Fibonacci Numbers

In 1175 AD, one of the greatest European mathematicians was born. His birth name was Leonardo Pisano. Pisano is Italian for the city of Pisa, which is where Leonardo was born. Leonardo wanted to carry his family name so he called himself Fibonacci, which is pronounced fib-on-arch-ee. Guglielmo Bonnacio was Leonardo’s father. Fibonacci is a nickname, which comes from filius Bonacci, meaning son of Bonacci. However, occasionally Leonardo would us Bigollo as his last name. Bigollo means traveler. I will call him Leonardo Fibonacci, but if anyone who does any research work on him may find the other names listed in older books.

Guglielmo Bonaccio, Leonardo’s father, was a customs officer in Bugia, which is a Mediterranean trading port in North Africa. He represented the merchants from Pisa that would trade their products in Bugia. Leonardo grew up in Bugia and was educated by the Moors of North Africa. As Leonardo became older, he traveled quite extensively with his father around the Mediterranean coast. They would meet with many merchants. While doing this Leonardo learned many different systems of mathematics. Leonardo recognized the advantages of the different mathematical systems of the different countries they visited.

But he realized that the Hindu-Arabic system of mathematics had many more advantages than all of the other systems combined. Leonardo stopped travelling with his father in the year 1200. He returned to Pisa and began writing. Leonardo wrote numerous books regarding mathematics. The books include his own contributions, which have become very significant, along with ancient mathematical skills that needed to be revived. Only four of his books remain today. His books were all handwritten so the only way for a person to obtain one in the year 1200 was to have another handwritten copy made.

The four books that still exist are Liber abbaci, Practica geometriae, Flos, and Liber quadratorum. Leonardo had written several other books, which unfortunately were lost. These books included Di minor guisa and Elements. Di minor guisa contained information on commercial mathematics. His book Elements was a commentary to Euclids Book X. In Book X, Euclid had approached irrational numbers from a geometric perspective. In Elements, Leonardo utilized a numerical treatment for the irrational numbers. Practical applications such as this made Leonardo famous among his contemporaries.

Leonardos book Liber abbaci was published in 1202. He dedicated this book to Michael Scotus. Scotus was the court astrologer to the Holy Roman Emperor Fredrick II. Leonardo based this book on the mathematics and algebra that he had learned through his travels. The name of the book Liber abbaci means book of the abacus or book of calculating. This was the first book to introduce the Hindu-Arabic place value decimal system and the use of Arabic numerals in Europe. Liber abbaci is predominately about how to use the Arabic numeral system, but Leonardo also covered linear equations in this book.

Many of the problems Leonardo used in Liber abacci were similar to problems that appeared in Arab sources. Liber abbaci was divided into four sections. In the second section of this book, Leonardo focused on problems that were practical for merchants. The problems in this section relate to the price of goods, how to calculate profit on transactions, how to convert between the various currencies in Mediterranean countries and other problems that had originated in China. In the third section of Liber abbaci, there are problems that involve perfect numbers, the Chinese remainder theorem, geometric series and summing arithmetic.

But Leonardo is best remembered today for this one problem in the third section: A certain man put a pair of rabbits in a place surrounded on all sides 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? This problem led to the introduction of the Fibonacci numbers and the Fibonacci sequence, which will be discussed in further detail in section II. Today almost 800 years later there is a journal called the Fibonacci Quarterly which is devoted to studying mathematics related to the Fibonacci sequence.

In the fourth section of Liber abbaci Leonardo discusses square roots. He utilized rational approximations and geometric constructions. Leonardo produced a second edition of Liber abbaci in 1228 in which he added new information and removed unusable information. Leonardo wrote his second book, Practica geometriae, in 1220. He dedicated this book to Dominicus Hispanus who was among the Holy Roman Emperor Fredrick IIs court. Dominicus had suggested that Fredrick meet Leonardo and challenge him to solve numerous mathematical problems. Leonardo accepted the challenge and solved the problems.

He then listed the problems and solutions to the problems in his third book Flos. Practica geometriae consists largely of geometry problems and theorems. The theorems in this book were based on the combination of Euclids Book X and Leonards commentary, Elements, to Book X. Practica geometriae also included a wealth of information for surveyors such as how to calculate the height of tall objects using similar triangles. Leonardo called the last chapter of Practica geometriae, geometrical subtleties; he described this chapter as follows:

Among those included is the calculation of the sides of the pentagon and the decagon from the diameter of circumscribed and inscribed circles; the inverse calculation is also given, as well as that of the sides from the surfacesto complete the section on equilateral triangles, a rectangle and a square are inscribed in such a triangle and their sides are algebraically calculated In 1225 Leonardo completed his third book, Flos. In this book Leonardo included the challenge he had accepted from the Holy Roman Emperor Fredrick II.

He listed the problems involved in the challenge along with the solutions. After completing this book he mailed it to the Emperor. Also in 1225, Leonardo wrote his fourth book titled Liber quadratorum. Many mathematicians believe that this book is Leonardo’s most impressive piece of work. Liber quadratorum means the book of squares. In this book he utilizes different methods to find Pythagorean triples. He discovered that square numbers could be constructed as sums of odd numbers. An example of square numbers will be discussed in section II regarding root finding. In this book Leonardo writes:

I thought about the origin of all square numbers and discovered that they arose from the regular ascent of odd numbers. For unity is a square and from it is produced the first square, namely 1; adding 3 to this makes the second square, namely 4, whose root is 2; if to this sum is added a third odd number, namely 5, the third square will be produced, namely 9, whose root is 3; and so the sequence and series of square numbers always rise through the regular addition of odd numbers. Leonardo died sometime during the 1240s, but his contributions to mathematics are still in use today.

Now I would like to take a closer look at some of Leonardos contributions along with some examples. Decimal Number System vs. Roman Numeral System Decimal Number System vs. Roman Numeral System As previously mentioned Leonardo was the first person to introduce the decimal number system or also known as the Hindu-Arabic number system into Europe. This is the same system that we use today, we call it the positional system and we use base ten. This simply means we use ten digits, 0,1,2,3,4,5,6,7,8,9, and a decimal point. In his book, Liber abbaci, Leonardo described and illustrated how to use this system.

Following are some examples of the methods Leonardo used to illustrate how to use this new system: 174174174174?28 = 6 remainder 6 + 28 – 28 x28 202 146 3480 + 1392 4872 It is important to remember that until Leonardo introduced this system the Europeans were using the Roman Numeral system for mathematics, which was not easy to do. To understand the difficulty of the Roman Numeral System I would like to take a closer look at it. In Roman Numerals the following letters are equivalent to the corresponding numbers:

In using Roman Numerals the order of the letters was important. If a smaller value came before the next larger value it was subtracted, if it came after the larger value it was added. For example: This system as you can imagine was quite cumbersome and could be confusing when attempting to do arithmetic. Here are some examples using roman numerals in arithmetic: (174) (28) (146) The order of the numbers in the decimal system is very important, like in the Roman Numeral System. For example 23 is very different from 32.

One of the most important factors of the decimal system was the introduction of the digit zero. This is crucial to the decimal system because each digit holds a place value. The zero is necessary to get the digits into their correct places in numbers such as 2003, which has no tens and no hundreds. The Roman Numeral System had no need for zero. They would write 2003 as MMIII, omitting the values not used. Leonardos Elements, commentary to Euclids Book X, is full of algorithms for geometry. The following information regarding Algorithm was obtained from a report by Dr.

Ron Knott titled Fibonaccis Mathematical Contributions: An algorithm is defined as any precise set of instructions for performing a computation. An algorithm can be as simple as a cooking recipe, a knitting pattern, or travel instructions on the other hand an algorithm can be as complicated as a medical procedure or a calculation by computers. An algorithm can be represented mechanically by machines, such as placing chips and components at correct places on a circuit board. Algorithms can be represented automatically by electronic computers, which store the instructions as well as data to work on. page 4) An example of utilizing algorithm principles would be to calculate the value of pi to 205 decimal places. Leonardo amazingly calculated the answer to the following challenge posed by Holy Roman Emperor Fredrick II: What causes this to be an amazing accomplishment is that Leonardo calculated the answer to this mathematical problem utilizing the Babylonian system of mathematics, which uses base 60. His answer to the problem above was: 1, 22, 7, 42, 33, 4, 40 is equivalent to: Three hundred years passed before anyone else was able to obtain the same accurate results.

As discussed earlier, the Fibonacci sequence is what Leonardo is famous for today. In the Fibonacci sequence each number is equal to the sum of the two previous numbers. For example: Leonardo used his sequence method to answer the previously mentioned rabbit problem. I will restate the rabbit problem: A certain man put a pair of rabbits in a place surrounded on all sides 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?

I will now give the answer to the problem, which I discovered in the Mathematics Encyclopedia. It is easy to see that 1 pair will be produced the first month, and 1 pair also in the second month (since the new pair produced in the first month is not yet mature), and in the third month 2 pairs will be produced, one by the original pair and one by the pair which was produced in the first month. In the fourth month 3 pairs will be produced, and in the fifth month 5 pairs.

After this things expand rapidly, and we get the following sequence of numbers: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 235, This is an example of recursive sequence, obeying the simple rule that two calculate the next term one simply sums the preceding two. Thus 1 and 1 are 2, 1 and 2 are 3, 2 and 3 are 5, and so on. (page 1) Leonardo Fibonacci was a mathematical genius of his time. His findings have contributed to the methods of mathematics that are still in use today. His mathematical influence continues to be evident by such mediums as the Fibonacci Quarterly and the numerous internet sites discussing his contributions.

Many colleges offer classes that are devoted to the Fibonacci methods. Leonardos dedication to his love of mathematics rightfully earned him a respectable place in world history. A statue of him stands today in Pisa, Italy near the famous Leaning Tower. It is a commemorative symbol that signifies the respect and gratitude that Italy endures toward him. Many of Leonardos methods will continue to be taught for generations to come. Fibonaccis Mathematical Contributions www. ee. surrey. ac. uk/personal/R. Knott/Fibonacci/fibBio. html (Feb. 10, 1999) www. mathacademy. com/platonic_realms/encyclop/articles/fibonac. html (March 23, 1999)


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