Fibonacci

Leonardo was born in Pisa around 1170, when the maritime republic was one of the wealthiest trading powers on the Mediterranean ^[1]. His father, Guglielmo Bonacci, worked as a customs official and notary representing Pisan merchants at the trading colony of Bugia (modern Bejaia in Algeria) on the North African coast ^[2]. The name Fibonacci, a contraction of filius Bonacci, meaning son of Bonacci, was attached to him centuries later by historians; in his own lifetime he was called Leonardo Pisano, Leonardo of Pisa ^[1].

As a boy, Leonardo joined his father in Bugia. There he received instruction in calculation from teachers trained in the Arabic mathematical tradition, and he encountered the nine Indian figures and the sign for zero that Arab scholars had adopted from Indian mathematics ^[2]. In the prologue to his great book he later recalled this schooling and described how the new numerals struck him as far superior to anything used in Italy ^[3].

European commerce at the time still ran on Roman numerals and the counting board. Multiplication and division with Roman numerals were slow and awkward, and merchants depended on the abacus for anything beyond simple sums. The young Pisan recognized that positional notation, in which the value of a digit depends on its place, made written arithmetic fast and reliable ^[1].

Fibonacci's education did not end in Bugia. His commercial travels took him around the Mediterranean, to Egypt, Syria, Greece, Sicily, and Provence, and in each place he sought out practitioners of calculation and compared their methods ^[2]. This long apprenticeship exposed him to the arithmetic of al-Khwarizmi and other Arabic authors, to Euclidean geometry, and to the practical reckoning of merchants from many traditions ^[3].

He returned to Pisa and, in 1202, completed Liber Abaci (the Book of Calculation), a comprehensive Latin manual of arithmetic and algebra built on the Hindu-Arabic numeral system ^[1]. Despite its title, the book argued against dependence on the abacus: it showed readers how to perform every commercial computation on paper. Its chapters covered the reading and writing of the new numerals, operations with whole numbers and fractions, currency conversion, profit sharing among partners, alloying of coinage metals, and methods for solving equations ^[3].

The book found its audience. Italian merchant houses needed exactly this kind of applied mathematics, and Liber Abaci circulated widely in manuscript. A revised edition appeared in 1228, dedicated to the scholar Michael Scot, and it is this second version that survives today ^[2]. Through it, and through the reckoning schools that later grew up in Italian cities, Hindu-Arabic numerals gradually displaced Roman numerals in European bookkeeping ^[4].

Among the many exercises in Liber Abaci sits the problem that made Fibonacci a household name. It asks how many pairs of rabbits will exist after a year if each pair produces a new pair every month and new pairs begin breeding in their second month. The answer generates the sequence 1, 1, 2, 3, 5, 8, 13, 21, and so on, in which each term is the sum of the two before it ^[1]. The nineteenth-century French mathematician Edouard Lucas attached the name Fibonacci sequence to these numbers, and the label stuck ^[4]. The sequence had appeared earlier in Indian scholarship on Sanskrit poetic meters, but Leonardo introduced it to European readers ^[4].

His achievements went well beyond the rabbits. In 1220 he published Practica Geometriae, a treatise applying geometry and trigonometry to surveying and measurement, drawing on Euclid and on Arabic geometric writing ^[2]. His most technically demanding work, Liber Quadratorum (the Book of Squares) of 1225, examined problems in number theory involving square numbers and what are now called congruent numbers. Historians of mathematics rank it among the most sophisticated works on number theory produced between antiquity and the seventeenth century ^[3].

A public test of his skill came at the court of the Holy Roman Emperor Frederick II, who kept learned men around him and enjoyed setting mathematical challenges. The court philosopher Johannes of Palermo posed several problems, including finding a solution to a particular cubic equation. Fibonacci answered them in his short work Flos, giving an approximate root of the cubic that was accurate to a remarkable degree for the era ^[2]. He also addressed Liber Quadratorum to Frederick, a sign of the emperor's regard ^[3].

Very little survives about Leonardo the man. No contemporary portrait exists, no letters, and no record of a wife or children. What historians know comes almost entirely from remarks in his own prologues and from a handful of Pisan documents ^[1]. The statue of him that stands in the Camposanto in Pisa was carved in the nineteenth century and is an artist's invention rather than a likeness ^[4].

His writings suggest a practical, patient teacher. Liber Abaci works through hundreds of examples in careful steps, anticipating the mistakes a merchant's clerk might make. He wrote in Latin for a learned audience but chose problems from everyday trade: prices of goods, division of profits, exchange of currencies between cities ^[3]. That grounding in commerce, unusual for a medieval mathematical author, reflects the customs-house world in which he grew up ^[2].

Fibonacci remained associated with Pisa through his later decades. In 1240 the commune of Pisa granted him an annual salary, recorded in a surviving decree, in recognition of his service to the city as an advisor on accounting and related matters. The decree refers to him as a discreet and learned man, and it is the last documentary trace of his life ^[1].

The date of his death is not recorded. Scholars generally place it around 1240 or in the decade that followed, and most assume he died in Pisa, the city of his birth ^[2]. Because so few records survive, even his approximate birth year of 1170 rests on inference from the chronology of his father's posting to Bugia and the dates of his books ^[4].

Any list of Fibonacci facts begins with the numerals we use every day. His books were the most influential channel through which the Hindu-Arabic system, including zero and positional notation, entered Western Europe ^[1]. The change took generations: some Italian cities initially restricted the new figures in official records because they were unfamiliar and thought easy to falsify, but by the fifteenth century the digits had won, aided by the spread of reckoning schools and later by printing ^[4].

The Fibonacci sequence turned out to matter far beyond a puzzle about rabbits. The ratio of successive terms approaches the golden ratio, roughly 1.618, and related patterns appear in the arrangement of leaves on plant stems, the spirals of sunflower heads and pinecones, and the geometry of certain shells ^[5]. In the modern era the numbers surface in computer algorithms, data structures, financial market analysis, and the mathematics of search and sorting. A dedicated research journal, the Fibonacci Quarterly, has published work on the sequence and its generalizations since 1963 ^[6].

Every serious Fibonacci biography also records what he meant to the discipline itself. Between the decline of ancient Greek mathematics and the European revival of the sixteenth century, few Western figures produced original mathematical work of lasting depth; Leonardo of Pisa is the outstanding exception ^[3]. His achievements bridged the Arabic and Latin worlds at a moment when that exchange reshaped European learning. An asteroid, a street in Pisa, and the sequence itself carry his name, a quiet sort of fame for a customs official's son who learned his arithmetic on the North African coast ^[2].