|WikiProject Mathematics||(Rated B-class, Mid-importance)|
- 1 Transcendental
- 2 Could the name be changed?
- 3 Irrationality
- 4 Practical use for this constant
- 5 Symbol for Eulers constant
- 6 Claim by anon contributor
- 7 Continued fraction representation in infobox
- 8 Rational??
- 9 Natural Logarithm?
- 10 Label for Integrals
- 11 Irrational?
- 12 incorrect formula?
- 13 Removal of infobox
- 14 Symbol γ, Mascheroni (1790), Bretschneider (1835) and the article of Glaisher (1872)
- 15 Unverifiable/unpublished claims
- 16 Solving for Gamma
- 17 Proof of irrationality?
- 18 Why gamma may be rational
- 19 On the inclusion of the hurwitz zeta function in the rational zeta expansion
- 20 ℇ (U+2107 EULER CONSTANT)
- 21 Gamma is approximately equal to 1/sqrt(3)
- 22 Two more Appearances
- 23 Existence
- 24 Euler-Mascheroni constant in probabilistic prime counting formula
Is there a name for the conjecture that γ is transcendental? --noösfractal 01:26, 7 August 2005 (UTC)
- No, it has no name. However, if you prove it is transcendental it will probably be named AFTER you, like Apery's constant. Whether its irrational is probably a more "answerable" question and is what most people are gunning for. Hope you enjoyed my edits on this awesome number! More is coming once I get my EDM.--Hypergeometric2F1[a,b,c,x] 14:22, 22 December 2005 (UTC)
Could the name be changed?
There's no reason to call this the Euler-Mascheroni constant instead of just Euler's constant. Euler defined it, proved that the limit in its definition exists, and then calculated it to 16 decimal places. Mascheroni eked out 3 more decimal places and gave it a new name; not exactly enough to have it named after him. Mathworld calls it the Euler-Mascheroni constant, but they are in the minority on this. Plenty of books and other sites call it simply Euler's constant, so there won't be any extra confusion by Wikipedia changing the name and then mentioning that some misguiding people mistakenly attribute the constant to Mascheroni. I don't know how to make changes to the title of an entry; maybe it's not possible without admin intervention. One option is to move the text over to Euler's constant and then redirect Euler-Mascheroni constant to there.
- But isn't Euler's constant more often used to refer to e rather than gamma ? Gandalf61 09:44, 2 March 2006 (UTC)
- Hmm, if the use denotation of e as Euler's constant is that common, then I withdraw my objection. Possibly a disambig page would still make sense? JoshuaZ 01:14, 3 March 2006 (UTC)
- It looks like there are three constants named for Euler on that page: e, γ, and Ca. But even if there are more, it seems to make more sense to call them all "Euler's number" and refer to them with different symbols, rather than misattribute one of them to someone who didn't do anything significant with it. The only reason I could see to keep it this way is overwhelming convention (like with Venn diagrams), but as I said, it appears to be mostly just Mathworld and Wikipedia doing this at the moment, and authors who got their information from Mathworld or Wikipedia. --Pexatus 06:30, 3 March 2006 (UTC)
- It's not just Mathworld and Wikipedia. Havil writes that "Its full accepted name is the Euler-Mascheroni constant" (p. 90), despite acknowledging that Mascheroni's primary contribution was to cause other mathematicians trouble with his erroneous calculation. I prefer "Euler's constant" myself, but I think the current title is more appropriate for Wikipedia, not least for disambiguation purposes. Fredrik Johansson 08:47, 5 April 2006 (UTC)
Sorry to reopen this after 3 years. However, Wikipedia policy on this is quite clear: the naming convention says that the title of an article should be that which "the greatest number of English speakers would most easily recognize". There is no doubt at all that the usual name for this number is "Euler's Constant". I do not remember ever having come across the use of that term for e. Google searches are not reliable sources of usage statistics, but for what it is worth one confirmed my own experience: "Euler's Constant" got substantially more hits than "Euler-Mascheroni constant", and checking every hit on the first few pages failed to reveal a single use of "Euler's constant" to refer to e, nor to anything else other than gamma.
It follows that, by Wikipedia policy, the title of the article should be "Euler's constant". Certainly it would be a good idea to put a link to the page on e, as people may sometimes confuse "Euler's constant" with "Euler's number". JamesBWatson (talk) 21:16, 16 July 2009 (UTC)
I read somewhere that it is not known wether or not γ is irrational is disputed. Would its irrationality not follow from the first limit definition at the beginning of the article? the sum of the reciprocals of any natural number of numbers will of course be rational, and following from the irrationality of e, the natural logatihm of any integer is irrational (except zero), so wouldn't their differenc be irrational? -- He Who Is[ Talk ] 21:56, 8 July 2006 (UTC)
- It would not follow. The sum of any finite number of rational terms would of course be rational; but here an infinite number of diminishing rational terms is being summed. Your question is a useful one, though. It hints at why most mathematicians would swear to the irrationality of γ, I think, even in the absence of a proof. Noetica 23:15, 8 July 2006 (UTC)
- In other words, each term in the sequence whose limit is gamma is indeed irrational, but the limit of a sequence of irrational terms can be rational e.g.
- Gandalf61 10:53, 10 July 2006 (UTC)
Thank you. I suppose I shouuld have thought that over more. I also apologise for the number of grammatical errors in my previous post. Looking at it now almost makes me cringe. -- He Who Is[ Talk ] 12:43, 10 July 2006 (UTC)
http://arxiv.org/ftp/math/papers/0310/0310404.pdf claims to be a proof that this constant is irrational. That was written in 2003. So what's wrong with this picture? Should this article be updated? -- not-so-anonymous, 2012-03-30 — Preceding unsigned comment added by 18.104.22.168 (talk) 03:34, 31 March 2012 (UTC)
Practical use for this constant
Reading this article through and through, one gets the feeling that Gamma is some sort of bizarre useless constant, with an artifical definition and connections to other mysterious and highly complex mathematical function.
Wouldn't it be nice to see in the article some practical use for this constant? Its practical use comes from its definition, being able to approximate the sum of 1/k by using the log function and Gamma.
Here is an example use, that my father showed me as a kid, taken from the book "Fifty Challenging Problems in Probability With Solutions". The question is: There are N different coupons in cereal boxes, and a set of one of each is required to get a prize. In each box there is one coupon. How many boxes on the avarage do you have to buy to win the prize?
A quick Google found a copy of the answer (I don't have the book here..) in . In short, the number of boxes you need to open is
N * (1/(N-1) + 1/(N-2) + ... + 1/2 + 1)
Can we, for large N, approximate this sum with some basic functions found in everyone's calculator? It turns out the answer is yes: for large N, it can be approximated by
= N * (ln(N) + γ)
Nyh 13:14, 18 March 2007 (UTC)
- Don't brush of Gamma so quickly. Do you not see the beauty in this number? Practical use? Please...have you ever read A Mathematician's Apology? The important things in this Universe have to practical use. Gamma's importance shows itself in the way it delightfully appears in all sorts of formulas, integrals, and so on. Just look at all those pretty forumlas. (BTW, I virtually created this page, and put all these formulas on here about 1.5 years ago).
- That was a neat problem though.--Hypergeometric2F1(a,b,c,x) 04:35, 29 April 2007 (UTC)
Symbol for Eulers constant
I'm doing some work on Unicode related articles and I want to include a linkk from the Unicode character Eulers constant (ℇ U+2107) to the appropriate artilce. However, upon seeing no mention of this notation in the article, I thought maybe I have the wrong constant. Any thoughts on this? Perhaps the article should mention this other notation (which in my fonts has no resemblence to gamma). Unicode includes Eulers constant as one of only three explicitly named constants with sa Unicode character devoted to it. Please let me know if there's another article that I should be looking for. Indexheavy 07:29, 1 May 2007 (UTC)
- Unicode 2107 looks like a curly capital E in my browser, so I suspect it is actually intended to denote Euler's number, not the Euler-Mascheroni constant. Not sure though why Unicode feel they need to introduce a special symbol for this - the standard notation is a lower case e. Gandalf61 10:10, 1 May 2007 (UTC)
Claim by anon contributor
however the most mathematicians in world as : sorin radulescu ,marius radulescu, ene horia , dorel homentcovschi,lazar dragos ,univ,bucharest with mathematics faculty , albu thoma , t.zevedei , irina olteanu , solved the main problems as euler and radon- nikodym ,together optimal maitenance policy , in others trep ,problems ;this can be verified —Preceding unsigned comment added by 22.214.171.124 (talk • contribs)
- Then please provide a reliable source for your claim, and also please take the time to write a clear and gramatical explanation, in English, of what you claim has been solved, how and by whom. Gandalf61 10:48, 2 June 2007 (UTC)
Continued fraction representation in infobox
I think the continued fraction representation in the infobox is more confusing then helpful. Only four places can fit in, and the way it's written makes it seems like there should be an obvious rule continuing this expansion, which of course there isn't. In the decimal/binary expansions, there are enough digits to make it obvious that the reader is not expected to be able to continue the sequence on his own. --Zvika 08:01, 5 September 2007 (UTC)
- I have gone ahead and added a disclaimer, as in Pi. I still think the continued fraction does not contribute much, but at least now it's not as misleading. --Zvika 05:06, 6 September 2007 (UTC)
Any post-2004 mathematician who thinks this number is likely rational?? Any numbers once conjectured to be irrational but now known to be rational?? Georgia guy 22:17, 6 October 2007 (UTC)
- see #Why gamma may be rational --126.96.36.199 (talk) 09:45, 6 June 2013 (UTC)
- 2nd question: maybe Legendre's constant? --188.8.131.52 (talk) 09:48, 6 June 2013 (UTC)
It says in the definition it is the difference between the harmonic series and the natural log, ln. Yet in the equation, it is represented as log(n). What's up with that? Nonagonal Spider (talk) 08:04, 24 November 2007 (UTC)
- The article follows the convention used by mathematicians, which is that log(x) means the natural logarithm unless another base is specifically mentioned. Gandalf61 (talk) 21:55, 24 November 2007 (UTC)
- I've changed ln to log in the Asymptotic expansions section, so that at least this article is now internally consistent (I don't think we should change the occurences in the references section, as these are in the actual titles of published articles). There doesn't seem to be a consistent convention across Wikipedia mathematics articles for representing the natural logarithm function - Euler's totient function uses log; harmonic number uses ln; and Riemann zeta function uses both ! Wikipedia:WikiProject Mathematics/Conventions does not set a standard. Maybe this wider issue should be raised at Wikipedia talk:WikiProject Mathematics. Gandalf61 (talk) 11:11, 9 January 2008 (UTC)
- "Mathematicians" do not have a convention on the issue. The meaning of log varies with the local custom (across universities, departments, research environments and so on). As a matter of clarity, one should use either ln or loge for the natural logarithm, and log10 for the base-ten logarithm. Clarity is the name of the game. 184.108.40.206 (talk) 20:11, 16 January 2008 (UTC)
- Our logarithm article says "Mathematicians generally understand both "ln(x)" and "log(x)" to mean loge(x) and write "log10(x)" when the base-10 logarithm of x is intended". For another reference, see this Math Forum answer. This also coincides with my own experience - a typical mathematician will assume that log(x) means loge(x), whereas a typical engineer will assume that it means log10(x). I agree with you that it would be clearer if the base were always stated explicitly, but the correct place to suggest this in Wikipedia is at Wikipedia talk:WikiProject Mathematics. Gandalf61 (talk) 12:00, 17 January 2008 (UTC)
Label for Integrals
In the section on properties, in the subsection on integrals, the article says "Indefinite integrals in which". The integrals are not indefinite. Perhaps the author means improper? —Preceding unsigned comment added by 220.127.116.11 (talk) 05:36, 17 December 2007 (UTC)
One of the formulas (referenced to Kraemer, 2005, which is now a dead link) appears to be false. See http://www.artofproblemsolving.com/Forum/viewtopic.php?t=159891 Suggest simply removing that paragraph? --18.104.22.168 (talk) 23:14, 27 July 2009 (UTC)
- OK, something's wrong, indeed. The first n terms of the sum give the value with error , so taking n = 2000000 I should get 6 decimal places of the sum correctly: it's ≈2.195811. OTOH, values of π and γ are readily available, γ + π2/6 = 2.22214973…. The other Vacca series is also wrong, and if fact seems to give the same value. I'm cutting the paragraph out, here it is if anyone feels like fixing it. Catalan's integral as well as the Vacca 1910 series appear in MathWorld (not that it would error free) and the series appears numerically to work, so I'll leave them alone. — Emil J. 10:41, 28 July 2009 (UTC)
Deleted stuff from the article
In 1926, Vacca found:
(see Krämer, 2005)
[The first] Vacca's series may be obtained by manipulation of Catalan's 1875 integral.
The correct series of Vacca is:
- Well, the verification of that series (but not its source) is available at http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1578117#1578117 . I think that the person who added that is likely Stefan Kraemer, and the source is again his 2005 thesis (or its extensions). Incidentally, the link to that work is no longer dead. --22.214.171.124 (talk) 12:51, 29 July 2009 (UTC)
- When I wrote "source", I meant a reliable source according to WP standards which could be put in the article. I'm afraid the Art of Problem Solving forum does not meet the criteria. I have convinced myself that the derivation is correct, but a formal source is still needed. Krämer's thesis would be good enough, but we need to check that it really states it. You seem to suggest that it is available online; could you post the exact link? I gather that his website is back online, but I couldn't find a copy of his thesis there. — Emil J. 13:16, 29 July 2009 (UTC)
The source was given with "Vacca 1926". Vacca wrote only one paper in 1926: JFM 52.0360.01. I'm sorry, I see now that a lot of mathematicians are not familar with the famous JFM, but you can quickly have a look at the online-version:
Please note: the online-version has a lot af typos!
My thesis and its extensions are not online (because the interest of the famous publisher Springer Germany was zero and he published a german translation of Havil's book). --Lagerfeuer (talk) 18:19, 29 July 2009 (UTC)
Removal of infobox
Based upon a discussion at Wikipedia talk:WikiProject Mathematics#"Infoboxes" on number articles, I've removed the infobox from the article. If anyone disagrees, could you please join the discussion there. Thanks, Paul August ☎ 14:50, 18 October 2009 (UTC)
- I have suggested centralizing this discussion to Wikipedia_talk:WikiProject_Mathematics#Irrational_numbers_infobox and Wikipedia_talk:WikiProject_Mathematics#Infobox_with_various_expansions as it refers to an infobox occurring in several articles. Please go there to build consensus on this edit. RobHar (talk) 19:35, 18 October 2009 (UTC)
Symbol γ, Mascheroni (1790), Bretschneider (1835) and the article of Glaisher (1872)
I found out in 1993, that there is a typo in the well known article of James Whitbread Lee Glaisher (1872). He wrote:
Euler's constant which throughout this note will be called γ after Mascheroni, De Morgan, &c. [...]
It is clearly convenient that the constant should generally be denoted by the same letter. Euler used C and O for it; Legendre, Lindman, &c., C; De Haan A; and Mascheroni, De Morgan, Boole, &c., have written it γ, which is clearly the most suitable, if it is to have a distinctive letter assigned to it. It has sometimes (as in Crelle, t.57, p.128) been quoted as Mascheroni's constant, but it is evident that Euler's labours have abundantly justified his claim to its being named after him.
— Glaisher: On the history of Euler's constant, 1872, p.25,30
Mascheroni has written only one paper about Euler's Constant (1790). He use only A. Comparing de:David Bierens de Haan using A, Glaisher should be corrected with "De Haan and Mascheroni A; De Morgan, Boole, &c., have written it γ". There are a lot of early papers about the Integrallogarithm. Around 1800 there was a famous discussion about the constant of the Logarithmic integral function (which is equal with γ). For example, the german mathematician Carl Anton Bretschneider (1808-1878) use in his 1837-paper (1835) the symbol γ.
It is indefensible that Havil unaudited the claim of Glaisher has taken over with
Mascheroni's permanent contribution to γ's story (apart from making a mistake that led to at least eight subsequent recalculations of the number) was to name it γ [...] By such serendipity, its full accepted name is the Euler-Mascheroni constant.
Glaisher, J.W.L. 1972 On the history of Euler's constant
— Julian Havil, 2003, p.90,256
An even more indefensible is that repeat these mistakes in the German translation (2007).
Article contained a table of decimal expansions of this constant, but several entries in the table did not meet Wikipedia's requirements for verifiable source citations, so I removed them. Basically, a few people were claiming to have calculated hundreds of millions or billions of digits and linking to their personal websites as sources. Please cite real, published references for these sorts of claims. -Mike —Preceding unsigned comment added by 126.96.36.199 (talk) 08:21, 8 February 2010 (UTC)
- True, they aren't fully verifiable sources. But don't take them out of the table. At the most, flag them as "unverified". By removing them from the table it implies that only 172,000 digits are known when in fact billions of digits are known.
- Also, the references (though not fully verifiable), come from at least 3 different and independent sources which all seem to agree with each other as far as the digits go.
- Here's are some alternate sources for some of those entries:
- http://numbers.computation.free.fr/Constants/Miscellaneous/Records.html (List of the current world records.)
- http://numbers.computation.free.fr/Constants/constants.html (This has records of when each record has been broken.)
- http://www.northbynorthwestern.com/2009/01/19235/nu-student-outwits-computing-experts-breaks-record/ (News article for the 14.9 billion digit computation.)
- http://www.ginac.de/~kreckel/news.html (This one mentions the 2 and 5 billion digit computations by Shigeru Kondo.)
- —Preceding unsigned comment added by 188.8.131.52 (talk) 18:15, 9 February 2010 (UTC)
- I'm not trying to take away your world record or something, but come on, this does not belong in an encyclopedia if the only sources that exist are a student newspaper article and two other websites linking to your personal homepages. Have a read through the Wikipedia verifiability standard http://en.wikipedia.org/wiki/Wikipedia:Verifiability. (Guinness Book might be a better vehicle for you?)
- Is there a way to flag this for some higher level / more experienced Wikipedia editor to mediate the dispute? If I'm wrong, fine, I'm wrong, but it seems silly to me for us to keep reversing each other's changes. At the same time, I believe you guys are abusing Wikipedia for your own self-promotional purposes, and that really hurts the Wikipedia project because it confirms the stereotype (some) people have that Wikipedia is amateurish and unreliable.
Someone pointed me to this page after noticing something peculiar. I believe I'm the one you should be talking to.
First of all, I am in complete agreement with you that some of the references don't meet the Wikipedia standard. However, the section is called "Known Digits". Although I agree that those entries can be removed, I believe that there should be at least a mention of the purported larger computations. Otherwise, the reader is left with the impression that fewer than a million digits are known - though I wouldn't go as far as to call it "misleading".
Secondly, I don't see why numbers.computation.free.fr/Constants/constants.html fails to meet standards. It is a well-respected and well-referenced 3rd party math site that keeps track of these records. Furthermore, it has no affiliation with the past 10 years of record computations.
If you're looking for a Guinness Book entry, or an IEEE publication of the recent records, that is unlikely to happen. These kind of records are far too obscure for anyone to be able to publish beyond a webpage. I understand that this is irrelevant to the topic of verifiability, but it is the reason why the recent computations have not been published any further.
And lastly, nobody here is using Wikipedia for promotional purposes of any kind. I understand your concern for Wikipedia, but to accuse multiple people of "self-promotional purposes" is completely uncalled for. Wikipedia isn't a place for unfounded accusations of any kind.
Also, your tone of writing suggests contempt towards the record holders. Is there a reason for this?
I agree, can we get a mod here? I'll accept whatever decision they come to.
The thought that strikes me, reading the above discussion, is "why does anyone care?" Frankly, whether anyone with time on their hands and nothing useful to do with it has calculated γ to thousands of millions of decimal places or not is of no significance to anyone, and if someone has done so then who that was is of no significance either. JamesBWatson (talk) 16:59, 12 February 2010 (UTC)
Solving for Gamma
Just to show how to solve for gamma in a cited article:
e^zeta=2.5188167*(1+2+3+4+...n) as n goes to infinity
where 2.51881667... = sqrt(2)*e^gamma
Doing an ln to both sides:
zeta = ln(2.5188167) + ln(1+2+3+4+...n)
zeta - ln(1+2+3+4+...n) = ln(2.51881667..)
Substitution sqrt(2)*e^gamma for 2.51881667
zeta - ln(1+2+3+4+...n) = ln(sqrt(2)*e^gamma))
zeta - ln(1+2+3+4+...n) = ln(sqrt(2)) + gamma
- There is no reference to a published reliable source in the OEIS link, and neither it nor your derivation seem to make much sense. There is no such thing as zeta(1), to begin with. The final result is wrong, it is easy to see that the limit is divergent: and , hence .—Emil J. 14:31, 6 April 2010 (UTC)
Emil is correct, it appears a square root is missing. The correct formula should read:
Proof of irrationality?
I came across this paper, which claims to prove that is irrational:
It doesn't seem to be referenced anywhere, is it bogus?
- It appears to have never been published in a peer-reviewed journal. ArXiv does not review correctness. I say, "bogus". Justin W Smith talk/stalk 22:20, 21 April 2010 (UTC)
Why gamma may be rational
On the inclusion of the hurwitz zeta function in the rational zeta expansion
I was wondering if there was any particular reason why the hurwitz zeta function is used here instead of either bernouilli numbers or the values of the zeta function at the negative integers? From what I can tell, Euler would have derived this from the Euler-maclaurin formula to begin with, which would have involved bernouilli numbers rather than the hurwitz zeta. I have no particular problems with the hurwitz zeta, I'm just wondering why it seems to have been given preference over what would be a more recognisable expansion. — Preceding unsigned comment added by 184.108.40.206 (talk) 21:13, 9 June 2011 (UTC)
After "fixing a double redirect" this mysterious character redirects here. But its appearance is not like γ. Any thoughts, which of Euler's constants it should signify? Incnis Mrsi (talk) 15:43, 9 January 2012 (UTC)
- Perhaps they were thinking of Euler's number? —Mark Dominus (talk) 17:02, 9 January 2012 (UTC)
Gamma is approximately equal to 1/sqrt(3)
- J.R., I do understand what you're trying to say... it's just that numbers don't necessarily "have to" agree to any number of digits in order for them to be `blood-related`. For instance, 0, 1 and e don't have any agreement with each other, yet the 2 in e comes from its first two terms: 0!-1 + 1!-1. Likewise, the .7 that follows doesn't have anything in common with 2, but 70% of its value it is obviously obtained by adding 2!-1 to the previous two terms.
- Beginning with a weak coincidence and looking for a cause for it (a "blood-relationship" as you call it) is not a line of inquiry that I would expect to be fruitful because the probability is that there is no such relationship. It is much better to simply work with the equations you have and try to derive more of them by transformations of the series, integrals, derivatives, or what-have-you. JRSpriggs (talk) 20:35, 18 February 2013 (UTC)
- So you're basically pretty certain that no other good can come of it that could provide us with deeper insight into Euler's constant, except a simple and helpful mnemonic or approximation ? No infinite sum, infinite product, nested fraction, or nested radical, etc. can ultimately be derived from it ? — Preceding unsigned comment added by 220.127.116.11 (talk) 21:56, 18 February 2013 (UTC)
No, I said no such thing. I know very little about the constant. I was just saying that your approach is one which I feel, based on my experience, is unlikely to work.
If you want to try to transform one of the known expressions for into an expression for and try to do something with that, be my guest. JRSpriggs (talk) 06:57, 19 February 2013 (UTC)
- Apparently it's explained by using the Gaussian quadrature for 2 points, and applying it to the integral definition of the constant. — 18.104.22.168 (talk) 03:59, 6 March 2013 (UTC)
Two more Appearances
I think it would be nice to include at least single proof of the existence and finiteness of the limit in the definition. — Preceding unsigned comment added by 22.214.171.124 (talk) 10:07, 20 December 2013 (UTC)
- I agree, this would be a good idea. One way I like to explain it to students is using an image similar to the first image of the article, showing that the blue area is finite by using upper and lower sum approximations to the Riemann integral of as a cute application of those ideas as well. I don't really have the time to write this up myself, but it would definitely make a worthy addition to the article. Sławomir Biały (talk) 16:20, 20 December 2013 (UTC)
Euler-Mascheroni constant in probabilistic prime counting formula
The Euler-Mascheroni constant appears also in the probabilistic prime counting formula which can be represented as follows:
where can be very closely approximated as:
The origin of this formula can be found here.