Wikipedia:Reference desk/Archives/Mathematics/2009 May 25

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May 25

New Elementary Suggestive Results Found By Birthday Coincidences Related To History

I wish to request that an EXPERT write a short article covering my new results found by happenstance. I suggest the title "The Main Fact". The coincidences need not be considered as a part of the matter, aside from having provided original motivation for making a calculation. The calculation first made was that (365+1/4)^4=17797577732+7^2/2^8. A related simple equation is (365+1/4)^2=3^7*61+9/16. This second equation's finding was motivated by curiosity as to whether there might be a routine avenue to a finding primordially of the first, and I posit that a number-game question such as "How can one most prettily arrange the result of a calculation--perhaps one limited to mixed fractions raised to a power--involving the first six digits?" followed by something as simple as hitting an x-squared button while looking at the result could have done it. At any rate, the results are here and it is no great stretch to provide a kind of heuristic argument giving something like, but not quite, a probability that there is not a rational number as "special" as 365+1/4, independent of what we know it signifies as an approximation. Obviously, I am not equipped right now to write the article in a way that would hold up against an edit, if for no other reason than there is (not quite absolute) stricture against new results being presented by a non-authoritative source. —Preceding unsigned comment added by Julzes (talk • contribs) 00:30, 25 May 2009 (UTC)

Apart from whatever mathematical merits there may be in your findings, the writing of such an article would contravene our rules outlawing original research. Get it published in a reputable journal, then it might (stress "might") have a chance, but only if it's considered notable. -- JackofOz (talk) 00:36, 25 May 2009 (UTC)

Do you think it might simply appear added to such an article as Mathematical Coincidences then?Julzes (talk) 00:43, 25 May 2009 (UTC)

Dobro'e Utro, Ozzie.Julzes (talk) 00:48, 25 May 2009 (UTC)

So, if a public intellectual of the right bearing--perhaps John Allen Paulos--were to write it, it could not appear in wikipedia?Julzes (talk) 00:54, 25 May 2009 (UTC)

I am still having problems distinguishing easily verifiable facts aquired by happenstance from "Original Research". Why not just say new facts however easy to check must be in print before they are accepted, while hypotheses are judged on a case-by-case basis by the community?Julzes (talk) 01:06, 25 May 2009 (UTC)

Mostly because "the community" here is not equipped to judge the merits of new hypotheses, and anyway we shouldn't have to. Wikipedia collects information that has been previously published in reliable third-party sources. Please read Wikipedia:No original research and Wikipedia:Verifiability; these are two of the core policies of Wikipedia. If John Allen Paulos were to write about it outside Wikipedia, then perhaps it might also be included in Wikipedia, citing his book or article. However, if Mr. Paulos were to write about it for the first time in a Wikipedia article, it would fall afoul of Wikipedia:No original research, because it could not be attributed to a third-party source.

I should also say something about "easily verifiable facts." While there are many mathematical facts that are easily verifiable—for example, the fact that $(\pi \cdot e^{\pi })^{2}$ is within 0.1% of 5280, the number of feet in a mile—they too are required to be verifiable by reference to reliable third-party sources. In this case it is not necessarily the truth of the fact that is in question, but its notability. See also Wikipedia is not an indiscriminate collection of information. —Bkell (talk) 05:15, 25 May 2009 (UTC)

Reliable third-party sources would include a calculator, though, would it not?Julzes (talk) 07:35, 25 May 2009 (UTC)

Yes, to establish truth, but not to establish notability. That's what I was saying above. There are infinitely many true mathematical facts (for example, 831 + 519 = 1350), but very few of them, relatively speaking, are notable. Your calculator can verify that a fact is true, but it certainly can't establish that a fact is notable. —Bkell (talk) 09:32, 25 May 2009 (UTC)

I certainly understand this point. I invite you to establish your own sense of notability. Is where you are at school where Michael Larsen is now? I followed his high school success with awe.Julzes (talk) 21:42, 25 May 2009 (UTC)

My "own sense of notability" is irrelevant here. Wikipedia has developed guidelines for notability. Or are you asking about my personal notability? I freely acknowledge that I am not notable enough for a Wikipedia article (and consequently I don't have one). I don't know who Michael Larsen is. —Bkell (talk) 22:19, 25 May 2009 (UTC)

Irrelevant to whom? Wikipedia? True, I assume, to a large degree at least. Oh, never mind. I was just being a little too friendly. You are almost being offensive by seeming to take offense. I MADE A MISTAKE BY HAVING WIKIPEDIA AS A PRIORITY FOR WHEN I GOT BACK ON THE INTERNET, AND BY TRYING TO EDIT THERE BEFORE BECOMING THOROUGHLY FAMILIAR WITH ITS INTENTS AND PRACTICES. I hope that is good enough, but also please judge whether I have or have not improved the introduction to the Mathematical coincidences article. I want to stop you from warring with its editors via some sort of blanket erasure, but I won't undo whatever you do there. That is up to other people. You have given fair warning, but I think you are asking too much of what is already in print and that SOME new stuff SHOULD appear in wikipedia. On the other hand, the article in question really could use some more relevant attribution.Julzes (talk) 07:25, 26 May 2009 (UTC)

And isn't there some sort of common knowledge exception? Consider: 365+1/4 is the commonly used approximation to the average year-length in days. And I am claiming happenstance, not research, in any case. As for its real-world significance, why wouldn't my suggestion of an expert opinion be reasonable? Paulos would have to change his publicly stated views for example, to even consider writing the article.Julzes (talk) 07:49, 25 May 2009 (UTC)

Again, please read Wikipedia:Verifiability, which says in part:

The burden of evidence lies with the editor who adds or restores material. All quotations and any material challenged or likely to be challenged must be attributed to a reliable, published source…

You are the editor attempting to add material. This material has obviously been challenged. Therefore, you need to provide a reliable, previously published source. (Wikipedia does not count as a source.) This is a fundamental policy of Wikipedia and is not negotiable. I remind you again that the numerical correctness of what you have written is not in question—it is the notability that must be established. —Bkell (talk) 09:48, 25 May 2009 (UTC)

OK, OK. So, I understand that the significance, if any, of my happenstance calculations must be asserted by peer review before it can be a part of an encyclopedia article. I'm still wondering about whether you are treating certain TYPES of articles differently in this regard. There is plenty of original thought in wikipedia articles, so I think it's a little dishonest to posit a blanket claim. Also, I do note that allowance for exceptions is a part of policy.Julzes (talk) 21:42, 25 May 2009 (UTC)

I am stating Wikipedia policy. It is true that there are articles that violate this policy by including original thought, and these articles are a problem and need to be fixed. Wikipedia is certainly not perfect, and the fact that some articles contain original thought or research is not an authorization for more original thought to be inserted. It is also true that "Ignore all rules" is a part of Wikipedia policy, but you're going to have to have a damn good justification for ignoring fundamental policies such as Wikipedia:Verifiability and Wikipedia:Notability. The purpose of "Ignore all rules" is to facilitate the building of a free encyclopedia. Inserting unsourced claims does not further that end. —Bkell (talk) 22:13, 25 May 2009 (UTC)

Also, I made the suggestion on hypotheses being judged by the community because I have the experience of suggesting a new hypothesis in the Fermi Paradox article and its then simply being edited for style rather than discarded from the article as you imply it should have been.Julzes (talk) 08:02, 25 May 2009 (UTC)

WP:OTHERCRAPEXISTS is not a valid argument. Also, give it some time... --Stephan Schulz (talk) 08:44, 25 May 2009 (UTC)

Title suggestion changed to "The Main Fact?" to reflect that is one person's opinion that a new term should be introduced.Julzes (talk) —Preceding undated comment added 08:34, 25 May 2009 (UTC).

I note that you have already attempted to add your observation to the numerology article, where your additions were reverted by two editors. In the subsequent discussion at Talk:Numerology, the "no original research" policy has been explained to you several times. Opening a new discussion here, especially when you did not give the full background, could be interpreted as "forum shopping", a type of behaviour that is frowned on in Wikipedia. Gandalf61 (talk) 08:57, 25 May 2009 (UTC)

And also Talk:Mathematical coincidence#Simple Numerology and Talk:Existence of God#Do These New Facts Settle Question?. —Bkell (talk) 09:36, 25 May 2009 (UTC)

Well, please at least recognize my conundrum. I'm just back on the internet after some years of cloistering myself, and I have these strange new results that I don't want to horde.Julzes (talk)

We are in no way suggesting you hoard them. We are simply stating that Wikipedia is not the best location to put them. Wikipedia is not about new anything. As an encyclopedia, it's a tertiary source. It's not the place where things are first published. Wikipedia only summarizes things other people have written about in WP:Reliable Sources. We don't want to stop you from posting your results, we just want you to realize Wikipedia is the wrong place to do it, and you need to take your results to some other website/publication which welcomes original research. -- 128.104.112.37 (talk) 01:16, 26 May 2009 (UTC)

I hope nobody believes I have ignored policy. I am new, and I did find certain exceptions detailed in policy.Julzes (talk) 07:25, 26 May 2009 (UTC)

If the only thing written in books about the value of Pi agreed with the Indiana Pi Bill then wiki would not be able to put in any other value than 3.2 and it wouldn't matter how many series you summed or circumferences of circles you measured. Dmcq (talk) 16:07, 25 May 2009 (UTC)

Proving inequality

Hi, can you help me to solve this problem about proving inequality?

Assume a, b, c > 0, prove:

${\begin{cases}{\frac {a^{3}}{a^{2}+8bc}}+{\frac {b^{3}}{b^{2}+8ac}}+{\frac {c^{3}}{c^{2}+8ab}}\geq (a+b+c)\\{\frac {a}{\sqrt {a^{2}+8bc}}}+{\frac {b}{\sqrt {b^{2}+8ac}}}+{\frac {c}{\sqrt {c^{2}+8ab}}}\geq 1\end{cases}}$

Thanks. --221.7.145.140 (talk) 10:33, 25 May 2009 (UTC

Aren't they inequalities? And are you sure you wrote the right ones? IN the first one, the first second and third term lhs is less than the corresponding term in the rhs (so it's never true).--131.114.72.215 (talk) 11:04, 25 May 2009 (UTC)

For a=b=c=1, the first inequality is false. twma 11:36, 25 May 2009 (UTC)

These problems look pretty much like homework to me, assuming the sign for the first inequality is <= . We would be glad to help if you showed some effort in doing these problems. Wikipedia does not do your homework. Rkr1991 (talk) 11:39, 25 May 2009 (UTC)

There is a Unicode ≤ sign for 'less or equal', see the frame with 'Insert' droplist below Save/Preview/Changes push-buttons when in editing mode. --CiaPan (talk) 12:44, 25 May 2009 (UTC)

Sorry for the mistake. The first one should be ${\frac {a^{3}}{a^{2}+8bc}}+{\frac {b^{3}}{b^{2}+8ac}}+{\frac {c^{3}}{c^{2}+8ab}}\geq {\frac {a+b+c}{9}}$
One of my friends asked me to help him to ask this problem somewhere and I chose here. I cannot solve this problem either. His math is better than mine. :) --221.7.145.138 (talk) 14:14, 25 May 2009 (UTC)

I haven't tried it, so I don't know if it will actually get you anywhere, but the first thing I would try would be to multiply through by all the denominators to clear the fractions. You can then simplify it as much as possible and see what happens (I can see an (a+b+c) will appear on the LHS, which is hopeful). --Tango (talk) 19:22, 25 May 2009 (UTC)

For the first inequality: Assume without loss of generality that $1=c\leq b\leq a$, so $c=1$,$b=1+s$ and $a=1+s+t$ where $s\geq 0,t\geq 0$. Multiply out like Tango suggested, expand and move everything to LHS. You'll get a sum of only non-negative terms.

This may also work for the second inequality. -- Meni Rosenfeld (talk) 11:49, 26 May 2009 (UTC)

perfect squares

are perfect squares limited only to integers? i mean, is 16/9 a perfect square, becasue that's equal to (4/3)²???? —Preceding unsigned comment added by 117.197.241.17 (talk) 15:36, 25 May 2009 (UTC)

In normal usage, it refers only to integers (see Square number). It is not unthinkable, though, that someone would want to use the term for the square of any rational number, but he should explicitly state this meaning. -- Meni Rosenfeld (talk) 18:47, 25 May 2009 (UTC)

It's fairly common to use the phrase "perfect square" to refer to squares of rationals, but if one does so one should be clear about what one means by "perfect square", because "perfect square" without further explanation means integers. Eric. 131.215.159.91 (talk) 20:29, 25 May 2009 (UTC)

Indeed, you can say "perfect square in the ring R", or whatever. It's perfectly well defined. Without qualification, though, it always means in the integers (to be more precise, in the rational integers). --Tango (talk) 21:03, 25 May 2009 (UTC)

Are these really called "perfect squares"? Apart from integers, I've only seen them being denoted as "squares". What does the "perfect" in "perfect square" mean anyway? Are there any imperfect squares? — Emil J. 13:29, 26 May 2009 (UTC)

Yes, "square" is more common. I think, in this context, "perfect" means "complete", as in you don't need to "complete the square". --Tango (talk) 17:19, 27 May 2009 (UTC)

conformal bijective map

The article unit disc contains the following:

"There is however no conformal bijective map between the open unit disk and the plane. Considered as a Riemann surface, the open unit disk is therefore different from the complex plane."
How does one prove this? Can anyone give me a reference? Thanks. 128.232.228.74 (talk) 15:58, 25 May 2009 (UTC)

It follows from Liouville's theorem: a conformal bijection from C to the unit disk is a bounded holomorphic function, therefore it is constant, which contradicts its being 1-to-1. — Emil J. 16:13, 25 May 2009 (UTC)

Finding cube root

Hi, Is there any method to manually compute the cube root of a number, at least approximately? —Preceding unsigned comment added by 116.68.76.122 (talk) 16:37, 25 May 2009 (UTC)

See Cube root#Numerical methods. -- Meni Rosenfeld (talk) 18:13, 25 May 2009 (UTC)

If by "manually" the OP means by hand, pencil and paper, then yes, there is an algorithm, as these pages show. Pallida  Mors 21:10, 25 May 2009 (UTC)

Addition, multiplication and division - the operations required to carry out the iterative methods - can all be done by hand, pencil and paper. -- Meni Rosenfeld (talk) 11:07, 26 May 2009 (UTC)

Fair enough. It's not the first idea that comes to my mind when thinking of manually, but you are perfectly right. Well, now the OP has at least two methods by hand to choose from :). Pallida  Mors 14:30, 26 May 2009 (UTC)

You could use a printed logarithms table. If it has a table of cube roots (of precision enough for you), use that, otherwise if it has a table of cubes, use that, otherwise use the plain logarithms and antilogarithms table. You may also use a slide rule if you have one. For quick estimates, I usually use approximations based on the powers of 8/5 (aka R5):

n	1.0	1.4	2.5	4.0	6.3	10	14	25	40	63	100	140	250	400	630	1000
n1/3	1.0			1.4			2.5			4.0			6.3			10

– b_jonas 11:10, 26 May 2009 (UTC)

There's an easy algorithm for (integer) cube roots that you can do on an abacus. Staecker (talk) 11:57, 26 May 2009 (UTC)

The clever person could recreate the standard manual algorithm as an inversion of the binomial theorem applied to the power three (and have the wherewithal to do the same sort of thing for higher powers). Alternatively--not recommended--you could use Newton's generalized binomial theorem applied to the power 1/3.Julzes (talk) 14:51, 26 May 2009 (UTC)

There is a clunky pencil and paper method involving chopping the number into groups of 3 digits and doing a messy procedure resembling long division only worse. It is similar to another such method for square roots. These things are pretty much forgotten lore by now. Usually (especially if you have a calculator) you would just take a reasonable guess at the cube root, then improve it with a few passes of Newton's method, which converges very fast for this type of function. You are trying to solve the equation $x^{3}-n=0$, so the Newton iteration would be

$x_{k+1}=x_{k}-{(x_{k}^{3}-n) \over {3x_{k}^{2}}}={1 \over 3}{\left(2x_{k}-{n \over x_{k}^{2}}\right)}$

where $x_{0}$ is your initial guess. 67.122.209.126 (talk) 23:49, 31 May 2009 (UTC)