Leonardo Fibonacci, also called Leonardo Pisano, was an Italian number theorist. He introduced the world to such wide-ranging mathematical concepts as what is now known as the Arabic numbering system, the concept of square roots, number sequencing, and even math word problems.

Background

Leonardo Fibonacci was born around 1170, in Pisa, Italy. He was the son of Guilichmus Bonacci, the secretary of the Republic of Pisa. The name of his mother is unknown. He is also known to have had a brother named Bonaccinghus, but nothing of him is known to be recorded, save for his name.

Contrary to what many believe, his name originally wasn't Fibonacci at all, but Leonardo Pisano. Fibonacci translates to "son of Bonacci," which would explain the reason for his nickname. This nickname wasn't used until 1838, when Guillaume Libri, a great historian first used the name.

Education

Although born in Italy, Fibonacci was educated in North Africa where his father was a diplomat who represented the merchants of the Republic of Pisa who were trading in Bugia, now called Bejaia (a city in Algeria, Africa), and was also a customs official.

Fibonacci began learning mathematics in Bejaia, and, as he traveled to other countries with his father, Fibonacci marveled at how each country had a different mathematical system, and that each had its own advantages and disadvantages. Below is a direct quote by Fibonacci recording his experience:

"When my father, who had been appointed by his country as a public notary in the customs at Bugia acting for the Pisan merchants going there, was in charge, he summoned me to him while I was still a child, and having an eye to usefulness and future convenience, desired me to stay there and receive instruction in the school of accounting. There, when I had been introduced to the art of the Indians' nine symbols through remarkable teaching, knowledge of the art very soon pleased me above all else and I came to understand it, for whatever was studied by the art in Egypt, Syria, Greece, Sicily, and Provence, in all its various forms."

Career

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. Fibonacci lived in the days before printing, so his books were handwritten and the only way to have a copy of one of his books was to have another hand-written copy made. Of his books, there are copies of Liber abaci (1202), Practica geometriae (1220), Flos (1225), and Liber quadratorum. Given that relatively few hand-made copies would ever have been produced, we are fortunate to have access to his writing in these works. However, we know that he wrote some other texts which, unfortunately, 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 which Euclid had approached from a geometric point of view.

One might have thought that at a time when Europe was little interested in scholarship, Fibonacci would have been largely ignored. This, however, is not so and widespread interest in his work undoubtedly contributed strongly to his importance. Fibonacci was a contemporary of Jordanus but he was a far more sophisticated mathematician and his achievements were clearly recognized, although it was the practical applications rather than the abstract theorems that made him famous to his contemporaries.

The Holy Roman emperor was Frederick II. He had been crowned king of Germany in 1212 and then crowned Holy Roman Emperor by the Pope in St Peter's Church in Rome in November 1220. Frederick II supported Pisa in its conflicts with Genoa at sea and with Lucca and Florence on land, and he spent the years up to 1227 consolidating his power in Italy. State control was introduced on trade and manufacture, and civil servants to oversee this monopoly were trained at the University of Naples which Frederick founded for this purpose in 1224.

Frederick 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 who was the court astrologer, Theodorus Physicus the court philosopher, and Dominicus Hispanus who suggested to Frederick that he meet Fibonacci when Frederick's court met in Pisa around 1225.

Johannes of Palermo, another member of Frederick II's court, presented a number of problems as challenges to the great mathematician Fibonacci. Three of these problems were solved by Fibonacci and he gives solutions in Flos which he sent to Frederick II. We give some details of one of these problems below.

After 1228 there is only one known document which refers to Fibonacci. This is a decree made by the Republic of Pisa in 1240 in which a salary is awarded to "the serious and learned Master Leonardo Bigollo."

This salary was given to Fibonacci in recognition of the services that he had given to the city, advising on matters of accounting and teaching the citizens.

Liber abaci, published in 1202 after Fibonacci's return to Italy, was dedicated to Scotus. 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. Indeed, although mainly a book about the use of Arab numerals, which became known as algorism, simultaneous linear equations are also studied in this work. Certainly, many of the problems that Fibonacci considers in Liber abaci were similar to those appearing in Arab sources.

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.

Thanks to his works, Fibonacci helped introduce Europe to concepts which are nowadays taken for granted, like using the numbers 0 - 9 in calculations, operations using integers and fractions, the extraction of roots, and how to apply this math to transactions.

Fibonacci believed the Indian number system had huge advantages over the Roman system and believed the people of Europe should adopt it. In 1202 he published Liber Abaci - the Book of Calculation - which began the spread of the modern number system in the West. Fibonacci updated the book and published a new edition in 1228.

Near the beginning of the Book of Calculation, he wrote: "I received an excellent education in the methods of the nine Indian numbers; the knowledge of these methods pleased me more than anything else... Therefore strictly embracing the Indian method, and adding some of my own ideas, and more still from Euclid’s geometry, I assembled them in this book as understandably as I could."

His Book of Calculation showed how calculations in commerce, finance, and pure mathematics could be carried out with the new number system.

Fibonacci's book was vital in planting a seed in European minds. Popularizing the new numbers was a long process; widespread adoption began only after the twin events of Gutenberg's invention of the printing press in 1440 (only hand-written copies of Fibonacci's works had been available before then) and the fall of Constantinople in 1453.

The fall of Constantinople resulted in its refugees arriving in Italy. Some of the refugees brought with them Ancient Greek texts that had been locked away for many centuries in Constantinople. These Greek texts helped trigger the Renaissance in Italy.

Fibonacci's Book of Calculation was also important for European commerce and finance. In Arab lands, the new number system had been used only by mathematicians and scientists. Fibonacci saw the superiority of the new system for businesses and devoted several chapters of his book to show calculations of profit, interest, and currency conversions. In fact, the book's immediate impact on the commercial world was much greater than on the scientific world.

Some of the topics Fibonacci considered in his book were: the new numbers; multiplication and addition; subtraction; division; fractions; rules for money; accounting; quadratic and cube roots; quadratic equations; binomials; proportion; rules of algebra; checking calculations by casting out nines; progressions; and applied algebra.

The algebra in the Book of Calculation was principally influenced by work published by the mathematicians Al-Khwarizmi from Persia; Abu-Kamil from Egypt; and Al-Karaji from Baghdad.

Fibonacci also famously considered the rabbit problem, which gave rise to the Fibonacci Sequence.

"A man places a pair of rabbits into a garden surrounded by a wall. How many pairs of rabbits can be produced in a year if every month each pair produces a new pair which from the second month on becomes productive?"

The month-by-month solution to the problem became known as the Fibonacci Sequence. It involves adding the preceding two terms to one another to generate the next term: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

This remarkable sequence, which was already known in Indian mathematics, occurs repeatedly in mathematics and also in the natural world, where, for example, the scales of pine cones run in spirals arranged in ratios determined by the Fibonacci Sequence.

Even in art, the Fibonacci Sequence is prominent. If one divide one term in the sequence by the previous term, the result gets closer and closer to the golden ratio - loved by artists and architects - as the terms get larger.

Quotations:
"If by chance I have omitted anything more or less proper or necessary, I beg forgiveness, since there is no one who is without fault and circumspect in all matters."

Personality

The only key personality noted towards Fibonacci was linked to his nickname "Bigollo." This term at the time would be used to call someone lazy, suggesting that, in his youth, he probably took no interest in theoretical mathematics (which was what he was educated in as a child), and probably had more interests in other areas of science. "Bigollo" is also known to translate to "Well-traveled." This latter translation is almost certainly a better fit, however, neither description has yet to be proven. Some other suggestions on the name exist but have little evidence.

Connections

Fibonacci had no children but was married to a woman named Ariel Tulared.