The Italian merchant Leonardo of Pisa (1180-1240), also known as Fibonacci, was by far the most important mathematician of the Middle Ages. The role of his books in the development of mathematics and the dissemination of mathematical knowledge in Europe can hardly be overestimated.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 ...
In the age of Fibonacci, the renaissance was still far away, but history gave Italy a brief period of time that could well be called a rehearsal for the impending Renaissance. This rehearsal was led by Frederick II, Emperor (since 1220) of the Holy Roman Empire. Brought up in the traditions of southern Italy, Frederick II was internally deeply far from European Christian chivalry.
Frederick II did not recognize the jousting tournaments so beloved by his grandfather. Instead, he cultivated much less bloody math competitions, in which opponents exchanged problems rather than blows.
At such tournaments, the talent of Leonardo Fibonacci shone. This was facilitated by a good education, which was given to his son by the merchant Bonacci, who took him with him to the East and assigned Arab teachers to him.
Frederick's patronage stimulated the release of Fibonacci's scientific treatises:
- The book of the abacus (Liber Abaci), written in 1202, but which has come down to us in its second version, which dates back to 1228.
- Geometry Practices" (1220)
- Book of squares (1225)
According to these books, superior in their level to Arabic and medieval European works, mathematics was taught almost until the time of Descartes (XVII century).
As documented in 1240, admiring citizens of Pisa said he was "a sensible and erudite man", and not so long ago, Joseph Gies (Joseph Gies), editor-in-chief of the Encyclopædia Britannica, declared that future scientists at all times "will do their duty Leonardo of Pisa as one of the world's greatest intellectual pioneers."
His works, after many years, are only now being translated from Latin into English. For those who are interested, the book titled Lenardo of Pisa and the New Mathematics of the Middle Ages by Joseph and Frances Gies is an excellent treatise on the Fibonacci age and his works.
Although he was the greatest mathematician of the Middle Ages, the only monuments to Fibonacci are a statue opposite the Leaning Tower of Pisa across the Arno River and two streets that bear his name, one in Pisa and the other in Florence. It seems strange that so few visitors to the 179-foot Leaning Tower have ever heard of Fibonacci or seen his statue. Fibonacci was a contemporary of Bonanna, the architect of the Leaning Tower of Pisa, whose construction he began in 1174. Both of them made contributions to world history, but one whose contribution is far superior to the other is almost unknown.
Fibonacci sequence, Fibonacci numbers
Of greatest interest to us is the work "The Book of the Abacus" ("Liber Abaci"). This book is a voluminous work containing almost all the arithmetic and algebraic information of that time and played a significant role in the development of mathematics in Western Europe over the next few centuries. In particular, it was from this book that Europeans got acquainted with Hindu (Arabic) numerals.
In "Liber Abaci" Fibonacci gives his sequence of numbers as a solution to a mathematical problem - finding the formula for the reproduction of rabbits. The numerical sequence is as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 (then ad infinitum).
On pages 123-124 of this manuscript, Fibonacci placed the following problem: "Someone placed a pair of rabbits in a certain place, fenced on all sides by a wall, in order to find out how many pairs of rabbits would 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.
The Fibonacci sequence has some very interesting features, not the least of which is the almost constant relationship between numbers.
- The sum of any two adjacent numbers is equal to the next number in the sequence. For example: 3 + 5 = 8; 5 + 8 = 13 etc.
- The ratio of any number in the sequence to the next one approaches 0.618 (after the first four numbers).
For example: 1: 1 = 1; 1:2=0.5; 2:3=0.67; 3: 5 = 0.6; 5: 8 = 0.625; 8: 13 = 0.615; 13:21 = 0.619 .
Note how the value of the ratios fluctuates around 0.618, with the range of fluctuations gradually narrowing; as well as on the values: 1.00; 0.5; 0.67. - The ratio of any number to the previous one is approximately equal to 1.618 (the reciprocal of 0.618). For example: 13:8 = 1.625; 21: 13 = 1.615; 34:21 = 1.619.
.The higher the numbers, the more they approach the value of 0.618 and 1.618. - The ratio of any number to the next one through one approaches 0.382, and to the previous one through one - 2.618. For example: 13:34 = 0.382; 34:13 = 2.615.
The Fibonacci sequence contains other interesting ratios, or coefficients, but the ones we have just given are the most important and well-known. As we emphasized above, in fact, Fibonacci is not the discoverer of his sequence. The fact is that the coefficient 1.618 or 0.618 was known to the ancient Greek and Egyptian mathematicians, who called it the "golden coefficient" or the "golden section". We find traces of it in music, visual arts, architecture and biology. The Greeks used the principle of the "golden section" in the construction of the Parthenon, the Egyptians - the Great Pyramid of Giza. The properties of the "golden ratio" were well known to Pythagoras, Plato and Leonardo da Vinci.
The proportions of Fibonacci numbers provide guidance not only for possible rollback levels, but also indicate the possible magnitude of the move if the trend continues. If, after a move, the market pulls back and then continues to move in the same direction, then in a typical case, the value of the continued move could be 1.618.
It will be interesting to see how the Fibonacci numbers are reflected in human proportions. In the figures we see that even our nature is proportional, and these relationships can be expressed using the Fibonacci sequence.
Whoever is the architect of our world works perfectly and harmoniously. The model of our world is so complex in all relationships and exceptions that it can only be described by mathematics.
> Thoughts for thoughts
The longest testament was written by one of the founding fathers of the United States, Thomas Jefferson. Indications regarding property were interspersed in the document with discourses on the history of America. Under this will, Jefferson's heirs received their shares of the inheritance only on the condition that they set free all their slaves.
Most offensive. One medieval farmer left 100 livres for his wife, but ordered that if she married, add another 100 livres, arguing that the poor man who would become her husband would need this money. Alas, divorce was forbidden in those days.
The most historically useful testament was left by William Shakespeare. He turned out to be a rather petty type and made an order regarding all his property, from furniture to shoes. The will is almost the only indisputable document that proves the existence of Shakespeare.
The shortest will was written by a banker from London. It contained three words: "I am completely broke."
The most indecent will in history was written by a shoemaker from Marseilles. Of the 123 words written in this will, 94 are impossible to pronounce even in relatively decent society.
The most difficult testament to understand was drawn up by the laboratory assistant of the famous physicist Niels Bohr. There were so many technical terms and complex phraseological turns in the will that experts-linguists had to be called in to decipher it.
The largest amount of cash ever bequeathed by a single person. Henry Ford bequeathed to distribute $ 500 million among 4157 educational and charitable institutions.
The most famous testament was left by Alfred Nobel. It was disputed by relatives. They received only half a million crowns, and the remaining 30 million were given to establish the famous Nobel Prize.
Billionaire Michel Rothschild left the most secret testament. In particular, it says: "... I categorically and unequivocally forbid any inventory of my inheritance, any judicial intervention and disclosure of my fortune ..." So the real size of the fortune is still not known.
The largest fortune left to an animal. The most stupid story about the inheritance is connected with the same will. Millionaire and film producer Roger Dorcas left all his $65 million to his beloved dog Maximilian. The court recognized such a decision as legal, since during his lifetime the millionaire straightened completely human documents to Maximilian. Dorcas left 1 cent for his wife. But she, according to the same dog documents, married a dog and, after his death, calmly entered into inheritance rights, since the dog, of course, did not leave a will.
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.
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 the repayment of taxes and customs fees, a certain Leonardo Fibonacci, applied Arabic numerals to banking accounting, thus introducing them to Europe.
Article "Banker" // ENE (ESBE)
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. Books VIII-X set out 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.
"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 + 2x + 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.
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).
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 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.
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). )
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