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Leonardo Pisano was the first great mathematician of medieval

Christian Europe. He played an important role in reviving

ancient mathematics and made great contributions of his own.

After his death in 1240, Leonardo Pisano became known as Leonardo

Fibonacci.

Leonardo Fibonacci was born in Pisa in about 1180, the son

of a member of the government of the Republic of Pisa. When he

was 12 years old, his father was made administer of Pisa\'s

trading colony in Algeria. It was in Algeria that he was taught

the art of calculating. His teacher, who remains completely

unknown seemed to have imparted to him not only an excellently

practical and well-rounded foundation in mathematics, but also a

true scientific curiosity.

In 1202, two years after finally settling in Pisa, Fibonacci

produced his most famous book, Liber abaci (the book of the

Calculator). The book consisted of four parts, and was revised by

him a quarter of a century later (in 1228). It was a thorough

treatise on algebraic methods and problems which strongly

emphasized and advocated the introduction of the Indo-Arabic

numeral system, comprising the figures one to nine, and the

innovation of the "zephirum" the figure zero. Dealing with

operations in whole numbers systematically, he also proposed the

idea of the bar (solidus) for fractions, and went on to develop

rules for converting fraction factors into the sum of unit

factors.

At the end of the first part of the book, he presented

tables for multiplication, prime numbers and factor numbers. In

the second part he demonstrated mathematical applications to

commercial transactions.

In part three he gave many examples of recreational

mathematical problems, much like the type which are enjoyed

today. Next he prepared a thesis on series from which was derived

what is now called the "Fibonnaci series." The "Fibonacci

Sequence" is also named after Fibonacci. The Fibonacci sequence

is a sequence in which each term is the sum of two terms

immediately preceding it. The Fibonacci Sequence that has one as

its first term is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. . . . The

numbers may also be referred to as Fibonacci numbers. Fibonacci

sequences have proven useful in number theory, geometry, the

theory of continued fractions, and genetics. They also arise in

many unrelated phenomena, for example, the Golden Section, (whose

value is 1.6180) a shape valued in art and architecture because

of its pleasing proportions, and spiral arrangement of petals and

branches on certain types of flowers and plants.

In the final part of the book Fibonnaci, a student of

Euclid, applied the algebraic method. Fibonacci\'s book, the Liber

abaci remained a standard text for the next two centuries.

In 1220 he published Practica geometriae, a book on geometry

that was very significant to future studies of the subject. In it

he uses algebraic methods to solve many arithmetical and

geometrical problems. He also published Flos (flowers) in 1224.

In this work he combined Euclidean methodology with techniques of

Chinese and Arabic origin in solving determinate problems.

Liber quadratorum was published in 1225("Book of Square

Numbers") was dedicated to the Holy Roman emperor, Frederick II.

This book was devoted entirely to Diophantine equations of the

second degree (i.e., containing squares). The Liber quadratorum

may be considered Fibonacci\'s masterpiece. It is a systematically

arranged collection of theorems, many invented by the author, who

used his own proofs to work out general solutions. Probably his

most creative work was in congruent numbers- numbers that give

the same remainder when divided by a given number. He worked out

an original solution for finding a number that, when added to or

subtracted from a square number, leaves a square number.

Leonardo\'s statement that X + Y and X - Y could not both be

squares was of great importance to the detemination of the area

of rational right triangles. Although the Liber abaci was more

influential and broader in scope, the Liber quadratorum alone

ranks its author as the major contributor to number theory

between Diophantus and Pierre de Fermat, the 17th-century French

mathematician.

Except for his roll of spreading the use of the Hindu-Arabic

numerals, Fibonacci\'s contribution to mathematics has been

largely overlooked. His name is known to modern mathematicians

mainly because of the Fibonacci Sequence dervived from a problem

in the Liber abaci:

A certain man puts 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