Talk:Square root of 2/Archive 1

I am not sure the sentence The discovery of the irrational numbers is usually attributed to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2 is quite exact.

If Hippasus has discovered the inrrationnality, that is with the pentagram and certainly not with a square. There is a hypothesis attributing the discovery to Hippasus, this hypothesis is not usual, neither Szabó^[1], nor Becker^[2] thinks that this attribution is correct. No one attributes a proof to Hippasus, proofs occur much later. He is sometime credited for the discovery, not the proof^[3]. Jean-Luc W (talk) 07:24, 6 April 2010 (UTC)

A french contributor with a poor level of english.

You can't discover the irrationality of the square root of 2 without a proof. Dmcq (talk) 10:09, 6 April 2010 (UTC)

As a matter of fact, not only you can, but the historians agree about a long time between these two events. The grecs knew, from the mesopotamian how to aproximate √2 by fractions with bigger and bigger numerators. This algorithm gives you two sequences one decreasing one increasing converging both to √2 but they never stop to a rational value (see Árpád Szabó). The process was called dynamis. This algorithm gives a hint, but is not a proof. This algorithm is the simplest for the pentagon, the next numerator is the last denominator, the next denominator is the sum of the last denominator and the last numerator, this argument of simplicity is used by Fritz.

This idea probably inspired the Zeno of Elea with his paradox, which was for him a proof that irationality does not exist. A real proof of the existence of incommensurable has been found only after, when the idea of a demonstration by reductio as absurdum has been discovered (probably around -450).

Becker thought that the first proof is based on a different principle called the even and the odd, and used with a rectangle isocele triangle. If all sides are commensurable, you choose the biggest possible unit such that all sides are multiple of the units. The hypotenuse is even according to Pythagoras's theorem. Then the other sides are odd, otherwise you can double the length of the unit. Then it is easy to proove that the other sides are also even. A number is therefore even and odd, which is the essence of the reductio as absurdum. Jean-Luc W (talk) 11:43, 6 April 2010 (UTC)

So are you saying they just suspected it might not be a rational ratio? Why did they not 'suspect' the irrationality of pi in anyway the same way then? I get the feeling there's some history revisionism at work here like I saw in another maths article a little while ago.Dmcq (talk) 12:27, 6 April 2010 (UTC)

I am not sure the word suspected is adequat. The idea of proof like we have it now has been elaborated during the V century (between Pythagoras and Plato). For instance, in the pythagoras time, a real proof of the theorem having its name did not exist^[4]. It does not mean that they just suspected the theorem to be true, but just that the idea of a proof was just not considered as a necessity. Burker supposes that they were able to show that a rectangle triangle of side 3 and 4 units would necessary have its hypotenuse of length 5 units. We call it now monstration and not demonstration. If you trust the testimony of Aristotle, Eudemus of Rhodes or Iamblichus, it seems that they really trusted in the existence of irrationality. But it seems also sure that Thales trusted that two angles in an isocele triangle are equal, and for sure, on Thales time, the idea of mathematical proof was not invented.

To suspect the irrationality of pi suppose that you are able to compute some equivalent sequence, which is in fact difficult. You can imagine to do so with regular polygons, but computation is not easy, Archimedius stops à 92 sides. And any increasing and converging sequence does not necesseraly converge to an irrationnal (look at the one of Zeno of Elea, for instance).

I don't know what you exactly mean by revisionism. If there is an evolution in the historian conception, you are right. Neugebauer^[5] thought in 1942 the discovery was very late and very near the proof time (first quarter of the fourth century and he also thought that Pythagoras was more a myth than a real mathematician). Knorr^[6] thought in 1945 (this reference is a reedition) the time between discovery and proof was only two or three decades, in the Árpád Szabó you will see p 25 that he defends the idea that the gap is longer, and his book is newer. If you think that people like Neugebauer, Knorr, Szabó or Von Fritz are not serious historians of the main stream, you are wrong. Jean-Luc W (talk) 13:23, 6 April 2010 (UTC)

Something is wrong with your dates. I don't think Knorr thought much in 1945; it was the year he was born. —David Eppstein (talk) 04:56, 7 April 2010 (UTC)

If they argued from the evenness and oddness of the numbers it was a proof. If they said we have tried lots of numbers and don't seem to be able to do it then it was not a proof. If there are books by reputable historians saying silly things otherwise then this is wikipedia and they should be noted as saying that. Saying that what they said is actually true is altogether different though. This is what I mean by revisionism, some historian makes up a definition of proof, finds that people a long time ago don't follow his idea of what proof is and then say they didn't prove things. Dmcq (talk) 14:36, 6 April 2010 (UTC)

I've raised this at Wikipedia_talk:WikiProject_Mathematics#Hippasus_of_Metapontum_and_the_square_root_of_2 as there may be somebody there who is also interested in the history and can access some of the books so as to get a proper weight. Dmcq (talk) 14:55, 6 April 2010 (UTC)

Mathematics have not been invented in one second. The notion of proof and the necessity of logic did not arise one morning of a specific day. To say that Neugebauer, Von Fritz, Szabó or Burker are silly because they were interested by this period of awakening is maybe not my opinion. On Pythagoras's time and before, proofs in your sens did not exist. I fear that your criteria is too rigid too allow an understanding of history of mathematics. Even much later, your point could be raised. Lambert prooved the irrationality of pi without prooving the convergence of the continued fraction he used, but he is always credited of the proof. Newton could not make any logical theory about his infinitesimal calculus with modern criteria, which does not mean that historians are just saying silly things about him, or that they should start history of infinitesimal calculus with Hilbert, after the rigourous construction of R.

I invite you to check if the historians I have quoted are reputable. They all think Hyppasus has nothing to do with the square root of two and a demonstration. I also invite you to find any reputable historian saying the opposite, you will find it extremly difficult.

By the way, they don't all agree. To say that Hyppasus has discovered irrationality is clearly contreversial (but not for the reason you describe). If Von Fritz thinks that, neither Neugebauer nor Knorr nor Becker will say that Hyppasus has anything to do with irrationality. But no specialist says that Hyppasus has proved irrationality or is a specialist of the square root of 2.Jean-Luc W (talk) 16:01, 6 April 2010 (UTC)

I've added a fact tag, and deleted some poorly sourced information. The Washington Post is not a reliable source for the history of mathematics, and Weisstein provides no citation. Heath points out that there are different versions of the legend about drowning Hippasus. -- Radagast3 (talk) 02:55, 7 April 2010 (UTC)

I am a bit at a loss about what is being argued here, but Neugebauer and Knorr are real authorities on history of mathematics; Weisstein is not (whether by assertion or omission). Arcfrk (talk) 04:14, 7 April 2010 (UTC)

What is claimed is :

-The idea that Hyppasus of Metaponte has discovered incommensurability is not usual. Neither Neugebauer, not Knorr think that's true. Hyppasus is supposed to be an early pythagorician, Neugebauer thinks that discovery happend during the first quarter of the fourth century (see second foot note and Knorr before -450 (p 37 of the reference I gave for Knorr could be checked on google book).

-If Hyppasus is the author of the discovery, then it is with a pentagone and not with a square (with the golden ratio and not with square root of two). This could be checked by Von Fritz, not accessible under Google but this fact is so well known that I am sure it could be checked with google in english.

-Hyppasus is sometime credited of the discovery, but never of the proof. For instance, p 37 you can read that for Knorr discovery time is sometimes before -450 and proof happend after (a decade or more). Jean-Luc W (talk) 06:24, 7 April 2010 (UTC)

If Neugebauer is a real authority on history of mathematics, he thinks that Pythagoras is more a myth than a real mathematician. This point of view has not really been followed by the contempory main stream. The more recent historian Ruckert is, up to my understanding, more a reference on this subject. Ruckert is extremly cautious about the discovery : The only certainty about the discovery of irrationality is that Theodorus of Cyrene proved that √n (for n = 3, ... 17 and not a perfect square) is irrational. p 439 of the given reference. —Preceding unsigned comment added by Jean-Luc W (talk • contribs) 06:47, 7 April 2010 (UTC)

The story of Hippasus being drowned for revealing the existence of irrational numbers is famous but doubtful. The sources for that period in history are almost always second hand and unreliable, so what really happened is primarily a matter of speculation. I think the best way to handle these situations is to just state which authority has which opinion, sprinkling liberally with weasel words.--RDBury (talk) 14:07, 7 April 2010 (UTC)

I propose something like :

Contrary to a common received idea, there is no certainty that √2 was the first irrational ever discovered^[1]. Anyway, there is a consensus among historians stating that the first proof^[2] of irrationality concerned √2 and was found during the Vth century BC^[3]. This discovery had a major influence, not only in mathematics, but also in logic^[4] and in philosophy^[5].

And I apologize for my poor english. Jean-Luc W (talk) 15:49, 7 April 2010 (UTC)

I've cut the knot. Hippasus is a proper place to go into details about what he did or did not discovered, according to various authorities (some of whom, apparently, disagreed with themselves, to say nothing of disagreeing with each other!) Arcfrk (talk) 06:32, 8 April 2010 (UTC)

I think Jean-Luc W hs a fair point about there being no certainty that SQRT(2) was the first irrational discoverd. It is after all only a presumption (based essentially on its absence in Theaetetus' list of what he had proved, by implication thus that it was known before Theaetetus) that the Pythagorean's knew it. I think an edited section could reasonably assert that with reference to (say) T. Heath. I also agree with Arcfrk that further speculation about Hippasus is more usefully placed in his article and not here. But what surely can't be controversial is that the phrase "kept as an official secret" and the bald statement that Hippasus was murdered is without any real foundation and should be edited out. The only really significant source for all this is the Scholium on Euclid X variously attributed to Proclus or Pappus (it's a Syriac manuscript) and the relevant passage from T. Heath is 'The scholium quotes further the legend according to which" the first of the Pythagoreans who made public the investigation of these matters perished in a shipwreck," conjecturing that the authors of this story" perhaps spoke allegorically, hinting that everything irrational and formless is properly concealed, and, if any soul should rashly invade this region of life and lay it open, it would be carried away into the sea of becoming and be overwhelmed by its unresting currents."'

I propose to do some editing of this whole article in a while and will make the relevant modifications here. If ou don't agree speak now or for ever hold your peace... Rinpoche (talk) 03:19, 12 September 2010 (UTC)

Just to correct myself on re-reading the relevant passage of Heath. It was Theodorus whp provided prooofs of the irrationality of the sqaure roots of 3,5 .. 17 (implying 2 had already been accomplished) and his pupil Theaetetus who established the general case which became Euclid X, 9. Apologies Rinpoche (talk) 03:30, 12 September 2010 (UTC)

Since Lichtenberg ratio is just a recently proposed neologism for the square root of 2, we should merge the meager content to here, and give Lichtenberg his credit without enshrining the newly proposed name as an article; and redirect Lichtenberg ratio to Square root of 2#Paper size; yes? Dicklyon (talk) 05:52, 17 November 2010 (UTC)

No, I believe the appropriate article is ISO 216] or maybe paper sizes, both places where it is already mentioned. Dmcq (talk) 09:40, 17 November 2010 (UTC)

One's about a number, the other of a paper format. So very strong oppose. "Lichtenberg format" might be a better name for that page however. Headbomb {talk / contribs / physics / books} 10:47, 17 November 2010 (UTC)

What is it you are opposing and which page were you suggesting renaming? Dmcq (talk) 11:26, 17 November 2010 (UTC)

OK, I'll propose a merge to ISO 216. Dicklyon (talk) 22:44, 15 January 2011 (UTC)

There are missing pieces to what is visible for the equation for Viete's formula, namely that m is also the number of square roots.Naraht (talk) 19:22, 16 May 2012 (UTC)

It says that just underneath the formula. Dmcq (talk) 20:12, 16 May 2012 (UTC)

Isn't this section a little laboured (and actually a non sequitur at some point)? It's an immediate consequence of unique factorisation but then the unique factorisation is a big theorem. All that really needs to be observed is that having decomposed a and b into products of primes, passing to the squares introduces no new primes to cancel and unique factorisation precludes finding any new ones. Thus b=1 and the square root of 2 must be an integer which is not so. The same argument shows that the square root of any integer not a perfect square is irrational, likewise the cube root of any integer not a perfect cube, the fourth root of any integer not a fourth power ... and so on. As per my remark above I will edit this section as well on my return unless slapped on the wrist here. Rinpoche (talk) 04:33, 10 September 2010 (UTC)

Any comments here? Okay for me to edit? My own approach would be to point out that the theorem is a trivial consequence of unique factorisation (ultimately a consequence of well-ordering), then go on to remark that you don't need to go as far as a developed theory of unique factorisation, just as far as Euclid's lemma to show reduction of a rational to lowest terms is unique and use that to show the square root of a non-square integer is irrational (which is Euclid X, 9), go on to discuss the classic proof (where we only in fact need to 'cast out twos') and finally point out the proof by infinite descent which doesn't make assumptions at all about reducing the rational. Would this be acceptable? Rinpoche (talk) 02:45, 12 September 2010 (UTC)

The proof is a bit laboured okay. But in fact there is a little bit extra I would like in really which is to point out explicitly where the fundamental theorem of arithmetic is needed. Dmcq (talk) 09:16, 12 September 2010 (UTC)

The proof says in step 3: "there must be a prime" but it should say that all the primes in the factorization of b must be different from each prime in the factorization of a so that the fraction is reduced. But I'm not going to change it. I'm going to let somebody else do it who is a math major at least. (Daniel) — Preceding unsigned comment added by 108.46.6.145 (talk) 21:29, 23 June 2012 (UTC)

Isn't the presentation of the Proof by infinite descent quite a tad wrong-headed? Because it postulates 'reduced to lowest terms' but that's a whole new bag of tricks and if you're using that you don't need 'infinite descent' or, what amounts to the same thing in this context, the Well-ordering principle (the point is that the idea of reducing to lowest terms does depend on well-ordering). There's a nice rigorous discussion of the infinite descent in Infinite descent. I don't want to tread on the toes of those who maintain this page but I'll look back a few days hence and if hasn't been appropiately edited (or I've been slapped on the wrist here) take it on myself to edit. Fellow arithmancers and fanciers of all things arithmetical will recognise the issues involved as absolutely fundamental in the divine art and it would be nice if Wikipedia was respectfully accurate on the topic. Rinpoche (talk) 01:08, 10 September 2010 (UTC)

Any comments here? OK for me to edit? Rinpoche (talk) 02:30, 12 September 2010 (UTC)

Yes it would be better without the reduced to lowest terms bit. Dmcq (talk) 09:05, 12 September 2010 (UTC)

The proof there is Euclid's proof not the proof by infinite descent. This is an example why citation is so necessary in Wikipedia so things don't get messed up like this. Dmcq (talk) 09:12, 12 September 2010 (UTC)

In re Infinite Descent Not Involving Factoring: Would it be too obvious to state "[Multiply both sides by n]" and, "[Multiply both sides by sqrt(2)]" (respectively) on the first step? It's not critics looking up the proof in Wikipedia; some people actually don't know the proof. I just don't have the html/nerve to edit the page myself. — Preceding unsigned comment added by Hamiltek (talk • contribs) 15:34, 9 October 2012 (UTC)

I'd like to add that, if one multiplies both sides of m/n = sqrt(2) by sqrt(2) and solve, one "proves" that sqrt(2) = 2, and therefore 2=1. I don't know how much qualification (m/n is irreducible IS needed, 2 > sqrt(2) > 0?) this proof actually needs. This is the proof my Linear Algebra Prof. put on the board in '05 (?) — Preceding unsigned comment added by Hamiltek (talk • contribs) 16:03, 9 October 2012 (UTC)

Why can't it be 1???? —Preceding unsigned comment added by 66.167.177.92 (talk) 02:06, 6 February 2010 (UTC)

Radical 2 is approxomatly 1.41421356237309584957343 — Preceding unsigned comment added by Dakoolst (talk • contribs) 13:25, 7 June 2013 (UTC)

This proof does not fall into infinite descent category. Infinite descent is not immediate. It requires "if we start from some value, we have a smaller value, and then smaller..." "if we keep on we will reach the point that require even smaller element, but we have no smaller element in the observed set". Here smaller can mean any property. The proof in Geometric proof section is simpler, it is non-existence based on presupposition of the property of an element, in this case its minimality, but it has no recursion or induction required for descent part. Also, sample values to be used for "Figure 1" would be useful. — Preceding unsigned comment added by Aperisic (talk • contribs) 21:25, 23 March 2014 (UTC)

The article currently mentions the approximation 99/70. For certain applications (e.g. line widths of diagonals in bitmap art), this is a bit unwieldy. Perhaps it would be worthwhile to have a section stating the following common approximations and their errors:

7/5, -1.005%
10/7, +1.015%
17/12, +0.173%
24/17, -0.173%
41/29, -0.030%
58/41, +0.030%
99/70, +0.005%
140/99, -0.005%
239/169, -0.001%
338/239, +0.001%
et cetera...

These approximations actually form two sequences, and so can be extended to find even more accurate approximations - the rule being that if your last term was N/D, your next will be (N+2D)/(N+D) (there's probably a proof of this somewhere online). The sequences relate to eachother in that you can flip each fraction and double it (because we're approximating root-2). It may be worth noting that the errors of each term in one sequence is NOT actually the negative of the corresponding term in the other sequence (they're just very much in the same ballpark).

So should there be a section on approximations (I find them useful and interesting, but then again I'm biased)? A435(m) 15:24, 30 December 2006 (UTC)

What are you waiting for? Go ahead and put it in. Sympleko (Συμπλεκω) 01:17, 26 June 2007 (UTC)

Two things. First, why bother with 24/17 when it's no better than 17/12 yet involves larger integers? Second these ratios (less the suboptimal ones, namely every second item in your list) are what you get by taking only the first n terms of the continued fraction expansion given in the article. So the logical thing to do is simply list the values of these truncations after the expansion itself, namely the odd numbered lines of your table (which should start out 1/1, 3/2, 7/5, 17/12, ...) --Vaughan Pratt (talk) 23:14, 16 March 2008 (UTC)

Vaughan, 17/12 only appears to be just as good an approximation as 24/17 because the cited error is truncated 3 digits after the decimal point.

(Though it is true that 24/17 and in fact every second fraction do not belong in this sequence, according to the N/D → (N+2D)/(N+D) definition.)

Actually, a much faster-converging sequence is to start with 1/1 and then go from N/D to (N² + 2D²)/(2ND). This is Newton's method, used to find the positive root of x² - 2. Or what is exactly the same: letting the last approximation be x_n, then the next one is x_n+1 := (x_n + 2/x_n)/2.

The first 6 convergents for Newton's method, starting from 1/1, are 1, 3/2, 17/12, 577/408, 665857/470832, 886731088897/627013566048. (The last one when squared begins as 2 with 23 zeroes after the decimal point: 2.0000000000000000000000025... .)Daqu (talk) 21:59, 24 March 2014 (UTC)

From the article: The square root of two is also the only real number whose infinite tetrate is equal to its square.

I admit I haven't heard of tetration before reading this article... but doesn't 1 also share this property?

142.30.227.14 (talk) 20:03, 2 June 2009 (UTC) Dart

The word "tetration" and its back-formation "tetrate" are recently made-up words that have not gained currency among mathematicians. They should therefore be avoided, especially when much simpler words will suffice. Instead, the meaning should be stated in words that most people can understand. For example, instead of

The square root of two is also the only real number whose infinite tetrate is equal to its square

it would be much better to say

If for c > 1 we define x₁ = c and x_n+1 = c^x_n for n > 1, we will call the limit of x_n as n → ∞, if this limit exists, by the name f(c). Then sqrt(2) is the only number c > 1 for which f(c) = c².

More words, but also more clarity.Daqu (talk) 23:42, 3 April 2014 (UTC)

I made a comment in an edit summary about Sqrt(2)/2 = .707... being the source of the name for the jet liner. That is a widely circulated story, but Boeing says it isn't true. http://www.boeing.com/news/frontiers/archive/2004/february/i_history.html None the less, Sqrt(2)/2 is an important constant, e.g. its sin (45 deg) and cos (45 deg) and it warrants a mention in the article.--agr (talk) 00:48, 22 May 2015 (UTC)

@ArnoldReinhold: sorry, I missed this note and left a message on your talk page. I do not agree with your estimation and interpretation of importance of selected numbers. Do you know the fake-proof of the claim that there are no unimportant numbers? (If there were, there must be a smallest, and that number would be important therefore!) For the same reason ${\displaystyle \pi /2,\;\pi /3,\;\pi /6,\;...,\;{\sqrt {3}}/2,\;...}$ and countably more would warrant a mention in Wikipedia.

What ever, I do not care that much. All the best. Purgy (talk) 16:29, 22 May 2015 (UTC)

I've added mention of the trig functions and the OEIS reference. I suspect a search of math and physics texts would find this number has a high hit count, comparable to square root of two by itself. And it is just getting a small mention at the end of a long article. Note that the "proof" that there are no unimportant numbers fails for real numbers as there is no guarantee a set of reals will have a smallest element. But in the set of issues facing Wikipedia ranked by importance, this could be the smallest element. Best to you as well.--agr (talk) 02:56, 26 May 2015 (UTC)

Just because I'm nitpicky: I.did.not.talk.about.a.proof. :) The method would work however for "all numbers contained in Wikipedia". Purgy (talk) 08:34, 26 May 2015 (UTC)

All of those proofs assume that square root of 2 is rational number, and contradicts the it to prove that square root of 2 is an irrational number. However, that's only under an assumption that square root of 2 is a real number. If proposition "square root of 2 is a real number" is not proven, any of those proofs are insufficient to prove that square root of 2 is irrational number. Could have been an imaginary number. — Preceding unsigned comment added by 69.65.95.5 (talk • contribs)

Every rational number is real. So if it's not real, it's also automatically not rational. —David Eppstein (talk) 02:22, 10 February 2016 (UTC)

Just because it's not rational, it doesn't mean it's irrational. It could have been an imaginary number. It never proves that sqrt of 2 is irrational number. With that logic, I could easily prove that sqrt of -2 is an irrational number using the same proofing system. — Preceding unsigned comment added by 169.139.8.21 (talk) 12:03, 10 February 2016 (UTC)

Doesn't this boil down to lingo? I agree an 'not rational' being a more apt term, and on putting some emphasis on 'square root of two' being real (completeness, or whatever argument?). The introductory paragraph might not count as general premise. Purgy (talk) 07:22, 11 February 2016 (UTC)

Write m^2=2n^2 and consider the exponent k of 2 in the unique prime factorization. It should be both even (because m^2 has even number of 2's) and odd (because 2n^2 has an odd number of 2's), so we have a contradiction.

It is the quickest proof, why we do not mention it?--Pokipsy76 (talk) 12:18, 1 May 2016 (UTC)

Yes, that's nice. Assuming you have a theorem about unique prime factorization already. Dicklyon (talk) 15:18, 1 May 2016 (UTC)

You don't need unique prime factorization for this argument, only unique factorization as a product of an odd number and a power of two, along with product-of-odds-is-odd. This is even weaker than unique representation as a binary numeral since it doesn't depend on representability of odd numbers, and certainly doesn't depend on the computationally infeasible concept of prime factorization, or even the much more feasible concept of coprimality for that matter. It is Proof 3' of about 30 proofs at this website. It is highly constructive in any logic that admits the Principle of Explosion in arguments about roots of rational-coefficient quadratics like √2---apartness follows from the fact that a² = ms and 2b² = nt are at least 1 apart when m and n are odd and s and t are distinct powers of two.

The "short" proof at the beginning of this section of the article relies on a 9-line proof of a much stronger theorem. If no one has any objection I'll replace it with a suitable paraphrase of Proof 3'. Vaughan Pratt (talk) 20:32, 6 May 2016 (UTC)

That the quick rational approximation proposed here is quite good may be seen as follows: to say that

${\displaystyle {\frac {99}{70}}\approx {\sqrt {2}}}$

is to say that

${\displaystyle 99^{2}\approx 70^{2}\cdot 2.\,}$

Now notice that

${\displaystyle {\begin{aligned}99^{2}&{}=9801,\\{\text{and}}\\70^{2}\cdot 2&{}=9800.\end{aligned}}}$

Michael Hardy (talk) 15:43, 31 July 2008 (UTC)

√2=99/70 can also be obtained by considering √2=1+Δ, with Δ from a recursive formula Δ_(k+1)=1/[2+Δ_(k)] and Δ_(0)=1/2. In the case concerned, Δ=29/70. The iterative algorithm has the advantage of allowing a ratio between two numbers of any desired digits to be obtained in a systematic manner.184.146.31.190 (talk) 05:13, 7 December 2017 (UTC)

Nice. I'm convinced. Anton Mravcek (talk) 21:00, 31 July 2008 (UTC)

Denote √2=1+Δ. Δ is a correction to 1. We can calculate Δ from a recursive relationship: Δ_(k+1)=1/[2+Δ_(k)], starting with Δ_(0)=1/2. If we are interested in a ratio of two three-digit numbers, Δ=408/985 and √2=1+408/985. The estimate is accurate to seven significant figures.184.146.31.190 (talk) 04:39, 7 December 2017 (UTC)

Is the formula equal obviously? Is there some rigorous proof? Thanks. —Preceding unsigned comment added by 222.66.117.10 (talk) 14:23, 14 April 2010 (UTC)

The first number you give above is bigger than the second one. Michael Hardy (talk) 17:56, 14 April 2010 (UTC)

More specifically, the left hand side is 2 while the right hand side is 1.96157056080646... (This assumes we're always taking the positive square root.) --Vaughan Pratt (talk) 03:24, 21 April 2010 (UTC)

The expression ${\displaystyle {\sqrt {2+{\sqrt {2+{\sqrt {2+\dots }}}}}}}$ means that if the implied sequence converges to anything at all (call it x), then x has the property that x² = 2 + x. Thus it is a solution of the equation x² - x - 2 = 0. Since x² - 2 - x can be factored as (x+1)(x-2), x must be equal to either -1 or 2. Since the implied sequence consists of only positive numbers, it cannot converge to a negative number. So if there is any limit, it must be x = 2.

To see that the implied sequence does converge, notice that if in the sequence x_n is followed by x_n+1, then starting from x₀ = 0 (or in fact any other positive number), we have x_n+1 = sqrt(2 + x_n).

If we show that for any positive number y, we have the inequality |sqrt(y+2) - 2| < |y-2|/2, then we have shown that for each iteration x_n → x_n+1, the distance |x_n+1 - 2| of x_n+1 to 2 is less than half the previous distance |x_n - 2|, of x_n to 2. This is sufficient to prove that the sequence {x_n} converges to 2. By considering the cases y < 2, and y > 2, separately, the fact that for any positive number y ≠ 2 we have |sqrt(y+2) - 2| < |y-2|/2 can be proven by simple manipulation of inequalities. This (along with the fact that sqrt(2+2) = 2) proves convergence.Daqu (talk) 23:50, 24 March 2014 (UTC)

The expression can be obtained by successive substitutions of 4=2+√4 into the 4 under the square root sign on the right hand side and finally subtracting 2 from both sides of the equal sign.184.146.31.190 (talk) 05:26, 7 December 2017 (UTC)

Proposition: In base ${\displaystyle 2}$ any square must end in an even number of trailing zeros.

The proposition comes directly from, for example, multiplying a binary number with itself using the standard algorithm or simply by squaring

${\displaystyle \displaystyle \sum _{k=0}^{N}b_{k}2^{k}}$

If we can represent ${\displaystyle {\sqrt {2}}={\frac {p}{q}}}$ then

${\displaystyle \displaystyle 2q^{2}=p^{2}}$

Multiplying by ${\displaystyle 2}$ is shifting all bits of a binary number to the left, so if the number was ending in ${\displaystyle m}$ trailing zeros after multiplication by ${\displaystyle 2}$ it will end in ${\displaystyle m+1}$ zeros.

This means that ${\displaystyle 2q^{2}}$ is ending in odd while ${\displaystyle p^{2}}$ is ending in an even number of zeros. Thus, these two cannot be equal.

(Notice that this proof does not care about the relative primality of ${\displaystyle p}$ and ${\displaystyle q}$.) — Preceding unsigned comment added by 51.175.86.30 (talk) 10:35, 4 November 2018 (UTC)

51.175.86.30 (talk) 10:50, 4 November 2018 (UTC)

This is essentially the same as the first proof, the Proof by infinite descent. Except that proof doesn't assume anything about an odd or even number of zeros, that is done by the descent, and doesn't bring in base 2 so is simpler. Dmcq (talk) 17:33, 4 November 2018 (UTC)

Plus you didn't provide a reliable source. This page is for discussing possible improvements to the article. Proofs without a reliable source can't go in so talking about them does nothing towards improving the article. Dmcq (talk) 17:36, 4 November 2018 (UTC)

This is not an proof by contradiction, it is perfectly constructive. We assume that ${\displaystyle {\sqrt {2}}}$ is rational and from this we derive a contradiction. But this is just the definition of irrational (i.e. not rational). We never derive ${\displaystyle P}$ from showing that ${\displaystyle \neg P}$ is false. I would like to remove the reference to "proof by contradiction" because it is misleading! See also Proof of negation and proof by contradiction

— Preceding unsigned comment added by Txa (talk • contribs) 18:33, 13 March 2019 (UTC)

It has exactly the form of a standard proof by contradiction: assume the opposite, derive a contradiction, conclude the opposite. Perhaps it can be written another way but "can be" ≠ "is". And validity in constructive mathematics is another issue entirely. —David Eppstein (talk) 15:33, 14 March 2019 (UTC)

From the article: This proof can be generalized to show that any root of any natural number is either a natural number or irrational. Where can I find such a proof? --Steerpike (talk) 19:06, 18 December 2007 (UTC)

Take the proof by infinite descent and plug in ${\displaystyle {\sqrt {n}}}$ for any positive integer ${\displaystyle n}$ in place of ${\displaystyle {\sqrt {2}}}$. --69.91.95.139 (talk) 02:15, 26 January 2008 (UTC)

The most obvious proof is to start from the other direction. The square of a non integer rational must be a non integer rational because squaring introduces no new prime factors to either the numerator or the denominator. Therefore the squre root of an integer must be either an integer or irrational (since for it to be a non integer rational would be a contradiction). Plugwash (talk) 02:29, 26 January 2008 (UTC)

Oh, interesting. Never thought about it that way. You learn something new every day. --69.91.95.139 (talk) 03:15, 6 February 2008 (UTC)

I propose the simple and easy to understand proof that any r-th root of any natural number (${\displaystyle {\sqrt[{r}]{q}}}$ where q and r are any natural number>1) is either a natural number or irrational. It can be used as a good alternative to the specific ${\displaystyle {\sqrt {2}}}$ proof.

1. Factorization into primes:

m = m1 * m2 * m3  ...  mx
mr = m1r * m2r * m3r * ... * mxr
n = n1 * n2 * n3 * ... * ny
nr = n1r * n2r * n3r * ... * nyr
q = q1 * q2 * q3 * ... * qz

2. Assume that ${\displaystyle {\sqrt[{r}]{q}}}$ is a rational number 1 ( i.e. ${\displaystyle {\sqrt[{r}]{q}}}$ = ${\displaystyle {n \over m}}$, where ${\displaystyle {n \over m}}$ is a rational number):

  q=(${\displaystyle {n \over m}}$)r
  
  m1r * m2r * m3r * ... * mxr * q1 * q2 * q3 * ... * qz = n1r * n2r * n3r * ... * nyr

On the left and right side we have sets of prime numbers: any qi ∈ Q, any mi ∈ M (on the left side of the equation) and any ni ∈ N (on the right side of the equation). We can see that the sets Q and M are disjoint.

First let’s look on sets M and N. Every element in each set must have r repetitions. It means that every particular prime number in each set must have natural number multiplied by r repetitions.

3. According to fundamental theorem of arithmetic, the both sets [Q + M] (the left side) and N (the right side) must be identical. It is highly intuitive because the same numbers cannot have a different prime factorization. It means that the M set must be a subset of the N set.

It leads to interesting characterization of set Q. To wit: in the Q set we must have also a natural number multiplied by r repetitions of each particular prime number.

In conclusion of this

${\displaystyle {n \over m}}$ must be integer for any rational ${\displaystyle {\sqrt[{r}]{q}}}$ number

what means that
The ${\displaystyle {\sqrt[{r}]{q}}}$ for q > 1 and  r > 1 can be only an irrational or natural number. Q.E.D. Wojciech M Dobkowski (talk) 11:00, 16 March 2019 (UTC)

@Wojciech M Dobkowski: I am somehow bothered for disturbing your impetus, but I also feel obliged to turn your attention to the -sometimes brute- facts that

• WP-Talk pages are no place to discuss your wishes, thoughts and achievments,
• you are adding to a thread that dates from 2008(!)
• WP-articles are not allowed to contain Original Research, neither yours nor other's,
• you must find everywhere a WP-acceptable reason to include something in WP, and, finally, that
• you should put a "signature and timestamp" at the end of each of your contributions on a TP by using the third button from left at the top of the edit window, or by placing yourself four tildes "~" at the end of your contribution.

Here comes my signature: Purgy (talk) 07:26, 16 March 2019 (UTC)

It is a commonly agreed demand in mathematics to use only those facts in a proof which cannot be omitted without spoiling the proof. However, the subsections "Proof by infinite descent" and "Proof by unique factorization" do not adhere to this principle as they resort to the Euclidean Algorithm, the concept of irreducibility, the notion of co-primes (which involves the notion of the greatest common divisor and properties of it in turn) and the Unique Prime Factorization Theorem (!), none of which is needed to prove the irrationality of √2. Instead, only basic knowledge about addition, multiplication and integer division is needed for a proof, as can be seen by the new section "Proof by induction" which I have just added.

I would therefore recomend to remove the subsections "Proof by infinite descent" and "Proof by unique factorization".

The proof in subsection "Proof by infinite descent, not involving factoring" is also an indirect induction proof (=> This procedure can be iterated ...). Why is it called "infinite descend". This notion is non-standard and normaly not used in textbooks and articles. It gives the impression for less experienced readers, that this is a proof technique different from induction. But it is only an indirect induction proof, nothing else. Understandability of an article is spoiled if different terms are used for the same notion.

Jack Rusell (talk) 18:39, 16 November 2019 (UTC)

I have removed all unsourced proofs, including yours, from the article. All material in Wikipedia, including proofs, must be supported by reliable sources. This is not the place to publish original research. Incidentally, the formatting of your mathematics was very far from MOS:MATH, but that was not why I removed it. —David Eppstein (talk) 19:01, 16 November 2019 (UTC)

I put the section back, this time providing a reference for the proof. It was not that easy to find a reference as the proof is so trivial for people familiar with the subject that it takes more time to find a reference than to write it down by heart. The proof is so elementar that it is only termed "Example" in the referenced book rather than a "Lemma" or a "Theorem".

Your "This is not the place to publish original research" made me assume that you do not have a scientific background. But then I learned from your home page that you are a professional scientist (with a huge publication record as DBLP reveals). Given this background, your assumption that a proof which is folklore for mathematians would be "original research" is even more confusing to me. Jack Rusell (talk) 23:09, 16 November 2019 (UTC)

If you made it up yourself, and don't supply a source for it, it's original research. Also, your proof is still horribly formatted. And I don't see how it is different in any substantial way from the proof by descent that follows; it is conceptualized bottom-up (induction) instead of top-down (descent) but otherwise makes more or less the same manipulations of the same quantities. What do you intend for a reader to get out of it that they wouldn't get from the other one? —David Eppstein (talk) 02:35, 17 November 2019 (UTC)

I explained my motivation already at the beginning of this talk, but I can give a brief summary here:

- "... but otherwise makes more or less the same manipulations of the same quantities." => I agree.

- "What do you intend for a reader to get out of it that they wouldn't get from the other one?" => neither one needs to know the Euclidean algorithm or what it is good for nor one needs to know what co-primes are when trying to understand "my" proof. No more math as already learned in school is required. I believe that "keep it as simple as possible" is beneficial for novices to the subject. But feel free to delete my contribution again if you disagree. Jack Rusell (talk) 17:07, 17 November 2019 (UTC)

A school-child does not need to know Euclid's algorithm or the word "coprime". They get drilled on putting fractions "in lowest terms" or "in simplest form", which is exactly what's required here. I fail to see what value the proof by induction is bringing. And if the word "coprime" is bad, I can't see how the assertion "This induction schema is sound" can be good. XOR'easter (talk) 20:07, 17 November 2019 (UTC)

"They get drilled on putting fractions "in lowest terms" or "in simplest form", which is exactly what's required here."

=> It is not required in "my" proof (see below for the reason).

"I fail to see what value the proof by induction is bringing."

=> Then give us a proof which does not use induction. The "Proof by infinite descent" is not a witness, as all proofs by infinite descent are indirect induction proofs. This is another point of critique: Why is an indirect induction proof called "proof by descent"? Look to any textbook of Formal Logic and compare what is written about induction and what about infinite descent.

The difference to "my" proof is that the "Proof by infinite descent" uses

    (a, b) ≪ (c, d) iff ∃ f ∈ ℕ. f ≥ 2 ∧ f | c ∧ f | d ∧ a = c / f ∧ b = d / f

as the well-founded relation for the (indirect) induction, whereas "my" proof uses the (simpler) well-founded relation

    (a, b) < (c, d) iff a + b < c + d .

By demanding that a and b are co-prime it is demanded that (a, b) is minimal wrt. ≪ and by inferring that both a and b are even, a contradiction is obtained as (a/2, b/2) ≪ (a, b).

By using ≪ for the induction, the notions of irreducibility and co-primes creep into the proof. In other words: The notions of irreducibility and co-primes are not necessarily required to prove the statement, but are a consequence of using ≪ instead of <.

"And if the word "coprime" is bad, I can't see how the assertion "This induction schema is sound" can be good"

=> It is only good in conjunction with the subsequent "because"-argument, since this provides the bridge between the indirect and the direct proof (both use the same well-founded relation). But the remark about soundness can be removed if one feels that this is irritating, it is not required to understand the proof.

The whole discussion is only about presentation. The "Proof by infinite descent" is so correct as a proof can be, and there is nothing at all to complain from the logic point of view. But David Eppstein is right in his "... makes more or less the same manipulations of the same quantities", so presenting both proofs would introduce to much redundancy into the article. Jack Rusell (talk) 23:24, 17 November 2019 (UTC)

The information "Shigeru Kondo calculated 1 trillion decimal places in 2010" needs a proper citation or needs to be removed. The cited reference has no mention of square root of 2 (not the live page and not the archive.org one either): "Constants and Records of Computation". Numbers.computation.free.fr. 2010-08-12. Archived from the original on 2012-03-01. Retrieved 2012-09-07.

-Paul- (talk) 18:30, 2 December 2019 (UTC)