Talk:Pascal's triangle

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Cronholm144 06:54, 21 May 2007 (UTC)

[edit] Code

Hello, I've just replaced the program with a much shorter, and, I hope, more comprehensible program. The goal here is to help people understand Pascal's triangle. I don't think the gyrations needed to get the numbers to line up were aiding that understanding. I also don't think the standard Java cruft (class name, imports, try/catch) was helping either. Anyway I'm sure that the floodgates are open now, and we're going to have 200 versions of this program in different languages... which is OK by me, as long as everybody is striving towards comprehension of Pascal's triangle. I claim the existing program is an improvement over the previous one. I don't claim it's the best possible. Happy editing, Wile E. Heresiarch 05:57, 8 Jun 2004 (UTC)

As I mentioned on Wikipedia: Wikicode/Specification, I really don't mind you reverting to the Python code; in fact, I probably shouldn't have converted it in the first place, since it's against my policy of leaving real-language code alone. Sorry about that. Derrick Coetzee 17:57, 10 Oct 2004 (UTC)

More about the code. I see the C code has been restored. I don't think this is an improvement. Why don't we just cut out the code altogther (in any language). Having an algorithm to print out some rows of the triangle doesn't have much to do with Pascal's triangle. I originally put in the Python because it replaced a Java program that was about 10 times as long; but in any event having a program is optional, so let's just cut it and avoid the language wars. Wile E. Heresiarch 06:47, 6 Mar 2005 (UTC)

So algorithms (though tied to a specific language) are not encyclopedia worthy? Cburnett 08:36, 6 Mar 2005 (UTC)

I agree with Wile that the code does not add any information on Pascal's triangle. The algorithm, based on Pascal's identity is already explained in English in the lead section, and it is straightforward to translate it in a specific programming language. -- Jitse Niesen 15:23, 6 Mar 2005 (UTC)

I've cut the section with the computer code. As I said in the edit summary, "source code doesn't shed any light on Pascal's triangle, and it's a language war magnet". For what it's worth, Wile E. Heresiarch 15:14, 7 Mar 2005 (UTC)

[edit] Two Questions

Are the schemes for row numbering American and Candian??
Has anyone proposed a new scheme that has names with no ambiguity? 66.32.251.248 23:12, 17 Oct 2004 (UTC)
The History of Pascal's triangle as Pingala's work is highly dubious, the source cited is http://www.anaphoria.com/, i dont think thats a trustworthy source ?

1, 2. "Row n" contains the values of C(n,k) 3. It looks as though the commentary is being confused with the work itself so I've altered it. By the way, you are allowed to change the heading above! Xanthoxyl 19:53, 19 January 2007 (UTC)

i thought you only had 2 questions??? nvm joeldudesx20:14,19 February 2008(UTC) sry its 15 feb Joeldudesx (talk) 12:21, 15 February 2008 (UTC) joeldudes20:21 15 February 2008

[edit] Question about a listed property

The page lists the following claim:

"the sum of the squares of the elements of the n^th row equals the middle element of the 2n^th."

But it appears to me that the claim should be "the sum of the squares of the elements of the n^th row equals the middle element of the 2n^th-1."

Am I wrong? This assumes that the row numbering starts at one (which is consistent with other parts of the text). In particular, only the odd numbered rows have a "middle element", and odd numbered rows must be of the form 2n-1, not 2n.

I believe you're right — this is an off-by-one error. Deco 08:37, 24 Mar 2005 (UTC)

Some further explanations prompted by the edits of Chad.nezar: The article assumes that the row with just one 1 is row number one. It then follows that row number n has the binomial coefficients

${n-1 \choose 0}, {n-1 \choose 1}, {n-1 \choose 2}, \ldots, {n-1 \choose n-1}.$

Hence, the text "the sum of the squares of the elements of the n^th row equals the middle element of the (2n - 1)^th" means

$\sum_{k=0}^{n-1} {n-1 \choose k}^2 = {2n-2 \choose n-1}.$

Substituting

m = n - 1

yields

$\sum_{k=0}^{m} {m \choose k}^2 = {2m \choose m},$

which is the formula given in the text. -- Jitse Niesen 10:57, 6 Apr 2005 (UTC)

Jitse, thanks for that example, which helps clarify some things for me. However, it also points out the other inconsistencies of this article. The crux of the problem is that the text uses the letter n to indicate rows as the n^th row, where the row numbering starts at 1. However, clearly the math formulas and notation require that n start at 0. In the example you gave above, you have to define

m = n - 1

for just this purpose.

This truth is demonstrated by the example in this section, showing that squares of the terms of the 5th row (where row numbering starts from 1) add up to 70. However, the math formula demonstrating this sums over k which starts at zero. Thus, if n is equal to 5 (as the text would indicate), that sum has a total of 6 terms, not 5. Clearly it is written assuming that the n^th row numbering starts with the 0^th row.

In the first section, there is also some confusion in that the page states "for positive integers n and k where n ≥ k", however the examples require k to start at 0, and implicitly that n start at zero (otherwise, the case of ${n \choose n}$ is irrelevant). I therefore think the wording should be "for non-negative integers n and k where n ≥ k", with all the math formulas based on n starting at zero, and the text should be updated to not use the phrase "n^th row", but something more appropriate. Either the text should be changed to indicate that we consider the first row to be the 0^th row, or use another variable (like m) to indicate the row number and give it's relationship to n. Or just change n^th row in the text, to

(n + 1)

^th row, where needed. Comments? Chad.netzer 22:07, 8 Apr 2005 (UTC)

I think your (Chad.netzer's) edits make the article clearer, so thanks for that. -- Jitse Niesen 11:51, 11 Apr 2005 (UTC)

[edit] Khayyam-Pascal's triangle?

Any evidence that this is the "internationally recognized name" as stated at the beginning of the article? I grew up in Soviet Union, where it was the good old Pascal triangle, and the same in Europe and US. Oleg Alexandrov 21:37, 5 October 2005 (UTC)

Its always been Pascal's triangle in my part of the world. Paul August ☎ 23:19, 5 October 2005 (UTC)

Together with Charles at Wikipedia talk:WikiProject Mathematics#.5B.5BKhayyam-Pascal.27s_triangle.5D.5D, that makes three well-respected mathematicians, to which I add my own voice. Regardless of which name is more historically correct, we should use the name that is actually used. So I renamed it back to Pascal's triangle. I also put the history section in chronological order again, without removing the new contents. -- Jitse Niesen (talk) 23:26, 5 October 2005 (UTC)

In italian schools it is usually nameded Tartaglia's triangle.

The following is copied from my talk page:--Niels Ø 16:23, 9 October 2006 (UTC) Dear Noe I'm sending you this message to prevent Edit war. I added "or Kayyam-Pascal triangle " because by seeing this page at the first time I fell in doubt if it was that. I'm from Iran and we always call this triangle "Khayyam triangle" or "Khayyam-Pascal triangle". As you can see this in Persian(Farsi) Wikipedia[1] and tajik one [2]. And the other revalant languages(Such as Urdu, Pashto,Azeri,Kurdi,Uzbek,...) don't have this article yet. You also can watch Keeper Movie (The legend of Omar Khayyam)[3][4][5]. In addition you can have a look at its invention history. Khayyam had invented it sevral centuries before Pascal and applied it for the other applications. Pascal just showed new applications of this triangle in the absence of the knowing about khayyam invension. I think "or Kayyam-Pascal triangle " is strongly needed for removing ~~dis~~ambiguation for hundred millions people. ~~--81.31.160.22 07:29, 9 October 2006 (UTC)~~--Soroush ^{☺talk | ☼Contributions} 07:32, 9 October 2006 (UTC)

My reply to User:Soroush_Mesry: I still disagree. This is an encyclopedia in English. The name of the triangle in English is Pascal's triangle. I suggest you create the relevant article (as a stub, for a start) in wikipedias in other relevant languages, and use interwiki links to connect things.

As for the history, I do not know whether Omar Khayyam invented the triangle. I don't think Pascal did. What we know for sure is that others invented it before any of them was born. So both names are misnomers, or at least give credit to the wrong people, but it's not our job to fix that - I mean, the name. What we can and should fix is giving credit to the right people when describing the history of the triangle, including the roles played by Khayyam, Pascal and others.

If names used in other languages must be listed at the top, the one used where you live is not the only one to be added.

PS. I am a bit of a Khayyam fan, but that does not affect the above.--Niels Ø 16:23, 9 October 2006 (UTC)

As can be gathered from my revert, I fully agree with Niels. -- Jitse Niesen (talk) 03:59, 10 October 2006 (UTC)

Agree too. Oleg Alexandrov (talk) 15:36, 10 October 2006 (UTC)

(See also the section #Recent changes to the lead further down on this talk page.)--Noe (talk) 13:26, 17 April 2008 (UTC)

[edit] Modifyed triangle "vandalism"

Recently, there was an edit to the modifyed triangle, by myself, that was referred to as vandalism and reverted as such. It is actually a mathematical correction. I invite anyone to check the math and find out for sure, and I will refrain from redoing the edit until someone can back me up, as I do not want to anger either the mod who made the edit nor the community as a whole. Aristotle2600 20:37, 22 October 2005 (UTC)

I reverted it. Thanks for bringing it up on the talk page, I should have done that myself. The change was first made several days ago here [6] by an anonymous user with no edit summary. To be perfectly honest, at least half if not more of anonymous edits are pure vandalism. So I checked the history of the page and the "84" value had been there for a long time (since the original version IIRC). On the balance of probabilities I therefore made the deciscion to revert the page on the likelihood that it was simple vandalism. Your change was the same as the anon's, only a day later, and I therefore decided to revert that also. Please please do use edit summaries if you make changes.

Understood. I'm still kinda new to Wikipedia, learning the finer details of etiquette and stuff; will try to remember edit summaries. Aristotle2600 20:22, 23 October 2005 (UTC)

If you just change a value with no justification in the edit summary it does look a bit suspicious. I am not mathematically inclined enough to calculate the value myself, so if you are happen to engage in dialog and are confident that it's correct value that's fine by me. I hope that explains things. chowells 21:36, 22 October 2005 (UTC)

I checked it and it indeed has to be 84. The rule in that triangle is that every value is twice the number to the upper left plus the number to the upper right, which makes 2 * 12 + 60 = 84. Thanks to Aristotle2600 for finding the mistake and to Chowells for doing RC patrol (though I wish you were a bit more into maths as a CompSci student ;) ). -- Jitse Niesen (talk) 00:31, 23 October 2005 (UTC)

Fine, thanks a lot for working it out. Yes I could have worked it out myself. I'm just lazy :) chowells 00:35, 23 October 2005 (UTC)

[edit] Exponents

In the second equation of the 'Uses of Pascal's triangle' section the exponents are partially not in superscript. But this has been so from the very beginning. I'm no mathematician, but has this been overlooked all the time or am I missing something? DirkvdM 10:19, 11 November 2005 (UTC)

The equation look fine to me. The numbers immediately after the a should be subscripts (they indicate that a₀ and a₁ are different and unrelated variables, see the first paragraph of Index (mathematics)) and the numbers after the x and y should be superscripts, and that is how they appear to me. Perhaps a browser problem? -- Jitse Niesen (talk) 12:34, 11 November 2005 (UTC)

You're right, it's a browser problem. I used Konqueror, but in Mozilla it works ok. However, the font isn't too clear because they're in italic. Removing them also solves the problem with Konqueror. The italics don't seem necessary, so I've removed them. Correct me if I'm wrong. DirkvdM 11:32, 12 November 2005 (UTC)

It is customary to write variables in italics; look at any maths book and at Wikipedia:Manual of Style (mathematics). I never heard of Konqueror having these problems. I tried it out, using Konqueror 3.4.2 as included in Debian testing, but it renders the page correctly in my computer. Sorry, but it seems I can't help you. Groetjes, Jitse Niesen (talk) 18:12, 12 November 2005 (UTC)

I use Konqueror 3.2.1, so I suppose I need to update. It's quite weird. Number-exponents render just fine, as does (x+1)^{row number}, but with (x+1)ⁿ⁺¹ the 'n' isn't superscripted, but the '+1' is and with (x+1)ⁿ⁺¹ they're both not superscripted, so it's purely a matter of italics. DirkvdM 08:12, 13 November 2005 (UTC)

[edit] question

Where does (n+1) come from in the following line?

Note that the first row therefore corresponds to the binomial ${0 \choose 0}$ , and can also be referred to as row $(n + 1)$ .

That's a good question. The answer is that I don't know. It has been there since 8 April. I removed the whole sentence. -- Jitse Niesen (talk) 12:46, 8 December 2005 (UTC)

In school, I haven't reached all of the complicated formulas. So I find it very confusing reading this article. There should really be some more explanation as to what the formulas mean. I'm just doing a patterning project and Pascal's triangle came up. I can't use any of the formulas, so there should be some morre explanation in words. I would have done this myself, but I don't know what any of this means. If someone could provide an explanation it would really help.

Yo, today 23:31, 5 February 2007 (UTC)

[edit] Added the matrix-exponential gimmick

I just added the paragraph concerning the matrix-exponential. Since I'm not reading wikipedia daily, please mail a copy of any questions/critizism to me, so I could answer in time. Gottfried Helms --Gotti 12:03, 26 December 2005 (UTC)

What about moving that section to the Pascal matrix entry? The help page Wikipedia:Merging and moving pages does not seem to mention how to do this. Haseldon 18:22, 9 November 2006 (UTC)

[edit] Properties of the triangle

Has anyone noticed this property?

2^0= 1 = 1

2^1= 1 + 1 = 2

2^2= 1 + 2 + 1 = 4

2^3= 1 + 3 + 3 + 1 = 8

2^4= 1 + 4 + 6 + 4 + 1 = 16

2^5= 1 + 5 + 10+10 + 5 + 1 = 32

-anon

That's in the article, at the bottom of the section on properties. Oleg Alexandrov (talk) 00:36, 29 December 2005 (UTC)

[edit] patterns

i have noted and seen many patterns in pascal's triangle, but have not seen them documented in this article. i am at present not capable of cataloging or posting all or any of these, but i would appreciate it if someone could do that.

also, in searching the triangle, i noticed some particular parterns regarding perfect numbers, and wondered whether i was alone or not. i will list the patterns in a following post.

some help could be useful, thx

spartan60

[edit] Mistake in triangle value?

I think the middle number in the last row should be 12870 (=16!/8!^2), not 12810. I've never edited anything here so I don't want to correct this myself.

You're right; I have fixed that.--Niels Ø 20:15, 5 November 2006 (UTC)

As I have noted at Image:Yanghui triangle.gif, there's also a mistake in that figure - an amusing little exercise to find it!--Niels Ø (noe) 20:18, 19 January 2007 (UTC)

[edit] Relation to Sierpinski Triangle?

Could someone explain how this triangle is related to the Sierpinski Triangle here, and add a section on it to the article?--AeomMai 04:50, 2 January 2007 (UTC)

I think a "See also" reference would do, perhaps to Sierpinski Triangle#Properties.--Niels Ø (noe) 10:26, 2 January 2007 (UTC)

Isnt there an easier, more user friendly discription of the formulas on this page?

[edit] Recent changes to the lead

I've made a post User_talk:Sangak/Archive3#Pascal.27s_triangle about the recent addition of Omar Khayyam to the lead. I guess that post really should have been here, but at least I've now linked to it.--Niels Ø (noe) 15:05, 25 February 2007 (UTC)

I did not notice the existence of a history section at the end of the long article. I've just noticed that some Indians and Italians also proposed similar ideas. I think the current version of the article is OK, although I see no point to have the history section at the end of the article. This is the first article I see with such a format. Sangak 15:22, 25 February 2007 (UTC)

I absolutely agree. It takes a bit more than just moving the history section, though, and I will not do it, at least not right now - go ahead if you are in the mood...--Niels Ø (noe) 15:31, 25 February 2007 (UTC)

[edit] please remove links to jstor and other closed-shop-information

Hi,

I find it annoying to click at a link in wikipedia and get pointed to a commercial, closed-shop site like jstor.

At least the author could have introduced a separate section like "commercial links" or the like. Please remove that.

Gottfried Helms --Gotti 07:04, 26 February 2007 (UTC) —The preceding unsigned comment was added by Druseltal2005 (talk • contribs) 07:02, 26 February 2007 (UTC).

[edit] Another property of the triangle or sequences

1/(1/999+1/999^2+1/999^3+1/999^4+1/999^5+1/999^6+1/999^7+1/999^8+.........infinity)=99800000....

                 1    9    45   165   495   1287  3003   6435   12870...
               1    8   36   120   330   792   1716   3432   6435.....             
             1   7    28   84   210   462   924   1716   3003...  
           1   6   21   56   126   252   462   792   1287...
         1   5   15   35   70   126   210   330   495.......
      1    4   10  20   35   56    84    120   165..........
    1    3   6   10   15   21   28    36    45..............
  1  2    3   4    5    6    7     8    9...................
1   1   1   1   1    1    1    1     1......................

  1   2    4   8   16   32   64   128   256

—Preceding unsigned comment added by Twentythreethousand (talk • contribs) 16:55, 24 May 2007

I'm not sure what the purpose of this post may be. The formula -

$\frac{1}{\frac{1}{999}+\frac{1}{999^2}+\frac{1}{999^3}+\ldots}=\frac{1}{\sum_{n=1}^{\infty}\frac{1}{999^n}}=998$

- is not obviously connected to Pascal's triangle. It follows from this:

The formula for a geometric series, $a+ar+ar^2+\ldots = \sum_{n=1}^{\infty}a\times r^{n-1} = \frac{a}{1-r}$

implies $\sum_{n=1}^{\infty}r^n = \frac{r}{1-r}$ .

With $r=\frac{1}{999}$ , this gives the sum $\frac{\frac{1}{999}}{1-\frac{1}{999}} = \frac{1}{999-1} = \frac{1}{998}$ .

The arrangement of the triangle shown next in the post seems to add nothing new.

The line sums being powers of two is covered in the article.--Niels Ø (noe) 09:58, 25 May 2007 (UTC)

Fibonacci number

1/999^1=0.001001001001001001001001001001001001001001001001001001001001001001001001001.....multiple of one

1/999^2=0.000001002003004005006007008009010011012013014015016017018019020021022023024.....

1/999^3=0.000000001003006010015021028036045055066078091105120136153171190210231253276.....

1/999^4=0.000000000001004010020035056084120165220286364455560680816970141331541773026.....

1/999^5=0.000000000000001005015035070126210330495716002366822383063880850992323865638.....

1/999^6=0.000000000000000001006021056126252462793289005007374196579643524375367691557.....

1/999^7=0.000000000000000000001007028084210462925719008013020394591170814338714081773.....

1/999^8=0.000000000000000000000001008036120330793719438446459479874465636450789503585.....

1/999^9=0.000000000000000000000000001009045165496290009447894353833708173810261050554.....

---1/998=0.001002004008016032064128256.....................................................

Pascal's triangle originated from one. 18:52,02 June 2007 twentythreethousand

square root of 98 twice:

3.14634628364578862062189264228281381561856023806624462402239289082033739605... close to Pi

Pi to the power of four equals 98-->two digits

98 to the power of two equals 9604-->four digits

9604 to the power of two equals 92236816-->eight digits

92236816 to the power of two equals 8507630225817856-->sixteen digits

The sequence keeps on going for every power of two.

01:02,03 June 2007 twentythreethousand

But

π 4 = 97.4

, not 98.

And if we continue,

850763 0225817856² = 72 3797720592 4958347626 4088436736

(32 digits)

72 3797720592 4958347626 4088436736² = 5238 8314033489 2668972443 7407788362 7107149416 9665108966 4274333696

(64 digits)

5238 8314033489 2668972443 7407788362 7107149416 9665108966 4274333696² = 27445354 4727148846 0806819005 1661236507 3947642680 2443831829 2390078008 4434721144 6634073687 9050118395 5547012413 9598029526 4761020416

(128 digits)

27445354 4727148846 0806819005 1661236507 3947642680 2443831829 2390078008 4434721144 6634073687 9050118395 5547012413 9598029526 4761020416² = 75324 7482132970 9217345275 9423346606 6515260444 6475346020 2222838532 9110949199 7633094461 9885532021 3811976016 8733739899 9296698028 1048683891 3751212180 5911904487 0320727730 0983919820 4004316869 4824738055 1808978241 7363969167 8243422064 3303915197 5580337172 1568813056

(255 digits, not 256)

- anyway, what's the point? Are you suggesting a change to the article on Pascal's triangle here?--Niels Ø (noe) 09:15, 2 July 2007 (UTC)

twentythreethousand23:24 22 July 2007

Dear twentythreethousand. If you have changes to the article in mind, please either be bold and make them, or try to state precisely what you're suggesting.--Niels Ø (noe) 08:56, 23 July 2007 (UTC)

I maybe a long way from achieving what the numbers mean in the properties of Pascal's triangle, so saying that I wouldn't urge me to change the article on Pascal's triangle because there is not much to say other than the product of the polynomials equals to the inverse of eight.

twentythreethousand17:37 23 July 2007

[edit] Relationship To Normal Distribution

Shouldn't we add something about how each row in Pascal's Triangle tends towards a normal distribution as we tend to the infinitieth row?

Martin Packer 09:40, 28 October 2007 (UTC)

[edit] Vandalism

Just pointing out the vandalism at the end of section 3, immediately before the section on History. It's relatively minor, but should be removed.

Chris Cornwell 09:56, 11 November 2007 (UTC)

Can you please be more specific? I don't see any vandalism. -- Jitse Niesen (talk) 12:08, 12 November 2007 (UTC)

It was removed shortly after I posted, so it's gone. --Chris Cornwell 10:20, 12 November 2007 (UTC) —Preceding unsigned comment added by 24.11.210.232 (talk)

i think that would be a long time ago —Preceding unsigned comment added by Joeldudesx (talk • contribs) 12:10, 15 February 2008 (UTC)

[edit] Two comments by joeldudesx

[edit] pascals triangle

who put that stuff there??? at pascals triangle that fibonacci numbers thingy or whatever and that supposed pic? of blaise pascal??? i hope someone will remove it...(Joeldudesx (talk) 12:19, 15 February 2008 (UTC)) joeldudesx/joeldudesx20:19 15 February 2008

[edit] everybody listen up!!!

ok, can someone pls change the headline of square pyramidal numbers to pyramid numbers??? i had problem finding it! —Preceding unsigned comment added by Joeldudesx (talk • contribs) 12:23, 15 February 2008 (UTC)

[edit] Replies (sort of)

Re "pascals triangle": What "stuff"? The Fibonacci numbers are really there; if you have a specific suggestion, question, criticism or whatever about that part of the article - well, please be more specific.

Re "everybody listen up": What headline are you talking about here? Again, please be more specific.

To sign your posts, place the cursor right after your post in the edit box, and click the signature button (the one with the mouse-over tip ""Your signature with timestamp") above the box.--Noe (talk) 16:36, 15 February 2008 (UTC)

[edit] multiples of numbers

is it true that if i were to shade the multiples of each natural number in turn, would the pattern generated be of triangles every time? i tried out a few and they were all so, but i just wanted to make sure

[edit] Another observation to pascal's triangle

The sum of the entries in the nth row of binomial terms is the nth power of 2.The length of the binomial terms in the nth row(row 1 is 1, row 2 is 2,row 3 is 3 etc...)
The sum of the entries in the nth row of trinomials is the nth power of 3.The length of the trinomial terms to every row to the power of nth follows a triangular number)
The sum of the entries in the nth row of poynomials with four terms is the nth power of 4.The length of polynomials with four terms to the power of nth shows a Tetrahedral number.

etc... Twentythreethousand (talk) 19:54, 14 April 2008 (UTC)

Please write posts here as suggested changes, as discussion of specific parts of the article, or the like. What is it you want?--Noe (talk) 20:52, 14 April 2008 (UTC)

[edit] Pascal's triangle or Tartaglia's triangle?

Here in Italy we always refer to it as Tartaglia's triangle and I'm adding a reference to it. I'm wondering if the whole article should be renamed to Tartaglia's triangle or Tartaglia-Pascal's triangle --82.48.35.113 (talk) 20:51, 11 June 2008 (UTC)

In English it is called Pascal's triangle, so we keep the most widespread usage. In Iran, it is called after a Persian mathematician, for example. See the history section in the article for more. Oleg Alexandrov (talk) 02:21, 12 June 2008 (UTC)