Fibonacci Sequence

FibonacciLeonardo Fibonacci, born around 1175 in the present-day Pisa, Italy, is known by various names. Being of Pisa, he is called Leonardo of Pisa, which in Italian is Leonardo Pisano. His full name was Leonardo Pisano Bigollo. Historians are not sure what “bigollo” means. It could mean “traveller” or “good-for-nothing” (see “Did his countrymen…”). Fibonacci’s father’s name was Guglielmo Bonaccio. As such, in 1828, centuries after Fibonacci’s time, Guillaume Libri invented the name “Fibonacci” from “filius Bonacci,” latin for “the son of Bonacci.” Fibonacci, as he is called by most today, is therefore, just a short version of “filius Bonacci.”

Fibonacci was born in Italy but was educated in North Africa where his father, Guilielmo, held a diplomatic post. His father’s job was to represent the merchants of the Republic of Pisa who were trading in Bugia, later called Bougie and now called Bejaia. Bejaia is a Mediterranean port in northeastern Algeria. The town lies at the mouth of the Wadi Soummam near Mount Gouraya and Cape Carbon. Fibonacci was taught mathematics in Bugia and travelled widely with his father and recognised the enormous advantages of the mathematical systems used in the countries they visited. Fibonacci writes in his famous book Liber abaci

The Fibonacci sequence is a set of numbers that starts with a one or a zero, followed by a one, and proceeds based on the rule that each number (called a Fibonacci number) is equal to the sum of the preceding two numbers. If the Fibonacci sequence is denoted F ( n ), where n is the first term in the sequence, the following equation obtains for n = 0, where the first two terms are defined as 0 and 1 by convention:

F (0) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 …

In some texts, it is customary to use n = 1. In that case the first two terms are defined as 1 and 1 by default, and therefore:

F (1) = 1, 1, 2, 3, 5, 8, 13, 21, 34 …

Fibonacci numbers are of interest to biologists and physicists because they are frequently observed in various natural objects and phenomena. The branching patterns in trees and leaves, for example, and the distribution of seeds in a raspberry are based on Fibonacci numbers.

The Fibonacci sequence is named for Leonardo Pisano (also known as Leonardo Pisano or Fibonacci ), an Italian mathematician who lived from 1170 – 1250. Fibonacci used the arithmetic series to illustrate a problem based on a pair of breeding rabbits:

“How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on?” The result can be expressed numerically as: 1, 1, 2, 3, 5, 8, 13, 21, 34 …

Another of Fibonacci’s books is Practica geometriae, which contains a large collection of geometry problems arranged into 8 chapters with theorems based on Euclid’s Elements and On Divisions. In addition to geometrical theorems with precise proofs, the book includes practical information for surveyors, including a chapter on how to calculate the height of tall objects using similar triangles. The final chapter presents the calculation of the sides of the pentagon and the decagon from the diameter of circumscribed and inscribed circles.

The Holy Roman emperor, Frederick II, became aware of Fibonacci’s work. A member of Frederick’s court presented a number of problems as challenges to the great mathematician Fibonacci. In Flos, 3 of these problems were solved by Fibonacci. In one, he approximates the solution to 10x + 2×2 + x3 = 20 to 9 decimal places.

Liber quadratorum, written in 1225, is Fibonacci’s most impressive piece of work, although not the work for which he is most famous. The book’s name means the “Book of Squares” and it is a number theory book which, among other things, examines methods to find Pythogorean triples. Fibonacci first notes that square numbers can be constructed as sums of odd numbers, essentially describing an inductive construction using the formula n2 + (2n+1) = (n+1)2. Fibonacci also proves many interesting number theory results such as: there is no x, y such that x2 + y2 and x2 – y2 are both squares, and x4 – y4 = z2 has no non-trivial integral solutions.

He also defined the concept of a congruum, a number of the form ab(a + b)(a – b), if a + b is even, and 4 times this if a + b is odd. Fibonacci proved that a congruum must be divisible by 24 and he also showed that for x, c such that x2 + c and x2 – c are both squares, then c is a congruum. He also proved that a square cannot be a congruum. The Liber quadratorum alone ranks Fibonacci as the major contributor to number theory between Diophantus and the seventeenth century French mathematician Pierre de Fermat.

In 1240, the Republic of Pisa gave him a salary in recognition for the services that he had given to the city, advising on matters of accounting and teaching the citizens.

The views and opinions expressed in this article are those of the authors/source and do not necessarily reflect the position of CSGLOBE or its staff.

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