Fibonacci number

In mathematics, the Fibonacci numbers or Fibonacci sequence are the numbers in the following integer sequence:

or (often, in modern usage):

By definition, the first two numbers in the Fibonacci sequence are either 1 and 1, or 0 and 1, depending on the chosen starting point of the sequence, and each subsequent number is the sum of the previous two.

In mathematical terms, the sequence F_n of Fibonacci numbers is defined by the recurrence relation

with seed values

or

The Fibonacci sequence is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier as Virahanka numbers in Indian mathematics. By modern convention, the sequence begins either with F₀ = 0 or with F₁ = 1. The sequence described in Liber Abaci began with F₁ = 1.

Fibonacci numbers are closely related to Lucas numbers in that they form a complementary pair of Lucas sequences and . They are intimately connected with the golden ratio; for example, the closest rational approximations to the ratio are 2/1, 3/2, 5/3, 8/5, ... .

Fibonacci

Leonardo Bonacci (c. 1170 – c. 1250)—known as Fibonacci (Italian: [fiboˈnattʃi]), and Leonardo of Pisa, Leonardo Pisano Bigollo, Leonardo Fibonacci—was an Italian mathematician, considered to be "the most talented Western mathematician of the Middle Ages".

Fibonacci popularized the Hindu–Arabic numeral system to the Western World primarily through his composition in 1202 of Liber Abaci (Book of Calculation). He also introduced to Europe the sequence of Fibonacci numbers which he used as an example in Liber Abaci.

Life

Fibonacci was born around 1170 to Guglielmo Bonacci, a wealthy Italian merchant and, by some accounts, the consul for Pisa. Guglielmo directed a trading post in Bugia, a port in the Almohad dynasty's sultanate in North Africa. Fibonacci travelled with him as a young boy, and it was in Bugia (now Béjaïa, Algeria) that he learned about the Hindu–Arabic numeral system.

Fibonacci travelled extensively around the Mediterranean coast, meeting with many merchants and learning of their systems of doing arithmetic. He soon realised the many advantages of the Hindu-Arabic system. In 1202 he completed the Liber Abaci (Book of Abacus or Book of Calculation) which popularized Hindu–Arabic numerals in Europe.

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