[過去ﾛｸﾞ] ガロア第一論文と乗数イデアル他関連資料スレ13 (1002ﾚｽ)
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722(2): 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2025/02/11(火) 19:45:53.55 ID:zr+dFWV7(13/15) AAS
つづき

https://en.wikipedia.org/wiki/Proofthat%CF%80isirrational
Proof that π is irrational
In the 1760s, Johann Heinrich Lambert was the first to prove that the number π is irrational, meaning it cannot be expressed as a fraction
a/b, where
a and b are both integers. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus. Three simplifications of Hermite's proof are due to Mary Cartwright, Ivan Niven, and Nicolas Bourbaki. Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich. Many of these are proofs by contradiction.
In 1882, Ferdinand von Lindemann proved that
π is not just irrational, but transcendental as well.[1]

Lambert's proof
略

Hermite's proof
略

Cartwright's proof
略

Niven's proof
略

Bourbaki's proof
略

Laczkovich's proof
略
以上

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723: １３２人目の素数さん [] 2025/02/11(火) 19:48:42.26 ID:MW1+hP7T(56/61) AAS
>>721-722　数学のスの字もわからん馬鹿素人は口をはさむなｗ

肝心なことは全部略のくせにｗｗｗ

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736(1): 現代数学の系譜 雑談 ◆yH25M02vWFhP [] 2025/02/11(火) 23:27:40.41 ID:zr+dFWV7(15/15) AAS
>>724
> https://manabitimes.jp/math/2697

ご苦労さまです
それ >>722 https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational
Proof that π is irrational
にあるよ Niven, Ivan (1947)だね

Niven's proof
This proof uses the characterization of
π as the smallest positive zero of the sine function.[9]
Suppose that
π is rational, i.e.
π=a/b
for some integers
a and b
which may be taken without loss of generality to both be positive. Given any positive integer
n, we define the polynomial function:
f(x)=x^{n}(a-bx)^{n}/{n!}
and, for each
x∈R let
F(x)=f(x)-f''(x)+f^4(x)+・・・ +(-1)^nf^2n(x).
Claim 1:
F(0)+F(π)} is an integer.
以下略す

References
9. Niven, Ivan (1947), "A simple proof that π is irrational" (PDF), Bulletin of the American Mathematical Society, vol. 53, no. 6, p. 509, doi:10.1090/s0002-9904-1947-08821-2

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