Talk:Basel problem
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[edit] Headline text
I'm thinking of renaming this article "The Basel Problem" and adding a bunch of historical remarks and putting in broader context. I still think the proof can stay here, it would just be a part (about half) of the whole article. I don't think it would make it too long, and people not interested in the proof could just skip that section. I thought of having 2 articles, but it seemed redundant and just giving the proof without any remarks about the problem seemed strange. Revolver 15:31, 26 Feb 2004 (UTC)
I manually renamed this article Basel problem, (since the system wouldn't let me do this directly) so this article should be deleted. Revolver 20:28, 26 Feb 2004 (UTC)
- Possibly...I find it hard to imagine someone linking to it, but it can't hurt. Revolver 02:41, 29 Feb 2004 (UTC)
That's a good idea...I forget simple things like, "why does this sum converge in the first place", assuming everyone has seen p-series in calculus, or something. There may be some other simple ways to approximate this that people used before Euler that might be interesting. Revolver 22:18, 3 Mar 2004 (UTC)
[edit] Overlooking the obvious?
It isn't clear to me from the article why this problem is called the Basel problem. Dataphile 19:49, Aug 22, 2004 (UTC)
- Basel is the name of a town, somehow the town is related to the origin of the problem. I don't know exactly -- in math, so many things have so many names, math people often don't know why something's called what it is. Revolver 17:51, 24 Aug 2004 (UTC)
The Bernoulli family, which worked on the problem for a long time, were located in Basel. Johann Bernoulli taught Euler at the University of Basel.
[edit] Picture as proof
"First note that 0 < sin x < x < tan x. This can be seen by considering the following picture:"
It's definitely a cool picture. Can its owner change the 'theta' to an x? Do we need to point out that the angle is equal to the length of the circular arc AD?
I can 'see' that sin x < x, but I can only convince myself that x < tan x using calculus (e.g., the derivative of (tan x - x) is tan2x, which is > 0 for x in (0, pi/2), ...). Is that a failure of imagination on my part? Perhaps it's obvious that tan x grows faster than x.
- The area of the triangle OAE is tan(θ)/2. The area of the circle sector OAD is θ/2. The circle sector is contained within the triangle, so θ < tan(θ). Fredrik Johansson - talk - contribs 18:18, 24 January 2006 (UTC)
Thanks. I've added this to the article. Buster79 23:08, 2 September 2006 (UTC)
[edit] You can prove it easily by using Fourier transform
I will not go into details, but the proof starts with the function absolute value of x on the interval [-pi,+pi] You calculate the Fourier transform of the function. Later you can seperate the sum into odd and even numbers and find out that the sum equals pi^2/6.
The Riemann zeta formula can be approached by Parseval's theorem.
The point of the proof in this article is that it needs only elementary methods. There are several proofs of this problem (there is an article called "Six ways to sum a series" which gives 6 of them), but most of them require more advanced machinery.
[edit] Basel problem extended
The Basel problem can be extended to find the closed forms for every N. An approximate sequence can be found in the OEIS, A111510. Included is an expression of Pi where the odd and even terms of Triangular(n)define the differences. Would the contributers to the Basel problem pages care to comment and to suggest the best way to include this? Marc M. 20-6-06
[edit] Irrelevant topic
The last topic in the article on the Basel problem reads as follows:
"Use in the calculation of π: In 1881, Ernesto Cesaro showed that the probability of two integers being relatively prime equals 6 / π2, which is the reciprocal of ζ(2). By the above proof, Cesaro's theorem thus allows a value for π to be calculated from a large collection of random integers, by determining the proportion of them which are relatively prime."
I don't see how this uses the result of the Basel problem, ζ(2) = π2 / 6, at all; it just mentions it in passing. Likewise, the phrase "By the above proof" is unnecessary, since the application of Cesaro's theorem goes along just fine without using the above proof at all. Moreover, it's very misleading to say that π can be calculated by this method. If you pick 100 pairs of random numbers from 1 to 10, and every possible pair shows up once, you will "calculate" π ≈ 3.086. (There are 63 pairs that are relatively prime.) No matter how many pairs are selected, the result will be an algebraic number, and π is transcendental. I suggest the entire paragraph be removed from the article. Did I miss anything? Gwil 20:03, 27 September 2006 (UTC)
[edit] Fundamental Theorem of Algebra unnecessary
I object to the following:
If p(t) is a polynomial of degree m, then p has no more than m distinct roots. Proof: This is a consequence of the fundamental theorem of algebra.
The statement "If p(t) has degree m, then p has no more than m distinct roots" can be proven by elementary methods, whereas the FTA is much more difficult (it's usually proven with complex analysis). Specifically, the "no more than m distinct roots" theorem follows from the factor theorem ( (x - r)|p(x) if and only if p(r) = 0, a consequence of polynomial division), together with an induction argument to show that the product of (x - rj) over all the roots rj of p(x) divides p(x), and an application of the fact that degree(f * g) = degree(f) + degree(g) for nonzero polynomials f and g.
Moreover, the "no more than m distinct roots" theorem holds in the polynomial ring of any integral domain, whereas the FTA is a very special fact about the complex numbers.
I would recommend that the line "Proof: This is a consequence of the fundamental theorem of algebra." be replaced by an outline of the actual proof from the factor theorem. --Gene496 08:35, 27 July 2007 (UTC)
While Euler is credited with the solution, i.e., an exact answer, is pi^2/6 an exact number? pi itself is the sum of an infinite series as is the original Basel series! SEIBasaurus 70.118.127.94 17:41, 7 August 2007 (UTC)
[edit] What you need to know
Just a comment on the "What you need to know" sub-section... love this, extremely helpful, should be present in more articles! Error792 03:20, 6 August 2007 (UTC)
[edit] Coprime?
Would it be worthwhile to add a wikilink to coprime, regarding the application of ζ(2) in evaluating the probability that two randomly chosen numbers are coprime? I just added a wikilink in the other direction, from that article to here. —David Eppstein (talk) 22:11, 15 December 2007 (UTC)
