Leonardo Pisano Fibonacci | Biography

Leonardo Pisano Fibonacci is considered one of the most talented mathematicians of the Middle Ages. He is credited with introducing the Hindu-Arabic numbering system into western European culture at the beginning of the 13th century. His series of books on mathematical subjects helped revive the tradition of ancient mathematics and laid the foundation for the development of number theory. While Fibonacci's introduction of the modern day numbering system had a profound impact on the subsequent history of mathematics, his fame as a mathematician is perhaps more often associated with his development of the Fibonacci sequence, a series of numbers that he derived in order to solve a riddle about the reproduction of rabbits.

Also known as Leonardo of Pisa, Fibonacci, was born sometime during the latter half of the 12th century in Pisa, Italy. His father, William Bonacci, was a merchant and a government representative of Pisa, then an independent city-state. The name "Fibonacci" is believed to have been derived from the contraction of Filiorum Bonacci, or possibly, Filius Bonacci,meaning respectively, "of the family of Bonacci" and "Bonacci's son." What is known of Fibonacci's life has been gleaned mostly from the brief autobiographical notes he included in the introduction to his first mathematical treatise, Liber Abaci, in 1202, from the dedication in his Liber Quadratorum, written in 1225, and from his only surviving letter, written to the Emperor Frederick II's philosopher, Magister Theodoris. Based on incidents described in these writings, Fibonacci's year of birth is approximated at 1170.

During the 12th century, Pisa had about 10,000 inhabitants, and was an important center of commerce. Its merchants traded throughout the Mediterranean region and maintained warehouses in the coastal cities. When Fibonacci was a boy, his father was appointed as the head of a warehouse in the city of Bugia, on the North African coast. Fibonacci traveled there to join his father and to receive a business education. He studied arithmetic under the tutelage of a Moorish schoolmaster, and learned to make calculations using the Hindu-Arabic numerals 0 through 9. Through Italy and western Europe the seven Roman symbols, I, V, X, L, C, D, and M were still used in their various combinations to express all possible integers. These symbols represented respectively: one, five, ten, fifty, one hundred, five hundred, and one thousand. Because the Roman numeral system lacked the concepts of zero and place-value, multiplication and division were cumbersome and virtually impossible operations. Performing such calculations required the use of an abacus, a mechanical device which allowed for no written verification of the result.

Fibonacci was undoubtedly taught the merits of the Hindu-Arabic numbering system and the methods of using it to perform various mathematical operations. It is not clear how much more advanced his studies were, or how long Fibonacci remained in Bugia. He returned to Italy in about 1200, after having traveled extensively throughout the Mediterranean. Fibonacci's travels brought him in contact with leading scholars of the day and exposed him to the monetary systems of Egypt, Syria, Constantinople, Greece, Sicily, and France.

In 1202, Fibonacci wrote Liber abaci ("Book of Calculations"). Its intent was to introduce Hindu-Arabic numerals to Western culture and to explain the utility of the new numbering system in business transactions. Divided into 15 chapters, Fibonacci's book covered a broad range of topics. There was a chapter on how to read and write the numerals, as well as separate chapters on how the numerals could be used practically to make additions, subtractions, multiplications, and divisions. The concepts of fractions and of squared and cubic roots were explained. A series of chapters dealt with such business practices as pricing, bartering, and partnership. His final and perhaps most important chapter dealt with the more sophisticated topics of geometry and algebra. Throughout the text, Fibonacci posed and showed solutions for various mathematical puzzles and riddles.

Fibonacci wrote Practica geometriae in 1220. This manuscript dealt with practical problems in geometry and the measurement of objects. It also covered algebraic and trigonometric operations and the use of square and cubic roots. These writings revealed Fibonacci's familiarity with the works of Euclid and other mathematicians of antiquity.

Fibonacci's reputation as an influential mathematician became widespread. The Holy Roman Emperor Frederick II had read and was impressed with Liber Abaci, and in 1225 he traveled to Pisa to conduct a mathematical tournament as a test of Fibonacci's skills. Johannes of Palermo, a member of the emperor's staff, composed three tournament questions, sent in advance to Fibonacci and several competitors. At the emperor's court in Pisa, Fibonacci demonstrated his mathematical ability by deriving correct answers to each of the questions. His competitors withdrew, unable to provide any of the solutions.

The first question posed by Johannes of Palermo was a second-degree problem, that is, one involving squares. Specifically, the contestants were asked to determine values of x and y, such that x² + 5 = y², when x² - 5 = y². The next question was one of the third degree, involving cubes. The third question, a first-degree problem, was posed in the form of a riddle. Three men owned, respectively a half, a third and a sixth of an unknown quantity of money. Each man took an unspecified amount of the money, leaving none left. Then each man, respectively, returned a half, a third, and a sixth of what he had first taken. The returned money was divided into equal thirds and redistribute to the men. This resulted in each man acquiring his fair share. The contestants were asked to determine the quantity of money owned by each man.

Having displayed his skills as a mathematician before the emperor, Fibonacci went on that same year to write his Liber quadratorum. The manuscript was dedicated to the emperor and in it Fibonacci described both their meeting and the particulars of the contest. Liber quadratorum contained a collection of theorems about indeterminate analysis, specifically relating to second degree equations, and its introduction included a description of the second-degree contest problem. The first and third-degree contest problems were described in a separate manuscript of Fibonacci's, entitled Flos. The originality of his work and the power of his methods for solution have caused Fibonacci to be ranked among the most important seminal figures in the field of number theory.

Fibonacci revised Liber abaci in 1228, and it is this text that eventually became most widely distributed in Europe. It is dedicated to Michael Scot, a friend of Fibonacci's who wrote science texts and served as chief astrologer to Emperor Frederick II. As in the first edition, Fibonacci argues strongly for the adoption of the Hindu-Arabic numbering system.

Tourists to Pisa, Italy, are today most frequently drawn to the city by its famous leaning tower, designed and partially built by a contemporary of Fibonacci's, Bonnano Pisano. Across the Arno River from the leaning tower is a lesser known monument, a statue representing Fibonacci. Since no drawings of Fibonacci has survived, the statue was created in the likeness of a "generic" Pisan of the 12th century. While the potential lack of resemblance might strike Fibonacci as an odd legacy, the enduring association of his name with a sequence of numbers he generated, in order to solve a puzzle about rabbits, might seem even more peculiar. In his Liber abaci, he described his solution to a well-known mathematical problem of his time. A pair of rabbits is kept in an enclosure and begins producing offspring at the rate of one pair per month, beginning in the second month. Each new pair reproduces at that same rate after its second month. Assuming there is no mortality, how many rabbits will exist at the end of a year? The numbers in the Fibonacci sequence represent the quantity of rabbit pairs at the end of each month, namely 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on. Fibonacci was aware that the sequence was recursive, that is, one in which the relationship between successive terms can be expressed with a formula. Various modern-day mathematical societies bearing Fibonacci's name, have devoted themselves to exploring the interesting features of this sequence.

Fibonacci is believed to have died around 1250, when Pisa was defeated by Genoa in a naval battle.