Pythagoras pants in all directions are equal to why. Interesting facts about the Pythagora theorem: we learn a new one about the famous theorem (15 photos)

The Roman architect Vitruvius highled the theorem of Pyphagora "from the numerous discoveries that had services to the development of human life", and urged to treat her with the greatest reverence. It was still in the first century to n. e. At the turn of the XVI-XVII centuries, the famous German astronomer Johann Kepler called it one of the treasures of geometry, comparable to a measure of gold. It is unlikely that in the whole mathematics there will be a more significant and significant approval, because by the number of scientific and practical applications, the Pythagore Theorem has no equal.

Pythagora theorem for the case of an equifiable rectangular triangle.

Science and life // illustration

Illustration for the Pythagore Theorem from the "Treatment of Measuring Six" (China, III century BC) and the proof reconstructed on its basis.

Science and life // illustration

S. Perkins. Pythagoras.

Drawing to the possible proof of Pythagora.

Pythagore Mosaic and the splitting of an AN-Nationals of three squares in the proof of the Pythagora theorem.

P. de Heh. Mistress and maid in the courtyard. Around 1660.

I. OXTERVELT. Stray musicians in the doors of a rich house. 1665 year.

Pythagora pants

The Pythagore Theorem is almost the most recognizable and, undoubtedly, the most famous in the history of mathematics. In geometry, it is applied literally at every step. Despite the simplicity of the wording, this theorem is by no means obvious: looking at the rectangular triangle with the parties 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 big squareor two small ones. What do you choose? " Opinions were divided in half, a lively discussion arose. What was the surprise of students when the teacher explained to them that there was no difference! But it is only necessary to require kartettes to be equal - and the statement of the theorem will become apparent (Fig. 1). And who after that will doubt that "Pythagora pants" are equal in all directions? But the same "pants", only in the "folded" form (Fig. 2). Such a drawing was used by the hero of one of the dialogues of Plato called "Menon", the famous philosopher Socrates, sacrificing with a slave boy to build a square whose area is twice the area of \u200b\u200bthis square. His arguments, in fact, were reduced to the proof of the Pythagore's theorem, albeit for a specific triangle.

Figures depicted in fig. 1 and 2, resemble the simplest ornament from squares and their equal parts - Geometric pattern, known since time immemorial. They can be completely covered with a plane. Mathematics would call such a plane coating by polygons parquet, or a mixing. What is the Pythagore? It turns out that he first decided the task of the right parquets, with which the study of the inspections of various surfaces began. So, Pythagoras showed that the plane around the point can be covered without spaces equal regular polygons of only three species: six triangles, four squares and three hexagons.

4000 years later

The history of the Pythagora theorem goes into deep antiquity. The mention of it is still contained in the Babylonian clinical texts of the Tsar Hammurapi (XVIII century BC), that is, 1200 years before the birth of Pythagora. The theorem was used as a ready-made rule in many tasks, the simplest of which is to find the diagonal of the square on its side. It is possible that the ratio A 2 + B 2 \u003d C 2 for an arbitrary rectangular triangle Babylonians received, simply "summarizing" the equality A 2 + A 2 \u003d C 2. But they are fortunate - for the practical geometry of the ancient, which reduced to measurements and calculations, not required strict justifications.

Now, almost 4,000 years later, we are dealing with the record holder in the number of all sorts of evidence. By the way, their collecting is a long tradition. Peak of interest in the Pythagora theorem came on the second half XIX - The beginning of the XX century. And if the first collections contain no more than two-three dozen evidence, then eND XIX. The century their number approached 100, and after half a century exceeded 360, and these are only those that managed to collect on different sources. Who just did not take for the solution of this unstasive task - from the famous scientists and popularizers of science to congressmen and schoolchildren. And what is noteworthy, in originality and simplicity of solving other lovers did not inferior professionals!

The most ancient of the proofs of the Pythagora theorem of about 2300 years have reached us. One of them is strict axiomatic - belongs to the ancient Greek mathematics Euclide, who lived in the IV-III centuries BC. e. In I, the book "Benefits" the Pythagore Theorem is as "offer 47". The most visual and beautiful proofs are built at the Pythagorean Pants Painting. They look like a cunning puzzle for cutting squares. But make the shapes move correctly - and they will open you the secret of the famous theorem.

This is what elegant proof is obtained on the basis of the drawing from one ancient Chinese treatise (Fig. 3), and immediately clarifies its connection with the task of doubling the square of the square.

It was such a proof that was trying to explain to his younger friend Seven-year-old Guido, not by the years, an intelligent hero of the novel of the English writer Oldhos Huxley "Little Archimedes". It is curious that the narrator who observed this picture, noted the simplicity and persuasiveness of evidence, therefore he attributed it ... Pythagora himself. And here the main character The fantastic story of Evgenia Wellistov "Electronics - a boy from the suitcase" knew 25 evidence of the Pythagora theorem, including this by Euclide; True, mistakenly called him the simplest, although in fact in the modern edition "began" it takes one and a half pages!

First mathematician

Pythagora Samossky (570-495 BC), whose name has long been and inextricably linked with a wonderful theorem, in a certain sense can be called the first mathematician. It is from him that mathematics begins as an exact science, where every new knowledge is the result of not visual ideas and the rules issued from experience, but the result of logical reasoning and conclusions. Only so you can forever establish the truth of any mathematical proposal. Before Pythagora, the deductive method was used only an ancient Greek philosopher and scientist Falez Miletsky, who lived at the turn of the VII-VI centuries to N. e. He suggested the idea of \u200b\u200bevidence, but applied it not systematically, selectively, as a rule, to obvious geometric statements like "Diameter divides the circle in half." Pythagoras advanced much further. It is believed that he introduced the first definitions, axioms and methods of evidence, and also created the first course of geometry, known to the ancient Greeks called "Tradition of Pythagora". He also stood at the origins of the theory of numbers and stereometry.

Another important merit of Pythagora is the foundation of the glorious school of mathematicians, which for more than a century determined the development of this science in ancient Greece. The term "mathematics" is associated with his name (from the Greek word μαθημA - the teaching, science), which united four relative disciplines created by Pythagoras and his adherents - Pythagoreans - knowledge systems: geometry, arithmetic, astronomy and harmonic.

It is impossible to separate the achievements of Pythagore from achievements: Following the custom, they attributed their own ideas and opening their teacher. No essays left early Pythagoreans left all the information they passed to each other orally. So 2500 years later, historians do not have anything else, except for reconstructing lost knowledge on transfers of other, later authors. We will give tribute to the Greeks: they though they surrounded the name of Pythagora many legends, but did not attribute anything such that he could not open or develop into the theory. And bearing his name theorem is no exception.

Such a simple proof

It is not known, Pythagoras himself discovered the ratio between the lengths of the sides in a rectangular triangle or borrowed this knowledge. Antique authors claimed that he himself, and loved retell the legend about how Pythagoras brought to sacrifice the bull in honor of his opening. Modern historians tend to believe that he learned about theorem, having acquainted with Mathematics Babylonian. We also do not know about what kind of Pythagoras formulated theorem: arithmetic, as accepted today, - the square of hypotenuses is equal to the sum of the squares of the cathets, or geometrically, in the spirit of the ancients, the square built on the hypotenneus of the rectangular triangle is equal to the sum of the squares built on His customs.

It is believed that it was Pythagoras who gave the first proof of the theorem that bears his name. It is certainly not preserved. According to one of the versions, Pythagoras could take advantage of the proportions developed at his school. It was based on, in particular, the theory of similarity on which reasoning is based. We draw in a rectangular triangle with Catetics A and B height to hypotenuze C. We get three similar triangles, including the original. Their appropriate parties are proportional to, a: c \u003d m: a and b: c \u003d n: b, from where a 2 \u003d c · m and b 2 \u003d c · n. Then a 2 + b 2 \u003d c · (m + n) \u003d C 2 (Fig. 4).

This is just a reconstruction proposed by one of the historians of science, but proof, agree, very simple: it takes only a few lines, it is not necessary to drag anything, repain, calculate ... It is not surprising that it has been rebounded more than once. It is contained, for example, in the "geometry practice" Leonardo Pisansky (1220), and it is still leading in textbooks.

Such evidence did not contradict the views of the Pythagoreans on the Summary: Initially, they believed that the ratio of lengths of any two segments, and therefore the areas of rectilinear figures can be expressed using natural numbers. They did not consider any other numbers, did not even allow fractions, replacing their relations 1: 2, 2: 3, etc. However, the irony of fate, it was the Pythagora theorem that led the Pythagoreans to the opening of incommensurability of the diagonal of the square and its part. All attempts to numerically present the length of this diagonal - in a single square, it is equal to √2 - they have not led to anything. It was easier to prove that the task is unresolved. At such a case, mathematicians have a proven method - proof from nasty. By the way, and he is attributed to Pythagora.

The existence of a relationship, not expressed by natural numbers, put an end to many Pythagorean ideas. It became clear that the numbers known to them are not enough to solve even simple tasks, what to say about all geometry! This discovery has become a turning point in the development of Greek mathematics, its central problem. At first, it led to the development of teachings on incommensurable values \u200b\u200b- irrationalities, and then to the expansion of the concept of the number. In other words, the centuries-old history of the study of many valid numbers began.

Mosaic Pythagora

If you cover the plane with the squares of two different sizes, surrounding each small square to four large, it turns out a Pythagore mosaic parquet. Such a drawing has long been decorated with stone floors, reminding the ancient evidence of Pythagore's theorem (hence its name). Differently overlapping a square grid on the parquet, you can get a splitting of squares built on the sides of the rectangular triangle, which were offered to different mathematicians. For example, if you arrange the grid so that all its nodes coincide with the right upper vertices of small squares, fragments of the drawing will be shown to the proof of the medieval Persian mathematics of An-Nairzi, which he placed in the comments to the "beginning" Euclidea. It is easy to see that the sum of the areas of large and small squares, the initial elements of the parquet, is equal to the area of \u200b\u200bone square superimposed mesh. And this means that the specified partition is really suitable for placing the parquet: connecting the resulting polygons into squares, as shown in the figure, you can fill with them without spaces and overlaps the entire plane.

»Honored Professor of Mathematics of the University of Warika, the famous popularizer of the Science of Ian Stewart, dedicated to the role of numbers in the history of mankind and the relevance of their study in our time.

Pytagorova Hypotenuse

Pythagora triangles have a direct angle and integer sides. In the simplest of them, the longest side has a length of 5, the remaining - 3 and 4. There are only 5 correct polyhedra. The fifth degree equation is impossible to solve with the help of the roots of the fifth degree - or any other roots. The lattices on the plane and in three-dimensional space do not have five-point symmetry of rotation, therefore such symmetries are not absent in crystals. However, they can be in the lattices in four-dimensional space and in advanced structures known as quasicrystals.

Hypotenuse of the smallest Pythagorough Three

The Pythagoreo Theorem states that the longest side of the rectangular triangle (notorious hypotenuse) correlates with two other sides of this triangle very simple and beautiful: the square of the hypotenuse is equal to the sum of the squares of the two other sides.

Traditionally, we call this theorem of Pythagora, but in fact the story of her is quite foggy. Clay plates suggest that the ancient Babylonians knew the theorem of Pythagora long before Pythagora itself; The fame of the discoverer brought him a mathematical cult of Pythagoreans, whose supporters believed that the universe was based on numerical laws. Ancient authors were attributed to the Pythagoreans - and therefore, and Pythagora is a variety of mathematical theorems, but in fact we have no idea about what mathematics Pythagores himself was engaged. We do not even know if the Pythagoreans could prove the theorem of Pythagore or just believed that she was true. Or, most likely, they had convincing data on its truth, which nevertheless would not have enough for what we consider evidence today.

Proof of Pythagora

The first provese proof of the Pythagore Theorem we find in the "beginning of" Euclidea. This is quite complex proof using the drawing, in which Victorian schoolchildren would immediately recognize "Pythagora Pants"; The drawing and the truth is reminded by drying the entrusters drying on the rope. Literally hundreds of other evidence are known, most of which make proven approval more obvious.

// Fig. 33. Pythagora pants

One of the simplest evidence is a kind of mathematical puzzle. Take any rectangular triangle, make four copies of it and collect them inside the square. At one laying, we see the square on the hypotenuse; With the other, the squares on the other two sides of the triangle. It is clear that the square is equal in the same case.

// Fig. 34. Left: Square on hypotenuse (plus four triangles). Right: the sum of the squares on the other two sides (plus the same four triangles). And now exclude triangles

Making perigal - another proof-puzzle.

// Fig. 35. Dissection perigal

There is also proof of the theorem using square laying on the plane. Perhaps this is how the Pythagoreans or their unknown predecessors opened this theorem. If you look at how the oblique square overlaps two other squares, you can see how to cut a large square into pieces, and then fold two smaller squares of them. You can also see the rectangular triangles, the sides of which give the size of the three squares involved.

// Fig. 36. Proof of paving

There are interesting evidence using similar triangles in trigonometry. It is known at least fifty different evidence.

Pythagora Troika

In the theory of numbers, Pythagorea Theorem has become a source of fruitful idea: to find integer solutions for algebraic equations. Pytagorova Troika is a set of integers a, b and c, such that

Geometrically, such a tripler defines a rectangular triangle with integer sides.

The smallest hypothenus of the Pythagoras Troika is 5.

The other two sides of this triangle are equal to 3 and 4. Here

32 + 42 = 9 + 16 = 25 = 52.

The next largest hypotenuse is equal to 10, because

62 + 82 = 36 + 64 = 100 = 102.

However, this is essentially the same triangle with doubled parties. The following largest and truly other hypotenuse is 13, for her

52 + 122 = 25 + 144 = 169 = 132.

Euclidean knew that there was an infinite number of different variants of Pythagora Trok, and gave what could be called the formula for finding them all. Later, Diofant Alexandrian offered a simple recipe, mainly the coincident with Euclidean.

Take any two natural numbers and calculate:

their double work;

the difference between their squares;

the sum of their squares.

Three numbers received will be the sides of the Pythazhov triangle.

Take, for example, numbers 2 and 1. Calculate:

double-wing work: 2 × 2 × 1 \u003d 4;

square differences: 22 - 12 \u003d 3;

summary of squares: 22 + 12 \u003d 5,

and we got the famous triangle 3-4-5. If you take the number 3 and 2 instead, we get:

twoful work: 2 × 3 × 2 \u003d 12;

square differences: 32 - 22 \u003d 5;

square summary: 32 + 22 \u003d 13,

and we get the following triangle 5 - 12 - 13, try to take numbers 42 and 23 and get:

udfieldy: 2 × 42 × 23 \u003d 1932;

square differences: 422 - 232 \u003d 1235;

squares sum: 422 + 232 \u003d 2293,

no one ever heard of the triangle 1235-1932-2293.

But these numbers also work:

12352 + 19322 = 1525225 + 3732624 = 5257849 = 22932.

In the Diophanty Rule, there is another feature, which has already hinted: having received three numbers, we can take another arbitrary number and multiply them on it. Thus, the triangle 3-4-5 can be turned into a triangle 6-8-10, multiplying all sides by 2, or in a triangle 15-20-25, multiplying everything on 5.

If you go to the language of algebra, the rule is becoming the following form: Let U, V and K be natural numbers. Then the rectangular triangle with the parties

2kuv and k (u2 - v2) has hypotenuse

There are other ways of presenting the main idea, but they all reduce the described above. This method allows you to get all the Troika Pythagoras.

Right polyhedra

There is a smooth account five correct polyhedra. The correct polyhedron (or polyhedron) is volume Figure With a finite number of flat faces. The edges converge with each other on the lines called the ribs; Ribs are found at the points called the vertices.

The culmination of Euclidean "Benefits" is proof that there can be only five right polyhedra, that is, polyhedra, which each face is right polygon (equal parties, equal angles), all the edges are identical and all the vertices are surrounded by an equal number of the same faces. Here are five right polyhedra:

tetrahedron with four triangular edges, four vertices and six ribs;

cube, or hexahedr, with 6 square faces, 8 vertices and 12 ribs;

octahedron with 8 triangular faces, 6 vertices and 12 ribs;

dodecahedron with 12 pyranioral glands, 20 vertices and 30 ribs;

ikosahedron with 20 triangular faces, 12 vertices and 30 ribs.

// Fig. 37. Five right polyhedra

Right polyhedra can be found in nature. In 1904, Ernst Geckel published drawings of tiny organisms known as radolaria; Many of them resemble the very five right polyhedra. It may be true, he corrected a little nature, and the drawings do not fully reflect the form of specific living beings. The first three structures are also observed in crystals. Dodecahedron and Ikosahedra in crystals you will not find, although the wrong dodecahedra and Ikosahedra sometimes come across there. Real dodecahedra can occur in the form of quasicrystals, which are similar to crystals in everything, except that their atoms do not form a periodic lattice.

// Fig. 38. Pictures of Geckel: Radiolaries in the form of the right polyhedra

// Fig. 39. Scanners of the correct polyhedra

It is interesting to make models of the correct polyhedra from paper, cutting the pre-set of the interconnected faces - this is called a polyhedron scan; The scan is folded along the ribs and glue the corresponding ribs among themselves. It is useful to add an extra charge for glue to one of the edges of each such pair, as shown in Fig. 39. If there is no such platform, you can use a sticky tape.

Fifth degree equation

There is no algebraic formula for solving the 5th degree equations.

IN general The fifth degree equation looks like this:

aX5 + BX4 + CX3 + DX2 + EX + F \u003d 0.

The problem is to find a formula for solutions of such an equation (it can have up to five solutions). The experience of circulation of square and cubic equations, as well as with the fourth degree equations suggests that such a formula must exist for the equations of the fifth degree, and in it, in theory, should appear the roots of the fifth, third and second degree. Again, it can be bolden to assume that such a formula, if it exists, will be very and very difficult.

This assumption ultimately turned out to be erroneous. In fact, no such formula exists; At least there is no formula consisting of coefficients A, B, C, D, E and F, composed using addition, subtraction, multiplication and division, as well as root extraction. Thus, among the 5 5 there is something completely special. The reasons for such unusual behavior of the five are very deep, and it took a lot of time to deal with them.

The first sign of the problem was the fact that, as if maths, he tried to find such a formula, no matter how smart they were, they invariably failed. For some time everyone believed that the reasons would lie in the incredible complexity of the formula. It was believed that no one would simply be able to figure out this algebra. However, over time, some mathematics began to doubt that such a formula exists at all, and in 1823 Niels Hendrik Abel managed to prove the opposite. This formula does not exist. Shortly thereafter, Galua's Evarister found a way to determine whether the equation of one way or another - the 5th, 6th, 7th, in general any - using this kind of formula.

Conclusion from all this is simple: the number 5 is special. You can solve algebraic equations (with the help of roots nth degree For different values \u200b\u200bn) for degrees 1, 2, 3 and 4, but not for the 5th degree. Here, the obvious pattern ends.

No one surprises that the degrees equations are more than 5 behave even worse; In particular, the same difficulty is connected with them: there are no general formulas for solving them. This does not mean that the equations do not have solutions; This does not mean that it is impossible to find very accurate numerical values \u200b\u200bof these solutions. The whole thing is limited to traditional algebra tools. It reminds the impossibility of the trisection of angle with the help of a ruler and a circulation. The answer exists, but the listed methods are insufficient and do not allow you to determine what it is.

Crystallographic limit

The crystals in two and three dimensions do not have a 5-beam symmetry of rotation.

Atoms in the crystal form a grid, that is, a structure that is periodically repeated in several independent directions. For example, the drawing on the wallpaper is repeated along the length of the roll; In addition, it is usually repeated in a horizontal direction, sometimes with a shift from one piece of wallpaper to the next. Essentially, wallpapers are a two-dimensional crystal.

There are 17 varieties of wallpaper drawings on the plane (see chapter 17). They differ in the type of symmetry, that is, according to methods, move hard drawing in such a way that it will definitely leave himself in its original position. Symmetry types include, in particular, various variants of the symmetry of rotation, where the drawing should be rotated to a certain angle around a certain point - the center of symmetry.

The order of symmetry of rotation is how many times you can turn the body to the complete circle so that all the details of the drawing returned to the initial positions. For example, the rotation of 90 ° is the symmetry of the rotation of the 4th order *. The list of possible types of symmetry of rotation in the crystal lattice again indicates the unusual of the number 5: it is not there. There are variants with the symmetry of rotation 2, 3, 4 and 6th orders, but no wallpaper drawing has the symmetry of rotation of the 5th order. The symmetry of the rotation of the order more than 6 in the crystals is also no case, but the first violation of the sequence is nevertheless, among the number 5.

The same happens with crystallographic systems in three-dimensional space. Here the grille repeats itself in three independent areas. There are 219 different types of symmetry, or 230, if you consider a mirror reflection of the pattern by a separate option, despite it, that in this case there is no mirror symmetry. Again, the symmetry of rotation of orders 2, 3, 4 and 6 is observed, but not 5. This fact is called the name of the crystallographic limit.

In the four-dimensional lattice space with 5th order symmetry exist; In general, for the lattices of a sufficiently high dimension possible, any advanced order of symmetry of rotation is possible.

// Fig. 40. Crystal lattice of the table salt. Dark balls depict sodium atoms, light - chlorine atoms

Quasicrystals

Although the symmetry of rotation of the 5th order in two-dimensional and three-dimensional lattices is impossible, it may exist in a slightly less regular structures known as quasicrystals. Taking advantage of the sketches of Kepler, Roger Penrose opened flat systems with a more common type of five-time symmetry. They got the name of quasicrystals.

Quasicrystals exist in nature. In 1984, Daniel Shechtman discovered that aluminum and manganese alloy can form quasicrystals; Initially, the crystallographs met his message with some skepticism, but later the discovery was confirmed, and in 2011 Shechtman was awarded Nobel Prize in chemistry. In 2009, a team of scientists under the leadership of Luke Bindi discovered quasicrystals in a mineral from the Russian Koryak Highlands - the combination of aluminum, copper and iron. Today, this mineral is called IkosaDritis. Measuring with the help of a mass spectrometer, the content in the mineral of different isotopes of oxygen, scientists have shown that this mineral originated on Earth. It formed about 4.5 billion years ago, at a time when solar system Only emerged, and spent most of the time in the belt of asteroids, turning around the sun, until some indignation changed his orbit and did not lead it in the end to the ground.

// Fig. 41. Left: One of the two quasicrystalline lattices with accurate five-time symmetry. Right: atomic model of icosahedral aluminum-palladium-manganese quasicrystal

Joking proof of Pythagore's theorem; Also a joke about baggy pants of a friend.

- Troika integers positive numbers x, y, z satisfying the x2 + equation 2 \u003d z2 ...
Mathematical encyclopedia
- Troika of such natural numbers that a triangle, the length of the side of the sidelines are proportional to these numbers, is rectangular, for example. Three numbers: 3, 4, 5 ...
Natural science. encyclopedic Dictionary
- See Rescue Rocket ...
Marigree
- Troika natural numbers such that triangle, whose lengths of which are proportional to these numbers, is rectangular ...
Great Soviet Encyclopedia
- MIL. Neism. The expression used when listed or opposing two facts, phenomena, circumstances ...
Training Frame Dictionary
- From the novel-anti-nightopy "Bottom courtyard" of the English writer George Orwell ...
- First meeting in the Satire "Liberal Diary in St. Petersburg" Mikhail Evgrafovich Saltykov-Shchedrin, who figuratively described the dual, cowardly position of Russian liberals - his ...
Dictionary of winged words and expressions
- It is said in the case when the interlocutor long and unbelievously tried to tell something, clutching the main idea of \u200b\u200bsecondary details ...
Dictionary of folk phraseology
- The number of buttons is known. Why should I closely? - About pants and men's sexual authority. . To prove it, you need to remove and show 1) about the theorem of Pythagora; 2) about wide pants ...
Living speech. Vocabulary spoken expressions
- cf. There is no immortality of the soul, so there is no virtue, "it means everything is allowed" ... a seductive theory of scoundrels ... Bushroom, and the essence is all: on the one hand, it is impossible not to admit, but on the other - you can not confess ...
Mikhilson's intelligent-phrase dictionary
- Piñagoras Pants in Austice. About human mancock. Cf. This is undoubtedly a sage. In ancient times, he would have invented Piñagorov's pants ... Saltykov. Festing Letters ...
- Came of one side - the other side. Cf. NѣTh Mesmless Soul, so NѣT and Dobyurators, "So, everything is allowed" ... Seductive theorem of the scoundrel .....
The intelligent-phraseological dictionary of Michelson (Orig. ORF)
- the comic name of the Pythagores Theorem, which arose due to the fact that the squares built on the sides of the rectangle and divergers in different directions are reminded by covering the pants ...
- ON THE ONE HAND ON THE OTHER HAND. Book ...
Fraseological dictionary of the Russian literary language
- See the title -...
IN AND. Dal. Proverbs of the Russian nation
- Zharg. shk. Jelly Pythagoras. ...
Big Dictionary Russian sayings

"Pythagoras pants in all directions are equal" in books

11. Pythagora pants

From the book Friedle Author Makarova Elena Grigorievna

11. Pythagoras pants My good girl! First of all, the most hot gratitude for the courtyard; It is very interesting, it's not so easy to read, but I am very glad to him. I will write to you more when you read several chapters. You can't imagine what joy gives me yours

III "Is not all places equal?"

From the book Batyushkov Author Sergeeva-Klyatis Anna Yurevna

III "Is not all places equal?" At the end of the post, without waiting for Easter, which in 1815 followed on April 18, Batyushkov on a passionate sadmice left St. Petersburg to the estate of Father Danilovskoye. However, before that, another event occurred, which are not mentioned in the letters of Batyushkov,

Pythagora pants

From the book from Doberman to Hooligan. From the names of their own in nominal Author Blau Mark Grigorievich

Pythagoras pants that "Pythagoras pants are equal in all directions," they knew more pre-revolutionary gemnasists, they also composed this poetic cheat sheet. What are the gymnasists there! Probably already great Lomonosov, who studied geometry in his Slavic Greco-Latin

1.16. Security measures both by the tax authorities and from taxpayers

From the book Tax checks. How with dignity to withstand the visit of inspectors Author Semenihin Vitaly Viktorovich

1.16. Security measures both by the tax authorities and on the part of taxpayers taxpayers rarely agree with the conclusions of tax authorities made on the results of tax audits. And at the same time most disputes in the courts are allowed in favor

Before credit all are equal

From the book money. Credit. Banks: Lecture Abstract Author Shevchuk Denis Aleksandrovich

Before the loan, the official history of emergency lending in America is counting since 1968, when a law on consumer loan was adopted there. In particular, it establishes fair rules for the provision of a loan, the upper limits of rates, rules

SWOT analysis (strengths, weaknesses, opportunities, threats)

From the book Tarning. Table book coach Author Thorn Kay.

SWOT analysis (strengths, weaknesses, opportunities, threats) This method is an addition to the "brainstorming" structure. Divide the flip chart sheet into four parts and enclocent them: strengths, weaknesses, opportunities, threats. The group can analyze the business,

Not all buyers are equal

From the book how to work for four hours a week by Ferris Timothy

Not all buyers are equal as soon as you reach the third stage and the influx of funds will become more or less established, it is time to appreciate the composition of your buyers and to rine this bed. Everything in the world is divided into good and bad: good and bad food, movies, sex. That's

Chapter VII "Pythagora Pants" - Opening of Assiro-Babylonian mathematicians

From the book when I started talking Author Matveev Konstantin Petrovich

Chapter VII "Pythagora Pants" - the opening of Assiro-Babylonian mathematicians Mathematics at Assyrians and Babylonians, as well as astronomy, was necessary primarily in practical life - in the construction of houses, palaces, roads, drawing up calendars, conducting channels,

"Under the mask all the ranks are equal"

From the book Petersburg Arabesque Author Aspidov Albert Pavlovich

"Under the mask all the ranks are equal," among New Year's shopping - Christmas toys and other things - may also be a mask. Putting it, we immediately become different - as in magic fairy tale. And who does not want to touch the magic at least once a year - to his joyful and harmless sides,

Pythagora numbers

From the book Big Soviet Encyclopedia (PI) author BSE

All are equal, but some are equal to more

From the book Encyclopedic dictionary of winged words and expressions Author Serov Vadim Vasilyevich

All are equal, but some are equal to more than others from the novel-anti-nightopy "Bottom Court" (1945) of the English writer George Orwell (alias Eric Blair, 1903-1950). Animals of a certain farm once overthrew their cruel owner and established the republic, proclaiming the principle: "All

Participation in negotiations as a party or assistant side

From the book of the Reader of Alternative Dispute Resolution Author Collective authors

Participation in the negotiations as a party or assistant side of another forms of negotiations published from mediation is the participation of the mediator together with the party (or without it) in the negotiations as a representative of the party. The method is fundamentally different from

Forces were equal

From book Great War Not finished. Results of the First World War Author Mlechin Leonid Mikhailovich

Forces were equal to no one assumed that the war would delay. But the work thoroughly developed by the General Stations collapsed in the first months. The forces of opposing blocks turned out to be approximately equal. The flourishing of a new military equipment has multiplied the number of victims, but did not allow to crush the enemy and

All animals are equal, but some are more equal than others

From the book of fascizophrenia Author Sysoev Gennady Borisovich

All animals are equal, but some are more equal than others finally, I would like to remember people who think that Kosovo can become some kind of precedent. Like, if the population of Kosovo "The World Community" (ie, the United States and the EU) will give the right to solve their destiny on

Almost equal

From the book Literary newspaper 6282 (No. 27 2010) Author Literary newspaper

Almost equal to the club 12 chairs are almost equal to the ironic prose. Death went to one poor man. And that deaf was. So normal, but slightly deaf ... and saw bad. I haven't seen nothing. - Oh, visit us! Please pass. Death says: - Wait to rejoice,

Pants - get on academician working coupon for a discount paper shop or profitable pants to buy with free shipping on the sale in Paper Shop

Jarg. shk. Jelly The Pythagoreo Theorem, which establishes the ratio between the squares of squares built on hypotenuze and the cate of the rectangular triangle. BTS, 835 ... Large dictionary of Russian sayings

Pythagora pants - The comic name of the Pytagora theorem, which arose due to the fact that the squares built on the sides of the rectangle and the squares diverged in different directions resemble the pants. I loved geometry ... And I received at the entrance exam in the university ... ... ... Fraseological dictionary of the Russian literary language

pythagora pants - the joking name of the Pythagoree theorem establishing the relationship between squares built on hypotenuses and cate of the rectangular triangle, which looks outwardly in the drawings looks like storks ... Dictionary of many expressions

Inloid: about man Darovite Wed. This is a misery. In ancient times, he would probably have invented Pythagorov's pants ... Saltykov. Pestus letters. Pythagoras pants (geom.): In the rectangle, the square of the hypotenuse is equal to the squares of the cathets (teaching ... ... Mikhelson's Big Thick-Frazological Dictionary

Pythagora pants for all sides are equal - The number of buttons is known. Why should I closely? (Rough) about pants and men's sexual organs. Pythagora pants for all sides are equal. To prove it, you need to remove and show 1) about the theorem of Pythagora; 2) about wide pants ... Living speech. Dictionary of spoken expressions

PiñaGorovy Pants (invent) inustice. About human mancock. Cf. This is undoubtedly a sage. In ancient times, he would have invented Piñagorov's pants ... Saltykov. Walking letters. PiñaGorovy pants (geom.): In the rectangle, the square of hypotenuse ... ... Large intelligent-phraseological dictionary of Michelson (original spelling)

Pythagoras pants in all directions are equal - joking proof of Pythagore's theorem; Also in a joke about baggy pants of a friend ... Dictionary of folk phraseology

Sat down, rude ...

Pythagoras pants on all directions are equal (the number of buttons is known. Why is it closely? / To prove it, you need to remove and show) - satisfied, rude ... Dictionary modern conversational phrases and progress

SUM., MN., UPOTR. compared. Often morphology: mn. what? Pants, (no) What? Pants, what? Pants, (see) What? Pants than? Pants, what? About pants 1. Pants This is a piece of clothing that has two short or long pants and closes the bottom ... ... Explanatory dictionary Dmitrieva

Books

Pythagora pants ,. In this book you will find fiction and adventure, wonders and fiction. Funny and sad, ordinary and mysterious ... And what else is needed for entertaining reading? The main thing is to be ...
Wonders on wheels, Markush Anatoly. Millions of wheels are spinning throughout the land - roll cars, measure the time in the clock, tapping under trains, perform countless work in the machines and a variety of mechanisms. They are…

In one one can be confident one hundred percent, which is to the question, what is equal to the square of hypotenuses, any adult will feel free to safely: "The sum of the squares of the cathets." This theorem firmly populated in the minds of each educated person, but it is enough to prove someone enough, and difficulties may arise. So let's remember and consider different methods Proof of the Pythagorean theorem.

Brief overview of the biography

Pythagore's theorem is familiar with almost everyone, but for some reason the biography of a person who made it on the light is not so popular. This is fixable. Therefore, before studying different ways of evidence of the Pythagora theorem, you need to briefly get acquainted with his personality.

Pythagoras - philosopher, mathematician, the thinker comes from today it is very difficult to distinguish his biography from the legends who have developed in memory of this great man. But as follows from the works of his followers, Pythahor Samos was born on the island of Samos. His father was the usual Kamneris, but the mother came from a noble family.

Judging by the legend, the appearance of Pythagora's light predicted a woman named Pythia, in whose honor and called the boy. According to her prediction, a born boy was supposed to bring a lot of benefit and good to humanity. What actually he did.

Birth of Theorem

In his youth Pyfagor moved to Egypt to meet there with famous Egyptian wise men. After a meeting with them, he was admitted to learning, where he knew all the great achievements of Egyptian philosophy, mathematics and medicine.

It is probably in Egypt Pythagoras inspired by Majesty and the beauty of the pyramids and created his great theory. It can shock readers, but modern historians believe that Pythagoras did not prove his theory. But only transferred his knowledge to followers who later completed all the necessary mathematical calculations.

Whatever it was, today there is not one method of evidence of this theorem, but at once several. Today it remains only to guess exactly the ancient Greeks produced their calculations, so here we consider different ways of evidence of the Pythagora theorem.

Pythagorean theorem

Before starting any calculations, you need to find out what the theory to prove. The Pythagore Theorem sounds like this: "In a triangle, in which one of the corners is 90 o, the sum of the squares of the cathets is equal to the square of the hypotenuse."

In total, there are 15 different ways to proof the Pythagora theorem. This is quite a big figure, so we will pay attention to the most popular of them.

Fashion first

First, we denote what we are given. These data will be distributed to other ways of evidence of the Pythagore's theorem, so it is necessary to immediately remember all the meanings.

Suppose, a rectangular triangle is given, with Catetics A, B and hypotenuse, equal to. The first way of proof is based on the fact that from the rectangular triangle you need to try the square.

To do this, you need to cathet in length and draw a segment of equal cathetu in, and vice versa. So there should be two equal side of the square. It remains only to draw two parallel straight, and the square is ready.

Inside the resulting figure you need to draw another square with a side of equal hypothenuze of the source triangle. To do this, there are two parallel segments of equal from. Thus, it turns out three sides of the square, one of which is the hypotenuse of the initial rectangular triangles. It remains only to dare the fourth segment.

Based on the resulting figure, it can be concluded that the area of \u200b\u200bthe outer square is equal to (A + B) 2. If you look inside the shape, you can see that in addition to the inner square there are four rectangular triangles. Each area is 0.5AV.

Therefore, the area is equal: 4 * 0.5Av + C 2 \u003d 2AV + C 2

Hence (a + c) 2 \u003d 2AV + C 2

And, therefore, from 2 \u003d a 2 + in 2

Theorem is proved.

Method two: similar triangles

This formula of the proof of the Pythagorean Theorem was derived on the basis of approval from the geometry section of similar triangles. It states that the roll of the rectangular triangle is the mean proportional to its hypotenuse and the segment of the hypotenuse, emanating from the top of the angle of 90 o.

The initial data remains the same, so let's start immediately with the proof. We will carry out the perpendicular side of the CD segment. Based on the above-described approval of the Kartets of Triangles are equal:

AC \u003d √AV * AD, SV \u003d √AV * DV.

To answer the question of how to prove the theorem of Pythagora, the proof must be built into the square of both inequalities.

AC 2 \u003d AV * AD and SV 2 \u003d AV * DV

Now you need to fold the resulting inequalities.

AC 2 + SV 2 \u003d AV * (hell * dv), where hell + dv \u003d av

Turns out that:

AC 2 + SV 2 \u003d AV * AV

And, therefore:

AC 2 + SV 2 \u003d AB 2

The proof of the Pythagore's theorem and various ways to solve it need a versatile approach to this task. However, this option is one of the simplest.

Another method of calculations

A description of various ways of proof of the Pythagore's theorem may not say anything, until the time you do not go to practice. Many techniques provide not only mathematical calculations, but also the construction of new figures from the initial triangle.

In this case, it is necessary to complete another rectangular triangle for another rectangular triangle from the Cate. Thus, now there are two triangles with a common cathet.

Knowing that the area of \u200b\u200bsuch figures have the ratio as the squares of their similar linear dimensions, then:

S AVC * C 2 - S AVD * B 2 \u003d S AVD * A 2 - S IT * A 2

S AVS * (C 2 -B 2) \u003d a 2 * (s AVD -S IT)

c 2 -B 2 \u003d a 2

c 2 \u003d a 2 + in 2

Since, from different ways of evidence of the Pythagora theorem for the 8th grade, this option is hardly suitable, you can use the following method.

The easiest way to prove the theorem of Pythagora. Reviews

As historians believe, this method was first used to proof theorem still in ancient Greece. It is the easiest way, as it does not require absolutely no calculations. If you correctly draw the drawing, then the proof of the statement is that and 2 + in 2 \u003d C 2 will be visible.

Conditions for this method will differ slightly from the previous one. To prove theorem, suppose that the rectangular triangle ABC is a waste.

Hypotenuse of the speakers accepted by the side of the square and suicide the three sides. In addition, you need to spend two diagonal direct in the resulting square. So that there are four ineffective triangles inside it.

It is also necessary to daughter on the square and sv, it is also necessary to spend on one diagonal direct in each of them. The first direct blacks from the vertex A, the second - from S.

Now you need to carefully look at the resulting drawing. Since the AU hypotenuse lies four triangles equal to the initial one, and on two categories, this indicates the truthfulness of this theorem.

By the way, thanks to this method, the proof of the Pythagora theorem and the famous phrase appeared: "Pythagoras pants in all directions are equal."

Proof J. Garfield

James Garfield is the twentieth president of the United States of America. In addition, he left his mark in history as the US ruler, he was also gifted self-taught.

At the beginning of his career, he was an ordinary teacher in a folk school, but soon became the director of one of the highest educational institutions. The desire for self-development and allowed him to offer new theory Proof of the Pythagorean theorem. Theorem and an example of its solution looks like this.

First you need to draw two rectangular triangles on a sheet of paper in such a way that the catt on one of them was a continuation of the second. The vertices of these triangles need to be combined so that the trapeze is ultimately it turns out.

As is known, the area of \u200b\u200bthe trapezium is equal to the work of half the grounds for height.

S \u003d a + in / 2 * (a + c)

If we consider the resulting trapeze, as a figure consisting of three triangles, then its area can be found like this:

S \u003d AV / 2 * 2 + C 2/2

Now it is necessary to equalize two source expressions.

2AV / 2 + C / 2 \u003d (A + B) 2/2

c 2 \u003d a 2 + in 2

On the theorem of Pythagora and the methods of its evidence, you can write not one volume tutorial. But is there a point in it when this knowledge can not be applied in practice?

Practical application of Pythagora theorem

Unfortunately, modern school programs provide for the use of this theorem only in geometric tasks. Graduates will soon leave the school walls, and without learning, and how they can apply their knowledge and skills in practice.

In fact, to use the theorem of Pythagora in his everyday life Maybe everyone. And not only in professional activity, but also in ordinary home affairs. Consider several cases when the Pythagoreo Theorem and the ways of its evidence may be extremely necessary.

Communication Theorem and Astronomy

It would seem that stars and triangles on paper may be related. In fact, astronomy is scientific sphereIn which the Pythagoreo Theorem is widely used.

For example, consider the movement of the light beam in space. It is known that the light moves in both directions at the same speed. The trajectory of the AV, which moves the beam of the world let's call l.. And half the time that the light is necessary to get from the point A to the point B, let's call t.. And the speed of the beam - c.. Turns out that: c * T \u003d L

If you look at this very ray of another plane, for example, from a cosmic liner, which moves at a speed V, then with this observation of the bodies, their speed will change. At the same time, even fixed elements will move at a speed V in the opposite direction.

Suppose a comic liner floats to the right. Then the points A and B, between which the beam rushes, will move to the left. Moreover, when the beam moves from point A to point in, point A has time to move and, accordingly, the light will arrive at the new point S. To find half the distance to which the point A is shifted, you need to multiply the speed of the ray at half time (T ").

And to find how during this time it was able to go beam light, you need to designate half the way of the new bucken S and get the following expression:

If you imagine that the points of light C and B, as well as the space liner - these are vertices equal triangle, then the segment from point A to the liner will divide it into two rectangular triangles. Therefore, thanks to the Pythagora theorem, you can find the distance that the beam of light can be passed.

This example, of course, is not the most successful, since only units can strive to try it in practice. Therefore, we consider more landed options for using this theorem.

Mobile signal transmission radius

Modern life is no longer possible to imagine without the existence of smartphones. But how much would have been added from them if they could not connect subscribers through mobile communications?!

Mobile quality directly depends on what height is the antenna of the mobile operator. In order to calculate how the distance from the mobile tower, the phone can receive a signal, you can apply the Pytagora theorem.

Suppose you need to find the approximate height of the stationary tower so that it can distribute the signal within a radius of 200 kilometers.

AB (tower height) \u003d x;

Sun (signal transmission radius) \u003d 200 km;

OS (radius of the globe) \u003d 6380 km;

Ov \u003d OA + AVOV \u003d R + x

Applying the Pythagora theorem, find out that the minimum tower height should be 2.3 kilometers.

Pythagore's theorem in everyday life

Oddly enough, the Pythagorean Theorem may be useful even in household matters, such as determining the height of the cabinet, for example. At first glance, there is no need to use such complex calculations, because you can simply remove the measurements using a roulette. But many are surprised why certain problems arise in the assembly process if all the measurements were removed more than exactly.

The fact is that the wardrobe is assembled in a horizontal position and only then rises and installed to the wall. Therefore, the sidewall of the cabinet in the process of lifting the structure should be freely undergo in height, and the diagonal of the room.

Suppose there is a wardrobe depth of 800 mm. The distance from the floor to the ceiling is 2600 mm. An experienced furniture maker will say that the height of the cabinet should be 126 mm less than the height of the room. But why exactly 126 mm? Consider on the example.

With the ideal cabinet size, we check the action of the Pythagora theorem:

AC \u003d √AV 2 + √ WP 2

AC \u003d √2474 2 +800 2 \u003d 2600 mm - everything converges.

Suppose, the height of the cabinet is not 2474 mm, but 2505 mm. Then:

AC \u003d √2505 2 + √800 2 \u003d 2629 mm.

Consequently, this wardrobe is not suitable for installation in this room. Since when lifting it in a vertical position, it is damaged to its corpus.

Perhaps, considering various ways to proof the Pythagora theorem with different scientists, it can be concluded that it is more than truth. Now you can use the information received in your daily life and be completely confident that all calculations will not only be useful, but also true.