In one thing, you can be one hundred percent sure that when asked what the square of the hypotenuse is, any adult will boldly answer: “The sum of the squares of the legs.” This theory is firmly planted in the minds of everyone. educated person, but it is enough just to ask someone to prove it, and then difficulties may 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 own 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 way of proof is based on the fact that right triangle you need to draw a square.
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 ac and s vertices, you need to draw two parallel segment equal with. 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.
AT this case it is necessary to complete one more 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.
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 picture. 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 was born: "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 new theory 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, in modern school programs This theorem is intended to be used 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 activity but also in normal 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.
» Honored Professor of Mathematics at the University of Warwick, a well-known popularizer of science Ian Stewart, dedicated to the role of numbers in the history of mankind and the relevance of their study in our time.
Pythagorean hypotenuse
Pythagorean triangles have a right angle and integer sides. In the simplest of them, the longest side has a length of 5, the rest are 3 and 4. There are 5 regular polyhedra. A fifth-degree equation cannot be solved with fifth-degree roots - or any other roots. Lattices in the plane and in three-dimensional space do not have a five-lobe rotational symmetry; therefore, such symmetries are also absent in crystals. However, they can be at the gratings in four-dimensional space and in curious structures known as quasicrystals.
Hypotenuse of the smallest Pythagorean triple
The Pythagorean theorem states that the longest side of a right triangle (the proverbial hypotenuse) is related to the other two sides of that triangle in a very simple and beautiful way: the square of the hypotenuse is equal to the sum squares of the other two sides.
Traditionally, we call this theorem after Pythagoras, but in fact its history is rather vague. Clay tablets suggest that the ancient Babylonians knew the Pythagorean theorem long before Pythagoras himself; the glory of the discoverer was brought to him by the mathematical cult of the Pythagoreans, whose supporters believed that the universe was based on numerical patterns. Ancient authors attributed to the Pythagoreans - and therefore to Pythagoras - a variety of mathematical theorems, but in fact we have no idea what kind of mathematics Pythagoras himself was engaged in. We don't even know if the Pythagoreans could prove the Pythagorean Theorem, or if they simply believed it was true. Or, more likely, they had convincing data about its truth, which nevertheless would not have been enough for what we consider proof today.
Evidence of Pythagoras
The first known proof of the Pythagorean theorem is found in Euclid's Elements. This is a rather complicated proof using a drawing that Victorian schoolchildren would immediately recognize as "Pythagorean pants"; the drawing really resembles underpants drying on a rope. Literally hundreds of other proofs are known, most of which make the assertion more obvious.
// Rice. 33. Pythagorean pants
One of the simplest proofs is a kind of mathematical puzzle. Take any right triangle, make four copies of it and collect them inside the square. With one laying, we see a square on the hypotenuse; with the other - squares on the other two sides of the triangle. It is clear that the areas in both cases are equal.
// Rice. 34. Left: square on the hypotenuse (plus four triangles). Right: the sum of the squares on the other two sides (plus the same four triangles). Now eliminate the triangles
The dissection of Perigal is another puzzle piece of evidence.
// Rice. 35. Dissection of Perigal
There is also a proof of the theorem using stacking squares on the plane. Perhaps this is how the Pythagoreans or their unknown predecessors discovered this theorem. If you look at how the oblique square overlaps the other two squares, you can see how to cut the large square into pieces and then put them together into two smaller squares. You can also see right-angled triangles, the sides of which give the dimensions of the three squares involved.
// Rice. 36. Proof by paving
There are interesting proofs using similar triangles in trigonometry. At least fifty different proofs are known.
Pythagorean triplets
In number theory, the Pythagorean theorem became the source of a fruitful idea: to find integer solutions to algebraic equations. A Pythagorean triple is a set of integers a, b and c such that
Geometrically, such a triple defines a right triangle with integer sides.
The smallest hypotenuse of a Pythagorean triple is 5.
The other two sides of this triangle are 3 and 4. Here
32 + 42 = 9 + 16 = 25 = 52.
The next largest hypotenuse is 10 because
62 + 82 = 36 + 64 = 100 = 102.
However, this is essentially the same triangle with doubled sides. The next largest and truly different hypotenuse is 13, for which
52 + 122 = 25 + 144 = 169 = 132.
Euclid knew that there were an infinite number of different variations of Pythagorean triples, and he gave what might be called a formula for finding them all. Later, Diophantus of Alexandria offered a simple recipe, basically the same as Euclidean.
Take any two natural numbers and calculate:
their double product;
difference of their squares;
the sum of their squares.
The three resulting numbers will be the sides of the Pythagorean triangle.
Take, for example, the numbers 2 and 1. Calculate:
double product: 2 × 2 × 1 = 4;
difference of squares: 22 - 12 = 3;
sum of squares: 22 + 12 = 5,
and we got the famous 3-4-5 triangle. If we take the numbers 3 and 2 instead, we get:
double product: 2 × 3 × 2 = 12;
difference of squares: 32 - 22 = 5;
sum of squares: 32 + 22 = 13,
and we get the next famous triangle 5 - 12 - 13. Let's try to take the numbers 42 and 23 and get:
double product: 2 × 42 × 23 = 1932;
difference of squares: 422 - 232 = 1235;
sum of squares: 422 + 232 = 2293,
no one has ever heard of the triangle 1235-1932-2293.
But these numbers work too:
12352 + 19322 = 1525225 + 3732624 = 5257849 = 22932.
There is another feature in the Diophantine rule that has already been hinted at: having received three numbers, we can take another arbitrary number and multiply them all by it. Thus, a 3-4-5 triangle can be turned into a 6-8-10 triangle by multiplying all sides by 2, or into a 15-20-25 triangle by multiplying everything by 5.
If we switch to the language of algebra, the rule takes the following form: let u, v and k be natural numbers. Then a right triangle with sides
2kuv and k (u2 - v2) has a hypotenuse
There are other ways of presenting the main idea, but they all boil down to the one described above. This method allows you to get all Pythagorean triples.
Regular polyhedra
There are exactly five regular polyhedra. A regular polyhedron (or polyhedron) is a three-dimensional figure with a finite number of flat faces. Facets converge with each other on lines called edges; edges meet at points called vertices.
The culmination of the Euclidean "Beginnings" is the proof that there can be only five regular polyhedra, that is, polyhedra in which each face is a regular polygon (equal sides, equal angles), all faces are identical and all vertices are surrounded by an equal number of equally spaced faces. Here are five regular polyhedra:
tetrahedron with four triangular faces, four vertices and six edges;
cube, or hexahedron, with 6 square faces, 8 vertices and 12 edges;
octahedron with 8 triangular faces, 6 vertices and 12 edges;
dodecahedron with 12 pentagonal faces, 20 vertices and 30 edges;
icosahedron with 20 triangular faces, 12 vertices and 30 edges.
// Rice. 37. Five regular polyhedra
Regular polyhedra can also be found in nature. In 1904, Ernst Haeckel published drawings of tiny organisms known as radiolarians; many of them are shaped like the same five regular polyhedra. Perhaps, however, he slightly corrected nature, and the drawings do not fully reflect the shape of specific living beings. The first three structures are also observed in crystals. You will not find a dodecahedron and an icosahedron in crystals, although irregular dodecahedrons and icosahedrons sometimes come across there. True dodecahedrons can occur as quasicrystals, which are like crystals in every way, except that their atoms do not form a periodic lattice.
// Rice. 38. Drawings by Haeckel: radiolarians in the form of regular polyhedra
// Rice. 39. Developments of Regular Polyhedra
It can be interesting to make models of regular polyhedra out of paper by first cutting out a set of interconnected faces - this is called a polyhedron sweep; the scan is folded along the edges and the corresponding edges are glued together. It is useful to add an additional area for glue to one of the edges of each such pair, as shown in Fig. 39. If there is no such platform, you can use adhesive tape.
Equation of the fifth degree
There is no algebraic formula for solving equations of the 5th degree.
AT general view The 5th equation looks like this:
ax5 + bx4 + cx3 + dx2 + ex + f = 0.
The problem is to find a formula for solving such an equation (it can have up to five solutions). Experience in dealing with quadratic and cubic equations, as well as with equations of the fourth degree, suggests that such a formula should also exist for equations of the fifth degree, and, in theory, the roots of the fifth, third and second degree should appear in it. Again, one can safely assume that such a formula, if it exists, will turn out to be very, very complex.
This assumption ultimately turned out to be wrong. Indeed, no such formula exists; at least there is no formula consisting of the coefficients a, b, c, d, e and f, composed using addition, subtraction, multiplication and division, and taking roots. Thus, there is something very special about the number 5. The reasons for this unusual behavior of the five are very deep, and it took a lot of time to figure them out.
The first sign of a problem was that no matter how hard mathematicians tried to find such a formula, no matter how smart they were, they always failed. For some time, everyone believed that the reasons lie in the incredible complexity of the formula. It was believed that no one simply could understand this algebra properly. However, over time, some mathematicians began to doubt that such a formula even existed, and in 1823 Niels Hendrik Abel was able to prove the opposite. There is no such formula. Shortly thereafter, Évariste Galois found a way to determine whether an equation of one degree or another - 5th, 6th, 7th, generally any - is solvable using this kind of formula.
The conclusion from all this is simple: the number 5 is special. You can solve algebraic equations (using roots of the nth degrees for different values of n) for degrees 1, 2, 3 and 4, but not for the 5th degree. This is where the obvious pattern ends.
No one is surprised that equations of powers greater than 5 behave even worse; in particular, they have the same difficulty: no general formulas for their solution. This does not mean that the equations have no solutions; it does not mean also that it is impossible to find very precise numerical values of these solutions. It's all about the limitations of traditional algebra tools. This is reminiscent of the impossibility of trisecting an angle with a ruler and a compass. There is an answer, but the listed methods are not sufficient and do not allow you to determine what it is.
Crystallographic limitation
Crystals in two and three dimensions do not have 5-beam rotational symmetry.
The atoms in a crystal form a lattice, that is, a structure that repeats periodically in several independent directions. For example, the pattern on the wallpaper is repeated along the length of the roll; in addition, it is usually repeated in the horizontal direction, sometimes with a shift from one piece of wallpaper to the next. Essentially, the wallpaper is a two-dimensional crystal.
There are 17 varieties of wallpaper patterns on the plane (see chapter 17). They differ in the types of symmetry, that is, in the ways of rigidly shifting the pattern so that it lies exactly on itself in its original position. The types of symmetry include, in particular, various variants of rotational symmetry, where the pattern should be rotated through a certain angle around a certain point - the center of symmetry.
The order of symmetry of rotation is how many times you can rotate the body to a full circle so that all the details of the picture return to their original positions. For example, a 90° rotation is 4th order rotational symmetry*. The list of possible types of rotational symmetry in the crystal lattice again points to the unusualness of the number 5: it is not there. There are variants with rotational symmetry of 2nd, 3rd, 4th and 6th orders, but no wallpaper pattern has 5th order rotational symmetry. There is also no rotational symmetry of order greater than 6 in crystals, but the first violation of the sequence still occurs at the number 5.
The same happens with crystallographic systems in three-dimensional space. Here the lattice repeats itself in three independent directions. There is 219 various types symmetry, or 230 if you count mirror reflection drawing as a separate version of it - moreover, in this case there is no mirror symmetry. Again, rotational symmetries of orders 2, 3, 4, and 6 are observed, but not 5. This fact is called the crystallographic constraint.
In four-dimensional space, lattices with 5th order symmetry exist; in general, for lattices of sufficiently high dimension, any predetermined order of rotational symmetry is possible.
// Rice. 40. Crystal cell table salt. Dark balls represent sodium atoms, light balls represent chlorine atoms.
Quasicrystals
While 5th order rotational symmetry is not possible in 2D and 3D lattices, it can exist in slightly less regular structures known as quasicrystals. Using Kepler's sketches, Roger Penrose discovered flat systems with more common type fivefold symmetry. They are called quasicrystals.
Quasicrystals exist in nature. In 1984, Daniel Shechtman discovered that an alloy of aluminum and manganese can form quasi-crystals; initially, crystallographers greeted his message with some skepticism, but later the discovery was confirmed, and in 2011 Shekhtman was awarded Nobel Prize in chemistry. In 2009, a team of scientists led by Luca Bindi discovered quasi-crystals in a mineral from the Russian Koryak Highlands - a compound of aluminum, copper and iron. Today this mineral is called icosahedrite. By measuring the content of various oxygen isotopes in the mineral with a mass spectrometer, scientists showed that this mineral did not originate on Earth. It formed about 4.5 billion years ago, at a time when solar system was in its infancy, and spent most of its time in the asteroid belt orbiting the Sun until some disturbance altered its orbit and brought it eventually to Earth.
// Rice. 41. Left: one of two quasi-crystalline lattices with exact fivefold symmetry. Right: Atomic model of an icosahedral aluminum-palladium-manganese quasicrystal
The Pythagorean theorem has been known to everyone since school days. An outstanding mathematician proved a great conjecture, which is currently used by many people. The rule sounds like this: the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. For many decades, not a single mathematician has been able to argue this rule. After all, Pythagoras walked for a long time towards his goal, so that as a result the drawings took place in everyday life.
- A small verse to this theorem, which was invented shortly after the proof, directly proves the properties of the hypothesis: "Pythagorean pants are equal in all directions." This two-line was deposited in the memory of many people - to this day the poem is remembered in calculations.
- This theorem was called "Pythagorean pants" due to the fact that when drawing in the middle, a right-angled triangle was obtained, on the sides of which there were squares. In appearance, this drawing resembled pants - hence the name of the hypothesis.
- Pythagoras was proud of the developed theorem, because this hypothesis differs from its similar ones by the maximum amount of evidence. Important: the equation was listed in the Guinness Book of Records due to 370 truthful evidence.
- The hypothesis was proved by a huge number of mathematicians and professors from different countries in many ways. The English mathematician Jones, soon after the announcement of the hypothesis, proved it with the help of a differential equation.
- At present, no one knows the proof of the theorem by Pythagoras himself. The facts about the proofs of a mathematician today are not known to anyone. It is believed that the proof of the drawings by Euclid is the proof of Pythagoras. However, some scientists argue with this statement: many believe that Euclid independently proved the theorem, without the help of the creator of the hypothesis.
- Current scientists have discovered that the great mathematician was not the first to discover this hypothesis.. The equation was known long before the discovery by Pythagoras. This mathematician managed only to reunite the hypothesis.
- Pythagoras did not give the equation the name "Pythagorean Theorem". This name was fixed after the "loud two-line". The mathematician only wanted the whole world to recognize and use his efforts and discoveries.
- Moritz Kantor - the great greatest mathematician found and saw notes with drawings on an ancient papyrus. Shortly thereafter, Cantor realized that this theorem had been known to the Egyptians as early as 2300 BC. Only then no one took advantage of it and did not try to prove it.
- Current scholars believe that the hypothesis was known as early as the 8th century BC. Indian scientists of that time discovered an approximate calculation of the hypotenuse of a triangle endowed with right angles. True, at that time no one could prove the equation for sure by approximate calculations.
- The great mathematician Bartel van der Waerden, after proving the hypothesis, concluded an important conclusion: “The merit of the Greek mathematician is considered not the discovery of direction and geometry, but only its justification. In the hands of Pythagoras were computational formulas that were based on assumptions, inaccurate calculations and vague ideas. However, the outstanding scientist managed to turn it into an exact science.”
- A famous poet said that on the day of the discovery of his drawing, he erected a glorious sacrifice to the bulls.. It was after the discovery of the hypothesis that rumors spread that the sacrifice of a hundred bulls "went wandering through the pages of books and publications." Wits joke to this day that since then all the bulls are afraid of a new discovery.
- Proof that Pythagoras did not come up with a poem about pants in order to prove the drawings he put forward: during the life of the great mathematician there were no pants yet. They were invented several decades later.
- Pekka, Leibniz and several other scientists tried to prove the previously known theorem, but no one succeeded.
- The name of the drawings "Pythagorean theorem" means "persuasion by speech". This is the translation of the word Pythagoras, which the mathematician took as a pseudonym.
- Reflections of Pythagoras on his own rule: the secret of what exists on earth lies in numbers. After all, a mathematician, relying on his own hypothesis, studied the properties of numbers, revealed evenness and oddness, and created proportions.
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Pythagorean pants The comic name of the Pythagorean theorem, which arose due to the fact that the squares built on the sides of a rectangle and diverging in different directions resemble the cut of trousers. I loved geometry ... and on entrance exam to the university even received praise from Chumakov, professor of mathematics, for explaining the properties of parallel lines and Pythagorean pants(N. Pirogov. Diary of an old doctor).
Phrasebook 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 Russian sayings
Pythagorean pants- A playful name for the Pythagorean theorem, which establishes the ratio between the areas of squares built on the hypotenuse and the legs of a right-angled triangle, which looks like the cut of pants in the drawings ... Dictionary of many expressions
Pythagorean pants (invent)- foreigner: about a gifted person Cf. This is the certainty of the sage. In ancient times, he probably would have invented Pythagorean pants ... Saltykov. Motley letters. Pythagorean pants (geom.): in a rectangle, the square of the hypotenuse is equal to the squares of the legs (teaching ... ... Michelson's Big Explanatory Phraseological Dictionary
Pythagorean pants are equal on all sides- The number of buttons is known. Why is the dick cramped? (roughly) about pants and the male sexual organ. Pythagorean pants are equal on all sides. To prove this, it is necessary to remove and show 1) about the Pythagorean theorem; 2) about wide pants ... Live speech. Dictionary of colloquial expressions
Pythagorean pants invent- Pythagorean pants (invent) foreigner. about a gifted person. Wed This is the undoubted sage. In ancient times, he probably would have invented Pythagorean pants ... Saltykov. Motley letters. Pythagorean pants (geom.): in a rectangle, the square of the hypotenuse ... ... Michelson's Big Explanatory Phraseological Dictionary (original spelling)
Pythagorean pants are equal in all directions- Joking proof of the Pythagorean theorem; also in jest about buddy's baggy trousers... Dictionary of folk phraseology
Adj., rude...
PYTHAGOREAN PANTS ARE EQUAL ON ALL SIDES (NUMBER OF BUTTONS IS KNOWN. WHY IS IT CLOSE? / TO PROVE THIS, IT IS NECESSARY TO REMOVE AND SHOW)- adj., rude ... Dictionary contemporary colloquial phraseological units and sayings
pants- 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 the bottom ... ... 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…
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 a 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 the plane around a point can be covered without gaps by equal regular polygons only three types: 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 fell on the second half of XIX- the beginning of the XX century. And if the first collections contained no more than two or three dozen proofs, then to late XIX 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 main character 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. This is where mathematics begins. 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 associated with his name (from Greek wordμαθημa - teaching, science), which united 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 expressible 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 issue. 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.
