FIBONACCI SEQUENCE
Melissa Abeyta
May 5, 2005
The Fibonacci sequence is extremely important and appears in many different places. This paper examines many aspects of the sequence. It is interesting to note that the sequence was not that important to Fibonacci however, since it has been discovered there has been a tremendous amount of research done on it (Evansville website). There is a Fibonacci Association in San Jose, California that was established in 1963 and the Fibonacci Quarterly is a journal that is devoted to the study of integers with special properties. It is amazing how one problem that surfaced in 1202 could make such an impact on the world.
HISTORY OF FIBONACCI AND THE FIBONACCI SEQUENCE
Fibonacci’s real name was Leonardo Pisano. He was born in Italy around 1170. Little is known about his life and what we do know is found in a few sentences in the 1228 edition of his famous Liber Abbaci (sometimes spelt Liber Abaci) (Evansville website). We know that his father, Guilielmo, held a diplomatic post in the Republic of Pisa. “His father’s job was to represent the merchants of the Republic of Pisa who were trading in Bugia,” (St Andrews website). Bugia is now called Bejaia and is located in Northeastern Algeria. Fibonacci went there with his father and was taught mathematics and accounting. He traveled with his father and visited Egypt, Syria, Greece, Sicily, and Provence. He was able to recognize “the enormous advantages of the mathematical systems used in the countries he visited,” (St Andrews website).
Around the turn of the century Fibonacci returned to Pisa and for the next twenty-five years he wrote books. The texts that he wrote “played an important role in reviving ancient mathematical skills and he made significant contributions of his own,” (St Andrews website). Only five works by Fibonacci have been preserved:
1. Liber Abbaci (1202, 1228)
2. Practica Geometriae (1220/1221)
3. A writing entitled Flos (1225)
4. An undated letter to Theodorus, the imperial philosopher
5. Liber quadratorum (1225)
We know that he wrote some other books, however they were lost. He wrote a book on commercial arithmetic, Di mino guisa, that was lost as well as his commentary on Book X of Euclid’s Elements “which contained a numerical treatment of irrational numbers which Euclid had approached from a geometric point of view,” (St Andrews website).
Fibonacci was well known for being a great mathematician and caught the attention of the Holy Emperor, Frederick II. The emperor was called “Stuper Mundi” (Wonder of the World). Frederick was also the King of the Two Sicilies and was considered to be one of the most remarkable men on the Middle Ages. He “encouraged learning and scholarship of every kind, having a special interest in mathematics and science,” (Evansville website). [Evansville website]
“Fibonacci corresponded with two of the Emperor’s scholars, Master Theodore and Michael Scott (Scotus) (whom Dante, in his Divina Commedia, consigned to hell as a wizard for perpetuating “magical frauds”),” (Evansville website). Fibonacci and Frederick met at the Emperor’s palazzo in Pisa. Master John of Palemo, who was a scholar, proposed mathematical questions for Fibonacci to solve. Some writers state that a mathematical tournament between Fibonacci and other mathematicians took place, but this is not likely. “At the time of his meeting with Frederick in the 1220’s, Fibonacci was probably at the height of his prowess,” (Evansville website). [Evansville website]
It is not known when Fibonacci died or how he died. The last time he was mentioned in a document was in 1240. He was a remarkable mathematician and has been called “the first great mathematician of the Christian West,” (Dictionary of Scientific Biography, 1971, p. 604).
The famous Fibonacci sequence came out of Liber Abbaci. The first edition was written in 1202 after he returned to Italy from traveling. It was based on arithmetic and algebra that he had accumulated during his travels with his father. He released a revised edition in 1228 and dedicated it to Michael Scott (Scotus), one of Frederick II scholars. It has been stated that the 2nd edition added new material and removed material that was superfluous (Dictionary of Scientific Biography, 1971). The book was divided into 15 chapters, which were analyzed in four sections.
In section 1 (Chapters 1-7), Fibonacci introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe. He began Liber Abacci with the following statement that was simple but profound.
“The nine Indian figures are: 9 8 7 6 5 4 3 2 1 With these nine figures, and with the sign 0…any number may be written, as is demonstrated below” (Evansville website). The new numbers were favorable in comparison to the traditional Roman numerals (Sigler, 2002). He also introduced the fraction bar. In this section he developed rules for the factoring of fractions into sums of unit factors. “Numerous tables (for multiplication, prime numbers, factoring numbers, etc) complete the text,” (Dictionary of Scientific Biography, 1971, p. 606).
Section 2 (Chapters 8-11), contained problems revolving around merchants, such as price of goods, calculation of profits, how to convert between the various currencies used in the Mediterranean countries, and some Chinese mathematical problems (St Andrews website).
Section 4 (Chapter 14 and 15) showed Fibonacci “to be a master in the application of algebraic methods and an outstanding student of Euclid,” (Dictionary of Scientific Biography, 1971, p. 607). Chapter 14 begins with a few formulas of general arithmetic and is devoted to calculations with radicals (Dictionary of Scientific Biography, 1971). Chapter 15 shows how Leonardo had complete control over the geometrical and algebraic methods for solving quadratic equations and had great skill in using them in applied problems (Dictionary of Scientific Biography, 1971).
Section 3 (Chapters 12 and 13) is being taken out of order for a reason. This section gives us the Fibonacci sequence. It is the most extensive section and contains many questions and puzzles. In Chapter 12 Fibonacci states the following problem:
“A certain man put a pair of rabbits in a place surrounded 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” (Evansville website).
The problem has been stated a little differently in many sources. Here is a more simple explanation of the problem.
- Start with a pair of rabbits (one male and one female)
- Assume that all months are of equal length
- Rabbits begin to reproduce two months after their own birth
- After reaching two months of age, each pair produces a mixed pair (one male and one female) and then another mixed pair each month thereafter
- No rabbit dies
How many pairs of rabbits will there be after one year? [Morris, n.d.]
The following figure is a visual representation of the rabbits being added each month.
Figure 1.1 taken from the Rabbits, Cows, and Bees Family Tree section on the University of Surrey website.
The following is a table representation of the results of the problem.
Month / Pairs of Rabbit1 / 1
2 / 1
3 / 2
4 / 3
5 / 5
6 / 8
7 / 13
… / …
The solution of this problem is the famous Fibonacci sequence.
I found a discrepancy in my research. One source said, “although it is almost certain that he knew, Fibonacci never wrote that each term was found by adding two previous terms,” (Ballew, n.d.). It states that the first record of such a statement was made almost 400 years after Fibonacci by Kepler (Ballew, n.d.). Contradicting this is the book A History of Mathematics written by Victor J. Katz in 1998 which states that Fibonacci listed the sequence in the margin and “notes that each number is found by adding the two previous numbers and ‘thus you can do it in order for an infinite number of months’,” (p. 309). I believe the Katz book to be more reliable and think that he was well aware of the pattern that the problem produced. Fibonacci was a highly regarded mathematician and it seems unreasonable to believe that he did not see the pattern and note it.
The resulting Fibonacci sequence was the first recurrent series (Dictionary of Scientific Biography, 1971). Each number within the sequence is called a Fibonacci number. The recursive definition for producing Fibonacci numbers is:
Fn = F(n-1) + F(n-2) where n>2
The following is a proof of the formula.
Proof: Let S = {n Fn = F(n-1) + F(n-2) where n>2}
Note that 3 S
F3 = F2 + F1 = 1+1 = 2
Suppose that {2,….,m}S for m>2
Consider Fm+1
In the m+1 stage we need to have all of the rabbits for the previous stage because they are still alive and then we need to add all of the rabbits from two months previous because they are now able to produce children and represent the number of newborns
Therefore Fm+1 = Fm + Fm-1
Thus, m+1 S, so by the PCI S =
FIBONACCI SEQUENCE AND THE GOLDEN RATIO
The story of the golden ratio begins with Ptolemy I. Ptolemy I was a successor to Alexander the Great who rose to power in 323 B.C. He established a school known as the ‘Museum’ in Alexandria, Egypt, which was founded around 300 B.C. The “father of formal deductive geometry,” Euclid, was one of the teachers (Livio, 2003, p. 52).
Euclid defined geometry and number theory in his famous book called the Elements. In this book, Euclid stated that a line can be divided into what he called its “extreme and mean ratio” (Livio, 2003, p. 52). In Book IV of the Elements, “a straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segments, so is the greater to the lesser,” (Livio, 2003, p. 52) Simply stated, “Euclid proposed the possibility of dividing a line into two segments such that the ratio of the whole line to the larger segment is the same as the ratio between the larger and the smaller segments. If we call the endpoints of Euclid’s straight line A and C, and its dividing point B, then mathematically the ratio of the line AC to its larger segment, AB, is equal to the ratio of line segments BC to AB,” (Livio, 2003, p. 53) We can write it numerically as:
AB + BC = AC and AB/AC = BC/AB
The following is a visual representation. Notice that we can also say that the line is 1, where the line AB equals b and the line BC equals 1-b.
This line was taken from the ‘What is the golden section (or Phi)?’ section of the University of Surrey website.
From this we get b/1 = (1-b)/b
b2 = 1-b or b2 + b –1 = 0
This result is a quadratic equation, which can be solved using the quadratic formula
x = (-b +- b2 – 4ac)/2a
Using the quadratic equation we get the following two answers for x.
x = (1 + 5 )/2 or (1 – 5 )/2
The values of these two numbers are 1.6180339887 and –0.618033987. The first number is called the golden ratio or Phi. The second value is called – phi. We use the first letter of each to tell us which one is bigger. Phi is 1.618… and – phi –0.618… [Knotts, 2004, What is the golden section (or Phi)?]
For centuries following Euclid’s publication of Elements mathematicians found it difficult to find an appropriate name to describe the ratio. In 1509, Luca Pacioli called the ratio the “divine proportion” in a book he had published (Livio, 2003). The book contains drawings made by Leonardo da Vinci. “It is probably Leonardo who first called it the section aurea (which is Latin for the golden ratio),” (Knotts, 2004, What is the golden section (or Phi)?). Finally, the 20th century mathematician Mark Burr gave the ratio the designation Phi. “Phi is the first letter in the name of the Greek sculptor Phidias (ca 490 – 430 B.C.) who many people believe used the golden ratio in proportioning his sculptures,” (Livio, 2003, p. 53)
The golden ratio comes up in more than mathematics and art. We will see later that the golden ratio, as well as the Fibonacci numbers, appears in many aspects of nature.
We can find the golden ratio in the Fibonacci sequence as well by taking the ratio of two successive numbers in Fibonacci’s sequence. By dividing each Fibonacci number by the previous number we get the following:
1/1 = 1 2/1 = 2 3/2 = 1.5 5/3 = 1.6666 8/5 = 1.6 13/8 = 1.625
21/13 = 1.61538
We can see what is happening if we plot the points on a graph.