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MiszaBot I (talk | contribs) m Archiving 2 thread(s) from Talk:Fibonacci number. |
MiszaBot I (talk | contribs) m Archiving 2 thread(s) from Talk:Fibonacci number. |
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I removed the sunflower image since when I tried to verify the spiral count I kept getting different results. I think the particular sunflower in the image had irregularities that made the spirals too unstable to make the counts meaningful. The other sunflower images I found weren't much better but I did find a chamomile image where I could draw in the spirals. I also removed the seashell image, the connection to the Fibonacci number has been claimed by some authors but my understanding is that this has been debunked.--[[User:RDBury|RDBury]] ([[User talk:RDBury|talk]]) 02:05, 28 April 2011 (UTC)
== First number ==
As the article notes, the Fibonacci sequence is sometimes said to start with 0, sometimes with 1. Strictly speaking, if you are calling this sequence the "Fibonacci" sequence, you should begin with 1 because that is how Fibonacci started the sequence. Might I suggest you start with 1 and note that "some sources start with 0". In addition to historical accuracy, I suspect starting with 1 is far more popular in the literature, despite the OEIS entry. Or perhaps it would require too much rewriting of the article to make it worth the trouble to be historically accurate? --[[User:Seberle|seberle]] ([[User talk:Seberle|talk]]) 06:06, 26 February 2011 (UTC)
I admit this is probably minor, but it still bugs me that starting with zero is not, very strictly speaking, the "Fibonacci" sequence, nor is it the sequence that most readers will come across in other readings. At the very least, could we replace "Some sources omit the initial 0" in the introduction with "Fibonacci originally started his sequence with two 1s instead of 0 and 1, and many still define the sequence that way instead"? The way it's written now makes it look as if starting with zero were the definitive way and starting with two 1s is an infrequent variation. --[[User:Seberle|seberle]] ([[User talk:Seberle|talk]]) 17:12, 27 February 2011 (UTC)
:I don't think Fibonacci called it "the Fibonacci sequence". It was named later, and named objects often deviate from the work of the person they were named after. Including 0 is mathematically important for many of the formulas and should be "our" definition, but we can reformulate the description of the alternative definition. It currently says: "Some sources omit the initial 0, instead beginning the sequence with two 1s". That can indeed give a misleading impression of the history and prevalence. I suggest: "Many sources do not include an initial 0, instead beginning the sequence with two 1s". [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 17:54, 27 February 2011 (UTC)
:There are two slightly different issues here:
::(A) Should the enumeration of the sequence start with
:::<math>F_0 = 0 \quad\text{and}\quad F_1 = 1\, ,</math>
::or with
:::<math>F_0 = 1 \quad\text{and}\quad F_1 = 1\, .</math>
::(B) Should the quoted sequence of values start 0,1,1,... or start 1,1,2,...
:On (A), it is clear that the enumeration must start with <math>F_0 = 0</math>; if we change this, many of the quoted properties of the sequence do not hold (for example, ''F''<sub>''kn''</sub> is no longer divisibel by ''F''<sub>''n''</sub>). On (B), I have no preference. On either point, I think that the argument "this is not how Fibonacci defined the sequence" does not carry much weight - many mathematical objects and terms have modern definitions that differ widely from those given by their originators. Fibonacci's original formulation may be of historical interest but it should not dictate how we write our article. [[User:Gandalf61|Gandalf61]] ([[User talk:Gandalf61|talk]]) 10:39, 28 February 2011 (UTC)
I agree. Clearly the best way to define the Fibonacci sequence is as it is done in Wikipedia. However, we are still dealing with two problems. (This is all rather minor really, a question of wording in the introduction.) (1) The sequence was historically defined as beginning 1, 1, ... This goes beyond how Fibonacci defined the sequence (and yes of course he didn't name it after himself -- he didn't name it all as far as I know) because (2) many sources (and I strongly suspect the majority of popular sources) still begin with 1, 1, ... today; this is not some archaic historical distinction. Therefore the remark "Some sources omit the initial 0", as if starting with 1 was an afterthought, is rather misleading. Though not intentionally stated, the wording gives the impression that those who start with 1 are doing something wrong. PrimeHunter's proposal is an improvement, though it still implies something is being left out, when actually 0 is being added to the original sequence for the reasons Gandalf61 points out. And it would still be nice to include that this is how the sequence was historically defined (and not vice versa). It really shouldn't be hard to come up with a better wording for this one sentence. --[[User:Seberle|seberle]] ([[User talk:Seberle|talk]]) 16:30, 28 February 2011 (UTC)
:The separation into (A) and (B) is good. I agree with the first choice in (A). This should not be too controversial though it isn't done by everyone.
:Here's what I remember from what I've seen in math and science, which may be correct: "the Fibonacci numbers" usually start with 1, 1 and then usually they're called ''F''<sub>1</sub>, ''F''<sub>2</sub>. The math people would generally be happy to include ''F''<sub>0</sub> = 0. Some would like to say ''F''<sub>0</sub> = 1, ''F''<sub>1</sub> = 1. Some experts would say it depends on what suits the context.
:I do think most people who know the sequence would start it 1,1,....
:Here is an argument for not including 0 in "the" (basic) sequence. The number ''f''<sub>''n''</sub> of sequences of 1's and 2's of total length ''n'' starts with ''f''<sub>0</sub> = 1, not 0. The value 0 doesn't have any place in this count. Many other things counted by Fibonacci numbers behave the same way. I don't doubt there are counterexamples; I'm just mentioning that this is one justification, in addition to the original Fibonacci problem, for preferring to start at 1, or at least not preferring to start at 0. In my experience the single best reason for including 0 in the sequence is the attractiveness of working backwards via the recurrence relation.
:My suggestion would be to revise the first few lines to say the sequence is 1,1,2,3,..., then say some prefer 0,1,1,2,3,..., but mathematically the conventional recurrence may start with initial conditions either ''F''<sub>0</sub> = 0, ''F''<sub>1</sub> = 1 or ''F''<sub>1</sub> = 1, ''F''<sub>2</sub> = 1; and furthermore some people displace the sequence and say ''F''<sub>0</sub> = 1, ''F''<sub>1</sub> = 1. This would be true to real life, which would probably be more useful to ordinary users than a single rigid definition. [[User:Zaslav|Zaslav]] ([[User talk:Zaslav|talk]]) 03:06, 26 March 2011 (UTC)
I believe that the history of how the Fibonacci series came about informs this discussion. That history is already in the article under the heading Origins. But Origins should start with the Fibonacci story if we are to retain the word Fibonacci in the title of the entry. The Indian mathematics which developed the same series at an earlier time should appear but not before the Fibonacci history is told. To do this in no way diminishes the Indian achievement - it just places it in the context of an article on the Fibonacci series. Fibonacci himself developed the series to do a job. That job was to calculate how many rabbits you would have after a year if you took as your data the number of kittens rabbits usually have, and how often they are fertile and assuming that on average they have equal numbers of male and female offspring. The problem thus starts with a pair of rabbits. So the series as conceived by Fibonacci starts with 1. (one pair of rabbits) Some might think that 0 should be the first number so that the series can start using the rule ''F''<sub>n</sub> = ''F''<sub>n-1</sub> + ''F''<sub>n-2</sub>. If we use this rule and start as Fibonacci started then we start with 1 and add the number which came before it. No number came before it so add zero. 1+0=1 the second number. And so on. But if we start with zero then the series gets nowhere if we wish to find the second number by the rule because if we start with 0 then the next number must be 0+0=0 and the series gets nowhere.
Even the Indian series starts with a whole number greater than zero. So starting with zero is really not only historically incorrect but conceptually incorrect and it is thus misleading as well.
This leads me to the idea that the article might be renamed The Fibonacci Series or Fibonacci Numbers. In order to recognise the common but historically and conceptually incorrect way of developing the series a footnote or a comment might be made that "some people after Fibonacci used to start and some still start the series with zero" and then the article could go on to comment that this is historically and conceptually correct and misleading.
[[User:Robarc|Robarc]] ([[User talk:Robarc|talk]]) 17:22, 12 June 2011 (UTC)
:In view of the discussion above I am changing the introduction to state that the series begins 1, 1 or 0, 1, depending on personal preference. [[User:Zaslav|Zaslav]] ([[User talk:Zaslav|talk]]) 05:52, 28 June 2011 (UTC)
:It seemed best to allow that the sequence may begin 1,1 or 0,1, and the numbering of the sequence may begin ''F''<sub>0</sub>,''F''<sub>1</sub> in either case or ''F''<sub>1</sub>,''F''<sub>1</sub> in the former case. There is no general agreement about which to use. I stated that in this article 0,1 is used. [[User:Zaslav|Zaslav]] ([[User talk:Zaslav|talk]]) 06:15, 28 June 2011 (UTC)
== Relation to Golden Ratio Information ==
I am not a mathematician, but the following sentence in the information beneath the image in the Golden Ratio section does not properly describe the equations in the image: "The length of the side of one square divided by that of the next smaller square is the golden ratio." Could someone that knows better fix either the image or the text?--[[User:Bainst|Bainst]] ([[User talk:Bainst|talk]]) 19:00, 1 May 2011 (UTC)
:The quoted caption at [[Fibonacci number#Relation to the golden ratio]] looks right to me. 1 / (1/φ) = φ (this holds for any non-zero number and not just φ). And more generally, (1/φ<sup>''n''</sup>) / (1/φ<sup>''n+1''</sup>) = φ (also holds for all non-zero numbers). In addition, φ has the special property 1/φ = φ−1. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 01:15, 2 May 2011 (UTC)
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