The Fabulous Fibonacci Numbers - Hardcover

Posamentier, Alfred S.; Lehmann, Ingmar

Hardcover

ISBN 10: 1591024757 ISBN 13: 9781591024750

Verlag: Prometheus Books, 2007

Inhaltsangabe

The most ubiquitous, and perhaps the most intriguing, number pattern in mathematics is the Fibonacci sequence. In this simple pattern beginning with two ones, each succeeding number is the sum of the two numbers immediately preceding it (1, 1, 2, 3, 5, 8, 13, 21, ad infinitum). Far from being just a curiosity, this sequence recurs in structures found throughout nature - from the arrangement of whorls on a pinecone to the branches of certain plant stems. All of which is astounding evidence for the deep mathematical basis of the natural world.

With admirable clarity, two veteran math educators take us on a fascinating tour of the many ramifications of the Fibonacci numbers. They begin with a brief history of a distinguished Italian discoverer, who, among other accomplishments, was responsible for popularizing the use of Arabic numerals in the West. Turning to botany, the authors demonstrate, through illustrative diagrams, the unbelievable connections between Fibonacci numbers and natural forms (pineapples, sunflowers, and daisies are just a few examples).

In art, architecture, the stock market, and other areas of society and culture, they point out numerous examples of the Fibonacci sequence as well as its derivative, the "golden ratio." And of course in mathematics, as the authors amply demonstrate, there are almost boundless applications in probability, number theory, geometry, algebra, and Pascal's triangle, to name a few.

Accessible and appealing to even the most math-phobic individual, this fun and enlightening book allows the reader to appreciate the elegance of mathematics and its amazing applications in both natural and cultural settings.

Über diesen Titel

Über die Autorin bzw. den Autor

Alfred S. Posamentier is dean of the School of Education and professor of mathematics education at Mercy College in Dobbs Ferry, New York. Previously, he had the same positions at the City College of the City University of New York for forty years.  He has published over fifty-five books in the area of mathematics and mathematics education, including  Pi: A Biography of the World's Most Mysterious Number (with Ingmar Lehmann). 
 
Ingmar Lehmann is retired from the mathematics faculty at Humboldt University in Berlin. For many years he led the Berlin Mathematics Student Society for gifted secondary-school students, with which he is still closely engaged today.  He is the coauthor with Alfred S. Posamentier of The Secrets of Triangles, The Glorious Golden Ratio, and three other books.

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

The (Fabulous) FIBONACCI Numbers

By Alfred S. Posamentier Ingmar Lehmann

Prometheus Books

Copyright © 2007 Alfred S. Posamentier
All right reserved.
ISBN: 978-1-59102-475-0

Contents

Acknowledgments........................................................................................................9Introduction...........................................................................................................11Chapter 1: A History and Introduction to the Fibonacci Numbers........................................................17Chapter 2: The Fibonacci Numbers in Nature............................................................................59Chapter 3: The Fibonacci Numbers and the Pascal Triangle..............................................................77Chapter 4: The Fibonacci Numbers and the Golden Ratio.................................................................107Chapter 5: The Fibonacci Numbers and Continued Fractions..............................................................161Chapter 6: A Potpourri of Fibonacci Number Applications...............................................................177Chapter 7: The Fibonacci Numbers Found in Art and Architecture........................................................231Chapter 8: The Fibonacci Numbers and Musical Form.....................................................................271Chapter 9: The Famous Binet Formula for Finding a Particular Fibonacci Number.........................................293Chapter 10: The Fibonacci Numbers and Fractals.........................................................................307Epilogue...............................................................................................................327Afterword by Herbert A. Hauptman.......................................................................................329Appendix A: List of the First 500 Fibonacci Numbers, with the First 200 Fibonacci Numbers Factored.....................343Appendix B: Proofs of Fibonacci Relationships..........................................................................349References.............................................................................................................371Index..................................................................................................................375

Chapter One

A History and Introduction to the Fibonacci Numbers

With the dawn of the thirteenth century, Europe began to wake from the long sleep of the Middle Ages and perceive faint glimmers of the coming Renaissance. The mists rose slowly as the forces of change impelled scholars, crusaders, artists, and merchants to take their tentative steps into the future. Nowhere were these stirrings more evident than in the great trading and mercantile cities of Italy. By the end of the century, Marco Polo (1254-1324) had journeyed the Great Silk Road to reach China, Giotto di Bondone (1266-1337) had changed the course of painting and freed it from Byzantine conventions, and the mathematician Leonardo Pisano, best known as Fibonacci, changed forever Western methods of calculation, which facilitated the exchange of currency and trade. He further presented mathematicians to this day with unsolved challenges. The Fibonacci Association, started in 1963, is a tribute to the enduring contributions of the master.

Leonardo Pisano-or Leonardo of Pisa, Fibonacci1-his name as recorded in history, is derived from the Latin "filius Bonacci," or a son of Bonacci, but it may more likely derive from "de filiis Bonacci," or family of Bonacci. He was born to Guilielmo (William) Bonacci and his wife in the port city of Pisa, Italy, around 1175, shortly after the start of construction of the famous bell tower, the Leaning Tower of Pisa. These were turbulent times in Europe. The Crusades were in full swing and the Holy Roman Empire was in conflict with the papacy. The cities of Pisa, Genoa, Venice, and Amalfi, although frequently at war with each other, were maritime republics with specified trade routes to the Mediterranean countries and beyond. Pisa had played a powerful role in commerce since Roman times and even before as a port of call for Greek traders. Early on it had established outposts for its commerce along its colonies and trading routes.

In 1192 Guilielmo Bonacci became a public clerk in the customs house for the Republic of Pisa, which was stationed in the Pisan colony of Bugia (later Bougie, and today Bejaia, Algeria) on the Barbary Coast of Africa. Shortly after his arrival he brought his son, Leonardo, to join him so that the boy could learn the skill of calculating and become a merchant. This ability was significant since each republic had its own units of money and traders had to calculate monies due them. This meant determining currency equivalents on a daily basis. It was in Bugia that Fibonacci first became acquainted with the "nine Indian figures," as he called the Hindu numerals and "the sign 0 which the Arabs call zephyr." He declares his fascination for the methods of calculation using these numerals in the only source we have about his life story, the prologue to his most famous book, Liber Abaci. During his time away from Pisa, he received instruction from a Muslim teacher, who introduced him to a book on algebra titled Hisb al-jabr w'almuqablah by the Persian mathematician al-Khowarizmi (ca. 780-ca. 850), which also influenced him.

During his lifetime, Fibonacci traveled extensively to Egypt, Syria, Greece, Sicily, and Provence, where he not only conducted business but also met with mathematicians to learn their ways of doing mathematics. Indeed Fibonacci sometimes referred to himself as "Bigollo," which could mean good-for-nothing or, more positively, traveler. He may have liked the double meaning. When he returned to Pisa around the turn of the century, Fibonacci began to write about calculation methods with the Indian numerals for commercial applications in his book, Liber Abaci. The volume is mostly loaded with algebraic problems and those "real-world" problems that require more abstract mathematics. Fibonacci wanted to spread these newfound techniques to his countrymen.

Bear in mind that during these times, there was no printing press, so books had to be written by the hands of scribes, and if a copy was to be made, that, too, had to be hand written. Therefore we are fortunate to still have copies of Liber Abaci, which first appeared in 1202 and was later revised in 1228. Among Fibonacci's other books is Practica geometriae (1220), a book on the practice of geometry. It covers geometry and trigonometry with a rigor comparable to that of Euclid's work, and with ideas presented in proof form as well as in numerical form, using these very convenient numerals. Here Fibonacci uses algebraic methods to solve geometric problems as well as the reverse. In 1225 he wrote Flos (on flowers or blossoms) and Liber quadratorum (or "Book of Squares"), a book that truly distinguished Fibonacci as a talented mathematician, ranking very high among number theorists. Fibonacci likely wrote additional works that are lost. His book on commercial arithmetic, Di minor guisa, is lost, as is his commentary on Book X of Euclid's Elements, which contained a numerical treatment of irrational numbers as compared to Euclid's geometrical treatment.

The confluence of politics and scholarship brought Fibonacci into contact with the Holy Roman Emperor Frederick II (1194-1250) in the third decade of the century. Frederick II, who had been crowned king of Sicily in 1198, king of Germany in 1212, and then crowned Holy Roman Emperor by the pope in St. Peter's Cathedral in Rome (1220), had spent the years up to 1227 consolidating his power in Italy. He supported Pisa, then with a population of about ten thousand, in its conflicts with Genoa at sea and with Lucca and Florence on land. As a strong patron of science and the arts, Frederick II became aware of Fibonacci's work through the scholars at his court, who had corresponded with Fibonacci since his return to Pisa around 1200. These scholars included Michael Scotus (ca. 1175-ca. 1236), who was the court astrologer and to whom Fibonacci dedicated his book Liber Abaci; Theodorus Physicus, the court philosopher; and Dominicus Hispanus, who suggested to Frederick II that he meet Fibonacci when Frederick's court met in Pisa around 1225. The meeting took place as expected within the year.

Johannes of Palermo, another member of Frederick II's court, presented a number of problems as challenges to the great mathematician Fibonacci. Fibonacci solved three of these problems. He provided solutions in Flos, which he sent to Frederick II. One of these problems, taken from Omar Khayyam's (1048-1122) book on algebra, was to solve the equation: [x.sup.3] + 2[x.sup.2] + 10x = 20. Fibonacci knew that this was not solvable with the numerical system then in place-the Roman numerals. He provided an approximate answer, since he pointed out that the answer was not an integer, nor a fraction, nor the square root of a fraction. Without any explanation, he gives the approximate solution in the form of a sexagesimal number as 1.22.7.42.33.4.40, which equals:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, with today's computer algebra system we can get the proper (real) solution-by no means trivial! It is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Another of the problems with which he was challenged is one we can explore here, since it doesn't require anything more than some elementary algebra. Remember that although these methods may seem elementary to us, they were hardly known at the time of Fibonacci, and so this was considered a real challenge. The problem was to find the perfect square that remains a perfect square when increased or decreased by 5.

Fibonacci found the number 41/12 as his solution to the problem. To check this, we must add and subtract 5 and see if the result is still a perfect square.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We have then shown that 41/12 meets the criteria set out in the problem. Luckily the problem asked for 5 to be added and subtracted from the perfect square, for if he were asked to add or subtract 1, 2, 3, or 4 instead of 5, the problem could not have been solved. For a more general solution of this problem, see appendix B.

The third problem, also presented in Flos, is to solve the following:

Three people are to share an amount of money in the following parts: 1/2, 1/3, and 1/6. Each person takes some money from this amount of money until there is nothing left. The first person then returns 1/2 of what he took. The second person then returns 1/3 of what he took, and the third person returns 1/6 of what he took. When the total of what was returned is divided equally among the three, each has his correct share, namely, 1/2, 1/3, and 1/6. How much money was in the original amount and how much did each person get from the original amount of money?

Although none of Fibonacci's competitors could solve any of these three problems, he gave as an answer of 47 as the smallest amount, yet he claimed the problem was indeterminate.

The last mention of Fibonacci was in 1240, when he was honored with a lifetime salary by the Republic of Pisa for his service to the people, whom he advised on matters of accounting, often pro bono.

The Book Liber Abaci

Although Fibonacci wrote several books, the one we will focus on is Liber Abaci. This extensive volume is full of very interesting problems. Based on the arithmetic and algebra that Fibonacci had accumulated during his travels, it was widely copied and imitated, and, as noted, introduced into Europe the Hindu-Arabic place-valued decimal system along with the use of Hindu-Arabic numerals. The book was increasingly widely used for the better part of the next two centuries-a best seller!

He begins Liber Abaci with the following:

The nine Indian figures are: 9 8 7 6 5 4 3 2 1. With these nine figures, and with the sign 0, which the Arabs call zephyr, any number whatsoever is written, as demonstrated below. A number is a sum of units, and through the addition of them the number increases by steps without end. First one composes those numbers, which are from one to ten. Second, from the tens are made those numbers, which are from ten up to one hundred. Third, from the hundreds are made those numbers, which are from one hundred up to one thousand ... and thus by an unending sequence of steps, any number whatsoever is constructed by joining the preceding numbers. The first place in the writing of the numbers is at the right. The second follows the first to the left.

Despite their relative facility, these numerals were not widely accepted by merchants, who were suspicious of those who knew how to use them. They were simply afraid of being cheated. We can safely say that it took the same three hundred years for these numerals to catch on as it did for the completion of the Leaning Tower of Pisa.

Interestingly, Liber Abaci also contains simultaneous linear equations. Many of the problems that Fibonacci considers, however, were similar to those appearing in Arab sources. This does not detract from the value of the book, since it is the collection of solutions to these problems that makes the major contribution to our development of mathematics. As a matter of fact, a number of mathematical terms-common in today's usage-were first introduced in Liber Abaci. Fibonacci referred to "factus ex multiplicatione," 10 and from this first sighting of the word, we speak of the "factors of a number" or the "factors of a multiplication." Another example of words whose introduction into the current mathematics vocabulary seems to stem from this famous book are the words "numerator" and "denominator."

The second section of Liber Abaci includes a large collection of problems aimed at merchants. They relate to the price of goods, how to convert between the various currencies in use in Mediterranean countries, how to calculate profit on transactions, and problems that had probably originated in China.

Fibonacci was aware of a merchant's desire to circumvent the church's ban on charging interest on loans. So he devised a way to hide the interest in a higher initial sum than the actual loan, and based his calculations on compound interest.

The third section of the book contains many problems, such as:

A hound whose speed increases arithmetically chases a hare whose speed also increases arithmetically. How far do they travel before the hound catches the hare? 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? Calculate the amount of money two people have after a certain amount changes hands and the proportional increase and decrease are given.

There are also problems involving perfect numbers, problems involving the Chinese remainder theorem, and problems involving the sums of arithmetic and geometric series. Fibonacci treats numbers such as [square root of 10] in the fourth section, both with rational approximations and with geometric constructions.

Some of the classical problems, which are considered recreational mathematics today, first appeared in the Western world in Liber Abaci. Yet the technique for solution was always the chief concern for introducing the problem. This book is of interest to us, not only because it was the first publication in Western culture to use the Hindu numerals to replace the clumsy Roman numerals, or because Fibonacci was the first to use a horizontal fraction bar, but because it casually includes a recreational mathematics problem in chapter 12 that has made Fibonacci famous for posterity. This is the problem on the regeneration of rabbits.

The Rabbit Problem

Figure 1-2 shows how the problem was stated (with the marginal note included):

Figure 1-2

Beginning 1 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. Because the above written pair in the first month bore, you will double First 2 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 Second 3 month 3 pairs; of these in one month two are pregnant and in Third 5 the third month 2 pairs of rabbits are born and thus there are 5 pairs in the month; in this month 3 pairs are pregnant and in the Fourth 8 fourth month there are 8 pairs, of which 5 pairs bear another 5 Fifth 13 pairs; these are added to the 8 pairs making 13 pairs in the fifth month; these 5 pairs that are born in this month do not mate in this month, but another 8 pairs are pregnant, and thus there are in the Sixth 21 sixth month 21 pairs; to these are added the 13 pairs that are born Seventh 34 in the seventh month; there will be 34 pairs in this month; to this are added the 21 pairs that are born in the eighth month; there will Eight 55 be 55 pairs in this month; to these are added the 34 pairs that are Ninth 89 born in the ninth month; there will be 89 pairs in this month; to these are added again the 55 pairs that are born in the tenth month; Tenth 144 there will be 144 pairs in this month; to these are added again the Eleventh 233 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 Twelfth 377 born in the last month; there will be 377 pairs and this many pairs are produced from the above-written pair in the mentioned place at the end of one year. You can indeed see in the margin how we operated, namely, that we added the first number to the second, namely the 1 to the 2, and the second to the third and the third to the fourth and the fourth to the fifth, and thus one after another until we added the tenth to the eleventh, namely the 144 to the 233, and we had the above-written sum of rabbits, namely 377 and thus you can in order find it for an unending number of months."

(Continues...)

Excerpted from The (Fabulous) FIBONACCI Numbersby Alfred S. Posamentier Ingmar Lehmann Copyright © 2007 by Alfred S. Posamentier. Excerpted by permission.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

Details zur Ausgabe

• Verlag: Prometheus Books
• Erscheinungsdatum: 2007
• Sprache: Englisch
• ISBN 10: 1591024757
• ISBN 13: 9781591024750
• Einband: Tapa dura
• Anzahl der Seiten: 364
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar