Talk:Euler-Mascheroni constant

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Contents 1 Transcendental 2 Could the name be changed? 3 Irrationality 4 Practical use for this constant

[edit] Transcendental

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)

[edit] 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)

Some people do this, however, it is incorrect. Having a clear cut note to that effect and a link to e would probably make sense. JoshuaZ 14:02, 2 March 2006 (UTC)

Agree with Gandalf61, disagree with JoshuaZ. See List of topics named after Leonhard Euler; these are not the only two constants named after Euler. linas 00:47, 3 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 C_a. 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)

Practically speaking though, I think most people call this one Euler's Constant and e Euler's Number. Holomorph 12:50, 18 May 2006 (UTC)

Whatever the "official" terms are, some users looking for this Euler-Mascheroni constant would have trouble finding it in Wikipedia. For this reason I have just put a link to this article at the e article, and I hope it stays there. Noetica 23:27, 17 June 2006 (UTC)

[edit] Irrationality

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)

[edit] 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 [1]. 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)