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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.
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)
I moved this from the main page:
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)
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)
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. 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)
Isn't the first proof for the irrationality of overly complicated? It basically states that when you transform to , 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)
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)
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)
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)
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)
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:
Here is a rephrasing:
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)
Michael Hardy wrote:
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)
...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)
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)
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.
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
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)
In the numerical example, the old version ends with this:
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)
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: . I can very well imagine a situation when someone wants to look up a quick definition of irrational number, sees a huge in the lead paragraph and then walks away thinking that he got it. Grue 20:22, 13 October 2006 (UTC)
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)
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)
Would it be possible to reproduce a version of this proof in the article without violating copyright ? Robert2957 20:33, 27 October 2006 (UTC)
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)
"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)
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)
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 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
"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
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)
Should this page be converted to use <math> 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)
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 Cat Hetware 23:24, 4 June 2007 (UTC)
I would like to know whether e raised to any natural number is irrational? —Preceding unsigned comment added by Sumitagarai ( talk • contribs)
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)
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)
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)
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 ( talk) 16:14, 11 October 2007 (UTC)
That's easy: 1/1 = 0.99999... Michael Hardy 19:28, 11 October 2007 (UTC)
1|1
.9 1|1 9 1
.999 1|1 9 1 9 1 9 1
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)
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:
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)
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)
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)
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)
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)
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.
For natural numbers n and integers a, b, the nth Root of is irrational for n > 2. . Hence this formula can be used to generate an infinite number of irrational numbers.
Assume that the nth Root of [ is rational, then so is nth Root
Hence, nth Root [] = c/q . . . . . . for some integers c and q
So,
And . . . . . let d = q * a and e = q * b,
Thus, , which since d and e are integers, contradicts “Fermat’s Last Theorem” which has recently been proved by Andrew Wiles. Hence nth Root 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)
This article says:
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)
"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)
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:
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)
can someone put some words about the the synbol of the irrational ( or ) 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)
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)
What is Wikipedia policy about just lifting text verbatim from out of copyright sources?-- Gentlemath ( talk) 02:18, 17 March 2010 (UTC)
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)
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 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 √2√2 is rational, then take a = b = √2. Otherwise, take a to be the irrational number √2√2 and b = √2. Then ab = (√2√2)√2 = √2√2·√2 = √22 = 2 which is rational.
Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem implies that √2√2 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)
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)
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)
I was referring to the geometrical interpretation of a number. kawaikx15 Saurabh ( talk) 09:54, 29 September 2012 (UTC)
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 ( talk • contribs) 21:29, 9 June 2014 (UTC)
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)
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
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)
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)
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)
Algebraic irrational numbers are not rational, we do have infinite of them, like one times √2, two times √2, and so on. Moreover, we also have √prime, and we also have the square root of the non-squared composite integers. Can someone points me the proof that the transcendental number is more likely to be picked rather than algebraic irrational numbers, so that the "almost all" phrase makes sense?
The first image here, File:Real numbers.svg, was removed because it is potentially misleading because, "it implies there are real numbers that are neither rational or irrational. Could also mislead one about the relative "sizes" of these sets," (a description which I added to the file page). The second image, File:Subsets of Numbers.png, may also be misleading because it depicts the real numbers on a number line dividing a circle in half and a large unnamed space. Hyacinth ( talk) 13:55, 22 May 2016 (UTC)
Most of the Venn diagrams of number systems, Commons:Category:Venn diagrams of numbers sets, do not seem include irrational numbers. Hyacinth ( talk) 13:57, 22 May 2016 (UTC) File:AlgebIrrat.svg and File:AlgebIrrat2.PNG both depict irrational numbers, but their text is in French. Hyacinth ( talk) 14:37, 22 May 2016 (UTC)
The second sentence is:
When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that there is no length such that each of them could be "measured" as being a certain integer multiple of it.
If the ratio is irrational, couldn't one line segment be rational (even an integer) while the other is irrational? -- Dan Griscom ( talk) 12:14, 24 June 2017 (UTC)
... meaning that there is no length such that both of them could be "measured" as being certain integer multiples of this single length.
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I put this video in the article adjacent to a proof which it explains, both in terms of its context and overall meaning and giving the same proof more-or-less in a "blackboard format". The video is from TED (conference), TED-Ed in particular, which has a fairly good reputation. There is no question of copyright violation in the link. It has a reference going back to the TED website, which also gives the video, but the main link is to YouTube, which is likely easier to use for most people. Having 2 links to the video (one in the video template, one in the ref) also helps prevent link rot.
Another editor reverted the addition of the video with edit summary of something like "YouTube videos aren't reliable sources." I disagree and will put back the video. I hope folks will at lest view the video before removing it. Smallbones( smalltalk) 18:33, 22 October 2017 (UTC)
References
There are some editors who can not seem to control the urge to uselessly extend any list that they see. In the third paragraph of this article there is a representation of the first few digits of π that some have mindlessly extended. There are a few editors who would like to put an end to this practice, but we have a slight disagreement on exactly how to do this. User:Purgy Purgatorio has suggested the minimalist approach and wanted to display just 3.1, while I have countered with 3.14159. Both of these are much shorter than what had been displayed. My choice was deliberate and involved the following considerations. First of all, I think that unless at least 3.14 is displayed, some readers would not recognize that π was being represented. Other readers (hopefully few) are under the impression that π is 3.14, so I felt that more digits would be necessary to bring the point of the sentence to the forefront. My choice is somewhat arbitrary, a little more that 3.14 but not too much more. Ultimately, it probably doesn't matter since neither of our approaches deals with the underlying problem. My concern is that I think the shorter 3.1 would be more likely to irritate the "extenders", but neither representation will discourage them. -- Bill Cherowitzo ( talk) 19:24, 10 February 2018 (UTC)
So by "is distinct from" your saying:
- Let DEToR x mean x's decimal expansion terminates or reports. - All x, rational x implies DEToR x - But All x, DEToR x does not imply rational x - Therefore exist x, DEToR x and not rational x
Wow really?! — Preceding unsigned comment added by 110.22.70.186 ( talk) 11:42, 10 March 2018 (UTC)
@ CiaPan: I found a list denoting the official names of the Berlin Journals in several periods. I am unsure, whether this disagrees with a recent edit regarding 1761. Just FYI. Purgy ( talk) 16:29, 18 September 2018 (UTC)
This article refers to "complex quadratic irrational numbers." However, its first sentence says that "a quadratic irrational number ... is an irrational number," and the first sentence of the article on irrational numbers says that "the irrational numbers are ... real numbers." Therefore, if quadratic irrational numbers are real, how can some of them be complex? I think that some of the wording / terminology needs to be clarified so that everything is consistent. 2604:2000:EFC0:2:4DF6:6328:1154:9482 ( talk) 02:30, 29 October 2019 (UTC)
@ Melikamp, Michael Hardy, and Arthur Rubin:
Hi, I just found that old thread from 2009 & 2014 above ( #The set of all irrationals) and I thought I'll post another continuous piece-wise map with two pieces:
The inverse map is
Best regards. -- CiaPan ( talk) 14:09, 29 October 2019 (UTC)
I removed "It is not known if either of the tetrations or is rational for some integer " since there is no consensus on what tetration even means for non-integer heights so speculation on irrationality is premature. 08:23, 18 December 2019 (UTC) — Preceding unsigned comment added by 2A00:23C6:1489:9900:1421:3227:6B97:7E02 ( talk)
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 — Preceding unsigned comment added by Wbaker716 ( talk • contribs) 00:37, 31 January 2020 (UTC)
Can we provide a source for this? I want to read the proof Immanuelle ( talk) 23:41, 4 April 2022 (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. Question is, did he really prove that this constant cannot be expressed as a/b with a and b being positive integers? Kwékwlos ( talk) 21:49, 19 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. D.M. from Ukraine ( talk) 13:33, 29 June 2023 (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? 2402:A00:401:B896:4923:CB28:C2C3:C3A7 ( talk) 17:01, 4 July 2024 (UTC)
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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.
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)
I moved this from the main page:
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)
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)
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. 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)
Isn't the first proof for the irrationality of overly complicated? It basically states that when you transform to , 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)
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)
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)
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)
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)
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:
Here is a rephrasing:
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)
Michael Hardy wrote:
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)
...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)
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)
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.
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
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)
In the numerical example, the old version ends with this:
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)
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: . I can very well imagine a situation when someone wants to look up a quick definition of irrational number, sees a huge in the lead paragraph and then walks away thinking that he got it. Grue 20:22, 13 October 2006 (UTC)
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)
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)
Would it be possible to reproduce a version of this proof in the article without violating copyright ? Robert2957 20:33, 27 October 2006 (UTC)
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)
"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)
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)
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 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
"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
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)
Should this page be converted to use <math> 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)
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 Cat Hetware 23:24, 4 June 2007 (UTC)
I would like to know whether e raised to any natural number is irrational? —Preceding unsigned comment added by Sumitagarai ( talk • contribs)
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)
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)
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)
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 ( talk) 16:14, 11 October 2007 (UTC)
That's easy: 1/1 = 0.99999... Michael Hardy 19:28, 11 October 2007 (UTC)
1|1
.9 1|1 9 1
.999 1|1 9 1 9 1 9 1
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)
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:
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)
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)
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)
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)
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)
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.
For natural numbers n and integers a, b, the nth Root of is irrational for n > 2. . Hence this formula can be used to generate an infinite number of irrational numbers.
Assume that the nth Root of [ is rational, then so is nth Root
Hence, nth Root [] = c/q . . . . . . for some integers c and q
So,
And . . . . . let d = q * a and e = q * b,
Thus, , which since d and e are integers, contradicts “Fermat’s Last Theorem” which has recently been proved by Andrew Wiles. Hence nth Root 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)
This article says:
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)
"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)
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:
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)
can someone put some words about the the synbol of the irrational ( or ) 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)
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)
What is Wikipedia policy about just lifting text verbatim from out of copyright sources?-- Gentlemath ( talk) 02:18, 17 March 2010 (UTC)
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)
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 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 √2√2 is rational, then take a = b = √2. Otherwise, take a to be the irrational number √2√2 and b = √2. Then ab = (√2√2)√2 = √2√2·√2 = √22 = 2 which is rational.
Although the above argument does not decide between the two cases, the Gelfond–Schneider theorem implies that √2√2 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)
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)
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)
I was referring to the geometrical interpretation of a number. kawaikx15 Saurabh ( talk) 09:54, 29 September 2012 (UTC)
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 ( talk • contribs) 21:29, 9 June 2014 (UTC)
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)
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
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)
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)
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)
Algebraic irrational numbers are not rational, we do have infinite of them, like one times √2, two times √2, and so on. Moreover, we also have √prime, and we also have the square root of the non-squared composite integers. Can someone points me the proof that the transcendental number is more likely to be picked rather than algebraic irrational numbers, so that the "almost all" phrase makes sense?
The first image here, File:Real numbers.svg, was removed because it is potentially misleading because, "it implies there are real numbers that are neither rational or irrational. Could also mislead one about the relative "sizes" of these sets," (a description which I added to the file page). The second image, File:Subsets of Numbers.png, may also be misleading because it depicts the real numbers on a number line dividing a circle in half and a large unnamed space. Hyacinth ( talk) 13:55, 22 May 2016 (UTC)
Most of the Venn diagrams of number systems, Commons:Category:Venn diagrams of numbers sets, do not seem include irrational numbers. Hyacinth ( talk) 13:57, 22 May 2016 (UTC) File:AlgebIrrat.svg and File:AlgebIrrat2.PNG both depict irrational numbers, but their text is in French. Hyacinth ( talk) 14:37, 22 May 2016 (UTC)
The second sentence is:
When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that there is no length such that each of them could be "measured" as being a certain integer multiple of it.
If the ratio is irrational, couldn't one line segment be rational (even an integer) while the other is irrational? -- Dan Griscom ( talk) 12:14, 24 June 2017 (UTC)
... meaning that there is no length such that both of them could be "measured" as being certain integer multiples of this single length.
External videos | |
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I put this video in the article adjacent to a proof which it explains, both in terms of its context and overall meaning and giving the same proof more-or-less in a "blackboard format". The video is from TED (conference), TED-Ed in particular, which has a fairly good reputation. There is no question of copyright violation in the link. It has a reference going back to the TED website, which also gives the video, but the main link is to YouTube, which is likely easier to use for most people. Having 2 links to the video (one in the video template, one in the ref) also helps prevent link rot.
Another editor reverted the addition of the video with edit summary of something like "YouTube videos aren't reliable sources." I disagree and will put back the video. I hope folks will at lest view the video before removing it. Smallbones( smalltalk) 18:33, 22 October 2017 (UTC)
References
There are some editors who can not seem to control the urge to uselessly extend any list that they see. In the third paragraph of this article there is a representation of the first few digits of π that some have mindlessly extended. There are a few editors who would like to put an end to this practice, but we have a slight disagreement on exactly how to do this. User:Purgy Purgatorio has suggested the minimalist approach and wanted to display just 3.1, while I have countered with 3.14159. Both of these are much shorter than what had been displayed. My choice was deliberate and involved the following considerations. First of all, I think that unless at least 3.14 is displayed, some readers would not recognize that π was being represented. Other readers (hopefully few) are under the impression that π is 3.14, so I felt that more digits would be necessary to bring the point of the sentence to the forefront. My choice is somewhat arbitrary, a little more that 3.14 but not too much more. Ultimately, it probably doesn't matter since neither of our approaches deals with the underlying problem. My concern is that I think the shorter 3.1 would be more likely to irritate the "extenders", but neither representation will discourage them. -- Bill Cherowitzo ( talk) 19:24, 10 February 2018 (UTC)
So by "is distinct from" your saying:
- Let DEToR x mean x's decimal expansion terminates or reports. - All x, rational x implies DEToR x - But All x, DEToR x does not imply rational x - Therefore exist x, DEToR x and not rational x
Wow really?! — Preceding unsigned comment added by 110.22.70.186 ( talk) 11:42, 10 March 2018 (UTC)
@ CiaPan: I found a list denoting the official names of the Berlin Journals in several periods. I am unsure, whether this disagrees with a recent edit regarding 1761. Just FYI. Purgy ( talk) 16:29, 18 September 2018 (UTC)
This article refers to "complex quadratic irrational numbers." However, its first sentence says that "a quadratic irrational number ... is an irrational number," and the first sentence of the article on irrational numbers says that "the irrational numbers are ... real numbers." Therefore, if quadratic irrational numbers are real, how can some of them be complex? I think that some of the wording / terminology needs to be clarified so that everything is consistent. 2604:2000:EFC0:2:4DF6:6328:1154:9482 ( talk) 02:30, 29 October 2019 (UTC)
@ Melikamp, Michael Hardy, and Arthur Rubin:
Hi, I just found that old thread from 2009 & 2014 above ( #The set of all irrationals) and I thought I'll post another continuous piece-wise map with two pieces:
The inverse map is
Best regards. -- CiaPan ( talk) 14:09, 29 October 2019 (UTC)
I removed "It is not known if either of the tetrations or is rational for some integer " since there is no consensus on what tetration even means for non-integer heights so speculation on irrationality is premature. 08:23, 18 December 2019 (UTC) — Preceding unsigned comment added by 2A00:23C6:1489:9900:1421:3227:6B97:7E02 ( talk)
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 — Preceding unsigned comment added by Wbaker716 ( talk • contribs) 00:37, 31 January 2020 (UTC)
Can we provide a source for this? I want to read the proof Immanuelle ( talk) 23:41, 4 April 2022 (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. Question is, did he really prove that this constant cannot be expressed as a/b with a and b being positive integers? Kwékwlos ( talk) 21:49, 19 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. D.M. from Ukraine ( talk) 13:33, 29 June 2023 (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? 2402:A00:401:B896:4923:CB28:C2C3:C3A7 ( talk) 17:01, 4 July 2024 (UTC)