Talk:Square root of 2

Why this simple proof that is not listed?

Write m^2=2n^2 and consider the power 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 queckest proof, why we do not mention it?--Pokipsy76 (talk) 12:18, 1 May 2016 (UTC)

Historical evidence

Certainly this number seems to be widely believed by mathematicians to be the first known irrational number. But what is the historical evidence? Michael Hardy 23:25, 15 Apr 2005 (UTC)

I suppose you mean "the first number known to be irrational". I guess it's hard to document that something is really a "first" like that. I've seen quasi-serious speculations suggesting the golden ratio was the first number known to be irrational. Both numbers were known, in geometrical form, to the Pythagoreans, who were fond of the pentagram (full of golden ratios). I suppose the squareroot of 2 is just the most likely candidate. Anyway, reliable historical evidence (sources) seems to be a problem with most things involving the Pythagoreans.--Niels Ø 17:19, 29 October 2006 (UTC)

What it should of been

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)

Usage?

As I see it,

Every number except 1

means something different from

Every number except one

Accordingly, I think this page should be called square root of 2.

Michael Hardy 23:25, 15 Apr 2005 (UTC)

Agreed. Fredrik | talk 23:53, 15 Apr 2005 (UTC)

Image title

As from the above disussion, normally any number which is written less than (<) 10 is written in its letter form; and anything that is written greater than (>) 10 is written as their actual number. So what made you change it to "Square root of 2". I find it misleading. --Kilo-Lima 17:08, 12 November 2005 (UTC)

That may be the convention in journalism, but not in mathematics. 84.70.26.165 11:46, 29 October 2006 (UTC)

Indeed. In fact, I think even most journalistic style guides prescribe the use of the numeral when referring to the number itself instead of to a quantity. —Caesura^(t) 01:43, 31 October 2006 (UTC)

Redirect Link

Down at the bottom where I added something about silver means, the redirect link from silver means goes to the Plastic Number article, I think that it would be much more useful to have it go to the article about the Silver Ratio --Carifio24 15:58, 7 July 2006 (UTC)

Factual accuracy

Article :« The first approximation of this number was given in ancient Indian mathematical texts, the Sulbasutras (800 B.C. to 200 B.C.) as follows: Increase a unit length by its third and this third by its own fourth less the thirty-fourth part of that fourth. »

The Babylonian clay tablet YBC 7289 (1700 ± 100 BCE) displays an approximation of √2 with an accuracy of 6 × 10^-7 (1.24 51 10 in sexagesimal base). See for instance Square root approximations in Old Babylonian mathematics : YBC 7289 in context. The exact date of the Salbasutra is too imprecise to be the first approximation ever, even compared to 3/2 in Meno :-). Lachaume 21:55, 29 August 2006 (UTC)

Yes you are quite right, I've updated the article accordingly. Thanks. Paul August ☎ 20:04, 30 August 2006 (UTC)

Approximations

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)

much simpler proof

the proof in the article seems overly complex, a much simpler proof would be

1. Assume that √2 is a rational number, meaning that there exists an integer a and an integer b such that a / b = √2.
2. Then √2 can be written as an irreducible fraction (the fraction is reduced as much as possible) a / b such that a and b are coprime integers and (a / b)² = 2.
3. Clearly √2 is not an integer so b > 1
4. If a and b are coprime then a² and b² must also be coprime since squaring introduces no new prime factors
5. therefore a²/b² must be an irriducible fraction with a denominator greater than 1.
6. therefore a²/b² cannot be 2 which contridicts the initial assumption

comments? Plugwash 22:31, 10 July 2007 (UTC)

Does this argument presuppose uniqueness of prime factorization, rather than only the simpler fact about divisibility by 2? Michael Hardy (talk) 01:05, 17 March 2008 (UTC)

I think you mean contradicts

Irreducible

DarkestMoonlight (talk) 19:37, 20 March 2008 (UTC)

"Positive"

"The square root of 2... is the positive real number that, when multiplied by itself, gives the number 2"

No, I may have only taken up to intermediate algebra, but I'm pretty damn certain that there's a positive and negative square root of 2.

—Preceding unsigned comment added by Mqduck (talk • contribs)

True, but when we talk about "the" square root of two, as a real number, we mean the positive one. —David Eppstein 15:05, 7 October 2007 (UTC)

Generalized proof

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 ${\sqrt {n}}$ for any positive integer $n$ in place of ${\sqrt {2}}$. --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)

Suggested merge

The overlap with Irrational number is so great (strong emphasis there on sqrt(2)) that this article could be merged with it with negligible loss, leaving just a redirect at this article. The bit about continued fractions can be generalized to the observation that every positive algebraic number has a periodic branching continued fraction expansion (2006 observation of N.R. Zakirov), with the quadratic irrationals such as sqrt(2) being exactly the nonbranching periodic continued fractions (sqrt(2) as a simple example). --Vaughan Pratt (talk) 23:40, 16 March 2008 (UTC)

If there's some need to merge this with another article, wouldn't silver ratio be the more obvious choice? —David Eppstein (talk) 23:55, 16 March 2008 (UTC)

99/70

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

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

is to say that

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

Now notice that

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

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

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

Tetration?

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)

Removal of infobox

Based upon a discussion at Wikipedia talk:WikiProject Mathematics#"Infoboxes" on number articles, I've removed the infobox from the article. If anyone disagrees, could you please join the discussion there. Thanks, Paul August ☎ 13:57, 18 October 2009 (UTC)

I have suggested centralizing this discussion to Wikipedia_talk:WikiProject_Mathematics#Irrational_numbers_infobox and Wikipedia_talk:WikiProject_Mathematics#Infobox_with_various_expansions as it refers to an infobox occurring in several articles. Please go there to build consensus on this edit. RobHar (talk) 19:34, 18 October 2009 (UTC)

Hippasus of Metapontum and the square root of 2

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, nor Becker 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. 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. 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 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 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 [1] 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. Anyway, there is a consensus among historians stating that the first proof of irrationality concerned √2 and was found during the Vth century BC. This discovery had a major influence, not only in mathematics, but also in logic and in philosophy.

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)

Unsourced material

This article contains a hodge-podge of various facts about sqrt(2), many of them unsourced. For example, I don't recall seeing the expression "Pythagoras' constant" from the opening line before. Weisstein and OEIS both refer to Mathematical constants by Steven Finch, did it originate with him? Of course, thanks to google and multiple wiki mirrors, now it has gotten a disproportionately larger weight because that's how this article starts! I am not at all convinced that all of these formulas and proofs add value to the article, but for those that are deemed worthwhile, it would be appropriate to give precise references to scholarly sources. Arcfrk (talk) 06:47, 8 April 2010 (UTC)

Searching Google books for "Pythagoras' constant" finds the Finch reference from 2003. Searching before 2003 finds only a journal paper that calls π by that name, and Google scholar didn't find anything else. So: I think it originates with Finch, in 2003. But one can find other reliable sources that copy him or us and call it that. I'm tempted to take it out as an unimportant neologism. But if it has caught on, maybe we shouldn't? —David Eppstein (talk) 06:58, 8 April 2010 (UTC)

Thank you, David! Nothing in Math Reviews, either. I've removed it from the opening sentence, summarized our findings in the Pythagoras' constant (that used to redirect to this article), and linked it from See also. Arcfrk (talk) 04:27, 13 April 2010 (UTC)

Is ${\sqrt {2+{\sqrt {2+{\sqrt {2+\dots }}}}}}=\dots {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}$?

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 ${\sqrt {2+{\sqrt {2+{\sqrt {2+\dots }}}}}}$ 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)

Constructive proof

The proof says 'Errett Bishop (1985)' I would like to see a proper citation including page number please. Does this proof use 'valuations' as used here? The reason I'd like to see that there is a specific proof there is that there are other simpler proofs here that can be easily changed to start with p/q and end up with that it can't be the square root of 2 which cuts out the business of the exclude middle. So I'd like to see that this specific method was used. Dmcq (talk) 07:15, 12 July 2010 (UTC)

OK, done. Which modification of which proof are you thinking about? Tkuvho (talk) 12:42, 12 July 2010 (UTC)

Any of them. Just take the infinite descent one to start with

Take any rational number a/b

reduce it to lowest terms.

compare a² and 2b², both are integers

show that a b can't be in lowest terms as in the proof if these integers are equal as then a would be even

therefore the two expressions are different integers

therefore they differ by at least 1

therefore a²/b² differs from 2 by at least 1/b²

therefore one can show a/b differs from sqrt(2) by at least 1/3b²

As to citations it is best to stick <ref>...</ref> around them at where they are referred to and then the citation gets stuck automatically in the references with a link. Please see WO:CITE or look through the source for some examples to copy. I looked up Bishop initially and found a different book by him in 1985 rather than what yoiu put in for 1985. I can't see the text but does it actually use valuations of p-adic numbers to prove the square root of 2 is irrational as shown? Dmcq (talk) 11:01, 13 July 2010 (UTC)

Your proof is fine. Just one remark: there is no need to reduce to lowest terms. The infinite descent proof as stated in the article differs from what you presented in two essential ways, which make it constructively suspect: (1) it relies on an argument by contradiction, hence on LEM; (2) it does not contain any explicit lower bound for the difference with a/b, hence not constructively convincing. Other than that, your proof is a good paraphrase of the constructive proof. Bishop does not use the term "valuation"; I used it because the page I linked it to discusses parity and defines the valuation in the context of p=2. Certainly there is no need to speak of p-adics here. Tkuvho (talk) 12:52, 13 July 2010 (UTC)

Clarification required

I would appreciate a clarification of WTF this is all about. It looks like petitio principii to me. To get your inequalities (standard looking continued fraction stuff sort thing) you need to assume in the first place that a^2 - 2*b^2 = 0 is not possible (but of course that begs the question) to pass to a^2 - 2*b^2 >= 1 to get the inequalities you arrive at but I don't call that 'direct' (which would be indeed be distinctly disconcerting). To be quite frank I call it extremely (and I suspect wilfully) stupid. I suggest the entire selection is deleted. Meanwhile I've inserted the nearest thing I can find to a WTF template. Rinpoche (talk) 04:54, 10 September 2010 (UTC)

I now understand what the contributor is expressing with his remark about valuations. He is using the Unique Factorisation Theorem to assert that the number of prime factor twos in p^2 is even whereas those in 2*q^2 is odd, a contradiction which establishes the integers are distinct (but that is nevertheless equivalent to saying the SQRT(2) is irrational even from a constructive point of view), and then uses this to prove a result, which in the context of real numbers would generally be regarded as stronger from a constructivist point of view, to show that p/q and SQRT(2) are 'apart'.

The proof still use contradiction and it is wrong to suppose that constructive proofs never use contradiction. Moreover 'apartness' is not a valid distinction comparing integers as there is a decidability criteria for testing equality. These web references discuss the issues 1 proofs that don't depend on contradiction 2 Proof of negation and proof by contradiction. They both assert that it is accepted that the classic proof of the irrationality of the SQRT(2) is in fact constructive.

The section doesn't attempt to describe what a contructive proof is or even link it to the Wiki entry (and which would indeed read very strangely to a newcomer as it describes a constructive proof as one which demonstrates the existence of a thing, but how can that be accomplished in a theorem which asserts the non-existence of a thing?)

Finally the reference provided is a relatively obscure one. While Errett Bishop is a noted mathematician I suspect the contributor has misunderstood the reference.

I suggest the section should be at the very least rewritten to discuss the issue of what a constructive proof of the irrationality of SQRT(2) really amounts to (and addressing the issue that the classical proof is regarded by some at least as constructive anyway) and presents the existing material more transparently with a less obscure reference.

I suggest further that the issue really isn't notable for this article and should be deleted, perhaps transferred to Constructive proof. Rinpoche (talk) 00:48, 11 September 2010 (UTC)

I agree with that, it was more a demonstration of constructive proof than anything new about proving the square root of 2 is irrational. Dmcq (talk) 18:21, 11 September 2010 (UTC)

David Corfield explained the issues there (at the first linke) well enough. Constructively speaking, applying LEM to an efffectively decidable predicate over Z is acceptable. That's why the 2p^2-q^2>0 step is OK. Tkuvho (talk) 20:01, 11 September 2010 (UTC)

but don't you agree that failing to conclude irrationality after demonstrating p^2-2*q^2>0 would be very puzzling for most readers and needs clarification? Is it in fact non-constructive to do so? I can't see that it really can be but the whole constructive thing is not an issue for me and I've never taken much of an interest in it. Most of all, and what is really at issue here, is whether the section (and certainly as it stands) really notable for the article. I think Dmcq is quite right to say it's more about constructive proofs than SQRT(2). I think it should go. Rinpoche (talk) 02:28, 12 September 2010 (UTC)

Is it puzzling to most readers that one does not stop after demonstrating that p^2-2*q^2>0 ? Certainly. As explained in detail at the page you listed above, at this stage you only know that sert{2} is not rational. To conclude that it is irrational, you need an explicit lower bound on the distance from a rational p/q, e.g. the bound 1/3q^2 currently mentioned here. Is it in fact non-constructive to conclude irrationality without the explicit lower bound? That depends on what you mean by "constructive". In Bishopian constructivism, considered the most influential branch of constructivism today, it is in fact not enough. I understand though what you say about this not being an issue for you. However, as a general rule, persecuting alternative viewpoints is unencyclopedic. Tkuvho (talk) 04:57, 12 September 2010 (UTC)

Oh well, I stand properly corrected I'm sure. I am obliged. Perhaps all these remarks would be better off in a page on logic at which I confess I'm indeed all at sixes and sevens and no ambition at all to make it eights and better. All I can say is that as the merest of fancier of things arithmetical I hadn't the faintest idea of what the section was on about at first sight and as a matter of fact still don't and that after some effort honest. If that's what being encylopedic is all about well then persecute and be damned I say. At least the section should surely explain to simple folk other than you like me that a distinction is made between not being rational and irrational. Ever helpful I've offered a provisional edit - by all means feel free to expand on it. Rinpoche (talk) 06:04, 12 September 2010 (UTC)

Before I found this thread on the talk page, I was thoroughly mystified by this section "Constructive proof". It turns out that Rinpoche has articulated some of my concerns, which are:

1. It begins In a constructive proof, one distinguishes between on the one hand not being rational,.... This sounds like all constructive proofs are about rationals and irrationals. Is that true? If not, it should say In a constructive proof of irrationality.... or In this constructive proof....

2. As Rinpoche points out above, the section doesn't attempt to describe what a constructive proof is as the term is used here; it now links to the Wiki entry constructive proof, which as Rinpoche said describes a constructive proof as one which demonstrates the existence of a thing, which is not done here. So as it stands it uses "constructive proof" one way and implicitly defines it (via wikilink) another way.

3. It uses the (to me unfamiliar, despite my often having used the concept) term "valuation" without giving a quick definition right here for the reader who wants to be able to understand with a minimal amount of clicking around. Given that it's easy to quickly define, I think it should be defined here.

4. It says that we are applying the law of trichotomy in the context of an effectively computable predicate over N. While "trichotomy" is linked, "effectively computable predicate" is neither linked (and there is no such article to link to) nor defined. For intelligibility to the reader who doesn't already know the material, I think "effectively computable predicate" should be defined.

5. It says An easy calculation then yields a lower bound of ${\frac {1}{3b^{2}}}$ for the difference $|{\sqrt {2}}-a/b|$. The calculation may be easy, but I don't see what it is -- if it's easy, it should be given here, since this is not a proof without a proof of this assertion. Duoduoduo (talk) 18:03, 15 March 2012 (UTC)

I tried. Tkuvho (talk) 18:16, 15 March 2012 (UTC)

Thanks very much!! I'll put in the simple derivation of the bound, along with a specific section reference in the source. Duoduoduo (talk) 20:31, 15 March 2012 (UTC)

New version

I propose a new version of the constructive proof, which looks clearer to me. See below:

Constructive proof

Let $a$ and $b$ be two positive non-zero integers. The constructive proof shows that there is a finite difference between ${\sqrt {2}}$ and the rational number $a/b$, without using reductio ad absurdum. Since ${\sqrt {2}}<2$ we can assume $a/b<2$, which implies ${\sqrt {2}}+a/b<4$. Moreover, since 2 divides $2b^{2}$ an odd number of times and $a^{2}$ an even number of times, $a^{2}$ and $2b^{2}$ must be distinct, i.e. $|2b^{2}-a^{2}|\geq 1$. It clearly follows that

$|{\sqrt {2}}-a/b|={\frac {|2b^{2}-a^{2}|}{b^{2}({\sqrt {2}}+a/b)}}\geq {\frac {1}{b^{2}({\sqrt {2}}+a/b)}}>{\frac {1}{4b^{2}}}.$

This proof constructively exhibits a discrepancy between ${\sqrt {2}}$ and any rational.

Superhiggs (talk) 15:49, 4 October 2012 (UTC)

Proof by infinite descent

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)

Proof by unique factorisation

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)

Analytic proof

Shouldn't this section attempt to describe in the first place what an 'analytic' proof is (and for that matter address the nice question of whether the preceding proofs are or are not 'analytic' - whatever they are their dependence on well-ordering ensures they aren't merely algebraic)?

It's quite a substantial and I doubt particularly easy to follow discussion (I mean frankly I can't be arsed myself to read through it). The suspicion must arise it's either a plagiarism or original research and there's no reference provided either by way of peer review.

Who needs it? It's plainly out of place and I'm going to delete it when I return to edit this page unless it's deficiencies have been corrected. Rinpoche (talk) 15:15, 11 September 2010 (UTC)

It certainly looks like the usual idea of an analytic proof to me where they depend on inequalities and limits. The proof looks like a variant of the one described in Liouville number, there must be a proper name around somewhere for that as it is quite important.

I think this proof is an intrinsically interesting one, but I really would like citations like Wikipedia is supposed to provide for sections like that. Dmcq (talk) 18:14, 11 September 2010 (UTC)

Actually another tweak on Liouville's theorem gives a lower bound on the difference between any approximation and the square root of 2 like in the constructive proof. Dmcq (talk) 18:26, 11 September 2010 (UTC)

Well, I'll have a read through then (but it looks desperately tedious ... oh dear). Liouville numbers are about transcendental quanities, surely? Needles to say SQRT(2) isn't transcendental. It certainly needs a reference surely.

I've noticed that the 'geometric proof' is likewise desperately laboured. In general I don't come away with a very good impression of this article. To be plain I think it's rather weak. I do think much of it should be rewritten. However I don't want to tread on any one's toes. If the page is being actively maintained and the contributors are satisfied with their work then it's not for me to do more than point out what I consider it's deficiencies. Rinpoche (talk) 02:17, 12 September 2010 (UTC)

THe theorem provides a bound on how well one can approximate thee solution of an nth degree polynomial and proves the Lioville number is trancendental because it can be approximated better than any any such bound. The n=2 and n=1 is what applies here. Dmcq (talk) 09:02, 12 September 2010 (UTC)

Merge proposal: from Lichtenberg ratio to square root of 2

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)

Viète's formula needs to be fixed.

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)

Geometric proof

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. — Preceding unsigned comment added by Aperisic (talk • contribs) 21:25, 23 March 2014 (UTC)

Sqrt(2)/2

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 $\pi /2,\;\pi /3,\;\pi /6,\;...,\;{\sqrt {3}}/2,\;...$ 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)

Proof is not enough

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)