# Talk:Irrational number

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## Request

It would be helpful to have a list of irrational numbers, probably as a seperate entry. I have not made this edit as I would not be able to go much beyond the obvious, but I am sure there are others that could.

That might be a long list, pal. ;^> DavidCBryant 18:19, 27 November 2006 (UTC)

How about examples of some of the more famous rational numbers, like Pi and e? 118.210.107.10 (talk) 06:39, 1 January 2013 (UTC)

Already included (assuming you meant to say "irrational" instead of "rational"; pi and e are irrational). See the fourth paragraph of the opening section (beginning with "Perhaps the best-known irrational numbers...")
-Hatster301 (talk) 23:32, 1 January 2013 (UTC)

## Moved from article page

I moved this from the main page:

[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 √2=m/n, then √2=(2n-m)/(m-n), 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 n and m are respectively the leg and the hypotenuse of an isosceles right triangle, then m-n and 2n-m 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.]

and changed the paragraph about Pythagoras's discovery accordingly. AxelBoldt 03:30 Oct 23, 2002 (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.

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. BL 22:58, 16 Sep 2003 (UTC)

That is ignorant nonsense; just look at Euclid's Elements and you will see that irrationality was know to the ancient Greeks. Michael Hardy 22:12, 17 Aug 2004 (UTC)
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. --Starx 01:09, 18 Aug 2004 (UTC)
In other words, as I said, it is ignorant nonsense to say it was not done until the middle ages. Michael Hardy 01:52, 18 Aug 2004 (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. --Starx

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? Michael Hardy 21:40, 18 Aug 2004 (UTC)

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? --Starx 01:10, 19 Aug 2004 (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. Michael Hardy 02:31, 19 Aug 2004 (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. --Starx 03:41, 19 Aug 2004 (UTC)

I will get the references.

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. Michael Hardy 18:35, 19 Aug 2004 (UTC)

Ok, to add my understanding (based on lecture notes given to me by a lecturer of Mathematical History). Firstly, it wasn't irrational numbers which were discovered, but incommensurable magnitudes i.e. a relationship, not a kind of number. The way the Greeks would have described it is roughly as follows;
a:b::c:d "a enjoys a relationship in size with b that is equal in size to the relationship shared by c and d"

It's looking at it from a very modern viewpoint to see these relationships as actually being irrational numbers if c and d are not both integers, and not at all how the ancients would have viewed it. They wouldn't have thought of these relationships as occupying a space on a number line for example. He then goes on to say that the length of the diagonal compared to the side of a square (i.e. $\sqrt 2$ in modern notation) wasn't really talked about until quite far into the 4th century BC. The first relationship found to be incomensurable was probably that of the diagonal of a pentagon in the 5th century BC - not much earlier than 410-420 BC (based on research by Wilbur Knorr). He also mentions that it wasn't really until the late 16th century AD that what we'd now call an irrational number was beginning to be discussed properly Richard B 00:06, 2 December 2005 (UTC)

## Complicated proof?

Isn't the first proof for the irrationality of $\sqrt{2}$ overly complicated? It basically states that when you transform $\left(\frac{p}{q}\right)^2 = 2$ to $p^2 = 2q^2$, the multiplicity of prime factor 2 is even on the left side, and odd on the right side -> contradiction.

Aragorn2 21:00, 17 Sep 2003 (UTC)

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. BL 21:27, 26 Sep 2003 (UTC)

Aragorn, you're assuming that a number has only one prime factorization. But that's much harder to prove than the special case that says the product of two odd numbers is odd, which is all that this proof needs. Michael Hardy 21:56, 13 October 2006 (UTC)

While that's all the proof needs, it is needlessly wordy thus obscuring its essential simplicity. My hope in tightening it up as I did just now is that it is easier to read than the far longer version it replaced without being any less accessible. --Vaughan Pratt (talk) 02:38, 17 March 2008 (UTC)

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

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

Charles Matthews 16:50, 29 Jan 2004 (UTC)

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.

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. --Henrygb 23:28, 13 Feb 2004 (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. Paul Beardsell 05:03, 20 Feb 2004 (UTC)

Thank you, Paul. I think you just answered a question of mine before I even got around to asking it. To be sure though, are some (or all) irrational numbers simply artifacts of the decimal system? That is, could a number which is irrational in base 10 be expressed rationally in, for example, base 9 or base 17? --Zaklog 05:56, 21 March 2006 (UTC)

No. Irrational numbers are irrational in every natural base. The definition of irrational number has nothing to do with how this number is written by humans.  Grue  06:39, 21 March 2006 (UTC)
The relevant theorem here is, for all integers r,s ≥ 2 and real x, x is ultimately periodic in radix r if and only if it is ultimately periodic in radix s. Vaughan Pratt 03:05, 7 May 2006 (UTC)

I didn't really like this line in the proof: "Since a:b is in its lowest terms, b must be odd."
Can't it be replaced with "Assume b is odd?" —Preceding unsigned comment added by 76.172.43.73 (talk) 07:27, 12 June 2008 (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? --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.) 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. 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. --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. 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. --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. Michael Hardy 23:22, 30 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. 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! 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. Oleg Alexandrov 03:04, 4 Apr 2005 (UTC)
I believe that the mathematical notation of the set of irrational numbers is not mentioned anywhere in the text, although it is quite obviously under this title, is it not?

## 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? --Jay (Histrion) 16:50, 26 October 2005 (UTC)

One way is by algebra: If √2 = m/n, then n√2 = m, and then we have
${2n - m \over m - n} = {2n - n\sqrt{2} \over n\sqrt{2} - n} = {n\sqrt{2}(\sqrt{2} - 1) \over n(\sqrt{2} - 1)} = \sqrt{2}.$
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!" --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 simple ruler-and-compass construction one creates a smaller isosceles right triangle in which the the respective lengths of diagonal and side are 2n − m and m − n. Someone's added a diagram showing this to the article titled square root of 2. Michael Hardy 18:59, 26 October 2005 (UTC)
Says we have a fixed point of the mapping x → (-x + 2)/(x - 1). Well, easy to check the fixed points are √2 and −√2. 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 √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. Oleg Alexandrov (talk) 23:28, 26 October 2005 (UTC)
(Whoops, yeah, that last one was me. Thanks, Oleg.) OK, then is there any stuff in this article that might be best moved to square root of two (or cut from this article because it's duplicated in that one, like the algebra proof that √2 is irrational)? --Jay (Histrion) (talkcontribs) 00:03, 27 October 2005 (UTC)
Why do you like to be so efficient? :) I think square root of two was the first real number to be proved to be irrational. The proof is also very simple. It makes sense to me that it would show up in both articles, it greatly clarifies (in my view) what irrationals are all about. Oleg Alexandrov (talk) 02:47, 27 October 2005 (UTC)
Efficiency? Can't help it. Copy editor for my high school newspaper, and I tutor at the local community college, so I proofread papers now and then. Proofread enough beginning writers, you start to look for redundancy. Anyway, I dunno, I just think it might be better if, rather than having the entire proof in both entries, one of them just pointed to that subsection of the other. But I'm not feeling bold enough to do it myself. Maybe tomorrow, when I've slept some. ;) --Jay (Histrion) (talkcontribs) 03:57, 28 October 2005 (UTC)

I read both Irrational number and Square root of 2 now. I would say that there are too many proofs of irrationality of square root of two at irrational number. I would agree with removing all proofs of that except for the first and referring for more detail to Square root of 2. I would disagree with removing all the proofs of irrationality of square root of two from there. I believe that proof is important enough in illuminating the article that it better be inline rather than referring the reader to a different article. Oleg Alexandrov (talk) 05:18, 28 October 2005 (UTC)

## Drowning of Hippasus

Can someone provide a cite for "he [Pythagoras] sentenced Hippasus to death by drowning."? The Wikipedia article on Hippasus calls the story a rumor, other sites use the word "legend". Others say that Hippasus died accidentally and the Pythagoreans were guilty only of tactless amusement at the fact. About.com says that there are many legends, and no one knows for sure. In any case, I haven't seen any claims that Hippasus was "sentenced", in the sense of receiving some kind of process.

Here is my version of the accident. Pythagoras was the smartest guy of the bunch, so it was him who proved the irrationality of sqrt(2). He told his disciples and Hippasus snitched the result to general public. That's why he was killed.  Grue  06:57, 27 May 2006 (UTC)
I reworded that part, to keep the story and at the same time make clear it's just that - a story. And also that Pythagoras is only :-) a mathematician and is in no position to sentence anybody. Haonhien 15:53, 22 September 2006 (UTC)
I changed "drowned to death" to simply "drowned", since the former is really quite tautologous. You drown, you die. :-) 81.159.229.181 10:39, 24 September 2006 (UTC)
Pythagoras was not only a mathematician, he was the founder of a fanatical religious cult mixed with science. Maybe he did it as a joke, then got rich, and couldn't stop the joke. Oops, that's another guy.13:05, 23 October 2006 (UTC)

I know of no source earlier than Kline for the claim that Pythagoras sentenced Hippasus to dweath by drowing for discovering irrationality. Much oldr sources claim that Hippasus was sentenced to death for divulging the secrect of irrationaility, something very different. I don't think Kline ios very reliable here. Hardicanute (talk) 16:15, 4 June 2011 (UTC)Hardicanute

## Galois theory

The following I removed on the grounds that it is irrelevant and sometimes erroneous:

Paolo Ruffini (1799) first proof, (largely ignored) of Abel–Ruffini theorem that the general quintic or higher equations cannot be solved by a general formula involving only arithmetical operations and roots. Évariste Galois (1831) sends a memoir to the French Academy of Science: On the condition of solvability of equations by radicals, later developed into Galois theory which has been central to the proof that π and e are transcendental. Joseph Liouville (1840) showed that neither e nor e2 can be a root of an integral quadratic equation. Niels Henrik Abel (1842) partially proves the Abel–Ruffini theorem. Gene Ward Smith 00:36, 13 July 2006 (UTC)

Maybe the removal of Galois Theory left a hole in the History part; sometime during Math History people thought that every irrational number was expressible by radicals, then transcendals were found, then the unsolvability of the 5th degree polynomial was found. Albmont 13:11, 23 October 2006 (UTC)

## Decimal expansions

In the numerical example, the old version ends with this:

$A=\frac{715.5}{999}=\frac{7155}{9990}=\frac{135\times 53}{135\times 74}=\frac{53}{74}.$
(The "135" above can be found quickly via Euclid's algorithm.)

I omitted the reduction of the fraction and also the reference to Euclid's algorithm, because the proof ends when we have a fraction - any fraction; and also because introducing the new concept of Euclid's algorithm unduly complicates what we're discussing here, and that is only about the repeating/terminating decimal expansion. (If you disagree you can copy the above back into the text.) Haonhien 15:46, 22 September 2006 (UTC)

## Formula in the lead

I don't understand. Why is it so important that it should be on its own line? It's not a definition of irrational number, in fact it's exact opposite. If there should be a formula, it should be something like that: $x\in\mathbf{I}\iff\not\exists a, b\in \mathbf{Z}: x=\frac{a}{b}$. I can very well imagine a situation when someone wants to look up a quick definition of irrational number, sees a huge $\frac{a}{b}$ in the lead paragraph and then walks away thinking that he got it.  Grue  20:22, 13 October 2006 (UTC)

"Irrational number" is inherently a negatively defined concept: an irrational number is a real number that is not rational. And if you can't understand why the formula you propose to display is inferior to expressing the same thing in words, then you'll probably never be any good at expressing things in words. It is absurd to think anyone will see the fraction and think they've got it---they'd need to read the words "where a and b are integers", and by then they'd have read the preceeding words too. What audience are you contemplating here? Those who readily understand such symbols as $\not\exists a, b\in \mathbf{Z}: x=\frac{a}{b}$ will not be easily confused, and those who don't already know what an irrational number is will not understand such a thing as what you propose. And look: your proposed displayed formula requires the reader to understand such things as sets and membership, which is subtler and less basic---and obviously FAR newer---than the concept of irrational number. Would you seriously propose that only those who understand such modern (19th-, 20th-, and 21st-century) concepts should be told what irrational numbers (an ancient Greek idea) are? Haven't you heard that the so-called "new-math" pedagogical approach of the 1960s is discredited, for excellent reasons? If not, what planet have you been on for the last four decades? Michael Hardy 21:54, 13 October 2006 (UTC)
"And if you can't understand why the formula you propose to display is inferior to expressing the same thing in words..." Huh? I was in fact arguing for making the formula inline or removing it altogether (replacing it with words). There is absolutely no reason for that formula to stay on its own line. It doesn't define irrational number. My formula does, but it's "new math" so you don't like it. That's fine with me, as long as the wrong formula is not displayed in such a prominent place.  Grue  07:30, 14 October 2006 (UTC)
I agree with Michael here. The formula does not define the irrationals, but comes as close as one can to that. Oleg Alexandrov (talk) 17:53, 14 October 2006 (UTC)
It comes as far as one can get from the formula defining the irrational numbers. It defines rational numbers, which are exactly opposite (wouldn't have thought I'll have to explain that to the two mathematicians!)  Grue  11:57, 15 October 2006 (UTC)
But the concept of irrational number is defined by saying what it is the opposite of! Michael Hardy 00:06, 16 October 2006 (UTC)

## Definition independant of a real number

I think we should find a definition that is independant of the definition of a real number - Since the definition of a real number depends on the definition of an irrational number. The current definition is circular, but according to the editors that reverted my edit, the second paragraph in the intro can't be used as a definition. We need a better one anyway. Fresheneesz 21:50, 25 October 2006 (UTC)

A reasonable definition of real number does not depend on the concept of irrational number. A reasonable definition of irrational number does depend on the concept of real number: an irrational number is any real number that is not a rational number. The suggested circularity needs to be avoided by giving a good definition of "real number"---one that does not presuppose knowledge of what an irrational number is. Michael Hardy 02:34, 26 October 2006 (UTC)
PS: the correct spelling is "independent". Michael Hardy 02:34, 26 October 2006 (UTC)

## SHOULD THIS BE INCLUDED ?

Someone has removed [this weblink to the proof that Richard Palais relates] from this article because (he says) : "I think that this inline reference to an external proof is not as helpful, and kind of distracting." Where could it be included? Should it go on another page? It is the simplest and nicest proof by descent I have ever seen. Robert2957 16:21, 27 October 2006 (UTC)

Possibly in the "External links" section of this article, or that of the article titled square root of 2. But it seems you need to fix the URL: it should be <http://www.mathacademy.com/pr/prime/articles/irr2>. Michael Hardy 19:35, 27 October 2006 (UTC)

Would it be possible to reproduce a version of this proof in the article without violating copyright ? Robert2957 20:33, 27 October 2006 (UTC)

It should be possible, but credit should be given. There is a question of whether that proof is actually novel. Michael Hardy 22:37, 27 October 2006 (UTC)

## General cleanup

Hi, all!

Overall this is a very nice article, but it needs a bit of cleanup (poorly constructed sentences, overuse of the verb "to see", too many id ests, etc). I've put that on my schedule of things to do this week, but thought I'd put this note up here first, to give fair notice to those who may have excessive emotional capital invested in the existing verbiage. DavidCBryant 19:36, 27 November 2006 (UTC)

## Don't Use The Word Obvious

"Therefore a2 is even because it is equal to 2 b2 which is obviously even."

How is it obvious? I didn't know it was even. If you say something is obvious and the reader didn't know it, it makes them feel stupid.

(b·talk·contribs) 22:57, 4 December 2006 (UTC)

It's obvious because it's 2 times another integer. That's what "even" means. Michael Hardy 23:46, 4 December 2006 (UTC)

Whether it is obvious is not always transferable to another reader. An objective word for this kind of situation is 'trivial'. One can say. "It is trivial by the definition of an even number." If this is understood, it becomes obvious. Not before. 72.234.3.249 (talk) 21:40, 30 May 2011 (UTC)

## Informal statement in introduction

I had considered your point that you should include m<>0, but have decided to leave it out to make the introduction clearer for non-mathematicians. Here is my reason. It is true that a rational number is of the form n/m with m<>0, and something that is NOT rational is NOT of the form n/m with m<>0. However, the mathematical definition is an irrational number is a REAL number that is NOT RATIONAL. I feel that by adding m<>0, you are clarifying what is meant by a rational number, but there is no mention of what a real number is. So if you were to put "m<>0", a non-mathematician might think "what if m=0?". By leaving out the m<>0 condition, you are defining an "irrational number" as a number that is not of the form n/m for ANY integers. I might not be getting my message across clearly but I hope you understand that I have kept it this way for clarity. AbcXyz 13:06, 4 January 2007 (UTC)

I understand your purpose, but I think your edit makes the text more confusing than illuminating. I will leave it they way you put it, but if somebody else reverts you also, then please don't revert. Let us see what others say. Oleg Alexandrov (talk) 16:42, 4 January 2007 (UTC)

## `The set of all irrationals' is badly written

I find these last two paragraphs to be badly written and obscurely organized.

1) Better would be: "Since the rationals are countable and the reals uncountable, the irrationals are uncountable."

2) This does not seem to be the right place to mention the uncountability of transcendental reals.

3) "form a metric space" is better than "become a metric space"

4) "a homeomorphism", not "the homeomorphism" -- there is more than one!

5) "This shows that the Baire category theorem applies to the space of irrational numbers." But this was not in doubt: the complement of a countable set of closed points in a Baire space is, immediately from the definition, again a Baire space.

6) Rather than "Whereas..." why not just say, if necessary, that the space of irrationals is totally disconnected.

7) "If removing the rationals from the continuum...one might imagine that...would connect it even better than with one copy." This is a horrible sentence: I am a mathematician and I can't quite parse it. (What is "it"?) Ditto the following sentence: "just as totally disconnected"?!?

The paragraph doesn't get any better from here, and I gave up. The sentiments expressed here, if they can be written so as to make sense (even) to a mathematical audience, would be more suitable on a topology page. 22:51, 27 January 2007 (UTC)Plclark

## Cantor displaces Liouville??

"the proof being subsequently displaced by Georg Cantor"

The space formally occupied by Liouville's proof was, starting in 1873, occupied by Georg Cantor??

Presumably what is trying to be discussed here is that Liouville's proof constructs explicit transcendental numbers, whereas Cantor's proof shows, more easily, that all but countably many real (or complex) numbers are transcendental. I don't know what it means for one proof to displace the other: these are the first two theorems in transcendence theory. Plclark 23:00, 27 January 2007 (UTC)Plclark

### History to the end

I hesitate to suggest at the bottom of such a passionate talk page, but could the history section be moved to the end as it is not core to the explanation. Diggers2004 07:15, 11 April 2007 (UTC)

Regarding the history section, the following sentence is curious: "Weierstrass's method has been completely set forth by Salvatore Pincherle in 1880,[10] and Dedekind's has received additional prominence through the author's later work (1888) and the recent endorsement by Paul Tannery (1894)." All of the sources listed in the References section are either from 1945 and later or 1880 and earlier. So I don't know which source could have referred to a "recent endorsement" in 1894. It sounds like something from the Encyclopedia Britannica, but that is not listed as a source for the article. —Preceding unsigned comment added by 75.3.22.86 (talk) 05:12, 7 December 2007 (UTC)

## [itex]

Should this page be converted to use [itex] 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. --Simetrical 01:13, 30 Dec 2004 (UTC)

Please note that there is no consensus about using [itex] 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. 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? --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 $2^x\,$ getting centered that should not -- the bottom of the "2" should be at the same level as the bottoms of the letters, as in 2x. Also, in $M+N\,$ the variables appear much bigger than they should, whereas in M + N they look good. Various other problems like those, too. 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—√2. Unless that shows up funny in some browsers, I don't see any reason not to. --Simetrical 00:38, 31 Dec 2004 (UTC)
Using "Bitstream Vera Sans" in Firefox 2.0.0.3, it shows up weird. There's space between the overline and the 2. Inline MathML is better IMHO. However currently both is used in the article. --Ysangkok 14:35, 17 May 2007 (UTC)

## reductio ad absurdum is inappropriate

The use of the term reductio ad absurdum in the discussion of proofs that the square root of 2 is irrational is inappropriate. Those are simply examples of proof by contradiction. For an example of reductio ad absurdum, see Schrödinger's CatHetware 23:24, 4 June 2007 (UTC)

I have always understood reductio ad absurdum to be the same thing as proof by contradiction. What do you understand the difference to be? Michael Hardy 00:39, 5 June 2007 (UTC)
There is a difference if it comes down to non-classical logic. In an intuitionistic way, "reductio ad absurdum" would mean the show that A implies contradisction, thus A is wrong, whereas "proof by contradiction" could mean to assume not-A implies contradiction, thus A is true. The latter is a much more complicated one which isn't even possible in some systems (you would need something like that the only logical sates are "true" and "false", so if it isn't false, it must be true). Also note that here we are dealing with the more simple first case: We show that "x is rational" cannot be true, so "x is irrational" (defined by: not-"x is rational"). --91.11.96.72 (talk) 13:22, 18 April 2009 (UTC)
I agree that the use of the term reductio ad absurdum is inappropriate in this case. Assuming the text of the article has not changed since the original comment (which is over three years old now), what is being proven is that 2 is irrational. This is being proven by showing that assuming 2's rationality leads to a contradiction. This is, by definition, a proof by contradiction: proving the validity of a statement by showing that its alternative leads to a contradiction. The important distinction to make between the two styles of proof is this: reductio ad absurdum is used to disprove a proposition, while proof by contradiction is used to prove a proposition's validity. --Dominickpastore (talk) 17:20, 9 November 2010 (UTC)
Reductio ad absurdum shows that sqrt{2} is not rational, which is an argument that is valid even intuitionistically. A constructively acceptable proof of irrationality that does not rely on a proof by contradiction is currently given in the article. Tkuvho (talk) 17:33, 9 November 2010 (UTC)

## Question

I would like to know whether e raised to any natural number is irrational? —Preceding unsigned comment added by Sumitagarai (talkcontribs)

Yes. — Loadmaster 23:33, 5 October 2007 (UTC)

I seem to recall a statement from Ivan Niven's book Irrational Numbers that the only rational point on the graph of y = ex is (0, 1). Consequently e raised to any rational power except 0 is irrational. Michael Hardy 03:03, 6 October 2007 (UTC)

This follows from the trancendence of e. If k is a rational number, and e^k=l is rational (apart from the special case given), then e would be the l^(1/k) which would imply e is algebraic. —Preceding unsigned comment added by 81.153.227.62 (talk) 23:19, 27 July 2008 (UTC)

## Complex and imaginary numbers

Since an irrational is a number that is not a rational, are imaginary and non-real complex numbers considered irrational? — Loadmaster 16:32, 2 October 2007 (UTC)

You'll notice the definition says an irrational number is any real number that is not rational. However, if one were to speak of rational and irrational complex numbers, then rational would probably generally be taken to mean a quotient of two Gaussian integers, so many non-real complex numbers would, by that definition, be rational. If you were to say that 2 + 3i is irrational, I doubt you'd find many mathematicians who would think that was weird. Michael Hardy 03:11, 6 October 2007 (UTC)
I ask because of a recent thread on sci.math. In particular, this post states that:
A Google Books search quickly confirms that this modern usage is widespread among eminent mathematicians, e.g. Conway, Gelfond, Manin, Ribenboim, Shafarevich, Waldschmidt (esp. in diophantine approximation, e.g. Thue-Siegel-Roth theorem, Gelfond-Schneider theorem, etc).
I specifically wanted to know whether i was considered irrational or not among mathematicians. Apparently some theorems involving complex numbers are easier to state if it is. — Loadmaster 16:04, 9 October 2007 (UTC)

I find it hard to imagine anyone considering i irrational. That doesn't mean it's included within the usual meaning of "rational number". It is, however, a Gaussian integer and a fortiori it is a "Gaussian rational". Michael Hardy 04:46, 10 October 2007 (UTC)

Whereas I see some reasons only to allow reals, i see no reason why the gaussian integers have any right to be more "rational" than e.g. the Eisensteinian integers. So if one really wants to extend that definition, then the ground field should stay the same: the rationals and nothing more. --91.11.96.72 (talk) 13:22, 18 April 2009 (UTC)

## .(9)

I keep hearing that .999... is a rational number, i cannot come up with 2 numbers that when divided by each other equals .999... 71.74.154.252 16:14, 11 October 2007 (UTC)

See 0.999... (talk page). — Loadmaster 17:12, 11 October 2007 (UTC)

That's easy: 1/1 = 0.99999... Michael Hardy 19:28, 11 October 2007 (UTC)

• Sorry, but this assertion does not prove anything. It remains just what it is: an identity previously stated. BeremizCpa? 23:30, 12 November 2010 (UTC)
That is not an assertion, but a true answer. Consider dividing 1 by 1:

1|1
If you choose the first digit of a result being 9 just after a decimal point, then 0.1 reminds for further dividing:
.9
1|1
9
1
That corresponds to $\textstyle \frac 11 = 0.9 + \frac {0.1}1$
Next steps follow the same pattern:
.999
1|1
9
1
9
1
9
1
$\textstyle \frac 11 = 0.9 + 0.09 + 0.009 + \frac {0.001}1 = 0.999 + \frac {0.001}1$
resulting eventually in continued fraction 1/1 = 0.99999999...and...so...on...forever...
You can see it is not an assertion or assumption; it is a true result of dividing two integers. --CiaPan (talk) 20:30, 13 September 2013 (UTC)

## Famous irrational numbers

The "most famous" or "best known" irrational numbers have been added to the lead, but now that they're sourced, the sources don't support the list. I think the list should go. — Arthur Rubin | (talk) 15:38, 24 October 2007 (UTC)

Please explain how "the sources do not support the list".
Please also explain why the list should go. (Because "the sources don't support the list", or because of some other reason? Perhaps we just need better sources?) My personal view is that some good, commonly known examples should be in the article, identified as such, and be placed near the top.
Also, please know that I view your use of italics on the word "now" as unnecessarily antagonistic to me, as were your reversions, as I do not believe they were in line with the guidelines at Help:Reverting. I prefer constructive criticism, please, and I am actually surprised that an administrator such as yourself would engage in such behavior. Please let me know if I've done something to personally offend you. Thank you in advance.
JonathanFreed 16:46, 24 October 2007 (UTC)
The two sources I moved to "perhaps" are separate lists of ten or fifteen "famous" irrational numbers, which include π and $\sqrt 2$, but do not identify a "most famous". I really don't think the list should be in the lead, but regardless of the where the list occurs, e is also a likely candidate for "most famous", and the list needs to have a reliable source. As for now, you inserted the text twice without a respectable source. The "current" sources which I moved to "perhaps" don't fully support the text. I still doubt a WP:RS can be found, without adding a full section, including φ. — Arthur Rubin | (talk) 17:11, 24 October 2007 (UTC)

## Explaining the name "irrational"

I contributed two explanations today: The first was to make it clear that "irrational" numbers are Not "numbers lacking in rational reasoning". The other contribution was in a similar vein to make it clear that "imaginary" numbers are Not "numbers lacking meaning in the real world".

Both contributions were hastily reverted. Here was the reason given for the revert:

"irrattional means not rational, and imaginary numbers are useful -- but don't exist. And also, unsourced."

Any encyclopedia entry that does a thorough job in explaining what these types of numbers are will make those points perfectly clear. Regarding the complaint that these fundamental points are not referenced, the entire 'history' section talks about how the set of numbers were thought to be non-rational (outside of the realm of sound logic) and therefore doomed to be excised out of the discipline of mathematics.

If after this anyone still has a problem with the comment being unsourced, then just google ["irrational number" misnomer] and you will find plenty of sources that make the exact same point.

If anyone has a substantial rebuttal to these points, we can all scrutinize that point of view for merit. However, if all objections are found to be lacking, then the proper action for improving the Wikipedia articles in question would be reinstatement of the contributions that were reverted today.

ChrisnHouston 19:13, 29 October 2007 (UTC)

Google search is not a viable sourcing technique -- see WP:V and WP:RS. But obviously, that's besides the point. I can't find where it states that irrational means "outside rational reasoning," rather than just "non-rational," but if it does then please point out the sentences -- I may have missed it. But if this is supposed to make the reader understand that the concept that irrational numbers are outside reasoning is a misnomer, it belongs after the point it's refuting -- i.e. the end of the history section, or preferably weaved into it -- not as a part of the lead. Gscshoyru 19:53, 29 October 2007 (UTC)
A point of confusion here is that there are two distinct meanings to the word "irrational". One means "non-logical". The other means "non-ratio-able" (if you will). I don't know the full etymology of the first (more common) meaning of the word, but it is quite possible that it came from the mathematical analysis that concluded that if a number cannot be expressed as a ratio of integers, then that number falls "outside of logical reasoning". The irony, of course, is that numbers with non-repeating decimals are perfectly logical, and in fact dominate the field of "real" numbers. (Which leads to the other point that "imaginary" numbers are quite real themselves.)
I consider it important to state this up front in order to reduce the obvious potential for confusion.
ChrisnHouston 22:24, 1 November 2007 (UTC)
But the part about non-repeating decimals should not be treated as if it were the essence of the idea. Michael Hardy (talk) 03:33, 17 March 2008 (UTC)

I would just like to point out that the Greek word alogos also carries the two meanings: illogical and incalculable/inexpressible. The meaning of ratio in Latin is reckoning. The trouble is that over the ages as our mathematical knowledge has grown, we have come to separate the terms irrational/rational into two meanings, one referring to the logical-ness of something and one referring to the calculablility (maybe the better word is commensurability) of something. It is highly likely that this distinction is merely a modern one that is a direct result of our 'coming to terms' with irrational numbers in recent centuries and that both meanings of the word were intended. Of course, we have no real way of knowing if there was a distinction or not in the minds of ancient Greeks, but it makes sense that to them anything that wasn't commensurable also wasn't logical. Personally, I feel that when people go out of their way to state that irrational numbers are perfectly logical and that imaginary numbers are just as tangible as any other number they are neglecting their etymology and therefor the mathematical history behind their birth. Tyler Haslam (talk) 17:17, 7 March 2015 (UTC)

## maybe i'm just insane, as this has no useful integration to real life.

i had a thought one day, about using decimals as a form of formulaic production (it's more of a sum really.) anyhow, i created an idea I originally called Diades (said as though plural) and using this i ran into a stump. Diades work through the use of multiple decimals, two specifically, and by using these decimals a number can be shared, so i guess it's more of a notation. it works by placing three numbers side by side like so: 2.4.8 or a.b.c Diades are performed by using this sequence (ac.(b/c)) so in words it is: a times b with a decimal value of b over c. making the aforementioned Diade equal to 8.8 the problems i encountered were pi, and remainders. using pi would render the following п.x and remainders could render a 4.9.5. as i thought i decided that pi shall count as three for a and 1415926... for b. and the remainder issue was solved as a x10 shift to allow the decimals to line up. so 4.9.5 would be 21.8. that just left pi's issue to be resolved as the problem became 3.1415926....x would equal infinite.infinite over 3 (I don't know how to write that in a proper manner. i'll work on that.) and as that became, on my paper, i wondered, is this number irrational? imaginary? or something else? —Preceding unsigned comment added by 24.187.112.51 (talk) 08:18, 15 February 2008 (UTC)

## Proof that there exists a pair of irrational numbers that a^b is rational

The article proves that either sqrt(2)^sqrt(2) or (sqrt(2)^sqrt(2))^sqrt(2) is such a pair; is it known whether or not sqrt(2)^sqrt(2) is rational? Or is it like the open questions from the following paragraph, where numbers like pi+e are strongly suspected to be irrational, but never conclusively proven? - Mike Rosoft (talk) 18:52, 15 September 2008 (UTC)

$\sqrt{2}^{\sqrt{2}}$ is transcendental by the Gelfond-Schneider Theorem. --Zundark (talk) 19:09, 15 September 2008 (UTC)

## The square roots of other numbers proof wrong

The proof at the start of this section is wrong. It can be fixed either to something a little shorter using the Fundamental theorem of arithmetic or else using Richard Dedekind's proof in [1]. I prefer the latter as it is more self contained and assumes less, the fundamental theorem wasn't properly proved till Gauss came along. I'll fix it in the next day or so if no-one else does and there's no objection. Dmcq (talk) 17:34, 16 September 2008 (UTC)

Assuming that you were referring to second proof I added and not the first, can you explain the flaw? --Bowlhover (talk) 19:42, 20 September 2008 (UTC)
The proof falls down on 'where p and q are integers and $\frac{p^2}{q^2}$ can be assumed to be in lowest terms'. It can't be assumed to be of this form when they are in lowest terms, this has to be proved. This can probably be done using Euclid's first theorem which grounds the thing a bit further back than the Fundamental theorem of arithmetic. In fact the whole business of lowest terms is iffy without rather a lot more proof at this level but wouldn't affect the proof too much I think. Dmcq (talk) 23:38, 20 September 2008 (UTC)
The square root of 10 proof could be fixed by showing that (10n+m)2 does not end in 0 for any m between 1 and 9 which can be done by enumerating the cases. It can't just be stated for any radix in general though without some proof otherwise it is assuming in effect what it is trying to prove.Dmcq (talk) 23:53, 20 September 2008 (UTC)
In that case, assume that p/q is in lowest terms instead. If p/q is not, one can always choose another p and another q so that p/q is in lowest terms. --Bowlhover (talk) 04:28, 21 September 2008 (UTC)
A couple of counterexamples to the reasoning are perhaps in order.
If we declare the number 3 unclean then our acceptable natural numbers are 1,2,4,5,6,7,8,9,10... and our prime numbers are 1,2,5,6,7,9,11,13,15 etc. Multiplying acceptable numbers together always gives another acceptable number. Then we can see that 9 is not the square of an acceable number, however its square root can be represented as the acceptable rational number 6/2.
Addition and subtraction have to be brought in to make the proof work, this is done in the proof of Euclid's first theorem. Dmcq (talk) 06:47, 21 September 2008 (UTC)
As to the proof using radixes try using the radix 8. Then 4*4=20 has an odd number of zeros at the end but is still a square.
The reasoning can be tricky. There was probably a very good reason why Theodorus of Cyrene stopped at 17 when proving the surds of non-square numbers are irrational. See What is a number? and Square root of 2 is irrational from cut-the-knot for some good proofs. In general one should follow Wikipedia:No original research - Wikipedia does not publish original research or original thought. You need to get proofs like this from published sources unless they are blindingly obvious which this isn't. Dmcq (talk) 06:47, 21 September 2008 (UTC)

Under the sub title "General roots", you have stated (although the proof is not clearly demonstrated), that "if an integer is not an exact kth power of another integer then its kth root is irrational" . What follows below is a proof that shows that this is generally true even for the more difficult case of fractions, a fact that was not apparent until a proof for Fermat’s Last Theorem was recently found.

## Theorem

For natural numbers n and integers a, b, the nth Root of $[(b/a)^n + 1]$ is irrational for n > 2. . Hence this formula can be used to generate an infinite number of irrational numbers.

## Proof

Assume that the nth Root of [$(b/a)^n + 1]$ is rational, then so is nth Root $[(b/a)^n + 1]*a$

Hence, nth Root [$(b/a)^n + 1$] = c/q . . . . . . for some integers c and q

So, $a^n + b^n = (c/q)^n$

And $q^n * a^n + q^n * b^n = c^n$ . . . . . let d = q * a and e = q * b,

Thus, $d^n + e^n = c^n$, which since d and e are integers, contradicts “Fermat’s Last Theorem” which has recently been proved by Andrew Wiles. Hence nth Root $[(b/a)^n + 1]$ must be irrational, for n > 2.

NB this result was already known for the case where b/a is actually a whole number (due to the fundamental theorem of arithmetic and the fact that the nth root of primes are irrational), and in this respect provides an alternative proof.. However this was not previously known to be true for fractions, as demonstrated above. For example if we take 16, which is a square of 4 and add 1, the square root is irrational. However if we take a = 4 and b =3, the fraction 3/4 when squared and added to one, does not yield an irrational number when square rooted. This result can only occur when n = 2 but the process will always produce an irrational number for n > 2. Indeed, if the above theorem could be shown using an alternative method, it would supply a rather quick proof of Fermat’s Last Theorem. --Pgb23 (talk) 19:29, 7 November 2008 (UTC)

I see how it would follow from FLT, but how you would prove this without FLT I don't see.
(And notice that your can write things like
$\sqrt[n]{\left(\frac{b}{a}\right)^n + 1},$
so you don't need this typewriter-style notation.) Michael Hardy (talk) 20:30, 7 November 2008 (UTC)
Not sure why you wrote all that.
It is original research without any citation and I can't find notability.
I don't see that it is interesting.
There's quite enough easily made up irrationals there already
Dmcq (talk) 20:23, 7 November 2008 (UTC)
I get the feeling all this business about an irrational generator was because I stuck in a reference to unique factorization into the proof that the nth root of an number is irrational if it isn't the nth power of an integer. The reference is necessary but an extension to rational numbers is trivial, if a rational in lowest terms is not of the form pn/qn then its nth root is irrational. So a simple generator of irrationals is for instance (2k/(2k+1))1/n. One can generate any number of these without going anywhere near anything like Fermat's Last Theorem. Dmcq (talk)

## Could someone cite such a book??

This article says:

Equivalently, irrational numbers are real numbers that cannot be represented as terminating or repeating decimals, and this presentation is often used in textbooks in elementary mathematics.[citation needed]

I put the "fact" tag there. I've long wondered why the erroneous belief that this is the definition persists over decades without ever being taught. If it is in fact taught, that would answer the question, although it raises another question: why don't mathematicians step in to correct the error?

Can someone cite one or more such books? Michael Hardy (talk) 18:20, 19 October 2009 (UTC)

Huh? I don't recall right off hand having seen any elementary textbook that didn't mention that irrational numbers in decimal form are non-repeating. The issue as to whether the non-repeating decimal concept is a desirable definition of an irrational number is another question. My question is simply: Why is there a "[citation needed]" appended to that sentence? If I were to write, for example, "Euclid's proof of the infinitude of primes is often presented in textbooks of elementary mathematics", why would I need a citation? Where would even I find a citation? Worldrimroamer (talk) 19:13, 10 November 2009 (UTC)
You'd find plenty of such citations by using Google Books. Michael Hardy (talk) 23:58, 10 November 2009 (UTC)
Either you did not read my posted comment or you somehow failed completely to miss the point. To give you yet another example (duh?): If I were to state that it is commonly taught in elementary math textbooks that 2 + 2 = 4 (even though this is not true in some groups, rings, etc.) what would I cite to validate that "most elementary math textbooks state that 2 + 2 = 4"? What is it that I'm failing to communicate here? Worldrimroamer (talk) 22:17, 11 November 2009 (UTC)
"Failed completely to miss the point" is an amusing locution.
But look: It says "Equivalently, irrational numbers are real numbers that cannot be represented as terminating or repeating decimals, and this presentation is often used in textbooks[...]". It doesn't say that fact is mentioned or proved in textbooks; it says "this presentation is used" in textbooks. It doesn't make sense to phrase it that way unless it means that's the definition. I don't believe ANY textbook does that. So I challenged someone to cite one. Michael Hardy (talk) 23:54, 12 November 2009 (UTC)
Wait, I just deleted my previous post. I see your point: It says it was "used" in the textbooks. OK, I see your point. Then, instead of saying "citation needed", why not change "used" to "mentioned"? Whatever, I don't care any more. Worldrimroamer (talk) 15:26, 15 November 2009 (UTC)

"This presentation is used...." appears to mean something different from "This corollary of the definition is presented". It makes it look as if they're using that as the definition. Michael Hardy (talk) 23:57, 10 November 2009 (UTC)

The word presentation perhaps wasn't a good one but the remark is an entirely trivial consequence of the division algorithm taught in schools and doesn't need citation because it's unlikely to be challenged. It is often mentioned because you can use it to write down irrationals immediately: any non-terminating non-repeating decimal expansion such as 0.101001000100001... provides an example. Rinpoche (talk) 04:47, 13 September 2010 (UTC)

## The set of all irrationals

the space of irrationals is topologically complete: that is, there is a metric on the irrationals inducing the same topology as the restriction of the Euclidean metric, but with respect to which the irrationals are complete. One can see this without knowing the aforementioned fact about G-delta sets: the continued fraction expansion of an irrational number defines a homeomorphism from the space of irrationals to the space of all sequences of positive integers, which is easily seen to be completely metrizable.

What is this homeomorphism? (What happened to negative numbers?) And also, there is a bigger problem. We know how to define a metric on a set of sequences (end of this section), but it is far from clear (to me, at least) how to give a metric that would induce the standard topology. I am working on an alternative way to completely metrize this space at the moment, using an enumeration of rationals. I stumbled upon this puzzle the other day and actually thought of continued fractions first. I was delighted to see them mentioned here, but after trying to work out the details, I really doubt that this route is easy or even feasible. Let me know what you think! melikamp (talk) 23:52, 28 December 2009 (UTC)

The space of all irrationals is homeomorphic to the space of positive irrationals if there is a strictly increasing bijection between them. And that exists of there is a strictly monotone bijection between all rationals and the positive rationals. And that exists by Cantor's back-and-forth method, although I think one could find simpler methods. Such as:

$x \mapsto x - \frac{1}{x}.$

That takes the set of positive rationals to the set of all rationals; it's bijective and strictly monotone.

How about this metric. If two sequences' first disagreement is in the nth place, then the distance between them is 1/2n. Maybe that will work. Michael Hardy (talk) 00:00, 29 December 2009 (UTC)

It's been a while since I looked at this article, but that map is injective, not bijective. The inverse map would be
$y \mapsto \frac 1 2 \left( y + \sqrt{y^2+4} \right)$
However, a piecewise linear map (with an infinite number of pieces) would work. — Arthur Rubin (talk) 20:14, 25 July 2014 (UTC)
Or, perhaps:
$x \mapsto \frac {x-1}{\min(1,x)} = \begin{cases} x-1 & x \ge 1 \\ 1 - \frac 1 x & x \le 1 \end{cases}$
Arthur Rubin (talk) 20:36, 25 July 2014 (UTC)
Function : $f: x \mapsto x - \frac{1}{x}$ maps $\Bbb R^+ \to \Bbb R$,
it is continuous and differentiable,
its derivative $f^\prime = 1+\frac 1{x^2}$ is positive everywhere
and its values span from $\lim_{x\to 0^+}f = -\infty$
to $\lim_{x\to \infty}f = \infty$
....and it is not bijective? --CiaPan (talk) 09:18, 28 July 2014 (UTC)
It's not bijective as a map from $\Bbb Q^+ \to \Bbb Q$, which is what was needed. — Arthur Rubin (talk) 12:14, 28 July 2014 (UTC)
Oh, now I see. --CiaPan (talk) 22:19, 28 July 2014 (UTC)

## The symbol of the Irrational number

can someone put some words about the the synbol of the irrational ($\scriptstyle\mathbb{J}$ or $\scriptstyle\mathbb{P}$) and it's source? since it was removed from here it's can't bee found anywhere. Yisrael Krul (talk) 19:16, 14 January 2010 (UTC)

## Plagarism

The well written section on modern developments with its mention of "recent" work by Tannery(1894) looks to be lifted from the 1896 book History of Modern Mathematics [1] by David Eugene Smith. It may be in the public domain but still.. --Gentlemath (talk) 21:16, 16 March 2010 (UTC)

That'll be out of copyright but it should still have some attribution, I'm not sure how one best does that. Dmcq (talk) 00:33, 17 March 2010 (UTC)

What is Wikipedia policy about just lifting text verbatim from out of copyright sources?--Gentlemath (talk) 02:18, 17 March 2010 (UTC)

That one must not do it. It causes a lot of trouble as it may be necessary to clean out the history as well as just the offending content. See Wikipedia:Copyright violations. People who do it are a pain in the neck and liable to be permanently blocked if they continue after being warned. Dmcq (talk) 11:29, 17 March 2010 (UTC)
Small excerpts may be used in some special circumstances, see Wikipedia:Non-free content. Dmcq (talk) 11:34, 17 March 2010 (UTC)
Sorry misread your sentence as being about lifting copyright sources! Swapped over the out and from. I believe the only policy is that all such stuff must be cited and anything substantial, which I take as enough to be a copyright violation if it was copyright, should have something explicitly saying it was copied. That's to avoid accusation of plagiarism which still holds even if a work isn't copyright. Dmcq (talk) 13:14, 17 March 2010 (UTC)
How much of this article was lifted out of that 1894 work? There should be a bigger acknowledgment than just a straightforward citation if it is more than a couple of sentences. Dmcq (talk) 12:43, 17 March 2010 (UTC)

## History (to Medieval Ages)

I haven't read the entire article but this section is extremely poor and actually nonsense in places.

The sentence "The then-current Pythagorean method would have claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other." is incomprehensible in the context provided and the use of the qualifier "evenly" as unfortunate as it is possible to imagine were one actually set out deliberately to muddle the issues involved - I do wonder whether its author can have any competence in mathematics whatsoever.

The presentation of the classical proof of the irrationality of SQRT(2) is distinctly laboured and the assertion that a^2=2*b^2 is a consequence of the Pythagoras' theorem just plain wrongheaded. It is a consequence of the assumption that SQRT(2) is a rational a/b. Moreover, while it's common to see the premis that a/b is reduced to its lowest terms in textbooks offering the classical proof as a stand-alone proof of the irrationality of SQRT(2), it shouldn't be premissed in an article discussing its historic basis. The Pythagoreans certainly didn't have sufficient arithmetic to prove you can uniquely reduce a fraction to lowest terms - this came later, possibly from Euclid himself, in the form of Euclid VII, 3 ('Euclid's algorithm' to extract the highest common factor of a pair of integers) and VII,21 and 22 (a/b is reduced to lowest terms iff (a,b) = 1) demonstrating uniqueness. They would have known how 'to cast out twos' reducing a/b to the point where not both a and b were even, and that is all that needs to be premissed (I see a later section 'Square roots' has an accurate proof).

The article repeats the usual nonsense about Hippasus. The reality is that virtually nothing is known about him and just two classical authors mention him. The stuff about the pentagram a novel fantasy I think.

But it is the section describing Eudoxus' work which is really lamentable here. It simply a travesty of his theory of proportion which, as is often remarked, leads to a description of the real numbers essentially the same as that provided by the Dedekind section.

I don't want to step on anyone's toes maintaining this page (well not in the first place anyway) but I will edit the section myself a few weeks hence if some effort hasn't been made in the meantime to correct its deficiencies (I would much rather existing editors of the page undertook this then have to spend the significant time involved myself).

Who is the editor providing these bulleted, almost syllogistic, mathematical proofs as found here and a number of other related pages I have noticed? These are uniformally weak, sometimes risibly so, and it's difficult to imagine the editor is adequately equipped mathematically to fulfill the task he/she has appointed for himself/herself. I notice the language is also somewhat archaic and I wonder if this is the plagiarism Dmcq has noticed. At any rate the editor involved needs to be discouraged. It is very far from helpful and a positive mischief to persist. Rinpoche (talk) 04:02, 13 September 2010 (UTC)

I hadn't really looked at the history section, I only answered above about how much credit one should give when copying something in the public domain. I did do some work on the different proofs below the history because some editors had stuck in their own working assuming the fundamental theorem of arithmetic without saying so. The stuff about Hipassus may or may not be nonsense but having two classical authors mention him is quite enough to establish that he should be mentioned even if his role is disputed. I haven't the foggiest what all the stuff about complex numbers is here for and it seems funny the way Liouville is just an also mentioned. By the way one only needs to assume numbers are ordered to get a/b in lowest terms, there is no assumption there isn't another p/q where p and q are larger and coprime yet p/q is equal to a/b. Dmcq (talk) 13:19, 13 September 2010 (UTC)
Of course Hippasus should be mentioned as the traditonal discoverer but it should also be made quite clear, if Wikipedia is not merely to propagate legend as so many inadequately researched textbooks do, that stories about him being murdered for his trouble are just that - stories. What you say about reducing a/b to lowest terms is of course quite true and it's as well known to me and others as it is to you that it depends on well-ordering (i.e. is not a purely 'algebraic' property of the integers) but you still need to show it's unique (I'm not in fact sure from your English what you mean to imply from your following remark about p/q) and Euclid did do this explicitly. The Pythagoreans simply didn't have the necessary arithmetic to base a proof on reduction to lowest terms. The classical proof first appears in a work by Aristotle some two centuries later than the Pythagoreans.
All this doesn't detract from my contention that the piece really sucks and needs to be rewritten. I don't at all want to spend the time doing this but I give fair warning that I will be sweeping a pretty clean broom if I do have to.Rinpoche (talk) 20:30, 13 September 2010 (UTC)
I was just talking about the 'uniquely reduce to lowest terms'. They certainly would know that for any given rational number there would be a unique a/b equal to it where b was smallest, however they wouldn't have been able to prove there wasn't a larger p/q equal to the same fraction but where q wasn't a multiple of b which is where Euclid's proof came in. I think it's a pretty advanced bit of mathematical thought though that someone realized there was actually a problem that had to be solved, so many people nowadays just assume it without thinking. Dmcq (talk) 20:55, 13 September 2010 (UTC)
Euclid didn't know (or at any rate couldn't state) unique factorisation of integers. But he did know his lemma and that's pretty close. What he did know about the highest common factor and divisibility is sufficient to provide a rigorous proof of the irrationality of the square root of a non-square integer by pointing out that such a presumed rational root reduced to lowest terms would still be reduced to its lowest terms on squaring and so have its denominator equal to unity thus implying the integer was a square, the contradiction. But he actually accomplished it by quite different arguments involving 'parts' and 'proportions' and I don't know the details. Sunmmarising it for the Wikipedia would be notable I think but I'm definitely not voluteering :-) The Wiki article on his lemma gives a convoluted (and unsourced - it really should be deleted) proof that doesn't depend on Bézout's identity. That could usefully be replaced I think with a sketch of Euclid's proof. Rinpoche (talk) 23:45, 13 September 2010 (UTC)

The reference about Kurt Von Fritz article is incomplete, which made it hard to me to find the article. It should be: Annals of Mathematics, Second Series, Vol.46, No.2 (Apr., 1945), pp. 242-264. I tried to edit it but when I click "edit" I only see: Reflist|2. I don't no how to do it. Also, for James R. Choike article: "The Two-Year College Mathematics Journal Vol. 11, No. 5 (Nov., 1980), pp. 312-316" Alithilatis (talk) 12:23, 4 January 2013 (UTC)

## Miscellaneous section?

"Miscellaneous Here is a famous pure existence or non-constructive proof:

There exist two irrational numbers a and b, such that ab is rational. Indeed, if √22 is rational, then take a = b = √2. Otherwise, take a to be the irrational number √22 and b = √2. Then ab = (√22)2 = √22·√2 = √22 = 2 which is rational.

Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem implies that √22 is transcendental, hence irrational."

Although I am not versed in mathematics, I do know that this is a non-sourced section that is certainly not written like an encyclopaedia (Here is ... ?). Besides that, miscellaneous information is not supposed to be part of the article. As such, I am moving this here until such time as it can be decided whether or not the above text is necessary, and what the section title should be. Crisco 1492 (talk) 10:06, 18 January 2011 (UTC)

Language is normally just something to be fixed if you don't like it. That proof is quite famous and is repeated in a number of places though only a few seem to know who wrote it originally. I'll stick in a citation. Dmcq (talk) 12:12, 18 January 2011 (UTC)
If it were just a language problem, I would have fixed it on my own. However, there were 3 things wrong with that section, hence why I moved it here for discussion. I didn't want to delete it out of the blue and look like a vandal, nor did I want to delete somebody's hard work. Good edit, by the way. Crisco 1492 (talk) 23:00, 18 January 2011 (UTC)

## The first sentence

In mathematics, an irrational number (lacks characteristics that are true of a rational number) and is therefore not a rational number. To the layperson, this is non sequitur and self-referential. It is not obvious that the set of rational numbers excludes irrational numbers, and it is not helping anyone by defining it in terms of itself (or its inverse). Perhaps a better definition would start by describing why it is necessary to divide numbers into the two groups rational and irrational. --MoonLichen (talk) 05:09, 29 January 2011 (UTC)

There is no circularity as rational numbers are not defined in terms of irrational numbers. However circular someone might get the feeling they are that is how they are normally defined. I think though saying they cannot be expressed as terminating or repeating decimals is more obscure than saying they cannot be expressed as the quotient of two integers. It was a major proof by the early Greeks that irrational numbers existed. Dmcq (talk) 10:36, 29 January 2011 (UTC)

Shouldn't we also mention that it is the choice of unit length that creates irrational number, indicating the inherent limitations of any number system. An irrational number, representing a certain length on the real line in a given number system, could become rational in another number system if we adjust the unit length in consideration. Kawaikx15 (talk) 04:19, 25 September 2012 (UTC)

Nope. Numbers are NOT defined in terms of length (unless you are an ancient Greek), so they don't depend on any choice of unit length. --CiaPan (talk) 05:34, 26 September 2012 (UTC)
And I think it should be admitted, Greeks developed an advanced system of measurements (see Systems of measurement#History lead) and used many lenght units (Ancient Greek units of measurement#Length). We may suppose they realized the choice of a unit is arbitrary, and the same physical distance would be expressed by different numbers with different units. So their 'numbers' were not just lenghts but rather ratios of lenghts - a 'number' was either a natural number telling how many times one line segment fits in another, or a pair of natural numbers telling how many times some common part fits in one and in the other line segment. This allows to consider quantities (lenghts, areas...) which appear in geometric constructions irrespective of the size of an actual drawing. --CiaPan (talk) 08:09, 28 September 2012 (UTC)

I was referring to the geometrical interpretation of a number. kawaikx15 Saurabh (talk) 09:54, 29 September 2012 (UTC)

## counterclimactic history

The first subsection in the history section deals with what is described here as claims "unlikely to be true". Apparently a scholarly controversy on the subject exists. As it currently appears, it makes for a rather awkward start for the history section. I would suggest postponing it until after the greek section, or deleting it altogether if credible sources say it is in fact not true. Regardless of how we decide to present the scholarly debate, the current opening for the history section is not very informative. Tkuvho (talk) 08:13, 30 January 2011 (UTC)

A contemporary view of "Indians" about Indian mathematics is the need of hour. Dr. Boyer no doubt was a great math historian, died in 1976. After his death lot of things have changed including a zeal among Indians about researching ancient contributions and rigorously analyzing it in scientific manner. The phrase "unlikely to be true" and word "claim" in Dr. Boyer reference reeks of Personal Conclusions seeded with doubt and therefore does not need to BOLDLY highlighted. Though it must be stated that people might have opposing views, I do not see any strong references stating Indian contributions to understanding Irrational numbers as totally untrue. In 1980s-1990s, Dr. TS Bhanu Murthy, a retired Director of Ramanujan Institute for Advanced studies in Mathematics produced a book A Modern Introduction to Ancient Indian Mathematics. This book was not only a mathematical revision but a historical examination of ancient contributions. The author is authentic, details can be found here at University of Madras, India official website. http://www.unom.ac.in/index.php?route=department/department/about&deptid=48 . The author Dr. Bhanu murthy comes from a mathematics academic world. He worked under Dr. Gelfand (well known Russian mathematician) and Dr. Harishchandra (Princeton who died in 1983 and was an I.B.M. von Neumann Professor). A link to one of the works of Dr. Murthy can be found here. http://www.ams.org/mathscinet-getitem?mr=MR23:A2481. therefore, the evidence suggests He is a real person with credible math academic background and has capacity to analyze ancient treatise on mathematics. His Book "A Modern Introduction to Ancient Indian Mathematics" is therefore a seminal contribution from a Indian Mathematician towards giving a glimpse of ancient mathematical treatise. I therefore rest my case that this should be acknowledged as a legitimate view opposing that of Dr. Boyer. PS: I would extend such arguments to other works in Wikipedia which treat western sources as authentic interpretations about ancient Indian works and try to play down contributions and genuine reexaminations from Indians. — Preceding unsigned comment added by Sudhee26 (talkcontribs) 21:29, 9 June 2014 (UTC)

You replaced a concrete and direct statement ("such claims are not well substantiated and unlikely to be true") that was well sourced to an imminent historian, with an almost meaningless statement ("propose different views"). The source that you added is also a bit questionable being held by only 41 libraries worldwide according to worldcat, compared to thousands of libraries for the Boyer source, and published by a publisher that doesn't even have a Wikipedia entry. If his work "A Modern Introduction to Ancient Indian Mathematics" is seminal, there is not much evidence of it yet. Sławomir Biały (talk) 23:12, 9 June 2014 (UTC)

## b non-zero?

In the definition of irrational numbers, is the requirement "with b non-zero" necessary? The inclusion of this statement implies that we WANT numbers like 2/0 included in the definition of irrational numbers. Since we are only considering the real numbers, values such as 2/0 are excluded immediately, so it doesn't do any harm, but I think the "with b non-zero" requirement is redundant. —Preceding unsigned comment added by 150.101.29.94 (talk) 23:47, 30 January 2011 (UTC)

## Removed a contribution

The following has been added twice to the medieval section. It doesn't make much sense to me there. I think what they're trying to say perhaps is the Indians dealt with irrationals just like rationals. I'm not sure the Indians actually knew there was a problem in the first place. Is there something salvageable? Dmcq (talk) 19:46, 18 December 2011 (UTC)

Brahmagupta was the first to compute with irrational numbers. “The readiness with which the Hindus passed from number to magnitude and vice versa. If we define algebra as the application of arithmetical operations to both rational and irrational numbers, then the Brahmans are the real inventors of algebra”. Herman Hankel, The Encyclopedia Brittanica, page 607, 1910 Available free of charge from Google books.

“Indians were the first to reckon with irrational square roots as with numbers” Henry Fine, “The Number System of Algebra”, Dean of Mathematics, Princeton University, page 106, 1897

It appears that the cited sources are the ones that can be found here: [2] and [3] Isheden (talk) 20:49, 18 December 2011 (UTC)

## Back to basics

The non-repeating infinite decimal expansion of an irrational number is represented by ellipsis. This fundamental fact is omitted from the article. To make matters worse: (the last time I checked), precomposed ellipsis  is denoted by three dots (...); yet when used in this article (without explanation) it is four dots, unless the four dots have some other meaning. Does one need a citation from a "reliable source" to modify the article accordingly, or would this fall under "common knowledge". Note: it might not be common knowledge for a user coming to this page to simply find out what an irrational number is, and how to identify one when found in text. ~E 74.60.29.141 (talk) 09:26, 24 October 2012 (UTC)

That is a matter of notation and not a property of irrational numbers themselves. That explanation probably belongs in Decimal representation. With regards to the TeX images, it simply appears to be the way the version of TeX Wikipedia uses does things. I changed it to \ldot which in LaTeX has only three dots but it still shows four dots. If you can find the right version of dots, feel free to change it. Transcendence (talk) 19:53, 24 October 2012 (UTC)
Actually it turns out that \dots does the right thing, it's just that someone put a period there after it for some reason. I've removed the period. Transcendence (talk) 19:58, 24 October 2012 (UTC)
Thanks (Sorry about getting a little cranky lately) ~E 74.60.29.141 (talk) 22:29, 24 October 2012 (UTC)

It's quite simple, Transcendence, the dots you removed here were obviously 'full stops', used to close sentences. --CiaPan (talk) 12:48, 4 January 2013 (UTC)

If an ellipsis (three dots) ends a sentence, it is to be followed by the appropriate punctuation symbol (the fourth dot of the above discussion). I'll put the periods back in, but while I'm at it I noticed that this is not the standard algorithm. The standard algorithm has two nice features; it makes the computation make sense and it avoids any unnecessary manipulations. The current algorithm, while mathematically correct, suffers on both counts – I'll fix that as well. Bill Cherowitzo (talk) 20:20, 4 January 2013 (UTC)

## Zeno "Proof" statement is nonsense

Currently one sentence in the History section reads """In the minds of the Greeks, disproving the validity of one view did not necessarily prove the validity of another""", which implies that this was erroneous thinking. Yet logically it is correct: disproving one hypothesis certainly *does not* prove another hypothesis: this is known as False dilemma. The only case that it is true is when is has been *proved* that there are only two cases. 80.254.148.123 (talk) 07:54, 10 April 2015 (UTC)