fibonacci-numbers

NATIONAL UNIVERSITY 

Fibonacci Number 

Munjal Patel 

January 30, 2010 

Fibonacci numbers are sequence of numbers developed in ancient India and later introduced to 

the western European mathematicians by Leonardo of Pisa also known as Fibonacci in 1202. 

This paper serves as an introduction to the Fibonacci numbers, their properties, applications 

and relation to the mother nature.

Leonardo Pisano Bogollo, (c. 1170 – c. 1250) known as Leonardo of Pisa, or simply 

Fibonacci was an Italian mathematician (Anderson, Frazier, & Popendorf, 1999). He is 

considered as the most talented mathematician of the middle ages (Eves, 1990). Fibonacci was 

first introduced to the number system we currently use with symbols from 0 to 9 along with the 

Fibonacci sequence by Indian merchants when he was in northern Africa (Anderson, Frazier, & 

Popendorf, 1999). He then introduced the Fibonacci sequence and the number system we 

currently use to the western Europe In his book Liber Abaci in 1202 (Singh, Acharya 

Hemachandra and the (so called) Fibonacci Numbers, 1986) (Singh, The So‐called Fibonacci 

numbers in ancient and medieval India, 1985). 

Fibonacci sequence was well known in India and was applied to the metrical sciences 

(prosody). Pingala (200 BC), Virahanka (6 th century AD), Gopāla (c.1135 AD), and Hemachandra 

(c.1150 AD) are given credits for the development of this sequence (Goonatilake, 1999). In 

Sanskrit prosody, long syllables have duration of 2 and short syllables have duration of 1. For 

this reason, it is possible to form a pattern of duration by adding a short syllable to the 

pattern of duration 1 or a long syllable to the pattern of duration 2. Sanskrit prosodists 

proved that the number of patterns of duration is the same as the sum of the previous two 

numbers in the sequence. It is believed that this is where the motivation for the Fibonacci 

sequence came from. Later, algorithms for finding pattern of duration were developed 

and the higher‐order Fibonacci numbers were discovered. This work has been reviewed by 

Donald Knuth in The Art of Computer Programming (Knuth, The Art of Computer Programming, 

2006) (Hall, 2007). 

Fibonacci numbers Page 2

In order to study Fibonacci numbers, Fibonacci investigated a problem about growth 

rate of rabbit population in a biologically unrealistic situation. The puzzle Fibonacci posed was: 

assume that a newly born pair of rabbit, a male and a female is to be mate at the age of one 

month to produce another pair of rabbit with a male and a female on the second month. 

Assume that rabbits never die and the same cycle repeats every month so that all pairs give 

birth to a new pair every month and a newly born pair can start mating from the second month. 

How many pairs of rabbit will there be at the end of the year The solution of the problem 

looks like the following: 

 

 

 

 

At the end of the first month, they mate, but there is still one only 1 pair. 

At the end of the second month the female produces a new pair making 2 pairs of rabbits. 

At the end of the third month, the original female produces a second pair, making 3 pairs. 

At the end of the fourth month, the original female has produced another new pair, the 

female born two months ago produces her first pair as well, making 5 pairs. 

If this cycle continues, at the end of n month, number of pairs will be the sum of number of 

pairs in the month 2 and 1. This is same as the Fibonacci number (Knott, Fibonacci 

Numbers and Nature, 2008). 

The Fibonacci numbers are represented by the numbers in the following sequence. By 

definition, the first two numbers in the Fibonacci series are 0 and 1, and the rest are the sum of 

the previous two numbers. 

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … 


The first 18 Fibonacci numbers denoted by for 0,1,2, … ,17 are the following (Knott, The 

Fibonacci numbers, 2005): 

 

0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 

The Fibonacci sequence can be represented by a recurrence relation , 

where seeds 0 and 1. The same sequence can also be extended to negative index . 

The sequence with negative index …,8,5,3,2,1,1,0,1,1,2,3,5,8,… can be satisfied by 

1 . 

One of the interesting properties of Fibonacci numbers is that every third number in the 

sequence is even and every element of the sequence is always a multiple of . for this 

reason, Fibonacci sequence satisfies a stronger divisibility property described by (Su) 

(Ribenboim, 2000) (Renault, 1996): 

gcd , , 

This unremarkable recursive sequence has yet another interesting pattern. The sides of 

the squares in the figure 1 correspond to the numbers in the Fibonacci sequence and squares 

are arranged in the outwardly spiraling pattern. Each rectangle in the image has roughly the 

same shape and ratio for their length and width. It is also very interesting to note that as the 

sequence of rectangles develops outwards, the ratio of length to width on every step is the 


atio of two successive terms (Platonic Realms). The ratios of these consecutive Fibonacci 

numbers form another sequence: 

1 

1 , 2 1 , 3 2 , 5 3 , 8 5 , 13 

8 , 21 

13 ,… 

In 1753, a Scottish mathematician Robert Simson proved that the ratios of successive Fibonacci 

numbers ⁄ converges to a real number known as the golden ratio () as approaches to 

infinity (Wells, 1987). 

1 

lim 

 

→ 

 

√ 

⁄ 

√ 

, where is the golden ratio. 

√ 

 

1.6180339887 … (Sloane) 

Figure 1 A tiling with squares whose sides are successive 

Fibonacci numbers in length 

Figure 2 Approximate and true golden spirals. The green 

spiral is made from quarter‐circles tangent to the interior of 

each square, while the red spiral is a Golden Spiral, a special 

type of logarithmic spiral. Overlapping portions appear 

yellow. The length of the side of one square divided by that 

of the next smaller square is the golden ratio. 


Because |1 | ⁄ √5 1⁄ 2for all 0, the closest integer to ⁄ √5 can be obtained 

by the rounding as following: 

 

, 0 (Tattersall, 2005) 

√ 

It is possible to test any number whether it belongs to the Fibonacci series of not. As we 

know, that the closest integer to is ⁄ √5, the most straightforward and brute‐force test 

would be the following identity with is valid if and only if is a Fibonacci number (Posamentier 

& Lehmann, 2007). 

log √5 1 

2 

Alternatively, if and only if either 5 4 or 5 4 is a perfect square, where is a positive 

integer, than is also a Fibonacci number (Posamentier & Lehmann, 2007). It is possible to 

conduct a more sophisticated test by considering the fact that the convergent of the continued 

fraction representation of are ratios of successive Fibonacci numbers. The following 

inequality is true if and only if ( and are coprime) and and are successive Fibonacci 

numbers (Posamentier & Lehmann, 2007). 

1 

From this, it is possible to derive that is a Fibonacci number if and only if the interval 

, contains a positive integer (Möbius, 1998). 

 


There are many other properties and identities related to Fibonacci numbers and people 

have written books about it. Due to the nature of this paper and a limited scope, it is hard to 

cover everything there is about for the Fibonacci numbers. For this reason, in this final section 

of the paper, I will introduce some real world applications related to Fibonacci numbers and 

their relationship with nature. 

Fibonacci numbers are important to perform a run‐time analysis of Euclid’s algorithm to 

find the greatest common divisor (GCD) of two integers. A pair of two consecutive Fibonacci 

numbers makes a worst case input for this algorithm (Knuth, Art of Computer Programming, 

Volume 1: Fundamental Algorithms, 1997). Fibonacci numbers have their application in the 

polyphase version of the Merge Sort algorithm. This algorithm divides an unsorted list in two 

lists such that the length of lists corresponds to two sequential Fibonacci numbers. The ratio of 

the lengths of the lists is an approximately same as (Knuth, Art of Computer Programming, 

Volume 1: Fundamental Algorithms, 1997). Fibonacci numbers are essential in the analysis of 

the Fibonacci heap data structures. A network topology for parallel computing uses a Fibonacci 

cube which is an undirected graph with Fibonacci number of nodes. The Fibonacci search 

technique is a one‐dimensional optimization method and is developed on the basis of Fibonacci 

numbers and their properties (Avriel & Wilde, 1966). IFF 8SVX audio file format in Amiga 

computers uses Fibonacci sequence to compand the original audio wave for optional lossy 

compression (Addison‐Wesley, 1991). The conversion factor from miles to kilometers 

1.609344 . When Fibonacci numbers are replaced by their successors, the sum of the 

decomposition of distance in miles into a sum of Fibonacci numbers is approximately same as 

kilometer sum. This can be achieved by shifting a radix 2 number register in golden ratio base 


. Shifting the register down the Fibonacci number results into the conversion from kilometers 

to miles (Hazewinkel, 2002) (Knott, Using the Fibonacci numbers to represent whole numbers, 

2009). 

A close observation of our nature revels that Fibonacci numbers are often found in two 

consecutive Fibonacci numbers in biological setting (Douady & Couder, 1995), branching in 

trees, and arrangement of leaves on a stem, and the fruitlets of a pineapple (Jones & Wilson, 

2006) for example. The flowering of artichoke and arrangements of a pine cone is yet another 

example (Brousseau, 1969). The Fibonacci numbers have also been observed in the family tree 

of honeybees (Thimbleby). H. Vogel in 

1979 proposed a model for the pattern of 

florets found in the head of a sunflower 

(Vogel, 1979) as below: 

2 , √ 

∅ Here, is the index number of the floret 

while is a constant scaling factor. 

Figure 3 Sunflower head displaying florets in spirals of 34 and 55 

around the outside 

Therefore, florets lie on Fermat's spiral. The divergence angle is approximately 137.51 ° which is 

known as the golden angle because this angle divides a circle in the golden ratio. The reason 

sunflower florets pack so efficiently is since this ratio is an irrational number; no floret has a 

neighbor at exactly the same angle from the center. The rational approximations to the golden 

ratio are of form : 1. For this reason, the nearest possible neighbor of floret number 

are those at for some index which is dependent of , the distance from the center. 


It is often observed that sunflowers and similar natural arrangements have 55 spirals in one 

direction while 89 in the other (Prusinkiewicz & Lindenmayer, 1991). 

Once again I would like to stress that this paper shall only be considered as the 

introduction to the Fibonacci numbers and there is still much more to Fibonacci numbers not 

covered in this paper. 


References 

Addison‐Wesley. (1991). Amiga ROM Kernel Reference Manual. Addison‐Wesley. 

Anderson, M., Frazier, J., & Popendorf, K. (1999). Leonardo Fibonacci (ca.1175 ‐ ca.1240). Retrieved 

January 28, 2010, from ThinkQuest: http://library.thinkquest.org/27890/biographies1.html 

Avriel, M., & Wilde, D. J. (1966). Optimality of the Symmetric Fibonacci Search Technique. Fibonacci 

Quarterly, (pp. 265–269). 

Brousseau, A. (1969). Fibonacci Statistics in Conifers. Fibonacci Quarterly , 525–532. 

Douady, S., & Couder, Y. (1995). Phyllotaxis as a Dynamical Self Organizing Process. Journal of 

Theoretical Biology , 255–274. 

Eves, H. (1990). An Introduction to the History of Mathematics (6th ed.). Brooks Cole. 

Goonatilake, S. (1999). Toward a Global Science: Mining Civilizational Knowledge (Race, Gender, and 

Science). Indiana University Press. 

Hall, R. W. (2007, October 31). Math for Poets and Drummers. Retrieved January 30, 2010, from Saint 

Joseph’s University: http://www.sju.edu/~rhall/mathforpoets.pdf 

Hazewinkel, M. (Ed.). (2002). Encyclopaedia of Mathematics. 

Jones, J., & Wilson, W. (2006). An Incomplete Education, 3,684 Things You Should Have Learned But 

probably Didn't. Ballantine Books. 

Knott, R. (2008, December 16). Fibonacci Numbers and Nature. Retrieved January 30, 2010, from The 

University of Surrey: http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/fibnat.html#Rabbits 


Knott, R. (2005, August 19). The Fibonacci numbers. Retrieved January 30, 2010, from The University of 

Surrey: http://www.maths.surrey.ac.uk/hosted‐sites/R.Knott/Fibonacci/fibtable.html 

Knott, R. (2009, October 23). Using the Fibonacci numbers to represent whole numbers. Retrieved 

January 30, 2010, from The University of Surrey: http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/fibrep.html#kilos 

Knuth, D. E. (1997). Art of Computer Programming, Volume 1: Fundamental Algorithms (3rd ed.). 

Addison‐Wesley Professional. 

Knuth, D. E. (2006). The Art of Computer Programming. Addison‐Wesley Professional. 

Möbius, M. (1998). Wie erkennt man eine Fibonacci Zahl 

Platonic Realms. (n.d.). The Fibonacci Sequence. Retrieved January 30, 2010, from Platonic Realms: 

http://www.mathacademy.com/pr/prime/articles/fibonac/index.asp 

Posamentier, A. S., & Lehmann, I. (2007). The Fabulous Fibonacci Numbers. Prometheus Books. 

Prusinkiewicz, P., & Lindenmayer, A. (1991). The Algorithmic Beauty of Plants (1st ed.). Springer. 

Renault, M. (1996). Properties of the Fibonacci Sequence. Retrieved January 30, 2010, from Temple 

University: http://www.math.temple.edu/~renault/fibonacci/thesis.html 

Ribenboim, P. (2000). My Numbers, My Friends. Springer‐Verlag. 

Singh, P. (1986). Acharya Hemachandra and the (so called) Fibonacci Numbers (Math. Ed. Siwan ed., Vol. 

20). 

Singh, P. (1985). The So‐called Fibonacci numbers in ancient and medieval India (Vol. 12). Historia 

Mathematica. 


Sloane, N. J. A001622: Decimal expansion of golden ratio phi (or tau) = (1 + sqrt 5 )/2. AT&T Labs. 

Su, F. E. (n.d.). Fibonacci GCD's, please. Retrieved January 30, 2010, from Mudd Math Fun Facts: 

http://www.math.hmc.edu/funfacts/ffiles/20004.5.shtml 

Tattersall, J. J. (2005). Elementary Number Theory in Nine Chapters (2nd ed.). Cambridge University 

Press. 

Thimbleby, H. (n.d.). “B–” for The da Vinci Code. Retrieved January 30, 2010, from Swansea University: 

http://www.cs.swansea.ac.uk/~csharold/dvc/dvc.pdf 

Vogel, H. (1979). A better way to construct the sunflower head. Mathematical Biosciences , 179–189. 

Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: 

Penguin Books.

fibonacci-numbers

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?