The life and work of Pythagoras. Presentation "School of Pythagoras" methodological development (grade 8) on the topic Pythagorean school presentation on philosophy

Section 2. Use of historical material on the topic “School of Pythagoras” outside of class hours.

Form of organization of extracurricular activities –math club lesson.

Forms of presenting historical material:student message, math newspaper, presentation display.

Types of educational activities:

– introduce students to historical facts from the life of Pythagoras and his school;

– introduce students to what was studied at the school of Pythagoras;

– develop skills of independent work with a large amount of information;

– learn to present the results of work using modern information technologies.

Planned educational results:

– acquire knowledge about Pythagoras and his school;

– will acquire knowledge about the merits of Pythagoras to humanity in various fields;

– update knowledge in the field of information and communication technologies, Internet technologies, programming.

Without knowledge of the past it is impossible to understand the present and
It is absolutely impossible to imagine the future correctly.

Historical reference.

In the list of the greatest mathematicians of antiquity and our days, Pythagoras should certainly come first. It was he who carried out a radical transformation of mathematics, turning it from a set of useful rules into an abstract deductive science.

Mathematician Proclus, who lived in the 5th century. AD, wrote: “Pythagoras transformed this science into a form of free education. He studied this science based on its first principles, and tried to obtain theorems using purely logical thinking, without concrete ideas.”

The only fragmentary information available about the life of Pythagoras has been preserved. He was born around 570 AD. e. on the Greek island of Samos (presentation slide No. 1-4).

As a young man striving for knowledge, Pythagoras left his native island. He visited all Hellenic and many foreign countries, studied with famous scientists and admired the wonders of the East (presentation slide Nos. 5-8).

When Pythagoras returned to the island of Samos, Polycrates ruled there. His tyranny was so strong that, as the ancient historian writes, “a free man could not endure arbitrariness and despotism with dignity.” Pythagoras moved to Croton, a city in southern Italy. There he founded the famous Pythagorean Union, which set itself not only scientific, but also religious, ethical and political goals. Pythagoras's fame as a teacher was so great that all the young men wanted to become his students, and their fathers preferred that they spend time with him rather than mind their own affairs. Plato, in his only mention of Pythagoras, calls him “the leader of youth”, who created a special Pythagorean way of life.

The activities of the union were secret. Access to it was not open to everyone (slide No. 9-17).

One could not share one’s discoveries with those who were not members of the union. The Pythagoreans distinguished four areas of science: the doctrine of numbers (arithmetic), figures and measurements (geometry), astronomy and the doctrine of harmony (music theory).

According to Pythagoras, it is the science of numbers that may possess the key to life and the essence of being.Penetrating into the properties of numbers and explaining their various combinations, Pythagoras tried to create a science of all sciences.

Number for the Pythagoreans is the main object of mathematics. They treated it as a collection of units, that is, they studied only positive integers. With their help, the Pythagoreans wanted to explain the entire world surrounding man, the structure of the universe. The statement “everything is number” belongs to Pythagoras himself, and was the basis of his teaching.

The units that make up positive integers were considered indivisible and depicted as dots. They looked at "triangular" numbers

1, 1+2=3, 1+2+3=6, 1+2+3+4=10,…,

1+2+3+…+ n = .

He divided all numbers into two types: even and odd, and with amazing sensitivity revealed the properties of the numbers of each group. Even numbers have the following properties: any number can be divided into two equal parts, both of which are either even or odd. For example, 14 is divided into two equal parts 7 + 7, where both parts are odd; 16 = 8 + 8, where both sides are even. The Pythagoreans considered the even number, the prototype of which was the dyad, to be indefinite and feminine.

Pythagoras divided even numbers into 3 classes: even-even, even-odd, odd-odd. The first class consists of numbers, which represent the doubling of numbers, starting from one. Thus, these are 1,2,4,8,16,32,64,128,512 and 1024. Pythagoras saw the perfection of these numbers in the fact that they can be divided in half and again, and so on until one is obtained. Even-even numbers have some unique properties. The sum of any number of terms1 except the last one is always equal to the last one minus one. For example, the sum of four terms (1+2+4+8) is equal to the fifth term - 16 minus one, that is, 15. A series of even-even numbers also has the following property: the first term, multiplied by the last, gives the last one in the series with An odd number of terms will not leave one number, which, when multiplied by itself, will give the last number in the series. Even-odd numbers are numbers that cannot be divided when divided in half. They are formed as follows: take an odd number, multiply by 2, and so on for the entire series of odd numbers. In this process, 1,3,5,7,9,11 gives the even-odd numbers 2,6,10,14,18,22. Thus, each such number is divisible by two once and cannot be divided more. Another feature of this class of numbers is that if the divisor is an odd number, the quotient will always be even, and vice versa. For example, if 22 is divided by 2, an even divisor, the quotient of 11 will be odd.

Even numbers are divided into three other classes: superperfect, imperfect and perfect. Superperfect numbers are those numbers, the sum of fractional parts that are greater than themselves. For example, 24 has the sum of its fractional parts 12+6+4+8+3+2+1 number 33, which is greater than 24, the original number. Pythagoras called numbers imperfect, the sum of fractional parts that are smaller than himself. For example, the number 14 is the sum of its fractional parts 7+2+1=10, which is less than 14. A perfect number is a number whose sum of fractional parts is equal to the number itself. Such numbers are extremely rare. There is only one number between 1 and 10, namely 6; one between 10 and 100 is the number 28, one between 100 and 1000 is 496, one between 1000 and 10000 is 8128. Perfect numbers are found as follows: the first number of a series of even-even numbers is added to the second number of the series, and if the result is a prime number, it is multiplied by the last number of a series of even-even numbers participating in the formation of the sum. If adding even-even numbers does not result in a non-composite number.

The Pythagoreans developed their philosophy from the science of numbers. Perfect numbers, they believed, are beautiful images of virtues. They represent the middle ground between excess and deficiency. They are very rare and are generated by perfect order. In contrast to this, superabundant and imperfect numbers, of which there are as many as possible, are not arranged in order and are not generated for some specific purpose. And therefore they have a great resemblance to vices, which are numerous, disordered and uncertain.

The Pythagoreans considered the odd number, the prototype of which was the monad, to be definite and masculine, although there was some disagreement among them about 1 (one). Some considered it positive because if it is added to an odd number, it becomes even and is thus seen as an androgenic number, combining both masculine and feminine attributes, making it both even and odd.

The custom of the Pythagoreans was to offer an odd number of objects to the high gods, while offering an even number to the goddesses and underground spirits.

Odd numbers are divided into 3 general classes: non-composite, composite and non-composite - composite. Non-composite numbers are those numbers that have no divisors other than themselves and one. These numbers are 3,5,7,11,13,17, etc. Composite numbers are numbers that are divisible not only by themselves, but also by some other numbers. Such numbers are those of the odd numbers that are not included in the group of non-composite numbers. These numbers are 9,15,21,25,27,33,39, etc. Non-composite numbers are numbers that do not have a common divisor, although each of them is divisible. If you take two numbers and find that they do not have a common factor, such numbers can be called non-composite-composite numbers. For example, the numbers 9 and 25. 9 is divisible by 3, and 25 by 5, but neither of them is divisible by the divisor of the other, they do not have a common divisor. They are called non-composite-composite because each of them has an individual divisor, and since these numbers do not have a common divisor, they are called non-composite. Thus, non-composite-composite numbers are found only in pairs with each other.

"Square" numbers were also considered

1, 1+3=4, 1+ 3 +5 = 9,…,

1 + 3 + 5+ … + (2n – 1) = n 2 (slide No. 18-26).

The Pythagoreans also determined “cubic” numbers

1,8,27,64,…,n 3.

The main achievement of the Pythagorean school was the construction of the theory of divisibility. They divided all natural numbers into even and odd, into prime and composite. They formulated a theorem: the product of two numbers is divisible by 2 if and only if at least one of the factors is divisible by 2. Then any even natural number can be represented as N = 2 k N 1 , where N 1_ - odd, k – non-negative integer.

The Pythagoreans posed the problem of finding perfect numbers, i.e. those that are equal to the sum of their divisors (excluding the number itself). For example: 6 = 1 + 2 + 3, 28 = 1 +2 + 4 +7 +14, etc.

One was considered the mother of all numbers, the number 2 expressed a line, 3 a triangle, 4 a pyramid. These arguments connected arithmetic with geometry. The unit could be interpreted as a point, the number 2 is a line, i.e. a one-dimensional image, the triangle defines a plane, and the number 4 is a three-dimensional image.

The Pythagoreans believed so deeply in the miraculous properties of the number 10 that they came up with a new planet and called it Counter-Earth. The fact is that at that time there were 9 celestial spheres (sky, Sun, Moon, Earth, Mercury, Mars, Jupiter, Saturn). They believed that there was another 10th sphere, and the Counter-Earth revolved around it.

They had an “oath numbering 36”. Special properties were attributed to him in connection with the fulfillment of the relations

36 = 1 3 + 2 3 + 3 3 ; 36 = (2 + 4 + 6 +8) + (1 + 3 + 5 + 7).

Exploring the set of natural numbers 1, 2, 3, ..., n, ..., the ancient Greeks were the first to realize the idea of the infinity of objects studied by mathematics.

They knew how to perform arithmetic operations with rational numbers m/n, where m and n are natural numbers.

The turning point in the development of ancient mathematics was the discovery of incommensurable segments, or, in other words, the discovery of irrational numbers.

Pythagoras proved the theorem

X 2 + Y 2 = Z 2,

where X, Y are the legs of a right triangle, and Z is the hypotenuse (slide No. 27,28).

According to legend, as a sign of gratitude, he sacrificed 100 bulls to the gods.

Triples of numbers that satisfy this equation are called “Pythagorean”,

(3, 4, 5), (5, 12, 13), (7, 24, 25), …

X=1/2(m 2 – 1), Y=m, Z=1/2(m 2 + 1), where m is an odd natural number.

But they only knew rational numbers. The Pythagoreans decided not to tell anyone about their paradoxical results.

According to legend, Hippasus divulged the secret and died under mysterious circumstances (it was believed that the gods punished him).

At the school of Pythagoras they studied not only mathematics (slide No. 29 -31).

Great attention was paid to philosophy and politics.

At the beginning of the 5th century. BC. After an unsuccessful performance in the political arena, the Pythagoreans were expelled from the cities of Southern Italy, their union collapsed.

The merits of Pythagoras are undoubtedly great and it is simply impossible to underestimate them (slide Nos. 32-34).Pythagoras lived in Croton for 30 years. During this time, he managed to realize what remained the dream of many initiates: he created, on top of political power, a wise power of higher knowledge, similar to the ancient Egyptian priesthood. The Council of Three Hundred, created and headed by Pythagoras, was the regulator of the political life of Croton and extended its influence to other cities of Greece for a quarter of a century. No reliable information has been preserved about the time and place of death of Pythagoras himself. The memories of the Great Teacher and his teaching were preserved by those few who managed to escape to Greece. We find it in the Golden Verses of Lysias, in the commentaries of Heraclitus, in passages by Philolaus and Archytas, and in Plato's Timaeus. The beautiful, harmonious system given to the world by Pythagoras has never been forgotten. It became the basis of Plato’s metaphysics and was revived in the Alexandrian school and in the works of many later ancient philosophers.

Material prepared: Isaeva E.P., Senina S.U.

Sources of information used:

1. Dorofeev A.V. Pages of history in mathematics lessons. – Lvov, “Kvantor” magazine, 1991.

2. Alexandrov A.F. Numerological matrix. Secrets of magic numbers and codes. – M.: RIPOL classic, 2008.

3.. Voloshinov A.V. Pythagoras: Union of truth, goodness and beauty. - M.: Education, 1993.

4. Zhmud L.Ya. Pythagoras and his school, - Science, 1990.

5. Losev A. Myth, number, essence, - M.: 1994.

6. Perepelitsin M.L. Philosopher's Stone, - 1990.

7Asmus V.F: Ancient philosophy, -1971.

8. Shure E. Great Initiates, volume 1, translation by E. Pisareva. - Kaluga: 1914.

9. Internet resources.

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Pythagoreans sing the Hymn to the Sun

Mathematicians are “cognizers”

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Pythagoras and his school. The work was carried out by: Isaeva E.P. Senina S. U. Pugachev – 2013

“All things are numbers” Pythagoras

Purpose of the study What is the essence of the teachings of Pythagoras? Who are the Pythagoreans? What is the connection between Pythagoras and the word “cosmos”?

Pythagoras of Samos (c. 580 - c. 500 BC) - ancient Greek philosopher, religious and political figure, founder of Pythagoreanism, mathematician. Pythagoras is credited with studying the properties of integers and proportions, proving the Pythagorean theorem, etc.

Biography of Pythagoras The parents of Pythagoras were Mnesarchus and Parthenides from Samos. Mnesarchus was a stone cutter; according to Porphyry, he was a rich merchant from Tyre, who received Samian citizenship for distributing grain in a lean year. Parthenida, later renamed Pyphaida by her husband, came from the noble family of Ankeus, the founder of the Greek colony on Samos. The birth of a child was supposedly predicted by the Pythia in Delphi, which is why Pythagoras received his name, which means “the one whom the Pythia announced.”

Years of study Iamblichus writes that Pythagoras at the age of 18 left his native island and, having traveled around the sages in different parts of the world, reached Egypt, where he stayed for 22 years, until he was taken to Babylon as a captive by the Persian king Cambyses, who conquered Egypt in 525 BC e. Pythagoras stayed in Babylon for another 12 years, communicating with magicians, until he was finally able to return to Samos at the age of 56, where his compatriots recognized him as a wise man.

School of Pythagoras The school was founded by Pythagoras and existed until the beginning of the 4th century. BC, although persecution began almost immediately after the death of Pythagoras in 500.

Pythagoreans sing the Hymn to the Sun

First stage Pythagoras usually sent the candidate back, advising him to wait and come again in three years. This outwardly very stern technique was filled with deep meaning - after all, any impulse, even the most beautiful and pure, must pass the test of time.

Second stage During this period, a person was not yet considered a student of the School and was called an acousmatic (“listener”). He listened, absorbed, realized - and all this happened in silence. Pythagoras “prescribed five years of silence for acousmatics, testing their ability to abstain, since silence is the most difficult type of abstinence.”

The third stage Only after many years of such work did the acousmatician become a real Pythagorean student. Now he bore the title of mathematician - “cognizing”. In classes taught by Pythagoras himself or his closest students, mathematicians were given a holistic picture of the world, the structure of Nature and man was revealed. The training of mathematicians took place over a long period of time, but this too was only preparation.

Mathematicians are “cognizers”

Fourth stage Devoting oneself to serving people, society, everyone who needs help and protection is a natural step for a mature philosopher. And when the mathematics students were ready for this, the choice of those directions and forms in which this service would be carried out took place, and then the final training in the chosen “specialty”. Some studied economics, others studied medicine, etc.

Fifth stage The highest level in the Pythagorean school was considered to be the training of politicians - people capable of managing society. The task is to lead people based on the common good, without being led by either one’s own or others’ interests. Later, Plato reworked and expanded the Pythagorean theory of the state - “Plato’s ideal state model.” Many of Pythagoras' students became famous as legislators and fair guardians of the laws. The years when the Pythagoreans participated in state affairs were prosperous,

Even and odd The Pythagoreans divided all numbers into two categories - even and odd. Later it turned out that the Pythagorean “even - odd”, “right - left” have deep and interesting consequences in quartz crystals, in the structure of viruses and DNA, in Pasteur’s famous experiments, in parity violation of elementary particles and other theories.

Even... Odd... The Pythagoreans considered even numbers to be feminine, and odd numbers to be masculine. Marriage is five equal to three plus two. For the same reason, they called a right triangle with sides three, four, five “the figure of the bride.”

Tetrad The numbers 1, 2, 3 and 4 made up the famous "tetrad". Geometrically, the tetrad was depicted as a “perfect triangle”, arithmetically - as a “triangular number” 1+2+3+4 = 10. The Pythagoreans swore “to those who put the tetrad into our soul, the source and root of eternal nature.”

Ideal number The sum of the numbers included in the tetrad is equal to ten, which is why ten was considered by the Pythagoreans to be an ideal number and symbolized the Universe. Since ten is the ideal number, they reasoned, there should be exactly ten planets in the sky. It should be noted that at that time only the Sun, Earth and five planets were known. They named the tenth planet Counter-Earth.

Ten Ten can be expressed by the sum of the first four numbers (1+2+3+4=10), where one is the expression of a point, two is a line and a one-dimensional image, three is a plane and a two-dimensional image, four is a pyramid, that is, a three-dimensional image. Why not Einstein's four-dimensional Universe?

Justice and equality The Pythagoreans saw justice and equality in the square of a number. Their symbol of constancy was the number nine, since all multiples of nine numbers have the sum of their digits again being nine. 9*2=18 1+8=9; 7*9=63 6+3=9; 11*9=99 9+9=18 1+8=9; 25*9= 225 2+2+5=9.

The number eight symbolized death among the Pythagoreans, since multiples of eight have a decreasing sum of digits. 8*2=16 1+6=7; 8*3=24 2+4=6; 8*4=32 3+2=5; 8*5+40 4+0=4; 8*6=48 4+8=12 1+2=3

“Bad numbers” In addition to the numbers that evoked admiration and admiration, the Pythagoreans also had so-called bad numbers. These are numbers that did not have any merit, and even worse if such a number was surrounded by “good” numbers. The famous number thirteen is the devil's dozen. The number seventeen, which caused particular disgust among the Pythagoreans.

More about numbers The Pythagoreans had an “oath by the number 36.” Special properties were attributed to him: 36=(2+4+6+8)+(1+3+5+ 7)

“COSMOS” Pythagoras introduced this word into science, meaning by it something harmonious and whole, subject to the laws of harmony and numbers.

WHAT IS PEACE? “The world is a limited sphere, rushing in infinity... The movement of heavenly bodies is an inaudible harmony of singing cosmic spheres...”

The merits of Pythagoras are undoubtedly great and it is simply impossible to underestimate them. Pythagoras lived in Croton for 30 years. During this time, he managed to realize what remained the dream of many initiates: he created, on top of political power, a wise power of higher knowledge, similar to the ancient Egyptian priesthood. The Council of Three Hundred, created and headed by Pythagoras, was the regulator of the political life of Croton and extended its influence to other cities of Greece for a quarter of a century. The beautiful, harmonious system given to the world by Pythagoras was never forgotten. It became the basis of Plato’s metaphysics and was revived in the Alexandrian school and in the works of many later ancient philosophers.

Information sources. Alexandrov A.F. Numerological matrix. Secrets of magic numbers and codes. – M.: RIPOL classic, 2008. 2. Dorofeeva A.V. Pages of history in mathematics lessons. Lvov, 1991. 3. 3..Voloshinov A.V. Pythagoras: Union of truth, goodness and beauty. - M.: Education, 1993. 4. Zhmud L.Ya. Pythagoras and his school, - Science, 1990. 5. Losev A. Myth, number, essence, - M.: 1994. 6. Perepelitsin M.L. Philosopher's Stone, - 1990. 7Asmus V.F: Ancient philosophy, -1971. 8. Shure E. Great Initiates, volume 1, translation by E. Pisareva. - Kaluga: 1914. 9. Internet resources.

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Presentation on the topic: Pythagorean school

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Pythagoras (from the Greek “persuasive speech”) is an ancient Greek philosopher and mathematician, the creator of the religious and philosophical school of the Pythagoreans. Born in Sidon, Phenicia around 570 BC. He studied in several temples in Greece. His first teachers were Pherikid of Syros and Elder Hermodamant. At a young age, Pythagoras went to Egypt. Pythagoras (from the Greek “persuasive speech”) is an ancient Greek philosopher and mathematician, the creator of the religious and philosophical school of the Pythagoreans. Born in Sidon, Phenicia around 570 BC. He studied in several temples in Greece. His first teachers were Pherikid of Syros and Elder Hermodamant. At a young age, Pythagoras went to Egypt.

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Pythagoras was one of the first to state that the Earth has the shape of a ball, and the Sun, Moon and other planets have their own trajectory of movement. Pythagoras was one of the first to state that the Earth has the shape of a ball, and the Sun, Moon and other planets have their own trajectory of movement. Pythagoras is credited with studying the properties of integers and proportions and proving the Pythagorean theorem. The Pythagoreans compiled a table of 10 opposites; Aristotle gives it in his “Metaphysics”: limit - infinite odd - even one - many right - left male - female rest - straight movement - crooked light - darkness good - evil square - elongated rectangle

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In Crotona (Southern Italy) Pythagoras founded a school - the Pythagorean Union. Pythagoras calls only those who have gone through many stages of knowledge his closest students. Pythagoreans study geometry, mathematics, harmony, and astronomy. In Crotona (Southern Italy) Pythagoras founded a school - the Pythagorean Union. Pythagoras calls only those who have gone through many stages of knowledge his closest students. Pythagoreans study geometry, mathematics, harmony, and astronomy. The activity of Pythagoras as a religious innovator of the 6th century. BC e. consisted in the creation of a secret society, which not only set itself political goals, but, mainly, the liberation of the soul through moral and physical purification with the help of secret teachings (mystical teachings about the cycle of migration of the soul). According to Pythagoras, the eternal soul moves from heaven into the mortal body of a person or animal and undergoes a series of migrations until it earns the right to return back to heaven.

Pythagoras and his school “All things are numbers.” Pythagoras

Mathematics teacher, Municipal Educational Institution "Secondary School in the village of Dinamovsky, Novoburassky District, Saratov Region"

Kuzmichev Sergey Mikhailovich

Pythagoras Pythagoras of Samos (c. 580 - c. 500 BC) - ancient Greek philosopher, religious and political figure, founder of Pythagoreanism, mathematician. Pythagoras is credited with studying the properties of integers and proportions, proving the Pythagorean theorem, etc. The Pythagorean school

The school was founded by Pythagoras and existed until the beginning of the 4th century. BC, although persecution began almost immediately after the death of Pythagoras in 500.
Admission to the school took place

School of Athens. First stage

Pythagoras usually sent the candidate back, advising him to wait and come again in three years. This outwardly very stern technique was filled with deep meaning - after all, any impulse, even the most beautiful and pure, must pass the test of time.

Second phase

During this period, a person was not yet considered a student of the School and was called an acousmatic (“listener”). He listened, absorbed, realized - and all this happened in silence.
Pythagoras “prescribed five years of silence for acousmatics, testing their ability to abstain, since silence is the most difficult type of abstinence.”

Third stage

Only after many years of such work did the acousmatician become a real Pythagorean student
Now he bore the title of mathematician - “cognitive”.
In classes taught by Pythagoras himself or his closest students, mathematicians were given a holistic picture of the world, the structure of Nature and man was revealed.
The training of mathematicians took place over a long period of time, but this too was only preparation.

Fourth stage

Devoting oneself to serving people, society, and everyone who needs help and protection is a natural step for a mature philosopher.
And when the mathematics students were ready for this, the choice of those directions and forms in which this service would be carried out took place, and then the final training in the chosen “specialty”.
Some studied economics, others studied medicine, etc.

Fifth stage

The highest level in the Pythagorean school was considered to be the training of politicians - people capable of managing society.
The task is to lead people based on the common good, without being led by either one’s own or others’ interests,
Plato later reworked and expanded the Pythagorean theory of the state - "Plato's model of the ideal state."
Many of Pythagoras' students became famous as legislators and fair guardians of the laws.
The years when the Pythagoreans participated in state affairs were prosperous,

It is also known that in addition to the spiritual and moral development of Pythagoras’s students, he was concerned about their physical development. He not only participated in the Olympic Games and won fist fights twice, but also trained a galaxy of great Olympians. The scientist devoted about forty years to the school he created and, according to one version, at the age of eighty Pythagoras was killed in a street fight during a popular uprising. After his death, the students surrounded the name of their teacher with many legends. Aphorisms of Pythagoras

Do not do anything shameful, either in front of others or in secret. Your first law should be self-respect.
To learn the customs of any people, try to first learn their language.
If you can be an eagle, do not strive to be the first among the jackdaws.
During anger one should neither speak nor act.
Life is like games: some come to compete, others come to trade, and the happiest come to watch.
No matter how short the words “yes” and “no” are, they still require the most serious consideration.

Wash the insult received not in blood, but in Lethe, the river of oblivion.

Wash the insult received not in blood, but in Lethe, the river of oblivion.
Drunkenness is an exercise in madness.
Ask a drunkard how he could stop drinking. I will answer for him: let him remember more often the things he does while drunk.
Friends have everything in common, and friendship is equality.

The great science of living happily is to live only in the present

The great science of living happily is to live only in the present
What is smarter than everyone else? Time is the wisest of all. Keeps the past, and the future - a seed.
What is most essential? - The light of hope. It exists where nothing else exists.
Don't judge your greatness by your shadow at sunset.

The Pythagoreans made many important discoveries in arithmetic and geometry:

theorem on the sum of interior angles of a triangle;
constructing regular polygons and dividing the plane into some of them;
geometric methods for solving quadratic equations;
dividing numbers into even and odd, simple and composite; introduction of figured, perfect and friendly numbers;
the creation of a mathematical theory of music and the doctrine of arithmetic, geometric and harmonic proportions and much more.

Even or odd

The Pythagoreans divided all numbers into two categories - even and odd.
Later it turned out that the Pythagorean “even - odd”, “right - left” have deep and interesting consequences in quartz crystals, in the structure of viruses and DNA, in Pasteur’s famous experiments, in parity violation of elementary particles and other theories.

Even... Odd...

The Pythagoreans considered even numbers to be feminine and odd numbers to be masculine. Marriage is five, equal to three plus two.
For the same reason, they called a right triangle with sides three, four, five “the figure of the bride.”

Ten

Ten can be expressed as the sum of the first four numbers (1+2+3+4=10), where one is the expression of a point, two is a line and a one-dimensional image, three is a plane and a two-dimensional image, four is a pyramid, that is, a three-dimensional image. Well, why not a four-dimensional Universe? Einstein?

Tetrad

The numbers 1, 2, 3 and 4 made up the famous "tetrad".
Geometrically, the tetrad was depicted as a “perfect triangle”, arithmetically - as a “triangular number” 1+2+3+4 = 10.
The Pythagoreans swore "to those who put the tetrad into our soul - the source and root of eternal nature."

Ideal number

The sum of the numbers included in the tetrad is equal to ten, which is why ten was considered by the Pythagoreans to be an ideal number and symbolized the Universe.
Since ten is the ideal number, they reasoned, there should be exactly ten planets in the sky. It should be noted that at that time only the Sun, Earth and five planets were known.

Justice and equality

The Pythagoreans saw justice and equality in the square of a number.
Their symbol of constancy was the number nine, since all multiples of nine numbers have the sum of their digits again being nine.

The number eight symbolized death among the Pythagoreans, since multiples of eight have a decreasing sum of digits.

8*6=48 4+8=12 1+2=3

"Bad numbers"

In addition to numbers that evoked admiration and admiration, the Pythagoreans also had so-called bad numbers. These are numbers that did not have any merit, and even worse if such a number was surrounded by “good” numbers.
The famous number thirteen is the devil's dozen
The number seventeen, which caused particular disgust among the Pythagoreans.

Number of the beast

The very concept of “number of the beast” first appears in the Revelations of John the Theologian, which first appeared probably in the 1st century AD.
Interestingly, the problem has been known for a long time - already in the 2nd century, Bishop Irenaeus argued that 616 is false, and the true number of the beast is 666.
What is the meaning of the “number of the beast”? It is believed that this is the encrypted name of the persecutor of Christians - Emperor Nero. Hebrew spelling “ Nero Kaisar” in total gives just 666, but the Latin “ Nero Caesar” just gives 616.
This is a palindrome
This is the Smith number, that is, the sum of its digits is equal to the sum of the digits of its prime factors
666 is the sum of the squares of the first seven prime numbers
In China, the number 6 is, on the contrary, lucky and on 06/06/06 a record number of marriages were concluded there.

The main Pythagorean identification mark was the symbol of health - pentagram or Pythagorean star.
The drawn pentagram was a secret sign by which the Pythagoreans recognized each other.

The famous Pythagorean theorem

The Pythagorean theorem is one of the main and, one might say, the most important theorem of geometry. Its significance lies in the fact that most of the theorems of geometry can be deduced from it or with its help. The Pythagorean theorem is also remarkable because in itself it is not at all obvious.

The truth will remain eternal, as soon as

A weak person knows everything!

And now the Pythagorean theorem

True, as in his distant age.

The sacrifice was abundant

To the gods from Pythagoras. A hundred bulls

He gave it up to be slaughtered and burned

For the light of the ray that came from the clouds.

Therefore, always since then:

The truth is just being born,

The bulls roar, sensing her, and follow her.

They are unable to stop the light,

Or they can only close their eyes and tremble...

From the fear that Pythagoras instilled in them.

From the history of the theorem

Pythagorean theorem in China

In Ancient China 1100 BC. a visual proof of this theorem was established, contained in the ancient Chinese treatise “Zhou-bi”.

Pythagorean theorem in Egypt

2000 BC The ancient Egyptians knew that a triangle with sides 3, 4, 5 is rectangular and used this ratio to construct right angles when constructing buildings.

Among the Pythagoreans, a method of proving the theorem “without words” was widespread. Listeners were presented with a drawing depicting two equal squares with sides a+b, after which they wrote one word “Look.”

Pythagorean pants are equal in all directions.

The Pythagorean theorem has long been widely used in various fields of science, technology and practical life. The Roman architect and engineer Vitruvius, the Greek moralist writer Plutarch, the 5th century mathematician Proclus and others wrote about it in their works.

Application of the Pythagorean theorem today

Construction
Astronomy
Lightning rod
mobile connection

what maximum height should the antenna have so that the transmission can be received within a certain radius (for example, radius R = 200 km?, if it is known that the radius of the Earth is 6380 km.)

Solution:

OB = OA + AB

OB = r + x

Answer:

Multiplication table

Do only what will not upset you later and will not force you to repent

"Golden Poems" of Pythagoras

Don't neglect the health of your body.
Bring him the food and drink and exercise he needs on time.

"Golden Poems" of Pythagoras

Don’t close your eyes when you want to sleep, without having sorted out all your actions of the past day.

"Golden Poems" of Pythagoras

Never do what you don’t know, but learn everything you need to know, and then you will lead a quiet life.

this is interesting

“...that Jesus and Pythagoras were natives of almost the same area in Sicily...”

“...their fathers were prophetically informed that sons would be born to them who would be benefactors of mankind...” “...that both were born while their parents were away from home...”

The truth will remain eternal, as soon as a weak person knows everything! And now the Pythagorean theorem is true, as in his distant age. A. Chamisso

Biography of Pythagoras Pythagoras is not only the most popular scientist, but also the most mysterious person. It is difficult to restore the true picture of his life and achievements, since there are no written documents about Pythagoras. From the shores of the Mediterranean, the cradle of European civilization, from those ancient times, called the “spring of humanity,” the name has come down to us

Biography of Pythagoras It is known that Pythagoras was born on the island of Samos, located in the Aegean Sea, in 576 BC. e. On the advice of Thales, he gained wisdom for 22 years in Egypt. He did not come to Babylon of his own free will. During the conquest of Egypt, he was captured and sold into slavery. For more than 10 years he lived in Babylon, studying ancient culture and scientific achievements of different countries.

Pythagorean school Returning to his homeland, Pythagoras organized a circle of youth from representatives of the aristocracy. They were accepted into the circle with great ceremonies after long trials. Each entrant renounced his property and swore an oath to keep the founder's teachings secret. Thus, in the south of Italy, which was then a Greek colony, the Pythagorean school arose.

Pythagorean school The Pythagoreans studied mathematics, philosophy, and natural sciences. They made many important discoveries in arithmetic and geometry. There was a decree at school according to which the authorship of all mathematical works was attributed to Pythagoras. The star pentagon, or pentagram, is a Pythagorean symbol of health and a secret identification sign

“Golden Poems” of Pythagoras Pythagorean moral teachings: Never do what you do not know, but learn everything you want to know Do not do anything shameful either in the presence of others or in secret Either remain silent or say what is more valuable than silence Before you become speak, let your thoughts ripen under your tongue Be with the one who shoulder the burden, and not with the one who dumps the burden Pythagoras taught that you need to start the day with poetry: “Before getting up from the sweet dreams evoked at night, spread out your soul: What Have you got anything to do for the day?” And the day had to end with verses: “Do not allow lazy sleep to fall on your tired eyes until you answer three questions about the day’s business: What did I do? What didn't you do? So what do I have left to do?

Numerical mysticism... “Everything is a number” “Numbers rule the world” “Everything is beautiful thanks to number” All these are the statements of Pythagoras and he believed that all the laws of the world can be expressed through numbers. It was Pythagoras who introduced even and odd numbers. The Pythagoreans deified numbers and geometric figures Number 1 meant fire, 2 - earth 3 - water 4 - air The sum of these numbers 10 the whole world 5 - love 6 - cold 7 - mind, light

Pythagoras and music It is noteworthy that the starting point in the Pythagorean doctrine of number was music. According to legend, Pythagoras himself established that pleasant to the ear consonances are obtained only when the length of the strings producing these sounds are related as integers of the first four 1: 2 2: 3 3: 4. The musical octave and scale appeared. Pythagoras with his students. Illustration from the book “Theory of Music” by Franchino Gafurio. Milan. 1492. Engraving depicts acoustic experiments of Pythagoras on vessels and pipes in the ratios 4: 6: 8: 9: 12: 16

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Pythagorean school

Slide 2

Biography of Pythagoras

Slide 3

What did Pythagoras do?

Pythagoras was one of the first to state that the Earth has the shape of a ball, and the Sun, Moon and other planets have their own trajectory of movement. Pythagoras is credited with studying the properties of integers and proportions and proving the Pythagorean theorem. The Pythagoreans compiled a table of 10 opposites; Aristotle gives it in his “Metaphysics”: limit - infinite odd - even one - many right - left male - female rest - straight movement - crooked light - darkness good - evil square - elongated rectangle

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In Crotona (Southern Italy) Pythagoras founded a school - the Pythagorean Union. Pythagoras calls only those who have gone through many stages of knowledge his closest students. Pythagoreans study geometry, mathematics, harmony, and astronomy. The activity of Pythagoras as a religious innovator of the 6th century. BC e. consisted in the creation of a secret society, which not only set itself political goals, but, mainly, the liberation of the soul through moral and physical purification with the help of secret teachings (mystical teachings about the cycle of migration of the soul). According to Pythagoras, the eternal soul moves from heaven into the mortal body of a person or animal and undergoes a series of migrations until it earns the right to return back to heaven.

The teachings of Pythagoras should be divided into two components: a scientific approach to understanding the world and a religious and mystical way of life

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Pythagoras symbol

The five-pointed star was considered a symbol of friendship in the school of Pythagoras; it was something like a talisman that was given to friends; a secret sign by which the Pythagoreans recognized each other.

The Pythagorean school gave Greece a whole galaxy of talented philosophers, physicists, and mathematicians. Such as Aristotle, Archytas from Tarentum, Philolaus from Croton, Hippasus from Metapontus. The scientific component of Pythagoras' teachings developed in the 5th century. BC e. through the efforts of his followers, but faded away in the 4th century. BC e., while the mystical-religious component received its development and rebirth in the form of neo-Pythagoreanism during the Roman Empire.

Slide 6

Thoughts and aphorisms

In the field of life, like a sower, walk with an even and constant step. The true fatherland is where there are good morals. Do not be a member of a learned society: the wisest, when they form a society, become commoners. Consider numbers, weight and measure sacred, as children of graceful equality. Measure your desires, weigh your thoughts, count your words. Do not be surprised at anything: the gods were surprised. If they ask: what is more ancient than the gods? - answer: fear and hope. “Try to be wise first, and learn when you have free time.”