Liber Abaci Essay, Research Paper

Leonardo Pisano was the first great mathematician of medievalChristian Europe. He played an important role in revivingancient mathematics and made great contributions of his own. After his death in 1240, Leonardo Pisano became known as LeonardoFibonacci. 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, Fibonacciproduced his most famous book, Liber abaci (the book of theCalculator). The book consisted of four parts, and was revised byhim a quarter of a century later (in 1228). It was a thoroughtreatise on algebraic methods and problems which stronglyemphasized and advocated the introduction of the Indo-Arabicnumeral system, comprising the figures one to nine, and theinnovation of the “zephirum” the figure zero. Dealing withoperations in whole numbers systematically, he also proposed theidea of the bar (solidus) for fractions, and went on to developrules for converting fraction factors into the sum of unitfactors. At the end of the first part of the book, he presentedtables for multiplication, prime numbers and factor numbers. Inthe second part he demonstrated mathematical applications tocommercial transactions. In part three he gave many examples of recreationalmathematical problems, much like the type which are enjoyedtoday. Next he prepared a thesis on series from which was derived what is now called the “Fibonnaci series.” The “FibonacciSequence” is also named after Fibonacci. The Fibonacci sequenceis a sequence in which each term is the sum of two termsimmediately preceding it. The Fibonacci Sequence that has one asits first term is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. . . . Thenumbers may also be referred to as Fibonacci numbers. Fibonaccisequences have proven useful in number theory, geometry, thetheory of continued fractions, and genetics. They also arise inmany unrelated phenomena, for example, the Golden Section, (whosevalue is 1.6180) a shape valued in art and architecture becauseof its pleasing proportions, and spiral arrangement of petals andbranches 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 Liberabaci remained a standard text for the next two centuries. In 1220 he published Practica geometriae, a book on geometrythat was very significant to future studies of the subject. In ithe uses algebraic methods to solve many arithmetical andgeometrical problems. He also published Flos (flowers) in 1224.In this work he combined Euclidean methodology with techniques ofChinese and Arabic origin in solving determinate problems. Liber quadratorum was published in 1225(”Book of SquareNumbers”) was dedicated to the Holy Roman emperor, Frederick II.This book was devoted entirely to Diophantine equations of thesecond degree (i.e., containing squares). The Liber quadratorummay be considered Fibonacci’s masterpiece. It is a systematicallyarranged collection of theorems, many invented by the author, whoused his own proofs to work out general solutions. Probably hismost creative work was in congruent numbers- numbers that givethe same remainder when divided by a given number. He worked outan original solution for finding a number that, when added to orsubtracted from a square number, leaves a square number.Leonardo’s statement that X + Y and X – Y could not both besquares was of great importance to the detemination of the areaof rational right triangles. Although the Liber abaci was moreinfluential and broader in scope, the Liber quadratorum aloneranks its author as the major contributor to number theorybetween Diophantus and Pierre de Fermat, the 17th-century Frenchmathematician. Except for his roll of spreading the use of the Hindu-Arabicnumerals, Fibonacci’s contribution to mathematics has beenlargely overlooked. His name is known to modern mathematiciansmainly because of the Fibonacci Sequence dervived from a problemin 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 month on becomes productive? The resulting number sequence, 1,1,2,3,5,8,13,21,35,55(Leonardo himself omitted the first term), in which each numberis the sum of the two preceding numbers, is the first recursivenumber sequence (in which the relation between two or moresuccesive terms can be expressed by a formula) known in Europe. Fibonacci died in around 1240 and despite Fibonacci’simportance as the most orginal and capable mathematician of the medieval world, none of his work has been translated intoEnglish. In the 19th century, the term Fibonacci Sequence wascoined by the French mathematician, Edouard Lucas, and since thenscientists began to discover the numbers in nature which broughtabout a new interest in the topic. Although still relativelyunknown in the United States, there is a “Fibonacci Association”in California. The purpose of that association is to encourageresearch in the topics that this great man once mastered.