCHMYCw/s1600-h/image%255B12%255D"><img width="244" height="161" title="image" align="right" style="float: right; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-6TVNn_N0h1g/XIB1Fzt4UiI/AAAAAAAADNs/nPPCdh4E5JMa1ndE6hAlZm_2x6XTTJCKQCHMYCw/image_thumb%255B4%255D?imgmax=800" border="0"></a>Os números racionais surgiram da necessidade de representar partes de um inteiro. É representado pelo símbolo Q e formado pelos seguintes elementos:<p><a href="https://lh3.googleusercontent.com/-eVNVsLanZu8/XIEDlUkl4lI/AAAAAAAADWE/jHvo5hfiouog94SqI5PPMbzLMjNQ4k0-QCHMYCw/s1600-h/image%255B12%255D"><img width="168" height="52" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-rIaSIyqml00/XIEDlyAaoGI/AAAAAAAADWI/fC0wBYgme24RWM5-kboIi8IfAHRll7PaACHMYCw/image_thumb%255B6%255D?imgmax=800" border="0"></a><p><br>De maneira simplificada, os números racionais são aqueles que podem ser escritos na forma de fração, onde o numerador e o denominador são números inteiros.<p><br><p><strong>O conjunto dos números irracionais</strong><p><a href="https://lh3.googleusercontent.com/-d8gnz7daMac/XIB1GZlyAAI/AAAAAAAADNw/4fTMzuN3Y0I--2f5SjafNwJaib0oeawVgCHMYCw/s1600-h/image%255B31%255D"><img width="244" height="119" title="image" align="right" style="float: right; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-l3HDFv12I-w/XIB1G-Ks6uI/AAAAAAAADN0/cERU55F4qb4o3gFVvMyZMhTjM01ltLO-wCHMYCw/image_thumb%255B13%255D?imgmax=800" border="0"></a>Os números irracionais são aqueles que não podem ser escritos na forma de fração, sendo números decimais, infinitos e não-periódicos. É representado pelo símbolo I. <p>Como exemplos de números irracionais podemos citar:<p><a href="https://lh3.googleusercontent.com/-Y28MoRp57Is/XIB1HfAUp1I/AAAAAAAADN4/cL2nvy9TV5MlrUDNwLiadSrIDbkjDL8ygCHMYCw/s1600-h/image%255B35%255D"><img width="103" height="22" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-4O9Vbfbqh30/XIB1HxCimkI/AAAAAAAADN8/ZCKbiRBZ7rAaMlm3hu0yJVetp0o7doxSgCHMYCw/image_thumb%255B15%255D?imgmax=800" border="0"></a><p><a href="https://lh3.googleusercontent.com/-gUTbdAjSdEQ/XIB1IXDAiYI/AAAAAAAADOA/i6Dypbb_3YUuxKn6WIxJbwaESeSY8ROUACHMYCw/s1600-h/image%255B39%255D"><img width="92" height="20" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-qe4TV8UjQQc/XIB1I0ZjNLI/AAAAAAAADOE/6xvf-dFL2IgP6i-hYF770qUYc43NDnWHwCHMYCw/image_thumb%255B17%255D?imgmax=800" border="0"></a><p><a href="https://lh3.googleusercontent.com/-8oZKO7JmNXg/XIB1JY-YkHI/AAAAAAAADOI/_WCbxnnvlyM6lV8TlNcADLWlZDdsA8D7QCHMYCw/s1600-h/image%255B47%255D"><img width="220" height="24" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-Dagpwo9Dvi8/XIB1J7XpIeI/AAAAAAAADOM/kk5xrnJbu4I41mZDAwM6JtpbVaAmPD3SACHMYCw/image_thumb%255B21%255D?imgmax=800" border="0"></a><p><a href="https://lh3.googleusercontent.com/-YT6fXLBLBvk/XIB1KWcg48I/AAAAAAAADOQ/30egq6qc_94CPSfZM7xMyqoO-evmuCYswCHMYCw/s1600-h/image%255B51%255D"><img width="226" height="21" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-qWB5jYZ1UGQ/XIB1K1rTreI/AAAAAAAADOU/yKAwNYahyFIMlGjk7YFuZ-MoCX5E67O6wCHMYCw/image_thumb%255B23%255D?imgmax=800" border="0"></a><p><strong>O conjunto dos números reais</strong><p><a href="https://lh3.googleusercontent.com/-v3UsXKkRADQ/XIB1LduKrzI/AAAAAAAADOY/LVU3xXsUJe8Et5hxSSRzvPj5IEI_y9F_gCHMYCw/s1600-h/image%255B54%255D"><img width="244" height="178" title="image" align="right" style="float: right; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-hS_ffKmZHP4/XIB1MVs9FhI/AAAAAAAADOc/KYw3g4lv1bgccUq3YoNR4DzcSPGWrv6YQCHMYCw/image_thumb%255B24%255D?imgmax=800" border="0"></a>O conjunto dos números reais é uma expansão do conjunto dos números racionais que engloba não só os inteiros e os fracionários, positivos e negativos, mas também todos os números irracionais. É representado pelo símbolo R.<p><a href="https://lh3.googleusercontent.com/-CllGuHBQN1s/XIECJjhd0ZI/AAAAAAAADVw/KXGQeUj_yMMJcanl8tQgvkcp-gEsgmDrACHMYCw/s1600-h/image%255B4%255D"><img width="125" height="22" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-pWdMdgoHLPQ/XIECKQYxEMI/AAAAAAAADV0/mg2dudaVhi4kr61en9dnAKnrIlB_NF9LgCHMYCw/image_thumb%255B2%255D?imgmax=800" border="0"></a><p><br><p><br><p><br><p><strong>Resolvendo uma equação do 2º grau</strong><p>Vamos encontrar as raízes da equação: x<sup>2</sup>- 6x + 10 = 0<p>Utilizando a fórmula de Bhaskara:</p><blockquote><p> <a href="https://lh3.googleusercontent.com/-rMZsfVajdiA/XIB1M6hwEjI/AAAAAAAADOg/i7gjlTe_FdcmqG7BvLb5n1vsjgSlRf7uQCHMYCw/s1600-h/image%255B16%255D"><img width="128" height="48" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-DBMHo00TxY4/XIB1NseOwqI/AAAAAAAADOk/hmDwlFXPM3IRcma6OB3H_ehLRLxWtP7BQCHMYCw/image_thumb%255B6%255D?imgmax=800" border="0"></a></p></blockquote><blockquote><p><a href="https://lh3.googleusercontent.com/-jsm6-zmtAnA/XIB1N_PPw8I/AAAAAAAADOo/LvG-_p3yuYwsp4SKTyucJbe6jLk4GN8RACHMYCw/s1600-h/image%255B20%255D"><img width="197" height="54" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-eyafNJ0Tm0o/XIB1ORZkviI/AAAAAAAADOs/vOiBh6SeWlw3tnUhqwXlDNKrtxCxrYW9gCHMYCw/image_thumb%255B8%255D?imgmax=800" border="0"></a></p></blockquote><blockquote><p><a href="https://lh3.googleusercontent.com/-jym-xOQGEyA/XIB1OwvOR5I/AAAAAAAADOw/Ls7LENwxdscUqwxwNrJdTDQ_v1hk0No8QCHMYCw/s1600-h/image%255B24%255D"><img width="115" height="45" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-xRTTXveoKEo/XIB1PdfdpXI/AAAAAAAADO0/nbuWM5xFY7ghtg566SbI3jyqZxDcg9GqACHMYCw/image_thumb%255B10%255D?imgmax=800" border="0"></a></p></blockquote><blockquote><p><a href="https://lh3.googleusercontent.com/-56umGdOypM8/XIB1QJPnpFI/AAAAAAAADO4/Z2abUEuXiuA0BZKEUlK-PNiKhrkXX7h_ACHMYCw/s1600-h/image%255B28%255D"><img width="84" height="44" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-QSez1M8hWNg/XIB1QoAI7BI/AAAAAAAADO8/wQw_anKmjPU0oKRAVgqrRm0Z8zHDpJtCACHMYCw/image_thumb%255B12%255D?imgmax=800" border="0"></a></p></blockquote><p>Não há como encontrar as raízes dessa equação pois não existe no conjunto dos números reais um número que elevado ao quadrado resulte em – 4 !!!!<p><strong>O conjunto dos números complexos</strong><p>O conjunto dos números complexos surgiu da necessidade de se encontrar as raízes não reais de equações de grau maior ou igual a 2 e dar um significado matemático às raízes quadradas de números negativos.<p><a href="https://lh3.googleusercontent.com/-xq99lMpGFXU/XIB1RXhqw5I/AAAAAAAADPA/YYnVwHBc09MjGx-SRVtz20k1R4DPtgYhgCHMYCw/s1600-h/image%255B70%255D"><img width="168" height="244" title="image" align="right" style="float: right; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-h-eCEh5x2ug/XIB1SEwtfUI/AAAAAAAADPE/bj-hoIFTsZo2fEtgHuMzo_xNIPjbwE_ZQCHMYCw/image_thumb%255B36%255D?imgmax=800" border="0"></a>Em 1545, Girolamo Cardano (1501-1576) falou das raízes quadradas de números negativos e, em 1572, Rafael Bombelli (1526-1572) percebeu que equações do tipo x<sup>2</sup>+a = 0 só poderiam ser resolvidas com o auxílio de números que mais tarde viriam a ser designados de imaginários por Descartes.<p><br><p><br><p><br><p><br><p><a href="https://lh3.googleusercontent.com/--CFf8ANJDRg/XIB1Sq0InTI/AAAAAAAADPI/jM3svqL-Yao-IS9NojK8Cub5Ixnfldl7gCHMYCw/s1600-h/image%255B74%255D"><img width="168" height="244" title="image" align="right" style="float: right; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-niG-ZCdo-LM/XIB1TeWbM7I/AAAAAAAADPM/ZVBDVPYyyjIbXpwDicdgsxwFRjQYTpGsgCHMYCw/image_thumb%255B38%255D?imgmax=800" border="0"></a>Foi criada, então, uma unidade imaginária a que Leonard Euler (1707-1783), em 1777, passou a representar pela letra <strong>i</strong>, de tal maneira que:<p><a href="https://lh3.googleusercontent.com/-yujvecSwULs/XIB1UNqt08I/AAAAAAAADPU/vtdc_EYGJg0mmSA8fPxnrbnvaOfsBh0ewCHMYCw/s1600-h/image%255B78%255D"><img width="177" height="55" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-nycBzeH3GRo/XIB1Uq2AhYI/AAAAAAAADPY/QH79HsUFuCQSDRG7Et83iQTBTJHPN1YKgCHMYCw/image_thumb%255B40%255D?imgmax=800" border="0"></a><p><br><p><br><p><br><p><a href="https://lh3.googleusercontent.com/-LooZsD_wh7I/XIB1VCSTpTI/AAAAAAAADPc/9GUrlSLxjTYYh6BlOU1eXeLcbMALdyEjgCHMYCw/s1600-h/image%255B81%255D"><br></a></p><p><a href="https://lh3.googleusercontent.com/-LooZsD_wh7I/XIB1VCSTpTI/AAAAAAAADPc/9GUrlSLxjTYYh6BlOU1eXeLcbMALdyEjgCHMYCw/s1600-h/image%255B81%255D"><img width="180" height="218" title="image" align="right" style="float: right; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-AUkMs91b80Q/XIB1V0AoG6I/AAAAAAAADPg/bngUU30Ijo07n0QUIuA5imlCTpcV5ztyQCHMYCw/image_thumb%255B41%255D?imgmax=800" border="0"></a>Mas foi Caspar Wessel (1745-1818) que em 1797 utilizou pela primeira uma representação gráfica para os números complexos.</p><p><br><p><br><p><br><p><br><p><br><p><a href="https://lh3.googleusercontent.com/-6ImV_YTaQIo/XIB1WcE8juI/AAAAAAAADPk/izjXnOsUMQ40rWSH8vvuvvObkaF5wlFlQCHMYCw/s1600-h/image%255B84%255D"><br></a></p><p><a href="https://lh3.googleusercontent.com/-6ImV_YTaQIo/XIB1WcE8juI/AAAAAAAADPk/izjXnOsUMQ40rWSH8vvuvvObkaF5wlFlQCHMYCw/s1600-h/image%255B84%255D"><img width="162" height="220" title="image" align="right" style="float: right; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-ttZiAfnPOUk/XIB1XFiOCQI/AAAAAAAADPo/0JPLfek6Hukw4qq9BxFn0Y5JtVokVpVcgCHMYCw/image_thumb%255B42%255D?imgmax=800" border="0"></a>Entretanto, somente por volta de 1806, a partir de um trabalho publicado por Jean Robert Argand (1768-1822), a representação geométrica dos números complexos ganhou o devido respeito.</p><p><br></p><p><br></p><p><br></p><p><br></p><p><br></p><p><a href="https://lh3.googleusercontent.com/--SFnv_9MV6s/XIB1X9Of9qI/AAAAAAAADPs/I6QdtLvXWhkKYN3e1ud-Pu1NVLPP8CWVwCHMYCw/s1600-h/image%255B89%255D"><img width="165" height="231" title="image" align="right" style="float: right; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-GMeimK_3AXo/XIB1YhrgufI/AAAAAAAADPw/2tnsMB67bZ4H0VUH290qVKYAvkK9qPZCgCHMYCw/image_thumb%255B48%255D?imgmax=800" border="0"></a>Somente no século XVIII é que esses estudos foram formalizados pelo matemático Carl Friedrich Gauss (1777-1855) que, em 1832, utilizou pela primeira vez a expressão “número complexo”.<p><br><p><br><p><br><p><br><p><br><p>Retomando a resolução da equação do 2º grau<p>Vamos encontrar as raízes da equação: x<sup>2</sup>- 6x + 10 = 0<p>Utilizando a fórmula de Bhaskara:</p><blockquote><p> <a href="https://lh3.googleusercontent.com/-vbE3EhlGJiU/XIB1ZNZ2a7I/AAAAAAAADP0/1VlZOtNL47ILnXPVk7xNHoto5ZwLfUxwgCHMYCw/s1600-h/image%255B98%255D"><img width="128" height="48" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-9akWIFd7wgw/XIB1ZjiXFiI/AAAAAAAADP4/12eKgvQ2TUIfgAX0iFSI_DuL0s0RQ-aMQCHMYCw/image_thumb%255B49%255D?imgmax=800" border="0"></a></p></blockquote><blockquote><p><a href="https://lh3.googleusercontent.com/-eaowZAUE9JM/XIB1Z6Y57WI/AAAAAAAADP8/Td0R7V80sKsYnpxKFr0XlafnceRFM6G2QCHMYCw/s1600-h/image%255B99%255D"><img width="197" height="54" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-D8MJApgpeUg/XIB1aZKNO6I/AAAAAAAADQA/J9uK84DPUoI8dgzcTglq6Yi-C2SNZWsIQCHMYCw/image_thumb%255B50%255D?imgmax=800" border="0"></a></p></blockquote><blockquote><p><a href="https://lh3.googleusercontent.com/-g5Jnk71WEeI/XIB1a2AP-2I/AAAAAAAADQE/94jZv1nzR1UkhkztlrVTO8YPKxSNTgKaQCHMYCw/s1600-h/image%255B100%255D"><img width="115" height="45" title="image" style="margin: 0px; border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-hA5mgxj1MDI/XIB1bc6-AVI/AAAAAAAADQI/ITTuHqvlywonFX-YThqHOQ7FALlSW7dFACHMYCw/image_thumb%255B51%255D?imgmax=800" border="0"></a></p></blockquote><blockquote><p><a href="https://lh3.googleusercontent.com/-UxGb7QbpaY0/XIB1b9Dr9vI/AAAAAAAADQM/joRsvlNEXzI8kHuSYXLPcpCa6UAspJJSgCHMYCw/s1600-h/image%255B101%255D"><img width="84" height="44" title="image" style="margin: 0px; border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-NAgoFIsJXI8/XIB1cdCuH8I/AAAAAAAADQQ/f3_9JdtH0-YvlIfDj-SxRRH978VdzRn4gCHMYCw/image_thumb%255B52%255D?imgmax=800" border="0"></a></p></blockquote><blockquote><p><a href="https://lh3.googleusercontent.com/-SZaBXx6QQWo/XIB1c5PQb9I/AAAAAAAADQU/1bSkVYteAAc3D8YwDRv9afc2P0y-C-JSgCHMYCw/s1600-h/image%255B105%255D"><img width="109" height="46" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-1cGxoW6Fkxk/XIB1dY3ka-I/AAAAAAAADQY/BkJjFma2nB4kmKYRkTRgZYZ-z3hg3W7PwCHMYCw/image_thumb%255B54%255D?imgmax=800" border="0"></a></p></blockquote><blockquote><p><a href="https://lh3.googleusercontent.com/-nOzQpfsXIog/XIB1d66kzfI/AAAAAAAADQc/WY4RqYSEGv44fHy0Yk5zgo_X5vVm82rwQCHMYCw/s1600-h/image%255B109%255D"><img width="79" height="40" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-p4-pTbxc0Zg/XIB1ecjko0I/AAAAAAAADQg/1hAWoK0cWAgLygP2wNKwrY81DNBu1GaOgCHMYCw/image_thumb%255B56%255D?imgmax=800" border="0"></a></p></blockquote><blockquote><p><a href="https://lh3.googleusercontent.com/-Zau5eJJj-PY/XIB1e0oKkYI/AAAAAAAADQk/0JbWi5di36U_Wlr2YexzcXwAkHCTa5HDgCHMYCw/s1600-h/image%255B113%255D"><img width="158" height="49" title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-thvDwmBmMpA/XIB1fXi2CvI/AAAAAAAADQo/z-szX9gdt405ubNWZhMwPSHSiIKeHH4zwCHMYCw/image_thumb%255B58%255D?imgmax=800" border="0"></a></p></blockquote><p>Chamadas de raízes complexas da equação.</p><p><strong>Os conjuntos numéricos</strong><p><a href="https://lh3.googleusercontent.com/-clo93-Xj-Fk/XIB1f4Ck_mI/AAAAAAAADQs/-WTgEaF2PeU_-Cu5gmflmwJ7uXVihBO8ACHMYCw/s1600-h/image%255B122%255D"><img width="546" height="350" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-OLXzkN8zoF8/XIB1gie2h4I/AAAAAAAADQw/ByL-gHH2-FsfOEuGfhCO5ukvDLj8WIJjACHMYCw/image_thumb%255B63%255D?imgmax=800" border="0"></a><p><br><p>Francisco Ismael Reis<p><a href="https://lh3.googleusercontent.com/-zdICeCJACw0/XIB1hGgOwAI/AAAAAAAADQ0/XGRkgf5Pm_kIRW73VjDvRKe-o-VsazORgCHMYCw/s1600-h/Assinatura%255B3%255D"><img width="141" height="41" title="Assinatura" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" alt="Assinatura" src="https://lh3.googleusercontent.com/-p1qrAX-Gxjo/XIB1hqH74qI/AAAAAAAADQ4/QX9Hjd2AyJYGn4Jy44Ae-BJmWdbAqjFZACHMYCw/Assinatura_thumb%255B1%255D?imgmax=800" border="0"></a><p>06/03/2019Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-33841130346417192792019-03-05T17:55:00.001-03:002019-03-05T17:55:21.839-03:00O assassinato cometido para ocultar uma descoberta matemática 'perigosa'<p align="right">Dalia Ventura - BBC Mundo<h4><font color="#800000" face="Georgia" size="3"><font style="font-weight: normal;">Há quem a descreva como uma das mais fundamentais de toda a ciência, mas, em seu tempo e contexto, era uma ameaça de magnitudes incomensuráveis.</font></font></h4><h4><font color="#800000" face="Georgia" size="3" style="font-weight: normal;">Dizem que certa manhã, em meados do século VI a.C., um homem foi jogado em mar aberto no litoral da Grécia.</font></h4><p><a href="https://lh3.googleusercontent.com/-eXKvSKdmEHI/XH7F2mcQ6uI/AAAAAAAADLU/z6J5lrvjFrodFxtrWXoeXjTavylBNAvEgCHMYCw/s1600-h/image%255B38%255D"><img width="342" height="462" title="image" style="border: 0px currentcolor; border-image: none; margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-Ye9cU4KRrGs/XH7F4dBneiI/AAAAAAAADLY/vutrNrLs3ok8ioBQOofH_dSuQQ7QtxHagCHMYCw/image_thumb%255B20%255D?imgmax=800" border="0"></a></p><blockquote><p>Hipaso de Metaponto se tornou um herege pitagórico</p></blockquote><blockquote><p><small><em>Foto: BBC News Brasil </em></small><hr><p><small><em><br></em></small></p></blockquote><p><font color="#800000" face="Georgia" size="3">Seu nome era Hipaso de Metaponto, matemático, teórico da música e filósofo pré-socrático.</font><p><font color="#800000" face="Georgia" size="3">Ele foi abandonado à sua própria sorte - e ela não poderia ser outra senão sua morte.</font><p><font color="#800000" face="Georgia" size="3">Como frequentemente acontece com o que conhecemos sobre o mundo antigo, há aqueles que acreditam que isso aconteceu, enquanto outros questionam essa versão.</font><p><font color="#800000" face="Georgia" size="3">Ninguém ainda foi capaz de verificar se essa parte da história é verdadeira.</font><p><font color="#800000" face="Georgia" size="3">Mas a outra parte dela é a mais interessante: a razão pela qual eles queriam matá-lo.</font><p><font color="#800000" face="Georgia" size="3">E é que poucos assassinatos têm um motivo tão assombroso quanto a descoberta da incomensurabilidade e da irracionalidade, matematicamente falando.</font><p><a href="https://lh3.googleusercontent.com/-rImWP5e5FlM/XH7F57rh9WI/AAAAAAAADLc/9g9uy8jwdcI-rZJqib04-NLd4rajMQSRQCHMYCw/s1600-h/image%255B8%255D"><img width="421" height="341" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-ePGG6dzhdZ8/XH7F7hKdo5I/AAAAAAAADLg/29_Jxr6iW8IXFz1lX740sSV8e7BV8I-eQCHMYCw/image_thumb%255B4%255D?imgmax=800" border="0"></a><blockquote><p>Não se sabe muito sobre a vida de Pitágoras, aqui pintada por Rafael, mas diz-se que ele viajou extensivamente pelo Oriente Médio antes de retornar à Grécia</p></blockquote><blockquote><p><small><em>Foto: Getty Images / BBC News Brasil</em></small><hr></blockquote><h5><font color="#800000" face="Georgia" size="3">Uma estrela da antiguidade</font></h5><p><font color="#800000" face="Georgia" size="3">A história tem início com uma das celebridades da Grécia antiga, Pitágoras de Samos (c. 580-c. 500 aC), o criador do famoso teorema que você deve ter aprendido na escola.</font><p><font color="#800000" face="Georgia" size="3">Pitágoras é, na verdade, um personagem controverso. Como ele não deixou nada por escrito, muitos se perguntaram se ele realmente foi o autor de muitas das descobertas revolucionárias no campo da matemática que lhe foram atribuídas.</font><p><font color="#800000" face="Georgia" size="3">A única prova é de que ele fundou uma escola, embora seus ensinamentos fossem considerados suspeitos e seus seguidores, estranhos.</font><p><a href="https://lh3.googleusercontent.com/-wTjIstHAdsM/XH7F86mAunI/AAAAAAAADLk/QfjZKPQ3Zo8c98He0dwJjMfXZt2EIjNDwCHMYCw/s1600-h/image%255B12%255D"><img width="449" height="312" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-tGyLfhhLz9E/XH7F-tR30QI/AAAAAAAADLo/dQ1e8mNpLakkJ5642U4recfM9mQU3eKWgCHMYCw/image_thumb%255B6%255D?imgmax=800" border="0"></a><blockquote><p>De acordo com fontes antigas, Pitágoras convenceu seus seguidores de que ele havia descido para o inferno e tinha visto as almas torturadas dos poetas (incluindo Homero e Hesíodo), atormentadas por trair os segredos dos deuses. "Pitágoras emergindo do submundo", de Salvatore Rosa.<p><small><em>Foto: Getty Images / BBC News Brasil</em></small><hr></blockquote><h5><font color="#800000" face="Georgia" size="3">Os pitagóricos</font></h5><p><font color="#800000" face="Georgia" size="3">As escolas dos pitagóricos se assemelhavam mais a uma seita, porque elas não apenas compartilhavam conhecimento.</font><p><font color="#800000" face="Georgia" size="3">Os alunos, homens e mulheres, levavam uma vida estruturada de estudo e exercício, inspirados por uma filosofia baseada na matemática.</font><p><font color="#800000" face="Georgia" size="3">Os primeiros pitagóricos eram de classe média alta e politicamente ativos.</font><p><font color="#800000" face="Georgia" size="3">Eles formaram uma elite moral que se esforçou para aperfeiçoar sua forma física nesta vida para obter a imortalidade na próxima.</font><p><font color="#800000" face="Georgia" size="3">Segundo os pitagóricos, para libertar a alma e alcançar a imortalidade, o corpo mortal tinha que seguir uma rigorosa disciplina de forma a permanecer moralmente puro e livre da natureza básica.</font><p><font color="#800000" face="Georgia" size="3">Caso contrário, a alma reencarnaria repetidamente, ou "transmigraria", até que fosse liberada pelo mérito acumulado.</font><p><font color="#800000" face="Georgia" size="3">Os pitagóricos também acreditavam no cosmos, que na época se referia a uma ideia de perfeita ordem e beleza em todo o Universo.</font><p><font color="#800000" face="Georgia" size="3">Embora eles provavelmente acreditassem no politeísmo grego clássico, manifestavam fé em uma divindade superior, que estava acima de todas as outras.</font><p><font color="#800000" face="Georgia" size="3">Eles tinham uma série de tabus, que incluíam carne e feijão, e viviam de acordo com uma série de regras que governavam todos os aspectos da vida.</font><p><a href="https://lh3.googleusercontent.com/-D29oEkeEEuw/XH7F_8yQbHI/AAAAAAAADLs/iy7ONBiZawkzftiWpf5RqatrdaQ2V8SbQCHMYCw/s1600-h/image%255B16%255D"><img width="462" height="271" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-kJgp6UDRZ6w/XH7GBNjPyeI/AAAAAAAADLw/9Ys26nWEa88TKyb0xf5nl4zBwD6mG0I_wCHMYCw/image_thumb%255B8%255D?imgmax=800" border="0"></a><blockquote><p>Lembra do Teorema de Pitágoras?<p><small><em>Foto: Getty Images / BBC News Brasil</em></small><hr></blockquote><h5><font color="#800000" face="Georgia" size="3">Triângulos e quadrados</font></h5><p><font color="#800000" face="Georgia" size="3">Outra coisa é certa: quando falamos de Pitágoras, pensamos imediatamente nas relações entre os lados dos triângulos retângulos, descoberta que aludia a egípcios e babilônios.</font><p><font color="#800000" face="Georgia" size="3">O teorema de Pitágoras afirma que, se você pegar um triângulo retângulo e fizer quadrados em todos os lados, a área do maior quadrado será igual à soma dos quadrados dos dois lados menores.</font><p><font color="#800000" face="Georgia" size="3">Em outras palavras, a soma dos quadrados dos catetos corresponde ao quadrado da hipotenusa.</font><p><font color="#800000" face="Georgia" size="3">É um teorema que ilustra uma das características da matemática grega: em vez de depender apenas de números, apela-se à geometria.</font><p><a href="https://lh3.googleusercontent.com/-YCttxLl3N3w/XH7GDHwIkkI/AAAAAAAADL0/4SEBSB0mNmAQstxiu8f1ngnelWbo5cERgCHMYCw/s1600-h/image%255B20%255D"><img width="404" height="369" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-eX3p-JWtLDs/XH7GE84Z3tI/AAAAAAAADL4/acGi195jxWwClCG8fZSrM8FjkkxKV5IOQCHMYCw/image_thumb%255B10%255D?imgmax=800" border="0"></a><blockquote><p>Pitágoras com um grupo de pitagóricos, que eram vegetarianos porque não acreditavam que os animais deviam ser mortos para comer<p><small><em>Foto: Getty Images / BBC News Brasil</em></small><hr></blockquote><h5><font color="#800000" face="Georgia" size="3">Música</font></h5><p><font color="#800000" face="Georgia" size="3">Embora muitas descobertas que foram creditadas a Pitágoras tenham sido contestadas, existe uma teoria matemática que ainda lhe é atribuída e que tem a ver com a música.</font><p><font color="#800000" face="Georgia" size="3">Dizem que, ao passar por um ferreiro um dia, Pitágoras ouviu as notas produzidas pelos golpes nas bigornas e notou que elas soavam em perfeita harmonia.</font><p><font color="#800000" face="Georgia" size="3">Ao buscar uma explicação racional para entender essa melodia, ele recorreu à matemática e descobriu que os intervalos entre as notas musicais harmoniosas sempre apareciam em proporções de números inteiros.</font><p><font color="#800000" face="Georgia" size="3">Pitágoras teria ficado tão entusiasmado com a descoberta que chegou à conclusão de que todo o Universo havia sido construído a partir de números.</font><p><a href="https://lh3.googleusercontent.com/-XDvvPI7HMA8/XH7GGU4ouiI/AAAAAAAADL8/kxX0KonvNLw1JmK7oltiwmJag5GBLG5kQCHMYCw/s1600-h/image%255B24%255D"><img width="403" height="462" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-00308r0A1lo/XH7GHqVL5SI/AAAAAAAADMA/dr6W-6CUa2Me0TY_CYhgFe5oe-_9WgKZACHMYCw/image_thumb%255B12%255D?imgmax=800" border="0"></a><blockquote><p>Gravura ilustra demonstração de Pitágoras da relação matemática reconhecida entre o comprimento da corda vibrante e as notas da escala musical<p><small><em>Foto: Getty Images / BBC News Brasil</em></small><hr></blockquote><h5><font color="#800000" face="Georgia" size="3">'Tudo é um número'</font></h5><p><font color="#800000" face="Georgia" size="3">Sua doutrina de que "todas as coisas são números" era importante para a história da filosofia e da ciência.</font><p><font color="#800000" face="Georgia" size="3">Segundo ele, a essência e a estrutura de todas as coisas podem ser determinadas encontrando as relações numéricas que as expressam.</font><p><font color="#800000" face="Georgia" size="3">Originalmente, era uma ampla generalização baseada em observações como:</font><p><font color="#800000" face="Georgia" size="3">Mas os seguidores de Pitágoras tentaram aplicar seus princípios em todos os lugares com maior precisão.</font><p><font color="#800000" face="Georgia" size="3">Nessa tentativa, eles se depararam com um desafio, que emergiu de suas próprias fileiras e envolveu o teorema de Pitágoras.</font><p><a href="https://lh3.googleusercontent.com/-owNLUVYkY8U/XH7GJLL22bI/AAAAAAAADME/rfCOsrtDJU8mLJsMe-nQpnJGYcFevtWVACHMYCw/s1600-h/image%255B28%255D"><img width="445" height="443" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-uxgdR2YGQH8/XH7GKnx-fHI/AAAAAAAADMM/VEfTzGlXRA48MIv3AoQt-72xdZv0vgWiQCHMYCw/image_thumb%255B14%255D?imgmax=800" border="0"></a><blockquote><p>Raiz quadrada de 2 foi gravada na tábua de Yale<p><small><em>Foto: Getty Images / BBC News Brasil</em></small><hr></blockquote><h5><font color="#800000" face="Georgia" size="3">A ameaça</font></h5><p><font color="#800000" face="Georgia" size="3">Um dos membros de maior prestígio da Escola Pitagórica era precisamente Hipaso (que morreu afogado no Mar Mediterrâneo no início dessa reportagem).</font><p><font color="#800000" face="Georgia" size="3">Sem qualquer má intenção, Hipaso se lançou a encontrar o comprimento da diagonal de um triângulo retângulo com dois lados que medem uma unidade.</font><p><font color="#800000" face="Georgia" size="3">Talvez seja mais fácil entender dessa forma: imagine um quadrado em que cada um dos lados tem 1 unidade de comprimento.</font><p><font color="#800000" face="Georgia" size="3">Quanto mede a diagonal do quadrado?</font><p><font color="#800000" face="Georgia" size="3">Graças ao teorema de Pitágoras, podemos calcular o quadrado do comprimento do lado mais longo de um triângulo retângulo, acrescentando os quadrados dos outros dois lados.</font><p><font color="#800000" face="Georgia" size="3">Então o comprimento da diagonal ao quadrado é (1 × 1) + (1 × 1) = 2. Assim, o comprimento da diagonal é √2. Ou seja, o número que multiplicado por ele próprio totaliza 2.</font><h5><font color="#800000" face="Georgia" size="3">Mas qual é esse número?</font></h5><p><font color="#800000" face="Georgia" size="3">A raiz quadrada de 2 não é 1 porque 1 x 1 é 1.</font><p><font color="#800000" face="Georgia" size="3">E não é 2, porque 2 x 2 é 4.</font><p><font color="#800000" face="Georgia" size="3">É algo entre os dois.</font><p><font color="#800000" face="Georgia" size="3">Algo que os babilônios haviam registrado na tábua de Yale, embora não o tivessem entendido.</font><p><font color="#800000" face="Georgia" size="3">Tratava-se de um número irracional, (como π, o número de Euler e o número de ouro ou phi).</font><p><a href="https://lh3.googleusercontent.com/-6HWvhx949qk/XH7GL8HREXI/AAAAAAAADMQ/g4kveuBDLjwyFQKawkW9ZHPeNFCJc3tLACHMYCw/s1600-h/image%255B32%255D"><img width="427" height="247" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-12YJfwJiYMU/XH7GNSG0_mI/AAAAAAAADMU/CBX2iCLIwfAdXZiV8zBy5Nf_L0CpSIvGwCHMYCw/image_thumb%255B16%255D?imgmax=800" border="0"></a><blockquote><p>Eles estavam prestes a descobrir os "números perigosos", entre os quais está π, o número de Euler (2,7182818284590452353 ...) e o número de ouro ou phi (1,61803398874989484820 ...)<p><small><em>Foto: Getty Images / BBC News Brasil</em></small><hr></blockquote><p><font color="#800000" face="Georgia" size="3">Foi uma das descobertas mais fundamentais na história da ciência: o lado e a diagonal de figuras simples, como o quadrado e o pentágono regular não podem ser medidos, ou seja, a relação quantitativa não pode ser expressa por uma razão de números inteiros.</font><h5><font color="#800000" face="Georgia" size="3">O segredo</font></h5><p><font color="#800000" face="Georgia" size="3">Esses números irracionais não se encaixavam na cosmovisão pitagórica.</font><p><font color="#800000" face="Georgia" size="3">Além disso, a descoberta ameaçou destruir a própria base da filosofia de Pitágoras.</font><p><font color="#800000" face="Georgia" size="3">Basicamente, a revelação implicava que os seguidores do famoso filósofo e matemático não eram mais possuidores de uma verdade: o dogma de que tudo tem sua medida era falso e o poder que havia sido atribuído aos números também.</font><p><font color="#800000" face="Georgia" size="3">Se os números naturais, que para os pitagóricos constituíam a essência da realidade, nem sempre servissem para encontrar a medida das coisas, tampouco eram o meio de conquistar o conhecimento divino.</font><p><a href="https://lh3.googleusercontent.com/-ZQAYv1qEI1s/XH7GN5DFPAI/AAAAAAAADMY/6enqbCWMjCM2kXz2l3cFafwfZDU0zDHsgCHMYCw/s1600-h/image%255B36%255D"><img width="447" height="277" title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-hdq6k_JuYEk/XH7GOpyQDyI/AAAAAAAADMc/4opcYJoFwA8C5pRSuJmr2TdtzW9_aF6bwCHMYCw/image_thumb%255B18%255D?imgmax=800" border="0"></a><blockquote><p>Especialistas da Grécia Antiga contam que Pitágoras fez sua escola jurar que não revelaria a descoberta<p><small><em>Foto: BBC News Brasil</em></small><hr></blockquote><p><font color="#800000" face="Georgia" size="3">Especialistas da Grécia Antiga contam que Pitágoras fez sua escola jurar que não revelaria a descoberta.</font><p><font color="#800000" face="Georgia" size="3">No entanto, Hipaso insistiu em divulgar a natureza do comensurável e imensurável, o conhecimento dos números irracionais "perigosos".</font><p><font color="#800000" face="Georgia" size="3">Esse teria sido o motivo do suposto crime: silenciá-lo.</font><p><font color="#800000" face="Georgia" size="3">Provavelmente, nunca saberemos se esse foi realmente o fim da história de Hipaso de Metaponto.</font><p><font color="#800000" face="Georgia" size="3">O que sabemos é que, apesar disso, os números irracionais acabaram vindo à tona.</font><p><font color="#800000" face="Georgia" size="3">05/03/2019</font><p><font color="#800000" face="Georgia" size="3">Fonte:</font><p><a title="https://www.bbc.com/portuguese/geral-47436677" href="https://www.bbc.com/portuguese/geral-47436677">https://www.bbc.com/portuguese/geral-47436677</a>Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-40161954908707312622019-03-03T12:43:00.001-03:002019-03-04T13:54:12.886-03:00Duas séries infinitas<p>Já tive a oportunidade, em momentos distintos, <a href="https://topicosmatematicos.blogspot.com/2011/02/o-numero-de-ouro.html" target="_blank">de escrever neste blog</a>, sobre o número de ouro, representado pela letra grega φ (fi) e cujo valor é:</p> <p><a href="https://lh3.googleusercontent.com/-dRk2DRxCkc4/XHv198Oo0sI/AAAAAAAADGg/xaiUIdtgn20xeblz9uStqauTjRC0oXBPwCHMYCw/s1600-h/image%255B2%255D"><img title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" border="0" alt="image" src="https://lh3.googleusercontent.com/-XEdyTtMubvA/XHv1-v8zlYI/AAAAAAAADGk/tpYCHRGWxbMB3WXuxOVMS7ebj2sHxoZoQCHMYCw/image_thumb?imgmax=800" width="162" height="65" /></a></p> <p>Pois bem motivado pela postagem anterior – Uma equação irracional – , vou escrever sobre duas expressões matemáticas que tem o formato de séries infinitas e cujo valor é surpreendente. </p> <ul> <li>A primeira delas é:</li> </ul> <p><a href="https://lh3.googleusercontent.com/-vflSs59g9S0/XH1YLNaIlOI/AAAAAAAADJQ/qC4jcKTb_7gZ7BTAV_-NqlGDOsG_8CEsQCHMYCw/s1600-h/image3%255B1%255D"><img title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" border="0" alt="image" src="https://lh3.googleusercontent.com/-LNlqDdKaqNc/XH1YLl185eI/AAAAAAAADJU/bA3H3Xme5n88xYufUpL-2AyV2BYbAATgQCHMYCw/image3_thumb?imgmax=800" width="215" height="70" /></a></p> <p>Vamos calcular seu valor:</p> <p>Igualando essa expressão a x:</p> <p><a href="https://lh3.googleusercontent.com/-KL9mkqFjSC8/XHv2Ar1gU2I/AAAAAAAADGw/PcxnIxXl3bU09DrNZud38SELZyGKS_VngCHMYCw/s1600-h/image%255B9%255D"><img title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" border="0" alt="image" src="https://lh3.googleusercontent.com/-MRcp5L6dGW4/XHv2BHwMXYI/AAAAAAAADG0/1oGLWbp1lysEmTKkpTd3KwZlWHIbr2eqACHMYCw/image_thumb%255B3%255D?imgmax=800" width="218" height="51" /></a></p> <p>Elevando, agora, ambos os membros ao quadrado:</p> <p><a href="https://lh3.googleusercontent.com/-FPR6KkSLrFM/XHv2Bi-trQI/AAAAAAAADG4/g8KSlo0Z8Dk33YXQV9E9AQ8uoa87X-wwQCHMYCw/s1600-h/image%255B17%255D"><img title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" border="0" alt="image" src="https://lh3.googleusercontent.com/-8ZEaqWd7nQk/XHv2Cb1bU3I/AAAAAAAADG8/wkIkU4lbVz4IxXSCFAvv9u-UHBsCD9jbwCHMYCw/image_thumb%255B7%255D?imgmax=800" width="219" height="45" /></a></p> <p>Observe que nesta última equação a expressão inicial, aquela que chamamos de x e que aparece assinalada em vermelho, se repete infinitamente, assim sendo, podemos escrever que:</p> <p><a href="https://lh3.googleusercontent.com/-TRMaFbEx7Uk/XHv2C12mnaI/AAAAAAAADHA/cGYRusC-tx4gfgPPC8mD1uajjHNWKI7UwCHMYCw/s1600-h/image%255B21%255D"><img title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" border="0" alt="image" src="https://lh3.googleusercontent.com/-ZUaeV5TQGBo/XHv2DUyTUdI/AAAAAAAADHE/DY1qCzC-lRgpKGRfx47C9iCGyzkakn5kACHMYCw/image_thumb%255B9%255D?imgmax=800" width="233" height="32" /></a> </p> <p>Resolvendo essa equação do 2º grau com o auxílio da fórmula de Bhaskara:</p> <p><a href="https://lh3.googleusercontent.com/--3lFErbE5RI/XHv2D8WHMbI/AAAAAAAADHI/tbkMKc8A4pMJYewssulDEG0QmtpCBfouACHMYCw/s1600-h/image%255B25%255D"><img title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" border="0" alt="image" src="https://lh3.googleusercontent.com/-zdMvC7RdKVE/XHv2EdZrKLI/AAAAAAAADHM/ctKCQdBptQ0CAxW0y_pM58zCUGIKMjIcwCHMYCw/image_thumb%255B11%255D?imgmax=800" width="112" height="91" /></a></p> <p>Se considerarmos apenas a raiz positiva, temos que:</p> <p><a href="https://lh3.googleusercontent.com/-BssMSMXsMCE/XHv2Eyg37kI/AAAAAAAADHQ/iwkecogSsR0rfWwlCu9ZwNSJtLtH3RY8ACHMYCw/s1600-h/image%255B28%255D"><img title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" border="0" alt="image" src="https://lh3.googleusercontent.com/-GCbg4ZgNrdw/XHv2FO7lsDI/AAAAAAAADHU/otpbwKBChJUH6v6Nppys0OIlF1-STVjzwCHMYCw/image_thumb%255B12%255D?imgmax=800" width="163" height="65" /></a></p> <p>Que como vimos anteriormente, trata-se do número de ouro.</p> <ul> <li>A outra expressão é:</li> </ul> <p><a href="https://lh3.googleusercontent.com/-0rbQ7CfZ57M/XH1YMYTTRQI/AAAAAAAADJY/yhzetJs94S0G4VK73y2QORbdFlmuWHtRwCHMYCw/s1600-h/image29%255B4%255D"><img title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" border="0" alt="image" src="https://lh3.googleusercontent.com/-_m7WrOWOYpM/XH1YM3L7yyI/AAAAAAAADJc/96xNFiYFM7U73v3iUoGM3Uy4DfCQ8AUJACHMYCw/image29_thumb%255B3%255D?imgmax=800" width="205" height="156" /></a></p> <p>Também aqui, vamos chamar de x a essa expressão:</p> <p><a href="https://lh3.googleusercontent.com/-cfag7oq9KuE/XHv2Gq5GzfI/AAAAAAAADHg/C93VQXMA3ZoldbLrxxReSjZC_Yv0lOX6wCHMYCw/s1600-h/image%255B42%255D"><img title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" border="0" alt="image" src="https://lh3.googleusercontent.com/-61fmfKk-dLE/XHv2HMUZ8BI/AAAAAAAADHk/s-h89L3LNNAppqdWQPqqlnidVIwVSAyrgCHMYCw/image_thumb%255B18%255D?imgmax=800" width="196" height="139" /></a></p> <p>Repare que nesta última equação, a parte em vermelho é o próprio x, ou seja:</p> <p><a href="https://lh3.googleusercontent.com/-3o3kCJod5Qk/XHv2HkmfmzI/AAAAAAAADHo/Poi7-wH-i7MdtFDa81vsPnG3s6kSODRTQCHMYCw/s1600-h/image%255B46%255D"><img title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" border="0" alt="image" src="https://lh3.googleusercontent.com/-3IGhcJ5Rv1Q/XHv2IWoklhI/AAAAAAAADHs/74L21KJHQmcBdV_EQPpRtlCCr8gqzi1pwCHMYCw/image_thumb%255B20%255D?imgmax=800" width="91" height="56" /></a></p> <p>Multiplicando ambos os membros dessa equação por x, obtemos:</p> <p><a href="https://lh3.googleusercontent.com/-4QkiGZGOcyo/XHv2JV6U6kI/AAAAAAAADHw/8dV307CA45c8rYBIG3WGqj50x3w1XVndACHMYCw/s1600-h/image%255B50%255D"><img title="image" style="margin: 0px; border: 0px currentcolor; border-image: none; display: inline; background-image: none;" border="0" alt="image" src="https://lh3.googleusercontent.com/-0wJFDOO1KEg/XHv2J1irZUI/AAAAAAAADH0/OXEsoVXQG2ALmvokkyph3fIsQeXdMMhNQCHMYCw/image_thumb%255B21%255D?imgmax=800" width="233" height="32" /></a></p> <p>Que tem como raízes:</p> <p><a href="https://lh3.googleusercontent.com/-38qskXWpnfA/XHv2KbOB-cI/AAAAAAAADH4/J_ZZurAmH5wH0CqAC0woIpktpkWmTes3wCHMYCw/s1600-h/image%255B57%255D"><img title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" border="0" alt="image" src="https://lh3.googleusercontent.com/-M27epuxrvyo/XHv2K3NouSI/AAAAAAAADH8/XXwe9mgzf80kmIJ4mHzS7MHQ5oWZ1QRTgCHMYCw/image_thumb%255B24%255D?imgmax=800" width="95" height="61" /></a></p> <p>Considerando apenas a raiz positiva:</p> <p><a href="https://lh3.googleusercontent.com/-7MWkw76EeIM/XHv2LZEeAOI/AAAAAAAADIA/ukfSKDBMhwAM9pczn41DvYvnkTL0j1SVgCHMYCw/s1600-h/image%255B61%255D"><img title="image" style="margin-right: auto; margin-left: auto; float: none; display: block; background-image: none;" border="0" alt="image" src="https://lh3.googleusercontent.com/-R9eMuL64OQw/XHv2L48tr0I/AAAAAAAADIE/o7nPIBUHP1QPm8UJ6JKdBWPcvlcMwwNMQCHMYCw/image_thumb%255B25%255D?imgmax=800" width="163" height="65" /></a></p> <p>E mais uma vez, temos como solução o número de ouro.</p> <p>Francisco Ismael Reis.</p> <p><a href="https://lh3.googleusercontent.com/-7BpagreIsM8/XHv2Mtf_1dI/AAAAAAAADII/nK-PGeFQebsXoBhPfYnY6F0BHfBgFu9nACHMYCw/s1600-h/image%255B66%255D"><img title="image" style="border: 0px currentcolor; border-image: none; display: inline; background-image: none;" border="0" alt="image" src="https://lh3.googleusercontent.com/-B_v5vu3lj0I/XHv2NLT36qI/AAAAAAAADIM/DHAhQG8WuqYjon9keSJuPO_-Kv5ZUnAtwCHMYCw/image_thumb%255B28%255D?imgmax=800" width="170" height="58" /></a></p> <p>03/03/2019</p>Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-12727110038688613942019-02-28T23:17:00.001-03:002019-03-01T14:25:26.345-03:00Uma equação irracional<h4>Raphael Sakamoto Ligero, <font style="font-weight: normal;">u</font><font style="font-weight: normal;">m aluno meu do 3EM do Colégio Guilherme Dumont Villares bastante interessado pela Matemática, me propôs encontrar as soluções reais da seguinte equação irracional:</font><br></h4>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://4.bp.blogspot.com/-hlE0w3RliFw/XHiRWhNGnJI/AAAAAAAADDw/AXLSUr9rDhAjoYk7S34QGQ8ouAbm57UNwCLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr.jpg" imageanchor="1"><img style="border: 0px currentcolor; border-image: none; background-image: none;" src="https://4.bp.blogspot.com/-hlE0w3RliFw/XHiRWhNGnJI/AAAAAAAADDw/AXLSUr9rDhAjoYk7S34QGQ8ouAbm57UNwCLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr.jpg" border="0" data-original-width="181" data-original-height="42"></a></div>
<div align="center"><br></div>
Apresento a seguir minha solução:<br>
Observemos inicialmente que os valores de x encontram-se no intervalo real<br>
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<a style="text-align: center; margin-right: 1em; margin-left: 1em;" href="https://4.bp.blogspot.com/-24PmJ_t1914/XHiSzBvzabI/AAAAAAAADEE/fyx2gTJuRconIkteI7aHhAYQEgjNbBdFgCLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr0.jpg" imageanchor="1"><img src="https://4.bp.blogspot.com/-24PmJ_t1914/XHiSzBvzabI/AAAAAAAADEE/fyx2gTJuRconIkteI7aHhAYQEgjNbBdFgCLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr0.jpg" border="0" data-original-width="59" data-original-height="18"></a></div>
valores esses que definem a condição de existência da equação.<br>
Note que essa equação é equivalente a:<br>
<br>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://4.bp.blogspot.com/-jKtmgRTaybU/XHiRsJxUwiI/AAAAAAAADD4/gnAr5z8OwoETZwTAmg-6zUloBllohFJ6ACLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr2.jpg" imageanchor="1"><img src="https://4.bp.blogspot.com/-jKtmgRTaybU/XHiRsJxUwiI/AAAAAAAADD4/gnAr5z8OwoETZwTAmg-6zUloBllohFJ6ACLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr2.jpg" border="0" data-original-width="288" data-original-height="55"></a></div>
<br>
Que, por sua vez, é equivalente a:<br>
<br>
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<a style="clear: left; margin-right: 1em; margin-bottom: 1em; float: left;" href="https://4.bp.blogspot.com/-Hm5Q5M-wrm4/XHiTueZcRoI/AAAAAAAADEM/FwnRP6oPam4jenwBv56QuMqIC5-bcMjiwCLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr3.jpg" imageanchor="1"><img src="https://4.bp.blogspot.com/-Hm5Q5M-wrm4/XHiTueZcRoI/AAAAAAAADEM/FwnRP6oPam4jenwBv56QuMqIC5-bcMjiwCLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr3.jpg" border="0" data-original-width="116" data-original-height="32"></a></div><p>
<br></p><p><font size="1"></font><br>
Elevando ambos os membros ao quadrado, ficamos com:<br>
<br></p>
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<a style="clear: left; margin-right: 1em; margin-bottom: 1em; float: left;" href="https://1.bp.blogspot.com/-R5ftXG-veu4/XHiT3QYbGHI/AAAAAAAADEQ/X6JP6fnXtm8Q6w5d-EdiM02b6xAHuAJJACLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr4.jpg" imageanchor="1"><img width="400" height="180" src="https://1.bp.blogspot.com/-R5ftXG-veu4/XHiT3QYbGHI/AAAAAAAADEQ/X6JP6fnXtm8Q6w5d-EdiM02b6xAHuAJJACLcBGAs/s400/Equa%25C3%25A7%25C3%25A3oIrr4.jpg" border="0" data-original-width="383" data-original-height="173"></a></div>
<br>
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<br>
Essa equação pode ser fatorada como apresentado a seguir:<br>
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<a style="clear: left; margin-right: 1em; margin-bottom: 1em; float: left;" href="https://2.bp.blogspot.com/-BIZSEjvjuDI/XHiUKYFCAlI/AAAAAAAADEc/--4LzveH-CsYpVMrZK8H32FnZ8L0LafawCLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr5.jpg" imageanchor="1"><img src="https://2.bp.blogspot.com/-BIZSEjvjuDI/XHiUKYFCAlI/AAAAAAAADEc/--4LzveH-CsYpVMrZK8H32FnZ8L0LafawCLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr5.jpg" border="0" data-original-width="307" data-original-height="32"></a></div>
<br>
<br>
<br>
De onde sai que:<br>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://1.bp.blogspot.com/-P9B7YM5weBU/XHiXsvqfyuI/AAAAAAAADFU/GHawcz-sSykeua-9d5gWdgHLh0jANoAPACLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr6.jpg" imageanchor="1"><img style="border: 0px currentcolor; border-image: none; background-image: none;" src="https://1.bp.blogspot.com/-P9B7YM5weBU/XHiXsvqfyuI/AAAAAAAADFU/GHawcz-sSykeua-9d5gWdgHLh0jANoAPACLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr6.jpg" border="0" data-original-width="207" data-original-height="146"></a> <a style="margin-right: 1em; margin-left: 1em;" href="https://1.bp.blogspot.com/-4YXPkCiJ-YI/XHiUezLRlQI/AAAAAAAADEo/Gd7VcprC6igPtGvMX6CJbx1xVyw-z_XcACLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr7.jpg" imageanchor="1"><img width="219" height="147" src="https://1.bp.blogspot.com/-4YXPkCiJ-YI/XHiUezLRlQI/AAAAAAAADEo/Gd7VcprC6igPtGvMX6CJbx1xVyw-z_XcACLcBGAs/s1600/Equa%25C3%25A7%25C3%25A3oIrr7.jpg" border="0" data-original-width="243" data-original-height="163"></a></div>
<div class="separator" style="text-align: center; clear: both;"><br></div>
<div align="center"> </div>
As soluções x<sub>1</sub> e x<sub>3</sub>, não satisfazem à condição de existência da equação, logo:<br>
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<a href="https://lh3.googleusercontent.com/-zFJ7TQxQHMs/XHlq_m_o_bI/AAAAAAAADFw/Us8YT9YWwaI_56MGJMNmgY0pqIuwFuE3wCHMYCw/s1600-h/image%255B2%255D"><img width="150" height="51" title="image" style="margin: 0px; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-hR1HkEMvP24/XHlrABtg05I/AAAAAAAADF0/5KyPrprvHvIhpUQvIHSBrFkNHIAfl9zIQCHMYCw/image_thumb?imgmax=800" border="0"></a></div>
<br>
<div align="center"> e </div>
<div align="center"><br></div>
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<a href="https://lh3.googleusercontent.com/-u0_EF8U7L_M/XHlrAraPGEI/AAAAAAAADF4/EgrS83AyTHEXg36y525BCXV3uu6nFZBTwCHMYCw/s1600-h/image%255B5%255D"><img width="140" height="48" title="image" style="margin: 0px; display: inline; background-image: none;" alt="image" src="https://lh3.googleusercontent.com/-KyzI84CeJaA/XHlrBTt5VSI/AAAAAAAADF8/ekXriAc0G3gDqLziCI4UG1eQ1nO1j17_ACHMYCw/image_thumb%255B1%255D?imgmax=800" border="0"></a></div>
representam as soluções da equação.<br>
A comprovação gráfica das soluções dessa equação é apresentada no gráfico abaixo:<br>
<br>
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<a style="margin-right: 1em; margin-left: 1em;" href="https://1.bp.blogspot.com/-eBd_KvzHEW8/XHiU4JxeBZI/AAAAAAAADE8/1j1UjXqFWwU2MFUJiUsnjZhzwcTaD_MDwCLcBGAs/s1600/Image%2B1.jpg" imageanchor="1"><img src="https://1.bp.blogspot.com/-eBd_KvzHEW8/XHiU4JxeBZI/AAAAAAAADE8/1j1UjXqFWwU2MFUJiUsnjZhzwcTaD_MDwCLcBGAs/s1600/Image%2B1.jpg" border="0" data-original-width="403" data-original-height="504"></a></div>
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<br>
Francisco Ismael Reis<br>
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28/02/2019<br>
<a href="file:///C:/Users/Ismael/AppData/Local/Temp/OpenLiveWriter-1377794338/supfiles3598B/EquaoIrr31.jpg"><br></a>Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-4014751972725390212017-01-04T19:52:00.001-02:002019-02-27T20:07:43.061-03:00Arredondamentos<br>
<img width="181" height="245" align="right" style="float: right; display: inline;" alt="Resultado de imagem para arredondamentos em matemática" src="https://encrypted-tbn3.gstatic.com/images?q=tbn:ANd9GcTaRvooob6ZgIZBvXYEn2ONzmh5DTVsOlCFSxhnNnRm0-g9Q5IECA">Em muitas situações do dia a dia, principalmente quando trabalhamos com números muito grandes ou números com muitas casas decimais, não é necessário que o façamos com os valores exatos. Por vezes, para facilitar os cálculos e até mesmo a leitura desses números, é interessante que façamos arredondamentos.<br>
<br>
Vamos considerar dois casos:<br>
<br>
<ul>
<li><span style="color: rgb(192, 80, 77); font-size: medium;">Arredondamento com números decimais</span></li>
</ul>
Inicialmente devemos decidir com quantas casas decimais queremos trabalhar e eliminar aquelas que não nos interessam. Feito isso, vamos olhar para o primeiro algarismo à direita daqueles que permanecerão. Se esse algarismo for menor que 5, nada muda, caso contrário, aumentamos o último algarismo do número com que queremos ficar de 1 unidade.<br>
Acompanhe os exemplos:<br>
<table style="width: 454px; height: 157px;" border="1" cellspacing="0" cellpadding="2"><tbody>
<tr> <td width="100" valign="top"><div align="center">
Número</div>
</td> <td width="88" valign="top"><div align="center">
Casas decimais</div>
</td> <td width="112" valign="top">Arredondamento</td> <td width="100" valign="top"><div align="center">
Como fica</div>
</td> </tr>
<tr> <td width="100" valign="top"><div align="center">
82,274369</div>
</td> <td width="88" valign="top"><div align="center">
4</div>
</td> <td width="112" valign="top"><div align="center">
82,2743|<strong><span style="color: red;">6</span></strong>9</div>
</td> <td width="100" valign="top"><div align="center">
82,2744</div>
</td> </tr>
<tr> <td width="100" valign="top"><div align="center">
82,274369</div>
</td> <td width="88" valign="top"><div align="center">
3</div>
</td> <td width="112" valign="top"><div align="center">
82,274|<span style="color: red;"><strong>3</strong></span>69</div>
</td> <td width="100" valign="top"><div align="center">
82,274</div>
</td> </tr>
<tr> <td width="100" valign="top"><div align="center">
82,274369</div>
</td> <td width="88" valign="top"><div align="center">
2</div>
</td> <td width="112" valign="top"><div align="center">
82,27|<span style="color: red;"><strong>4</strong></span>369</div>
</td> <td width="100" valign="top"><div align="center">
82,27</div>
</td> </tr>
<tr> <td width="100" valign="top"><div align="center">
82,274369</div>
</td> <td width="88" valign="top"><div align="center">
1</div>
</td> <td width="112" valign="top"><div align="center">
82,2|<span style="color: red;"><strong>7</strong></span>4369</div>
</td> <td width="100" valign="top"><div align="center">
82,3</div>
</td> </tr>
</tbody></table>
<br>
<ul>
<li><span style="color: rgb(192, 80, 77); font-size: medium;">Arredondamento com números inteiros</span></li>
</ul>
Com números grandes, o arredondamento é feito de acordo com a ordem desejada, seguindo-se a mesma lógica utilizada no arredondamento dos números decimais.<br>
Relembremos como se classificam os números em classes e ordens.<br>
<br>
Observe agora os exemplos a seguir:<br>
<table style="width: 459px; height: 156px;" border="1" cellspacing="0" cellpadding="2"><tbody>
<tr> <td width="93" valign="top">Número</td> <td width="98" valign="top">Ordem</td> <td width="109" valign="top">Arredondamento</td> <td width="100" valign="top">Como fica</td> </tr>
<tr> <td width="93" valign="top">419.735.237</td> <td width="98" valign="top">dezena</td> <td width="109" valign="top">419.735.23|<span style="color: red;"><strong>7</strong></span></td> <td width="100" valign="top">419.735.240</td> </tr>
<tr> <td width="93" valign="top">419.735.237</td> <td width="98" valign="top">centena</td> <td width="109" valign="top">419.735.2|<span style="color: red;"><strong>3</strong></span>7</td> <td width="100" valign="top">419.735.200</td> </tr>
<tr> <td width="93" valign="top">419.735.237</td> <td width="98" valign="top">u. de milhar</td> <td width="109" valign="top">419.735.|<span style="color: red;"><strong>2</strong></span>37</td> <td width="100" valign="top">419.735.000</td> </tr>
<tr> <td width="93" valign="top">419.735.237</td> <td width="98" valign="top">u. de milhão</td> <td width="109" valign="top">419.|<strong><span style="color: red;">7</span></strong>35.237</td> <td width="100" valign="top">420.000.000</td> </tr>
</tbody></table>
<br>
Francisco Ismael Reis.<br>
<a href="https://lh3.googleusercontent.com/-WLsjUD9uakk/WG1uiQm6e3I/AAAAAAAACX8/XoitgzqQ870/s1600-h/AssinaturaFundoCla%25255B3%25255D.jpg"><img width="100" height="40" title="AssinaturaFundoCla" style="border: 0px currentcolor; border-image: none; padding-top: 0px; padding-right: 0px; padding-left: 0px; display: inline; background-image: none;" alt="AssinaturaFundoCla" src="https://lh3.googleusercontent.com/-bkAE8RsknyE/WG1ui8IDAkI/AAAAAAAACYA/Q6-U6mqJ4QI/AssinaturaFundoCla_thumb%25255B1%25255D.jpg?imgmax=800" border="0"></a><br>
04/01/2017.Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-77454571352157573462017-01-04T15:23:00.001-02:002019-02-23T22:14:16.301-03:00O número do sapato<br />
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<img align="right" alt="Resultado de imagem para sapatos pre históricos ilustrações" src="https://encrypted-tbn3.gstatic.com/images?q=tbn:ANd9GcQ5UI7wsFO3WGavr10lFe20PXOeg7_JJVQ2UcMAvQp-kKo_nos2Tw" style="display: inline; float: right;" />A história dos calçados começa quando o homem percebeu que seus pés eram sensíveis e precisavam de proteção para ajudá-lo a se locomover através dos diferentes tipos de solo, principalmente os mais pedregosos.<br />
Acredita-se que o ofício de sapateiro tenha nascido com os antigos egípcios por volta de 1500 A.C.<br />
<br />
Entretanto, foi no século XIII, na Inglaterra, que o rei Eduardo I estabeleceu que uma polegada (2,54 cm, o tamanho do seu polegar) seria o equivalente a 3 grãos de cevada secos e alinhados. Surgia, assim, a primeira padronização para se medir o tamanho dos sapatos e uma nova unidade métrica, o <strong><em>ponto</em></strong>. Dessa forma, ao pé de uma pessoa que medisse 40 grãos de cevada de comprimento, seria associado número 40. Esse padrão é utilizado, ainda hoje, na Inglaterra e nos Estados Unidos.<br />
Países, como o Brasil, adotam sistemas diferentes desse, mas baseados na ideia de <strong><em>ponto</em></strong>. Aqui, o ponto equivale a 2/3 de cm (0,66 cm).<br />
Para escrevermos a função que expressa o número do nosso sapato, vamos chamar de <strong><em>s</em></strong> à variável dependente que representa, em pontos, o número do sapato e de <strong><em>x</em></strong> à variável independente que representa, em centímetros, o comprimento do pé.<br />
Assim:<br />
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<a href="https://2.bp.blogspot.com/-S8qQHRifBeU/XHHn5sHYQII/AAAAAAAAC-8/-BkDQ-rkuRMk3mainmaZtXj6KzusJXZKACLcBGAs/s1600/Sapato.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="46" data-original-width="104" src="https://2.bp.blogspot.com/-S8qQHRifBeU/XHHn5sHYQII/AAAAAAAAC-8/-BkDQ-rkuRMk3mainmaZtXj6KzusJXZKACLcBGAs/s1600/Sapato.gif" /></a></div>
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Dessa maneira, a um pé cujo comprimento seja de 26 cm, corresponderá, em pontos, o número:<br />
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<a href="https://4.bp.blogspot.com/-5AgPDz3CQu0/XHHqurPcf2I/AAAAAAAAC_I/7ilgh87nIrYPKtaVUP8gRtbOj4uZTbEkwCLcBGAs/s1600/Sapato2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="128" data-original-width="130" src="https://4.bp.blogspot.com/-5AgPDz3CQu0/XHHqurPcf2I/AAAAAAAAC_I/7ilgh87nIrYPKtaVUP8gRtbOj4uZTbEkwCLcBGAs/s1600/Sapato2.gif" /></a></div>
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<br />
Francisco Ismael Reis<br />
<a href="https://lh3.googleusercontent.com/-8oLvGn6aQ4s/WG0vegFXbYI/AAAAAAAACXQ/fzecmx4NI_0/s1600-h/AssinaturaFundoCla%25255B3%25255D.jpg"><img alt="AssinaturaFundoCla" border="0" height="40" src="https://lh3.googleusercontent.com/-GOB86sMG__8/WG0ve8iF-MI/AAAAAAAACXU/U4P8Xt_L7vg/AssinaturaFundoCla_thumb%25255B1%25255D.jpg?imgmax=800" style="background-image: none; border-bottom-width: 0px; border-left-width: 0px; border-right-width: 0px; border-top-width: 0px; display: inline; padding-left: 0px; padding-right: 0px; padding-top: 0px;" title="AssinaturaFundoCla" width="100" /></a><br />
04/01/2017.Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-2879161202708079322016-12-31T10:39:00.002-02:002019-02-23T21:53:28.594-03:00Uma curiosidade com números de 3 algarismos<br />
Escolha um número qualquer de 3 algarismos e multiplique-o, sucessivamente, por 7, 11 e 13.<br />
Observe os exemplos:<br />
1) Escolhendo o número 217<br />
<blockquote>
217 x 7 = 1519<br />
1519 x 11 = 16709<br />
16709 x 13 = 217217</blockquote>
2) Escolhendo 539<br />
<blockquote>
539 x 7 = 3773<br />
3773 x 11 = 41503<br />
41503 x 13 = 539539</blockquote>
Experimente fazer o mesmo, escolhendo outros números.<br />
Note que, qualquer que seja o número de três algarismos escolhido, o resultado final é formado pelo número escolhido, escrito duas vezes.<br />
Tente achar uma explicação para isso.<br />
31/12/2016Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0tag:blogger.com,1999:blog-7649133033293598080.post-14384491748187034982016-12-30T20:25:00.001-02:002016-12-31T09:32:50.002-02:00Palíndromos<p> </p> <p> </p> <p><a href="https://lh3.googleusercontent.com/-gG5IK64h0qE/WGbetddNRMI/AAAAAAAACVQ/4UwD5m83VZM/s1600-h/image%25255B2%25255D.png"><img title="image" style="border-left-width: 0px; border-right-width: 0px; background-image: none; border-bottom-width: 0px; float: right; padding-top: 0px; padding-left: 0px; display: inline; padding-right: 0px; border-top-width: 0px" border="0" alt="image" src="https://lh3.googleusercontent.com/-zAc4zbsr640/WGbetws8xHI/AAAAAAAACVU/Linx7Go4bnk/image_thumb.png?imgmax=800" width="162" align="right" height="161" /></a></p> <p>Uma palavra, frase ou qualquer sequência de algarismos que tenha a propriedade de apresentar o mesmo resultado ao ser lida da direita para a esquerda ou da esquerda para a direita é chamada <b>palíndromo</b>.</p> <p>O número 1234321 é palíndromo ou capícua.</p> <p>Num palíndromo, são desconsiderados os sinais ortográficos, bem como os espaços entre palavras.</p> <p>Abaixo, alguns exemplos de frases palíndromas.</p> <p> </p> <p>1. A BASE DO TETO DESABA</p> <p>2. A CARA RAJADA DA JARARACA</p> <p>3. A DIVA EM ARGEL ALEGRA-ME A VIDA</p> <p>4. A GRAMA É AMARGA</p> <p>5. A MALA NADA NA LAMA</p> <p>6. A TORRE DA DERROTA</p> <p>7. ADIAS A DATA DA SAÍDA</p> <p>8. AME O POEMA</p> <p>9. AMOR A ROMA</p> <p>10. ANOTARAM A DATA DA MARATONA</p> <p>11. ANOTARAM A MARATONA</p> <p>12. APÓS A SOPA</p> <p>13. ARARA RARA</p> <p>14. ATÉ O POETA</p> <p>15. EVA ASSE ESSA AVE</p> <p>16. LÁ VOU EU EM MEU OVAL</p> <p>17. LIVRE DO PODER VIL</p> <p>18. LUZ AZUL</p> <p>19. O CASACO</p> <p>20. O CASO DA DROGA DA GORDA DO SACO</p> <p>21. O CÉU SUECO</p> <p>22. O GALO AMA O LAGO</p> <p>23. O LOBO AMA O BOLO</p> <p>24. O ROMANO ACATA AMORES A DAMAS AMADAS E ROMA ATACA O NAMORO</p> <p>25. ÓDIO DO DOIDO</p> <p>26. ÓTO COME DOCE SECO DE MOCOTÓ</p> <p>27. RIR, O BREVE VERBO RIR</p> <p>28. ROMA ME TEM AMOR</p> <p>29. SAIRAM O TIO SÁ E OITO MARIAS</p> <p>30. SOCORRAM-ME SUBI NO ONIBUS EM MARROCOS</p> <p> </p> <p><a href="https://lh3.googleusercontent.com/-4uy8_-TGw-4/WGbeuQp2ztI/AAAAAAAACVY/wltB-mGdOH4/s1600-h/image%25255B5%25255D.png"><img title="image" style="border-left-width: 0px; border-right-width: 0px; background-image: none; border-bottom-width: 0px; padding-top: 0px; padding-left: 0px; margin: 0px; display: inline; padding-right: 0px; border-top-width: 0px" border="0" alt="image" src="https://lh3.googleusercontent.com/-L3_3eVFSMT8/WGbeu9TeoAI/AAAAAAAACVc/sovPFuTKzHo/image_thumb%25255B1%25255D.png?imgmax=800" width="244" height="138" /></a></p> <p> </p> <p>30/12/2016</p>Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com2tag:blogger.com,1999:blog-7649133033293598080.post-71069758160115038262012-06-02T11:31:00.001-03:002019-02-23T21:56:46.395-03:00Fazendo o dinheiro circular<br />
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<a href="https://1.bp.blogspot.com/-zu_BpkVP83k/XHHruGHhC3I/AAAAAAAAC_U/377cjfobPMEO7E-OR05e8HuXAW9WzG_LgCLcBGAs/s1600/dineiro.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="183" data-original-width="275" src="https://1.bp.blogspot.com/-zu_BpkVP83k/XHHruGHhC3I/AAAAAAAAC_U/377cjfobPMEO7E-OR05e8HuXAW9WzG_LgCLcBGAs/s1600/dineiro.jpg" /></a></div>
Um viajante chega numa cidade e entra num pequeno hotel. Na recepção, entrega duas notas de R$ 100,00 e pede para ver um quarto. <br />
Enquanto o viajante inspeciona os quartos, o gerente do hotel sai correndo com as duas notas de R$ 100,00 e vai até o açougue pagar suas dívidas com o açougueiro. <br />
Este pega as duas notas e vai até um criador de suínos a quem, coincidentemente, também deve R$ 200,00 e quita a dívida. <br />
O criador, por sua vez, pega também as duas notas e corre ao veterinário para liquidar uma dívida de R$ 200,00. <br />
O veterinário, com as duas notas em mãos, vai até a farmácia quitar a sua dívida que coincidentemente era de R$ 200,00. <br />
O dono da farmácia sai com o dinheiro em direção ao hotel, lugar onde, às vezes, se hospeda e que ultimamente não havia pago pelas acomodações. Valor total da dívida: R$ 200,00. Ele avisa ao gerente que está pagando a conta e coloca as notas em cima do balcão. <br />
Nesse momento, o viajante retorna dos quartos e diz não ser o que esperava, pega as duas notas de volta, agradece e sai do hotel. <br />
Ninguém ganhou ou gastou um centavo; porém, agora, toda a cidade vive sem dívidas, com o crédito restaurado e começa a ver o futuro com confiança! <br />
<strong>MORAL DA HISTÓRIA: NÃO QUEIRA ENTENDER ECONOMIA!</strong>Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com2tag:blogger.com,1999:blog-7649133033293598080.post-23238954491659134832012-04-14T17:31:00.001-03:002012-04-14T17:42:43.671-03:00A verdade matemática<p><a href="http://lh3.ggpht.com/-qdEf3cLEvwQ/T4ngbALliMI/AAAAAAAABbc/8XFG1jp5Zsw/s1600-h/image%25255B4%25255D.png"><img style="background-image: none; border-bottom: 0px; border-left: 0px; margin: 0px 0px 0px 10px; padding-left: 0px; padding-right: 0px; display: inline; float: right; border-top: 0px; border-right: 0px; padding-top: 0px" title="image" border="0" alt="image" align="right" src="http://lh5.ggpht.com/-IWXR_gzNRpk/T4ngdAM7_NI/AAAAAAAABbg/4ZO1HW66HNA/image_thumb%25255B2%25255D.png?imgmax=800" width="176" height="187" /></a>Para averiguar qual das profissões era mais criativa, trancaram um engenheiro eletrotécnico, um matemático e um físico em celas separadas. As celas, de 4m x 3m, tinham apenas uma cama e uma geladeira fechada a cadeado.  <br />Os prisioneiros foram avisados de que as portas só seriam abertas dali a uma semana. <br />Passado o prazo, abriram a porta do engenheiro: estava deitado, gordo e bem disposto, a ouvir rádio. <br /> <br />– Como é que?…</p> <p>– Fácil. Com o canivete suíço com que ando sempre, abri o cadeado. Desmontei a geladeira para tirar peças que não lhe são essenciais, mas que deram para fazer o rádio. <br /> <br />Abriram a porta do físico. Estava deitado, gordo mas chateado por não ter nada que fazer.</p> <p>– Como é que?…</p> <p>– Isso foi fácil, bastou aplicar as leis da termodinãmica. Apliquei a serpentina ao cadeado, que congelado é facil de quebrar. Mas é um saco ficaraqui sem fazer nada! <br /> <br />Abriram a porta do matemático. Estava magérimo, a andar de um lado para o outro, e a murmurar:</p> <p>– Suponhamos que a porta da geladeira está aberta… Suponhamos que a porta da geladeira está aberta… Suponhamos que a porta da geladeira está aberta… Suponhamos que a porta da geladeira está aberta…</p> <p>Fonte: <a href="http://aalfacinha.net/2008/05/07/190/">http://aalfacinha.net/2008/05/07/190/</a></p> Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com1tag:blogger.com,1999:blog-7649133033293598080.post-47362377607801834052012-04-08T15:40:00.001-03:002019-02-28T22:30:33.855-03:00O astrolábio<br>
<div class="separator" style="text-align: center; clear: both;">
<a style="clear: right; margin-bottom: 1em; margin-left: 1em; float: right;" href="https://4.bp.blogspot.com/-FoLefP_-EvY/XHHsZW4NweI/AAAAAAAAC_c/796iiSaxBqoWIfYALcMA8X4_b49DI8kFACLcBGAs/s1600/Astrolabio1555.jpg" imageanchor="1"><img width="158" height="200" src="https://4.bp.blogspot.com/-FoLefP_-EvY/XHHsZW4NweI/AAAAAAAAC_c/796iiSaxBqoWIfYALcMA8X4_b49DI8kFACLcBGAs/s200/Astrolabio1555.jpg" border="0" data-original-width="199" data-original-height="251"></a></div>
<span style="font-size: xx-small;"><font size="3">O astrolábio era um antigo instrumento para medir a altura dos astros acima do horizonte, utilizado na Idade Média para fins astrológicos e astronômicos. Também era utilizado para resolver problemas geométricos, como calcular a altura de um edifício ou a profundidade de um poço. Era formado por disco de latão graduado na sua borda, num anel de suspensão e numa mediclina (espécie de ponteiro). O astrolábio náutico era uma versão simplificada do tradicional e tinha a possibilidade apenas de medir a altura dos astros para ajudar na localização em alto mar.</font></span><br>
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<a style="clear: left; margin-right: 1em; margin-bottom: 1em; float: left;" href="https://3.bp.blogspot.com/-oxhOHOeX0RE/XHHtCkgYB2I/AAAAAAAAC_o/O-ebN3tZON8xlHqZ_nxwivgHKXhDdge6QCLcBGAs/s1600/astrolabio1616.jpg" imageanchor="1"><img width="159" height="200" src="https://3.bp.blogspot.com/-oxhOHOeX0RE/XHHtCkgYB2I/AAAAAAAAC_o/O-ebN3tZON8xlHqZ_nxwivgHKXhDdge6QCLcBGAs/s200/astrolabio1616.jpg" border="0" data-original-width="160" data-original-height="201"></a></div><font size="2"><font size="3">
O quadrante era um astrolábio reduzido à quarta parte. Muitos exemplares espalhados pelo mundo foram fabricados em Portugal e têm o nome ou marca de seu fabricante, como Agostinho de Gois Raposo, Francisco Gois e João Dias. Poucos astrolábios náuticos chegaram até os nossos dias, mas com o desenvolvimento da arqueologia subaquática foi possível recuperar mais exemplares. Há atualmente cerca de 80 e são mundialmente registrados no Museu Marítimo de Greenwich. Além de um número de registro passaram também a serem conhecidos por um nome, normalmente relacionados com o navio ou local onde foram encontrados.</font><br>
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<a style="clear: right; margin-bottom: 1em; margin-left: 1em; float: right;" href="https://4.bp.blogspot.com/-1U2JLqOlVws/XHHt8rJD1qI/AAAAAAAAC_w/DX5o-Qw_wPwRiqdaKJMGQdLOw7CFuRGIgCLcBGAs/s1600/AstrolabioUtiliza%25C3%25A7%25C3%25A3o.jpg" imageanchor="1"></a></div>
<font size="2"><a style="clear: right; margin-bottom: 1em; margin-left: 1em; float: right;" href="https://4.bp.blogspot.com/-1U2JLqOlVws/XHHt8rJD1qI/AAAAAAAAC_w/DX5o-Qw_wPwRiqdaKJMGQdLOw7CFuRGIgCLcBGAs/s1600/AstrolabioUtiliza%25C3%25A7%25C3%25A3o.jpg" imageanchor="1"><img src="https://4.bp.blogspot.com/-1U2JLqOlVws/XHHt8rJD1qI/AAAAAAAAC_w/DX5o-Qw_wPwRiqdaKJMGQdLOw7CFuRGIgCLcBGAs/s1600/AstrolabioUtiliza%25C3%25A7%25C3%25A3o.jpg" border="0" data-original-width="241" data-original-height="166"></a><font size="3">Inicialmente tinha a forma da parte de trás do astrolábio </font><a style="clear: right; margin-bottom: 1em; margin-left: 1em; float: right;" href="https://4.bp.blogspot.com/-1U2JLqOlVws/XHHt8rJD1qI/AAAAAAAAC_w/DX5o-Qw_wPwRiqdaKJMGQdLOw7CFuRGIgCLcBGAs/s1600/AstrolabioUtiliza%25C3%25A7%25C3%25A3o.jpg" imageanchor="1"></a><font size="3">tradicional. </font><a style="clear: right; margin-bottom: 1em; margin-left: 1em; float: right;" href="https://4.bp.blogspot.com/-1U2JLqOlVws/XHHt8rJD1qI/AAAAAAAAC_w/DX5o-Qw_wPwRiqdaKJMGQdLOw7CFuRGIgCLcBGAs/s1600/AstrolabioUtiliza%25C3%25A7%25C3%25A3o.jpg" imageanchor="1"></a><font size="3">No entanto, com a experiência dos pilotos sua forma foi aperfeiçoada. Deixou de ser fabricado em chapa de metal ou madeira e passou a ser feito de cobre para que ficasse mais pesado, cerca de dois quilos e menos sujeito ao balanço do navio. O disco inicial foi parcialmente aberto para diminuir a resistência ao vento. O manejo do astrolábio exigia a participação de duas pessoas; consistia em grande círculo, por cujo interior corria uma régua; um homem suspendia o astrolábio na altura dos olhos, alinhando a régua com o sol enquanto outro lia os graus marcados no círculo.<br>
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<span style="font-size: xx-small;">Fonte: </span><a href="http://www.museutec.org.br/previewmuseologico/o_astrolabio.htm"><span style="font-size: xx-small;">http://www.museutec.org.br/previewmuseologico/o_astrolabio.htm</span></a>Francisco Ismael Reishttp://www.blogger.com/profile/11691776380126088126noreply@blogger.com0