Fibonacci numbers
Introduction
The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1. Named after Italian mathematician Leonardo of Pisa, also known as Fibonacci, the sequence has been recognized as early as the 6th century in Indian mathematics, but it was Fibonacci who introduced it to the western world in his 1202 book, Liber Abaci.
Historical Background
The Fibonacci sequence was first introduced to the western world by Fibonacci in his book Liber Abaci. However, the sequence had been previously described in Indian mathematics. The earliest known mention of the sequence is in the Chandaḥśāstra, a Sanskrit treatise on prosody written by the ancient Indian scholar Pingala between the 5th and 2nd century BC.
Mathematical Explanation
The Fibonacci sequence is defined by the recurrence relation:
F(n) = F(n-1) + F(n-2),
with seed values
F(0) = 0, F(1) = 1.
In other words, the sequence starts 0, 1, and each subsequent number is the sum of the previous two. This sequence results in the following series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so forth.
Properties
The Fibonacci sequence has numerous mathematical properties and has been discovered in various aspects of art, nature, and science.
Divisibility Properties
The Fibonacci numbers have intriguing number-theoretic properties. For example, the sequence of Fibonacci numbers modulo m (i.e., the remainder when divided by m) is periodic and repeats itself eventually. This is known as Pisano's period.
Golden Ratio
The ratio of consecutive Fibonacci numbers converges on a constant value, approximately 1.618033988749895, known as the golden ratio. This ratio appears in many areas of mathematics and science and is often associated with aesthetically pleasing proportions in art and architecture.
Fibonacci Primes
Fibonacci primes are prime numbers that are also Fibonacci numbers. The first few Fibonacci primes are 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, and so on.
Applications
The Fibonacci sequence is used in many different areas, including computer algorithms, financial market analysis, and the creation of realistic computer graphics.
Computer Algorithms
In computer science, the Fibonacci numbers are commonly used in algorithms, such as those for sorting and searching. For example, the Fibonacci search technique is a method of searching a sorted array using a divide and conquer algorithm that narrows down possible locations with the aid of Fibonacci numbers.
Financial Market Analysis
In financial markets, Fibonacci retracement levels are a method of technical analysis for determining support and resistance levels. They are named after their use of the Fibonacci sequence. Fibonacci retracement is based on the idea that markets will retrace a predictable portion of a move, after which they will continue to move in the original direction.
Nature and Biology
In nature, the Fibonacci sequence often appears in the arrangement of leaves on a stem or the structure of crystals. It is also seen in the arrangement of seeds on flower heads and fruits, and the branching patterns of trees.