Does Euler's Proof of Pi's Irrationality Meet Modern Rigor Standards?

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Discussion Overview

The discussion centers on the comparison of Lambert's and Euler's proofs of the irrationality of π, specifically examining whether Euler's approach meets modern standards of mathematical rigor. The scope includes historical context, mathematical reasoning, and the evaluation of proofs.

Discussion Character

  • Debate/contested
  • Historical
  • Mathematical reasoning

Main Points Raised

  • One participant notes that Lambert was the first to prove π's irrationality in 1761 using a continued fraction for the tangent function, while Euler presented a simpler derivation in 1780 based on Lagrange's binomial continued fraction.
  • Another participant points out the chronological order of the proofs, confirming that Euler's work came after Lambert's.
  • There is a question raised about whether Euler's manipulations of the continued fraction require additional justifications to align with modern rigor standards.
  • Some participants express that while Lambert's proof is more complex, they are curious about the sufficiency of Euler's methods.

Areas of Agreement / Disagreement

Participants generally agree on the historical timeline of the proofs but express differing views on the rigor of Euler's approach compared to Lambert's. The discussion remains unresolved regarding the adequacy of Euler's justifications.

Contextual Notes

Participants have not fully explored the specific mathematical steps involved in Euler's proof, nor have they clarified the definitions of rigor being applied in the modern context.

chaoky
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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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Isn't 1780 later than 1761?
 
AD yes, BC no.
 
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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