The Fibonacci Sequence in Real Life and its Applications

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ACKNOWLEDGEMENT

I would take this opportunity to thank my research supervisor, family and friends for their support and guidance without which this research would not have been possible

DECLARATION

I, [type your full first names and surname here], declare that the contents of this dissertation/thesis represent my own unaided work, and that the dissertation/thesis has not previously been submitted for academic examination towards any qualification. Furthermore, it represents my own opinions and not necessarily those of the University (Bardhan, 2001, 467).

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TABLE OF CONTENTS

TABLE OF CONTENTS4

INTRODUCTION5

LITERATURE REVIEW9

Fibonacci Sequence In real Life11

The Origin of the Series11

Fibonacci Goes Gold in Art and Architecture12

Fibonacci Spirals in Nature14

Fibonacci For Fun16

Views of Fibonacci dynamics16

Fixed points17

Escape time20

FIBONACCI SEQUENCE IN CLASS ROOM23

Fibonacci numbers and golden ratio and history of mathematics23

Case study24

Participants24

Data collection tools25

The implementation of data collecting tools26

Data analysis26

Results and Discussion26

The findings obtained from the performance test27

The findings obtained from the questionnaire form27

Fibonacci Numbers in Learning30

What If33

A Fibonacci Property35

Fibonacci in the Forest and Students40

Content Background40

Research Background41

Lesson Outline42

Lesson42

Procedure43

Evaluation47

Extensions47

CONCLUSION AND RECOMMENDATION50

INTRODUCTION

Leonardo Pisano (1170-1250), better known by his nickname Fibonacci, was born in Italy but was educated in North Africa where his father, Guilielmo, held a diplomatic post. His father's job was to represent the merchants of the Republic of Pisa who were trading in Bugia, later called Bougie and now called Bejaia (Algeria). Fibonacci was taught mathematics in Bugia and travelled widely with his father and recognized the enormous advantages of the mathematical systems used in the countries they visited (Fig. 1).

Fig. 1. Leonardo Pisano Fibonacci.

Fibonacci ended his travels around the year 1200 and at that time he returned to Pisa. There he wrote a number of important texts which played an important role in reviving ancient mathematical skills and he made significant contributions of his own (Peitgen, 1986).

The most important book of Fibonacci, Liber abaci (the book of calculations), was published in 1202 after Fibonacci's return to Italy. The book was based on the arithmetic and algebra that Fibonacci had accumulated during his travels. The book, which went on to be widely copied and imitated, introduced the Hindu-Arabic place-valued decimal system and the use of Arabic numerals into Europe.

The book begins with the following premise: “The nine Indian figures  are 9,8,7,6,5,4,3,2,1. With these figures, and with the sign 0... any number may be written, as is demonstrated below.”

Although Liber abaci is mainly a book about the use of Arab numerals, it contains also the solution of important problems. In fact, the second section of Liber abaci contains a large collection of problems aimed at merchants. They relate to the price of goods, how to calculate profit on transactions, how to convert between the various currencies in use in Mediterranean countries, and problems which had originated in China. A problem in the third section of Liber abaci led to the introduction of the Fibonacci numbers and the Fibonacci sequence for which Fibonacci is best remembered today: “A certain man puts a pair of rabbits in a place surrounded on all sides by a wall (Dickson, ...