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# Fibonacci sequence (formula, list)

The Fibonacci sequence is a set of numbers that has had an undeniable impact on the development of mathematics, art and biology. We talk about what it is, how and when it appeared, and where it is used.

## History of Fibonacci numbers

Fibonacci numbers are nothing more than the elements of a numerical linear sequence defined recursively. Recursion is a way of describing an object (not necessarily mathematical) through its own elements. So, for the Fibonacci sequence, only the first two numbers are given: 0 and 1 (in some sources, two units are indicated as the first two elements of the sequence), and the values of the remaining elements are determined as the sum of the two previous ones.

Thus, the sequence of Fibonacci numbers is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, ...

The sequence got its name in honor of one of the first major mathematicians of the Middle Ages - Leonardo Pisano (Leonardo Pisano), nicknamed Fibonacci (Fibonacci). By the way, Fibonacci numbers were well known long before the life of Leonardo (about 1170 - about 1250) and were widely used in the metric sciences of Ancient India, were familiar to Arabic and ancient Greek mathematicians. The Italian mathematician learned about this sequence during one of his many travels, set about studying its properties and presented the results of his work in the work "The Book of the Abacus" (Liber abaci) in 1202. In addition to information about this sequence, the book contained almost all the theoretical information from arithmetic and algebra known at that time.

Thus, the Fibonacci sequence was named after its popularizer among Western European mathematicians in the Middle Ages. At the same time, historians attribute the first use of this name only to the 19th century.

## Rabbit Breeding Problem

A significant hobby in the life of Leonardo of Pisa was mathematical tournaments, participation in which introduced him to many exciting problems and aroused in him an interest in solving them. The mathematician described the formulation and analysis of a number of similar problems in his works. One of these was the problem of the population of rabbits, considered in the "Book of the Abacus" and led just to the concept of the Fibonacci sequence.

The task is to consider the population model of rabbit reproduction. According to the condition at the beginning of the experiment, we have an isolated environment favorable for the life of animals and a pair of rabbits placed in it. Every month, starting from the second month of life, a pair of rabbits produces another pair, which in a month will produce another pair, and so on. Fibonacci considered ideal conditions under which rabbits do not die, so the increase in the number of pairs of rabbits in each month compared to the previous one is equal to the number of pairs of rabbits two months ago.

And finally, the question of the problem: "How many pairs of rabbits will we get a year after the start of the experiment?" Please note that the conditions of the problem completely repeat the rule by which the Fibonacci sequence is built - each element is equal to the sum of the two previous ones (in each month, the number of pairs of rabbits is equal to the sum of the number of pairs in the previous month and the number of pairs produced by rabbits that have reached puberty).

Mathematically, the solution to the riddle is described by the formula: F(n) = F(n−1) + F(n−2), where F(0) = 0, F(1) = 1, n ≥ 2 and is an integer .

The answer to the riddle is that in a year there will be 377 pairs of rabbits. Such rapid growth is due to the idealization of the conditions in which rabbits live and breed, unattainable in the real world. Of course, in order to consider the dynamics of changes in the number of any species, it is necessary to significantly refine the model presented by Fibonacci. However, its contribution to the development of science is difficult to overestimate: some researchers consider this task to be the first population model in history.

## Golden Ratio

Elements of the Fibonacci sequence have a number of interesting and practical characteristics. The fundamental and most important property of the Fibonacci numbers is the existence of a limit on the ratio of neighboring elements, with the growth of the number of elements tending to the golden ratio, approximately equal to 1.618.

The first mention of the golden section goes back to the works of Euclid on the construction of a regular triangle, and the concept of this proportion that is widely used today and familiar to us was introduced only in the 18th century.

The graphical reflection of the principles of the golden section and, accordingly, the Fibonacci numbers is a golden spiral, which is an arc of circles inscribed in squares, the sides of which are related to each other as elements of a sequence. The golden spiral has found wide application in many areas of life: from art and architecture to applied mathematics and programming.

Of course, the useful properties and applications of the Fibonacci sequence are not limited to those mentioned. The significance of Fibonacci numbers in people's lives with the development of civilization is only growing, so this discovery can truly be called great. Read more about the practical benefits of the sequence - you will undoubtedly discover something new for yourself.

## Interesting facts

- Leonardo of Pisa himself never called himself "Fibonacci". He usually signed Bonacci, sometimes using the name Leonardo Bigollo (the word bigollo in the Tuscan dialect - "wanderer, idler").
- The fingers of the human hand, with the exception of the thumbs, consist of three phalanges. The ratio of the top two to the entire length of the finger is equal to the golden ratio.
- The logos of Apple, Pepsi, and Twitter follow the golden ratio.
- The Milky Way rotates clockwise. Rotation is described by a spiral, which is based on the golden ratio of 1.6180339...

- Help
## What is the Fibonacci sequence and how does it work?

Many people tend to think of mathematics as boring and rather abstract. The Fibonacci sequence is one of the most striking proofs of the practicality and materiality of the world of mathematics.

### Fibonacci everywhere

Fibonacci numbers have a property that is very important for practical use - the existence of a limit on the ratio of two neighboring elements. This ratio tends to the golden ratio, equal to 1.618, as the serial number of numbers increases... This fact determines the vastness of the areas of use of Fibonacci numbers.

In fact, in the life of each of us, the influence of this sequence is much more significant than we can imagine. Fibonacci numbers describe both many natural phenomena (spirals on the fruits of trees and animal bodies, the arrangement of leaves and even the human structure), as well as a significant part of the theoretical and applied scientific achievements of our time (in the field of photography, architecture, cryptography, programming, trading). Let's expand on some of these examples.

### Art, architecture and design

The proportion of the golden spiral, based on the golden ratio, is one of the most common and well-known proportions due to its harmony and versatility. Artists and architects began to use the principle of the golden spiral in their works to achieve the ideal proportions of objects back in the Ancient East, and the golden section reached its greatest prosperity in the Renaissance.

At the same time, the idea is still relevant today: to create the most eye-pleasing composition of objects in the frame, many photographers and cameramen use a grid based on the principle of the golden section, and 3D modelers find application for Fibonacci numbers in creating virtual models and competent ratio of their parts.

### Pseudo-random number generation and cryptography

With the development of computer technologies and methods for studying real models, there is a need for researchers to generate random numbers. However, the computing power of modern computers is not enough to generate random numbers even in the most outstanding experiments.

Scientists have managed to find a compromise between saving material and time resources, and having some of the most important properties of random numbers - these are pseudo-random numbers. The sequence of such numbers is the result of the work of some algorithm and is cyclical (numbers begin to repeat from a certain moment), but it is quite suitable for solving a number of problems important for science. Of course, the determining factor in the quality of the pseudo-random number generator is the value of the period after which these numbers begin to repeat. This is most important in the field of cryptography (encryption and data integrity).

In the middle of the last century, scientists proposed a method for generating pseudo-random numbers, involving the use of Fibonacci numbers. The generator, which was based on the properties of the Fibonacci sequence, is still used today, including in applied (biology, bioinformatics) tasks.

In addition to those mentioned, there are many more examples of the practical use of Fibonacci numbers, many of which we do not pay attention to in everyday life. The scope of this sequence is indeed astonishingly wide. If this topic interests you, explore for yourself other areas in which Fibonacci numbers occur and discover entertaining and useful mathematics.

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