Talk:Mathematical constant/Archive 1

Computability / Definability

I think it would be nice to add one or two new rows for the computability / definability of each constant. What do you think? —Preceding unsigned comment added by 134.184.131.153 (talk) 10:40, 6 September 2007 (UTC)

Chaitin's constant?

Should Chaitin's "constant" be here? It's not actually a real number until you've chosen a computational machine, and I'm not aware of any canonical choices for that. For any such choice, on the other hand, we can make some statements about Ω. For example, for any machine for which the string "0" is a program that simply terminates, Ω > 0.5. We might even know the first digit.

Prumpf 14:37, 14 Aug 2004 (UTC)

Okay, I've removed it. I think readding a concrete entry for a particular machine's Ω would be great, but it should be a vaguely natural computational machine, and someone should actually do the math for those digits we can calculate.

Prumpf 13:27, 29 April 2006 (UTC)

Square roots of 2 and 3?

Is there any reason why ${\sqrt {2}}$ and ${\sqrt {3}}$ are listed as irrational rather than algebraic? Gkhan 16:45, Sep 7, 2004 (UTC)

Exactly what I was wondering. It may make more sense to use R = rational (none of the constants given are rational, I guess) A = algebraic (&irrational), T = transcendental --Andrew Kepert 07:05, 7 Oct 2004 (UTC)

I like this proposal ([R]ational, [A]lgebraic but not rational, [T]ranscendental). Also I would leave out the question marks. It's implicit that if we don't fill in the gap we just don't know. PizzaMargherita 21:23, 1 November 2005 (UTC)

1?

Is the number 1 not a mathematical constant? It is used to define the set of natural numbers. --Lambyuk 01:44, 13 May 2005 (UTC)

I'll second that. I think 0, 1 and i have a very interesting history behind them (which I didn't have time to write in my tentative entries), and deserve a place in the table for completeness. These numbers are not as obvious as you may think. PizzaMargherita 21:04, 20 November 2005 (UTC)

...so as above, zero, unity and the imaginary unit deserve a place in the table, on the basis that:

1. They are mathematical constants
2. They are not obvious at all—and I find that remark rather insulting to whom spent their lives studying them
3. They are part of Euler's identity. And I quote from the article: "the identity links five fundamental mathematical constants"
4. They are arguably more fundamental than many other constants in the list

PizzaMargherita 08:53, 26 February 2006 (UTC)

Hurrah, it was added today :) Lambyuk 12:47, 26 February 2006 (UTC)

Err, 1 and 0 are not constants. They are numbers. In software development, for example, 0 is a literal, but A=0 would mean A is a constant representing number 0. Secondly, if we have 0 and 1, why not 2, 3 and 4? How about 0x0a? How about 20? It is also very interesting...

Similarly, i is the same as 1 but for the complex plane. Just a unit, nothing special. —Preceding unsigned comment added by 216.55.199.86 (talk) 20:43, 14 February 2008 (UTC)

New table format: comments?

I've redone the table in (I hope) a better looking format. It is similar to format used on Table of mathematical symbols. Any comments? Paul August ☎

Some of the table rows need to be bigger. I would do it myself, but I don't want to mess it up. -Mihirgk

Golden ratio is irrational right?

Right? (Are all constants that are irrational-but-not-transcendent algebraic?)

All real numbers that are not transcendental are algebraic, because the definition of a transcendental number is a real number that is not algebraic. The golden ratio is irrational and algebraic, being the solution to the equation x² - x - 1 = 0 -GTBacchus 21:24, 1 November 2005 (UTC)

Mill's constant

Is Mill's constant symbolised by theta (as in the table) or phi (as in it's seperate article)? --Saboteur 01:11, 31 March 2006 (UTC)

I corrected the article--Saboteur 07:23, 31 March 2006 (UTC)

Hafner-Sarnak-McCurley constant

Accoring to this article,D(1)=6/pi^2,not the HSM constant. It uses sigma for the HSM. We should change the symbol.

I just changed the symbol.

Erdos-Borwein constant:algebraic?

Is the Erdos-Borwein constant really algebraic? You should make something called I. It will mean "known to be irrational,may be algebraic or transcendental." That would be a good extra symbol.

Landau's constant

Most precise does not equal most accurate. "Number of known digits" as used in this table means number of digits known to be correct, not number of digits that could be right. Fredrik Johansson 22:57, 6 September 2006 (UTC)

Fredrik, I understand your interpretation of "Number of known digits". Why don't we let the math community of WP decide? Either outcome will be fine with me. Giftlite 23:44, 6 September 2006 (UTC)

Cahen's constant

The article "http://en.wikipedia.org/wiki/Cahen%27s_constant" give the following value : 0.64341054629...

What is the true value ?

cf [1]

Papy77 15:41, 1 February 2007 (UTC)

Mathematical constint

Is Mathematical constint a good redirect? Constint 12:27, 27 February 2007 (UTC)

Red links and external links

There are so many red links in the table. We should create some pages and remove the external links. Math Maniac 11:46, 1 March 2007 (UTC)

I created Niven's constant. ^Over/_Under 13:29, 2 March 2007 (UTC)

Thanks. Math Maniac 12:07, 3 March 2007 (UTC)

MRB Constant

I would like to see if anyone can further expand upon my attempts of researching the following value. Sloan's A037077 ie. one of two real decimal expansions of 1^(1/1)-2^(1/2)+3^(1/3)... or a generalized sum to the divergent series. http://www.research.att.com/~njas/sequences/A037077[2] This constant remains a mystery. For instance, before 1998 what was the computation-history of the value that is presently called the MRB Constant? What is the closed form expression (assuming it exists) for the value of this constant? What relation does this constant of infinite dimensional “hypercubes," have with respect to "hyperspheres” of dimensions without bound? In what way might this infinite-dimensional constant be used in string theory? Most of my findings can be seen by following the links on Sloan's encyclopedia. From those links you will also come across a few references to more rigorous research done on that value's general form. If you endeavor to research this constant, I will try to help by answering any questions as to what I have already found in the past 9 years. last update--Marvin Ray Burns 19:54, 15 April 2007 (UTC)

One formula I didn't see in your documents (perhaps I didn't look hard enough) is the Euler transform of that series (with 1 subtracted from each term to force convergence):

$s=\sum _{n=1}^{\infty }2^{-n-1}\sum _{k=1}^{n+1}(-1)^{k}{n \choose {k-1}}k^{1/k}$

I also tried a few other sequence transformations, but didn't end up with anything fruitful. Fredrik Johansson 09:17, 22 April 2007 (UTC)

Fredrik Johansson, I have used your reference to the transform as an example of what is required in the article that I am commissioning.--75.2.16.2 01:32, 23 April 2007 (UTC)

A recent observation about and definition of this constant can be found at http://www.marvinrayburns.com/what_is_mrb.mht[3]. I hope that, through the input of interested and knowledgeable people, this constant will qualify for an article on Wikipedia. Marvin Ray Burns (talk) 01:45, 22 April 2009 (UTC)

Notice: Reward for Article about the MRB Constant

http://en.wikipedia.org/wiki/Wikipedia:Reward_board#MRB_Constant --Marvin Ray Burns 00:24, 17 April 2007 (UTC)

Mathematical constants

This page should be merged with Mathematical constants. Jaunt 16:50, 12 April 2007 (UTC)

Action by Marvin Ray Burns

It is obvious that this change needed to be made: http://en.wikipedia.org/w/index.php?title=Mathematical_constant&diff=122317608&oldid=122257451

The old symbols were larger than the row height of the table. —--Marvin Ray Burns 21:46, 12 April 2007 (UTC) Marburns (talk • contribs) 21:33, 12 April 2007 (UTC).

Approximate Value

${\sqrt {-1}}$ is exact and not an approximate value. $1.{\sqrt {-1}}$ is approximate. Better is the Cartesian representation0.0+1.i. However, the last approximation uses i to define i. I believe it is best to put " ${\sqrt {-1}}$exactly ." I'm going to post it and see if anyone has a better -- more accurate—way of displaying it.--Marvin Ray Burns 01:55, 15 April 2007 (UTC)

definition

Is the addition of complex number in the definition really appropriate? The only 'complex constant' I've head of is $i$, because it's a 'unit'. Randomblue (talk) 15:43, 11 December 2007 (UTC)

2 questioned constants and some suggestions...

Hi fellow wikipedians! I've been checking all the constants in the list and verified the existence of each red one (except two) via Wolfram MathWorld (entries in that list, or found by searching that site).

There are however two constants which I fail to verify via internet searching: Hughes constant ("Sh") and Jacevicius constant ("J²").

Does anyone else know of these constants and can verify their existence, i.e. justify that they are present in the list? Otherwise, they might be subject for deletion...

Furthermore I find that the following uncategorized constants could be placed in the following fields of mathematics (as far as I understand from checking the articles and/or Wolfram MathWorld);

• Cahen's constant : Number theory (derived from Sylvester's sequence in Number theory)
• Laplace limit: Calculus (isn't it something like Geometry->Celestial mechanics->Calculus, i.e. Calculus?)
• Apéry's constant: Number theory (categorized as (analytic) number theory)
• Backhouse's constant: Number theory (it's about prime numbers)

Any opinions on this? --Dna-Dennis (talk) 15:48, 3 February 2008 (UTC)

'Hughes constant' appears to be an emperical constant associated with leave thickness of plants and trees. Seems a rapid delete from list of math constants. -- JocK (talk) 18:41, 5 February 2008 (UTC)

No opinion on the suggestions. But I do have a request. Could you add citations for the sources you found to the table? That would be very helpful. If you don't want to add them to the article, adding them here on the talk page would also be helpful. Thanks in advance. Paul August ☎ 18:55, 5 February 2008 (UTC)

Paul, good idea! I've added them as refs to the article (which could help future editors to write entry articles on them). I will repeat them here, and also give links to the entries in OEIS:

• Backhouse's constant - MathWorld - OEIS

• Bernstein's Constant - MathWorld - OEIS

• Alladi-Grinstead Constant - MathWorld - OEIS

• Lengyel's Constant - MathWorld - OEIS

• Porter's Constant - MathWorld - OEIS

• Lieb's Square Ice Constant - MathWorld - OEIS

I will also see if I can optimize the list format in the article - it's a little tough to edit it as it is.

JocK, don't worry, I won't delete any constants on my own without consulting you other wikipedians. The problem is that I can't find neither Hughes constant nor Jacevicius constant in neither Wolfram MathWorld nor OEIS (The On-Line Encyclopedia of Integer Sequences), and I don't find any useful references either by googling. I don't necessarily question their existence, but, the following is important: (1) Are they noteworthy enough to credit an entry in this list? (There are many more other constants which are not currently present in the list). (2) We need to make sure that they have a pure mathematical origin, and are not Physical constants, i.e. measured. JocK, do you have any good links to Hughes constant? That would be most helpful. My regards, --Dna-Dennis (talk) 21:38, 5 February 2008 (UTC)

Here's a link to an article about the Hughes Constant: http://aob.oxfordjournals.org/cgi/content/full/89/5/537

It varies from species to species. I'm almost positive the entry here is immature vandalism. It and the Jacevicius constant were added at the same time by a non-registered user. Also, the value for the Hughes constant looks very similar to a date, possibly the birthday of whoever added it? The Jacevicius constant is a value squared. I've never seen a constant represented that way. Why not show the true value by just squaring it? I'd say given the evidence (or lack thereof) these are good to delete. sam (talk) 05:35, 16 February 2008 (UTC)

Why are they all pretty small?

Why are all these constants small? Now I know "small" isn't really well defined, but why do constants tend to be close to 0 or 1. Not the most well defined question, but you know what I mean. Has there ever been any discussion in the literature about this? Brentt (talk) 20:06, 6 May 2008 (UTC)

New section added

I just added a section about the decimal place similarity of irrational constants. More on this coming soon!

Quantum Anomaly (talk) 04:48, 1 July 2008 (UTC)

I really doubt that these "patterns" are statistically significant. There are more than 40 constants listed in this article. If you pick a random digit (0-9) 40 times, there is a 0.5% chance that you will get 10 or more 9s. Doesn't sound very high, but if you take 40 random numbers (between 0 and 1), the probability that 10 or more will contain the digit 9 at the same decimal place somewhere in their first 20 decimal places is 1-(1-0.005)²⁰ or about 10%. And if you don't specify in advance that it is the digit 9 you are looking for the probability of finding a "pattern" is even higher. Try a simulation - with 40 random numbers I think there is a greater than 50% chance of finding the same digit (not necessarily 9) at the same decimal place somwehere in the first 20 places in 10 or more of them.

This is an example of the Texas sharpshooter fallacy - if you look hard enough at a large enough volume of data, you will find some patterns. But this is meaningless unless you form an ex-ante hypothesis before looking at the data.

Even though these "patterns" may be just conincidence, they could still be mentioned in the article if you could cite a reliable third-party source where they are discussed. But I am afraid that at the moment your footnote "Observation first noted by Seth Lewis" does not qualify as a reliable source. Unless you can find a source, I think this section should be removed. Gandalf61 (talk) 12:49, 1 July 2008 (UTC)

I see you have removed the silver ratio because it is 1+sqrt(2), so obviously the decimal expansions of these two constants are related. Note that the golden ratio, phi, is (1+sqrt(5))/2, so anywhere there is a 9 in the decimal expansion of sqrt(5), the corresponding digit in the decimal expansion of phi can only be a 4 or a 9 - so the decimal expansions of these two constants are also related.

Unless you can quite soon provide an independent reference to show these coincidences in decimal expansions have been noted and discussed elsewhere, I am seriously considering removing this section on the grounds that it is apparently original research. Gandalf61 (talk) 08:28, 2 July 2008 (UTC)

Okay, now 3 days since my first post on this, and no response from anyone, so I am removing the new section on the grounds that it is unsourced, and so likely OR. Removed text is shown below. If someone can provide a reliable source that discusses these "patterns" then I have no objection to re-instating the section. Gandalf61 (talk) 08:32, 4 July 2008 (UTC)

I agree with the removal. Your analysis is correct, and any inclusion of examples of interesting patterns would need to be supported with reliable sources indicating why they are significant. Given that these decimal expansions go on forever, it seems inevitable that patterns like the ones removed will occur. --Johnuniq (talk) 01:01, 5 July 2008 (UTC)

Well it seems there is a consensus that the section doesn't belong. That's fine, but I think some have misunderstood my intentions in posting it. I never claimed to understand the reason for the frequent occurrence of nines. I made no statements that would require a "reliable third-party source". The nines are there and they speak for themselves. The purpose of the post was to draw attention to this trend so that the community could see this and share ideas, explore further, etc. If there is a better place for this than Wikipedia then I am open to suggestions, but I don't feel it should be written off as insignificant so quickly. When the decimals of these irrational constants are displayed in a line graph the numbers appear not to be random. Just because nobody has identified any specific patterns yet doesn't mean that the possibility should not be investigated. As far as I'm concerned it is an open question, so where does such a question belong if not here?

Quantum Anomaly (talk) 21:33, 8 July 2008 (UTC)

I think the issue is that Wikipedia is supposed to be a repository of verifiable information, not a forum or source of original research. Sure, you demonstrated an interesting pattern, but one can find lots of patterns that are essentially just outcomes of one particular roll of the dice. So, this section would need a reliable source showing (or at least claiming) that the particular patterns really are significant. I'm sure Google can find newsgroups and mailing lists discussing mathematical issues, and that is where these patterns could be pursued. --Johnuniq (talk) 03:22, 9 July 2008 (UTC)

---

Removed text

It is interesting to note the frequent occurrence of irrational constants having the number nine in the 12th and/or 14th decimal places. This pattern confirms a relationship between irrational constants and hints at the possibility of underlying self-similarity of a larger scale.

A working list of irrational constants with nine in their 12th decimal place:

Pi: Phi: Apéry's Constant: The Square Root of 5: Copeland-Erdös Constant: Landau-Ramanujan Constant: Sierpiński's Constant: e: Omega Constant:

03.141592653589793238462643383279 01.618033988749894848204586834365 01.202056903159594285399738161511 02.236067977499789696409173668731 00.235711131719232931374143475359 00.764223653589220662990698731250 02.584981759579253217065893587383 02.718281828459045235360287471352 00.567143290409783872999968662210

A working list of irrational constants with nine in their 14th decimal place:

Pi: Phi: The Square Root of 2: Erdős-Borwein Constant: Apéry's Constant: Feigenbaum Delta Constant: Feigenbaum Alpha Constant: 1st 0 of Riemann Zeta Function: Kepler-Bouwkamp Constant:

03.141592653589793238462643383279 01.618033988749894848204586834365 01.414213562373095048801688724209 01.606695152415291763783301523190 01.202056903159594285399738161511 04.669201609102990671853203820466 02.502907875095892822283902873218 14.134725141734693790457251983562 00.114942044853296200701040157469

• In the numbers above, the last decimal is truncated, not rounded.

• 14.134... is known as "the imaginary part of the first nontrivial zero of the Riemann zeta function". It has been abbreviated in the table above.

The significance of nine in an irrational constant is also a theme of the Feynman point.

Section "Equations" (removed)

A new section titled "Equations" was added below "Calculation". I was going to remove the "Equations" and add the following to the bottom of "Calculation". However, the new section was reverted, a decision that I support (but failed to do myself due to lack of boldness). The expressions are quite cute, so here they are:

This result can be rearranged

$\varphi ={\frac {1+{\sqrt {5}}}{2}}=.5+{\sqrt {5}}*.5={\sqrt {\frac {5+{\sqrt {5}}}{5-{\sqrt {5}}}}}$

--Johnuniq (talk) 04:13, 7 July 2008 (UTC)

Reverting "Anton's constant"

I have twice four times now reverted an anonymous contributor who has added "Anton's constant" (value=6) to the table of mathematical constants. No source, no Google hits, no article, wikilinks are to 6 (number) and rational number, so I am assuming this is complete nonsense. Gandalf61 (talk) 10:51, 15 October 2008 (UTC)

Golden ratio

The table in this article says that the golden ratio is known to 3.14x10^9 places, while 5^0.5 is known to only 10^6 places. But the golden ratio is equal to (1+5^0.5)/2 so they must both be known to the same number of places. Which is right? Ehrenkater (talk) 21:11, 22 November 2008 (UTC)

Still start class?

Looks pretty good to me... a big fat bold article in the middle of the mathematics vital articles is annoying. Leon math (talk) 22:12, 4 January 2009 (UTC)

Decimal expansions

Shouldn't we include the approximate decimal values of the constants? I'm surprised no one's said anything about that. I hope I didn't miss anyone's comment... Leon math (talk) 22:15, 4 January 2009 (UTC)

Decimal expansions of constants and, where appropriate, number of known decimal digits are given in a big table near the end of the article. Gandalf61 (talk) 09:38, 5 January 2009 (UTC)