The Fibonacci Sequence
By Leslie Quinn
April 12, 2005
MATH 4010
In the twelfth century, Europe was beginning to emerge, little by little, from
medieval thinking; an age known for a lax in concern for global education and rise in
religious crusades and conquests in general. Great strides were being made in Italy in
education, with the opening of many new institutes of higher learning1, and commerce
was booming in the Eastern hemisphere, especially along the Mediterranean where there
was a commercial revolution2.
While the conflict between the Papacy and the Holy Roman Empire was dieing
down, many Italian cities were left as independent city-states, and turned to trade and
other endeavors to sustain their economies. One such city-state was Pisa3. Pisa is located
conveniently on the Mediterranean which makes it easily accessible for trade by boat and
land. As Pisa was beginning to grow economically due to many great commercial
endeavors, it, as well as other parts of Western Europe, was also beginning to awake from
its intellectual slumber. While the great minds of Euclid and Ptolemy were resurfacing in
England4, in the city-state of Pisa another magnificent mind was being born.
Leonardo of Pisa was born around 1175 AD, though no-one knows for certain the
exact date. He is perhaps best known to mathematicians as Fibonacci, which is short for
Filius Bonacci, meaning son of Bonaccio5 (Bonacci being plural for Bonaccio, which is
assumed to be a family name). Leonardo sometimes referred to himself as Leonardo
Bigollo, which means traveler.6 Not much is known about his father, Guilielmo Bonacci,
but that he benefited occupationally from the commercial revolution by working as a
1
Horadam
bartleby
3
Horadam
4
timeline
5
Knott
6
JJ O’Connor
2
2
customs officer, or state official for merchants of Pisa by overseeing Pisan trade in Bugia,
present day BejaIa, Algeria, in northern Africa. While growing up in Northern Africa
Leonardo’s father instilled the importance of mathematics to him, and thus Leonardo was
educated there in the Hindu-Arabic mathematical system.
Later, it is said, Leonardo traveled across the Mediterranean, and joined up with
many merchants to learn their mathematical systems7. It was there in his youth that he
became fascinated with Hindu-Arabic notation, and its ease of use as compared to the
Roman numeral system that was being used in Pisa, and all other systems he came across.
Upon realizing the advantages to Hindu-Arabic notation and his return to Pisa in 1200, he
began writing texts which would prove to be a huge contribution to mathematics.
Because Leonardo lived before the invention of the printing press, his works were
hand copied, and for this reason only a few exist today. Among his existing works are:
Liber Abaci, 1202, Practica Geometriae, 1220, Flos, 1225, and Liber Quadratorum8.
Leonardo’s most famous and influential book is Liber Abaci, meaning “book of
calculations”. The original was published in 1202, but it is his revision that was
published 26 years later that we know today [7]. At the time Liber Abaci was written,
Roman and Greek mathematics were dominant in Europe, and Fibonacci, as well as some
others, felt that dealing with Roman numerals was extremely cumbersome. Liber Abaci
was intended to introduce Hindu-Arabic numerals to Europe in a manner that would
convince them to convert their current mathematical system. He was not the first to try
and accomplish this huge feat, but he is definitely credited with achieving it. Many
7
8
Knott
JJ O’Connor
3
credit the style in which the materials in Liber Abaci were presented that eventually
inspired a change.
This book consists of fifteen chapters of elementary arithmetic and algebra,
mostly taken from Arabic and Asian sources, in a basic how to manual of Hindu-Arabic
mathematical procedures. It opens:
Begin the first chapter. The nine Hindu figures are 9, 8, 7, 6, 5, 4, 3, 2, 1.
It is shown below (how) to write any number with (just) these nine figures
and the sign 0, which the Arabs called “cipher”[2]
It appears that Leonardo had a book from each of the fields of arithmetic and algebra to
use for reference in his book. His biggest influence was the book Algebra by Abū Kāmil,
written in 900 AD. He used 29 examples from Algebra in his writing of Liber Abaci [2].
The popularity of Liber Abaci was not only due to its persuasive descriptions of
Hindu-Arabic equations, but also due to one of the problems it contained that would end
up having a profound influence on mathematicians and scientists alike. This problem is
referred to as the famous rabbit problem which led to the popularization and global
awareness of the famous Fibonacci Sequence which is the topic of much of the remainder
of this paper. Though Leonardo put little effort into properties of the famous sequence
that bears his name, it has been found to contain in itself a seemingly infinite amount of
importance.
In the third section of Liber Abaci, chapter 12, Leonardo introduces the famous
Rabbit Problem. Most likely this problem was copied out of an Arabic or Eastern text,
but it was this publication that led to its popularity and in turn the popularity of Leonardo
and his Fibonacci Sequence.
4
Rabbit Problem
How Many Pairs of Rabbits Are Created by One Pair in Each Year
A certain man had one pair of Rabbits together in a certain enclosed place, and one
wishes to know how many are created from the pair in one year when it is the nature of
them in a single month to bear another pair, and in the second month those born to bear
also [10]9.
Leonardo’s original solution as translated is:
Because the abovewritten pair in the first month bore, you will double it;
there will be two pairs in one month.
One of these, namely the first, bears in the second month, and thus there are in the
second month 3 pairs;
Of these in one month two are pregnant and in the third month 2 pairs of rabbits are
born, and thus there are five pairs in the month;
…
there will be 144 pairs in this [the tenth] month;
to these are added again the 89 pairs that are born in the eleventh month;
there will be 233 pairs in this month,
To these are still added the 144 pairs that are born in the last month;
there will be 377 pairs, and this many pairs are produced from the abovewritten pair in
the mentioned place at the end of the one year[10] 10.
The following table illustrates population growth of the rabbits in one year’s time:
“Rabbit Problem” as represented in Fibonacci’s Liber Abaci translated into English for the first time by L
E Sigler (2002), and reproduced in [10]
9
5
Month
Original
January
February
March
April
May
June
July
August
September
October
November
December
Grown
Rabbits
1
1
2
3
5
8
13
21
34
55
89
144
233
New Babies
Total
Pairs
1
2
3
5
8
13
21
34
55
89
144
233
377
0
1
1
2
3
5
8
13
21
34
55
89
144
The Column to the far right, total pairs, represents what would later be labeled the
Fibonacci Sequence by Edouard Lucas in 1878. Since this sequence has been named it
has been found that this it was also discovered by such men as Euler, Virgil, and Johann
Kepler, all independent of each other.
At first glance one may wonder what is so special about this sequence and what
does it all mean? These questions will surely be answered in the following pages.
The Fibonacci Sequence in its original notation is as follows:
f 0f0=11,
f1  2 ,
f2  3,
f3  5,
f4  8 ,
f 5  13 ,
f 6  21 ,
and so on10…
By inspection you will notice that f n  2  f n 1  f n , for n ≥ 0. This is called a recursive
sequence, which means that each number in the sequence (from f 3 and on in this case) is
the sum of the previous two numbers. The Fibonacci Sequence is said to be the first
recursive sequence ever discovered.
10
Modern notation of the Fibonacci sequence is: 0,1,1,2,3,5,8,13,21…[7]
6
The rabbit problem is essentially asking the solver to find how many pairs of
rabbits will exist at the end of the year, or the value of f 12 which, as Fibonacci states and
is shown in the table, is 377. This problem was most likely originally tediously solved by
Fibonacci, in the arithmetic fashion shown above. However, since Fibonacci there has
been a lot of attention drawn to this sequence, and thus there now exists a formula to find
the nth Fibonacci number. This formula was discovered by a few mathematicians
including Euler and Bernoulli independently of each other, but today it is called Binet’s
formula:
fn 
(1  5) n  (1  5) n
2n 5
It can also be written in terms of  and  , where  =
n  n
fn 
 
1 5
1 5
and  =
:
2
2
[3]
But to better understand this formula let us discuss the value of  , commonly referred to
as  , also known as the golden ratio, golden mean, or golden section.
Phi
First we should observe that  and  are the roots to the equation:
x 2  x  1 , or x 2  x  1  0 (1)
One of the astonishing properties of the Fibonacci Sequence is that the sequence
of ratios of two successive terms converges, meaning that as you calculate the ratios of
the greater term to the previous lesser term these numbers become arbitrarily close to one
number, and that number is  . Observe:
7
Fractional Ratios:
f n 1
2 3 5 8 13 21 34 55 89 144 233 377 610
, , , ,
,
,
,
,
,
,
,
,
, …,
1 2 3 5
8
21 34 55 89 144 233 377
13
fn
Decimal Ratios:
2, 1.5, 1.6 , 1.6, 1.625, 1.615384615, …, 1.618037135, …, 1.618033989, …
Approximation of phi:
1 5
 1.618033989
2
The last number in the decimal sequence is the ratio of the 24th term and the 23rd
term in the Fibonacci Sequence and approximates  correctly to nine decimal places.
This number, as stated previously, is known as the golden ratio. Even as far back as
ancient Greece  :1 has been thought of as the most aesthetically pleasing ratio in art and
architecture, as well as in nature[14].
It is also interesting to observe in what other ways the Fibonacci sequence, and
other recursive sequences relate to equation (1) and the golden ratio:
Let us define a recursive function:
x n  2  x n 1  x n , for n  0
Say we divide through by a term such as x n , then we have:
x n  2 x n 1  x n

 x2  x 1
xn
xn
very interesting[17].
Not only is it fascinating that the Fibonacci sequence converges to the golden
ratio, which is a root of the above equation, but there are many fascinating properties of
8
 that are useful in proving Binet’s formula (I will use  in place of  in the following
properties to better relate to the form of Binet’s formula that will be used in the proof):
 2  1   (2)
 2  1   (3)
  1   (4)
 
1

(5)
    1 (6)
    5 (7)
 *   1 (8)
and so on… (There are many other interesting properties of  which are not addressed
here).
Now that we know some information about  and  we can see how Binet and
others related these terms to the Fibonacci sequence.
Proving Binet Right
Binet’s formula can be proven in many different ways. One of the most basic
ways is by using the principle of mathematical induction. To prove the formula in this
way let us start out by defining the Fibonacci sequence in modern notation, so:
f 0  0, f 1  1, f 2  2,..., f n  f n 1  f n  2 for all n  2 .
Theorem: f n 
1 5
1 5
n  n
, for all n  IN , where  =
and  =
2
2
 
9
0  0
1  1
Proof: Let n = 0, then f 0 
 0 . Now let n = 1, so f 1 
 1 . Both of
 

these are correct, so the hypothesis holds for n  0,1 . Now assume that the hypothesis
holds for all n  k , so:
fk 
f k 1
k  k
,

 k 1   k 1

 
(we are assuming that these are correct)
Using the definition of a recursive sequence we gather:
f k 1
 k 1   k 1  k   k


 
 
 k   k   k 1   k 1


 k   k 1   k   k 1



 k 1 (  1)   k 1 (  1)


 k 1 2   k 1 2
 
 k 1   k 1

 f k 1

So the hypothesis holds for k  1 and by the PMI for all n  IN .
Work In Progress
10
Bibliography
[1] Benjamin, Arthur T., and Jennifer J. Quinn. Proofs That Really Count: The Art of
Combinatorial Proof. United States of America: The Mathematical Association of
America, 2003. 1-16.
[2] Bridger, Clyde A. "Leonardo, His Rabbits, and Other Curiosa." The Two-Year
College Mathematics Journal 6.1 (1975): 14-20.
[3] Eggen, Maurice, Douglas Smith, and Richard St. Andre. A Transition to Advanced
Mathematics. 5th ed. United States: Brooks/Cole, 2001. 106.
[4] Garland, Trudi Hammel. Fascinating Fibonaccis: Mystery and Magic in Numbers.
New Jersey: Dale Seymour Publications, 1987. 1-95.
[5] Hendel, Russell Jay. "Approaches to the Formula for the nth Fibonacci Number."
The College Mathematics Journal 25.2 (1994): 139-142.
[6] Horadam, A F. "A Generalized Fibonacci Sequence." The American Mathematical
Monthly 68.5 (1961): 455-459.
[7] Horadam, A F. "Eight hundred years young." The Australian Mathematics Teacher
31. (1975): 123-134.
[8] Karpinski, Louis C. "Algebra." Modern Language Notes 28.3 (1913)
[9] Katz, Victor J. A History of Mathematics: An Introduction. 2nd ed. Massachusetts:
Addison Wesley Longman, Inc., 1998. 307-310.
[10] Knott, Dr Ron. Who Was Fibonacci? 1996. University of Surrey, Dept. of
Mathematics and Statistics. Mar. 2005
<http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibBio.html>.
[11] Koshy, Thomas. Fibonacci and Lucas Numbers With Applications. New York: John
Wiley & Sons, Inc., 2001. 1-196.
[12] Lovasz, L, J Pelikan, and K Vesztergombi. Discrete Mathematics: Elementary and
Beyond. New York: Springer-Verlag, 2003. 65-124.
[13] O'Connor, J J., and E F. Robertson. Leonardo Pisano Fibonacci. Oct. 1998. Mac
Tutor History of Mathematics Archives. Mar. 2005 <http://www-groups.dcs.stand.ac.uk/~history/Mathematicians/Fibonacci.html>.
11
[14] Renault, Marc. Properties of the Fibonacci Sequence Under Various Moduli. 1996.
Wake Forest University. Apr. 2005
<http://www.math.temple.edu/~renault/fibonacci/thesis.html>.
[15] Roberts, Andrew. Time line for the history of science and social science. Mar. 1999.
Middlesex University . Apr. 2005
<http://www.mdx.ac.uk/www/study/sshtim.htm#Ptolemy>.
[16] The Golden Mean. The NeWeb Group. Apr. 2005
<http://www.newebgroup.com/academy/phi/fib1.htm>.
[17] The Fibonacci Sequence . 2004. Math Academy Online. Apr. 2005
<http://www.mathacademy.com/pr/prime/articles/fibonac/>.
12