Talk:Fibonacci sequence/Archive 2: Difference between revisions

Revision as of 01:04, 3 April 2008

Revision as of 01:07, 29 April 2008

Line 45:

:I fixed the formula that you added for F_2''n''. The F_2''n''+''k'' formula looks fine to me - it is just a special case of Formula 47 from the MathWorld page. When ''k''=0 you have F_''k''=0, F_''k''-1=1 and F_''k''-2=-1, so you get F_2''n'' = 2F_''n''+1F_''n'' - F_''n''² as expected. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 17:17, 3 January 2008 (UTC)

== Proof by induction ==

Why haven’t you completed your Proof by induction of Binet’s formula?

You’ve shown its true for 0 and 1.I think you now need to show that ''if'' it’s true for n and n+1 ''then'' it is also true for n+2,the dominoes topple, and you’ve proved it for all the natural numbers.

I’ve just tried to do this on a bit of paper and I can’t.It certainly isn’t so obvious you can just leave it out! <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/91.107.165.60|91.107.165.60]] ([[User talk:91.107.165.60|talk]]) 21:23, 9 January 2008 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

Duh! I’ve just seen how to do it. It ''is'' pretty obvious but someone who can write Latex ought to put it up. <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/91.107.165.60|91.107.165.60]] ([[User talk:91.107.165.60|talk]]) 21:39, 9 January 2008 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

I’ve just tried to do it by cutting and pasting Lyx code but that doesn’t work. I get “parsing error”. I’m not going to learn all the bloody code, someone else will have to do it.

Anyone who thinks for 5 minutes will see how the proof works anyway. It just annoys me it isn’t completed.

[[User:Dave59|Dave59]] ([[User talk:Dave59|talk]]) 23:01, 9 January 2008 (UTC)

let P(n) be the variable proposition

<math>F_{n}=\frac{\phi^{n}-\psi^{n}}{\sqrt{5}}</math>

P(n+1) is

<math>F_{n+1}=\frac{\phi^{n+1}-\psi^{n+1}}{\sqrt{5}}</math>

P(n+2) is

<math>F_{n+2}=\frac{\phi^{n+2}-\psi^{n+2}}{\sqrt{5}}</math>

Now

<math>F_{n+2}=F_{n+1}+F_{n}</math>

Therefore

<math>F_{n+2}=\frac{\phi^{n}-\psi^{n}}{\sqrt{5}}+\frac{\phi^{n+1}-\psi^{n+1}}{\sqrt{5}}=\frac{\left(\phi^{n}+\phi^{n+1}\right)-\left(\psi^{n}+\psi^{n+1}\right)}{\sqrt{5}}</math>

we have allready shown <math>\phi^{n+2}=\phi^{n+1}+\phi^{n}\qquad and\qquad\psi^{n+2}=\psi^{n+1}+\psi^{n}</math>

Therefore

<math>F_{n+2}=\frac{\phi^{n+2}-\psi^{n+2}}{\sqrt{5}}</math>

So

<math>P(n)\qquad and\qquad P(n+1)\Rightarrow P(n+2)</math>

we have already shown P(0) and P(1) are true

Therfore by mathematical induction the proposition is proved for all

natural numbers.(or for all the natural numbers plus zero if you want to be ''really'' pedantic)

Code a damn site harder than the maths

Dave. <small>—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/91.105.18.197|91.105.18.197]] ([[User talk:91.105.18.197|talk]]) 12:42, 11 January 2008 (UTC)</small><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:This seems already explained in [[Fibonacci_number#Proof_by_induction]].--[[User:Patrick|Patrick]] ([[User talk:Patrick|talk]]) 13:46, 11 January 2008 (UTC)

It probably says enough for a mathematician to understand the drift straight away.

However it is not a formal proof by induction and I didn’t understand it the first time I read it.

This just dots the i’s and crosses the t’s.

This is pure maths and I feel we ought to be precise.

I have used slightly different notation to the main article. I’m unfamiliar with Latex and this took me ages to do. I’m not even going to try to integrate it into the main article. It is probably true that most of the people who are going to read the article don’t need it but it might be useful for people who are just learning proof by induction and want to see a few examples.

[[User:Dave59|Dave59]] ([[User talk:Dave59|talk]]) 15:37, 11 January 2008 (UTC)

== Fibonacci sequence ==

This article should be called ''Fibonacci sequence'' and not ''Fibonacci number''. A ''Fibonacci number'' is meaningless out of the context of its sequence. If I asked you "what is 21?", nobody would say "the Fibonacci number after 13". But if I asked "what is 1, 1, 2, 3, 5, 8, 13, 21...?, I'd have a much greater chance of hearing "Fibonacci sequence". This article should be moved to ''Fibonacci sequence'' over the redirect, and ''Fibonacci number'' should redirect to ''Fibonacci sequence''. <font style="background-color:#218921; color:blue;font-family: Monotype Corsiva">Table</font><font style="background-color:#121298; color:white;font-family: Monotype Corsiva">Manners</font>^{[[Special:Contributions/TableManners|C]]·[[User:TableManners|'''U''']]·[[User_talk:TableManners|T]]} 06:05, 18 January 2008 (UTC)

:: I was going to say it's commonly called "numbers" by everyone in the world, but then I looked at the interwiki links: bg, cs, eo, pt, ru - Numbers. ca, de, el, es, fr, it, scn, sk, tr, uk - Sequence. Still, I've mostly seen it as "numbers" in English - for example that's how it's called on the Integer Sequences site [http://www.research.att.com/~njas/sequences/A000045] and, for another example, Wolfram's Mathworld defines the Sequence [http://mathworld.wolfram.com/FibonacciSequence.html] as "see Fibonacci Number". The Marriam-Webster dictionary of the English Language has the entry for numbers [http://www.m-w.com/dictionary/Fibonacci%20number] but not sequence, while American Heritage Dictionary has both and essentially says "See Sequence" for Number: [http://www.bartleby.com/61/0/F0100000.html] and [http://www.bartleby.com/61/1/F0100100.html]. Doesn't look like there is an agreement. --[[User:Cubbi|Cubbi]] ([[User talk:Cubbi|talk]]) 12:23, 18 January 2008 (UTC)

:Though I'm not sure it's supported by Cubbi's post, isn't "sequence" a alightly technical mathematician's way of putting it, and "numbers" what the man in the street would say? Fibonacci numbers are rather insignificant in professional math, but play a quite significant role in popular math, recreational math. I'm for keeping the article at "numbers".--[[User:Noe|Niels Ø (noe)]] ([[User talk:Noe|talk]]) 13:19, 18 January 2008 (UTC)

: Conditions such as "if ''n'' is a Fibonacci number" naturally arise, independent of any overt connexion to the sequence, often enough that I disagree with TableManners: they are a meaningful set or class of numbers. &mdash;[[User:Tamfang|Tamfang]] ([[User talk:Tamfang|talk]]) 23:07, 22 January 2008 (UTC)

*Just passing by, figured I'd share my 2 cents. Google results:

::"Fibonacci sequence" (with quotes): ~186,000

::"Fibonacci number" (with quotes): ~86,000

::"Fibonacci numbers" (with quotes): ~216,000

:Therefore, I propose a move to [[Fibonacci numbers]] with [[Fibonacci sequence]] and [[Fibonacci number]] redirecting to that title. [[User:FireCrotch|FireCrotch]] ([[User talk:FireCrotch|talk]]) 15:18, 28 January 2008 (UTC)

::Current convention for Wikipedia articles on integer sequences is to name them ''xxx number'' or ''xxx sequence'' but never ''xxx numbers'' - see [[:Category:Integer sequences]] for many examples. This follows [[Wikipedia:Naming conventions]], which says "In general only create page titles that are in the singular, unless that noun is always in a plural form in English (such as scissors or trousers)". [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 15:32, 28 January 2008 (UTC)

::: Thank you for pointing that out, Gandolf61. In that case, I suggest that it be renamed to "Fibonacci sequence". Now that I think about it, of course "Fibonacci numbers" is going to have more results - it includes all pages that contain "Fibonacci number" as well! [[User:FireCrotch|FireCrotch]] ([[User talk:FireCrotch|talk]]) 04:00, 29 January 2008 (UTC)