FibonacciNumbersDefinition

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Fibonacci Numbers

Fibonacci numbers appeared as the solution of the following problem mentioned in the third section of Fibonacci’s Liber Abbaci (1202) (Book of Calculation)[1]:¹²
How Many Pairs of Rabbits Are Created by One Pair in One Year?
(Quot paria coniculorum in uno anno ex uno pario germinentur)
A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

Their name was given them by Édouard Lucas (1842-1891) in 1877.

The Fibonacci recurrence was studied by Indian scholars prior to Fibonacci (cf. [2], [3] ). Indian mathematician Virahanka (sometimes between 600 AD and 800 AD) gave method for their formulation.

In about 1135 another Indian named Gopala had also studied these numbers. The religious teacher Acharya Hemchandra (1089-1173) in a text written about 1150 considered the following problem:

assume that lines are composed of syllables which are either short or long
each long syllable takes twice as long to articulate as a short syllable.
a line of length n contains n units where each short syllable is one unit and each long syllable is two units.
a line of length n units takes the same time to articulate regardless of how it is composed.

Hemchandra problem asks: How many different combinations of short and long syllables are possible in a line of length n?
He also offers the solution:
If we denote by the possibilities for a line of length , then the line of length either ends in a short syllable or in a long syllable. In the former case there remains a line of length which can be composed in ways. In the alter case the line of length ends in a long syllable then the remaining line of length can be composed in ways. Consequently
.

In 1356 an important representative of the Kerala school Narayana Pandit (1340-1400) describes a relation involving multinomial coefficients of which the Fibonacci numbers are only a particular case.

In 1611 Kepler in his small monograph Strena seu de Nive Sexangula (A New Year Gift : On Hexagonal Snow)⁴ [4] rediscovered Fibonacci sequence, their recurrence property and made the remarkable discover about the ratios of consecutive terms of the Fibonacci sequence⁵⁶.

Definition

The mathematical solution for Fibonacci’s rabbit population is a series with terms 1, 1, 2, 3, 5, 8, 11, ... (Fibonacci did not start the sequence with as we do) expressed by he recursion

(1)

with the initial conditions

(2)

Therefore the total number of rabbits after the births at the start of month n is Thus the answer to Fibonacci's question is

the count at the start of year 2,
at the start of year 3,
at the start of year 4,
at the start of year 5,
etc.

To compute the nth Fibonacci number go to .

To compute segments of the sequence of the Fibonacci numbers go to .

It is possible to extend the definition of Fibonacci numbers to negative indices using the formula .

Table 1. Fibonacci numbers for negative indices

n	0	-1	-2	-3	-4	-5	-6	-7	-8	-9	-10	-11

F(n)	0	1	-1	2	-3	5	-8	13	-21	34	-55	89

There is even an extension to an real valued index using the relation, which is based on Binet’s formula .

The plot of this function with the corresponding vales of Fibonacci numbers looks as follows:

[Graphics:HTMLFiles/FibonacciNumbersDefinition_24.gif]

[5]

Notes

Liber Abbaci is one of the first books introducing the Hindu numerals to Europe. The first section of the book starts with: The nine Indian figures are: 987654321. With these figures, and with the sign 0, which Arabs call zephir any number whatsoever is written ... The Latinized Arabic word, zephirum, became zefiro in Italian, and in the Venetian dialect, zero, the name by which we know it in English today. The book also btigs the notion of an algorithm (derived from the name of the Persian scholar Abu 'Abd Allah Muhammad ibn Musa al-Khwarizmi (ca. 780--850)), and the subject of algebra, which comes from the title of al-Khwarizmi's book, Hisab Al-Jabr wal Mugabalah (Book of Calculations, Restoration and Reduction). In 2002, the 800th anniversary of Liber Abbaci, the book was republished for the first time in a modern language
The book was written in 1202 and a second revised version of Liber Abbaci was produced in 1228. In 1857 Baldassare Boncompagni prepared a Latin edition (unfortunately with errors) from several existing manuscripts. The book gives us an almost complete review of the mathematical knowledge that had accumulated in the Europe since the Greeks. The book can also be regarded as a comprehensive merchant’s handbook and its stated aim was to demonstrate to the merchants the advantages of the Hindu Arabic number system compared with the Roman system.

²	http://www.mathematik.uni-kl.de/~luene/miszellen/abbaci.html

The 19th century english puzzlist, Henry Dudeney, altered the problem to concern cows. He noticed that it is only important to consider the number of females, and simplified the problem to a more realistic:
"If a cow produces its first she-calf at age two years and after that produces another single she-calf every year, how many she-calves are there after 12 years, assuming none die?" http://people.bath.ac.uk/ma2jph/projectp1.html

⁴	This booklet is mainly concerned with the packing together of circles in a plane and spheres in space. It also describes the structure of flowers in quincuncial arrangements (i.e. arrangments of five objects at each of the four corners of a square).

⁵

J.Kepler: Vom sechseckigen Schnee, Leipzig 1987, p.34-35:

Die fünfzähligen regelmässigen Körper und ihr Ursprung aus der göttlichen Proportion (goldener Schnitt)

Es gibt zwei regelmässige Körper, das Dodekaeder und das Ikosaeder, die such aus dem Grundplan des Fünfecks ergeben. Jenes wird von Fünfecken umschlossen, dieses von gleichseitigen Dreiecken, aber stets sind fünf in der Form eines Fünfeks angeordnet. Die Konstruktion beider Körper, jedoch vor allem die des Fünfecks selbst, ist ohne jene Proportion nicht durchführbar, die die heutigen Mathematiker die göttliche nennen. Diese aber ist in ihrem Aufbau so, dass die beiden kleineren Glieder einer stetigen Proportion zusammen das dritte Glied ergeben, und so bildet immer die Summe zweier vorhergehender Glieder das folgende Glied. Dabei kann die gleiche Proportion ins Unendliche fortsgesetzt werden. In Zahlen kann man das Ergebnis nicht vollständig ausdrücken. Je weiter wir uns von der Einheit entfernen, desto besser wird die Annähering. Es seien die beiden kleinsten Zahlen gegeben 1 und 1, die Du als ungleich betrachten musst. Wenn Du sie addierst, dann ergeben 2. Dazu die (grössere) 1 addiert, ergibt 3, nun 2 dazu gezählt, ergibt 5. Addiert man weiter 3 dazu, so erhältst Du 8, dann 5 hinzugeben, ergibt 13, 8 dazu addiert, ergibt 21. Es verhält sich nun 5 zu 8 annähernd wie 8 zu 13 und 8 zu 13 annähernd wie 13 zu 21.
Entsprechend einer Analogie dieser sich selbst fortsetzenden Proportion ist, so glauch ich, das Zeugungvermögen versinnbildlicht. So wird die Blütte dem Zeugungsvermögen für die Früchte deutlich ein fünfzackiges Fähnlein vorangetragen. Ich übergebe manche Gedanken, die zur Bestärkung dieser Ansicht in einer ergötzlichen Betrachtung hinzugefügt werden könnten. Doch dafür braiche ich einen eigenen Platz. Hier hatte ich diese Gedanken nur als ein Beispiel aufgenommen, um zur Erforschung der secheckigen gestalt des Schnees besser geübt und vorbereitet zu sein.

⁶	Kepler also develops these ideas in his Welharmonik (Harmonices Mundi, 1619), III. Buch; Kapitel XV, translation by M. Caspar, Oldenbourg Verlag, Munich 1939, p. 165 - 166.

References

[1]	Pisano [Fibonacci], L. Liber Abbaci.

[2]	Singh, P. (1985). The so-called Fibonacci numbers in ancient and medevial India. Historia Mathematica , 12, 229-244.

[3]	Knuth, D. E. (2004). The Art of Computer Programming, Vol. 1. Boston - San Francisco - New York - Toronro - etc. : Addison Wesley.

[4]	Kepler, J. (1987). Vom sechseckigen Schnee (Strena seu de Nive sexangula). Leipzig: Akademische Verlagsgeselschaft, Geest & Portig K.-G..

[5]

Pisano, L. (2003). Fibonacci's Liber abaci. A translation into modern English of Leonardo Pisano's Book of calculation. Transl. from the Latin and with an introduction, notes and bibliography by L. E. Sigler. Paperback ed. (English) Sources and Studies in the History of Mathematics and Physical Sciences. . New York: Springer.

Cite this web-page as:

Štefan Porubský: Fibonacci Numbers.

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