Different ways of proving the Pythagorean theorem: examples, descriptions and reviews. Pythagorean pants Pythagorean theorem pants

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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 Supervisor: Dolotova Nadezhda Ivanovna, teacher of mathematics 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, we only pass one of the most important statements in the eighth grade. I decided to open the veil of secrecy and investigate the Pythagorean theorem.

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Tasks To study the biography of Pythagoras. Explore the history of the origin and proof of the theorem. Find out how the theorem is used in art. Find historical problems in the solution of which the Pythagorean theorem is applied. 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. The commandments and aphorisms of Pythagoras. Pythagorean theorem. History of the theorem. Why are "Pythagorean trousers equal in all directions"? Various proofs of the Pythagorean theorem by other scientists. Application of the Pythagorean theorem. Survey. 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. Received a good education. According to legend, Pythagoras, in order to get acquainted with the wisdom of Eastern scholars, 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 the scientific knowledge of the Babylonian priests. Legends attribute Pythagoras to visit India as well. This is very likely, since Ionia and India then had trade links. Returning to his homeland (c. 530 BC), Pythagoras tried to organize his own philosophical school. However, for unknown reasons, he soon leaves Samos and settles in Crotone (a 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, and a political party, and a religious brotherhood. The status of the Pythagorean union was very harsh. 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 supporters of democratic views had a very large influence in Ionia. In social matters, the Pythagoreans understood "order" as the rule of the aristocrats. They condemned ancient Greek democracy. Pythagorean philosophy was a primitive attempt to substantiate the rule of the slave-owning aristocracy. At the end of the 5th century. BC e. a wave of the democratic movement swept through Greece and its colonies. Democracy won in Crotone. Pythagoras, together with his students, leaves Croton and goes to Tarentum, and then to Metapont. The arrival of the Pythagoreans in Metapont coincided with the outbreak of a popular uprising there. In one of the night skirmishes, almost ninety-year-old Pythagoras died. His school ceased to exist. The disciples of Pythagoras, fleeing persecution, settled throughout Greece and its colonies. In order to earn their livelihood, they organized schools in which they taught mainly arithmetic and geometry. Information about their achievements is contained in the writings of the later scientists - Plato, Aristotle, etc.

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The commandments and aphorisms of Pythagoras Thought is above all among people on earth. Do not sit down on the measure of bread (i.e., do not live idly). When leaving, do not look back (i.e., before death, do not cling to life). Do not walk along the beaten track (that is, follow not the opinions of the crowd, but the opinions of a few who understand). Do not keep swallows in the house (that is, do not accept guests who are talkative and not restrained in language). Be with the one who loads the load, do not be with the one who dumps the load (that is, encourage people not to idleness, but to virtue, to work). On 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, making up a society, become commoners. Honor numbers, weight and measure as sacred as children of graceful equality. Measure your desires, weigh your thoughts, count your words. Do not be surprised at anything: surprise produced the gods.

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Statement of the theorem. In a right-angled 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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Proof of the theorem. At the 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 You are given a right-angled triangle with legs a, b and hypotenuse c. Let us prove that c² = a² + b² Let us complete the triangle to a square with side a + b. The area S of this square is (a + b) ². On the other hand, a square is made up of four equal right-angled 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 with a b

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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 proof, and the very relationship between the hypotenuse and the legs was established empirically on the basis of measurements. Pythagoras seems to have found proof of this relationship. An ancient legend has survived 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 have been found. Currently, there are more than a hundred of them, but the most popular is the theorem with the construction of a square using a given right-angled 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 Cantor (the largest German historian of mathematics) believes that the equality 3 ² + 4 ² = 5² was already known to the Egyptians around 2300 BC. BC, during the time of King Amenemhat (according to papyrus 6619 of the Berlin Museum). According to Cantor, the harpedonapts, or "rope pulls", built right angles using right-angled 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, was not the discovery of mathematics, but its systematization and substantiation. In their hands, computational recipes based on vague notions have become an exact science. "

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Why are "Pythagorean trousers equal in all directions"? For two millennia, the most common proof of the Pythagorean theorem was that of Euclid. It is included in his famous book "Beginnings". Euclid lowered the CH height from the vertex of the right angle to the hypotenuse and argued 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 to prove this theorem is jokingly called "Pythagorean pants." For a long time, it 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 very difficult by the students of the Middle Ages. Weak students, who had learned the theorems by heart, without understanding, and therefore called "donkeys", were unable to overcome the Pythagorean theorem, which served as an insurmountable bridge for them. Because of the drawings accompanying the Pythagorean theorem, the students also called it a "windmill", composed poems such as "Pythagorean pants are equal on all sides," and drew cartoons.

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Proofs of the Theorem The simplest proof of the theorem is obtained in the case of an isosceles right-angled triangle. Indeed, it is enough to simply look at the mosaic of isosceles right-angled triangles to verify the validity of the theorem. For example, for a triangle ABC: the square built on the hypotenuse AC contains 4 original triangles, and the squares built on the legs - two each.

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“Bride's chair” In the figure, the squares built on the legs are placed in steps one next to the other. This figure, which is found in evidence dating as early as the 9th century AD. e., Indians called "the bride's chair".

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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 of various areas of mathematics. An important condition for increasing production efficiency is the widespread introduction of mathematical methods into technology and the national economy, which presupposes the creation of new, effective methods of qualitative and quantitative research that make it possible 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 style, the upper parts of the windows are dissected 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 secure 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. Will 50 m of cable be enough to secure the mast?

“Pythagorean pants are equal on all sides.
To prove it, you need to film and show. "

This rhyme has been known to everyone since high school, since the time when we studied the famous Pythagorean theorem in geometry class: the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. Although Pythagoras himself never wore pants - in those days the Greeks did not wear them. Who is Pythagoras?
Pythagoras of Samos from lat. Pythagoras, Pythian broadcaster (570-490 BC) - ancient Greek philosopher, mathematician and mystic, the founder of the religious and philosophical school of the Pythagoreans.
Among the conflicting teachings of his teachers, Pythagoras was looking for a living connection, a synthesis of a single great whole. He set himself the goal - to find the path leading to the light of truth, that is, to know life in unity. For this purpose, Pythagoras visited the entire ancient world. He believed that he should broaden his already broad horizons by studying all religions, doctrines and cults. He lived among the rabbis and learned a lot about the secret traditions of Moses, the lawgiver of Israel. Then he visited Egypt, where he was initiated into the Mystery of Adonis, and, having managed to cross the Euphrates valley, he stayed for a long time with the Chaldeans in order to adopt their secret wisdom. Pythagoras visited Asia and Africa, including Hindustan and Babylon. In Babylon, he studied the knowledge of magicians.
The merit of the Pythagoreans was the advancement of ideas about the quantitative laws of the development of the world, which contributed to the development of mathematical, physical, astronomical and geographical knowledge. At the heart of things is Number, Pythagoras taught, to know the world means to know the numbers that govern it. Studying numbers, the Pythagoreans developed numerical relationships and found them in all areas of human activity. Pythagoras taught secretly and did not leave behind written works. Pythagoras attached great importance to number. His philosophical views are largely due to mathematical concepts. He said: "Everything is number", "All things are numbers", thus highlighting one side in understanding the world, namely, its measurability by numerical expression. Pythagoras believed that number owns all things, including moral and spiritual qualities. He taught (according to Aristotle): "Justice ... is a number multiplied by itself." He believed that in every object, in addition to its changeable states, there is an unchanging being, some unchanging substance. This is the number. Hence the main idea of Pythagoreanism: number is the basis of all that exists. The Pythagoreans saw in numbers and in mathematical relations an explanation of the hidden meaning of phenomena, the laws of nature. According to Pythagoras, objects of thought are more real than objects of sensory cognition, since numbers have a timeless nature, i.e. forever. They are a kind of reality that is higher than the reality of things. Pythagoras says that all properties of an object can be destroyed, or they can change, except for only one numerical property. This property is one. A unit is the existence of things, indestructible and indestructible, unchanging. Break up any object into tiny particles - each particle will be one. Asserting that numerical being is the only unchanging being, Pythagoras came to the conclusion that all objects are the essence of copies of numbers.
One is an absolute number. One has eternity. The unit does not have to be in any relation to anything else. It exists by itself. Two is only a relationship of one to one. All numbers are only
numerical relations Units, its modifications. And all forms of being are only certain aspects of infinity, and hence Units. The original One contains all the numbers, therefore it contains the elements of the whole world. Objects are real manifestations of abstract being. Pythagoras was the first to designate the cosmos with all things in it, as an order that is established by number. This order is available to the mind, is realized by it, which allows you to see the world in a completely new way.
The process of knowing the world, according to Pythagoras, is the process of knowing the numbers that govern it. After Pythagoras, the cosmos began to be viewed as ordered by the number of the universe.
Pythagoras taught that the human soul is immortal. He owns the idea of transmigration of souls. He believed that everything that happens in the world repeats itself over and over again after certain periods of time, and the souls of the dead after some time move into others. The soul, as a number, is a Unit, i.e. the soul is essentially perfect. But every perfection, since it comes into motion, turns into imperfection, although it seeks to regain its previous perfect state. Pythagoras called the deviation from the Unity imperfection; therefore, Two was considered a cursed number. The soul in a person is in a state of comparative imperfection. It consists of three elements: intelligence, intelligence, passion. But if animals also possess mind and passions, then only man is endowed with reason (reason). Any of these three sides in a person can prevail, and then a person becomes predominantly either reasonable, or sane, or sensual. Accordingly, he turns out to be either a philosopher, or an ordinary person, or an animal.
However, back to the numbers. Indeed, numbers are an abstract manifestation of the fundamental philosophical law of the Universe - the Unity of Opposites.
Note. Abstraction serves as the basis for the processes of generalization and the formation of concepts. She is a necessary condition for categorization. It forms generalized images of reality, which make it possible to single out connections and relations of objects that are significant for a certain activity.
The Unity of the Opposites of the Universe consists of Form and Content, Form is a quantitative category, and Content is a qualitative category. Naturally, numbers express quantitative and qualitative categories in abstraction. Hence the addition (subtraction) of numbers is a quantitative component of the abstraction of Forms, and multiplication (division) is a qualitative component of the abstraction of Content. The numbers of abstraction of Forms and Contents are inextricably linked with the Unity of Opposites.
Let's try to perform mathematical operations, establishing an inextricable connection between Form and Content over numbers.

So let's look at a number series.
1,2,3,4,5,6,7,8,9. 1 + 2 = 3 (3) 4 + 5 = 9 (9) ... (6) 7 + 8 = 15 -1 + 5 = 6 (9). Then 10 - (1 + 0) + 11 (1 + 1) = (1 + 2 = 3) - 12 - (1 + 2 = 3) (3) 13- (1 + 3 = 4) + 14 - (1 + 4 = 5) = (4 + 5 = 9) (9)… 15 - (1 + 5 = 6) (6)… 16- (1 + 6 = 7) + 17 - (1 + 7 = 8) ( 7 + 8 = 15) - (1 + 5 = 6) ... (18) - (1 + 8 = 9) (9). 19 - (1 + 9 = 10) (1) -20 - (2 + 0 = 2) (1 + 2 = 3) 21 - (2 + 1 = 3) (3) - 22- (2 + 2 = 4 ) 23- (2 + 3 = 5) (4 + 5 = 9) (9) 24- (2 + 4 = 6) 25 - (2 + 5 = 7) 26 - (2 + 6 = 8) - 7+ 8 = 15 (1 + 5 = 6) (6) Etc.
From here we observe a cyclic transformation of Forms, which corresponds to the cycle of Contents –1st –cycle - 3-9-6 - 6-9-3; 2-nd cycle - 3-9-6 -6-9-3, etc.
6
9 9
3

The cycles represent the eversion of the torus of the Universe, where the Opposites of the abstraction numbers of Forms and Contents are 3 and 6, where 3 defines Compression, and 6 - Stretch. The compromise for their interaction is the number 9.
Further 1,2,3,4,5,6,7,8,9. 1x2 = 2 (3) 4x5 = 20 (2 + 0 = 2) (6) 7x8 = 56 (5 + 6 = 11 1 + 1 = 2) (9), etc.
The cycle looks like this 2- (3) -2- (6) - 2- (9) ... where 2 is a constituent element of the cycle 3-6-9.
Next is the multiplication table:
2x1 = 2
2x2 = 4
(2+4=6)
2x3 = 6
2x4 = 8
2x5 = 10
(8+1+0 = 9)
2x6 = 12
(1+2=3)
2x7 = 14
2x8 = 16
(1+4+1+6=12;1+2=3)
2x9 = 18
(1+8=9)
Cycle -6.6 - 9 - 3.3 - 9.
3x1 = 3
3x2 = 6
3x3 = 9
3x4 = 12 (1 + 2 = 3)
3x5 = 15 (1 + 5 = 6)
3x6 = 18 (1 + 8 = 9)
3x7 = 21 (2 + 1 = 3)
3x8 = 24 (2 + 4 = 6)
3x9 = 27 (2 + 7 = 9)
Cycle 3-6-9; 3-6-9; 3-6-9.
4x1 = 4
4x2 = 8 (4 + 8 = 12 1 + 2 = 3)
4x3 = 12 (1 + 2 = 3)
4x4 = 16
4x5 = 20 (1 + 6 + 2 + 0 = 9)
4x6 = 24 (2 + 4 = 6)
4x7 = 28
4x8 = 32 (2 + 8 + 3 + 2 = 15 1 + 5 = 6)
4x9 = 36 (3 + 6 = 9)
Cycle 3.3 - 9 - 6.6 - 9.
5x1 = 5
5x2 = 10 (5 + 1 + 0 = 6)
5x3 = 15 (1 + 5 = 6)
5x4 = 20
5x5 = 25 (2 + 0 + 2 + 5 = 9)
5x6 = 30 (3 + 0 = 3)
5x7 = 35
5x8 = 40 (3 + 5 + 4 + 0 = 12 1 + 2 = 3)
5x9 = 45 (4 + 5 = 9)
Cycle -6.6 - 9 - 3.3 - 9.
6x1 = 6
6x2 = 12 (1 + 2 = 3)
6x3 = 18 (1 + 8 = 9)
6x4 = 24 (2 + 4 = 6)
6x5 = 30 (3 + 0 = 3)
6x6 = 36 (3 + 6 = 9)
6x7 = 42 (4 + 2 = 6)
6x8 = 48 (4 + 8 = 12 1 + 2 = 3)
6x9 = 54 (5 + 4 = 9)
Cycle - 3-9-6; 3-9-6; 3-9.
7x1 = 7
7x2 = 14 (7 + 1 + 4 = 12 1 + 2 = 3)
7x3 = 21 (2 + 1 = 3)
7x4 = 28
7x5 = 35 (2 + 8 + 3 + 5 = 18 1 + 8 = 9)
7x6 = 42 (4 + 2 = 6)
7x7 = 49
7x8 = 56 (4 + 9 + 5 + 6 = 24 2 + 4 = 6)
7x9 = 63 (6 + 3 = 9)
Cycle - 3.3 - 9 - 6.6 - 9.
8x1 = 8
8x2 = 16 (8 + 1 + 6 = 15 1 + 5 = 6.
8x3 = 24 (2 + 4 = 6)
8x4 = 32
8x5 = 40 (3 + 2 + 4 + 0 = 9)
8x6 = 48 (4 + 8 = 12 1 + 2 = 3)
8x7 = 56
8x8 = 64 (5 + 6 + 6 + 4 = 21 2 + 1 = 3)
8x9 = 72 (7 + 2 = 9)
Cycle -6.6 - 9 - 3.3 - 9.
9x1 = 9
9x2 = 18 (1 + 8 = 9)
9x3 = 27 (2 + 7 = 9)
9x4 = 36 (3 + 6 = 9)
9x5 = 45 (4 + 5 = 9)
9x6 = 54 (5 + 4 = 9)
9x7 = 63 (6 + 3 = 9)
9x8 = 72 (7 + 2 = 9)
9x9 = 81 (8 + 1 = 9).
The cycle is 9-9-9-9-9-9-9-9-9.

The numbers of the qualitative category of Content - 3-6-9, indicate the nucleus of an atom with a different number of neutrons, and the quantitative category indicates the number of electrons of the atom. A chemical element is nuclei, the masses of which are multiples of 9, and multiples of 3 and 6 are isotopes.
Note. Isotope (from the Greek "equal", "the same" and "place") - a variety of atoms and nuclei of one chemical element with a different number of neutrons in the nucleus. A chemical element is a collection of atoms with the same nuclear charges. Isotopes are varieties of atoms of a chemical element with the same nuclear charge, but different mass numbers.

All real things are composed of atoms, and atoms are defined by numbers.
Therefore, it is natural that Pythagoras was convinced that numbers are real objects, and not simple symbols. Number is a certain state of material objects, the essence of a thing. And in this Pythagoras was right.

Pythagorean pants - equal on all sides.
To prove this, you need to film and show.

To prove his theorem, Pythagoras drew a figure of squares on the sides of a triangle in the sand. The sum of the squares of the legs in a right-angled triangle is equal to the square of the hypotenuse A square plus B square is equal to C square. It was 500 BC. Today the Pythagorean theorem is being taught in high school. In the Guinness Book of Records, the Pythagorean Theorem is the theorem with the maximum number of proofs. Indeed, in 1940 a book was published containing three hundred and seventy proofs of the Pythagorean theorem. One of them was proposed by US President James Abram Garfield. Only one proof of the theorem is still unknown to any of us: the proof of Pythagoras himself. For a long time, it was believed that Euclid's proof was the Pythagorean proof, but now mathematicians think that this proof belongs to Euclid himself.

The classical proof of Euclid is aimed at establishing the equality of areas between rectangles formed by cutting a square above the hypotenuse with a height from a right angle with squares above the legs.

The construction used for the proof is as follows: for a right-angled triangle ABC with a right angle C, squares above the legs ACED and BCFG, and a square above the hypotenuse ABIK, build the height CH and the ray s extending it, splitting the square above the hypotenuse into two rectangles AHJK and BHJI. The proof is aimed at establishing the equality of the areas of the rectangle AHJK with the square above the leg AC; the equality of the areas of the second rectangle that makes up a square above the hypotenuse and the rectangle above the other leg is established in the same way.

The equality of the areas of the rectangle AHJK and ACED is established through the congruence of triangles ACK and ABD, the area of each of which is equal to half the area of rectangles AHJK and ACED, respectively, due to the following property: the area of a triangle is equal to half the area of a rectangle if the figures have a common side, and the height of the triangle is k the common side is the other side of the rectangle. The congruence of triangles follows from the equality of the two sides (sides of the squares) and the angle between them (composed of the right angle and the angle at A.

Thus, the proof establishes that the area of the square above the hypotenuse, made up of rectangles AHJK and BHJI, is equal to the sum of the areas of the squares above the legs.

German mathematician Karl Gauss proposed to cut giant Pythagorean trousers from trees in the Siberian taiga. Looking at these pants from space, aliens must make sure that intelligent beings live on our planet.

It's funny that Pythagoras himself never wore pants - in those days, the Greeks simply did not know about such a wardrobe item.

Sources:

sandbox.fizmat.vspu.ru
ru.wikipedia.org
kuchmastar.fandom.com

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

The Roman architect Vitruvius singled out the Pythagorean theorem "from the numerous discoveries that provided services to the development of human life," and called for it to be treated with the greatest respect. It was back in the 1st century BC. e. At the turn of the XVI-XVII centuries, the famous German astronomer Johannes Kepler called it one of the treasures of geometry, comparable to the measure of gold. It is unlikely that in all mathematics there will be a more weighty and significant statement, because in terms of the number of scientific and practical applications, the Pythagorean theorem has no equal.

Pythagoras' theorem for the case of an isosceles right triangle.

Science and Life // Illustrations

Illustration to the Pythagorean theorem from the "Treatise on the Measuring Pole" (China, III century BC) and the proof reconstructed on its basis.

Science and Life // Illustrations

S. Perkins. Pythagoras.

A blueprint for a possible proof of Pythagoras.

"Mosaic of Pythagoras" and an-Nayrizi tiling of three squares in the proof of the Pythagorean theorem.

P. de Hooch. A hostess and a maid in the courtyard. Around 1660.

J. Ohtervelt. Wandering musicians at the door of a rich house. 1665 year.

Pythagorean pants

Pythagoras' theorem is perhaps the most recognizable and undoubtedly the most famous in the history of mathematics. In geometry, it is used literally at every step. Despite the simplicity of its formulation, this theorem is by no means obvious: looking at a right-angled triangle with sides a< b < c, усмотреть соотношение a 2 + b 2 = c 2 невозможно. Однажды известный американский логик и популяризатор науки Рэймонд Смаллиан, желая подвести учеников к открытию теоремы Пифагора, начертил на доске прямоугольный треугольник и по квадрату на каждой его стороне и сказал: «Представьте, что эти квадраты сделаны из кованого золота и вам предлагают взять себе либо один большой квадрат, либо два маленьких. Что вы выберете?» Мнения разделились пополам, возникла оживлённая дискуссия. Каково же было удивление учеников, когда учитель объяснил им, что никакой разницы нет! Но стоит только потребовать, чтобы катеты были равны, - и утверждение теоремы станет явным (рис. 1). И кто после этого усомнится, что «пифагоровы штаны» во все стороны равны? А вот те же самые «штаны», только в «сложенном» виде (рис. 2). Такой чертёж использовал герой одного из диалогов Платона под названием «Менон», знаменитый философ Сократ, разбирая с мальчиком-рабом задачу на построение квадрата, площадь которого в два раза больше площади данного квадрата. Его рассуждения, по сути, сводились к доказательству теоремы Пифагора, пусть и для конкретного треугольника.

Figures shown in fig. 1 and 2, resemble the simplest ornament of squares and their equal parts - a geometric pattern known from time immemorial. They can completely cover the plane. A mathematician would call such a covering of a plane by polygons parquet, or tiling. What does Pythagoras have to do with it? It turns out that he was the first to solve the problem of regular parquets, which began the study of tilings of various surfaces. So, Pythagoras showed that the plane around a point can be covered without gaps by equal regular polygons of only three types: six triangles, four squares and three hexagons.

4000 years later

The history of the Pythagorean theorem goes back to ancient times. It is mentioned in the Babylonian cuneiform texts of the times of King Hammurabi (18th century BC), that is, 1200 years before the birth of Pythagoras. The theorem was used as a ready-made rule in many problems, the simplest of which is finding the diagonal of a square along its side. It is possible that the Babylonians obtained the ratio a 2 + b 2 = c 2 for an arbitrary right-angled triangle by simply "generalizing" the equality a 2 + a 2 = c 2. But it is forgivable for them - for the practical geometry of the ancients, which was reduced to measurements and calculations, no rigorous justification was required.

Now, almost 4,000 years later, we are dealing with a theorem that holds the record for the number of possible proofs. By the way, collecting them is a long tradition. The peak of interest in the Pythagorean theorem fell on the second half of the 19th - early 20th centuries. And if the first collections contained no more than two or three dozen proofs, then by the end of the 19th century their number was close to 100, and after another half a century it exceeded 360, and these are only those that were collected from various sources. Who has not undertaken the solution of this ageless problem - from eminent scientists and popularizers of science to congressmen and schoolchildren. And what is remarkable, in the originality and simplicity of the solution, some amateurs were not inferior to professionals!

The oldest surviving proof of the Pythagorean theorem is about 2300 years old. One of them - the strict axiomatic - belongs to the ancient Greek mathematician Euclid, who lived in the 4th-3rd centuries BC. e. In Book I of the Elements, the Pythagorean theorem is listed as Proposition 47. The most graphic and beautiful proofs are based on the reshaping of the "Pythagorean trousers". They look like a tricky square-cutting puzzle. But make the pieces move correctly - and they will reveal to you the secret of the famous theorem.

Here is an elegant proof obtained on the basis of a drawing from one ancient Chinese treatise (Fig. 3), and its connection with the problem of doubling the area of a square becomes immediately clear.

It was this proof that the seven-year-old Guido, the precocious hero of the short story "Little Archimedes" by the English writer Aldous Huxley, tried to explain to his younger friend. It is curious that the narrator, who observed this picture, noted the simplicity and persuasiveness of the proof, so he attributed it ... to Pythagoras himself. But the protagonist of Evgeny Veltistov's fantastic story "An Electronic - a Boy from a Suitcase" knew 25 proofs of the Pythagorean theorem, including those given by Euclid; true, he mistakenly called it the simplest, although in fact in the modern edition of the "Elements" it occupies one and a half pages!

First mathematician

Pythagoras of Samos (570-495 BC), whose name has long been inextricably linked with a remarkable theorem, in a sense can be called the first mathematician. It is with him that mathematics begins as an exact science, where any new knowledge is not the result of visual representations and rules derived from experience, but the result of logical reasoning and conclusions. This is the only way to establish once and for all the truth of any mathematical proposition. Before Pythagoras, the deductive method was used only by the ancient Greek philosopher and scientist Thales of Miletus, who lived at the turn of the 7th-6th centuries BC. e. He expressed the very idea of the proof, but applied it not systematically, selectively, as a rule, to obvious geometric statements such as “the diameter divides the circle in half”. Pythagoras went much further. It is believed that he introduced the first definitions, axioms and methods of proof, and also created the first course in geometry, known to the ancient Greeks under the name "The Tradition of Pythagoras". He also stood at the origins of the theory of numbers and stereometry.

Another important merit of Pythagoras is the founding of the glorious school of mathematicians, which for more than a century determined the development of this science in Ancient Greece. The term "mathematics" (from the Greek word μαθημa - doctrine, science) is also associated with his name, uniting four related disciplines of the knowledge system created by Pythagoras and his adherents, the Pythagoreans: geometry, arithmetic, astronomy and harmonics.

It is impossible to separate the achievements of Pythagoras from the achievements of his students: following custom, they attributed their own ideas and discoveries to their Teacher. The early Pythagoreans did not leave any compositions, they transmitted all information to each other orally. So 2500 years later, historians have no choice but to reconstruct the lost knowledge based on the transcriptions of other, later authors. Let's pay tribute to the Greeks: although they surrounded the name of Pythagoras with many legends, they did not ascribe anything to him that he could not discover or develop into a theory. And the theorem that bears his name is no exception.

Such a simple proof

It is not known whether Pythagoras himself discovered the relationship between the lengths of the sides in a right triangle or borrowed this knowledge. Ancient authors claimed that he himself, and loved to retell the legend of how, in honor of his discovery, Pythagoras sacrificed a bull. Modern historians tend to believe that he learned about the theorem by becoming acquainted with the mathematics of the Babylonians. We also do not know in what form Pythagoras formulated the theorem: arithmetically, as is customary today, - the square of the hypotenuse is equal to the sum of the squares of the legs, or geometrically, in the spirit of the ancients, - a square built on the hypotenuse of a right-angled triangle is equal to the sum of squares built on his legs.

It is believed that it was Pythagoras who gave the first proof of the theorem that bears his name. It, of course, has not survived. According to one version, Pythagoras could use the doctrine of proportions developed in his school. On it was based, in particular, the theory of similarity, on which the reasoning is based. Draw in a right-angled triangle with legs a and b the height to the hypotenuse c. We get three similar triangles, including the original one. Their respective sides are proportional, a: c = m: a and b: c = n: b, whence a 2 = c m and b 2 = c n. Then a 2 + b 2 = = c · (m + n) = c 2 (Fig. 4).

This is just a reconstruction proposed by one of the historians of science, but the proof, you see, is quite simple: it takes only a few lines, you do not need to complete, redraw, calculate anything ... It is not surprising that it was rediscovered more than once. It is contained, for example, in the "Practice of Geometry" by Leonardo of Pisa (1220), and it is still quoted in textbooks.

This proof did not contradict the ideas of the Pythagoreans about commensurability: initially they believed that the ratio of the lengths of any two segments, and hence the areas of rectilinear figures, can be expressed using natural numbers. They did not consider any other numbers, did not even allow fractions, replacing them with ratios 1: 2, 2: 3, etc. However, ironically, it was the Pythagorean theorem that led the Pythagoreans to the discovery of the incommensurability of the diagonal of a square and its side. All attempts to numerically represent the length of this diagonal - for the unit square it is equal to √2 - have led nowhere. It turned out to be easier to prove that the problem is unsolvable. For such a case, mathematicians have a proven method - proof by contradiction. By the way, he is also attributed to Pythagoras.

The existence of a relationship that is not expressed in natural numbers put an end to many ideas of the Pythagoreans. It became clear that the numbers they knew were not enough to solve even simple problems, let alone all geometry! This discovery was a turning point in the development of Greek mathematics, its central problem. First, it led to the development of the doctrine of incommensurable quantities - irrationalities, and then - to the expansion of the concept of number. In other words, the centuries-old history of the study of the set of real numbers began with him.

Mosaic of Pythagoras

If you cover the plane with squares of two different sizes, surrounding each small square with four large ones, you get the "Pythagoras mosaic" parquet floor. Such a pattern has long adorned stone floors, recalling the ancient proofs of the Pythagorean theorem (hence its name). By applying a square grid to the parquet in different ways, you can get the partitions of squares built on the sides of a right-angled triangle, which were proposed by different mathematicians. For example, if you arrange the grid so that all its nodes coincide with the upper right vertices of the small squares, fragments of the drawing will appear for the proof of the medieval Persian mathematician al-Nayrizi, which he placed in the comments to Euclid's Beginnings. It is easy to see that the sum of the areas of the big and small squares, the original elements of the parquet, is equal to the area of one square of the grid superimposed on it. And this means that the specified partition is really suitable for laying parquet: by connecting the resulting polygons into squares, as shown in the figure, you can fill the entire plane with them without gaps and overlaps.

A humorous proof of the Pythagorean theorem; also joking about buddy's baggy trousers.

- triples of positive integers x, y, z satisfying the equation x2 + y 2 = z2 ...
Encyclopedia of mathematics
- triples of natural numbers such that a triangle, the lengths of the sides of which are proportional to these numbers, is rectangular, for example. three numbers: 3, 4, 5 ...
Natural science. encyclopedic Dictionary
- see Rescue rocket ...
Marine vocabulary
- triples of natural numbers such that a triangle whose side lengths are proportional to these numbers is rectangular ...
Great Soviet Encyclopedia
- mil. Frenzy. An expression used when listing or opposing two facts, phenomena, circumstances ...
Educational phraseological dictionary
- From the dystopian novel Animal Farm by the English writer George Orwell ...
- It is first encountered in the satire "Diary of a Liberal in St. Petersburg" by Mikhail Evgrafovich Saltykov-Shchedrin, who so figuratively described the dual, cowardly position of Russian liberals - their own ...
Dictionary of winged words and expressions
- It is said in the case when the interlocutor tried to communicate something for a long time and indistinctly, cluttering the main idea with secondary details ...
Dictionary of folk phraseology
- The number of buttons is known. Why dick cramped? - about pants and the male genital organ. ... 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
- Wed There is no immortality of the soul, so there is no virtue either, "so everything is allowed" ... A seductive theory for scoundrels ... A braggart, but the point is, on the one hand, one cannot but confess, and on the other, one cannot but confess ...
Explanatory phraseological dictionary of Michelson
- Piѳagorov's trousers and socks. about a man gifted. Wed This is undoubtedly a wise man. In antiquity, he probably would have invented Piѳagor's pants ... Saltykov. Motley letters ...
- From one side - from the other side. Wed Nt immortality of the soul, so nѣt and virtues, "then, everything is allowed" ... Seductive theory of scoundrels .....
Michelson's Explanatory Phraseological Dictionary (original orph.)
- The comic name of the Pythagorean theorem, which arose due to the fact that the squares built on the sides of the rectangle and diverging in different directions resemble the cut of pants ...
- ON THE ONE HAND ON THE OTHER HAND. Book ...
Phraseological dictionary of the Russian literary language
- See RANKS -...
IN AND. Dahl. Russian proverbs
- Zharg. shk. Shuttle. Pythagoras. ...
A large dictionary of Russian sayings

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