#195591 - Thu Oct 02 2003 09:17 PM
Fibonacci Numbers
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Prolific
Registered: Fri Jun 06 2003
Posts: 1336
Loc: Mumbai India
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The usual set of numbers that come to mind when one thinks of 'Fibonacci Numbers' are the usual 1, 1, 2, 3, 5, 8.... I, however, visited a few sites and saw this particular sequence as 0, 1, 1, 2, 3, 5..., i.e., with a '0' at the beginning. So my question is - which is the more widely accepted and employed variety? 
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#195592 - Fri Oct 03 2003 02:05 PM
Re: Fibonacci Numbers
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Multiloquent
Registered: Fri Nov 23 2001
Posts: 3082
Loc:
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Logic says - if you start 1 on the first line then 1,2 on the second
then you would have to start 0 then 0,1 in the alternative version
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#195593 - Fri Oct 03 2003 06:24 PM
Re: Fibonacci Numbers
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Prolific
Registered: Fri Jun 06 2003
Posts: 1336
Loc: Mumbai India
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I guess you're right then, Fosse. However, if you begin directly with one, can you just assume that the previous number is zero (without writing down the zero), since 'nothing at all' can be represented by a zero? (Please pardon the run-on sentence!  )
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#195594 - Fri Oct 03 2003 09:18 PM
Re: Fibonacci Numbers
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Moderator
Registered: Mon Dec 03 2001
Posts: 20912
Loc: Sydney
NSW Australia
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Harish, I share your puzzlement- I have never seen that series start with a zero. It sounds a bit silly, sort of like counting your fingers, and starting with 'none, one, two...'
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#195596 - Sat Oct 04 2003 09:36 PM
Re: Fibonacci Numbers
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Enthusiast
Registered: Fri Jan 03 2003
Posts: 365
Loc: New Delhi India
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I always thought the Fibonacci series started with 0.
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#195597 - Sun Oct 05 2003 02:09 PM
Re: Fibonacci Numbers
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Multiloquent
Registered: Fri Nov 23 2001
Posts: 3082
Loc:
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The Fibonacci rule is 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
each number is a sum of the two previous numbers. It obviously can't start with a Zero as 0+0 = 0 and the next series would then be 0+0 = 0. You can't get to the starting point of 1. I suppose you could argue that the starting point is arrived at with 0 + 1 but where did the 0 come from using the same system ,
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#195598 - Sun Oct 05 2003 07:38 PM
Re: Fibonacci Numbers
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Multiloquent
Registered: Mon Dec 06 1999
Posts: 2742
Loc: Wyoming USA Way Out West
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I have a question. Does this Fibonacci number sequence have any practical application? If not, I have a sequence of numbers I would like to have named after me. 
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#195599 - Sun Oct 05 2003 07:42 PM
Re: Fibonacci Numbers
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Prolific
Registered: Fri Jun 06 2003
Posts: 1336
Loc: Mumbai India
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I heard that when this Leonardo Fibonacci guy thought of it, he'd used it to plot the growth of rabbit population.
Strange!
{Edit}
From Wikipedia: Quote:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657...
This sequence was first described by Leonardo of Pisa aka. Fibonacci (ca. 1200), to describe the growth of a rabbit population. The numbers describe the number of pairs in a (somewhat idealized) rabbit population after n months if it is assumed that
* the first month there is just one newly born pair,
* newly born pairs become productive from their second month on,
* we have no genetic problems whatsoever generated by inbreeding,
* each month every productive pair begets a new pair, and
* the rabbits never die
Hmm...
Edited by harish_256 (Sun Oct 05 2003 07:44 PM)
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#195600 - Sun Oct 05 2003 07:45 PM
Re: Fibonacci Numbers
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Prolific
Registered: Fri Jun 06 2003
Posts: 1336
Loc: Mumbai India
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#195601 - Sun Oct 05 2003 09:01 PM
Re: Fibonacci Numbers
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Multiloquent
Registered: Mon Dec 06 1999
Posts: 2742
Loc: Wyoming USA Way Out West
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Thanks, Harish, for the explanation and reference sites. It still appears to me that the examples given that fits the hypothesis that the string of numbers in a Fibonacci number sequence is a coincidental relationship, not one that can be repeated with certainty. How they derive the spiral of a nautilus shell from a series of squares is possible simply by doodling on a piece of paper. But, then again, I'm no mathematician. I'll bet the rabbits had fun with the experiment!
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#195603 - Tue Oct 07 2003 09:35 PM
Re: Fibonacci Numbers
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Enthusiast
Registered: Mon Sep 29 2003
Posts: 234
Loc: Philadelphia, PA
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Quote:
Fosse4:Logic says - if you start 1 on the first line then 1,2 on the second
then you would have to start 0 then 0,1 in the alternative version
Fosse4(later post):The Fibonacci rule is 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
each number is a sum of the two previous numbers. It obviously can't start with a Zero as 0+0 = 0 and the next series would then be 0+0 = 0. You can't get to the starting point of 1. I suppose you could argue that the starting point is arrived at with 0 + 1 but where did the 0 come from using the same system ,
There are several ways mathematicians play with the definition of the Fibonacci series, some include 0, and some don't. Some include negative numbers. and some don't. Here are some examples:
The common definition is that each number is the sum of the previous two integers, with an arbitrary starting point of 1. This starting point is arbitrary because the first "1" in the series does not equal (0 + -1). This definition does not include "0". Curiously, this definition creates a "mirror fibonacci", which goes into negative integers and arbitrarily begins at -1, and still doesn't include 0.
Another definition is that each number is equal to the absolute value of the sum of the two integers before it. By this definition, the starting point of "1" is not arbitrary, and "0" is still not included. The first "1" therefore is defined as |-1 + 0|, which equals 1, the second "1" is |1 + 0|, "2" is |1 + 1|, and this pattern continues along the series.
Still another definition is that each number is equal to the difference of the absolute value of both the integer after it and the integer before it, with the greater number being the first term. This definition would include 0. For example, 0 = |1| - |-1|, 1 = |1| - |0|, 1 = |2| - |1|, 2 = |3| - |1|, etc. The absolute value is only necessary if you wish the set to include 0.
There are many more ways to define the series that become increasingly more convoluted and complex. The truth is, like many things in mathematics, there are many ways to approach the problem.
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