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Irrationality of pi proof

  1. Dec 12, 2012 #1
    I was wondering about Lambert's proof of π's irrationality. Supposedly he was the first one to prove it in 1761 when he derived a continued fraction for the tangent function. Then I was reading through some of Euler's translated papers when I stumbled upon the same continued fraction in Euler's E750 paper (Commentatio in fractionem continuam, qua illustris La Grange potestates binomiales expressit) (English translation http://arxiv.org/abs/math/0507459). This was delivered to the St. Petersburg Academy of sciences in 1780 and Euler's derivation seems to be much simpler (based on Lagrange's binomial continued fraction) although Lambert's proof is more widely known (and a bit more involved). Do Euler's manipulation of the original continued fraction follow the modern standards of rigor? Or is there more justification needed when Euler was toying around with the fraction?
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  3. Dec 13, 2012 #2


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    Isn't 1780 later than 1761?
  4. Dec 18, 2012 #3
    AD yes, BC no.
  5. Dec 18, 2012 #4
    Well, yes, although Lambert's proof is much more involved. Are there any additional justifications to Euler's manipulation of Lagrage's fraction that are needed?
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