Why say Pythagorean pants. The Pythagorean theorem: background, evidence, examples of practical application

Almost Equal 12 Chairs Club Almost Equal IRONIC PROSE Death came to a poor man. And he was deaf. So normal, but a little deaf ... And he saw badly. I saw almost nothing. - Oh, we have guests! Please pass. Death says: - Wait to rejoice,

The Roman architect Vitruvius singled out the Pythagorean theorem "from the numerous discoveries that have rendered services to the development of human life", and called for treating it with the greatest respect. It was in the 1st century BC. e. At the turn of the 16th-17th centuries, the famous German astronomer Johannes Kepler called it one of the treasures of geometry, comparable to a measure of gold. It is unlikely that in all of mathematics there is a more weighty and significant statement, because in terms of the number of scientific and practical applications, the Pythagorean theorem has no equal.

The Pythagorean theorem for the case of an isosceles right triangle.

Science and life // Illustrations

An illustration of the Pythagorean theorem from the Treatise on the Measuring Pole (China, 3rd century BC) and a proof reconstructed on its basis.

Science and life // Illustrations

S. Perkins. Pythagoras.

Drawing for a possible proof of Pythagoras.

"Mosaic of Pythagoras" and division of an-Nairizi of three squares in the proof of the Pythagorean theorem.

P. de Hoch. Mistress and maid in the courtyard. About 1660.

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

Pythagorean pants

The Pythagorean 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 the formulation, this theorem is by no means obvious: looking at right 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 more area this square. His reasoning, in fact, came down to proving the Pythagorean theorem, albeit for a specific triangle.

The figures depicted 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 with polygons a parquet, or a tiling. Why is Pythagoras here? 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 only three types of equal regular polygons can cover the plane around a point without gaps: six triangles, four squares and three hexagons.

4000 years later

The history of the Pythagorean theorem goes back to ancient times. Mentions of it are contained in the Babylonian cuneiform texts of the times of King Hammurabi (XVIII century BC), that is, 1200 years before the birth of Pythagoras. The theorem has been applied 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 relation a 2 + b 2 = c 2 for an arbitrary right-angled triangle was obtained by the Babylonians simply by “generalizing” the equality a 2 + a 2 = c 2 . But this is excusable for them - for the practical geometry of the ancients, which was reduced to measurements and calculations, strict justifications were not required.

Now, almost 4000 years later, we are dealing with a record-breaking theorem in terms of the number of possible proofs. By the way, their collecting is a long tradition. The peak of interest in the Pythagorean theorem occurred in the second half of the 19th - early 20th century. And if the first collections contained no more than two or three dozen pieces of evidence, then by the end of the 19th century their number approached 100, and after another half a century it exceeded 360, and these are only those that were collected from various sources. Who just did not take up the solution of this ageless task - from eminent scientists and popularizers of science to congressmen and schoolchildren. And what is remarkable, in the originality and simplicity of the solution, other amateurs were not inferior to professionals!

The oldest proof of the Pythagorean theorem that has come down to us is about 2300 years old. One of them - 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 visual and beautiful proofs are built on the redrawing of "Pythagorean pants". They look like an ingenious square-cutting puzzle. But make the figures 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 immediately becomes clear.

It was this proof that the seven-year-old Guido, the bright-eyed 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 evidence, and therefore attributed it to ... Pythagoras himself. But the protagonist of the fantastic story by Evgeny Veltistov "Electronics - a boy from a suitcase" knew 25 proofs of the Pythagorean theorem, including that given by Euclid; True, he mistakenly called it the simplest, although in fact in the modern edition of the Beginnings 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 from him that mathematics begins as an exact science, where any new knowledge is not the result of visual representations and rules learned 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 ​​proof, but applied it unsystematically, selectively, as a rule, to obvious geometric statements like "the diameter bisects the circle." 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." And he stood at the origins of number theory and stereometry.

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

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

Such a simple proof

It is not known whether Pythagoras himself discovered the ratio 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 are inclined 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, the square built on the hypotenuse of a right triangle is equal to the sum of the squares built on his skates.

It is believed that it was Pythagoras who gave the first proof of the theorem that bears his name. It didn't survive, of course. 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 reasoning is based. Let's draw a height to the hypotenuse c in a right-angled triangle with legs a and b. 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 don’t need to finish building, reshaping, calculating 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 given in textbooks.

Such a 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 the square and its side. All attempts to numerically represent the length of this diagonal - for a unit square it is equal to √2 - did not lead to anything. It turned out to be easier to prove that the problem is unsolvable. In such a case, mathematicians have a proven method - proof by contradiction. By the way, it is also attributed to Pythagoras.

The existence of a relation not expressed by 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, to say nothing of 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 a Pythagorean mosaic parquet. Such a pattern has long adorned stone floors, reminiscent of the ancient proofs of the Pythagorean theorem (hence its name). By imposing a square grid on the parquet in different ways, one can obtain 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 small squares, fragments of the drawing will appear for the proof of the medieval Persian mathematician an-Nairizi, which he placed in the comments to Euclid's "Principles". It is easy to see that the sum of the areas of the large and small squares, the initial 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.

Some discussions amuse me immensely...

Hi what are you doing?
- Yes, I solve problems from a magazine.
-Wow! Didn't expect from you.
-What didn't you expect?
- That you will sink to problems. It seems smart, after all, but you believe in all sorts of nonsense.
-Sorry I dont understand. What do you call nonsense?
-Yes, all your math. It's obvious that it's complete bullshit.
-How can you say that? Mathematics is the queen of sciences...
-Just let's do without this pathos, right? Mathematics is not a science at all, but one continuous heap of stupid laws and rules.
-What?!
- Oh, well, don't make such big eyes, you yourself know that I'm right. No, I do not argue, the multiplication table is a great thing, it has played a significant role in the development of culture and the history of mankind. But now it's all irrelevant! And then, why complicate things? In nature, there are no integrals or logarithms, these are all inventions of mathematicians.
-Wait a minute. Mathematicians did not invent anything, they discovered new laws of the interaction of numbers, using proven tools ...
-Yes of course! And do you believe it? Don't you see what nonsense they are constantly talking about? Can you give an example?
-Yes, please.
-Yes please! Pythagorean theorem.
- Well, what's wrong with her?
-It's not like that! "Pythagorean pants are equal on all sides," you see. Do you know that the Greeks in the time of Pythagoras did not wear pants? How could Pythagoras even talk about something he had no idea about?
-Wait a minute. What's with the pants?
- Well, they seem to be Pythagorean? Or not? Do you admit that Pythagoras didn't have pants?
Well, actually, of course, it wasn't...
-Aha, so there is a clear discrepancy in the very name of the theorem! How then can one take seriously what it says?
-Wait a minute. Pythagoras didn't say anything about pants...
- You admit it, don't you?
- Yes... So, can I continue? Pythagoras did not say anything about trousers, and there is no need to attribute other people's nonsense to him ...
- Yeah, you yourself agree that this is all nonsense!
- I didn't say that!
- Just said. You're contradicting yourself.
-So. Stop. What does the Pythagorean theorem say?
-That all pants are equal.
-Damn, did you read this theorem at all?!
-I know.
-Where?
-I read.
-What did you read?!
-Lobachevsky.
*pause*
- Excuse me, but what does Lobachevsky have to do with Pythagoras?
- Well, Lobachevsky is also a mathematician, and he seems to be even a tougher authority than Pythagoras, you say no?
*sigh*
-Well, what did Lobachevsky say about the Pythagorean theorem?
- That the pants are equal. But this is nonsense! How can you wear pants like that? And besides, Pythagoras did not wear pants at all!
- Lobachevsky said so?!
*pause for a second, confidently*
-Yes!
- Show me where it's written.
- No, well, it's not written so directly ...
-What name has this book?
- It's not a book, it's a newspaper article. About the fact that Lobachevsky was actually a German intelligence agent... well, that's beside the point. Anyway, that's exactly what he said. He is also a mathematician, so he and Pythagoras are at the same time.
- Pythagoras didn't say anything about pants.
-Well, yes! That's what it's about. It's all bullshit.
-Let's go in order. How do you personally know what the Pythagorean theorem says?
-Oh, come on! Everyone knows this. Ask anyone, they will answer you right away.
- Pythagorean pants are not pants ...
-Oh, of course! This is an allegory! Do you know how many times I've heard this before?
-The Pythagorean theorem states that the sum of the squares of the legs is equal to the square of the hypotenuse. And EVERYTHING!
-Where are the pants?
- Yes, Pythagoras did not have any pants !!!
- Well, you see, I'm telling you about it. All your math is bullshit.
-And that's not bullshit! Take a look yourself. Here is a triangle. Here is the hypotenuse. Here are the skates...
-Why all of a sudden it’s the legs, and this is the hypotenuse? Maybe vice versa?
-Not. Legs are two sides that form a right angle.
Well, here's another right angle for you.
- He's not straight.
-And what is he, a curve?
- No, he's sharp.
Yes, this one is sharp too.
-He's not sharp, he's straight.
- You know, don't fool me! You just call things whatever you like, just to tailor the result to what you want.
-The two short sides of a right triangle are the legs. The long side is the hypotenuse.
-And who is shorter - that leg? And the hypotenuse, then, no longer rolls? You listen to yourself from the outside, what nonsense you are talking about. In the yard of the 21st century, the flowering of democracy, and you have some kind of Middle Ages. His sides, you see, are unequal ...
- Right triangle with equal sides does not exist...
-Are you sure? Let me draw you. Look. Rectangular? Rectangular. And all sides are equal!
- You drew a square.
-So what?
- A square is not a triangle.
-Oh, of course! As soon as he does not suit us, immediately "not a triangle"! Do not fool me. Count yourself: one corner, two corners, three corners.
-Four.
-So what?
-It's a square.
What about a square, not a triangle? He's worse, right? Just because I drew it? Are there three corners? There is, and even here is one spare. Well, here it is, you know...
- Okay, let's leave this topic.
-Yeah, are you giving up already? Nothing to object? Are you admitting that math is bullshit?
- No, I don't.
- Well, again, great again! I just proved everything to you in detail! If all your geometry is based on the teachings of Pythagoras, which, I'm sorry, is complete nonsense ... then what can you even talk about further?
- The teachings of Pythagoras are not nonsense ...
- Well, how! And then I have not heard about the school of the Pythagoreans! They, if you want to know, indulged in orgies!
-What's the matter here...
-And Pythagoras was generally a faggot! He himself said that Plato was his friend.
-Pythagoras?!
-You didn `t know? Yes, they were all fagots. And three-legged on the head. One slept in a barrel, the other ran around the city naked ...
Diogenes slept in a barrel, but he was a philosopher, not a mathematician...
-Oh, of course! If someone climbed into the barrel, then he is no longer a mathematician! Why do we need more shame? We know, we know, we passed. But you explain to me why all sorts of fagots who lived three thousand years ago and ran without pants should be an authority for me? Why should I accept their point of view?
- Okay, leave...
- No, you listen! After all, I listened to you too. These are your calculations, calculations ... You all know how to count! And ask you something to the point, right there right away: "this is a quotient, this is a variable, and these are two unknowns." And you tell me in oh-oh-oh-general, without particulars! And without any there unknown, unknown, existential... It makes me sick, you know?
-Understand.
- Well, explain to me why twice two is always four? Who came up with this? And why am I obliged to take it for granted and have no right to doubt?
- Doubt as much as you want...
- No, you explain to me! Only without these things of yours, but normally, humanly, to make it clear.
-Two times two equals four, because two times two equals four.
- Butter oil. What did you tell me new?
-Twice two is two times two. Take two and two and put them together...
So add or multiply?
-This is the same...
-Both on! It turns out that if I add and multiply seven and eight, it will also turn out the same thing?
-Not.
-And why?
Because seven plus eight doesn't equal...
-And if I multiply nine by two, it will be four?
-Not.
-And why? Multiplied two - it turned out, but suddenly a bummer with a nine?
-Yes. Twice nine is eighteen.
-And twice seven?
-Fourteen.
-And twice five?
-Ten.
- That is, four is obtained only in one particular case?
-Exactly.
-Now think for yourself. You say that there are some rigid laws and rules for multiplication. What kind of laws can we talk about here if in each specific case a different result is obtained?!
-That's not entirely true. Sometimes the result may be the same. For example, twice six equals twelve. And four times three - too ...
-Worse! Two, six, three four - nothing at all! You can see for yourself that the result does not depend on the initial data in any way. The same decision is made in two radically different situations! And this despite the fact that the same two, which we constantly take and do not change for anything, always gives a different answer with all the numbers. Where, you ask, is the logic?
-But it's just logical!
- For you - maybe. You mathematicians always believe in all sorts of transcendental crap. And these your calculations do not convince me. And do you know why?
-Why?
-Because I I know why do you really need your math. What is she all about? "Katya has one apple in her pocket, and Misha has five. How many apples should Misha give to Katya so that they have equal apples?" And you know what I'll tell you? Misha don't owe anything to anyone give away! Katya has one apple - and that's enough. Not enough for her? Let her go to work hard, and she will honestly earn for herself even for apples, even for pears, even for pineapples in champagne. And if someone wants not to work, but only to solve problems - let him sit with his one apple and not show off!

“Pythagorean pants are equal on all sides.
To prove it, it is necessary to remove and show.

This rhyme has been known to everyone since high school, ever since we studied the famous Pythagorean theorem in a geometry lesson: 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, creator of the religious and philosophical school of the Pythagoreans.
Among the contradictory 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. To this end, 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 much about the secret traditions of Moses, the lawgiver of Israel. Then he visited Egypt, where he was initiated into the Mysteries of Adonis, and, having managed to cross the valley of the Euphrates, 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 the idea of ​​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 the Number, Pythagoras taught, to know the world means to know the numbers that control it. By studying numbers, the Pythagoreans developed numerical relationships and found them in all areas human activity. Pythagoras taught in secret and left no written works behind him. Pythagoras attached great importance number. His philosophical views are largely due to mathematical concepts. He said: “Everything is a number”, “all things are numbers”, thus highlighting one side in understanding the world, namely, its measurability numerical expression. Pythagoras believed that the 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 changing states, there is an unchanging being, some kind of unchanging substance. This is the number. Hence the main idea of ​​Pythagoreanism: number is the basis of everything 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, the objects of thought are more real than the objects of sensory knowledge, since numbers have a timeless nature, i.e. are eternal. They are a reality that is higher than the reality of things. Pythagoras says that all properties of an object can be destroyed, or can change, except for only one numerical property. This property is Unit. The unit is the being of things, indestructible and indecomposable, immutable. Crush any object into tiny particles - each particle will be one. Arguing that numerical being is the only unchanging being, Pythagoras came to the conclusion that all objects are copies of numbers.
One is an absolute number One has eternity. The unit need not be in any relation to anything else. It exists on its own. Two is only the relation of one to one. All numbers are only
numerical relations Units, its modifications. And all forms of being are only certain sides of infinity, and hence the Unit. The original One contains all numbers, therefore, contains the elements of the whole world. Objects are real manifestations of abstract being. Pythagoras was the first to designate the cosmos, with all the things in it, as an order that is established by number. This order is available to the mind, it 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 control it. Cosmos after Pythagoras began to be regarded as ordered by the number of the universe.
Pythagoras taught that the human soul is immortal. He owns the idea of ​​the transmigration of souls. He believed that everything that happens in the world is repeated again and again after certain periods of time, and the souls of the dead, after some time, inhabit others. The soul, as a number, represents the Unit, i.e. the soul is perfect in essence. But every perfection, insofar as it comes into motion, turns into imperfection, although it strives to regain its former perfect state. Pythagoras called imperfection the deviation from Unity; therefore Two was considered a cursed number. The soul in man is in a state of comparative imperfection. It consists of three elements: reason, mind, passion. But if animals also have mind and passions, then only man is endowed with reason (reason). Any of these three sides in a person can prevail, and then the person becomes predominantly either rational, 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 main philosophical law of the Universe - the Unity of Opposites.
Note. Abstraction serves as the basis for the processes of generalization and concept formation. It is a necessary condition for categorization. It forms generalized images of reality, which make it possible to single out the connections and relations of objects that are significant for a certain activity.
The Unity of the Opposites of the Universe consist 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 the quantitative component of the Forms abstraction, and the multiplication (division) is the qualitative component of the Contents abstraction. Numbers of abstraction of Forms and Contents are inextricably linked by the Unity of Opposites.
Let's try to perform mathematical operations, establishing an inseparable connection between Form and Content over numbers.

So let's take a look at the numbers.
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). Further 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 the cyclic transformation of Forms, which corresponds to the cycle of Content - the 1st cycle - 3-9-6 - 6-9-3 2nd 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 numbers of abstraction of Forms and Contents are 3 and 6, where 3 determines Compression, and 6 - Stretching. The compromise for their interaction is the number 9.
Next 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 loop looks like this 2-(3)-2-(6)- 2-(9)… where 2 is the constituent element of the 3-6-9 loop.
Here 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
7х5=35 (2+8+3+5=18 1+8=9)
7x6=42 (4+2=6)
7x7=49
7х8=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 indicate the number of electrons of the atom. Chemical elements are nuclei whose masses are multiples of 9, and multiples of 3 and 6 are isotopes.
Note. Isotope (from the Greek "equal", "same" and "place") - varieties of atoms and nuclei of the same chemical element with different numbers of neutrons in the nucleus. An element is a collection of atoms with the same nuclear charge. Isotopes are varieties of atoms of a chemical element with the same nuclear charge but different mass numbers.

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

The potential for creativity is usually attributed to the humanities, leaving the natural scientific analysis, practical approach and dry language of formulas and numbers. Mathematics cannot be classified as a humanities subject. But without creativity in the "queen of all sciences" you will not go far - people have known about this for a long time. Since the time of Pythagoras, for example.

School textbooks, unfortunately, usually do not explain that in mathematics it is important not only to cram theorems, axioms and formulas. It is important to understand and feel its fundamental principles. And at the same time, try to free your mind from clichés and elementary truths - only in such conditions are all great discoveries born.

Such discoveries include the one that today we know as the Pythagorean theorem. With its help, we will try to show that mathematics not only can, but should be fun. And that this adventure is suitable not only for nerds in thick glasses, but for everyone who is strong in mind and strong in spirit.

From the history of the issue

Strictly speaking, although the theorem is called the "Pythagorean theorem", Pythagoras himself did not discover it. The right triangle and its special properties have been studied long before it. There are two polar points of view on this issue. According to one version, Pythagoras was the first to find a complete proof of the theorem. According to another, the proof does not belong to the authorship of Pythagoras.

Today you can no longer check who is right and who is wrong. It is only known that the proof of Pythagoras, if it ever existed, has not survived. However, there are suggestions that the famous proof from Euclid's Elements may belong to Pythagoras, and Euclid only recorded it.

It is also known today that problems about a right-angled triangle are found in Egyptian sources from the time of Pharaoh Amenemhet I, on Babylonian clay tablets from the reign of King Hammurabi, in the ancient Indian treatise Sulva Sutra and the ancient Chinese work Zhou-bi suan jin.

As you can see, the Pythagorean theorem has occupied the minds of mathematicians since ancient times. Approximately 367 various pieces of evidence that exist today serve as confirmation. No other theorem can compete with it in this respect. Notable evidence authors include Leonardo da Vinci and the 20th President of the United States, James Garfield. All this speaks of the extreme importance of this theorem for mathematics: most of the theorems of geometry are derived from it or, in one way or another, connected with it.

Proofs of the Pythagorean Theorem

School textbooks mostly give algebraic proofs. But the essence of the theorem is in geometry, so let's first of all consider those proofs of the famous theorem that are based on this science.

Proof 1

For the most simple proof Pythagorean theorem for a right triangle, you need to set ideal conditions: let the triangle be not only right-angled, but also isosceles. There is reason to believe that it was such a triangle that was originally considered by ancient mathematicians.

Statement "a square built on the hypotenuse of a right triangle is equal to the sum of the squares built on its legs" can be illustrated with the following drawing:

Look at the isosceles right triangle ABC: On the hypotenuse AC, you can build a square consisting of four triangles equal to the original ABC. And on the legs AB and BC built on a square, each of which contains two similar triangles.

By the way, this drawing formed the basis of numerous anecdotes and cartoons dedicated to the Pythagorean theorem. Perhaps the most famous is "Pythagorean pants are equal in all directions":

Proof 2

This method combines algebra and geometry and can be seen as a variant of the ancient Indian proof of the mathematician Bhaskari.

Construct a right triangle with sides a, b and c(Fig. 1). Then build two squares with sides equal to the sum of the lengths of the two legs - (a+b). In each of the squares, make constructions, as in figures 2 and 3.

In the first square, build four of the same triangles as in Figure 1. As a result, two squares are obtained: one with side a, the second with side b.

In the second square, four similar triangles constructed form a square with a side equal to the hypotenuse c.

The sum of the areas of the constructed squares in Fig. 2 is equal to the area of ​​the square we constructed with side c in Fig. 3. This can be easily verified by calculating the areas of the squares in Fig. 2 according to the formula. And the area of ​​​​the inscribed square in Figure 3. by subtracting the areas of four equal right-angled triangles inscribed in the square from the area of ​​\u200b\u200ba large square with a side (a+b).

Putting all this down, we have: a 2 + b 2 \u003d (a + b) 2 - 2ab. Expand the brackets, do all the necessary algebraic calculations and get that a 2 + b 2 = a 2 + b 2. At the same time, the area of ​​the inscribed in Fig.3. square can also be calculated using the traditional formula S=c2. Those. a2+b2=c2 You have proved the Pythagorean theorem.

Proof 3

The very same ancient Indian proof is described in the 12th century in the treatise “The Crown of Knowledge” (“Siddhanta Shiromani”), and as the main argument the author uses an appeal addressed to the mathematical talents and powers of observation of students and followers: “Look!”.

But we will analyze this proof in more detail:

Inside the square, build four right-angled triangles as indicated in the drawing. The side of the large square, which is also the hypotenuse, is denoted with. Let's call the legs of the triangle a and b. According to the drawing, the side of the inner square is (a-b).

Use the square area formula S=c2 to calculate the area of ​​the outer square. And at the same time calculate the same value by adding the area of ​​​​the inner square and the area of ​​\u200b\u200ball four right triangles: (a-b) 2 2+4*1\2*a*b.

You can use both options to calculate the area of ​​a square to make sure they give the same result. And that gives you the right to write down that c 2 =(a-b) 2 +4*1\2*a*b. As a result of the solution, you will get the formula of the Pythagorean theorem c2=a2+b2. The theorem has been proven.

Proof 4

This curious ancient Chinese proof is called the "Bride's Chair" - because of the chair-like figure that results from all the constructions:

It uses the drawing we have already seen in Figure 3 in the second proof. And the inner square with side c is constructed in the same way as in the ancient Indian proof given above.

If you mentally cut off two green right-angled triangles from the drawing in Fig. 1, transfer them to opposite sides of the square with side c and attach the hypotenuses to the hypotenuses of the lilac triangles, you will get a figure called “bride’s chair” (Fig. 2). For clarity, you can do the same with paper squares and triangles. You will see that the "bride's chair" is formed by two squares: small ones with a side b and big with a side a.

These constructions allowed the ancient Chinese mathematicians and us following them to come to the conclusion that c2=a2+b2.

Proof 5

This is another way to find a solution to the Pythagorean theorem based on geometry. It's called the Garfield Method.

Construct a right triangle ABC. We need to prove that BC 2 \u003d AC 2 + AB 2.

To do this, continue the leg AC and build a segment CD, which is equal to the leg AB. Lower Perpendicular AD line segment ED. Segments ED and AC are equal. connect the dots E and AT, as well as E and With and get a drawing like the picture below:

To prove the tower, we again resort to the method we have already tested: we find the area of ​​the resulting figure in two ways and equate the expressions to each other.

Find the area of ​​a polygon ABED can be done by adding the areas of the three triangles that form it. And one of them ERU, is not only rectangular, but also isosceles. Let's also not forget that AB=CD, AC=ED and BC=CE- this will allow us to simplify the recording and not overload it. So, S ABED \u003d 2 * 1/2 (AB * AC) + 1 / 2BC 2.

At the same time, it is obvious that ABED is a trapezoid. Therefore, we calculate its area using the formula: SABED=(DE+AB)*1/2AD. For our calculations, it is more convenient and clearer to represent the segment AD as the sum of the segments AC and CD.

Let's write both ways to calculate the area of ​​​​a figure by putting an equal sign between them: AB*AC+1/2BC 2 =(DE+AB)*1/2(AC+CD). We use the equality of segments already known to us and described above to simplify the right-hand side of the notation: AB*AC+1/2BC 2 =1/2(AB+AC) 2. And now we open the brackets and transform the equality: AB*AC+1/2BC 2 =1/2AC 2 +2*1/2(AB*AC)+1/2AB 2. Having finished all the transformations, we get exactly what we need: BC 2 \u003d AC 2 + AB 2. We have proved the theorem.

Of course, this list of evidence is far from complete. The Pythagorean theorem can also be proved using vectors, complex numbers, differential equations, stereometry, etc. And even physicists: if, for example, liquid is poured into square and triangular volumes similar to those shown in the drawings. By pouring liquid, it is possible to prove the equality of areas and the theorem itself as a result.

A few words about Pythagorean triplets

This issue is little or not studied in the school curriculum. Meanwhile, it is very interesting and is of great importance in geometry. Pythagorean triples are used to solve many mathematical problems. The idea of ​​them can be useful to you in further education.

So what are Pythagorean triplets? So called natural numbers, collected in threes, the sum of the squares of two of which is equal to the third number squared.

Pythagorean triples can be:

• primitive (all three numbers are relatively prime);
• non-primitive (if each number of a triple is multiplied by the same number, you get a new triple that is not primitive).

Even before our era, the ancient Egyptians were fascinated by the mania for the numbers of Pythagorean triplets: in tasks they considered a right-angled triangle with sides of 3.4 and 5 units. By the way, any triangle whose sides are equal to the numbers from the Pythagorean triple is by default rectangular.

Examples of Pythagorean triples: (3, 4, 5), (6, 8, 10), (5, 12, 13), (9, 12, 15), (8, 15, 17), (12, 16, 20) ), (15, 20, 25), (7, 24, 25), (10, 24, 26), (20, 21, 29), (18, 24, 30), (10, 30, 34), (21, 28, 35), (12, 35, 37), (15, 36, 39), (24, 32, 40), (9, 40, 41), (27, 36, 45), (14 , 48, 50), (30, 40, 50) etc.

Practical application of the theorem

The Pythagorean theorem finds application not only in mathematics, but also in architecture and construction, astronomy, and even literature.

First, about construction: the Pythagorean theorem is widely used in it in problems of different levels of complexity. For example, look at the Romanesque window:

Let's denote the width of the window as b, then the radius of the great semicircle can be denoted as R and express through b: R=b/2. The radius of smaller semicircles can also be expressed in terms of b: r=b/4. In this problem, we are interested in the radius of the inner circle of the window (let's call it p).

The Pythagorean theorem just comes in handy to calculate R. To do this, we use a right-angled triangle, which is indicated by a dotted line in the figure. The hypotenuse of a triangle consists of two radii: b/4+p. One leg is a radius b/4, another b/2-p. Using the Pythagorean theorem, we write: (b/4+p) 2 =(b/4) 2 +(b/2-p) 2. Next, we open the brackets and get b 2 /16+ bp / 2 + p 2 \u003d b 2 / 16 + b 2 / 4-bp + p 2. Let's transform this expression into bp/2=b 2 /4-bp. And then we divide all the terms into b, we give similar ones to get 3/2*p=b/4. And in the end we find that p=b/6- which is what we needed.

Using the theorem, you can calculate the length of the rafters for a gable roof. Determine how high a mobile tower is needed for the signal to reach a certain settlement. And even steadily install a Christmas tree in the city square. As you can see, this theorem lives not only on the pages of textbooks, but is often useful in real life.

As far as literature is concerned, the Pythagorean theorem has inspired writers since antiquity and continues to do so today. For example, the nineteenth-century German writer Adelbert von Chamisso was inspired by her to write a sonnet:

The light of truth will not soon dissipate,
But, having shone, it is unlikely to dissipate
And, like thousands of years ago,
Will not cause doubts and disputes.

The wisest when it touches the eye
Light of truth, thank the gods;
And a hundred bulls, stabbed, lie -
The return gift of the lucky Pythagoras.

Since then, the bulls have been roaring desperately:
Forever aroused the bull tribe
event mentioned here.

They think it's about time
And again they will be sacrificed
Some great theorem.

(translated by Viktor Toporov)

And in the twentieth century, the Soviet writer Yevgeny Veltistov in his book "The Adventures of Electronics" devoted a whole chapter to the proofs of the Pythagorean theorem. And half a chapter of the story about the two-dimensional world that could exist if the Pythagorean theorem became the fundamental law and even religion for a single world. It would be much easier to live in it, but also much more boring: for example, no one there understands the meaning of the words “round” and “fluffy”.

And in the book “The Adventures of Electronics”, the author, through the mouth of the mathematics teacher Taratara, says: “The main thing in mathematics is the movement of thought, new ideas.” It is this creative flight of thought that generates the Pythagorean theorem - it is not for nothing that it has so many diverse proofs. It helps to go beyond the usual, and look at familiar things in a new way.

Conclusion

This article has been created so that you can look beyond school curriculum in mathematics and learn not only those proofs of the Pythagorean theorem that are given in the textbooks "Geometry 7-9" (L.S. Atanasyan, V.N. Rudenko) and "Geometry 7-11" (A.V. Pogorelov), but and other curious ways to prove the famous theorem. And also see examples of how the Pythagorean theorem can be applied in everyday life.

Firstly, this information will allow you to claim higher scores in math classes - information on the subject from additional sources is always highly appreciated.

Secondly, we wanted to help you get a feel for how interesting mathematics is. To be convinced by specific examples that there is always a place for creativity in it. We hope that the Pythagorean theorem and this article will inspire you to do your own research and exciting discoveries in mathematics and other sciences.

Tell us in the comments if you found the evidence presented in the article interesting. Did you find this information helpful in your studies? Let us know what you think about the Pythagorean theorem and this article - we'd love to discuss it all with you.

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