The Pythagorean theorem: background, evidence, examples of practical application. Treasure of geometry Whose pants are equal in all directions

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Everyone knows the Pythagorean theorem school time. An outstanding mathematician proved a great conjecture, which is currently used by many people. The rule sounds like this: the square of the length of the hypotenuse right triangle is equal to the sum of the squares of the legs. For many decades, not a single mathematician has been able to argue this rule. After all, Pythagoras walked for a long time towards his goal, so that as a result the drawings took place in everyday life.

1. A small verse to this theorem, which was invented shortly after the proof, directly proves the properties of the hypothesis: "Pythagorean pants are equal in all directions." This two-line was deposited in the memory of many people - to this day the poem is remembered in calculations.
2. This theorem was called "Pythagorean pants" due to the fact that when drawing in the middle, a right-angled triangle was obtained, on the sides of which there were squares. In appearance, this drawing resembled pants - hence the name of the hypothesis.
3. Pythagoras was proud of the developed theorem, because this hypothesis differs from its similar ones by the maximum amount of evidence. Important: the equation was listed in the Guinness Book of Records due to 370 truthful evidence.
4. The hypothesis was proved by a huge number of mathematicians and professors from different countries in many ways. The English mathematician Jones, soon after the announcement of the hypothesis, proved it with the help of a differential equation.
5. At present, no one knows the proof of the theorem by Pythagoras himself. The facts about the proofs of a mathematician today are not known to anyone. It is believed that the proof of the drawings by Euclid is the proof of Pythagoras. However, some scientists argue with this statement: many believe that Euclid independently proved the theorem, without the help of the creator of the hypothesis.
6. Current scientists have discovered that the great mathematician was not the first to discover this hypothesis.. The equation was known long before the discovery by Pythagoras. This mathematician managed only to reunite the hypothesis.
7. Pythagoras did not give the equation the name "Pythagorean Theorem". This name was fixed after the "loud two-line". The mathematician only wanted the whole world to recognize and use his efforts and discoveries.
8. Moritz Kantor - the great greatest mathematician found and saw notes with drawings on an ancient papyrus. Shortly thereafter, Cantor realized that this theorem had been known to the Egyptians as early as 2300 BC. Only then no one took advantage of it and did not try to prove it.
9. Current scholars believe that the hypothesis was known as early as the 8th century BC. Indian scientists of that time discovered an approximate calculation of the hypotenuse of a triangle endowed with right angles. True, at that time no one could prove the equation for sure by approximate calculations.
10. The great mathematician Bartel van der Waerden, after proving the hypothesis, concluded an important conclusion: “The merit of the Greek mathematician is considered not the discovery of direction and geometry, but only its justification. In the hands of Pythagoras were computational formulas that were based on assumptions, inaccurate calculations and vague ideas. However, the outstanding scientist managed to turn it into an exact science.”
11. A famous poet said that on the day of the discovery of his drawing, he erected a glorious sacrifice to the bulls.. It was after the discovery of the hypothesis that rumors spread that the sacrifice of a hundred bulls "went wandering through the pages of books and publications." Wits joke to this day that since then all the bulls are afraid of a new discovery.
12. Proof that Pythagoras did not come up with a poem about pants in order to prove the drawings he put forward: during the life of the great mathematician there were no pants yet. They were invented several decades later.
13. Pekka, Leibniz and several other scientists tried to prove the previously known theorem, but no one succeeded.
14. The name of the drawings "Pythagorean theorem" means "persuasion by speech". This is the translation of the word Pythagoras, which the mathematician took as a pseudonym.
15. Reflections of Pythagoras on his own rule: the secret of what exists on earth lies in numbers. After all, a mathematician, relying on his own hypothesis, studied the properties of numbers, revealed evenness and oddness, and created proportions.

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MBOU Bondarskaya secondary school Student project on the topic: “Pythagoras and his theorem” Prepared by: Ektov Konstantin, student of grade 7 A Head: Dolotova Nadezhda Ivanovna, mathematics teacher 2015

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Annotation. Geometry is a very interesting science. It contains many theorems that are not similar to each other, but sometimes so necessary. I became very interested in the Pythagorean theorem. Unfortunately, one of the most important statements we pass only in the eighth grade. I decided to lift the veil of secrecy and explore the Pythagorean theorem.

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Tasks To study the biography of Pythagoras. Explore the history of the emergence and proof of the theorem. Find out how the theorem is used in art. Find historical problems in which the Pythagorean theorem is used. To get acquainted with the attitude of children of different times to this theorem. Create a project.

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Research progress Biography of Pythagoras. Commandments and aphorisms of Pythagoras. Pythagorean theorem. History of the theorem. Why are "Pythagorean pants equal in all directions"? Various proofs of the Pythagorean theorem by other scientists. Application of the Pythagorean theorem. Poll. Conclusion.

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Pythagoras - who is he? Pythagoras of Samos (580 - 500 BC) Ancient Greek mathematician and idealist philosopher. Born on the island of Samos. Got a good education. According to legend, Pythagoras, in order to get acquainted with the wisdom of Eastern scientists, went to Egypt and lived there for 22 years. Having mastered well all the sciences of the Egyptians, including mathematics, he moved to Babylon, where he lived for 12 years and got acquainted with scientific knowledge Babylonian priests. Traditions attribute to Pythagoras a visit to India. This is very likely, since Ionia and India then had trade relations. Returning to his homeland (c. 530 BC), Pythagoras tried to organize his philosophical school. However, for unknown reasons, he soon leaves Samos and settles in Croton ( Greek colony in northern Italy). Here Pythagoras managed to organize his own school, which operated for almost thirty years. The school of Pythagoras, or, as it is also called, the Pythagorean Union, was at the same time a philosophical school, a political party, and a religious brotherhood. The status of the Pythagorean union was very severe. In his philosophical views, Pythagoras was an idealist, a defender of the interests of the slave-owning aristocracy. Perhaps this was the reason for his departure from Samos, since in Ionia there is a very big influence had democratic views. In public matters, by "order" the Pythagoreans understood the rule of the aristocrats. They condemned ancient Greek democracy. Pythagorean philosophy was a primitive attempt to justify the dominance of the slave-owning aristocracy. At the end of the 5th century BC e. a wave of democratic movement swept through Greece and its colonies. Democracy won in Croton. Pythagoras leaves Croton with his disciples and goes to Tarentum, and then to Metapont. The arrival of the Pythagoreans at Metapont coincided with the outbreak of a popular uprising there. In one of the night skirmishes, almost ninety-year-old Pythagoras died. His school has ceased to exist. The disciples of Pythagoras, fleeing persecution, settled throughout Greece and its colonies. Earning their livelihood, they organized schools in which they taught mainly arithmetic and geometry. Information about their achievements is contained in the writings of later scientists - Plato, Aristotle, etc.

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Commandments and aphorisms of Pythagoras Thought is above all between people on earth. Do not sit down on a grain measure (i.e., do not live idly). When leaving, do not look back (that is, before death, do not cling to life). Do not go down the beaten road (that is, follow not the opinions of the crowd, but the opinions of the few who understand). Do not keep swallows in the house (i.e., do not receive guests who are talkative and not restrained in language). Be with the one who takes the load, do not be with the one who dumps the load (that is, encourage people not to idleness, but to virtue, to work). In the field of life, like a sower, walk with even and steady steps. The true fatherland is where there are good morals. Do not be a member of a learned society: the wisest, making up a society, become commoners. Revere sacred numbers, weight and measure, as a child of graceful equality. Measure your desires, weigh your thoughts, number your words. Be astonished at nothing: astonishment has produced gods.

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Statement of the theorem. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

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Proofs of the theorem. On the this moment 367 proofs of this theorem have been recorded in the scientific literature. Probably, the Pythagorean theorem is the only theorem with such an impressive number of proofs. Of course, all of them can be divided into a small number of classes. The most famous of them: proofs by the area method, axiomatic and exotic proofs.

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Pythagorean theorem Proof Given a right triangle with legs a, b and hypotenuse c. Let's prove that c² = a² + b² Let's complete the triangle to a square with side a + b. The area S of this square is (a + b)². On the other hand, the square is made up of four equal right triangles, each S equal to ½ a b, and a square with side c. S = 4 ½ a b + c² = 2 a b + c² Thus, (a + b)² = 2 a b + c², whence c² = a² + b² c c c c c a b

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The history of the Pythagorean theorem The history of the Pythagorean theorem is interesting. Although this theorem is associated with the name of Pythagoras, it was known long before him. In Babylonian texts, this theorem occurs 1200 years before Pythagoras. It is possible that at that time they did not yet know its evidence, and the very relationship between the hypotenuse and the legs was established empirically on the basis of measurements. Pythagoras apparently found proof of this relationship. An ancient legend has been preserved that in honor of his discovery, Pythagoras sacrificed a bull to the gods, and according to other testimonies, even a hundred bulls. Over the following centuries, various other proofs of the Pythagorean theorem were found. Currently, there are more than a hundred of them, but the most popular theorem is the construction of a square using a given right triangle.

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Theorem in Ancient China "If a right angle is decomposed into its component parts, then the line connecting the ends of its sides will be 5 when the base is 3 and the height is 4."

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Theorem in Ancient Egypt Kantor (the largest German historian of mathematics) believes that the equality 3 ² + 4 ² = 5² was already known to the Egyptians around 2300 BC. e., during the time of King Amenemhat (according to papyrus 6619 of the Berlin Museum). According to Cantor, harpedonapts, or "stringers", built right angles using right triangles with sides 3, 4 and 5.

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About the theorem in Babylonia “The merit of the first Greek mathematicians, such as Thales, Pythagoras and the Pythagoreans, is not the discovery of mathematics, but its systematization and substantiation. In their hands, computational recipes based on vague ideas have become an exact science.

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Why are "Pythagorean pants equal in all directions"? For two millennia, the most common proof of the Pythagorean theorem was that of Euclid. It is placed in his famous book "Beginnings". Euclid lowered the altitude CH from the top right angle on the hypotenuse and proved that its continuation divides the square completed on the hypotenuse into two rectangles, the areas of which are equal to the areas of the corresponding squares built on the legs. The drawing used in the proof of this theorem is jokingly called "Pythagorean pants". For a long time he was considered one of the symbols of mathematical science.

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The attitude of children of antiquity to the proof of the Pythagorean theorem was considered by students of the Middle Ages to be very difficult. Weak students who memorized theorems without understanding, and therefore called "donkeys", were not able to overcome the Pythagorean theorem, which served for them like an insurmountable bridge. Because of the drawings accompanying the Pythagorean theorem, students also called it a “windmill”, composed poems like “Pythagorean pants are equal on all sides”, and drew caricatures.

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Proofs of the theorem The simplest proof of the theorem is obtained in the case of an isosceles right triangle. Indeed, it is enough just to look at the tiling of isosceles right triangles to see that the theorem is true. For example, for triangle ABC: the square built on the hypotenuse AC contains 4 initial triangles, and the squares built on the legs contain two.

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"Chair of the bride" In the figure, the squares built on the legs are placed in steps one next to the other. This figure, which occurs in evidence dating no later than the 9th century CE, e., the Hindus called the "chair of the bride."

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Application of the Pythagorean theorem At present, it is generally recognized that the success of the development of many areas of science and technology depends on the development various directions mathematics. An important condition for increasing production efficiency is the widespread introduction mathematical methods in technology and National economy which involves the creation of new effective methods qualitative and quantitative research, which allow us to solve the problems put forward by practice.

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Application of the theorem in construction In buildings of the Gothic and Romanesque styles, the upper parts of the windows are divided by stone ribs, which not only play the role of an ornament, but also contribute to the strength of the windows.

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Historical tasks To fix the mast, you need to install 4 cables. One end of each cable should be fixed at a height of 12 m, the other on the ground at a distance of 5 m from the mast. Is 50 m of rope enough to secure the mast?

Pythagorean pants The comic name of the Pythagorean theorem, which arose due to the fact that the squares built on the sides of a rectangle and diverging in different directions resemble the cut of trousers. I loved geometry ... and at the entrance exam to the university I even received praise from Chumakov, a professor of mathematics, for explaining the properties of parallel lines and Pythagorean pants without a blackboard, drawing with my hands in the air(N. Pirogov. Diary of an old doctor).

Phraseological dictionary of the Russian literary language. - M.: Astrel, AST. A. I. Fedorov. 2008 .

See what "Pythagorean pants" are in other dictionaries:

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Pythagorean pants- ... Wikipedia

Pythagorean pants- Zharg. school Shuttle. The Pythagorean theorem, which establishes the relationship between the areas of squares built on the hypotenuse and the legs of a right triangle. BTS, 835... Big dictionary of Russian sayings

Pythagorean pants- A playful name for the Pythagorean theorem, which establishes the ratio between the areas of squares built on the hypotenuse and the legs of a right-angled triangle, which looks like the cut of pants in the drawings ... Dictionary of many expressions

Pythagorean pants (invent)- foreigner: about a gifted person Cf. This is the certainty of the sage. In ancient times, he probably would have invented Pythagorean pants ... Saltykov. Motley letters. Pythagorean pants (geom.): in a rectangle, the square of the hypotenuse is equal to the squares of the legs (teaching ... ... Michelson's Big Explanatory Phraseological Dictionary

Pythagorean pants are equal on all sides- The number of buttons is known. Why is the dick cramped? (roughly) about pants and the male sexual organ. Pythagorean pants are equal on all sides. To prove this, it is necessary to remove and show 1) about the Pythagorean theorem; 2) about wide pants ... Live speech. Dictionary of colloquial expressions

Pythagorean pants invent- Pythagorean pants (invent) foreigner. about a gifted person. Wed This is the undoubted sage. In ancient times, he probably would have invented Pythagorean pants ... Saltykov. Motley letters. Pythagorean pants (geom.): in a rectangle, the square of the hypotenuse ... ... Michelson's Big Explanatory Phraseological Dictionary (original spelling)

Pythagorean pants are equal in all directions- Joking proof of the Pythagorean theorem; also in jest about buddy's baggy trousers... Dictionary of folk phraseology

Adj., rude...

PYTHAGOREAN PANTS ARE EQUAL ON ALL SIDES (NUMBER OF BUTTONS IS KNOWN. WHY IS IT CLOSE? / TO PROVE THIS, IT IS NECESSARY TO REMOVE AND SHOW)- adj., rude ... Dictionary modern colloquial phraseological units and sayings

trousers- noun, pl., use comp. often Morphology: pl. what? pants, (no) what? pants for what? pants, (see) what? pants what? pants, what? about pants 1. Pants are a piece of clothing that has two short or long legs and covers lower part… … Dictionary of Dmitriev

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• Pythagorean pants, . In this book you will find fantasy and adventure, miracles and fiction. Funny and sad, ordinary and mysterious... And what else is needed for entertaining reading? The main thing is to be…

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Jarg. school Shuttle. The Pythagorean theorem, which establishes the relationship between the areas of squares built on the hypotenuse and the legs of a right triangle. BTS, 835... Big dictionary of Russian sayings

Pythagorean pants- The comic name of the Pythagorean theorem, which arose due to the fact that the squares built on the sides of a rectangle and diverging in different directions resemble the cut of trousers. I loved geometry ... and at the entrance exam to the university I even received from ... ... Phraseological dictionary of the Russian literary language

Pythagorean pants- A playful name for the Pythagorean theorem, which establishes the ratio between the areas of squares built on the hypotenuse and the legs of a right-angled triangle, which looks like the cut of pants in the drawings ... Dictionary of many expressions

Foreigner: about a gifted man Cf. This is the certainty of the sage. In ancient times, he probably would have invented Pythagorean pants ... Saltykov. Motley letters. Pythagorean pants (geom.): in a rectangle, the square of the hypotenuse is equal to the squares of the legs (teaching ... ... Michelson's Big Explanatory Phraseological Dictionary

Pythagorean pants are equal on all sides- The number of buttons is known. Why is the dick cramped? (roughly) about pants and the male sexual organ. Pythagorean pants are equal on all sides. To prove this, it is necessary to remove and show 1) about the Pythagorean theorem; 2) about wide pants ... Live speech. Dictionary of colloquial expressions

Pythagorean pants (invent) foreign language. about a gifted person. Wed This is the undoubted sage. In ancient times, he probably would have invented Pythagorean pants ... Saltykov. Motley letters. Pythagorean pants (geom.): in a rectangle, the square of the hypotenuse ... ... Michelson's Big Explanatory Phraseological Dictionary (original spelling)

Pythagorean pants are equal in all directions- Joking proof of the Pythagorean theorem; also in jest about buddy's baggy trousers... Dictionary of folk phraseology

Adj., rude...

PYTHAGOREAN PANTS ARE EQUAL ON ALL SIDES (NUMBER OF BUTTONS IS KNOWN. WHY IS IT CLOSE? / TO PROVE THIS, IT IS NECESSARY TO REMOVE AND SHOW)- adj., rude ... Explanatory dictionary of modern colloquial phraseological units and sayings

Exist., pl., use. comp. often Morphology: pl. what? pants, (no) what? pants for what? pants, (see) what? pants what? pants, what? about pants 1. Pants are a piece of clothing that has two short or long legs and covers the bottom ... ... Dictionary of Dmitriev

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• Miracles on wheels, Markusha Anatoly. Millions of wheels spin all over the earth - they roll cars, measure time in hours, tap under trains, perform countless jobs in machine tools and various mechanisms. They are…