Talk:Irrational number: Difference between revisions

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== Definition change ==

:''[I regret that I cannot carefully edit this paragraph at this time, but in good conscience I must question whether any responsible historian of mathematics ascribes the argument below to Pythagoras. An algebraic argument for the irrationality of the square root of 2 observes that if <var>&radic;2=m/n</var>, then <var>&radic;2=(2n-m)/(m-n)</var>, so that a fraction in lowest terms is reduced to yet lower terms. That is a contradiction, completing the reductio ad absurdum. To me it is plausible that Pythagoras or someone of his school discovered a geometric argument showing that if <var>n</var> and <var>m</var> are respectively the leg and the hypotenuse of an isosceles right triangle, then <var>m-n</var> and <var>2n-m</var> are respectively the leg and the hypotenuse of a smaller isosceles right triangle. An ancient Greek geometer would have constructed the smaller triangle from the larger one, rather than doing algebra, as we do today. I would recommend that any mathematician editing this page look at Thomas Heath's translations of the writings of ancient Greek geometers before ascribing anything to Pythagoras.]''

I think the definition of irrational numbers should be modified. My definition would be "Irrational numbers are those numbers that can be defined by a finite number of integers". I am sure I am not the first one to recommend this definition, but I want to elaborate on the effect of this change. First, this makes irrational numbers countable and makes rational numbers a proper subset of irrational numbers. Second, this opens up the possibility of another class of numbers I will call the structured set. This set is defined as "numbers that, when expressed in a digital form (in any base), knowing the first N digits allows us, in theory, to calculate the next digit”. Pi fits this definition and we can generate many other structured numbers as well. An example is the number formed in the following manner: .10100100010000… This number is unique in that it fits the definition regardless of the base! Of course, any number that fits the definition will also fit the definition when raised to a rational power. Finally, the only uncountable set is the continuous set, S. Interestingly, S is the only set we cannot define an entry that is not in the structured set. User:Infinitesets <!-- Template:Unsigned --><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Wbaker716|Wbaker716]] ([[User talk:Wbaker716#top|talk]] • [[Special:Contributions/Wbaker716|contribs]]) 00:37, 31 January 2020 (UTC)</small> <!--Autosigned by SineBot-->

:<small>Please put new talk page messages at the bottom of talk pages, provide headers for new sections, and sign your messages with four tildes (<nowiki>~~~~</nowiki>) — See [[Help:Using talk pages]]. Thanks.</small>

and changed the paragraph about Pythagoras's discovery accordingly. [[User:AxelBoldt|AxelBoldt]] 03:30 Oct 23, 2002 (UTC)

:{{rto|Wbaker716}} In Wikipedia everything must be based on [[wp:reliable sources]]. There is no place here for [[wp:original research]]. I don't think we'll ever find a source that supports your proposal, for the simple reason that, for example, the rational numbers are in no way a subset of the irrational numbers, let alone a proper one. Also, the irrational numbers are ''not'' countable. Etcetera. Please read the article before proposing to make changes to it. Also note that article talk pages are not for discussions about the ''subject'', but about the ''article'' — see [[wp:Talk page guidelines]] - [[User:DVdm|DVdm]] ([[User talk:DVdm|talk]]) 09:13, 31 January 2020 (UTC)

: {{re|Wbaker716}} As DVdm said above, Wikipedia is not a place for publishing your original research (see the policy at [[Wikipedia:No original research]]). Wikipedia is not for introducing new inventions and ideas nor for developing old ones (see [[WP:What Wikipedia is not]], especially the section [[WP:NOTESSAY]]), its aim is to summarize and present the common knowledge, as documented in reliable sources (see [[WP:Verifiability]] and [[WP:Reliable sources]]). So I'd suggest you publish the revolutionary ideas in some journal on the science branch (say the MDPI's Open Access journal on Mathematics – https://www.mdpi.com/journal/mathematics), not here. When it gains some significant recognition in science, then it will also find its way to Wikipedia. --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 10:44, 31 January 2020 (UTC)

I suspect the Greeks' argument might also have used Euclid's own version of Euclid's algorithm, involving repeated subtraction rather than the division used in today's optimised variant. PML.

== all square roots of natural numbers, other than of perfect squares, are irrational ==

About the Irrationality of the squareroot of 2. My math teacher said today that [[Pythagoras]] believed that sqrt(2) actually WAS a rational number and that that was a thought that his followers the [[Pythagoreans]] also thought. He also said that someone during the [[Middle Ages]] proved that sqrt(2) = irrational and that that guy subsequently was murdered. [[User:BL|BL]] 22:58, 16 Sep 2003 (UTC)

Can we provide a source for this? I want to read the proof [[User:MaitreyaVaruna|Immanuelle]] ([[User talk:MaitreyaVaruna|talk]]) 23:41, 4 April 2022 (UTC)

:That is ignorant nonsense; just look at Euclid's Elements and you will see that irrationality was know to the ancient Greeks. [[User:Michael Hardy|Michael Hardy]] 22:12, 17 Aug 2004 (UTC)

:Read the article and follow the link: "The square roots of all natural numbers that are not [[perfect squares]] are irrational and a proof may be found in [[quadratic irrational]]s". Specifically, [[Quadratic irrational number#Square root of non-square is irrational]]. [[User:Meters|Meters]] ([[User talk:Meters|talk]]) 23:57, 4 April 2022 (UTC)

== 2^e ==

::Euclid was born about a hundred years after pythagoris' death. The proof of the irrationality was discovered by one of pythagoris' followers, but if I remember my reading correctly he was banished, not killed. --[[User:Starx|Starx]] 01:09, 18 Aug 2004 (UTC)

It was stated by Alexandru Froda in his ''Sur l'irrationalité du nombre 2^e''. Just recently, Amiram Eldar claimed that the number is irrational. More at https://oeis.org/A262993.

:::In other words, as I said, it is ignorant nonsense to say it was not done until the middle ages. [[User:Michael Hardy|Michael Hardy]] 01:52, 18 Aug 2004 (UTC)

Question is, did he really prove that this constant cannot be expressed as a/b with a and b being positive integers? [[User:Kwékwlos|Kwékwlos]] ([[User talk:Kwékwlos|talk]]) 21:49, 19 June 2023 (UTC)

:It does say in "http://www.bdim.eu/item?fmt=pdf&id=RLINA_1963_8_35_6_472_0" (in French) that the irrationality of 2^e is assumed. But then he proceeds to demonstrate that the assumption that 2^(e-1) is rational leads to the same number being irrational, contradicting the hypothesis. He finishes by saying that since 2^(e-1) is irrational, 2* 2^(e-1) = 2^e is also irrational. [[User:Dhrm77|Dhrm77]] ([[User talk:Dhrm77|talk]]) 16:43, 20 June 2023 (UTC)

::::If Euclid was around ''after'' pythagoras, then the fact that he knew of the irrationality of the square root of 2 is not surprising, considering it was during pythagoras' time that it was first proven. Do you have any sources? Cause there are plenty documented sources saying it was, in fact, a follower of pythagoras. --[[User:Starx|Starx]]

::So, as far as I understand, we should change the section about open questions. Is this a reliable source with a proof that <math>2^e</math> is irrational? [[User:D.M. from Ukraine|D.M. from Ukraine]] ([[User talk:D.M. from Ukraine|talk]]) 12:30, 22 June 2023 (UTC)

:::If so, this would imply that any number of the form rational^transcendental, assuming that the transcendental is not a logarithm of a rational with the base being the aforementioned rational, is irrational. [[User:Kwékwlos|Kwékwlos]] ([[User talk:Kwékwlos|talk]]) 23:35, 23 June 2023 (UTC)

::::I don't think so. I think that proof (if correct) is only valid for that number. The same way you can form an integer from 2 rational numbers, or a rational number from 2 irrational numbers, I believe there might be cases where rational^transcendental is not irrational. But besides 1^e or 1^pi, I can't find a non-trivial example. [[User:Dhrm77|Dhrm77]] ([[User talk:Dhrm77|talk]]) 00:57, 26 June 2023 (UTC)

But the main question is whether the proof about just 2^e is situated in a source reliable enough for us to change the Wikipedia article. [[User:D.M. from Ukraine|D.M. from Ukraine]] ([[User talk:D.M. from Ukraine|talk]]) 13:33, 29 June 2023 (UTC)

So you and I both agree that it was known to the Pythagoreans and therefore to Euclid, who came later. And we both agree therefore that it is ignorant nonsense to say that it was not done until the middle ages. Right? As for sources, I've read some of [[Thomas Heath]]'s books, but it's been a while, so I cannot cite chapter and verse. On another matter, why do you keep deleting my assertion in the article that the conventional algebraic argument is not the one that the Pythagoreans used? [[User:Michael Hardy|Michael Hardy]] 21:40, 18 Aug 2004 (UTC)

== Irrational numbers ==

Because everything I've read has said that that was the proof. The only one I can think of off the top of my head is the golden ratio by mario livio, which has quite a bit on the history of math. If you have a better source that says otherwise then I'll concede, but all you've done so far is claim that it's ignorant nonsense. If we both agree that it was known to the pythagoreans, and I'm saying it was the pythagoreans who first discovered it, where do you get the middle ages?? Who brought that up? --[[User:Starx|Starx]] 01:10, 19 Aug 2004 (UTC)

If it is 5, you can write as 5/1(five upon one or five divided by 1). It is because every number multiplied by 1 is number itself. If π or √2 are irrational, can't we write them as ratio of π to 1 or √2 to 1? If it is wrong and ratio needs co-prime numbers, then what is ratio of 2 or 5? [[Special:Contributions/2402:A00:401:B896:4923:CB28:C2C3:C3A7|2402:A00:401:B896:4923:CB28:C2C3:C3A7]] ([[User talk:2402:A00:401:B896:4923:CB28:C2C3:C3A7|talk]]) 17:01, 4 July 2024 (UTC)

I did not "get middle ages"!! That is what I called "ignorant nonsense". I never said that it is "ignorant nonsense" to say that the relatively recent algebraic proof of irrationality is how the Pythagoreans did it. It is '''not''' how the Pythagoreans did it; it is how many mathematicians believe (and write) that the Pythagoreans did it; I never said that that error is "ignorant nonsense" -- only that it is an error. [[User:Michael Hardy|Michael Hardy]] 02:31, 19 Aug 2004 (UTC)

:We can write π as π/1, but for a number to be rational it is necessary to have possibility to write it as a ratio of two INTEGERS (WHOLE numbers). The number π is not an integer. It is not important for the numbers in the ratio to be co-prime, important is to be integers. The number π is irrational, 2/5 is rational. Do you ask about the notion "irrational number" or about possibility to divide numbers? :) [[User:D.M. from Ukraine|D.M. from Ukraine]] ([[User talk:D.M. from Ukraine|talk]]) 14:31, 11 July 2024 (UTC)

I'm not debating about anything that happened during the middle ages. I'm debating about whether or not the proof displayed on the page was done by one of pythagoras' followers. That's what our recent edits have concerned so I think it would be fairly obvious that that is what the discussion is about. I don't understand why you're still bringing up the comment another user made on the middle ages, that's not the subject of the debate and that's why I want to know where you're getting that from. I'm sorry if I was unclear. I'm asking what referances do you have pertaining to what proof pythagoras used to determine the irrationality of the square root of two. Because I have ''referances'' that say that what's displayed is the correct proof. I said this in my above post and I'll say it again: ''If you have a better source that says otherwise then I'll concede''. --[[User:Starx|Starx]] 03:41, 19 Aug 2004 (UTC)

== infinity as irrational ==

I will get the references.

i like the idea that since..

What I called "ignorant nonsense" was the statement about the middle ages. Then you attacked me for calling your statements about the Pythagoreans and Euclid "ignorant nonsense". That's why I brought up the matter of the middle ages. [[User:Michael Hardy|Michael Hardy]] 18:35, 19 Aug 2004 (UTC)

a*b = c therefore b = c/a, is our axiom..

== Complicated proof? ==

if infinity is in fact a valid number, and we treat it as such,

Isn't the first proof for the irrationality of <math>\sqrt{2}</math> overly complicated?

It basically states that when you transform <math>\left(\frac{p}{q}\right)^2 = 2</math> to <math>p^2 = 2q^2</math>, the multiplicity of prime factor 2 is even on the left side, and odd on the right side -> contradiction.

1/0 = inf,

[[User:Aragorn2|Aragorn2]] 21:00, 17 Sep 2003 (UTC)

1/inf = 0,

0 * inf = 1.. (identity)

No, because the proof builds on other proofs that has to be explicitly stated. Like that the square of an even number also is even. As it is on the page is how my math teacher described it. [[User:BL|BL]] 21:27, 26 Sep 2003 (UTC)

so infinitely scaling "nothing" is equal to ONE.

and infinity can not be expressed as p/q quotient of integers or natural numbers, therefore it is irrational.

----

we respect our axioms of algebra, we deem infinity an irrational number,

The recent posting on the history is directly taken from Article 3 of a 1906 book at www.gutenberg.net/etext05/hsmmt10p.pdf .

and we allow 0*inf = 1.

stating that infinity is irrational makes sense on all levels. and i think this is elegant and respects our logical foundations. [[Special:Contributions/2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|2A02:A46E:D6AB:0:E03E:572F:C70E:EA59]] ([[User talk:2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|talk]]) 02:07, 16 January 2025 (UTC)

I'll leave it there for the present; but in any case it would need a thorough edit.

:Infinity is not a valid number in the usual sense, so none of this applies.—[[User:Anita5192|Anita5192]] ([[User talk:Anita5192|talk]]) 03:30, 16 January 2025 (UTC)

[[User:Charles Matthews|Charles Matthews]] 16:50, 29 Jan 2004 (UTC)

::okay but who said that infinity is not a valid number? i mean how many natural or rational numbers exists? it must be a quantity, therefore we can respect our axioms and treat infinity as an irraional number.

::its a puzzle piece that fits perfectly, i mean all irrational numbers have infinitely continuing decimals and thus has brought infinity itself to the attention of mathematicians. its only elegant that infinity itself may be a kind of irrational. [[Special:Contributions/2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|2A02:A46E:D6AB:0:E03E:572F:C70E:EA59]] ([[User talk:2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|talk]]) 13:29, 16 January 2025 (UTC)

::i mean i understand it's not usual, but it's a very nice way to tie infinity formally into our mathematics without logical breaks. [[Special:Contributions/2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|2A02:A46E:D6AB:0:E03E:572F:C70E:EA59]] ([[User talk:2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|talk]]) 13:30, 16 January 2025 (UTC)

:Apart from what {{u|Anita5192}} said above, I would like to point out that rational neednt be a ratio of natural numbers, but a bit more generally a ratio of integer numbers. This allows us to have ''negative'' rational numbers, too. And if so, and you explicitly said yourself <math>\infty = 1/0</math>, thus including zero among valid denominators (opposite to what standard arithmetic assumes)... It clearly means you defined infinity as a quotient of two integer numbers, hence it certainly must be rational. --[[User:CiaPan|CiaPan]] ([[User talk:CiaPan|talk]]) 13:43, 16 January 2025 (UTC)

----

::hm, yeah perhaps the logic kind of breaks, it would have been nice though to accept infinity as a valid number so u can get algebraeic results from it.. such as 0*infinity=1.

::and i figured that maybe zero isn't really quite a natural number, or its a weird integer and so it doesnt really apply to say it can be expressed as a quotient of natural integers..

BL: a root of a [[natural number]] m (i.e. a positive/non-negative integer) is either a natural number or an irrational: Suppose we are looking at m^(1/n) and this was a/b (i.e. rational with a,b integers), so a^n=m*b^n. Then write m in terms of a product of powers of prime numbers (m=p^x * q^y * r^z * ...). Do the same with a and b, and then match exponents on each side.

::but it would also have weird implications on the infinite monkey theorem [[Special:Contributions/2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|2A02:A46E:D6AB:0:E03E:572F:C70E:EA59]] ([[User talk:2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|talk]]) 13:49, 16 January 2025 (UTC)

::isn't it possible that we made poor assumptions thousands of years ago though? maybe to think outside of the box can yield interesting results [[Special:Contributions/2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|2A02:A46E:D6AB:0:E03E:572F:C70E:EA59]] ([[User talk:2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|talk]]) 13:53, 16 January 2025 (UTC)

If all of x,y,z,... are multiples of n, we will be able to take the n-th root of m and get a natural number. If any of them are not, then we will not even be able to get a rational number because the [[LHS]] of a^n=m*b^n will be a product of powers of primes where all the exponents are multiples of n while the [[RHS]] will not be, which based on the [[fundamental theorem of arithmetic]] leads to a contraction of the hypothesis that m^(1/n) is rational. --[[User:Henrygb|Henrygb]] 23:28, 13 Feb 2004 (UTC)

::i just felt like maybe stamping "undetermined" on things is a workaround that doesnt really tie everything up perfectly.. if you used limits to consider these results it would have been trivial, but everything breaks exactly when hitting the bounds of the number line or when using zero as a divisor.. [[Special:Contributions/2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|2A02:A46E:D6AB:0:E03E:572F:C70E:EA59]] ([[User talk:2A02:A46E:D6AB:0:E03E:572F:C70E:EA59|talk]]) 14:14, 16 January 2025 (UTC)

----

:''The irrational numbers are precisely those numbers whose decimal expansion never ends and never enters a periodic pattern''

I know that is true but there is no need to invoke ''decimal'' when describing irrational numbers. I have witnessed confusion when irrational numbers are defined thus. People think that the set of irrational numbers are different in base-2 than they are in base-10 because of definitions like that. [[User:Psb777|Paul Beardsell]] 05:03, 20 Feb 2004 (UTC)

==Grammar==

From the article: ''(because none of its prime factors is 2)''

Factors is plural, so shouldn't it be ''are'' instead of ''is''? --[[User:Starx|Starx]] 01:51, 20 Dec 2004 (UTC)

No. "Its factors" is the object of the preposition "of". If I wrote "Not even one of its factors is prime", obviously it would be grossly wrong to write "are". Similarly if I wrote "Just one of these factors is prime", would you say I should have written "are", when I'm writing about only one, on the grounds that "factors" is plural? Traditionally, "none" is singular. Of course, recently many people have used "none" as plural, but even so, there can hardly be a grammatical objection to using a singular "none". (And somehow the misspelling of "grammar" in the edit summary doesn't inspire confidence either.) [[User:Michael Hardy|Michael Hardy]] 23:24, 20 Dec 2004 (UTC)

... and also, when you say "because ''factors'' is plural", I almost fear that next you'll write something like "One of these are correct". I actually hear people say that from time to time; it's as if the fact that ''these'' is plural means that the phrase ''one of these'' is plural. Obviously the phrase ''one of these'' is singular and should be followed by ''is'', not ''are''. [[User:Michael Hardy|Michael Hardy]] 23:57, 20 Dec 2004 (UTC)

*It's nice to see you take an honest question and be a dick about answering it. You sounded like a decent human being right up until the parenthetical remark in your first responce. But that one remark wasn't enough, you had to go back for a second responce. Kudos. --[[User:Starx|Starx]] 06:16, 21 Dec 2004 (UTC)

::I am in fact a decent human being. And I stand by what I wrote above: misspelling ''grammar'' two times running doesn't inspire confidence; it may be useful for you to know that. If you disagree with that or any of the other points above, you could argue the point instead of engaging in name-calling. What, specifically, do you object to in the second response? Writing "One of these are correct" is in fact grammatically parallel to the usage you raised a question about. [[User:Michael Hardy|Michael Hardy]] 21:44, 21 Dec 2004 (UTC)

:::It may be grammatically parallel in a technical sense, but the original case is far more obscure. Your example states '''''one''' of these factors'', it's obviously singular. The snippet from the article isn't so clear. In either event you very nicely explained things to me and should have stopped there. But instead you chose to make remarks about my spelling not inspiring confidence and how you "fear" I'll do something even stupider. I stand by what I wrote above, you're being a dick for no reason I can see other then possibly a superiority complex. --[[User:Starx|Starx]] 22:44, 21 Dec 2004 (UTC)

In discussions of politics or scientific controversies a rhetorical device such as "Since you're advocating X's theory, next I expect you'll be saying the Big Bang didn't happen" is not generally construed literally; people aren't so touchy. But when the topic is grammar, it seems they are. I don't understand why the difference. Let me rephrase my comment that was found offensive. Originally I wrote:

:''... and also, when you say "because ''factors'' is plural", I almost fear that next you'll write something like "One of these are correct". I actually hear people say that from time to time; it's as if the fact that ''these'' is plural means that the phrase ''one of these'' is plural. Obviously the phrase ''one of these'' is singular and should be followed by ''is'', not ''are''.''

Here is a rephrasing:

:''... and also, the phrase "because ''factors'' is plural", is syntactically parallel to, "One of these are correct". I actually hear people say that from time to time; it's as if the fact that ''these'' is plural means that the phrase ''one of these'' is plural. Obviously the phrase ''one of these'' is singular and should be followed by ''is'', not ''are''.''

If I had not thought that was obviously what was meant, I would have phrased it in that literal way originally. [[User:Michael Hardy|Michael Hardy]] 23:22, 30 Dec 2004 (UTC)

== &lt;math> ==

Should this page be converted to use &lt;math&gt; tags rather than radical symbols? IMO, the radicals with no overline look really ugly. If nobody objects within a few days, I'll switch it over. --[[User:Simetrical|Simetrical]] 01:13, 30 Dec 2004 (UTC)

:Please note that there is no consensus about using &lt;math&gt; tages, i.e. TeX, for '''''inline''''' symbols, rather than displayed formulae. It is generally preferred that inline MathML is left as such, until there is more agreement. How it appears may well be browser-dependent, so that changing it to suit one user may not have a good effect for another. It is often reported that inline TeX looks odd. [[User:Charles Matthews|Charles Matthews]] 07:14, 30 Dec 2004 (UTC)

::Hmm. Well, I very much prefer it even for inline, but I can see how some might have problems with it. Is there a talk page for discussing the use of inline TeX? --[[User:Simetrical|Simetrical]] 00:16, 31 Dec 2004 (UTC)

:::I think there is; I don't know the URL at this moment. Formerly it failed to get centered and looked terrible. Now, when it does get centered, you have things like <math>2^x\,</math> getting centered that should not -- the bottom of the "2" should be at the same level as the bottoms of the letters, as in 2^''x''. Also, in <math>M+N\,</math> the variables appear much bigger than they should, whereas in ''M'' + ''N'' they look good. Various other problems like those, too. [[User:Michael Hardy|Michael Hardy]] 00:25, 31 Dec 2004 (UTC)

::::I just found the URL myself: [[Wikipedia:WikiProject_Mathematics]]. Anyway, maybe we should at least overline the roots—&radic;<font style="text-decoration: overline">2</font>. Unless that shows up funny in some browsers, I don't see any reason not to. --[[User:Simetrical|Simetrical]] 00:38, 31 Dec 2004 (UTC)

==Repitend thing==

Michael Hardy wrote:

:''I think this should say either "the period is 3" or "the length of the repitend is 3", but NOT "the length of the period is 3"''.

You are right. My spell-checker gave me "repitend" as an option. I should have looked up a dictionary and confirm this is correct. I instead chose to replace it with "period" assuming it will be the same thing. I would actually appreciate a bit of clarification here, if it would not take too long. [[User:Oleg Alexandrov|Oleg Alexandrov]] 02:51, 4 Apr 2005 (UTC)

:: Actually, I think I figured it out myself. All it took is reading what you wrote, and actually thinking about it. Thanks! [[User:Oleg Alexandrov|Oleg Alexandrov]] 02:56, 4 Apr 2005 (UTC)

::: I would appreciate in general more feedback with my spelling. And sorry for "polluting" your watchlist, I believe quite a bit of my bot changes show up there. [[User:Oleg Alexandrov|Oleg Alexandrov]] 03:04, 4 Apr 2005 (UTC)

== I feel rather foolish for asking this... ==

...but I can't figure out the logic behind the statement, "if √2=m/n, then √2=(2n-m)/(m-n)." Can someone derive that, or point me to another site that has the derivation? --[[User:Histrion|Jay (Histrion)]] 16:50, 26 October 2005 (UTC)

:One way is by algebra: If &radic;2 = ''m''/''n'', then ''n''&radic;2 = ''m'', and then we have

::<math>{2n - m \over m - n} = {2n - n\sqrt{2} \over n\sqrt{2} - n}

= {n\sqrt{2}(\sqrt{2} - 1) \over n(\sqrt{2} - 1)} = \sqrt{2}.</math>

:::Well, I feel silly now — a straightforward substitution. I was trying to derive the expression, when I could have just simplified it. As Strong Bad might say, "Holy crap!" --[[User:Histrion|Jay (Histrion)]] 20:17, 26 October 2005 (UTC)

:Another way is by geometry: if ''m'' is the diagonal, and ''n'' the side, of an isosceles right triangle, then by a simpler [[ruler-and-compass construction]] one creates a smaller isosceles right triangle in which the the respective lengths of diagonal and side are 2''n''&nbsp;&minus;&nbsp;''m'' and ''m''&nbsp;&minus;&nbsp;''n''. Someone's added a diagram showing this to the article titled [[square root of 2]]. [[User:Michael Hardy|Michael Hardy]] 18:59, 26 October 2005 (UTC)

::Says we have a fixed point of the mapping x &rarr; (-x + 2)/(x - 1). Well, easy to check the fixed points are &radic;2 and &minus;&radic;2. [[User:Charles Matthews|Charles Matthews]] 19:32, 26 October 2005 (UTC)

::Actually, that brings up a good question -- how much of the material in this entry is repeated in that [[square root of 2|√2]] entry? Should we look at merging them -- or, alternately, moving some text from this entry to the other? -- written and unsigned at 13:21, 26 October 2005 by Histrion

::: I would oppose the merger. The main article for irrational numbers is this one. As for the [[square root of two]], it is an important enough example of irrational numbers to keep its own article. [[User:Oleg Alexandrov|Oleg Alexandrov]] ([[User talk:Oleg Alexandrov|talk]]) 23:28, 26 October 2005 (UTC)