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Curso

UMA INFINIDADE DE INFINITOS

Responsável: Jorge Rocha

Descrição: O universo da nossa intuição é finito. Quando contamos, medimos, calculamos áreas, pesamos, etc., obtemos valores necessariamente finitos, valores esses que podem até ser estimados por aproximações. Quando passamos para o mundo não finito (i.e infinito) podemos ser traídos pela intuição. Uma barra de dois metros pode ser medida usando primeiro uma régua de um metro (uma única vez) depois uma régua de meio metro (também uma só vez), seguida de uma régua de ¼ de metro.... na n-ésima medição usando uma régua de ½^n metros e assim sucessivamente.

No “fim” conseguimos medir a barra? Sim, no entanto usámos um número infinito de réguas  e o tempo para o fazer depende da nossa habilidade. Se não formos  infinitamente rápidos demoraremos literalmente uma eternidade!

Neste pequeno curso, necessariamente finito, abordaremos as seguintes questões:

  • Quantos elementos tem um conjunto?
  • Como é que sabemos se dois conjuntos têm o mesmo número de elementos?
  • Há mais números racionais do que números primos?
  • Há mais números reais do que números racionais?
  • Há muitos “infinitos”?
  • A soma de uma infinidade de números pode ser igual a um número real? Podemos trocar a ordem das parcelas?
  • Como é que podemos construir conjuntos estranhos (fractais)?
  • Qual é a grande vantagem de ser proprietário de um hotel com um número infinito de quartos?
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