Talk:Square root of 2

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)

Proposed: two additional (√2) series representations

Dear Readers:

I have never edited a Wikipedia page so I'm placing this proposed contribution on the Talk page for review by more experienced contributors. My proposed contribution to the Series and Product representation section:

The number can be represented by one plus an infinite sum of differences between alternating odd-numbered convergents of the ${\sqrt {2}}$ continued fraction, with denominators defined by the recurrence relation $a(n)=35a(n-1)-35a(n-2)+a(n-3);a(0)=5,a(1)=145,a(2)=4,901$ or $a(n)=34a(n-1)-a(n-2)-24;a(0)=5,a(1)=145.$

${\sqrt {2}}=1+\sum _{n=0}^{\infty }{\frac {2}{a(n)}}=1+{\frac {2}{5}}+{\frac {2}{145}}+{\frac {2}{4,901}}+{\frac {2}{166,465}}+{\frac {2}{5,654,885}}+{\frac {2}{192,099,601}}+\cdots$

The number can be represented by three-halves minus an infinite sum of differences between alternating even-numbered convergents of the ${\sqrt {2}}$ continued fraction, with denominators defined by the recurrence relation $a(n)=35a(n-1)-35a(n-2)+a(n-3);a(0)=24,a(1)=840,a(2)=28,560$ or $a(n)=34a(n-1)-a(n-2)+24;a(0)=24,a(1)=840.$

${\sqrt {2}}={\frac {3}{2}}-\sum _{n=0}^{\infty }{\frac {2}{a(n)}}={\frac {3}{2}}-{\frac {2}{24}}-{\frac {2}{840}}-{\frac {2}{28,560}}-{\frac {2}{970,224}}-{\frac {2}{32,959,080}}-{\frac {2}{1,119,638,520}}-\cdots$ 

References

I have created a page on my blog that provides more information about this contribution. Thank you for your attention. Meditate085 (talk) 01:16, 28 March 2025 (UTC)

Do you have reliable sources about this, e.g. a published journal paper, scholarly monograph, textbook, or the like? Wikipedia is not a good place for publishing original research, see WP:OR. –jacobolus (t) 02:50, 28 March 2025 (UTC)