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 it to be treated 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.
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 больше площади данного квадрата. Его рассуждения, по сути, сводились к доказательству теоремы Пифагора, пусть и для конкретного треугольника.
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 by simply “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 centuries. 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 main character of the fantastic story by Evgeny Veltistov "Electronics - a boy from a suitcase" knew 25 proofs of the Pythagorean theorem, including those given by Euclid; True, he mistakenly called it the simplest, although in fact in the modern edition of the 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 geometry course, known to the ancient Greeks under the name "The Pythagorean Tradition". 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 united four related disciplines created by Pythagoras and his followers - 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 attribute anything to him 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.
One thing you can be sure of one hundred percent, that when asked what the square of the hypotenuse is, any adult will boldly answer: "The sum of the squares of the legs." This theorem is firmly planted in the minds of every educated person, but it is enough just to ask someone to prove it, and then difficulties can arise. Therefore, let's remember and consider different ways of proving the Pythagorean theorem.
Brief overview of the biography
The Pythagorean theorem is familiar to almost everyone, but for some reason the biography of the person who produced it is not so popular. We'll fix it. Therefore, before studying the different ways of proving the Pythagorean theorem, you need to briefly get acquainted with his personality.
Pythagoras - a philosopher, mathematician, thinker, originally from Today it is very difficult to distinguish his biography from the legends that have developed in memory of this great man. But as follows from the writings of his followers, Pythagoras of Samos was born on the island of Samos. His father was an ordinary stone cutter, but his mother came from a noble family.
According to legend, the birth of Pythagoras was predicted by a woman named Pythia, in whose honor the boy was named. According to her prediction, a born boy was to bring many benefits and good to mankind. Which is what he actually did.
The birth of a theorem
In his youth, Pythagoras moved to Egypt to meet the famous Egyptian sages there. After meeting with them, he was admitted to study, where he learned all the great achievements of Egyptian philosophy, mathematics and medicine.
Probably, it was in Egypt that Pythagoras was inspired by the majesty and beauty of the pyramids and created his great theory. This may shock readers, but modern historians believe that Pythagoras did not prove his theory. But he only passed on his knowledge to his followers, who later completed all the necessary mathematical calculations.
Be that as it may, today not one technique for proving this theorem is known, but several at once. Today we can only guess how exactly the ancient Greeks made their calculations, so here we will consider different ways of proving the Pythagorean theorem.
Pythagorean theorem
Before you start any calculations, you need to figure out which theory to prove. The Pythagorean theorem sounds like this: "In a triangle in which one of the angles is 90 o, the sum of the squares of the legs is equal to the square of the hypotenuse."
There are 15 different ways to prove the Pythagorean Theorem in total. This is a fairly large number, so let's pay attention to the most popular of them.
Method one
Let's first define what we have. This data will also apply to other ways of proving the Pythagorean theorem, so you should immediately remember all the available notation.
Suppose a right triangle is given, with legs a, b and hypotenuse equal to c. The first method of proof is based on the fact that a square must be drawn from a right-angled triangle.
To do this, you need to draw a segment equal to the leg in to the leg length a, and vice versa. So it should turn out two equal sides of the square. It remains only to draw two parallel lines, and the square is ready.
Inside the resulting figure, you need to draw another square with a side equal to the hypotenuse of the original triangle. To do this, from the vertices ac and sv, you need to draw two parallel segments equal to c. Thus, we get three sides of the square, one of which is the hypotenuse of the original right-angled triangle. It remains only to draw the fourth segment.
Based on the resulting figure, we can conclude that the area of \u200b\u200bthe outer square is (a + b) 2. If you look inside the figure, you can see that in addition to the inner square, it has four right-angled triangles. The area of each is 0.5 av.
Therefore, the area is: 4 * 0.5av + s 2 \u003d 2av + s 2
Hence (a + c) 2 \u003d 2av + c 2
And, therefore, with 2 \u003d a 2 + in 2
The theorem has been proven.
Method two: similar triangles
This formula for the proof of the Pythagorean theorem was derived on the basis of a statement from the section of geometry about similar triangles. It says that the leg of a right triangle is the mean proportional to its hypotenuse and the hypotenuse segment emanating from the vertex of an angle of 90 o.
The initial data remain the same, so let's start right away with the proof. Let us draw a segment CD perpendicular to the side AB. Based on the above statement, the legs of the triangles are equal:
AC=√AB*AD, SW=√AB*DV.
To answer the question of how to prove the Pythagorean theorem, the proof must be laid by squaring both inequalities.
AC 2 \u003d AB * HELL and SV 2 \u003d AB * DV
Now we need to add the resulting inequalities.
AC 2 + SV 2 \u003d AB * (AD * DV), where AD + DV \u003d AB
It turns out that:
AC 2 + CB 2 \u003d AB * AB
And therefore:
AC 2 + CB 2 \u003d AB 2
The proof of the Pythagorean theorem and various ways of solving it require a versatile approach to this problem. However, this option is one of the simplest.
Another calculation method
Description of different ways of proving the Pythagorean theorem may not say anything, until you start practicing on your own. Many methods involve not only mathematical calculations, but also the construction of new figures from the original triangle.
In this case, it is necessary to complete another right-angled triangle VSD from the leg of the aircraft. Thus, now there are two triangles with a common leg BC.
Knowing that the areas of similar figures have a ratio as the squares of their similar linear dimensions, then:
S avs * s 2 - S avd * in 2 \u003d S avd * a 2 - S vd * a 2
S avs * (from 2 to 2) \u003d a 2 * (S avd -S vvd)
from 2 to 2 \u003d a 2
c 2 \u003d a 2 + in 2
Since this option is hardly suitable from different methods of proving the Pythagorean theorem for grade 8, you can use the following technique.
The easiest way to prove the Pythagorean theorem. Reviews
Historians believe that this method was first used to prove a theorem in ancient Greece. It is the simplest, since it does not require absolutely any calculations. If you draw a picture correctly, then the proof of the statement that a 2 + b 2 \u003d c 2 will be clearly visible.
The conditions for this method will be slightly different from the previous one. To prove the theorem, suppose that the right triangle ABC is isosceles.
We take the hypotenuse AC as the side of the square and draw its three sides. In addition, it is necessary to draw two diagonal lines in the resulting square. So that inside it you get four isosceles triangles.
To the legs AB and CB, you also need to draw a square and draw one diagonal line in each of them. We draw the first line from vertex A, the second - from C.
Now you need to carefully look at the resulting drawing. Since there are four triangles on the hypotenuse AC, equal to the original one, and two on the legs, this indicates the veracity of this theorem.
By the way, thanks to this method of proving the Pythagorean theorem, the famous phrase: "Pythagorean pants are equal in all directions."
Proof by J. Garfield
James Garfield is the 20th President of the United States of America. In addition to leaving his mark on history as the ruler of the United States, he was also a gifted self-taught.
At the beginning of his career, he was an ordinary teacher at a folk school, but soon became the director of one of the higher educational institutions. The desire for self-development and allowed him to offer a new theory of proof of the Pythagorean theorem. The theorem and an example of its solution are as follows.
First you need to draw two right-angled triangles on a piece of paper so that the leg of one of them is a continuation of the second. The vertices of these triangles need to be connected to end up with a trapezoid.
As you know, the area of a trapezoid is equal to the product of half the sum of its bases and the height.
S=a+b/2 * (a+b)
If we consider the resulting trapezoid as a figure consisting of three triangles, then its area can be found as follows:
S \u003d av / 2 * 2 + s 2 / 2
Now we need to equalize the two original expressions
2av / 2 + s / 2 \u003d (a + c) 2 / 2
c 2 \u003d a 2 + in 2
More than one volume can be written about the Pythagorean theorem and how to prove it study guide. But does it make sense when this knowledge cannot be put into practice?
Practical application of the Pythagorean theorem
Unfortunately, modern school curricula provide for the use of this theorem only in geometric problems. Graduates will soon leave the school walls without knowing how they can apply their knowledge and skills in practice.
In fact, everyone can use the Pythagorean theorem in their daily life. And not only in professional activities, but also in ordinary household chores. Let's consider several cases when the Pythagorean theorem and methods of its proof can be extremely necessary.
Connection of the theorem and astronomy
It would seem how stars and triangles can be connected on paper. In fact, astronomy is a scientific field in which the Pythagorean theorem is widely used.
For example, consider the motion of a light beam in space. We know that light travels in both directions at the same speed. We call the trajectory AB along which the light ray moves l. And half the time it takes for light to get from point A to point B, let's call t. And the speed of the beam - c. It turns out that: c*t=l
If you look at this same beam from another plane, for example, from a space liner that moves at a speed v, then with such an observation of the bodies, their speed will change. In this case, even stationary elements will move with a speed v in the opposite direction.
Let's say the comic liner is sailing to the right. Then points A and B, between which the ray rushes, will move to the left. Moreover, when the beam moves from point A to point B, point A has time to move and, accordingly, the light will already arrive at a new point C. To find half the distance that point A has shifted, you need to multiply the speed of the liner by half the travel time of the beam (t ").
And in order to find how far a ray of light could travel during this time, you need to designate half the path of the new beech s and get the following expression:
If we imagine that the points of light C and B, as well as the space liner, are the vertices of an isosceles triangle, then the segment from point A to the liner will divide it into two right triangles. Therefore, thanks to the Pythagorean theorem, you can find the distance that a ray of light could travel.
This example, of course, is not the most successful, since only a few can be lucky enough to try it out in practice. Therefore, we consider more mundane applications of this theorem.
Mobile signal transmission range
Modern life can no longer be imagined without the existence of smartphones. But how much would they be of use if they could not connect subscribers via mobile communications?!
The quality of mobile communications directly depends on the height at which the antenna of the mobile operator is located. In order to calculate how far from a mobile tower a phone can receive a signal, you can apply the Pythagorean theorem.
Let's say you need to find the approximate height of a stationary tower so that it can propagate a signal within a radius of 200 kilometers.
AB (tower height) = x;
BC (radius of signal transmission) = 200 km;
OS (radius the globe) = 6380 km;
OB=OA+ABOB=r+x
Applying the Pythagorean theorem, we find out that the minimum height of the tower should be 2.3 kilometers.
Pythagorean theorem in everyday life
Oddly enough, the Pythagorean theorem can be useful even in everyday matters, such as determining the height of a closet, for example. At first glance, there is no need to use such complex calculations, because you can simply take measurements with a tape measure. But many are surprised why certain problems arise during the assembly process if all the measurements were taken more than accurately.
The fact is that the wardrobe is assembled in a horizontal position and only then rises and is installed against the wall. Therefore, the sidewall of the cabinet in the process of lifting the structure must freely pass both along the height and diagonally of the room.
Suppose there is a wardrobe with a depth of 800 mm. Distance from floor to ceiling - 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? Let's look at an example.
With ideal dimensions of the cabinet, let's check the operation of the Pythagorean theorem:
AC \u003d √AB 2 + √BC 2
AC \u003d √ 2474 2 +800 2 \u003d 2600 mm - everything converges.
Let's say the height of the cabinet is not 2474 mm, but 2505 mm. Then:
AC \u003d √2505 2 + √800 2 \u003d 2629 mm.
Therefore, this cabinet is not suitable for installation in this room. Since when lifting it to a vertical position, damage to its body can be caused.
Perhaps, having considered different ways of proving the Pythagorean theorem by different scientists, we can conclude that it is more than true. Now you can use the information received in your daily life and be completely sure that all calculations will be not only useful, but also correct.
Pythagorean pants The comic name of the Pythagorean theorem, which arose due to the fact that the squares built on the sides of a rectangle and diverging in different directions resemble the cut of pants. I loved geometry ... and at the entrance exam to the university I even received praise from Chumakov, a professor of mathematics, for explaining the properties of parallel lines and Pythagorean pants without a blackboard, drawing with my hands in the air(N. Pirogov. Diary of an old doctor).
Phraseological dictionary of Russian literary language. - M.: Astrel, AST. A. I. Fedorov. 2008 .
See what "Pythagorean pants" are in other dictionaries:
Pants - get a working SuperStep discount coupon at Akademika or buy cheap pants with free shipping on sale at SuperStep
Pythagorean pants- ... Wikipedia
Pythagorean pants- Zharg. school Shuttle. The Pythagorean theorem, which establishes the relationship between the areas of squares built on the hypotenuse and the legs of a right triangle. BTS, 835... Big dictionary of Russian sayings
Pythagorean pants- A playful name for the Pythagorean theorem, which establishes the ratio between the areas of squares built on the hypotenuse and the legs of a right-angled triangle, which looks like the cut of pants in the drawings ... Dictionary of many expressions
Pythagorean pants (invent)- foreigner: about a gifted person Cf. This is the certainty of the sage. In ancient times, he probably would have invented Pythagorean pants ... Saltykov. Motley letters. Pythagorean pants (geom.): in a rectangle, the square of the hypotenuse is equal to the squares of the legs (teaching ... ... Michelson's Big Explanatory Phraseological Dictionary
Pythagorean pants are equal on all sides- The number of buttons is known. Why is the dick cramped? (roughly) about pants and the male sexual organ. Pythagorean pants are equal on all sides. To prove this, it is necessary to remove and show 1) about the Pythagorean theorem; 2) about wide pants ... Live speech. Dictionary of colloquial expressions
Pythagorean pants invent- Pythagorean pants (invent) foreigner. about a gifted person. Wed This is the undoubted sage. In ancient times, he probably would have invented Pythagorean pants ... Saltykov. Motley letters. Pythagorean pants (geom.): in a rectangle, the square of the hypotenuse ... ... Michelson's Big Explanatory Phraseological Dictionary (original spelling)
Pythagorean pants are equal in all directions- Joking proof of the Pythagorean theorem; also in jest about buddy's baggy trousers... Dictionary of folk phraseology
Adj., rude...
PYTHAGOREAN PANTS ARE EQUAL ON ALL SIDES (NUMBER OF BUTTONS IS KNOWN. WHY IS IT CLOSE? / TO PROVE THIS, IT IS NECESSARY TO REMOVE AND SHOW)- adj., rude ... Explanatory dictionary of modern colloquial phraseological units and sayings
trousers- noun, pl., use comp. often Morphology: pl. what? pants, (no) what? pants for what? pants, (see) what? pants what? pants, what? about pants 1. Pants are a piece of clothing that has two short or long legs and covers lower part… … Dictionary of Dmitriev
Books
- Pythagorean pants, . In this book you will find fantasy and adventure, miracles and fiction. Funny and sad, ordinary and mysterious... And what else is needed for entertaining reading? The main thing is to be…
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MBOU Bondarskaya secondary school Student project on the topic: “Pythagoras and his theorem” Prepared by: Ektov Konstantin, student of grade 7 A Head: Dolotova Nadezhda Ivanovna, mathematics teacher 2015
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Annotation. Geometry is a very interesting science. It contains many theorems that are not similar to each other, but sometimes so necessary. I became very interested in the Pythagorean theorem. Unfortunately, one of the most important statements we pass only in the eighth grade. I decided to lift the veil of secrecy and explore the Pythagorean theorem.
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Tasks To study the biography of Pythagoras. Explore the history of the emergence and proof of the theorem. Find out how the theorem is used in art. Find historical problems in which the Pythagorean theorem is used. To get acquainted with the attitude of children of different times to this theorem. Create a project.
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Research progress Biography of Pythagoras. Commandments and aphorisms of Pythagoras. Pythagorean theorem. History of the theorem. Why are "Pythagorean pants equal in all directions"? Various proofs of the Pythagorean theorem by other scientists. Application of the Pythagorean theorem. Poll. Conclusion.
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Pythagoras - who is he? Pythagoras of Samos (580 - 500 BC) Ancient Greek mathematician and idealist philosopher. Born on the island of Samos. Received a good education. According to legend, Pythagoras, in order to get acquainted with the wisdom of Eastern scientists, went to Egypt and lived there for 22 years. Having mastered all the sciences of the Egyptians, including mathematics, he moved to Babylon, where he lived for 12 years and got acquainted with the scientific knowledge of the Babylonian priests. Traditions attribute to Pythagoras a visit to India. This is very likely, since Ionia and India then had trade relations. Returning to his homeland (c. 530 BC), Pythagoras tried to organize his philosophical school. However, for unknown reasons, he soon leaves Samos and settles in Croton (a Greek colony in northern Italy). Here Pythagoras managed to organize his own school, which operated for almost thirty years. The school of Pythagoras, or, as it is also called, the Pythagorean Union, was at the same time a philosophical school, a political party, and a religious brotherhood. The status of the Pythagorean union was very severe. In his philosophical views, Pythagoras was an idealist, a defender of the interests of the slave-owning aristocracy. Perhaps this was the reason for his departure from Samos, since supporters of democratic views had a very large influence in Ionia. In public matters, by "order" the Pythagoreans understood the rule of the aristocrats. They condemned ancient Greek democracy. Pythagorean philosophy was a primitive attempt to justify the dominance of the slave-owning aristocracy. At the end of the 5th century BC e. a wave of democratic movement swept through Greece and its colonies. Democracy won in Croton. Pythagoras leaves Croton with his disciples and goes to Tarentum, and then to Metapont. The arrival of the Pythagoreans at Metapont coincided with the outbreak of a popular uprising there. In one of the night skirmishes, almost ninety-year-old Pythagoras died. His school has ceased to exist. The disciples of Pythagoras, fleeing persecution, settled throughout Greece and its colonies. Earning their livelihood, they organized schools in which they taught mainly arithmetic and geometry. Information about their achievements is contained in the writings of later scientists - Plato, Aristotle, etc.
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Commandments and aphorisms of Pythagoras Thought is above all between people on earth. Do not sit down on a grain measure (i.e., do not live idly). When leaving, do not look back (that is, before death, do not cling to life). Do not go down the beaten road (that is, follow not the opinions of the crowd, but the opinions of the few who understand). Do not keep swallows in the house (i.e., do not receive guests who are talkative and not restrained in language). Be with the one who takes the load, do not be with the one who dumps the load (that is, encourage people not to idleness, but to virtue, to work). In the field of life, like a sower, walk with even and steady steps. The true fatherland is where there are good morals. Do not be a member of a learned society: the wisest, making up a society, become commoners. Revere sacred numbers, weight and measure, as a child of graceful equality. Measure your desires, weigh your thoughts, count your words. Be astonished at nothing: astonishment has produced gods.
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Statement of the theorem. In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
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Proofs of the theorem. At the moment, 367 proofs of this theorem have been recorded in the scientific literature. Probably, the Pythagorean theorem is the only theorem with such an impressive number of proofs. Of course, all of them can be divided into a small number of classes. The most famous of them: proofs by the area method, axiomatic and exotic proofs.
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Pythagorean theorem Proof Given a right triangle with legs a, b and hypotenuse c. Let's prove that c² = a² + b² Let's complete the triangle to a square with side a + b. The area S of this square is (a + b)². On the other hand, the square is made up of four equal right triangles, each S equal to ½ a b, and a square with side c. S = 4 ½ a b + c² = 2 a b + c² Thus, (a + b)² = 2 a b + c², whence c² = a² + b² c c c c c a b
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The history of the Pythagorean theorem The history of the Pythagorean theorem is interesting. Although this theorem is associated with the name of Pythagoras, it was known long before him. In Babylonian texts, this theorem occurs 1200 years before Pythagoras. It is possible that at that time they did not yet know its evidence, and the very relationship between the hypotenuse and the legs was established empirically on the basis of measurements. Pythagoras apparently found proof of this relationship. An ancient legend has been preserved that in honor of his discovery, Pythagoras sacrificed a bull to the gods, and according to other testimonies, even a hundred bulls. Over the following centuries, various other proofs of the Pythagorean theorem were found. Currently, there are more than a hundred of them, but the most popular theorem is the construction of a square using a given right triangle.
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Theorem in Ancient China "If a right angle is decomposed into its component parts, then the line connecting the ends of its sides will be 5 when the base is 3 and the height is 4."
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Theorem in Ancient Egypt Kantor (the largest German historian of mathematics) believes that the equality 3 ² + 4 ² = 5² was already known to the Egyptians around 2300 BC. e., during the time of King Amenemhat (according to papyrus 6619 of the Berlin Museum). According to Cantor, the harpedonapts, or "stringers", built right angles using right triangles with sides 3, 4 and 5.
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About the theorem in Babylonia “The merit of the first Greek mathematicians, such as Thales, Pythagoras and the Pythagoreans, is not the discovery of mathematics, but its systematization and substantiation. In their hands, computational recipes based on vague ideas have become an exact science.
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Why are "Pythagorean pants equal in all directions"? For two millennia, the most common proof of the Pythagorean theorem was that of Euclid. It is placed in his famous book "Beginnings". Euclid lowered the height CH from the vertex of the right angle to the hypotenuse and proved that its continuation divides the square completed on the hypotenuse into two rectangles, the areas of which are equal to the areas of the corresponding squares built on the legs. The drawing used in the proof of this theorem is jokingly called "Pythagorean pants". For a long time he was considered one of the symbols of mathematical science.
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The attitude of children of antiquity to the proof of the Pythagorean theorem was considered by students of the Middle Ages to be very difficult. Weak students, who memorized theorems without understanding, and therefore nicknamed "donkeys", were not able to overcome the Pythagorean theorem, which served for them like an insurmountable bridge. Because of the drawings accompanying the Pythagorean theorem, students also called it a “windmill”, composed poems like “Pythagorean pants are equal on all sides”, and drew caricatures.
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Proofs of the theorem The simplest proof of the theorem is obtained in the case of an isosceles right triangle. Indeed, one need only look at the tiling of isosceles right-angled triangles to see that the theorem is true. For example, for triangle ABC: the square built on the hypotenuse AC contains 4 initial triangles, and the squares built on the legs contain two.
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"Chair of the bride" In the figure, the squares built on the legs are placed in steps one next to the other. This figure, which occurs in evidence dating no later than the 9th century CE, e., the Hindus called the "chair of the bride."
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Application of the Pythagorean theorem At present, it is generally recognized that the success of the development of many areas of science and technology depends on the development of various areas of mathematics. An important condition for increasing the efficiency of production is the widespread introduction of mathematical methods in technology and the national economy, which involves the creation of new, effective methods of qualitative and quantitative research that allow solving problems put forward by practice.
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Application of the theorem in construction In buildings of the Gothic and Romanesque styles, the upper parts of the windows are divided by stone ribs, which not only play the role of an ornament, but also contribute to the strength of the windows.
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Historical tasks To fix the mast, you need to install 4 cables. One end of each cable should be fixed at a height of 12 m, the other on the ground at a distance of 5 m from the mast. Is 50 m of rope enough to secure the mast?
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 has 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 finds wide application 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 a story about a 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 was created so that you can look beyond the 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 also 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 will be happy to discuss all this with you.
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