Fibonacci Number
Fibonacci numbers form a recursive sequence where each number is the sum of the two preceding ones, starting with 0 and 1. This sequence can be mathematically expressed as F0 = 0, F1 = 1, and Fn = Fn-1 + Fn-2 for n > 1, generating a series that begins 0, 1, 1, 2, 3, 5, 8, 13, 21, and so forth. Notably, as n increases, the ratio of consecutive Fibonacci numbers approaches the golden ratio (approximately 1.618). The concept was first introduced in 1202 in the book "Liber Abaci" by Leonardo of Pisa, also known as Fibonacci, who aimed to demonstrate the advantages of Arabic numerals over Roman numerals in calculations.
One famous problem from this book involves rabbit populations, illustrating how Fibonacci numbers can model growth patterns. Additionally, every positive integer can be expressed uniquely as a sum of nonadjacent Fibonacci numbers, a principle known as Zeckendorf's theorem. The Fibonacci sequence also connects to the Lucas numbers, a sequence with similar recurrence relations. Interestingly, these numbers appear in biological contexts, such as the ancestry of honeybees, highlighting their relevance in nature and mathematics.
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Fibonacci Number
The Fibonacci numbers are a sequence of numbers in which two adjacent numbers are added together to get the next number in the sequence: the sequence begins with 0 and 1, which are added together to get 1; 1 and 1 are then added together to get 2, 1 and 2 to get 3, and so on.
The Fibonacci numbers are a recursive sequence of numbers defined by F0 = 0, F1 = 1, and Fn = Fn - 1 + Fn - 2 for n > 1 (i.e., add together two adjacent Fibonacci numbers to get the next one). This gives a sequence that begins 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…. In many ways the Fibonacci numbers are associated with the number ( =
, often called the golden ratio. For example, for large values of n, the ratio Fn + 1/Fn is approximately (, and the larger that n is, the closer the ratio gets to (. Also, using the recursive definition it is impossible to calculate, say, F100 without knowing F99 and F98; but F99 and F98 are also not known without knowing the values of smaller Fibonacci numbers. But if we let ( = 
, then Fn = ((n - (n)/ 
. Zeckendorf’s theorem says that every positive integer is the sum of nonadjacent Fibonacci numbers in a unique way. For example, 81 = 5 + 21 + 55 = F5 + F8 + F10.Overview
The first occurrence of the Fibonacci numbers was in 1202 in the book Liber Abaci by Leonardo of Pisa, also known as Fibonacci. Liber Abaci is a book of mathematical problems and solutions that was written to introduce Arabic numbers to Europeans. At the time, Europeans used Roman numerals, and many calculations were done with an abacus. One of Fibonacci’s goals was to show how Arabic numerals made calculations much simpler, and how calculations could be done by hand without an abacus.
The Fibonacci numbers show up in the following problem from Liber Abaci: Suppose a pair of rabbits bears a young pair of rabbits every month for one year, and suppose each new pair of rabbits bears young in this fashion, but does so beginning the second month after they are born? How many rabbits will there be in a year? The answer is 377, which is F14, and, in general, in n months there will be Fn + 2 rabbits.
French mathematician Edouard Lucas is responsible for the modern interest in the Fibonacci numbers, and he gave these numbers this name. Lucas also studied the sequence of numbers 2, 1, 3, 4, 7, 11, 18…, a sequence that came to be known as the Lucas numbers, Ln.. L0 = 2, L1 = 1 (and, like the Fibonacci numbers, Ln = Ln - 1 + Ln - 2 for n > 1). The Fibonacci numbers are related to the Lucas numbers in seemingly endless ways. For example, if you add two Fibonacci numbers that are two apart, a Lucas number results: Fn + 1 + Fn - 1 = Ln. Also, Ln + 1 + Ln - 1 = 5Fn.
Fibonacci numbers show up in honeybee genealogy. A male bee comes from an unfertilized egg and has a mother but no father, and so one parent. But the mother has two parents, and in the next generation are three grandparents. The number at every generation is a Fibonacci number.
Bibliography
Devlin, Keith. The Man of Numbers: Fibonacci’s Arithmetic Revolution. New York: Walker, 2011. Print.
du Sautoy, Marcus. "What Is the Fibonacci Sequence?" Science Focus, 6 Mar. 2022, www.sciencefocus.com/science/what-is-the-fibonacci-sequence. Accessed 18 Nov. 2024.
Koshy, Thomas. Fibonacci and Lucas Numbers with Applications. New York: Wiley, 2001. Print.
Posamenter, Alfred S., and Ingmar Lehman. The Glorious Golden Ratio. Amherst: Prometheus, 2012. Print.