Biography of Leonardo of Pisa, aka fibonacci. Leonardo fibonacci - life under the auspices of Emperor Leonardo of Pisa short biography

Plan:

Introduction

• 1 Fibonacci, Arabic Numerals and Banking
• 2 Scientific activity
• 3 Fibonacci numbers
• 4 Fibonacci targets
Literature
Notes

Introduction

Leonardo of Pisa(lat. Leonardo Pisano, about 1170, Pisa - about 1250, ibid) - the first major mathematician of medieval Europe. Best known by the nickname Fibonacci (Fibonacci); There are different versions about the origin of this pseudonym. According to one of them, his father Guillermo had the nickname Bonacci (« well-intentioned”), and Leonardo himself was nicknamed Filius Bonacci("son of the Well-Meaning"). According to another Fibonacci comes from the phrase Figlio Buono Nato Ci, which means "a good son was born" in Italian.

Fibonacci's father was often in Algeria on business, and Leonardo studied mathematics there with Arab teachers. Later he visited Egypt, Syria, Byzantium, Sicily. Leonardo studied the works of mathematicians of Islamic countries (such as al-Khwarizmi and Abu Kamil); from Arabic translations, he also got acquainted with the achievements of ancient and Indian mathematicians. Based on the knowledge he acquired, Fibonacci wrote a number of mathematical treatises, which are an outstanding phenomenon of medieval Western European science.

In the 19th century, a monument to the scientist was erected in Pisa.

1. Fibonacci, Arabic Numerals and Banking

It is impossible to imagine modern accounting and financial accounting in general without the use of the decimal number system and Arabic numerals, the beginning of which was used in Europe by Fibonacci.

One of the Pisan bankers, who traded in Tunisia and was engaged there in loans and paying off taxes and customs fees, a certain Leonardo Fibonacci, applied Arabic numerals to banking accounting, thus introducing them to Europe.

Article "Banker" // ENE (ESBE)

2. Scientific activity

A significant part of the knowledge he acquired, he outlined in his outstanding "Book of the Abacus" ( Liber abaci, 1202; only the supplemented manuscript of 1228 has survived to this day). This book contains almost all the arithmetic and algebraic information of that time, presented with exceptional completeness and depth. The first five chapters of the book are devoted to integer arithmetic based on decimal numbering. In chapters VI and VII, Leonardo outlines operations on ordinary fractions. Chapters VIII-X present methods for solving commercial arithmetic problems based on proportions. Chapter XI deals with mixing problems. Chapter XII presents tasks for summing series - arithmetic and geometric progressions, a series of squares and, for the first time in the history of mathematics, a reciprocal series leading to a sequence of so-called Fibonacci numbers. Chapter XIII sets out the rule of two false positions and a number of other problems reduced to linear equations. In the XIV chapter, Leonardo, using numerical examples, explains how to approximate the extraction of square and cube roots. Finally, in the XV chapter a number of problems on the application of the Pythagorean theorem and a large number of examples on quadratic equations are collected.

The "Book of the abacus" rises sharply above the European arithmetic and algebraic literature of the 12th-14th centuries. the variety and strength of methods, the richness of tasks, the evidence of presentation. Subsequent mathematicians widely drew from it both problems and methods for solving them.

Fibonacci monument in Pisa

"The Practice of Geometry" ( Practica geometriae, 1220) contains various theorems related to measurement methods. Along with the classical results, Fibonacci gives his own - for example, the first proof that the three medians of a triangle intersect at one point (Archimedes knew this fact, but if his proof existed, it did not reach us).

In the treatise "Flower" ( Flos, 1225) Fibonacci explored the cubic equation x 3 + 2x 2 + 10x = 20 , offered to him by John of Palermo at a mathematical competition at the court of Emperor Frederick II. John of Palermo himself almost certainly borrowed this equation from Omar Khayyam's treatise On the Proofs of Problems in Algebra, where it is given as an example of one of the types in the classification of cubic equations. Leonardo of Pisa investigated this equation, showing that its root cannot be rational or have the form of one of the quadratic irrationalities found in Book X of Euclid's Elements, and then found the approximate value of the root in sexagesimal fractions, equal to 1; 22.07.42, 33,04,40, without indicating, however, the method of its solution.

"The Book of Squares" ( Liber quadratorum, 1225), contains a number of problems for solving indefinite quadratic equations. In one of the problems, also proposed by John of Palermo, it was required to find a rational square number, which, when increased or decreased by 5, again gives rational square numbers.

3. Fibonacci numbers

In honor of the scientist, a number series is named, in which each subsequent number is equal to the sum of the previous two. This number sequence is called the Fibonacci numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, … (OEIS sequence A000045)

This series was known in ancient India long before Fibonacci. The Fibonacci numbers got their current name due to the study of the properties of these numbers, carried out by the scientist in his work The Book of the Abacus (1202).

4. Fibonacci tasks

• "The Problem of Rabbit Breeding".
• “The problem of weights” (“The problem of choosing the best system of weights for weighing on a balance scale”):

1, 3, 9, 27, 81,... (degrees of 3, OEIS sequence A009244)

Literature

• History of mathematics from ancient times to the beginning of the 19th century (under the editorship of A.P. Yushkevich), volume II, M., Nauka, 1972, pp. 260-267.
• Karpushina N."Liber abaci" by Leonardo Fibonacci, Mathematics at School, No. 4, 2008.
• Shchetnikov A.I. On the reconstruction of an iterative method for solving cubic equations in medieval mathematics. Proceedings of the third Kolmogorov readings. Yaroslavl: Publishing House of YaGPU, 2005, p. 332-340.
• Yaglom I. M. Italian merchant Leonardo Fibonacci and his rabbits. // Kvant, 1984. No. 7. P. 15-17.
• Glushkov S. On approximation methods of Leonardo Fibonacci. Historia Mathematica, 3, 1976, p. 291-296.
• Sigler, L.E. Fibonacci's Liber Abaci, Leonardo Pisano's Book of Calculations" Springer. New York, 2002, ISBN 0-387-40737-5.

Notes

1. Karpushina N. M. "Liber abaci" by Leonardo Fibonacci, Mathematics at School, No. 4, 2008 http://n-t.ru/tp/in/la.htm - n-t.ru/tp/in/la.htm
2. A. P. Stakhov. Two famous Fibonacci problems http://www.goldenmuseum.com/1001TwoProblems_eng.html - www.goldenmuseum.com/1001TwoProblems_eng.html
3. Leonardo Pisano Fibonacci http://www.xfibo.ru/fibonachi/leonardo-pisano-fibonacci.htm - www.xfibo.ru/fibonachi/leonardo-pisano-fibonacci.htm

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Introduction

A person strives for knowledge, tries to study the world that surrounds him. In the process of observation, numerous questions arise, which, accordingly, need to be answered. A person is looking for these answers, and finding them, other questions appear.

Today, in the age of high technology, the study is carried out not only on our planet Earth, but also beyond its borders - in the Universe. But this does not mean that everything on Earth has been studied, but on the contrary, there remains a huge number of incomprehensible and inexplicable phenomena. But there are “answers” ​​that explain several such phenomena at once.

It turns out that the regularity of natural phenomena, the structure and diversity of living organisms on our planet, everything that surrounds us, striking the imagination with its harmony and orderliness, the laws of the universe, the movement of human thought and the achievements of science - all this can be tried to be explained by the Fibonacci sequence.

But let's talk about everything in order.

Biography

Leonardo of Pisa, aka Fibonacci.
Very little biographical information remains about Leonardo's life. As for the name Fibonacci, under which he entered the history of mathematics, it was fixed to him only in the 19th century.
Leonardo of Pisa never called himself Fibonacci; this pseudonym was given to him later, presumably by Guillaume Libri in 1838. The word Fibonacci is short for the two words "filius Bonacci" that appeared on the cover of The Book of the Abacus; they could mean either "son of Bonaccio" or, if the word Bonacci is interpreted as a surname, "son of Bonacci". According to the third version, the very word Bonacci must also be understood as a nickname meaning "lucky". He himself usually signed Bonacci; sometimes he also used the name Leonardo Bigollo - the word bigollo in the Tuscan dialect meant "wanderer", as well as "loafer".
Fibonacci was born in the Italian city of Pisa, presumably in the 1170s (some sources say 1180). His father, Guillermo, was a merchant. At that time, Pisa was one of the largest commercial centers actively cooperating with the Islamic East, and Fibonacci's father actively traded in one of the trading posts founded by the Italians on the northern coast of Africa. In 1192, he was appointed to represent the Pisan trading colony in North Africa and frequented Bejai, Algeria. Thanks to this, he managed to "arrange" his son, the future great mathematician Fibonacci, in one of the Arab schools, where he was able to receive an excellent mathematical education for that time. Leonardo studied the works of mathematicians from the countries of the Muslim faith (such as al-Khwarizmi and Abu Kamil); from Arabic translations, he also got acquainted with the achievements of ancient and Indian mathematicians.

Later Fibonacci visited Egypt, Syria, Byzantium, Sicily.

Based on the knowledge he acquired, Fibonacci wrote a number of mathematical treatises, which are an outstanding phenomenon of medieval Western European science.
In 1200, Leonardo returned to Pisa and set about writing his first work, The Book of the Abacus. At that time, very few people in Europe knew about the positional number system and Arabic numerals. In his book, Fibonacci strongly supported Indian methods of calculation and methods. According to the historian of mathematics A.P. Yushkevich, “The Book of the Abacus rises sharply above the European arithmetic and algebraic literature of the 12th-14th centuries by the variety and power of methods, the richness of problems, the evidence of presentation ... Subsequent mathematicians widely drew from it both problems and techniques their decisions." According to the first book, many generations of European mathematicians studied the Indian positional number system.

The work of Leonardo Fibonacci "The Book of the Abacus" contributed to the spread in Europe of a positional number system, more convenient for calculations than Roman notation; in this book, the possibilities of using Indian numerals, which had previously remained unclear, were studied in detail, and examples were given of solving practical problems, in particular, those related to trading. The positional system gained popularity in Europe during the Renaissance.

The book interested Emperor Frederick II and his courtiers, among whom was the astrologer Michael Scotus, the philosopher Theodorus Physicus and Dominicus Hispanus. The latter suggested that Leonardo be invited to court on one of the emperor's visits to Pisa around 1225, where he was given tasks by Johannes of Palermo, another court philosopher of Frederick II. Some of these problems appeared in Fibonacci's later work. Thanks to a good education, Leonardo managed to attract the attention of Emperor Frederick II during mathematical tournaments. Subsequently, Leonardo enjoyed the patronage of the emperor.
For several years Fibonacci lived at the emperor's court. His work The Book of Squares, written in 1225, dates back to this time. The book is devoted to Diophantine equations of the second degree and puts Fibonacci on a par with such scientists who developed the theory of numbers as Diophantus and Fermat. The only mention of Fibonacci after 1228 is in 1240, when he was awarded a pension for services to the city in the Republic of Pisa.
No lifetime portraits of Fibonacci have been preserved, and the existing ones are modern ideas about him. Leonardo of Pisa left virtually no autobiographical information; the only exception is the second paragraph of The Book of the Abacus, where Fibonacci lays out his reasons for writing the book:
“When my father was assigned the position of a customs officer in charge of the affairs of the Pisan merchants who flocked to him in Bejaia, in my adolescence he called me to him and offered to study the counting art for several days, which promised many conveniences and benefits for my future. Taught by the mastery of teachers the basics of Indian counting, I acquired a great love for this art, and at the same time I learned that something about this subject is known among the Egyptians, Syrians, Greeks, Sicilians and Provencals, who developed their methods. Later, during my trading journeys throughout these parts, I devoted much labor to a detailed study of their methods, and, moreover, mastered the art of scientific dispute. However, compared with the method of the Indians, all the constructions of these people, including the approach of algorismists and the teachings of Pythagoras, seem almost delusional, and therefore I decided, having studied the Indian method as carefully as possible, to present it in fifteen chapters as clearly as I can, with additions from my own mind. and with some useful notes from Euclid's geometry inserted along the way. In order that the inquisitive reader may study Indian reckoning in the most thoughtful way, I have accompanied nearly every statement with convincing evidence; I hope that from now on the Latin people will not be deprived of the most accurate information about the art of calculations. If, more than expected, I missed something more or less important, or maybe necessary, then I pray for forgiveness, because there is no one among people who would be sinless or have the ability to foresee everything.
However, the exact meaning of this paragraph cannot be considered fully known, because its text, like the entire Latin text of the book, has come down to us with errors introduced by scribes.

Scientific activity
Much of the knowledge he acquired, he outlined in his "Book of the abacus"(Liberabaci, 1202; only the amended manuscript of 1228 survives to this day). This book consists of 15 chapters and contains almost all the arithmetic and algebraic information of that time, presented with exceptional completeness and depth. The first five chapters of the book are devoted to integer arithmetic based on decimal numbering. In chapters VI and VII, Leonardo outlines operations on ordinary fractions. Chapters VIII-X present methods for solving commercial arithmetic problems based on proportions. Chapter XI deals with mixing problems. Chapter XII presents tasks for summing series - arithmetic and geometric progressions, a series of squares and, for the first time in the history of mathematics, a reciprocal series leading to a sequence of so-called Fibonacci numbers. Chapter XIII sets out the rule of two false positions and a number of other problems reduced to linear equations. In the XIV chapter, Leonardo, using numerical examples, explains how to approximate the extraction of square and cube roots. Finally, in the XV chapter a number of problems on the application of the Pythagorean theorem and a large number of examples on quadratic equations are collected. Leonardo was the first in Europe to use negative numbers, which he considered as debt. The book is dedicated to Mikael Scotus.
Another Fibonacci book "The Practice of Geometry"(Practicageometriae, 1220), consists of seven parts and contains various theorems with proofs relating to measurement methods. Along with the classical results, Fibonacci gives his own - for example, the first proof that the three medians of a triangle intersect at one point (Archimedes knew this fact, but if his proof existed, it did not reach us). Among the land surveying techniques to which the last section of the book is devoted is the use of a square marked in a certain way to determine distances and heights. To determine the number π, Fibonacci uses the perimeters of the inscribed and circumscribed 96-gon, which leads him to the value

3.1418. The book was dedicated to Dominicus Hispanus. In 1915

R. S. Archibald was engaged in the restoration of the lost work of Euclid on the division of figures, based on the "Practice of Geometry" by Fibonacci and the French translation of the Arabic version.
In the treatise "Flower"(Flos, 1225) Fibonacci studied the cubic equation x 3 + 2x 2 + 10 x = 20 offered to him by John of Palermo at a mathematical contest at the court of Emperor Frederick II. John of Palermo himself almost certainly borrowed this equation from Omar Khayyam's treatise On the Proofs of Problems in Algebra, where it is given as an example of one of the types in the classification of cubic equations. Leonardo of Pisa investigated this equation, showing that its root cannot be rational or have the form of one of the quadratic irrationalities found in the X book of Euclid's Elements, and then found the approximate value of the root in sexagesimal fractions, equal to 1; 22.07.42, 33,04,40, without indicating, however, the method of its solution.
"The Book of Squares"(Liberquadratorum, 1225) contains a number of problems for solving indefinite quadratic equations. Fibonacci worked on finding numbers that, when added to a square number, would again give a square number. He noted that the numbers x 2 + y 2 and x 2 − y 2 cannot be square at the same time, and also used the formula x 2 + (2 x + 1) = (x + 1) 2 to search for square numbers. In one of the tasks of the book,

also originally proposed by John of Palermo, it was required to find a rational square number, which, when increased or decreased by 5, again gives rational square numbers.

Among the works of Fibonacci that have not come down to us are Diminorguisa's treatise on commercial arithmetic, as well as commentaries on book X of Euclid's Elements.
What we now know as "Fibonacci numbers" was known to ancient Indian mathematicians long before they were used in Europe.

Fibonacci targets
Remaining true to mathematical tournaments, Fibonacci assigns the main role in his books to problems, their solutions and comments. Tasks for tournaments were proposed both by Fibonacci himself and by his rival, the court philosopher of Frederick II, Johannes of Palermo. Fibonacci problems, like their counterparts, continued to be used in various mathematical textbooks for several centuries. They can be found in Pacioli's "Sum of Arithmetic" (1494), in "Pleasant and Entertaining Problems" by Basche de Miziriac (1612), in Magnitsky's "Arithmetic" (1703), in Euler's "Algebra" (1768).
After Fibonacci, a large number of problems remained, which 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.
The rabbit problem
Fibonacci set the following 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 conditional rabbits are placed in a closed space and breed. It is also stipulated that no rabbit dies from some mysterious rabbit disease.

We need to calculate how many rabbits we will 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 (a pair has parents + 1 pair - 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 couple gives birth to a new couple, the second couple does not lose time and also gives birth to a new couple, the third couple is just mating. Total - 5 pairs of rabbits.

The number of rabbits in the nth month = the number of pairs of rabbits from the previous month + the number of newborn pairs (there are the same number of pairs of rabbits as there were 2 months before now). And all this is described by the formula that we have already given above: Fn = Fn-1 + Fn-2.

Thus, we get a recurrent (an explanation of recursion - below) numerical sequence. In which each next number is equal to the sum of the previous two:

233+ 144 = 377
You can continue the sequence for a long time: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987. But since we have set a specific period - a year, we are interested in the result obtained on the 12th "move". Those. 13th member of the sequence: 377.
The answer is in the problem: 377 rabbits will be obtained if all the stated conditions are met.
So, Reflecting on this topic, Fibonacci built the following series of numbers:

1,1,2,3,5,8,13,21,34,55,89,144,…

But as it turned out, this sequence has a number of remarkable properties.

Properties of the Fibonacci Sequence

1. The ratio of each number to the next more and more tends to 0.618 as the serial number increases. The ratio of each number to the previous one tends to 1.618 (reverse to 0.618).

2. When dividing each number by the next one, the number 0.382 is obtained through one; vice versa - respectively 2.618.

55: 144:55=2,618…

144=0,382…
3. Selecting ratios in this way, we obtain the main set of Fibonacci coefficients: … 4.235, 2.618, 1.618, 0.618, 0.382, 0.236.

One of the properties of the Fibonacci sequence is very curious. If you take two consecutive pairs from a series and divide the larger number by the smaller one, the result will gradually approach the golden ratio.

In the language of mathematics, "the limit of the ratios a n + 1 to a n is equal to the golden ratio."

An explanation about recursion
Recursion is a definition, description, image of an object or process that contains this object or process itself. That is, in fact, an object or process is a part of itself.
Recursion finds wide application in mathematics and computer science, and even in art and popular culture.
Fibonacci numbers are defined using a recursive relation. For a number n>2, the nth number is (n - 1) + (n - 2).

The golden ratio is the division of a whole (for example, a segment) into such parts that are correlated according to the following principle: a large part relates to a smaller one in the same way as the entire value (for example, the sum of two segments) to a larger part.
The first mention of the golden ratio can be found in Euclid's treatise "Beginnings" (about 300 BC). In the context of building a regular rectangle.
The term familiar to us in 1835 was introduced by the German mathematician Martin Ohm.
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 the number 1.6180339887.
The golden ratio finds practical application in the visual arts (paintings by Leonardo da Vinci and other Renaissance painters), architecture, cinema (S. Ezenstein's Battleship Potemkin) and other areas. For a long time it was believed that the golden ratio is the most aesthetic proportion. This view is still 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).

The length of the segment c \u003d 1, a \u003d 0.618, b \u003d 0.382.

The ratio c to a = 1.618.

Ratio c to b = 2.618

Now back to the Fibonacci numbers. Take two successive terms from its sequence. Divide the larger number by the smaller and get approximately 1.618. And now let's 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 = 1.618 and 233/377 = 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 for the beginning of the sequence. But on the other hand, as you move along the row and the numbers increase, it works fine.
And in order to calculate the entire series of Fibonacci numbers, it is enough to know three members of the sequence, following each other. You can see for yourself!
Kettlebell problems
The problem of choosing the best system of weights for weighing on a balance scale was first formulated by Fibonacci. Leonardo of Pisa offers two options for the task:
A simple option: you need to find five weights, with which you can find all weights less than 30, while weights can only be placed on one scale pan (Answer: 1, 2, 4, 8, 16).

The solution is built in the binary number system.

Difficult option: you need to find the smallest number of weights with which you can weigh all the weights less than the given one (Answer: 1, 3, 9, 27, 81, ...).

The solution is built in the base three number system and is generally the sequence A000244 in OEIS.

Problems in number theory
In addition to the rabbit problem, Fibonacci proposed a number of other problems in number theory:

Find a number that is divisible by 7 and has a remainder of 1 when divided by 2, 3, 4, 5 and 6;

Find the number whose product with seven gives the remainder 1, 2, 3, 4, 5 when divided by 2, 3, 4, 5, 6, respectively;

Find a square number (that is, a number equal to the square of an integer) that, when increased or decreased by 5, would give a square number.

Some other tasks
Find a number whose 19/20 is equal to the square of the number itself. (Answer: 19/20).

An alloy of 30 weight parts consists of three metals: the first metal is worth three coins per part, the second metal is worth two coins per part, and the third metal has one coin every two parts; the cost of the whole alloy is 30 coins. How many parts of each metal does the alloy contain? (Answer: 3 parts of the first metal, 5 parts of the second metal, 22 parts of the third). In such terms, Fibonacci reformulated the well-known problem about birds, which used the same numbers (30 birds of three different species cost 30 coins, at given prices, find the number of birds of each species).

"Joke problem about seven old women" who went to Rome, and each had seven mules, each of which had seven bags, each of which had seven loaves, each of which had seven knives, each of which had seven scabbard. You need to find the total number of items. This task went around many countries, the first known mention of it was in ancient Egypt in the papyrus of Ahmes. (Answer: 137256).
Problems in combinatorics
Fibonacci numbers are widely used in solving problems in combinatorics.
Combinatorics is a branch of mathematics that deals with the study of a selection of a given number of elements from a designated set, enumeration, etc.
Let's look at examples of combinatorics tasks designed for the high school level.
Task #1:
Lesha climbs a ladder of 10 steps. He jumps up either one step or two steps at a time. In how many ways can Lesha climb the stairs?
Solution:
The number of ways that Lesha can climb a ladder of n steps is denoted as n. It follows that a 1 = 1, a 2 = 2 (after all, Lesha jumps either one or two steps).
It is also stipulated that Lesha jumps on a ladder of n > 2 steps. Suppose he jumped two steps the first time. So, according to the condition of the problem, he needs to jump another n - 2 steps. Then the number of ways to complete the climb is described as a n–2 . And if we assume that for the first time Lesha jumped only one step, then we will describe the number of ways to complete the climb as a n–1 .
From here we get the following equality: a n = a n–1 + a n–2 (looks familiar, doesn't it?).
Since we know a 1 and a 2 and remember that there are 10 steps according to the condition of the problem, calculate all a n in order: 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 #2:
It is required to find the number of words with a length of 10 letters, which consist only of the letters "a" and "b" and should not contain two letters "b" in a row.
Solution:
Denote by a n the number of words of length n letters that consist only of the letters "a" and "b" and do not contain two letters "b" in a row. So a 1 = 2, a 2 = 3.
In the sequence a1, a2, a n, we will express each next term in terms of the previous ones. Therefore, the number of words of length n letters, which also do not contain a double letter "b" and begin with the letter "a", is a n-1. And if a word with a length of n letters 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" according to the condition of the problem). Therefore, the number of words of length n letters in this case will be denoted as a n–2 . In both the first and second cases, any word can follow (length of n - 1 and n - 2 letters, respectively) without doubled "b".
We were able to justify why a n = a n–1 + a n -2.
Let us now calculate a 3 = a 2 + a 1 = 3 + 2 = 5, a 4 = a 3 + a 2 = 5 + 3 = 8, a 10 = a 9 + a 8 = 144. And we get the familiar Fibonacci sequence.
Answer: 144.
Task #3:
Imagine that there is a tape divided into cells. It goes to the right and lasts indefinitely. Place a grasshopper on the first cell of the tape. On whichever of the cells of the tape he is, he can only move to the right: either one cell, or two. How many ways are there for a grasshopper to jump from the beginning of the tape to the nth cell?
Solution:
Let's denote the number of ways to move the grasshopper along the tape to the nth cell as a n . In this case, a 1 = a 2 = 1. Also, the grasshopper can get into the n + 1st cell either from the nth cell or by jumping over it. Hence a n + 1 = a n - 1 + a n . Whence a n \u003d F n - 1.
Answer: Fn - 1.
You can create similar problems yourself and try to solve them in math lessons with your classmates.

Fibonacci's works
Under the patronage of the emperor, Leonardo of Pisa wrote several books:

The Book of the Abacus (Liberabaci), 1202, supplemented in 1228;

"Practice of Geometry" (Practicageometriae), 1220;

"Flower" (Flos) 1225;

The Book of Squares (Liberquadratorum), 1225;

Diminorguisa, lost;

Commentary on Book X of Euclid's Elements, lost;

Letter to Theodorus, 1225.

Golden Rectangle and Fibonacci Spiral
Another curious parallel between the Fibonacci numbers and the golden ratio allows us to draw the so-called "golden rectangle": its sides are related in the proportion of 1.618 to 1. But we already know what the number 1.618 is, right?
For example, let's take two consecutive terms of the Fibonacci series - 8 and 13 - and build a rectangle with the following parameters: width = 8, length = 13.
And then we break the large rectangle into smaller ones. Mandatory condition: the lengths of the sides of the rectangles must correspond to the Fibonacci numbers. Those. the side length of the larger rectangle must be equal to the sum of the sides of the two smaller rectangles.
The way it 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 building from squares with a side of 1. To which, guided by the principle voiced above, figures with sides equal to the Fibonacci numbers are completed. 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, its special case is the Fibonacci spiral. It is characterized, in particular, by the fact that it has no boundaries and does not change shape.

Such a spiral is often found in nature. Mollusk shells are one of the most striking examples. Moreover, some galaxies that can be seen from 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 section rule - the corresponding pattern can be seen in the intervals of its bends.

Such amazing “coincidences” cannot but excite the minds and give rise to talk about some kind of single algorithm that all phenomena in the life of the Universe obey. Now you understand the doors to what amazing worlds mathematics can open for you?

Fibonacci numbers in nature
The connection between Fibonacci numbers and the golden ratio suggests curious patterns. So curious that it is tempting to try to find sequences like Fibonacci numbers in nature and even in the course of historical events. And nature indeed gives rise to such assumptions. But can everything in our life be explained and described with the help of mathematics?

It should be said that the Fibonacci spiral can be double. There are numerous examples of these double helixes found all over the place. This is how sunflower spirals always correlate with the Fibonacci series. Even in an ordinary pinecone, you can see this double Fibonacci spiral. The first spiral goes in one direction, the second - in the other. If we count the number of scales in a spiral rotating in one direction and the number of scales in the other spiral, we can see that these are always two consecutive numbers of the Fibonacci series. There may be eight in one direction and 13 in the other, or 13 in one and 21 in the other 3.

What is the difference between the Golden Ratio Spirals and the Fibonacci Spiral? The golden ratio spiral is perfect. It corresponds to the Primary source of harmony. This spiral has neither beginning nor end. She is endless. The Fibonacci spiral has a beginning, from which it starts “unwinding”. This is a very important property. It allows Nature, after the next closed cycle, to carry out the construction of a new spiral from “zero”.
So, examples of wildlife that can be described using the Fibonacci sequence:

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

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

location of 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.

Plants

Even Goethe emphasized the tendency of nature to spirality. The spiral and spiral arrangement of leaves on tree branches was noticed long ago. The spiral was seen in the arrangement of sunflower seeds, in pine cones, pineapples, cacti, etc. The joint work of botanists and mathematicians shed light on these amazing natural phenomena. It turned out that in the arrangement of leaves on a branch of sunflower seeds, pine cones, the Fibonacci series manifests itself, and therefore, the law of the golden section manifests itself.

Among the roadside grasses, an unremarkable plant grows - chicory. Let's take a closer look at it. A branch was formed from the main stem. Here is the first leaf. The process makes a strong ejection into space, stops, releases a leaf, but already shorter than the first one, again makes an ejection into space, but of less force, releases a leaf of an even smaller size and ejection again. If the first outlier is taken as 100 units, then the second is 62 units, the third is 38, the fourth is 24, and so on. The length of the petals is also subject to the golden ratio. In growth, the conquest of space, the plant retained certain proportions. Its growth impulses gradually decreased in proportion to the golden section.

Compositae plants

Platonic solids and the Fibonacci series

And now let's look at another remarkable property of the Fibonacci series.

There are only five unique forms that are of paramount importance. They are called Platanus bodies. Any Platonic solid has some special characteristics.

First, all faces of such a body are equal in size.

Secondly, the edges of the Platonic solid are of the same length.

Thirdly, the internal angles between its adjacent faces are equal.

And, fourthly, being inscribed in a sphere, the Platonic solid touches the surface of this sphere with each of its vertices.

There are only four shapes besides the cube that have all of these characteristics. The second body is a tetrahedron (tetra means "four"), having four faces in the form of equilateral triangles and four vertices. Another solid is the octahedron (octa means "eight"), whose eight faces are equilateral triangles of the same size. The octahedron contains 6 vertices. A cube has 6 faces and eight vertices. The other two Platonic solids are somewhat more complicated. One is called the icosahedron, which means "having 20 faces", represented by equilateral triangles. The icosahedron has 12 vertices. The other is called the dodecahedron (dodeca is "twelve"). Its faces are 12 regular pentagons. The dodecahedron has twenty vertices.

These bodies have the remarkable properties of being inscribed all in just two figures - a sphere and a cube. A similar relationship with the Platonic solids can be traced in all areas. So, for example, the system of orbits of the planets of the solar system can be represented as Platonic solids nested into each other, inscribed in the corresponding spheres, which determine the radii of the orbits of the corresponding planets of the solar system.

CONCLUSION

The Fibonacci series could have remained only a mathematical incident if it were not for the fact that all researchers of the golden division in the plant and animal world, not to mention art and architecture, invariably came to this series as an arithmetic expression of the golden division law.

Thus, the total Fibonacci sequence can easily interpret the pattern of manifestations of Golden numbers found in nature. These laws operate regardless of our knowledge, from someone's desire to accept or not accept them.
In my work, of course, I cannot state the essence of this issue to the smallest detail, but I tried to reflect the most interesting and significant aspects.

I am convinced that this topic will be relevant for a long time, and more and more new facts will be discovered confirming the presence and influence of the Fibonacci sequence on our lives.

I hope that I was able to tell you a lot of interesting and useful things today. For example, you can now look 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."
Although there is an opinion that almost all statements that find Fibonacci numbers in natural and historical phenomena are incorrect, this is a common myth, which often turns out to be an inexact fit to the desired result.

Republic of Pisa

Scientific activity

He set out a significant part of the knowledge he had acquired in his outstanding "Book of the abacus" ( Liber abaci, 1202; only the supplemented manuscript of 1228 has survived to this day). This book contains almost all the arithmetic and algebraic information of that time, presented with exceptional completeness and depth. The first five chapters of the book are devoted to integer arithmetic based on decimal numbering. In chapters VI and VII, Leonardo outlines operations on ordinary fractions. Chapters VIII-X present methods for solving commercial arithmetic problems based on proportions. Chapter XI deals with mixing problems. Chapter XII presents tasks for summing series - arithmetic and geometric progressions, a series of squares and, for the first time in the history of mathematics, a reciprocal series, leading to a sequence of so-called Fibonacci numbers. Chapter XIII sets out the rule of two false positions and a number of other problems reduced to linear equations. In the XIV chapter, Leonardo, using numerical examples, explains how to approximate the extraction of square and cube roots. Finally, in the XV chapter a number of problems on the application of the Pythagorean theorem and a large number of examples on quadratic equations are collected. Leonardo was the first in Europe to use negative numbers, which he considered as debt.

The "Book of the abacus" rises sharply above the European arithmetic and algebraic literature of the 12th-14th centuries. the variety and strength of methods, the richness of tasks, the evidence of presentation. Subsequent mathematicians widely drew from it both problems and methods for solving them. According to the first book, many generations of European mathematicians studied the Indian positional number system.

Fibonacci monument in Pisa

Another book by Fibonacci, The Practice of Geometry ( Practica geometriae, 1220), contains a variety of theorems related to measurement methods. Along with the classical results, Fibonacci gives his own - for example, the first proof that the three medians of a triangle intersect at one point (Archimedes knew this fact, but if his proof existed, it did not reach us).

In the treatise "Flower" ( Flos, 1225) Fibonacci investigated the cubic equation proposed to him by John of Palermo at a mathematical competition at the court of Emperor Frederick II. John of Palermo himself almost certainly borrowed this equation from Omar Khayyam's treatise On the Proofs of Problems in Algebra, where it is given as an example of one of the types in the classification of cubic equations. Leonardo of Pisa investigated this equation, showing that its root cannot be rational or have the form of one of the quadratic irrationalities found in the X book of Euclid's Elements, and then found the approximate value of the root in sexagesimal fractions, equal to 1; 22,07,42, 33,04,40, without indicating, however, the method of its solution.

"The Book of Squares" ( Liber quadratorum, 1225), contains a number of problems for solving indefinite quadratic equations. In one of the tasks, also proposed by John of Palermo, it was required to find a rational square number, which, when increased or decreased by 5, again gives rational square numbers.

Fibonacci numbers

In honor of the scientist, a number series is named, in which each subsequent number is equal to the sum of the previous two. This number sequence is called the Fibonacci numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, … (OEIS sequence A000045)

Fibonacci targets

1, 3, 9, 27, 81,… (degrees of 3, OEIS sequence A009244)

Fibonacci's works

• "The book of the abacus" (Liber abaci), 1202

see also

Notes

Literature

• History of mathematics from ancient times to the beginning of the 19th century (under the editorship of A.P. Yushkevich), volume II, M., Nauka, 1972, pp. 260-267.
• Karpushina N."Liber abaci" by Leonardo Fibonacci, Mathematics at School, No. 4, 2008.
• Shchetnikov A.I. On the reconstruction of an iterative method for solving cubic equations in medieval mathematics. Proceedings of the third Kolmogorov readings. Yaroslavl: Publishing House of YaGPU, 2005, p. 332-340.
• Yaglom I. M. Italian merchant Leonardo Fibonacci and his rabbits. // Kvant, 1984. No. 7. P. 15-17.
• Glushkov S. On approximation methods of Leonardo Fibonacci. Historia Mathematica, 3, 1976, p. 291-296.
• Sigler, L.E. Fibonacci's Liber Abaci, Leonardo Pisano's Book of Calculations" Springer. New York, 2002, ISBN 0-387-40737-5 .

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- (Fibonacci) Leonardo (c. 1170 c. 1240), Italian mathematician. Author of "Liber Abaci" (c. 1200), the first Western European work, which proposed the adoption of the Arabic (Indian) system of writing numbers. Developed mathematical... Scientific and technical encyclopedic dictionary

See Leonardo of Pisa... Big Encyclopedic Dictionary

fibonacci- (1170 1288) One of the early representatives of Italian accounting, whose main merit is the introduction and promotion of Arabic numerals in Europe (that is, the replacement of the additive Roman deduction system with positional decimal). )