Talk:Pascal's triangle

From Wikipedia, the free encyclopedia

Contents 1 Code 2 Two Questions 3 Question about a listed property 4 Khayyam-Pascal's triangle? 5 Modifyed triangle "vandalism" 6 Exponents 7 question 8 Added the matrix-exponential gimmick 9 Properties of the triangle 10 patterns 11 Mistake in triangle value?

[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

1. Are the schemes for row numbering American and Candian??
2. Has anyone proposed a new scheme that has names with no ambiguity??

66.32.251.248 23:12, 17 Oct 2004 (UTC)

[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

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

Substituting m = n − 1 yields

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 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 disambiguation 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)

[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 , 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)

[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)