B0609030714.pdf - International Journal of Engineering...

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International Journal of Engineering Science Invention ISSN (Online): 2319 6734, ISSN (Print): 2319 6726 ||Volume 6 Issue 9|| September 2017 || PP. 07-14 7 | Page The Fibonacci Numbers and Its Amazing Applications *Sudipta Sinha Assistant Professor Department of Mathematics Burdwan Raj College,Burdwan,West Bengal,713104 Corresponding Author: Sudipta Sinha Abstract: Fibonacci sequence of numbers and the associate d “Golden Ratio” are manifested in nature and in certain works of art. We observe that many of the natural things follow the Fibonacci sequence. It appears in biological settings such as branching in trees, phyllotaxis (the arrangement of leaves on a stem), the fruit sprouts of a pineappl e , the flowering of an artichoke, an uncurling fern and the arrangement of a pine cone's bracts etc. At present Fibonacci numbers plays very important role in coding theory. Fibonacci numbers in different forms are widely applied in constructing security coding. Keywords: Fibonacci Numbers, Golden ratio, Coding, Encryption, Decryption --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 05-08-2017 Date of acceptance: 13-09-2017 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction The Fibonacci numbers were first discovered by a man named Leonardo Pisano. He was known by his nickname, Fibonacci. The Fibonacci sequence is a sequence in which each term is the sum of the 2 numbers preceding it. The Fibonacci Numbers are defined by the recursive relation defined by the equations F n = F n-1 + F n-2 for all n 3 where F 1 = 1; F 2 = 1 where Fn represents the nth Fibonacci number (n is called an index). The Fibonacci sequence can elaborately written as {1,1,2,3,5,8,13,21,34,55,89,144,233 …….} . One of the most common experiments dealing with the Fibonacci sequence is his experiment with rabbits. Fibonacci put one male and one female rabbit in a field. Fibonacci supposed that the rabbits lived infinitely and every month a new pair of one male and one female was produced. Fibonacci asked how many would be formed in a year. Following the Fibonacci sequence perfectly the rabbits reproduction was determined...144 rabbits. Though unrealistic, the rabbit sequence allows people to attach a highly evolved series of complex numbers to an everyday, logical, comprehendible thought.Bortner and Peterson (2016) elaborately described the history and application of Fibonacci numbers. II. Fibonacci Sequence In Nature Fibonacci can be found in nature not only in the famous rabbit experiment, but also in beautiful flowers (Internet access, 12). On the head of a sunflower and the seeds are packed in a certain way so that they follow the pattern of the Fibonacci sequence. This spiral prevents the seed of the sunflower from crowding themselves out, thus helping them with survival. The petals of flowers and other plants may also be related to the Fibonacci sequence in the way that they create new petals (Internet access, 10).
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