Does someone know a proof (books, articles) that $\pi$ is not a quadratic irrational?
The proof should not use that $\pi$ is transcendental.
Any hints would be appreciated.
Does someone know a proof (books, articles) that $\pi$ is not a quadratic irrational?
The proof should not use that $\pi$ is transcendental.
Any hints would be appreciated.
As mentioned in the comments Lambert proved that $\pi$ is irrational (http://www.pi314.net/eng/lambert.php). Lambert showed that it's irrational by first demonstrating that the continued fraction expansion holds: $$\tan(x) = \cfrac{x}{1 - \cfrac{x^2}{3 - \cfrac{x^2}{5 - \cfrac{x^2}{7 - {}\ddots}}}}.$$ He then proved that if $x$ is non-zero and a rational number then the expression must be irrational. Consequently because $\tan(\frac {\pi}{4}) = 1$, it follows that $\frac {\pi}{4}$ is irrational and as a result that $\pi$ is irrational. (Refer to the Wikipedia article on the proof of $\pi$ being irrational by Lambert - Lambert's Proof)
Laczkovich's Proof is a simplification of Lambert's proof (refer here for more information).
I found a few resources detailing numerous proofs of $\pi$ being irrational, none of them utilising $\pi$ as transcendental. The Wikipedia article lists 6 different proofs (including Lambert's and Laczkovich's). There was also a question asked on this math forum about understanding the following proof of $\pi$ being irrational making use of polynomials and calculus (A simple proof that $\pi$ is irrational).
I hope this helps.