A convenient and very accurate method used by land surveyors to draw perpendicular lines on the ground is as follows. Let it be required to draw a perpendicular to the line MN through point A (Fig. 13). Set aside from A in the direction of AM three times some distance a. Then three knots are tied on the cord, the distances between which are equal to 4a and 5a. Attaching the extreme knots to points A and B, pull the cord over the middle knot. The cord will be located in a triangle in which corner A is right.

This ancient method, apparently used thousands of years ago by the builders of the Egyptian pyramids, is based on the fact that each triangle, the sides of which are related as 3: 4: 5, according to the well-known theorem of Pythagoras, is rectangular, since

3 2 + 4 2 = 5 2 .

In addition to the numbers 3, 4, 5, there is, as is known, an innumerable set of positive integers a, b, c satisfying the relation

A 2 + b 2 = c 2.

They are called Pythagorean numbers. According to the Pythagorean theorem, such numbers can serve as the lengths of the sides of some right-angled triangle; therefore a and b are called "legs" and c - "hypotenuse".

It is clear that if a, b, c are three Pythagorean numbers, then pa, pb, pc, where p is an integer factor, are Pythagorean numbers. Conversely, if the Pythagorean numbers have a common factor, then all of them can be canceled by this common factor, and again you get a triple of Pythagorean numbers. Therefore, we will first investigate only triples of relatively prime Pythagorean numbers (the rest are obtained from them by multiplying by an integer factor p).

Let us show that in each of these triples a, b, c, one of the "legs" must be even and the other odd. Let's talk "by contradiction." If both "legs" a and b are even, then the number a 2 + b 2 will be even, and hence the "hypotenuse". This, however, contradicts the fact that the numbers a, b, c do not have common factors, since three even numbers have a common factor 2. Thus, at least one of the "legs" a, b is odd.

There remains one more possibility: both "legs" are odd, and the "hypotenuse" is even. It is not difficult to prove that this cannot be. Indeed: if the "legs" are of the form

2x + 1 and 2y + 1,

then the sum of their squares is

4x 2 + 4x + 1 + 4y 2 + 4y + 1 = 4 (x 2 + x + y 2 + y) + 2,

that is, it is a number that, when divided by 4, gives a remainder of 2. Meanwhile, the square of any even number must be divisible by 4 without a remainder. Hence, the sum of the squares of two odd numbers cannot be the square of an even number; in other words, our three numbers are not Pythagorean.

So, of the "legs" a, b, one is even and the other is odd. Therefore, the number a 2 + b 2 is odd, which means that the "hypotenuse" c is also odd.

Suppose, for definiteness, that the "leg" a is odd and b is even. From equality

a 2 + b 2 = c 2

we easily get:

A 2 = c 2 - b 2 = (c + b) (c - b).

The factors c + b and c - b on the right side are relatively prime. Indeed, if these numbers had a common prime factor other than unity, then the sum

(c + b) + (c - b) = 2c,

and the difference

(c + b) - (c - b) = 2b,

and work

(c + b) (c - b) = a 2,

that is, the numbers 2c, 2b, and a would have a common factor. Since a is odd, this factor is different from two, and therefore the numbers a, b, c have the same common factor, which, however, cannot be. The resulting contradiction shows that the numbers c + b and c - b are coprime.

But if the product of coprime numbers is an exact square, then each of them is a square, i.e.


Having solved this system, we find:

C = (m 2 + n 2) / 2, b = (m 2 - n 2) / 2, and 2 = (c + b) (c - b) = m 2 n 2, a = mn.

So, the considered Pythagorean numbers have the form

A = mn, b = (m 2 - n 2) / 2, c = (m 2 + n 2) / 2.

where m and n are some relatively prime odd numbers. The reader can easily be convinced of the opposite: for any odd type, the written formulas give three Pythagorean numbers a, b, c.

Here are several triples of Pythagorean numbers obtained for different types:

For m = 3, n = 1 3 2 + 4 2 = 5 2 for m = 5, n = 1 5 2 + 12 2 = 13 2 for m = 7, n = 1 7 2 + 24 2 = 25 2 for m = 9, n = 1 9 2 + 40 2 = 41 2 with m = 11, n = 1 11 2 + 60 2 = 61 2 with m = 13, n = 1 13 2 + 84 2 = 85 2 with m = 5 , n = 3 15 2 + 8 2 = 17 2 with m = 7, n = 3 21 2 + 20 2 = 29 2 with m = 11, n = 3 33 2 + 56 2 = 65 2 with m = 13, n = 3 39 2 + 80 2 = 89 2 with m = 7, n = 5 35 2 + 12 2 = 37 2 with m = 9, n = 5 45 2 + 28 2 = 53 2 with m = 11, n = 5 55 2 + 48 2 = 73 2 with m = 13, n = 5 65 2 + 72 2 = 97 2 with m = 9, n = 7 63 2 + 16 2 = 65 2 with m = 11, n = 7 77 2 + 36 2 = 85 2

(All other triples of Pythagorean numbers either have common factors or contain numbers greater than one hundred.)

»Distinguished Professor of Mathematics at the University of Warwick, the famous 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. The simplest of them has the longest side of length 5, the others - 3 and 4. There are 5 regular polyhedra in total. A fifth-degree equation cannot be solved using fifth-degree roots - or any other roots. Lattices on the plane and in three-dimensional space do not have five-lobed symmetry of rotation; therefore, such symmetries are absent in crystals either. However, they can be found in lattices in four-dimensional space and in interesting structures known as quasicrystals.

Hypotenuse of the smallest Pythagorean triplet

The Pythagorean theorem says that the longest side of a right triangle (the notorious hypotenuse) relates to the other two sides of this triangle very simply and beautifully: the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Traditionally, we call this theorem by the name of 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 laws. The 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 doing. We don't even know if the Pythagoreans could prove the Pythagorean theorem or if they simply believed it to be true. Or, most likely, they had compelling evidence of its truth, which would nonetheless not be sufficient for what we consider to be evidence today.

Proofs of Pythagoras

We find the first known proof of the Pythagorean theorem in Euclid's Elements. This is a rather complex proof using a drawing in which Victorian schoolchildren would immediately recognize "Pythagorean pants"; the drawing really resembles underpants drying on a rope. There are literally hundreds of other pieces of evidence known, most of which make the claim being argued more obvious.


// Rice. 33. Pythagorean Pants

One of the simplest proofs is a kind of math puzzle. Take any right triangle, make four copies of it and collect them inside a square. With one stacking, we see a square on the hypotenuse; on the other, squares on the other two sides of the triangle. At the same time, 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 exclude the triangles.

Perigalle's dissection is another proof puzzle.


// Rice. 35. Dissection of Perigalle

There is also a proof of the theorem using the packing of squares in the plane. Perhaps this is how the Pythagoreans or their unknown predecessors discovered this theorem. If you look at how an oblique square overlaps two other squares, you can see how to cut a large square into pieces and then fold the two smaller squares out of them. You can also see right-angled triangles, the sides of which give the dimensions of the three involved squares.


// Rice. 36. Paving proof

There is interesting evidence using similar triangles in trigonometry. At least fifty different pieces of evidence 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 triplet defines a right-angled triangle with integer sides.

The smallest hypotenuse of the Pythagorean triplet 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 it

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

Euclid knew that there are an infinite number of different variants of Pythagorean triplets, and gave what can be called a formula for finding them all. Later, Diophantus of Alexandria proposed a simple recipe that basically coincided with the Euclidean one.

Take any two natural numbers and calculate:

their doubled work;

the difference between 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 triangle 3-4-5. 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 most 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 1235-1932-2293 triangle.

But these numbers work too:

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

There is another feature in the Diophantine rule, which 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 go to the language of algebra, the rule takes the following form: let u, v and k be natural numbers. Then a right-angled 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 triplets.

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. The faces converge on lines called edges; edges meet at points called vertices.

The culmination of 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:

a tetrahedron with four triangular faces, four vertices and six edges;

a 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;

an 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 resemble in shape the very 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 icosahedron in crystals, although irregular dodecahedrons and icosahedrons sometimes come across there. True dodecahedrons can arise in the form of quasicrystals, which are similar in every way to crystals, except that their atoms do not form a periodic lattice.


// Rice. 38. Haeckel's drawings: radiolarians in the form of regular polyhedra


// Rice. 39. Sweep of regular polyhedra

It can be interesting to make models of regular polyhedra out of paper by cutting out a set of interconnected faces beforehand - this is called a polyhedron unfolding; the reamer is folded along the edges and the corresponding edges are glued together. It is useful to add an additional glue pad to one of the edges of each such pair, as shown in fig. 39. If there is no such area, you can use adhesive tape.

Equation of the fifth degree

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

In general terms, the fifth-degree equation looks like this:

ax5 + bx4 + cx3 + dx2 + ex + f = 0.

The problem is to find a formula for the solutions of 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 degrees should appear in it. Again, we can safely assume that such a formula, if it exists, will be very, very difficult.

This assumption turned out to be wrong in the end. Indeed, no such formula exists; at least there is no formula of the coefficients a, b, c, d, e, and f, using addition, subtraction, multiplication and division, and extraction of roots. So 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 long time to figure them out.

The first sign of the problem was that no matter how hard mathematicians tried to find such a formula, no matter how smart they were, they invariably failed. For some time, everyone believed that the reasons lay in the incredible complexity of the formula. It was believed that no one simply can properly understand this algebra. However, over time, some mathematicians began to doubt that such a formula existed at all, and in 1823 Niels Hendrik Abel was able to prove the opposite. There is no such formula. Soon thereafter, Evariste Galois found a way to determine whether an equation of one degree or another - 5th, 6th, 7th, in general, any - was solvable using this kind of formula.

The takeaway from all this is simple: the number 5 is special. You can solve algebraic equations (using nth roots for different values ​​of n) for degrees 1, 2, 3, and 4, but not for 5th degree. This is where the obvious pattern ends.

It comes as no surprise to anyone that equations of powers greater than 5 behave even worse; in particular, the same difficulty is associated with them: there are no general formulas for their solution. This does not mean that the equations have no solutions; it also does not mean 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 compass. The answer exists, but the listed methods are insufficient 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 periodically repeats 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 horizontally, sometimes with a shift from one piece of wallpaper to the next. Essentially, wallpaper is a two-dimensional crystal.

There are 17 varieties of flat wallpaper (see chapter 17). They differ in the types of symmetry, that is, in the ways of rigidly shifting the drawing so that it exactly lies on itself in its original position. The types of symmetry include, in particular, various variants of symmetry of rotation, where the picture should be rotated by a certain angle around a certain point - the center of symmetry.

Rotation symmetry order is how many times the body can be rotated to a full circle so that all the details of the drawing return to their original positions. For example, a 90 ° rotation is a 4th order rotation symmetry *. The list of possible types of rotation symmetry in the crystal lattice again indicates the unusualness of the number 5: it is not there. There are 2, 3, 4, and 6th order rotational symmetries, but no wallpaper has 5th order rotational symmetry. Rotation symmetry of the order of more than 6 in crystals also does not exist, but the first violation of the sequence occurs nevertheless at the number 5.

The same thing happens with crystallographic systems in three-dimensional space. Here the grid repeats itself in three independent directions. There are 219 different types of symmetry, or 230, if we consider the mirror image of a drawing as a separate version of it - despite the fact that in this case there is no mirror symmetry. Again, rotation symmetries of orders 2, 3, 4, and 6 are observed, but not 5. This fact is called crystallographic constraint.

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


// Rice. 40. Crystal lattice of table salt. Dark balls represent sodium atoms, light ones - chlorine atoms

Quasicrystals

Although rotational symmetry of the 5th order in 2D and 3D lattices is not possible, it can exist in slightly less regular structures known as quasicrystals. Using Kepler's sketches, Roger Penrose discovered planar systems with a more general type of fivefold symmetry. They are called quasicrystals.

Quasicrystals exist in nature. In 1984, Daniel Shechtman discovered that an alloy of aluminum and manganese can form quasicrystals; initially, crystallographers greeted his message with some skepticism, but later the discovery was confirmed, and in 2011 Shechtman was awarded the Nobel Prize in Chemistry. In 2009, a team of scientists led by Luka Bindi discovered quasicrystals in a mineral from the Russian Koryak Highlands - a combination of aluminum, copper and iron. Today this mineral is called icosahedrite. Having measured the content of various oxygen isotopes in the mineral with a mass spectrometer, scientists have shown that this mineral did not originate on Earth. It formed about 4.5 billion years ago, at a time when the solar system was just nascent, and spent most of the time in the asteroid belt, orbiting the sun, until some disturbance changed its orbit and led it eventually to The earth.


// Rice. 41. Left: one of two quasicrystalline lattices with exactly fivefold symmetry. Right: atomic model of an icosahedral aluminum-palladium-manganese quasicrystal

Educational: to study a number of Pythagorean triplets, develop an algorithm for their application in various situations, draw up a memo on their use.
  • Educational: the formation of a conscious attitude to learning, the development of cognitive activity, the culture of educational work.
  • Developing: development of geometric, algebraic and numerical intuition, intelligence, observation, memory.

    • During the classes

    I. Organizational moment

    II. Explanation of the new material

    Teacher: The riddle of the attractive power of the Pythagorean triplets has long worried mankind. The unique properties of the Pythagorean triplets explain their special role in nature, music, mathematics. The Pythagorean spell, the Pythagorean theorem, remains in the brains of millions, if not billions, of people. This is a fundamental theorem that every student is forced to memorize. While the Pythagorean Theorem is understandable for ten-year-olds, it is an inspiring start to a problem that has fiasco the greatest minds in the history of mathematics, Fermat's Theorem. Pythagoras from the island of Samos (see. Annex 1 , slide 4) was one of the most influential and yet enigmatic figures in mathematics. Since no reliable reports of his life and work have survived, his life has become shrouded in myths and legends, and historians find it difficult to separate fact from fiction. There is no doubt, however, that Pythagoras developed the idea of ​​the logic of numbers and that it was to him that we owe the first golden age of mathematics. Thanks to his genius, numbers were no longer used only for counting and calculations and were appreciated for the first time. Pythagoras studied the properties of certain classes of numbers, the relationship between them and the figures that form the numbers. Pythagoras understood that numbers exist independently of the material world, and therefore the study of numbers is not affected by the inaccuracy of our senses. This meant that Pythagoras gained the ability to discover truths independent of anyone's opinion or prejudice. Truths are more absolute than any previous knowledge. Based on the studied literature concerning Pythagorean triplets, we will be interested in the possibility of using Pythagorean triplets in solving trigonometry problems. Therefore, we will set ourselves a goal: to study a number of Pythagorean triplets, develop an algorithm for their use, draw up a memo on their use, conduct research on their use in various situations.

    Triangle ( slide 14), whose sides are equal to the Pythagorean numbers, is rectangular. In addition, any such triangle is Heronic, i.e. such that all sides and area are integers. The simplest of them is the Egyptian triangle with sides (3, 4, 5).

    Let's compose a series of Pythagorean triplets by multiplying the numbers (3, 4, 5) by 2, by 3, by 4. We will receive a series of Pythagorean triplets, sort them in ascending order of the maximum number, select the primitive ones.

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

    III. During the classes

    1. Let's spin around the tasks:

    1) Using the relations between trigonometric functions of the same argument, find if

    it is known that .

    2) Find the value of the trigonometric functions of the angle?, If it is known that:

    3) The system of training tasks on the topic "Addition formulas"

    knowing that sin = 8/17, cos = 4/5, and are the angles of the first quarter, find the value of the expression:

    knowing that and are the angles of the second quarter, sin = 4/5, cos = - 15/17, find:.

    4) The system of training tasks on the topic "Double angle formulas"

    a) Let sin = 5/13, be the angle of the second quarter. Find sin2, cos2, tg2, ctg2.

    b) It is known that tg? = 3/4, is the angle of the third quarter. Find sin2, cos2, tg2, ctg2.

    c) It is known that, 0< < . Найдите sin, cos, tg, ctg.

    d) It is known that , < < 2. Найдите sin, cos, tg.

    e) Find tg (+) if it is known that cos = 3/5, cos = 7/25, where and are the angles of the first quarter.

    f) Find , Is the angle of the third quarter.

    We solve the problem in the traditional way using the basic trigonometric identities, and then we solve the same problems in a more rational way. To do this, we use an algorithm for solving problems using Pythagorean triples. We draw up a memo for solving problems using Pythagorean triples. To do this, recall the definition of sine, cosine, tangent and cotangent, an acute angle of a right-angled triangle, depict it, depending on the conditions of the problem on the sides of a right-angled triangle, correctly arrange the Pythagorean triples ( rice. 1). We write down the ratio and place signs. The algorithm has been worked out.

    Picture 1

    Algorithm for solving problems

    Review (study) theoretical material.

    Know by heart the primitive Pythagorean triplets and, if necessary, be able to construct new ones.

    Apply the Pythagorean theorem for points with rational coordinates.

    Know the definition of sine, cosine, tangent and cotangent of an acute angle of a right triangle, be able to draw a right triangle and, depending on the condition of the problem, correctly arrange the Pythagorean triplets on the sides of the triangle.

    Know the signs of sine, cosine, tangent and cotangent, depending on their location in the coordinate plane.

    Necessary requirements:

    1. know what signs sine, cosine, tangent, cotangent have in each of the quarters of the coordinate plane;
    2. know the definition of sine, cosine, tangent and cotangent of an acute angle of a right triangle;
    3. know and be able to apply the Pythagorean theorem;
    4. know the basic trigonometric identities, addition formulas, double angle formulas, half-argument formulas;
    5. know the casting formulas.

    Taking into account the above, fill in the table ( Table 1). It must be filled in by following the definition of sine, cosine, tangent and cotangent, or using the Pythagorean theorem for points with rational coordinates. In this case, it is always necessary to remember the signs of the sine, cosine, tangent and cotangent, depending on their location in the coordinate plane.

    Table 1

    Triples of numbers sin cos tg ctg
    (3, 4, 5) Part I
    (6, 8, 10) II part - -
    (5, 12, 13) III part - -
    (8, 15, 17) IV p. - - -
    (9, 40, 41) Part I

    For successful work, you can use the memo on the use of Pythagorean triplets.

    table 2

    (3, 4, 5), (6, 8, 10), (5, 12, 13) , (9, 12, 13), (8, 15, 17) , (12, 16, 20), (15, 20, 25), (7, 24, 25) , (10, 24, 26), (20, 21, 29) , (18, 24, 30), (16, 30, 34), (21, 28, 35), (12, 35, 37), (15, 36, 39), (24, 32, 40), (9, 40, 41) , (14, 48, 50), (30, 40, 50), …

    2. We solve together.

    1) Problem: find cos, tg and ctg if sin = 5/13 if is the angle of the second quarter.

    Vitaly worm

    Download:

    Preview:

    Competition of scientific projects for schoolchildren

    Within the framework of the regional scientific and practical conference "Eureka"

    Small Academy of Sciences for students of the Kuban

    Study of Pythagorean numbers

    Section mathematics.

    Worm Vitaly Gennadievich, grade 9

    MOBU SOSH №14

    Korenovsky district

    Art. Zhuravskaya

    Supervisor:

    Manko Galina Vasilievna

    Mathematic teacher

    MOBU SOSH №14

    Korenovsk 2011

    Wormyak Vitaly Gennadievich

    Pythagorean numbers

    Annotation.

    Research topic:Pythagorean numbers

    Research objectives:

    Research objectives:

    • Identification and development of mathematical abilities;
    • Expansion of mathematical representation on a given topic;
    • Formation of a sustainable interest in the subject;
    • Development of communicative and general educational skills of independent work, the ability to lead a discussion, reason, etc .;
    • Formation and development of analytical and logical thinking;

    Research methods:

    • Use of Internet resources;
    • Referring to reference literature;
    • Carrying out an experiment;

    Output:

    • This work can be used in a geometry lesson as additional material, for conducting elective courses or electives in mathematics, as well as in extracurricular work in mathematics;

    Wormyak Vitaly Gennadievich

    Krasnodar Territory, village Zhuravskaya, MOBU Secondary School No. 14, Grade 9

    Pythagorean numbers

    Scientific adviser: Manko Galina Vasilievna, teacher of mathematics MOBU secondary school №14

    1. Introduction ………………………………………………………………… 3
    2. Main part

    2.1 Historical page ………………………………………………… 4

    2.2 Proof of even and odd legs ... ... ... ............................. 5-6

    2.3 Derivation of a pattern for finding

    Pythagorean numbers ………………………………………………………… 7

    2.4 Properties of Pythagorean numbers ……………………………………………… 8

    3. Conclusion …………………………………………………………………… 9

    4.List of used sources and literature …………………… 10

    Applications ................................................. .................................................. ......eleven

    Appendix I …………………………………………………………………… 11

    Appendix II ……………………………………………………………… ..13

    Wormyak Vitaly Gennadievich

    Krasnodar Territory, village Zhuravskaya, MOBU Secondary School No. 14, Grade 9

    Pythagorean numbers

    Scientific adviser: Manko Galina Vasilievna, teacher of mathematics MOBU secondary school №14

    Introduction

    I heard about Pythagoras and his life in the fifth grade at a mathematics lesson, and I was interested in the statement "Pythagoras pants are equal in all directions." While studying the Pythagorean theorem, I was interested in the Pythagorean numbers.purpose of the study: Learn more about the Pythagorean theorem and "Pythagorean numbers".

    Relevance of the topic... The value of the Pythagorean theorem and Pythagorean triplets has been proven by many scientists of the world for many centuries. The problem that will be discussed in my work looks quite simple because it is based on a mathematical statement that everyone knows - the Pythagorean theorem: in any right-angled triangle, the square built on the hypotenuse is equal to the sum of the squares built on the legs. Now triples of natural numbers x, y, z, for which x 2 + y 2 = z 2 , it is customary to callPythagorean triplets... It turns out that the Pythagorean triplets were already known in Babylon. Gradually, Greek mathematicians also found them.

    The purpose of this work

    1. Explore Pythagorean numbers;
    2. Understand how Pythagorean numbers are obtained;
    3. Find out what properties Pythagorean numbers have;
    4. Experimentally, construct perpendicular straight lines on the ground using Pythagorean numbers;

    In accordance with the purpose of the work, a number of the following tasks:

    1. To study more deeply the history of the Pythagorean theorem;

    2. Analysis of the universal properties of Pythagorean triplets.

    3. Analysis of the practical application of Pythagorean triplets.

    Object of study: Pythagorean triplets.

    Subject of study: maths .

    Research methods: - Use of Internet resources; -Reference to reference literature; -Conducting an experiment;

    Theoretical significance:the role played by the discovery of Pythagorean triplets in science; practical application of the discovery of Pythagoras in human life.

    Applied valueresearch consists in the analysis of literary sources and systematization of facts.

    Wormyak Vitaly Gennadievich

    Krasnodar Territory, village Zhuravskaya, MOBU Secondary School No. 14, Grade 9

    Pythagorean numbers

    Scientific adviser: Manko Galina Vasilievna, teacher of mathematics MOBU secondary school №14

    From the history of Pythagorean numbers.

    • Ancient China:

    Chu-pei's math book:[ 2]

    "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".

    • Ancient Egypt: [2]

    Cantor (the largest German historian of mathematics) believes that equality 3 ² + 4 ² = 5² was already known to the Egyptians around 2300 BC. e., at the time of the king Amenemkhet (according to papyrus 6619 of the Berlin Museum). According to Kantor harpedonapts, or "rope tensioners", built right angles using right-angled triangles with sides 3; 4 and 5.

    • Babylonia: [3]

    “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 notions have become an exact science. "

    • History of the Pythagorean theorem:,

    Although this theorem is associated with the name of Pythagoras, it was known long before him.

    In Babylonian texts, it is found 1200 years before Pythagoras.

    Apparently, he was the first to find proof of it. In this regard, the following entry was made: "... when he discovered that in a right-angled triangle the hypotenuse has a correspondence with the legs, he sacrificed a bull made of wheat dough."

    Wormyak Vitaly Gennadievich

    Krasnodar Territory, village Zhuravskaya, MOBU Secondary School No. 14, Grade 9

    Pythagorean numbers

    Scientific adviser: Manko Galina Vasilievna, teacher of mathematics MOBU secondary school №14

    Study of the Pythagorean numbers.

    • Each triangle, the sides are related as 3: 4: 5, according to the well-known Pythagorean theorem, - rectangular, since

    3 2 + 4 2 = 5 2.

    • In addition to the numbers 3,4 and 5, there is, as is known, an infinite set of positive integers a, b, and c satisfying the relation
    • A 2 + b 2 = c 2.
    • These numbers are calledPythagorean numbers

    Pythagorean triplets have been known for a very long time. In the architecture of ancient Sopotamian tombstones, there is an isosceles triangle composed of two rectangular ones with sides of 9, 12 and 15 cubits. The pyramids of Pharaoh Snephru (XXVII century BC) were built using triangles with sides 20, 21 and 29, as well as 18, 24 and 30 dozen Egyptian cubits.[ 1 ]

    A right-angled triangle with legs 3, 4 and a hypotenuse 5 is called the Egyptian triangle. The area of ​​this triangle is equal to a perfect number 6. The perimeter is equal to 12 - a number that was considered a symbol of happiness and prosperity.

    Using a rope divided by knots into 12 equal parts, the ancient Egyptians built a right-angled triangle and a right angle. A convenient and very accurate method used by land surveyors to draw perpendicular lines on the ground. It is necessary to take a cord and three pegs, the cord is placed in a triangle so that one side consists of 3 parts, the second of 4 shares and the last of five such shares. The cord will be located in a triangle with a right angle.

    This ancient method, apparently used thousands of years ago by the builders of the Egyptian pyramids, is based on the fact that each triangle, the sides of which are related as 3: 4: 5, according to the Pythagorean theorem, is rectangular.

    Euclid, Pythagoras, Diophantus and many others were engaged in finding the Pythagorean triplets.[ 1]

    It is clear that if (x, y, z ) Is a Pythagorean triplet, then for any natural k triple (kx, ky, kz) will also be a Pythagorean triplet. In particular, (6, 8, 10), (9, 12, 15), etc. are Pythagorean triplets.

    As the numbers increase, Pythagorean triplets are less and less common and more and more difficult to find. The Pythagoreans invented a method of finding

    such triplets and, using them, proved that there are infinitely many Pythagorean triplets.

    Triples that have no common divisors greater than 1 are called simplest.

    Let's consider some properties of Pythagorean triplets.[ 1]

    According to the Pythagorean theorem, these numbers can serve as the lengths of some right-angled triangle; therefore a and b are called "legs", and c - "hypotenuse".
    It is clear that if a, b, c are a triple of Pythagorean numbers, then pa, pb, pc, where p is an integer factor, are Pythagorean numbers.
    The converse is also true!
    Therefore, we will first investigate only triples of relatively prime Pythagorean numbers (the rest are obtained from them by multiplying by an integer factor p).

    Let us show that in each of these triples a, b, c, one of the "legs" must be even and the other odd. We will argue "by contradiction." If both "legs" a and b are even, then the number a will be even 2 + in 2 , and hence the "hypotenuse". But this contradicts the fact that the numbers a, b and c do not have common factors, since three even numbers have a common factor of 2. Thus, at least one of the "legs" a and b is odd.

    There remains one more possibility: both "legs" are odd, and the "hypotenuse" is even. It is easy to prove that this cannot be, since if the "legs" are of the form 2x + 1 and 2y + 1, then the sum of their squares is

    4x 2 + 4x + 1 + 4y 2 + 4y +1 = 4 (x 2 + x + y 2 + y) +2, i.e. is a number that, when divided by 4, gives a remainder of 2. Meanwhile, the square of any even number must be divisible by 4 without a remainder.

    This means that the sum of the squares of two odd numbers cannot be the square of an even number; in other words, our three numbers are not Pythagorean.

    OUTPUT:

    So, from the "legs" a, into one even, and the other odd. Therefore, the number a 2 + in 2 is odd, which means that the "hypotenuse" is also odd.

    Pythagoras found formulas that in modern symbolism can be written as follows: a = 2n + 1, b = 2n (n + 1), c = 2 n 2 + 2n + 1, where n is an integer.

    These numbers are Pythagorean triplets.

    Wormyak Vitaly Gennadievich

    Krasnodar Territory, village Zhuravskaya, MOBU Secondary School No. 14, Grade 9

    Pythagorean numbers

    Scientific adviser: Manko Galina Vasilievna, teacher of mathematics MOBU secondary school №14

    Derivation of patterns for finding the Pythagorean numbers.

    Here are the following Pythagorean triplets:

    • 3, 4, 5; 9+16=25.
    • 5, 12, 13; 25+144=225.
    • 7, 24, 25; 49+576=625.
    • 8, 15, 17; 64+225=289.
    • 9, 40, 41; 81+1600=1681.
    • 12, 35, 37; 144+1225=1369.
    • 20, 21, 29; 400+441=881

    It is easy to see that multiplying each of the numbers of the Pythagorean triple by 2, 3, 4, 5, etc., we get the following triples.

    • 6, 8, 10;
    • 9,12,15.
    • 12, 16, 20;
    • 15, 20, 25;
    • 10, 24, 26;
    • 18, 24, 30;
    • 16, 30, 34;
    • 21, 28, 35;
    • 15, 36, 39;
    • 24, 32, 40;
    • 14, 48, 50;
    • 30, 40, 50, etc.

    They are also Pythagorean numbers /

    Wormyak Vitaly Gennadievich

    Krasnodar Territory, village Zhuravskaya, MOBU Secondary School No. 14, Grade 9

    Pythagorean numbers

    Scientific adviser: Manko Galina Vasilievna, teacher of mathematics MOBU secondary school №14

    Properties of Pythagorean numbers.

    • When looking at Pythagorean numbers, I saw a number of properties:
    • 1) One of the Pythagorean numbers must be a multiple of three;
    • 2) The other of them must be a multiple of four;
    • 3) And the third of the Pythagorean numbers must be a multiple of five;

    Wormyak Vitaly Gennadievich

    Krasnodar Territory, village Zhuravskaya, MOBU Secondary School No. 14, Grade 9

    Pythagorean numbers

    Scientific adviser: Manko Galina Vasilievna, teacher of mathematics MOBU secondary school №14

    Conclusion.

    Geometry, like other sciences, arose from the needs of practice. The word "geometry" itself is Greek, translated means "surveying".

    People very early on were faced with the need to measure land. Already 3-4 thousand years BC. every piece of fertile land in the valleys of the Nile, Euphrates and Tigris, the rivers of China was important for the lives of people. This required a certain amount of geometric and arithmetic knowledge.

    Gradually, people began to measure and study the properties of more complex geometric shapes.

    In both Egypt and Babylon, colossal temples were built, the construction of which could only be carried out on the basis of preliminary calculations. Water pipelines were also built. All this required drawings and calculations. By this time, special cases of the Pythagorean theorem were well known, they already knew that if we take triangles with sides x, y, z, where x, y, z are integers such that x 2 + y 2 = z 2 , then these triangles will be rectangular.

    All this knowledge was directly applied in many spheres of human life.

    So until now, the great discovery of the scientist and philosopher of antiquity Pythagoras finds direct application in our life.

    Construction of houses, roads, spaceships, cars, machine tools, oil pipelines, airplanes, tunnels, subways and much, much more. Pythagorean triplets find direct application in the design of many things that surround us in everyday life.

    And the minds of scientists continue to look for new versions of proofs of the Pythagorean theorem.

    • V As a result of my work, I managed to:
    • 1. Learn more about Pythagoras, his life, the brotherhood of the Pythagoreans.
    • 2. Get acquainted with the history of the Pythagorean theorem.
    • 3. Learn about Pythagorean numbers, their properties, learn to find them and apply them in practice.

    Wormyak Vitaly Gennadievich

    Krasnodar Territory, village Zhuravskaya, MOBU Secondary School No. 14, Grade 9

    Pythagorean numbers

    Scientific adviser: Manko Galina Vasilievna, teacher of mathematics MOBU secondary school №14

    Literature.

    1. Interesting algebra. ME AND. Perelman (p. 117-120)
    2. www.garshin.ru
    3. image.yandex.ru

    4. Anosov D.V. A look at mathematics and something from it. - M .: MTsNMO, 2003.

    5. Children's encyclopedia. - M .: Publishing house of the Academy of Pedagogical Sciences of the RSFSR, 1959.

    6. Stepanova L.L. Selected chapters of elementary number theory. - M .: Prometheus, 2001.

    7. V. Serpinsky Pythagorean triangles. - M .: Uchpedgiz, 1959.S. 111

    Research progress Historical page; Pythagorean theorem; Prove that one of the "legs" must be even and the other odd; Derivation of patterns for finding Pythagorean numbers; Reveal the properties of Pythagorean numbers;

    Introduction I heard about Pythagoras and his life in the fifth grade at a mathematics lesson, and I was interested in the statement "Pythagoras pants are equal in all directions." While studying the Pythagorean theorem, I was interested in the Pythagorean numbers. I set a research goal: to learn more about the Pythagorean theorem and "Pythagorean numbers".

    Truth will be eternal, how soon will a weak person cognize It! And now the theorem of Pythagoras Verne, as in his distant century

    From the history of Pythagorean numbers. Ancient China Mathematical book Chu-pei: "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".

    Pythagorean numbers among the ancient Egyptians Cantor (the largest German historian of mathematics) believes that the equality 3 ² + 4 ² = 5² was already known to the Egyptians around 2300 BC. BC, during the time of King Amenemhat (according to papyrus 6619 of the Berlin Museum). According to Cantor, the harpedonapts, or "rope-tensioners", built right angles using right-angled triangles with sides 3; 4 and 5.

    The theorem of Pythagoras in Babylonia “The merit of the first Greek mathematicians, such as Thales, Pythagoras and the Pythagoreans, was not the discovery of mathematics, but its systematization and substantiation. In their hands, computational recipes based on vague notions have become an exact science. "

    Each triangle, the sides are related as 3: 4: 5, according to the well-known theorem of Pythagoras, - rectangular, since 3 2 + 4 2 = 5 2. In addition to the numbers 3,4 and 5, there is, as you know, an infinite set of positive integers a , в and с, satisfying the relation А 2 + в 2 = с 2. These numbers are called Pythagorean numbers

    According to the Pythagorean theorem, these numbers can serve as the lengths of some right-angled triangle; therefore a and b are called "legs", and c - "hypotenuse". It is clear that if a, b, c are three Pythagorean numbers, then pa, pb, pc, where p is an integer factor, are Pythagorean numbers. The converse is also true! Therefore, we will first investigate only triples of coprime Pythagorean numbers (the rest are obtained from them by multiplying by an integer factor p)

    Output! So from the numbers a and to one is even, and the other is odd, which means that the third number is also odd.

    Here are the following Pythagorean triplets: 3, 4, 5; 9 + 16 = 25. 5, 12, 13; 25 + 144 = 169. 7, 24, 25; 49 + 576 = 625. 8, 15, 17; 64 + 225 = 289. 9, 40, 41; 81 + 1600 = 1681. 12, 35, 37; 144 + 1225 = 1369. 20, 21, 29; 400 + 441 = 841

    It is easy to see that multiplying each of the numbers of the Pythagorean triple by 2, 3, 4, 5, etc., we get the following triples. 6, 8, 10; 9,12,15. 12, 16, 20; 15, 20, 25; 10, 24, 26; 18, 24, 30; 16, 30, 34; 21, 28, 35; 15, 36, 39; 24, 32, 40; 14, 48, 50; 30, 40, 50, etc. They are also Pythagorean numbers.

    Properties of Pythagorean numbers When considering the Pythagorean numbers, I saw a number of properties: 1) One of the Pythagorean numbers must be a multiple of three; 2) one of them must be a multiple of four; 3) And the other of the Pythagorean numbers must be a multiple of five;

    Practical application of Pythagorean numbers

    Conclusion: As a result of my work, I managed to 1. Learn more about Pythagoras, his life, the brotherhood of the Pythagoreans. 2. Get acquainted with the history of the Pythagorean theorem. 3. Learn about Pythagorean numbers, their properties, learn to find them. Experimentally - experimentally postpone a right angle using Pythagorean numbers.

    Pythagorean triples of numbers

    Creative work

    pupil 8 "A" class

    MAOU "Gymnasium No. 1"

    Oktyabrsky district of Saratov

    Panfilov Vladimir

    Supervisor - a teacher of mathematics of the highest category

    Grishina Irina Vladimirovna


    Content

    Introduction ………………………………………………………………………………… 3

    The theoretical part of the work

    Finding the main Pythagorean triangle

    (formulas of ancient Hindus) ……………………………………………………………… 4

    The practical part of the work

    Compilation of Pythagorean triplets in various ways ..................... 6

    An important property of Pythagorean triangles …………………………………… ... 8

    Conclusion …………………………………………………………………………… .... 9

    Literature…. ……………………………………………………………………… ... 10

    Introduction

    This academic year, in mathematics lessons, we studied one of the most popular theorems in geometry - the Pythagorean theorem. The Pythagorean theorem is applied in geometry at every step; it has found wide application in practice and everyday life. But, in addition to the theorem itself, we also studied the theorem converse to the Pythagorean theorem. In connection with the study of this theorem, we got acquainted with the Pythagorean triples of numbers, i.e. with sets of 3 natural numbersa , b andc , for which the following relation is valid: = + ... Such sets include, for example, the following triplets:

    3,4,5; 5,12,13; 7,24,25; 8,15,17; 20,21,29; 9,40,41; 12,35,37

    I immediately had questions: how many Pythagorean triplets can you think of? How do you compose them?

    In our geometry textbook, after the presentation of the theorem inverse to the Pythagorean theorem, an important remark was made: it can be proved that the legsa andb and hypotenusewith right-angled triangles, the lengths of the sides of which are expressed in natural numbers, can be found by the formulas:

    a = 2kmn b = k ( - ) c = k ( + , (1)

    wherek , m , n - any natural numbers, andm > n .

    Naturally, the question arises - how to prove these formulas? And is it only by these formulas that Pythagorean triplets can be compiled?

    In my work, I made an attempt to answer the questions that I had.

    The theoretical part of the work

    Finding the main Pythagorean triangle (formulas of the ancient Hindus)

    First, we prove formulas (1):

    Let us denote the lengths of the legs throughNS andat , and the length of the hypotenuse throughz ... By the Pythagorean theorem, we have the equality:+ = .(2)

    This equation is called the Pythagorean equation. The study of Pythagorean triangles is reduced to solving equation (2) in natural numbers.

    If each side of some Pythagorean triangle is increased by the same number of times, then we get a new right-angled triangle, similar to this one with sides expressed in natural numbers, i.e. again the Pythagorean triangle.

    Among all such triangles, there is the smallest one, it is easy to guess that this will be a triangle, the sides of whichNS andat expressed by relative prime numbers

    (GCD (x, y )=1).

    We call such a Pythagorean trianglethe main .

    Finding the main Pythagorean triangles.

    Let the triangle (x , y , z ) - the main Pythagorean triangle. The numbersNS andat - are mutually simple, and therefore both cannot be even. Let us prove that they cannot be both and odd. To do this, note thatthe square of an odd number when divided by 8 gives a remainder of 1. Indeed, any odd natural number can be represented as2 k -1 , wherek belongsN .

    Hence: = -4 k +1 = 4 k ( k -1)+1.

    The numbers( k -1) andk - consecutive, one of them is necessarily even. Then the expressionk ( k -1) divided by2 , 4 k ( k -1) is divisible by 8, which means that the number dividing by 8 gives a remainder of 1.

    The sum of the squares of two odd numbers gives, when divided by 8, in the remainder of 2, therefore, the sum of the squares of two odd numbers is an even number, but not a multiple of 4, and therefore this numbercannot be the square of a natural number.

    So, equality (2) cannot hold ifx andat both are odd.

    Thus, if the Pythagorean triangle (x, y, z ) is the main one, then among the numbersNS andat one must be even and the other odd. Let the number y be even. The numbersNS andz odd (oddz follows from equality (2)).

    From the equation+ = we get that= ( z + x )( z - x ) (3).

    The numbersz + x andz - x as the sum and difference of two odd numbers - the numbers are even, and therefore (4):

    z + x = 2 a , z - x = 2 b , wherea andb belongN .

    z + x =2 a , z - x = 2 b ,

    z = a + b , x = a - b. (5)

    It follows from these equalities thata andb - coprime numbers.

    Let us prove this by arguing by contradiction.

    Let gcd (a , b )= d , whered >1 .

    Thend z andx , and hence the numbersz + x andz - x ... Then, based on equality (3) would be a divisor of the number ... In this cased would be a common divisor of numbersat andNS but numbersat andNS must be mutually simple.

    Numberat is known to be even, thereforey = 2c , wherewith - natural number. Equality (3) based on equality (4) takes the following form: = 2a * 2 b , or = ab.

    It is known from arithmetic thatif the product of two coprime numbers is the square of a natural number, then each of these numbers is also the square of a natural number.

    Means,a = andb = , wherem andn Are mutually prime numbers, since they are divisors of coprime numbersa andb .

    Based on equality (5), we have:

    z = + , x = - , = ab = * = ; c = mn

    Theny = 2 mn .

    The numbersm andn since are coprime, cannot be even at the same time. But they cannot be odd at the same time, because in this casex = - would be even, which is impossible. So one of the numbersm orn is even and the other is odd. Obviously,y = 2 mn is divisible by 4. Therefore, in every basic Pythagorean triangle at least one of the legs is divisible by 4. It follows that there are no Pythagorean triangles, all sides of which would be prime numbers.

    The results obtained can be expressed as the following theorem:

    All basic triangles in whichat is an even number, obtained from the formula

    x = - , y =2 mn , z = + ( m > n ), wherem andn - all pairs of coprime numbers, one of which is even and the other odd (it does not matter which one). Each basic Pythagorean triple (x, y, z ), whereat - even, - is uniquely determined in this way.

    The numbersm andn cannot be both even or both odd, since in these cases

    x = would be even, which is impossible. So one of the numbersm orn is even and the other is odd (y = 2 mn is a multiple of 4).

    The practical part of the work

    Composing Pythagorean triplets in various ways

    In the formulas of the Indiansm andn - coprime, but they can be numbers of arbitrary parity and it is rather difficult to compose Pythagorean triples from them. Therefore, we will try to find a different approach to compiling Pythagorean triples.

    = - = ( z - y )( z + y ), whereNS - odd,y - even,z - odd

    v = z - y , u = z + y

    = uv , whereu - odd,v - odd (coprime)

    Because the product of two odd coprime numbers is the square of a natural number, thenu = , v = , wherek andl - coprime, odd numbers.

    z - y = z + y = k 2 , whence, adding the equalities and subtracting the other from one, we get:

    2 z = + 2 y = - that is

    z = y = x = kl

    k

    l

    x

    y

    z

    37

    9

    1

    9

    40

    41 (szeros)*(100…0 (szeros) +1)+1 =200…0 (s-1zeros) 200…0 (s-1zeros) 1

    An important property of Pythagorean triangles

    Theorem

    In the main Pythagorean triangle, one of the legs is necessarily divisible by 4, one of the legs is necessarily divisible by 3, and the area of ​​the Pythagorean triangle is necessarily a multiple of 6.

    Proof

    As we know, in any Pythagorean triangle at least one of the legs is divisible by 4.

    Let us prove that one of the legs is also divisible by 3.

    For the proof, suppose that in the Pythagorean triangle (x , y , z x ory multiple of 3.

    Now we will prove that the area of ​​a Pythagorean triangle is divisible by 6.

    Every Pythagorean triangle has an area expressed by a natural number divisible by 6. This follows from the fact that at least one of the legs is divisible by 3 and at least one of the legs is divisible by 4. The area of ​​the triangle, determined by the half-product of the legs, should be expressed as a multiple of 6 ...

    Conclusion

    In work

    - proven formulas of ancient Hindus

    -A study was carried out on the number of Pythagorean triplets (there are infinitely many of them)

    - methods of finding Pythagorean triples are indicated

    -studied some properties of Pythagorean triangles

    It was a very interesting topic for me and finding answers to my questions became a very interesting exercise. In the future, I plan to consider the connection of Pythagorean triples with the Fibonacci sequence and Fermat's theorem and learn many more properties of Pythagorean triangles.

    Literature

      L.S. Atanasyan "Geometry. 7-9 grades" M .: Education, 2012.

      V. Serpinsky “Pythagorean Triangles” M.: Uchpedgiz, 1959.

    Saratov

    2014