Fibonacci sequence illustrated by nature. Fibonacci numbers and the golden ratio: the relationship

However, this is not all that can be done with the golden ratio. If we divide the unit by 0.618, then it turns out 1.618, if we square it, then we get 2.618, if we square it, we get the number 4.236. These are the Fibonacci expansion coefficients. What is missing is the number 3.236 proposed by John Murphy.

What do the specialists think about the sequence?

Someone will say that these numbers are already familiar, because they are used in technical analysis programs to determine the amount of correction and expansion. In addition, these same series play an important role in the Eliot wave theory. They are its numerical basis.

Our expert Nikolay Verified portfolio manager of Vostok investment company.

- Nikolay, do you think it is accidental that the appearance of Fibonacci numbers and its derivatives on the charts of various instruments? And can one say: “Fibonacci series practical application” takes place?
- I feel bad about mysticism. And on the exchange charts all the more. Everything has its own reasons. in the book “Fibonacci Levels” he beautifully told where the golden section appears, that he did not begin to be surprised that it appeared on the stock quotes charts. But in vain! In many of the examples he cited, the number Pi often appears. But for some reason it is not in price ratios.
“So you don’t believe in the validity of the Eliot wave principle?”
“No, that's not the point.” The wave principle is one thing. The numerical ratio is different. And the reasons for their appearance on price charts are the third
- What, in your opinion, are the reasons for the appearance of the golden section on stock charts?
- The correct answer to this question may be able to earn the Nobel Prize in economics. So far we can guess the true reasons. They are clearly not in harmony with nature. There are many models of exchange pricing. They do not explain the indicated phenomenon. But not understanding the nature of the phenomenon should not deny the phenomenon as such.
- And if this law is ever open, can it destroy the exchange process?
- As the same wave theory shows, the law of changing stock prices is pure psychology. It seems to me that knowledge of this law will not change anything and cannot destroy the exchange.

Material provided by the blog of the webmaster Maxim.

The coincidence of the foundations of the principles of mathematics in a variety of theories seems incredible. Maybe it's a fantasy or a fit to the end result. Wait and see. Much of what was previously considered unusual or was not possible: space exploration, for example, has become familiar and does not surprise anyone. Also, the wave theory, which may be incomprehensible, will eventually become more accessible and understandable. What was previously unnecessary in the hands of an analyst with experience will become a powerful tool for predicting further behavior.

Fibonacci numbers in nature.

Look

And now, let's talk about how you can refute the fact that the Fibonacci digital series is involved in any patterns in nature.

Take any other two numbers and build a sequence with the same logic as the Fibonacci numbers. That is, the next member of the sequence is equal to the sum of the two previous ones. For example, let's take two numbers: 6 and 51. Now we will build a sequence that we end with two numbers 1860 and 3009. Note that when dividing these numbers, we get a number close to the golden ratio.

Moreover, the numbers that were obtained by dividing the other pairs decreased from the first to the last, which suggests that if this series continues indefinitely, we will get a number equal to the golden ratio.

Thus, Fibonacci numbers do not stand out by anything. There are other sequences of numbers, of which there are an infinite number, which give as a result of the same operations the golden number phi.

Fibonacci was not an esoteric. He did not want to invest any mysticism in numbers, he simply solved the ordinary problem of rabbits. And he wrote a sequence of numbers that flowed from his task, in the first, second and other months, how many rabbits would be after breeding. Within a year, he received that very sequence. And did not make a relationship. No golden proportion, the Divine relation of speech did not go. All this was invented after him in the Renaissance.

Before mathematics, the virtues of Fibonacci are enormous. He adopted the system of numbers from the Arabs and proved its justice. It was a hard and long struggle. From the Roman numeral system: heavy and inconvenient for counting. She disappeared after the French Revolution. It has nothing to do with the Fibonacci golden ratio.

There are infinitely many spirals, the most popular: the natural logarithm spiral, the Archimedes spiral, the hyperbolic spiral.

Now let's take a look at the Fibonacci spiral. This piecewise-composite unit consists of several quarters of circles. And it is not a spiral as such.

Output

No matter how long we look for confirmation or refutation of the applicability of the Fibonacci series on the exchange, such a practice exists.

Huge masses of people operate according to the Fibonacci line, which is located in many user terminals. Therefore, whether we want to or not: Fibonacci numbers influence, and we can take advantage of this influence.

We read the article without fail.

The text of the work is posted without images and formulas.
The full version of the work is available in the tab "Files of work" in PDF format

Introduction

The highest assignment of mathematics is to find a hidden order in Chaos that surrounds us.

Wiener N.

A person strives for knowledge all his life, trying to study the world around him. And in the process of observation, he has questions that need to be answered. Answers are found, but new questions arise. In archaeological finds, in traces of civilization, distant from each other in time and space, one and the same element is found - a pattern in the form of a spiral. Some consider it a symbol of the sun and associate it with the legendary Atlantis, but its true meaning is unknown. What is common between the forms of the galaxy and the atmospheric cyclone, the arrangement of leaves on the stem and seeds in a sunflower? These patterns come down to the so-called “golden” spiral, the amazing Fibonacci sequence discovered by the great Italian mathematician of the 13th century.

Fibonacci numbers history

For the first time I heard about Fibonacci numbers from a mathematics teacher. But, in addition, how the sequence of these numbers is formed, I did not know. That's what this sequence is really famous for, how it affects a person, and I want to tell you. Little is known about Leonardo Fibonacci. There is not even an exact date of his birth. It is known that he was born in 1170 in the family of a merchant, in the city of Pisa in Italy. Fibonacci's father often visited Algeria on business matters, and Leonardo studied mathematics there from Arab teachers. Subsequently, he wrote several mathematical works, the most famous of which is The Abacus Book, which contains almost all the arithmetic and algebraic information of that time. 2

Fibonacci numbers are a sequence of numbers with a number of properties. Fibonacci discovered this numerical sequence by chance when he tried in 1202 to solve the practical problem of rabbits. “Someone placed a pair of rabbits in a place enclosed by a wall on all sides on all sides to find out how many pairs of rabbits will be born during the year, if the nature of the rabbits is such that in a month a pair of rabbits gives birth to another pair, and rabbits give birth from the second months after his birth. " In solving the problem, he took into account that each pair of rabbits gives rise to two more pairs during life, and then dies. So a sequence of numbers appeared: 1, 1, 2, 3, 5, 8, 13, 21, ... In this sequence, each next number is equal to the sum of the two previous ones. It was called the Fibonacci sequence. The mathematical properties of the sequence

I wanted to investigate this sequence, and I revealed some of its properties. This pattern is of great importance. The sequence is approaching more and more slowly to a certain constant ratio of approximately 1,618, and the ratio of any number to the next is approximately equal to 618.

You can notice a number of interesting properties of Fibonacci numbers: two adjacent numbers are coprime; every third number is even; every fifteenth ends in zero; every fourth is a multiple of three. If you select any 10 neighboring numbers from the Fibonacci sequence and add them together, you always get a multiple of 11. But that's not all. Each sum is equal to the number 11 times the seventh member of the sequence. And here is another curious feature. For any n, the sum of the first n members of the sequence will always be equal to the difference between the (n + 2) th and first members of the sequence. This fact can be expressed by the formula: 1 + 1 + 2 + 3 + 5 + ... + an \u003d a n + 2 - 1. Now we have the following trick: to find the sum of all terms

sequence between two given members, it is enough to find the difference of the corresponding (n + 2) -x members. For example, a 26 + ... + a 40 \u003d a 42 - a 27. Now look for the connection between Fibonacci, Pythagoras and the Golden Ratio. The most famous evidence of the mathematical genius of mankind is the Pythagorean theorem: in any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of its legs: c 2 \u003d b 2 + a 2. From a geometric point of view, we can consider all sides of a right-angled triangle, as sides of three squares built on them. The Pythagorean theorem says that the total area of \u200b\u200bsquares built on the legs of a right triangle is equal to the area of \u200b\u200ba square built on the hypotenuse. If the lengths of the sides of a right triangle are integers, then they form a group of three numbers called Pythagorean triples. Using the Fibonacci sequence, one can find such triples. Take any four consecutive numbers from the sequence, for example, 2, 3, 5 and 8, and construct three more numbers as follows: 1) the product of two extreme numbers: 2 * 8 \u003d 16; 2) the double product of two numbers in the middle: 2 * (3 * 5) \u003d 30; 3) the sum of the squares of two averages: 3 2 +5 2 \u003d 34; 34 2 \u003d 30 2 +16 2. This method works for any four consecutive Fibonacci numbers. Any three consecutive Fibonacci numbers behave in a predictable way. If you multiply the two extreme ones and compare the result with the square of the average, then the result will always differ by one. For example, for the numbers 5, 8 and 13 we get: 5 * 13 \u003d 8 2 +1. If you consider this property from the point of view of geometry, you will notice something strange. Divide the square

8x8 in size (a total of 64 small squares) into four parts, the lengths of the sides of which are equal to Fibonacci numbers. Now from these parts we will build a 5x13 rectangle. Its area is 65 small squares. Where does the extra square come from? The thing is that an ideal rectangle does not form, but tiny gaps remain, which in total give this additional unit of area. Pascal's triangle also has a connection with the Fibonacci sequence. You just need to write the lines of the Pascal triangle one below the other, and then stack the elements diagonally. The result is a Fibonacci sequence.

Now consider the “golden” rectangle, one side of which is 1.618 times longer than the other. At first glance, it may seem to us like an ordinary rectangle. However, let's do a simple experiment with two ordinary bank cards. We put one of them horizontally and the other vertically so that their lower sides are on the same line. If we draw a diagonal line in a horizontal map and extend it, we will see that it passes exactly through the upper right corner of the vertical map - a pleasant surprise. Maybe this is an accident, or maybe such rectangles and other geometric shapes that use the "golden ratio" are especially pleasing to the eye. Did Leonardo da Vinci think of the golden ratio while working on his masterpiece? This seems unlikely. However, it can be argued that he attached great importance to the connection between aesthetics and mathematics.

Fibonacci numbers in nature

The relationship of the golden ratio with beauty is not only a matter of human perception. It seems that nature itself singled out a special role. If squares are sequentially entered into the “golden” rectangle, then an arc is drawn in each square, an elegant curve is obtained, which is called a logarithmic spiral. It is not at all a mathematical curiosity. 5

On the contrary, this wonderful line is often found in the physical world: from the nautilus shell to the sleeves of galaxies, and in the elegant spiral of blossoming rose petals. The connections between the golden ratio and the Fibonacci numbers are numerous and unexpected. Consider a flower that looks very different from a rose - a sunflower with seeds. The first thing we see is that the seeds are arranged in two types of spirals: clockwise and counterclockwise. If we count the spirals of the hourly hand, we get two seemingly ordinary numbers: 21 and 34. This is not the only example when you can find Fibonacci numbers in the structure of plants.

Nature gives us numerous examples of the arrangement of homogeneous objects described by Fibonacci numbers. In various spiral arrangements of small parts of plants, two families of spirals can usually be seen. In one of these families, spirals curl clockwise, and in the other, counterclockwise. The numbers of spirals of one and the other type often turn out to be adjacent Fibonacci numbers. So, taking a young pine twig, it is easy to notice that the needles form two spirals, going from left to bottom right up. On many cones, the seeds are located in three spirals, hollowly winding on the stem of the cone. They are located in five spirals, steeply spiraling in the opposite direction. In large cones, it is possible to observe 5 and 8, and even 8 and 13 spirals. Fibonacci spirals are also clearly visible on the pineapple: usually there are 8 and 13 of them.

The shoot of chicory makes a strong release into space, stops, releases a leaf, but is already shorter than the first, again makes a release into space, but of lesser strength, releases a leaf of even smaller size and again a release. The impulses of its growth gradually decrease in proportion to the "golden" section. To appreciate the enormous role of Fibonacci numbers, just look at the beauty of the nature around us. Fibonacci numbers can be found in numbers

branches on the stem of each growing plant and among the petals.

Let’s recount the petals of some flowers — iris with its 3 petals, primrose with 5 petals, ragweed with 13 petals, nimbus with 34 petals, asters with 55 petals, etc. Is it accidental, or is it a law of nature? Look at the stems and flowers of yarrow. Thus, the total Fibonacci sequence can easily be interpreted as the pattern of manifestations of the "Golden" numbers found in nature. These laws operate regardless of our consciousness and the desire to accept them or not. The patterns of "golden" symmetry are manifested in the energy transitions of elementary particles, in the structure of certain chemical compounds, in planetary and space systems, in the gene structures of living organisms, in the structure of individual organs of the person and the body as a whole, and also manifest themselves in biorhythms and the functioning of the brain and visual perception.

Fibonacci numbers in architecture

The "Golden Section" is also manifested in many remarkable architectural creations throughout the history of mankind. It turns out that even ancient Greek and ancient Egyptian mathematicians knew these coefficients long before the Fibonacci and called them the "golden ratio". The Greeks used the principle of the "golden section" in the construction of the Parthenon, the Egyptians - the Great Pyramid in Giza. Advances in the field of construction equipment and the development of new materials have opened up new opportunities for architects of the twentieth century. American Frank Lloyd Wright was one of the main proponents of organic architecture. Shortly before his death, he designed the Solomon Guggenheim Museum in New York, which is an overturned spiral, and the interior of the museum resembles a nautilus shell. Polish-Israeli architect Zvi Hecker also used spiral designs in a 1995 Heinz Galinsky school project in Berlin. Hecker began with the idea of \u200b\u200ba sunflower with a central circle, from where

all architectural elements diverge. The building is a combination

orthogonal and concentric spirals, symbolizing the interaction of limited human knowledge and controlled chaos of nature. Its architecture mimics a plant that follows the movement of the sun, so classrooms are lit throughout the day.

In Quincy Park, located in Cambridge, Massachusetts (USA), the "golden" spiral can often be found. The park was designed in 1997 by artist David Phillips and is located near the Clay Institute of Mathematics. This institution is a famous center for mathematical research. In Quincy Park you can stroll among the "golden" spirals and metal curves, reliefs from two shells and a rock with a square root symbol. On the plate is written information about the "golden" proportion. Even bicycle parking uses the symbol F.

Fibonacci numbers in psychology

In psychology, turning points, crises, upheavals are marked, which signify the transformation of the structure and functions of the soul on the life path of a person. If a person has successfully overcome these crises, he becomes able to solve the tasks of a new class, which he had not even thought about before.

The presence of fundamental changes gives reason to consider life time as a decisive factor in the development of spiritual qualities. After all, nature does not measure our time generously, “no matter how much it will be, so much it will be”, but just enough so that the development process materializes:

in body structures;

in feelings, thinking and psychomotorics - until they acquire harmonynecessary for the emergence and start of the mechanism

creativity

in the structure of human energy potential.

The development of the body cannot be stopped: the child becomes an adult. The mechanism of creativity is not so simple. Its development can be stopped and its direction changed.

Is there a chance to catch up with time? Of course. But for this you need to do a lot of work on yourself. That which develops freely, naturally, does not require special efforts: the child freely develops and does not notice this enormous work, because the process of free development is created without violence against oneself.

How is the meaning of the life path in everyday consciousness understood? The layman sees it this way: at the foot - birth, at the top - the flowering of forces, and then - everything goes downhill.

The sage will say: everything is much more complicated. He divides the ascent into stages: childhood, adolescence, youth ... Why so? Few people are able to answer, although everyone is sure that these are closed, integral stages of life.

To find out how the mechanism of creativity is developing, V.V. Klimenko used mathematics, namely, the laws of Fibonacci numbers and the proportion of the "golden section" - the laws of nature and human life.

Fibonacci numbers divide our life into stages according to the number of years lived: 0 - the beginning of the count - the child was born. He still lacks not only psychomotor skills, thinking, feelings, imagination, but also operational energy potential. He is the beginning of a new life, a new harmony;

1 - the child has mastered walking and masters the immediate environment;

2 - understands speech and acts using verbal instructions;

3 - acts through the word, asks questions;

5 - “the age of grace” - the harmony of psychomotorism, memory, imagination and feelings that already allow the child to embrace the world in its entirety;

8 - feelings come to the fore. The imagination serves them, and thinking by the forces of its criticality is aimed at supporting the internal and external harmony of life;

13 - the talent mechanism begins to work, aimed at transforming the material acquired in the process of inheritance, developing its own talent;

21 - the mechanism of creativity has approached the state of harmony and attempts are being made to perform talented work;

34 — harmony of thinking, feelings, imagination and psychomotor skills: the ability to work brilliantly is born;

55 - at this age, subject to the preserved harmony of body and soul, a person is ready to become a creator. And so on…

What are serifs for the Fibonacci Numbers? They can be compared with dams on the path of life. These dams await each of us. First of all, it is necessary to overcome each of them, and then patiently raise your level of development until one day it falls apart, opening the way for the next to the free flow.

Now that we understand the meaning of these nodal points of age development, we will try to decipher how all this happens.

In 1 year the child takes possession of walking. Prior to this, he knew the world with the front of his head. Now he knows the world with his hands - the exclusive privilege of man. The animal moves in space, and he, knowing, takes possession of space and develops the territory on which he lives.

2 years - understands the word and acts in accordance with it. It means that:

the child learns the minimum number of words - meanings and modes of action;

it doesn’t yet separate itself from the environment and merged into integrity with the environment,

therefore, acts on someone else’s direction. At this age, he is the most obedient and pleasant for parents. From a sensual person, a child turns into a cognitive person.

3 years- action using your own word. This person has already separated from the environment - and he is learning to be an independent person. Hence he:

consciously confronts the environment and parents, kindergarten teachers, etc .;

recognizes its sovereignty and fights for independence;

tries to subordinate close and well-known people to his will.

Now for a child, a word is an action. The acting person begins with this.

5 years- "the age of grace." He is the personification of harmony. Games, dances, adroit movements - everything is saturated with harmony, which a person tries to master on his own. Harmonious psychomotorism helps to lead to a new state. Therefore, the child is aimed at psychomotor activity and strives for the most active actions.

Materialization of work products of sensitivity is carried out by:

ability to display the environment and ourselves as part of this world (we hear, see, touch, smell, etc. - all senses work on this process);

ability to design the outside world, including myself

(the creation of a second nature, hypotheses - to do both tomorrow, build a new machine, solve a problem), by the forces of critical thinking, feelings and imagination;

the ability to create a second, man-made nature, products of activity (the implementation of the plan, specific mental or psychomotor actions with specific objects and processes).

After 5 years, the mechanism of imagination comes forward and begins to dominate the rest. The child performs a gigantic work, creating fantastic images, and lives in a world of fairy tales and myths. The hypertrophy of the child’s imagination is surprising in adults, because the imagination does not correspond to reality.

8 years - feelings come to the fore and own measurements of feelings (cognitive, moral, aesthetic) arise when the child accurately:

evaluates the known and the unknown;

distinguishes moral from immoral, moral from immoral;

beautiful from that which threatens life, harmony from chaos.

13 years old - the mechanism of creativity begins to work. But this does not mean that it works at full capacity. One of the elements of the mechanism comes to the fore, and all the others contribute to its work. If harmony is maintained in this age period of development, which restructures its structure almost all the time, then the lad will painlessly get to the next dam, quietly overcome it and live at the age of a revolutionary. At the age of a revolutionary, the youth must take a new step forward: to separate from the closest society and live in it a harmonious life and activity. Not everyone can solve this problem that arises before each of us.

21 years old. If the revolutionary has successfully overcome the first harmonious peak of life, then his talent mechanism is capable of fulfilling the talented

work. Feelings (cognitive, moral or aesthetic) sometimes overshadow thinking, but in general, all elements work together: feelings are open to the world, and logical thinking is able to call and find measures of things from this peak.

The mechanism of creativity, developing normally, reaches a state that allows you to receive certain fruits. He starts to work. At this age, the mechanism of feelings comes forward. As the imagination and its products are evaluated by feelings and thinking, antagonism arises between them. Feelings prevail. This ability is gradually gaining power, and the lad begins to use it.

34 years- balance and harmony, productive effectiveness of talent. The harmony of thinking, feelings and imagination, psychomotorism, which is replenished with optimal energy potential, and the mechanism as a whole - the ability to perform brilliant work is born.

55 years - a person can become a creator. The third harmonious peak of life: thinking subjugates the power of feelings.

Fibonacci numbers call the stages of human development. Whether a person passes this path non-stop depends on the parents and teachers, the educational system, and then on himself and how the person will know and overcome himself.

On the path of life, a person discovers 7 objects of relationships:

From birthday to 2 years - the discovery of the physical and objective world of the immediate environment.

From 2 to 3 years - the discovery of oneself: "I am myself."

From 3 to 5 years - speech, an effective world of words, harmony and the "I - You" system.

From 5 to 8 years - the discovery of the world of other people's thoughts, feelings and images - the system "I - We".

From 8 to 13 years - the discovery of the world of tasks and problems solved by the geniuses and talents of mankind - the "I - Spirituality" system.

From 13 to 21 years - the discovery of the ability to independently solve the well-known tasks, when thoughts, feelings and imagination begin to work actively, the "I - Noosphere" system arises.

From 21 to 34 years - the discovery of the ability to create a new world or its fragments - the awareness of the self-concept "I am the Creator."

The life path has a spatio-temporal structure. It consists of age and individual phases, determined by many parameters of life. A person masters to a certain extent the circumstances of his life, becomes the creator of his history and the creator of the history of society. A truly creative attitude to life, however, does not appear immediately and not even in every person. Between the phases of the life path there are genetic connections, and this determines its regular nature. It follows that, in principle, future development can be predicted based on knowledge of its early phases.

Fibonacci numbers in astronomy

From the history of astronomy it is known that I. Titius, a German astronomer of the 18th century, using the Fibonacci series, found a pattern and order in the distances between the planets of the solar system. But one case, it would seem, was against the law: there was no planet between Mars and Jupiter. But after the death of Titius at the beginning of the XIX century. concentrated observation of this part of the sky led to the discovery of the asteroid belt.

Conclusion

In the process of research, I found out that Fibonacci numbers are widely used in the technical analysis of prices on the stock exchange. One of the simplest ways to apply Fibonacci numbers in practice is to determine the length of time through which an event will occur, for example, a change in price. The analyst counts a certain number of Fibonacci days or weeks (13,21,34,55, etc.) from a previous similar event and makes a prediction. But this is still too difficult for me to figure out. Although Fibonacci was the greatest mathematician of the Middle Ages, the only monuments of Fibonacci are a statue in front of the Leaning Tower of Pisa and two streets that bear his name: one in Pisa and the other in Florence. And yet, in connection with everything I saw and read, quite logical questions arise. Where did these numbers come from? Who is this architect of the universe who tried to make it perfect? What will be next? Finding the answer to one question, you get the following. You will solve it, you will get two new ones. You will deal with them, three more will appear. Having solved them, you will get five unresolved. Then eight, thirteen, etc. Do not forget that on two hands there are five fingers, two of which consist of two phalanges, and eight of three.

Literature:

Voloshinov A.V. "Mathematics and Art", M., Enlightenment, 1992.

Vorobyov N.N. "Fibonacci numbers", M., Science, 1984.

Stakhov A.P. The Da Vinci Code and the Fibonacci Series, Peter Format, 2006

F. Corvalan “The Golden Ratio. The mathematical language of beauty ”, M., De Agostini, 2014

Maksimenko S.D. "Sensitive periods of life and their codes."

Fibonacci numbers. Wikipedia

Transfer

Introduction

Fibonacci numbers programmers should already be fed up. Examples of their calculation are used everywhere. It all comes down to the fact that these numbers provide the simplest example of recursion. They are also a good example of dynamic programming. But is it necessary to calculate them like this in a real project? Do not. Neither recursion nor dynamic programming are ideal options. And not a closed formula using floating point numbers. Now I will tell you how to do it right. But first, let's go through all the known solutions.

The code is for Python 3, although it should go for Python 2 as well.

To begin with - I recall the definition:

F n \u003d F n-1 + F n-2

And F 1 \u003d F 2 \u003d 1.

Closed formula

We skip the details, but those who wish can familiarize themselves with the conclusion of the formula. The idea is to assume that there is a certain x for which F n \u003d x n, and then find x.

What does it mean

Cut x n-2

We solve the quadratic equation:

This is where the “golden section” grows ϕ \u003d (1 + √5) / 2. Substituting the initial values \u200b\u200band doing another calculation, we get:

Which we use to calculate F n.

From __future__ import division import math def fib (n): SQRT5 \u003d math.sqrt (5) PHI \u003d (SQRT5 + 1) / 2 return int (PHI ** n / SQRT5 + 0.5)

Good:
Fast and easy for small n
The bad:
Floating point operations are required. For large n, greater accuracy is required.
Evil:
Using complex numbers to calculate F n is beautiful from a mathematical point of view, but ugly from a computer.

Recursion

The most obvious solution that you have seen many times already is most likely as an example of what recursion is. I will repeat it again, for completeness. In Python, it can be written on one line:

Fib \u003d lambda n: fib (n - 1) + fib (n - 2) if n\u003e 2 else 1

Good:
Very simple implementation, repeating the mathematical definition
The bad:
Exponential lead time. For large n, it is very slow
Evil:
Stack overflow

Memorization

A recursive solution has a big problem: intersecting calculations. When fib (n) is called, fib (n-1) and fib (n-2) are counted. But when fib (n-1) is counted, it again independently counts fib (n-2) - that is, fib (n-2) is counted twice. If we continue the discussion, we will see that fib (n-3) will be counted three times, etc. Too many intersections.

Therefore, you just need to remember the results so as not to count them again. This solution consumes time and memory in a linear fashion. I use a dictionary in the solution, but a simple array could be used.

M \u003d (0: 0, 1: 1) def fib (n): if n in M: return M [n] M [n] \u003d fib (n - 1) + fib (n - 2) return M [n]

(In Python, this can also be done using the decorator, functools.lru_cache.)

Good:
Just turn recursion into a memorized solution. Converts exponential execution time to linear, for which it spends more memory.
The bad:
Spends a lot of memory
Evil:
Stack overflow possible, like recursion

Dynamic programming

After the decision with memorization, it becomes clear that we do not need all the previous results, but only the last two. In addition, instead of starting with fib (n) and going back, you can start with fib (0) and go forward. The following code has linear execution time, and memory usage is fixed. In practice, the speed of the solution will be even higher, since there are no recursive function calls and the work associated with this. And the code looks simpler.

This solution is often cited as an example of dynamic programming.

Def fib (n): a \u003d 0 b \u003d 1 for __ in range (n): a, b \u003d b, a + b return a

Good:
Fast for small n, simple code
The bad:
Still linear runtime
Evil:
Yes, nothing special.

Matrix algebra

And finally, the least illuminated, but the most correct solution, competently using both time and memory. It can also be extended to any homogeneous linear sequence. The idea of \u200b\u200busing matrices. Just see that

A generalization of this suggests that

Two values \u200b\u200bfor x that we obtained earlier, of which one was a golden section, are the eigenvalues \u200b\u200bof the matrix. Therefore, another way to derive a closed formula is to use a matrix equation and linear algebra.

So how useful is this formulation? The fact that exponentiation can be done in a logarithmic time. This is done by squaring. The bottom line is that

Where the first expression is used for even A, the second for odd. It remains only to organize the multiplication of matrices, and you're done. It turns out the following code. I organized a recursive implementation of pow as it is easier to understand. See the iterative version here.

Def pow (x, n, I, mult): "" "Returns x to the power of n. Assumes that I is the identity matrix that multiplies with mult, and n is a positive integer" "" if n \u003d\u003d 0: return I elif n \u003d\u003d 1: return x else: y \u003d pow (x, n // 2, I, mult) y \u003d mult (y, y) if n% 2: y \u003d mult (x, y) return y def identity_matrix (n): "" "Returns the identity matrix n by n" "" r \u003d list (range (n)) return [for j in r] def matrix_multiply (A, B): BT \u003d list (zip (* B) ) return [for row_a in A] def fib (n): F \u003d pow ([,], n, identity_matrix (2), matrix_multiply) return F

Good:
Fixed memory, logarithmic time
The bad:
More complicated code
Evil:
I have to work with matrices, although they are not so bad

Performance comparison

Only the option of dynamic programming and the matrix is \u200b\u200bworth comparing. If we compare them by the number of characters in the number n, it turns out that the matrix solution is linear, and the solution with dynamic programming is exponential. A practical example is calculating fib (10 ** 6), a number that has more than two hundred thousand characters.

N \u003d 10 ** 6
We calculate fib_matrix: fib (n) has a total of 208988 digits, the calculation took 0.24993 seconds.
We calculate fib_dynamic: fib (n) has a total of 208988 digits, the calculation took 11.83377 seconds.

Theoretical remarks

Not directly touching the code above, this comment still has some interest. Consider the following graph:

We count the number of paths of length n from A to B. For example, for n \u003d 1 we have one path, 1. For n \u003d 2 we again have one path, 01. For n \u003d 3 we have two paths, 001 and 101 It is quite simple to show that the number of paths of length n from A to B is exactly F n. Having written the adjacency matrix for the graph, we get the same matrix that was described above. This is a well-known result from graph theory that for a given adjacency matrix A, occurrences in A n are the number of paths of length n in the graph (one of the problems mentioned in the movie "Good Will Hunting").

Why are there such signs on the edges? It turns out that when you look at an infinite sequence of characters on a sequence of paths on a graph that is infinite on both sides, you get something called "subshifts of finite type", which is a type of system of symbolic dynamics. Specifically, this sub-shift of a finite type is known as the “shift of the golden section” and is defined by a set of “forbidden words” (11). In other words, we get binary sequences that are infinite in both directions and no pairs of them are adjacent. The topological entropy of this dynamical system is equal to the golden ratio ϕ. It is interesting how this number periodically appears in different areas of mathematics.

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Have you ever heard that mathematics is called the “queen of all sciences”? Do you agree with this statement? As long as mathematics remains a set of boring tasks for you in a textbook, you can hardly feel the beauty, versatility and even humor of this science.

But there are such topics in mathematics that help make curious observations of things and phenomena that are common to us. And even try to penetrate behind the veil of secrecy in the creation of our universe. There are curious patterns in the world that can be described using mathematics.

We present you the Fibonacci numbers

Fibonacci numbers called elements of a numerical sequence. In it, each next number in a row is obtained by summing the two previous numbers.

Example sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 ...

It can be written like this:

F 0 \u003d 0, F 1 \u003d 1, F n \u003d F n-1 + F n-2, n ≥ 2

You can start a series of Fibonacci numbers with negative values. n. Moreover, the sequence in this case is two-sided (i.e., covers negative and positive numbers) and tends to infinity in both directions.

An example of such a sequence: -55, -34, -21, -13, -8, 5, 3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55.

The formula in this case looks like this:

F n \u003d F n + 1 - F n + 2 or else you can do this: F -n \u003d (-1) n + 1 Fn.

What we now know under the name "Fibonacci numbers" was known to ancient Indian mathematicians long before they began to be used in Europe. And with this name in general one continuous historical joke. To begin with, Fibonacci himself never called himself Fibonacci during his lifetime - this name began to be applied to Leonardo of Pisa only a few centuries after his death. But let's talk about everything in order.

Leonardo of Pisa, aka Fibonacci

The son of a merchant who became a mathematician, and subsequently received the recognition of posterity as the first major mathematician in Europe during the Middle Ages. Not least thanks to the Fibonacci numbers (which then, recall, have not yet been called that). Which he described at the beginning of the 13th century in his work Liber abaci (Book of the Abacus, 1202).

Traveling with his father to the East, Leonardo studied mathematics with Arab teachers (and at that time they were in this business, and in many other sciences, some of the best specialists). He read the works of mathematicians of Antiquity and Ancient India in Arabic translations.

Having thoroughly comprehended everything read and connected his own inquiring mind, Fibonacci wrote several scientific treatises on mathematics, including the already mentioned Abacus Book. In addition to her, he created:

Practica geometriae (Practice of Geometry, 1220);
Flos (Flower, 1225 - study on cubic equations);
Liber quadratorum (Book of squares, 1225 - problems on indefinite quadratic equations).

He was a big fan of mathematical tournaments, so in his treatises he paid much attention to the analysis of various mathematical problems.

Very little biographical information is left about the life of Leonardo. As for the name Fibonacci, under which he entered the history of mathematics, it was entrenched in him only in the 19th century.

Fibonacci and its tasks

After Fibonacci there remained a large number of problems that were very popular among mathematicians in the following centuries. We will consider the problem of rabbits, in the solution of which the Fibonacci numbers are used.

Rabbits are not only valuable fur

Fibonacci set such conditions: there is a pair of newborn rabbits (male and female) of such an interesting breed that they regularly (starting from the second month) produce offspring - always one new pair of rabbits. Also, as you might guess, male and female.

These conditioned rabbits are placed in a confined space and breed with enthusiasm. It is also stipulated that no rabbit dies from any mysterious rabbit disease.

We need to calculate how many rabbits we get in a year.

At the beginning of 1 month we have 1 pair of rabbits. At the end of the month they mate.
The second month - we already have 2 pairs of rabbits (in a couple - parents + 1 couple - their offspring).
Third month: The first pair gives birth to a new pair, the second pair mates. Total - 3 pairs of rabbits.
Fourth month: The first pair gives birth to a new pair, the second pair does not lose time and also gives birth to a new pair, the third pair is only mating. Total - 5 pairs of rabbits.

The number of rabbits in nmonth \u003d the number of pairs of rabbits from the previous month + the number of newborn pairs (there are as many as pairs of rabbits 2 months before now). And all this is described by the formula that we have already given above: F n \u003d F n-1 + F n-2.

Thus, we obtain a recurrence (explanation of recursion - below) a numerical sequence. In which each following number is equal to the sum of the previous two:

1 + 1 = 2
2 + 1 = 3
3 + 2 = 5
5 + 3 = 8
8 + 5 = 13
13 + 8 = 21
21 + 13 = 34
34 + 21 = 55
55 + 34 = 89
89 + 55 = 144
144 + 89 = 233
233+ 144 = 377 <…>

The sequence can be continued for a long time: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987<…>. But since we set a specific deadline - a year, we are interested in the result obtained on the 12th “move”. Those. 13th member of the sequence: 377.

The answer in the problem: 377 rabbits will be obtained subject to all the stated conditions.

One of the properties of a sequence of Fibonacci numbers is very curious. If you take two consecutive pairs from a row and divide the larger number into the smaller one, the result will gradually approach the golden ratio (you can read more about it later in the article).

In the language of mathematics, “Relationship limit a n + 1to a nequal to the golden ratio ".

More problems in number theory

Find a number that can be divided by 7. In addition, if you divide it by 2, 3, 4, 5, 6, the remainder will be one.
Find the square number. It is known about him that if you add 5 to it or subtract 5, you will again get a square number.

We suggest that you search for the answers to these tasks yourself. You can leave your options to us in the comments to this article. And then we will tell you whether your calculations were correct.

Recursion Explanation

Recursion - definition, description, image of an object or process in which this object or process itself is contained. That is, in fact, an object or process is part of itself.

Recursion is widely used in mathematics and computer science, and even in art and popular culture.

Fibonacci numbers are determined using the recurrence relation. For the number n\u003e 2 n-e number is (n - 1) + (n - 2).

Golden Section Explanation

Golden ratio - dividing the whole (for example, a segment) into such parts that are correlated according to the following principle: the larger part refers to the smaller one as well as the entire value (for example, the sum of two segments) to the larger part.

The first mention of the golden ratio can be found at Euclid in his treatise "Beginnings" (about 300 years BC). In the context of building a regular rectangle.

The term familiar to us was introduced in 1835 by the German mathematician Martin Om.

If you describe the golden ratio approximately, it is a proportional division into two unequal parts: approximately 62% and 38%. In numerical terms, the golden ratio is a number 1,6180339887 .

The Golden Ratio finds practical application in the visual arts (paintings by Leonardo da Vinci and other Renaissance painters), architecture, cinema (“Battleship Potemkin” by S. Ezenshtein) and other fields. For a long time it was believed that the golden ratio is the most aesthetic proportion. This opinion is popular today. Although according to the results of research visually most people do not perceive such a proportion as the most successful option and consider it too elongated (disproportionate).

Cut length from = 1, but = 0,618, b = 0,382.
Attitude from to but = 1, 618.
Attitude fromto b = 2,618

Now back to the Fibonacci numbers. Take two successive members from its sequence. Divide the larger number by the smaller and get approximately 1.618. And now we use the same larger number and the next member of the series (i.e., an even larger number) - their ratio is early 0.618.

Here is an example: 144, 233, 377.

233/144 \u003d 1.618 and 233/377 \u003d 0.618

By the way, if you try to do the same experiment with numbers from the beginning of the sequence (for example, 2, 3, 5), nothing will work. Almost. The golden ratio rule is almost not respected to begin the sequence. But then, as you move along the row and increase in numbers, it works fine.

And in order to calculate the entire series of Fibonacci numbers, it is enough to know the three members of the sequence, one after the other. You can see for yourself!

Golden Rectangle and Fibonacci Spiral

Another interesting parallel between the Fibonacci numbers and the golden ratio allows us to draw the so-called “golden rectangle”: its sides are correlated in the proportion of 1.618 to 1. But we already know what the number 1.618 is, right?

For example, take two consecutive members of the Fibonacci series - 8 and 13 - and build a rectangle with the following parameters: width \u003d 8, length \u003d 13.

And then break the big rectangle into smaller ones. Prerequisite: the lengths of the sides of the rectangles must correspond to the Fibonacci numbers. Those. the side length of the larger rectangle should be equal to the sum of the sides of the two smaller rectangles.

So, as is done in this figure (for convenience, the figures are signed in Latin letters).

By the way, you can build rectangles in the reverse order. Those. start the construction with squares with side 1. To which, guided by the principle stated above, figures with sides equal to the Fibonacci numbers are added. Theoretically, this can be continued indefinitely - after all, the Fibonacci series is formally infinite.

If we connect the corners of the rectangles obtained in the figure with a smooth line, we get a logarithmic spiral. Rather, her special case is the Fibonacci spiral. It is characterized, in particular, by the fact that it has no boundaries and does not change shape.

A similar spiral is often found in nature. Mollusk shells are one of the most striking examples. Moreover, some galaxies that can be seen from the Earth have a spiral shape. If you pay attention to weather forecasts on TV, you may have noticed that cyclones have a similar spiral shape when shooting them from satellites.

It is curious that the DNA helix also obeys the golden ratio rule - the corresponding pattern can be seen in the intervals of its bends.

Such amazing “coincidences” cannot but excite the minds and do not generate talk about a certain unified algorithm to which all phenomena in the life of the Universe obey. Now do you understand why this article is called that way? And the doors to what amazing worlds can mathematics open for you?

Fibonacci numbers in wildlife

The relationship between the Fibonacci numbers and the Golden Ratio suggests thoughts about curious patterns. So curious that there is a temptation to try to find sequences similar to Fibonacci numbers in nature and even during historical events. And nature really gives rise to such assumptions. But can everything in our life be explained and described using mathematics?

Examples of wildlife that can be described using the Fibonacci sequence:

the arrangement of leaves (and branches) in plants - the distances between them are correlated with Fibonacci numbers (phyllotaxis);

arrangement of sunflower seeds (seeds are located in two rows of spirals twisted in different directions: one row clockwise, the other counterclockwise);

the location of the scales of pine cones;
flower petals;
pineapple cells;
the ratio of the lengths of the phalanges of the fingers on the human hand (approximately), etc.

Combinatorics Tasks

Fibonacci numbers are widely used in solving combinatorics problems.

Combinatorics - this is a branch of mathematics that studies the selection of a certain given number of elements from the indicated set, listing, etc.

Let's look at examples of combinatorics problems, designed for the high school level (source - http://www.problems.ru/).

Task number 1:

Alex goes up the stairs from 10 steps. At one time, he jumps up either one step or two steps. How many ways can Lesha climb the stairs?

The number of ways that Lesha can climb the stairs from n steps, we denote and n.It follows that a 1 = 1, a 2 \u003d 2 (after all, Lesha jumps either one or two steps).

It is also stipulated that Alex jumps up the stairs from n\u003e 2 steps. Suppose the first time he jumped two steps. So, by the condition of the problem, he needs to jump another n - 2 steps. Then the number of ways to finish the climb is described as a n – 2. And if we assume that the first time Lesha jumped only one step, then the number of ways to finish the climb will be described as a n – 1.

Hence we get the following equality: a n \u003d a n – 1 + a n – 2 (looks familiar, doesn't it?).

Once we know a 1and a 2and remember that the steps according to the conditions of problem 10, calculated in order all and n: a 3 = 3, a 4 = 5, a 5 = 8, a 6 = 13, a 7 = 21, a 8 = 34, a 9 = 55, a 10 = 89.

Answer: 89 ways.

Task number 2:

It is required to find the number of words 10 letters long, which consist only of the letters "a" and "b" and should not contain two letters "b" in a row.

Denote by a n number of words in length nletters that consist only of the letters “a” and “b” and do not contain two letters “b” in a row. Means a 1= 2, a 2= 3.

In sequence a 1, a 2, <…>, a nwe will express each next member through the previous ones. Therefore, the number of words in length nletters that also do not contain the double letter “b” and begin with the letter “a”, this a n – 1. And if the word is long nletters begins with the letter “b”, it is logical that the next letter in such a word is “a” (after all, there cannot be two “b” by the condition of the problem). Therefore, the number of words in length nletters in this case we denote a n – 2. In both the first and second cases, any word (length of n - 1and n - 2 letters, respectively) without double "b".

We were able to substantiate why a n \u003d a n – 1 + a n – 2.

We calculate now a 3= a 2+ a 1= 3 + 2 = 5, a 4= a 3+ a 2= 5 + 3 = 8, <…>, a 10= a 9+ a 8\u003d 144. And we get the familiar Fibonacci sequence.

The answer is 144.

Task number 3:

Imagine that there is a tape broken into cells. She goes to the right and lasts forever. Place a grasshopper on the first square of the ribbon. Whichever of the cells of the tape he is, he can only move to the right: either one cell, or two. How many ways a grasshopper can jump from the beginning of the tape to nth cell?

Let us denote the number of ways a grasshopper can move along the tape to nth cell like a n. In this case a 1 = a 2 \u003d 1. Also in n + 1-th cell the grasshopper can get either from nth cell, or jumping over it. From here a n + 1 = a n - 1 + a n. Where from a n = F n - 1.

Answer: F n - 1.

You can compose such problems yourself and try to solve them in math classes with classmates.

Fibonacci numbers in popular culture

Of course, such an unusual phenomenon as Fibonacci numbers cannot but attract attention. Nevertheless, there is something attractive and even mysterious in this strictly verified regularity. It is not surprising that the Fibonacci sequence somehow "lit up" in many works of modern mass culture of various genres.

We will tell you about some of them. And you try to search for yourself yet. If you find, share with us in the comments - we are also curious!

Fibonacci numbers are mentioned in Dan Brown's best-selling book The Da Vinci Code: the Fibonacci sequence serves as the code by which the main characters of the book open the safe.
In the 2009 American film “Mr. Nobody,” in one of the episodes, the address of the house is part of the Fibonacci sequence - 12358. In addition, in another episode, the protagonist must call the phone number, which is essentially the same, but slightly distorted (extra figure) after the number 5) sequence: 123-581-1321.
In the 2012 series “Communication,” the protagonist, a boy with autism, is able to distinguish patterns in what is happening in the world. Including through Fibonacci numbers. And manage these events also through numbers.
The developers of the Doom RPG java game for mobile phones placed a secret door on one of the levels. The code that opens it is the Fibonacci sequence.
In 2012, the Russian rock band Spleen released a concept album, Optical Illusion. The eighth track is called Fibonacci. In the verses of the leader of the group, Alexander Vasiliev, the sequence of Fibonacci numbers is beaten. For each of the nine consecutive terms, there is a corresponding number of lines (0, 1, 1, 2, 3, 5, 8, 13, 21):

0 The composition has set off

1 Snapped one joint

1 Jerked one sleeve

2 Get it stuff

Get it stuff

3 A request for boiling water

The train goes to the river

The train goes in the taiga<…>.

limerick (a short poem of a certain form - usually five lines, with a certain rhyming scheme, comic in content, in which the first and last lines are repeated or partially duplicate) James Lindon also uses a reference to the Fibonacci sequence as a humorous motive:

Dense Fibonacci wives

It only benefited them, not otherwise.

Wives weighed according to rumor

Each is like the previous two.

To summarize

We hope that we were able to tell you a lot of interesting and useful things today. For example, you can now search for the Fibonacci spiral in the nature around you. Suddenly, it is you who will be able to unravel the "secret of life, the universe and in general."

Use the formula for Fibonacci numbers when solving combinatorics problems. You can rely on the examples described in this article.

site, with full or partial copying of the material, a link to the source is required.

The Italian mathematician Leonardo Fibonacci lived in the 13th century and was one of the first in Europe to use Arabic (Indian) numbers. He came up with a somewhat artificial problem about rabbits that are raised on a farm, all of which are considered females, males are ignored. Rabbits begin to breed after they turn two months old, and then each month give birth to a rabbit. Rabbits never die.

It is necessary to determine how many rabbits will be on the farm through n months, if at the initial time there was only one newborn rabbit.

Obviously, the farmer has one rabbit in the first month and one rabbit in the second month. In the third month there will already be two rabbits, in the fourth - three, etc. Denote the number of rabbits in n month like. In this way,
,
,
,
,
, …

You can build an algorithm that allows you to find for any n.

According to the condition of the problem, the total number of rabbits
at n+1 month is decomposed into three components:

one-month-old rabbits incapable of breeding, in the amount of

;

Thus, we obtain

. (8.1)

Formula (8.1) allows you to calculate a series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ...

The numbers in this sequence are called fibonacci numbers .

If accept
and
, then using the formula (8.1) we can determine all the other Fibonacci numbers. Formula (8.1) is called recurrent the formula ( recurrence - “return” in Latin).

Example 8.1.Suppose there is a staircase in n steps. We can climb it in steps of one step, or - in steps of two steps. How many combinations of different lifting methods are there?

If n \u003d 1, there is only one solution to the problem. For n \u003d 2 there are 2 options: two single steps or one double. For n \u003d 3 there are 3 options: three unit steps, or one unit and one double, or one double and one unit.

In the following case n \u003d 4, we have 5 possibilities (1 + 1 + 1 + 1, 2 + 1 + 1, 1 + 2 + 1, 1 + 1 + 2, 2 + 2).

In order to answer the question at random n, denote the number of options as , and try to determine
by famous and
. If we start with a single step, then we have combinations for the remaining n steps. If we start with a double step, then we have
combinations for the remaining n–1 steps. Total number of options for n+1 steps equals

. (8.2)

The resulting formula as a twin resembles formula (8.1). However, this does not allow to identify the number of combinations with Fibonacci numbers . We see, for example, that
but
. However, the following relationship occurs:

This is true for n \u003d 1, 2, and also holds true for each n. Fibonacci numbers and the number of combinations are calculated using the same formula, however, the initial values
,
and
,
they differ.

Example 8.2.This example is of practical importance for error-correcting coding problems. Find the number of all binary words of length nnot containing multiple zeros in a row. Denote this number by . Obviously
, and the words of length 2 satisfying our restriction are: 10, 01, 11, i.e.
. Let be
- such a word from n characters. If symbol
then
may be arbitrary (
) is a letter word that does not contain several zeros in a row. So the number of words with a unit at the end is
.

If the symbol
then be sure
, and the first
character
can be arbitrary taking into account the considered restrictions. Therefore, there is
words of length n with zero at the end. Thus, the total number of words of interest to us is equal to

Given the fact that
and
, the resulting sequence of numbers is the Fibonacci numbers.

Example 8.3In example 7.6, we found that the number of binary words of constant weight t (and length k) is equal . Now find the number of binary words of constant weight tnot containing multiple zeros in a row.

You can reason like this. Let be
the number of zeros in the words in question. Any word has
gaps between the nearest zeros, each of which contains one or more units. It is assumed that
. Otherwise, there is not a single word without adjacent zeros.

If we remove exactly one unit from each gap, we get a word of length
containing zeros. Any such word can be obtained in the indicated way from some (and, moreover, only one) ka letter containing zeros, no two of which stand side by side. Hence, the desired number coincides with the number of all words of length
containing exactly zeros i.e. equally
.

Example 8.4.Let us prove that the sum
equal to the Fibonacci numbers for any integer . Symbol
denotes smallest integer greater than or equal to . For example, if
then
; what if
then
ceil ("ceiling"). The symbol also occurs.
which denotes largest integer less than or equal to . In English, this operation is called floor ("floor").

If
then
. If
then
. If
then
.

Thus, for the cases considered, the sum is really equal to the Fibonacci numbers. Now we give a proof for the general case. Since the Fibonacci numbers can be obtained using the recurrence equation (8.1), then the equality

And it really does:

Here we used the previously obtained formula (4.4):
.

Sum of Fibonacci numbers

We determine the sum of the first n Fibonacci numbers.

0+1+1+2+3+5 = 12,

0+1+1+2+3+5+8 = 20,

0+1+1+2+3+5+8+13 = 33.

It is easy to see that by adding one to the right side of each equation, we again get the Fibonacci number. The general formula for determining the sum of the first n Fibonacci numbers has the form:

We prove this using the method of mathematical induction. To do this, we write:

This amount should be equal to
.

Reducing the left and right sides of the equation by –1, we obtain equation (6.1).

Formula for Fibonacci numbers

Theorem 8.1. Fibonacci numbers can be calculated by the formula

Evidence. We verify the validity of this formula for n \u003d 0, 1, and then we prove the validity of this formula for an arbitrary n by induction. We calculate the ratio of the two nearest Fibonacci numbers:

We see that the ratio of these numbers fluctuates around 1.618 (if you ignore the first few values). By this property, the members of the geometric progression resemble Fibonacci numbers. Will accept
, (
) Then the expression

converted to

which after simplifications looks like this

We got a quadratic equation whose roots are equal:

Now we can write:

(Where c is a constant). Both members and don't give Fibonacci numbers for example
, while
. However the difference
satisfies the recurrence equation:

For n\u003d 0, this difference gives , i.e:
. However, when n\u003d 1 we have
. To obtain
, you must accept:
.

Now we have two sequences: and
that start with the same two numbers and satisfy the same recurrence formula. They should be equal:
. The theorem is proved.

With increasing n member getting very big while
, and the role of the member in the difference is reduced. Therefore, for large n approximately we can write

We ignore 1/2 (since the Fibonacci numbers increase to infinity with growth n to infinity).

Attitude
called golden ratio, it is used outside of mathematics (for example, in sculpture and architecture). The golden ratio is the ratio between the diagonal and the side regular pentagon (Fig. 8.1).

Fig. 8.1. Regular pentagon and its diagonals

To denote the golden ratio, it is customary to use the letter
in honor of the famous Athenian sculptor Phidias.

Prime numbers

All natural numbers, large units, fall into two classes. The first includes numbers that have exactly two natural divisors, one and oneself, the second - all the others. First class numbers are called simpleand the second - composite. Prime numbers within the first three dozen: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...

The properties of primes and their relationship with all natural numbers were studied by Euclid (3rd century BC). If you write down primes in a row, you will notice that their relative density decreases. There are 4 in the first ten, i.e. 40%, in the hundred - 25, i.e. 25%, per thousand - 168, i.e. less than 17%, per million - 78498, i.e. less than 8%, etc. .. However, their total number is infinite.

Among the primes there are pairs of such, the difference between which is equal to two (the so-called simple twins), however, the finiteness or infinity of such pairs has not been proved.

Euclid considered it obvious that by multiplying only primes all natural numbers can be obtained, and each natural number can be represented as a product of primes uniquely (up to the order of factors). Thus, primes form a multiplicative basis of the natural series.

Studying the distribution of primes has led to the creation of an algorithm that allows you to obtain tables of primes. Such an algorithm is sieve of eratosthenes (3rd century BC). This method consists in sifting out (for example, by striking through) those integers of a given sequence
which are divisible by at least one of the primes smaller
.

Theorem 8 . 2 . (Euclidean theorem). The number of primes is infinite.

Evidence. We prove the Euclidean theorem on the infinity of primes by the method proposed by Leonard Euler (1707–1783). Euler reviewed the product for all primes p:

at
. This product converges, and if it is revealed, then, due to the uniqueness of the decomposition of natural numbers into prime factors, it turns out that it is equal to the sum of the series , whence the Euler identity follows:

Since when
since the series on the right diverges (harmonic series), then the Euclidean theorem follows from Euler's identity.

Russian mathematician P.L. Chebyshev (1821–1894) derived a formula defining the limits in which the number of primes lies
not exceeding X:

where
,
.