Leonardo Fibonacci Leonardo Fibonacci was born in Pisa, Italy around 1175 to Guilielmo Bonacci. Leonardo's father was the secretary of the Republic of Pisa and directed the Pisan trading colony. His father intended on Leonardo becoming a merchant. His father enlisted him in the Pisan Republic, sending him to various countries. As Leonardo continued to travel with his father, he acquired mathematical skills while in Bugia. Fibonacci continued to study throughout his travels, which ended around the year 1200. Leonardo began writing books on number theory, practical problems of business mathematics, surveying, advanced problems in algebra and recreational mathematics. Leonardo's recreational problems became known as story problems and became mental challenges in the 13th century. Of all the books he wrote we still have copies of Liber abbaci (1202), Practica geometriae (1220), Flos (1225), and Liber Quadratorum. Sadly his books on commercial arithmetic Di minor guisa is lost as well as his commentary on Book X Euclid's Elements. One of Leonardo's contributions to mathematics was his introducing the Decimal Number system into Europe. He was one of the first people to introduce the HinduArabic number system into Europe. Fibonacci also introduced the Decimal Positional System, which originated from India and Arabia. Fibonacci wrote story problems in his book, Liber abbaci. Examples of those problems are, "A spider climbs so many feet up a wall each day and slips back a fixed number each night, how many days does it take him to climb the wall. These problems became quite popular. Another accomplishment was his forming the Fibonacci Series. It is a series of number in which each member is the sum of the two preceding numbers. For example, a series beginning 0, 1 continues as 1, 2, 3, 5, 8, 13, 21, and so forth. The exact period of this discovery is not known. Leonardo was a bright man, but left much of his solutions to his...
...patterns in nature focusing in the Fibonacci sequence as a main and looking for angles. What was first done was to count a pine cone’s pieces, a flower’s petals, a celery, and grapes to find the Fibbonacci sequence which not found only on the celey and on the flower, elsewhere the Fibonacci was there.
After finishing the experiment I started noticing more patterns relating to the Fibonacci sequence. For example, in a tree you start counting by the tree trunk; if you start going up there are two branches with three leaves, then five, them eight until there is no more to count you go to the next branch and do the same thing until you reach the top of the tree. I think math can be found practically everywhere you look if you can find the right sequence. When you are looking for patterns there is at least one for anything. Math can be very important and people can start caring more about it if they know it is all around them.
Introduction
In my science fair project I am going to try to find mathematical patterns in nature. The main pattern I am looking for is for the Fibonacci sequence, which consist of the numbers in the following order: 1,1,2,3,5,8,13,21,34… so forth and so on always adding the number before. I will try my experiments in trees, pine combs, flowers, fruits, seashells, and vegetables. I think that the Fibonacci sequence will only be found in a pine comb or in a flower. I am going...
...Leonardo Pisano(11701250) was an Italian number theorist, who was considered to be one of the most talented mathematicians in the Middle Ages.
However, He was better known by his nickname Fibonacci, as many famoustheorems were named after it. In addition to that, Fibonacci himself sometimes used the name Bigollo, which means goodfornothing or a traveller. Thisis probably because his father held a diplomatic post, and Fibonacci travelledwidely with him. Although he was born in Italy, he was educated in NorthAfrica and he was taught mathematics in Bugia. While being a 'bigollo', hediscovered the enormous advantages of the mathematical systems used in thecountries he visited.
Fibonacci's contributions to mathematics are remarkable. Even in the worldtoday, we still make daily use of his discovery. His most outstanding contributionwould be the replacement of decimal number system. Yet, few people realizedit. Fibonacci had actually replaced the old Roman numeral system with theHinduArabic numbering system, which consists of HinduArabic(09) symbols.
There were some disadvantages with the Roman numeral system: Firstly, it didnot have 0's and lacked place value; Secondly, an abacus was usually requiredwhen using the system. However, Fibonacci saw the superiority of using HinduArabic system and that is the reason why we have our numbering system today.
1He had included the explanation of...
...Leonardo of Pisa or Fibonacci and the Issue of Moneylenders
NFaly Konate
Texas A&M University – Central Texas
FIN 590
Dr. Mary Kelly
Summer 2012
Northern Italy in the early thirteen century was a land subdivided into multiple feuding citystates. Among the many remnants of defunct Roman Empire was a numerical system (I, ii, iii, iv…) singularly ill suited to complex mathematical calculation, let alone the needs of commerce. Nowhere was this more of a problem than in Pisa, where merchants also had to contend with seven different forms of coinage in circulation. By comparison, economical life in the Eastern world was far more advanced, just as it had been in the time of Charlemagne. To discover modern finance, Europe needed to import it. In this, a young mathematician called Leonardo of Pisa, or Fibonacci played a crucial role.
LeonardoFibonacci also known as Leonardo Pisano, Leonardo of Pisa, was the greatest European mathematician of the middle ages. He was born in Pisa in Italy circa 1170 and died sometime after 1240. Leonardo’s father, Gugliemo, was a customs official and engaged in commerce representing Pisa at Bougie on the north coast of Africa. Young Leonardo consequently received a Moorish education as well as the traditional European education and was introduced to HinduArabic numbers. Later on, he traveled about the...
...The Discovery of the Fibonacci Sequence
A man named Leonardo Pisano, who was known by his nickname, "Fibonacci", and named the series after himself, first discovered the Fibonacci sequence around 1200 A.D. The Fibonacci sequence is a sequence in which each term is the sum of the 2 numbers preceding it. The first 10 Fibonacci numbers are: (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89). These numbers are obviously recursive.
Fibonacci was born around 1170 in Italy, and he died around 1240 in Italy, but the exact dates of his birth and death are not known. He played an important role in reviving ancient mathematics and made significant contributions of his own. Even though he was born in Italy, he was educated in North Africa where his father held a diplomatic post. He published a book called Liber abaci, in 1202, after his return to Italy and it was in this book that the Fibonacci numbers were first discussed. It was based on bits of Arithmetic and Algebra that Fibonacci had accumulated during his travels with his father. Liber abaci introduced the HinduArabic placevalued decimal system and the use of Arabic numerals into Europe. Though people were interested, this book was somewhat controversial because it contradicted some of the foremost Roman and Grecian Mathematicians of the time, and even proved many of their calculations to be false....
...Leonardo Pisano (Fibonacci)
0,1,1,2,3,5,8,13… Does this sequence look familiar? If you thought for one second that this was the Fibonacci sequence then you’re right! The Fibonacci sequence was one of the few things created by Leonardo Pisano, considered the greatest European mathematician in the middle ages, that was a significant contribution to math. In order to gain a better understanding of the life ofLeonardo Pisano, better known as Fibonacci, and his contributions to the mathematical society let us first take a look at the brief history of what is known of Leonardo from birth to death.
Leonardo Pisano was born in Pisa, Italy roughly around 1175 as the son of Guglielmo Bonaccio. Guglielmo worked as a secretary and at a diplomatic post in numerous factories located on the southern and eastern coasts of the Mediterranean for merchants of Pisa and as a result Leonardo was educated in the Algerian city of Bejaia (then known as Bougie and Bugia) instead of in Italy. It was there where he was taught mathematics. Aside from this not much is known about his childhood. He was later able to tour the Mediterranean area with his father up until 1200 when he decided to stop. During the time spent in his travels, Leonardo learned about the advantages of the mathematical systems of each of the countries that he visited with...
...LeonardoFibonacci was born around 1170 A.D., and died around 1250 A.D. He was born in Pisa, Italy and died there too. Leonardo’s mom was Alessandra, and she died when he was nine. His father was Guglielmo Bonacci, who directed a trading post Bugia, Barbary. As a young boy, Leonardo traveled there to help him, and that’s where he learned about the HinduArabic numeral system. He recognized that arithmetic with HinduArabic numerals is simpler and more efficient that with Roman numerals and so he traveled throughout the Mediterranean world to study under the leading Arab mathematicians of the time. Leonardo returned from his travels around 1200 and in 1202, age 32, he published Liber Abaci. Through the Liber Abaci he introduced HinduArabic numerals to Europe.
Liber Abaci is a book that LeonardoFibonacci wrote in 1202. In it Fibonacci introduces the socalled modus Indorum (method of the Indians), today known as the Arabic numerals. It shows the practical importance of the new numeral system, using lattice multiplication and Egyptian fractions, by applying it to commercial bookkeeping, conversion of weights and measures, the calculation of interest, moneychanging, and other applications. Liber Abaci also posed, and solved, a problem involving the growth of a hypothetical population of rabbits based on idealized assumptions. The solution, generation by generation, was a...
...Fibonacci's Rabbits
The original problem that Fibonacci investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances.
Suppose a newlyborn pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never dieand that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was...
How many pairs will there be in one year?
1. At the end of the first month, they mate, but there is still one only 1 pair.
2. At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.
3. At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.
4. At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.
The number of pairs of rabbits in the field at the start of each month is 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
Another view of the Rabbit's Family Tree:
  
Both diagrams above represent the same information. Rabbits have been numbered to enable comparisons and to count them, as follows:
* All the rabbits born in the same month are of the same generation and are...
...Anatolia College 
Mathematics HL investigation

The Fibonacci sequence 
Christos Vassos

Introduction
In this investigation we are going to examine the Fibonacci sequence and investigate some of its aspects by forming conjectures and trying to prove them. Finally, we are going to reach a conclusion about the conjectures we have previously established.
Segment 1: The Fibonacci sequence
The Fibonacci sequence can be defined as the following recursive function:
Fn=un1+ un2
Where F0=0 and F1=1
Using the above we can find the first eight terms of the sequence. An example of calculations is given below:
F2=F1F0F2=1+0=1
We are able to calculate the rest of the terms the same way:
F0  F1  F2  F3  F4  F5  F6  F7 
0  1  1  2  3  5  8  13 
Segment 2: The Golden ratio
In order to define the golden ratio we need to examine the following sketch:
The line above is divided into two segments in such a way that ABAP=APPB
The ratio described above is called the golden ratio.
If we assume that AP=x units and PB=1 units we can derive the following expression:
x+1x=x1
By solving the equation x2x1=0 we find that: x=1+52
Segment 3: Conjecture of φn
In this segment we examine the following geometric sequence:
φ,φ2,φ3…
Since x=1+52 can simplify φ by replacing the value of x to the formula of the golden ratio we discussed before. Therefore:
φ=x+1x...