# Leonardo Pisano Fibonacci

# Leonardo Pisano Fibonacci

**c. 1170-c. 1250**

**Italian Mathematician**

Atowering figure in the history of mathematics—to say nothing of mathematical studies during the medieval period—Leonardo Pisano Fibonacci is credited with the introduction of the Hindu-Arabic numeral system to Europe. He is also remembered for the Fibonacci sequence, whose recursive quality has made it a continued subject of fascination for mathematicians.

The dates of Fibonacci's life are uncertain, and his name itself presents something of a challenge. He was born in Pisa, and thus is sometimes known as Leonard of Pisa. His father was William or Guillermo Bonacci, and the name Fibonacci appears to be a contraction meaning family of Bonacci, or son of Bonacci. Thus the mathematician became known to history by his given name, that of his city, and the contraction: Leonardo Pisano Fibonacci.

Pisa at that time was a great trading center, and when Fibonacci was a boy, his father received an appointment as director of a warehouse in the North African port of Bugia. There Fibonacci trained under a Moorish instructor, who used the system of numerals 0 through 9 developed centuries before in India and later adopted throughout the Arab world.

Though other European scholars aware of Eastern advances in numbering had attempted to introduce so-called "Arabic numerals" to the West, European mathematics—including business arithmetic—was still conducted using the cumbersome old Roman symbols I, V, X, L, C, D, and M. The latter system had no expression for zero, nor was there a concept of place value, and this meant that large numbers could only be calculated using an abacus. This in turn meant that it was impossible to provide written verification of a result, and thus mathematical progress was severely limited.

After leaving Burga and travelling to a number of destinations, including Egypt, Syria, Constantinople, Greece, Sicily, and France, Fibonacci returned to Italy. There in 1202 he produced *Liber abaci,* or *Book of Calculations,* in which he introduced Europeans to the wondrous numerical system he had learned in the East. The book explained the rudiments of reading and using Hindu-Arabic numerals, and went on to a discussion of fractions, squares, and cube roots before addressing more challenging applications in geometry and algebra. Fibonacci, himself trained in business mathematics, also included chapters on the practical uses of the numeral system.

The publication of *Practicae geometriae* in 1220, which examined a number of algebraic, geometric, and trigonometric questions, added to Fibonacci's growing reputation. Soon he attracted the attention of Holy Roman emperor Frederick II (1194-1250), a renowned patron of the sciences, who in 1225 visited Pisa and held a mathematical competition to test Fibonacci's talents.

Johannes of Palermo, a mathematician working for the emperor, presented three questions to Fibonacci and his challengers. The first involved a second-degree problem and the second a third-degree or cubic equation. The last was a mere first-degree problem, but it involved a complex riddle concerning three men dividing an unspecified sum of money unequally. Fibonacci solved all three equations and his competitors withdrew without providing a single solution.

Later in 1225, Fibonacci wrote *Liber quadratorum,* a work he dedicated to the emperor. The book discussed a number of theorems involving indeterminate analysis, and recounted the problems put before him in the earlier competition. In 1228, Fibonacci presented a revised edition of *Liber abaci,* which he dedicated to his friend Michael Scot (c. 1175-c. 1235), an astrologer in Frederick's court.

The revised *Liber abaci* contained the famous "Fibonacci sequence" problem, which concerned a pair of rabbits who produce offspring at the rate of one pair a month, beginning in the second month. Given the proposition that each pair will reproduce at the same rate, and no rabbits will die, the problem addressed the question of how many rabbits would be produced at the end of a year. The answer was the sequence 1, 1, 2, 3, 5, 8, 13, 21, and so on. This is a recursive series, that is, one in which the relationship between successive terms can be expressed by means of a formula.

Fibonacci, who enjoyed great acclaim during his time and whose reputation has only grown in subsequent centuries, died in about 1250, during a war between Pisa and Genoa.

**JUDSON KNIGHT**

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**Leonardo Pisano Fibonacci**