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

[chaoky]

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?

 

[arildno]

Isn't 1780 later than 1761?

 

[Diffy]

AD yes, BC no.

 

[chaoky]

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?

 

[CellCoree]

First of all, it is important to note that both Lambert and Euler were highly respected mathematicians in their time, and their contributions to the field cannot be underestimated. However, it is also important to recognize that mathematical proofs and standards of rigor have evolved and become more stringent over the years.

In terms of the specific proof of pi's irrationality, it is difficult to say definitively whether Euler's manipulation of the continued fraction meets modern standards of rigor. Without a detailed analysis of Euler's proof, it is impossible to make a definitive judgment. However, it is worth noting that the concept of a continued fraction was not fully formalized until the 19th century, so it is possible that Euler's proof may not meet modern standards of rigor.

Furthermore, it is important to remember that mathematical proofs are not always straightforward and can often involve creative and unconventional approaches. Just because Euler's proof may not meet modern standards of rigor does not necessarily mean it is incorrect or less valuable. In fact, many of Euler's proofs have stood the test of time and are still considered valid today.

Ultimately, the irrationality of pi has been proven by multiple mathematicians using different approaches, and each proof adds to our understanding and appreciation of this fundamental mathematical constant. While it is important to continue to strive for rigor and precision in mathematical proofs, it is also important to acknowledge and value the contributions of past mathematicians, even if their methods may not meet modern standards.