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

  • Thread starter Thread starter chaoky
  • Start date Start date
  • Tags Tags
    Pi Proof
Click For Summary
Lambert's proof of π's irrationality, established in 1761, is recognized for its complexity and the introduction of a continued fraction for the tangent function. Euler's simpler derivation, presented in 1780, also utilizes a continued fraction based on Lagrange's work, yet raises questions about its adherence to modern rigor standards. The discussion centers on whether Euler's methods require further justification compared to Lambert's more detailed approach. Despite Euler's later date of publication, his manipulation of the continued fraction prompts scrutiny regarding its mathematical validity. Ultimately, the conversation highlights the need for a critical evaluation of historical proofs in light of contemporary standards.
chaoky
Messages
2
Reaction score
0
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?
 
Mathematics news on Phys.org
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?
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

Mathematica This Week's Finds in Mathematical Physics (Week 244)
  • · Replies 1 ·
Replies
1
Views
3K
Pure Math Preparation (the ultimate high school / pre-university book thread)
  • · Replies 15 ·
Replies
15
Views
41K
Does Mass of an Object Truly Change with Velocity?
  • · Replies 1 ·
Replies
1
Views
11K
Prospects of spin foam formalism in loop quantum gravity [Alexandrov, Roche]
  • · Replies 9 ·
Replies
9
Views
6K
Mathematica This Week's Finds in Mathematical Physics (Week 266)
  • · Replies 1 ·
Replies
1
Views
4K