Talk:Square root of 2

"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)

There are only 115 propositions in book X of Elements

The article cites the proof of irrationality to Elements book X proposition 117, which doesn't exist. — Preceding unsigned comment added by 2601:647:C901:20C0:28AA:3E0A:5D6A:B040 (talk) 22:16, 21 August 2024 (UTC)

It exists, it is merely often numbered differently, because the consensus of scholars is that it is a later addition to Euclid (by other ancient Greek mathematicians): see [1]. —David Eppstein (talk) 23:02, 21 August 2024 (UTC)

Our discussion about the general topic of the Elements Book X, incommensurability, the Greek concept(s) of ratio and proportion, etc., could be much more complete. There are a couple of books by Knorr (1975) and Fowler (1987) as well as various papers by these authors and others, discussing the pre-Euclidean history, and there is also a long post-Euclidean history, none of which we do a very good job describing anywhere in Wikipedia. –jacobolus (t) 00:47, 22 August 2024 (UTC)

Constructive validity of the usual proof that sqrt(2) is irrational

So the proof is essentially "to prove ¬p, assume p and reach some contradiction" with p being "${\sqrt {2}}$ is rational". This is of course constructive valid because p ⇒ false is exactly the definition of ¬p.

On the other hand, the proof by contradiction is "to prove p, assume ¬p and reach some contradiction". This is not constructive valid because what we proved is, by definition, ¬¬p.

In practice we often call both "assume p to prove ¬p" and "assume ¬p to prove p" proof by contradiction, but if we want to be rigorous only the second one is the "real" proof by contradiction as the first one is constructive and logically uncontroversial. 129.104.241.224 (talk) 18:47, 2 December 2024 (UTC)