Talk:Square root of 2

Unclear scope of "coprime" in Pythagorean Triple proof of irrationality.
The Sierpinski lemma is quoted as If a, b, and c are coprime positive integers such that a2 + b2 = c2, then c is never even.

As written, it's not clear whether a, b, and c are SETWISE coprime (there exists no natural number k which evenly divides a and b and c) or PAIRWISE coprime (there exists no natural number k which evenly divides more than one of a, b, and c). Reading the sufficiency clause as requiring PAIRWISE coprime numbers causes a serious problem later in the proof when we arrive at b2 + b2 = a2, where the two left-hand side elements, here b and b, are self-evidently not coprime.

One can easily show that some natural number k which evenly divides both a and b (that is, gcd(a,b)=k) must evenly divide c2 (see note [1] below), meaning the situation where a and c are pairwise coprime but a and b aren't cannot arise; but this lemma goes unstated, and it certainly isn't so obvious as to escape comment.

I hope what I've written here is clear. My point isn't that the proof is incorrect as-such, just that it's missing one of the strings required to tie the result together.

[1] Rather, you can easily show k2|c2; it's then a bit more complicated to show that k2 | c2 ⇒ k | c, but it does follow from a comparison of their respective prime decompositions -- if there is a prime q number in the prime decomposition of k not in the prime decomposition of c, there is no square of that prime available in c2 to evenly divide it out. 2603:7000:A101:F9F1:3D23:B869:D654:A984 (talk) 22:09, 3 February 2023 (UTC)

Should I nominate this article for GA?
I feel this article can become a GA because,


 * 1) It is well-written and concise to the square root of 2.
 * 2) It is neutral (because it isn’t political)
 * 3) It is relatively stable, and does not break the three revert rule.
 * 4) It is well illustrated

However, I do not want to risk a drive-by nomination like I did at several articles, so that’s why I’m asking whether I should nominate this for GA status.  Brachy 08  (Never Gonna Give You Up, Never Gonna Let You Down)  02:56, 12 April 2023 (UTC)


 * There are many unsourced sentences, paragraphs, and even sections. I would likely quickfail it for that, even if it had a nominator who had put sufficiently many hours of work into the article and sufficiently much content to not count as drive-by. The lead does not properly summarize the body of the article, instead having its own content not repeated with more detail elsewhere. Several references appear unreliable and the reference formatting is not consistent. The long sections of formula after formula are also very dubious with respect to WP:GACR 3b, and the applications section looks thin and unbalanced. —David Eppstein (talk) 05:53, 12 April 2023 (UTC)
 * Thanks for letting me know.  Brachy 08  (Let’s Have A Kiki,  I Wanna Have a Kiki)  00:15, 17 April 2023 (UTC)

"Apostol's" proof is not Apostol's
Tom Apostal published a proof in the November 2000 issue of the American Mathematical Monthly, and people have since been calling it "Apostol's proof." I presented the same proof in classrooms several times before that, having learned it from a book published in about 1960. The attribution to Apostols is false. Michael Hardy (talk) 05:36, 10 September 2023 (UTC)

Rationalizing any digits of the (√2) by a single fraction of integers
Dear readers; I wish here only to demonstrate my recent new discovery that partly deals about the (square root 2); such as the following:

Rationalizing any digits of the (√2) by a single fraction of integers; is affirmatively possible in Circle to Circle: C2C, “CycLomeTrics”.

For instance, the next one is a source for the 32 digits equivalent of this issue:

(14398739476117879 / 10181446324101389) = 1.4142135623730950488016887242097

And c2c has also its own (√2) extraction pattern that operates infinitely than any known school thought…

I think this might be in help in this regard.

Thanks for your comments;

Abebaw Abebe Manaye (talk) 18:01, 24 December 2023‎ (UTC)


 * Hi Aboltek. I moved your comment from the article to the talk page, since it seems to be intended as a new discussion rather than a part of the article. I don't really understand what you are getting at with your comment. What does "circle to circle" mean? If you are interested in approximating √2 by rational numbers, there's some discussion in the article already, or you may want to look at Pell's equation. If your method is really a "new discovery" (not published previously) then it does not yet belong on Wikipedia, which only repeats material found in "reliable sources", and is not an appropriate place to share original research. –jacobolus (t) 08:59, 25 December 2023 (UTC)

Proof by infinite descent
The proof is not currently styled as a proof by infinite descent. Instead, it is styled as a proof using an "extremal element". The extremal element being the (a,b) relatively prime. Compare with the proof in the article on infinite descent. The small difference being that in the extremal element we appeal directly to the well-ordering principle, the set of solution is assumed non-empty and we take the minimum solution (the relatively prime a,b). In the infinite descent, we appeal to "there are no strictly decreasing sequences of natural numbers", which is equivalent to well-ordering, but the form of the proof actually produces a strictly decreasing sequence, instead of a contradiction with a minimum. In the case of the proof of the irrationality of sqrt(2), it is true that the two styles are pretty similar, but in some other cases, like in graph theory, a proof using an extremal element and a proof by infinite descent can look significantly different, even though one can always translate one into the other. Thatwhichislearnt (talk) 15:03, 10 March 2024 (UTC)

Opposite of rational not irrational
Please see https://math.andrej.com/2010/03/29/proof-of-negation-and-proof-by-contradiction/ and https://relatedwork.blogspot.com/2024/07/a-proof-of-proof-by-infinite-descent.html which are interesting reads on this. One point raised there that affects the article is that a proof that the square root of 2 is not rational does not imply it is irrational unless it is also proved to be real. After all the square root of minus 1 is not a rational number. NadVolum (talk) 23:30, 1 July 2024 (UTC)


 * Adding a proof that the square root of two is a real number seems excessive for the intended audience of this article. Maybe you could include or link one from a footnote though. (For instance a proof that the "Babylonian method" converges would suffice.) –jacobolus (t) 00:36, 2 July 2024 (UTC)