A newspaper for everyone interested in mathematics

MBOU TSSH No. 2 November 2013

In the room:

* The pride of Russian mathematics

*Entertaining tasks

*Math puzzles and entertainment

*Rebuses, crosswords

Kolmogorov

Andrey Nikolaevich

Andrei Nikolaevich was born on April 12 (25), 1903. in Tambov. Kolmogorov's mother, Maria Yakovlevna Kolmogorova, died during childbirth. Father Kataev Nikolai Matveevich, an agronomist by training, died in 1919.

Andrei's aunts organized a school in their house for children of different ages who lived nearby and taught them. A handwritten magazine “Spring Swallows” was published for the children. It published the creative works of students - drawings, poems, stories. Andrei’s “scientific works” also appeared in it - arithmetic problems he invented. Here the boy published his first scientific work in mathematics at the age of five. True, it was just a well-known algebraic pattern, but the boy noticed it himself, without outside help!

The outstanding Russian mathematician Academician Andrei Nikolaevich solved many complex problems and made more than one discovery in various branches of modern mathematics. The range of Andrei Nikolaevich’s life interests was not limited to pure mathematics. He was fascinated by philosophical problems and the history of science, painting, literature, and music.

Academician Kolmogorov is an honorary member of many foreign academies and scientific societies. In March 1963, the scientist was awarded the international Bolzano Prize, which is called the “Nobel Prize for mathematicians.”

In recent years, Kolmogorov headed the department of mathematical logic.

TASKS FOR CURIOUS KIDS

There is a cat sitting in each of the 4 corners of the room. Opposite each of these cats are three cats. How many cats are there in this room?

The father has 6 sons. Every son has a sister. How many children does the father have in total?

To dress my sons warmly, two socks are missing. How many sons are there in a family if there are six socks in the house?

Grandfather, woman, granddaughter, Bug, cat andthe mouse pulled and pulled the turnipand finally pulled it out. How many eyes were looking atturnip?

Near the dining room where skiers who came from hike, there were 20 skis, and in snow was stuck 20 sticks How many skiers went to hike?

In the proposed sayings there are missing numbers that you must fill in. Whoever inserts these numbers correctly and then adds them will get a total of 23.

1. Lied from... the box.

2. He has... Fridays in the week.

3. ... measure once, ... cut once.

4. They are waiting for the promised... year.

5. ... boots - a pair.

PUZZES WITH NUMBERS AND ABOUT NUMBERS

Crossword "Young mathematician"

Horizontally: 1. Measure of time. 2. Smallest even number. 3. Very poor assessment of knowledge. 4. A series of numbers connected by action signs.

5. Measure of land area. 6. Number within ten. 7. Part of an hour.

8. Signs that are placed when it is necessary to change the order of actions. 9. Smallest four-digit number. 10. Unit of the third category. 11. Centenary. 12. Arithmetic operation. 13. Name of the month.

Vertically: 7. Spring month. 8. Calculation device.

14. Geometric figure. 15. Small measure of time. 16. Measure of length.

17. Subject taught at school. 18. Measure of liquids. 19. Monetary unit. 20. Question to solve. 21. A certain number of units.

22. Name of the month. 23. First month of the year. 24. Last month of school holidays.

A cat is walking around the yard.

The horse stood at the gate.

The old dog is sleeping on the grass,

A goose is running along the path.

Five tiny ducklings

They are in a hurry to swim in the puddle.

Two goats chew burdock.

A rooster flew onto the fence.

Vasya went out onto the porch,

Going to the river.

How many legs are there?

The issue was prepared by students 5 "A" And 5 B" classes,

mathematic teacher Timolyanova O.V..

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Mathematics in Ancient Greece

The concept of ancient Greek mathematics covers the achievements of Greek-speaking mathematicians who lived between the 6th century BC. e. and V century AD e.

Until the 6th century BC. e. Greek mathematics was not famous for anything outstanding. As usual, counting and measurement were mastered. We know about the achievements of early Greek mathematicians mainly from the comments of later authors, mainly Euclid, Plato and Aristotle.

In the 6th century BC. e. The “Greek miracle” begins: two scientific schools appear at once: Ionians (Thales of Miletus) and Pythagoreans (Pythagoras).

Thales, a wealthy merchant, apparently learned Babylonian mathematics and astronomy well during his trading trips. The Ionians gave the first proofs of geometric theorems . However, the main role in the creation of ancient mathematics belongs to Pythagoreans.

Pythagoras, the founder of the school, like Thales, traveled a lot and also studied with Egyptian and Babylonian sages. It was he who put forward the thesis “Numbers rule the world", and worked on its justification.

The Pythagoreans made a lot of progress in the theory of divisibility, but were overly carried away by games with “triangular”, “square”, “perfect”, etc. numbers, to which, apparently, they attached mystical significance. Apparently, the rules for constructing “Pythagorean triplets” were already discovered then; comprehensive formulas for them are given by Diophantus. The theory of greatest common divisors and least common multiples is also apparently of Pythagorean origin. They probably also built a general theory of fractions (understood as ratios (proportions), since the unit was considered indivisible), learned to perform comparisons with fractions (reducing to a common denominator) and all 4 arithmetic operations.

Athens School of Pythagoras

From the history of mathematics

Mathematics in the East

Al-Khwarizmi or Muhammad ibn Musa Khwarizmi (c. 783 - c. 850) - great Persian mathematician, astronomer and geographer, founder of classical algebra.

Book about algebra and almukabal

Al-Khorezmi is best known for his “Book of Complementation and Opposition” (“Al-kitab al-mukhtasar fi hisab al-jabr wa-l-mukabala”), from the title of which the word “ algebra".

In the theoretical part of his treatise, al-Khwarizmi gives a classification equations 1st and 2nd degrees and distinguishes six types:

• squares are equal to roots (example 5 x 2 = 10 x);
• squares equal a number (example 5 x 2 = 80);
• the roots are equal to the number (example 4 x = 20);
• squares and roots are equal to a number (example x 2 + 10 x = 39);
• squares and numbers are equal to roots (example x 2 + 21 = 10 x );
• roots and numbers are equal to the square (example 3 x + 4 = x 2 ).

This classification is explained by the requirement that both sides of the equation contain positive members. Having characterized each type of equations and showing with examples the rules for solving them, al-Khwarizmi gives geometric proof of these rules for the last three species, when the solution is not reduced to simple extraction of the root.

To bring quadratic equational-Khwarizmi introduces two actions to one of the six canonical types. The first of them, al-jabr, consists of transferring negative member from one part to the other to obtain positive terms in both parts. The second action - al-mukabala - consists of bringing similar terms in both sides of the equation. In addition, al-Khwarizmi introduces the multiplication rule polynomials . He shows the application of all these actions and the rules introduced above using the example of 40 problems.

Persian Gulf

Euclidean geometry

Euclid
ancient Greek mathematician
(365-300 BC)

Almost nothing is known about Euclid, where he was from, where and with whom he studied.

The Pope of Alexandria (3rd century) claimed that he was very friendly to all those who made at least some contribution to mathematics. Correct, extremely decent and completely devoid of vanity. Once King Ptolemy I asked Euclid if there was a shorter way to study geometry than studying the Elements. To this Euclid boldly replied that “in geometry there is no royal road.” Euclid, like other great Greek geometers, studied astronomy, optics and music theory.

We know much more about the mathematical creativity of Euclid. First of all, Euclid is for us the author of the Elements, from which mathematicians all over the world studied. This amazing book has survived more than two millennia, but has still not lost its significance not only in the history of science, but also in mathematics itself. The system of Euclidean geometry created there is now studied in all schools of the world and underlies almost all practical activities of people. Classical mechanics is based on Euclid’s geometry, its apotheosis was the appearance in 1687 of “Newton’s mathematical principles of natural philosophy, where the laws of earthly and celestial mechanics and physics are established in the absolute Euclidean space.

"N The beginnings of Euclid consist of 15 books. The 1st formulates the initial provisions of geometry, and also contains the fundamental theorems of planimetry, including the theorem on the sum of the angles of a triangle and the Pythagorean theorem. The 2nd book sets out the foundations of geometric algebra. The 3rd book is devoted to properties of the circle, its tangents and chords. In the 4th book, regular polygons are considered, ...

Geometry of the Middle Ages

The geometry of the Greeks, today called Euclidean, or elementary, was concerned with the study of the simplest forms: straight lines, planes, segments, regular polygons and polyhedra, conic sections, as well as balls, cylinders, prisms, pyramids and cones. Their areas and volumes were calculated. The transformations were mainly limited to similarities.

Muse of Geometry, Louvre.

The Middle Ages gave a little to geometry, and the next great event in its history was the discovery by Descartes in the 17th century of the coordinate method (“Discourse on Method”, 1637). Sets of numbers are associated with points; this allows one to study the relationships between shapes using algebraic methods. This is how analytical geometry appeared, which studies figures and transformations that are specified in coordinates by algebraic equations. Approximately at the same time, Pascal and Desargues began research into the properties of plane figures that do not change when projected from one plane to another. This section is called projective geometry. The coordinate method underlies the differential geometry that appeared somewhat later, where figures and transformations are still specified in coordinates, but by arbitrary, fairly smooth functions.

In geometry we can roughly distinguish the following sections:

• Elementary geometry - the geometry of points, lines and planes, as well as figures on a plane and bodies in space. Includes planimetry and stereometry.
• Analytical geometry - geometry of the coordinate method. Studies lines, vectors, figures and transformations that are given by algebraic equations in affine or Cartesian coordinates, using algebraic methods.
• Differential geometry and topology studies lines and surfaces defined by differentiable functions, as well as their mappings.
• Topology is the science of the concept of continuity in its most general form.

The study of Euclid's axiom system in the second half of the 19th century showed its incompleteness. In 1899, D. Hilbert proposed the first sufficiently strict axiomatics of Euclidean geometry.

Lobachevsky geometry

Nikolai Ivanovich Lobachevsky (November 20, 1792 – February 12, 1856), great Russian mathematician

The reason for the invention of Lobachevsky’s geometry was Euclid’s V postulate: “Through a point not lying on a given line there passes only one straight line that lies with the given line in the same plane and does not intersect it" The relative complexity of its formulation gave rise to a feeling of its secondary nature and gave rise to attempts to derive it from the rest of Euclid’s postulates.

Attempts to prove Euclid's fifth postulate were carried out by such scientists as the ancient Greek mathematician Ptolemy (2nd century), Proclus (5th century), Omar Khayyam (11th - 12th centuries), and the French mathematician A. Legendre (1800).

Attempts were made to use proof by contradiction: the Italian mathematician G. Saccheri (1733), the German mathematician I. Lambert (1766). Finally, an understanding began to emerge that it was possible to construct a theory based on the opposite postulate:German mathematicians F. Schweickart (1818) and F. Taurinus (1825) (however, they did not realize that such a theory would be logically just as harmonious).

Lobachevsky in his work “On the Principles of Geometry” (1829), his first published work on non-Euclidean geometry, clearly stated that the V postulate cannot be proven on the basis of other premises of Euclidean geometry, and that the assumption of a postulate opposite to the postulate of Euclid allows one to construct a geometry so as meaningful as Euclidean, and free from contradictions.

In 1868, E. Beltrami published an article on Lobachevsky’s interpretations of geometry. Beltrami determined the metric of the Lobachevsky plane and proved that it has constant negative curvature everywhere. Such a surface was already known at that time - this is the Minding pseudosphere. Beltrami concluded that locally the Lobachevsky plane is isometric to a section of the pseudosphere.

The consistency of Lobachevsky's geometry was finally proven in 1871, after the appearance of Klein's model.

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Divisor VALUE

PRIVATE

PRIVATE

MULTIPLIER MULTIPLIER VALUE

WORKS

WORK

SUBTRACT VALUE

DIFFERENCES

DIFFERENCE

TERM TERM VALUE

AMOUNTS

SUM

1 km = 1000m

1m = 10 dm

1 dm = 10cm

1cm = 10mm

1m = 100cm =1000mm

1 century = 100 years

1 year = 12 months

1 year = 365(366) days

1 day = 24 hours

1 hour = 60 minutes

1 minute = 60 seconds

1 t = 1000kg

1kg = 1000g

1c = 100kg

1t = 10c

R straight. = a+b+a+b

R straight. = (a+b) 2

R straight. = a 2 + b 2

P square = a+a+a+a

P squared = a 4

a – length S = a b

b – width a = S b

S – area b = S a

(m, cm, etc.)

Increase

in time

Decrease

in time

How many times

More less

Increase

by... units

Decrease

by... units

How long

more less

1. ()

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Mathematical sophisms

Sophistry is a deliberately false conclusion that has the appearance of being correct. Whatever the sophistry, it necessarily contains one or more disguised errors. Especially often in mathematical sophisms “forbidden” actions are performed or the conditions of applicability of theorems, formulas and rules are not taken into account. Sometimes reasoning is carried out using an erroneous drawing or is based on “obviousness” leading to erroneous conclusions. There are sophisms containing other errors.

How are sophisms useful for students of mathematics? What can they give? Analysis of sophisms, first of all, develops logical thinking, that is, instills the skills of correct thinking. To discover an error in sophism means to realize it, and awareness of the error prevents it from being repeated in other mathematical reasoning. Analysis of sophisms helps the conscious assimilation of the mathematical material being studied, develops observation, thoughtfulness and a critical attitude towards what is being studied.

TRY YOUR STRENGTH

1) 4 rubles = 40,000 kopecks. Let’s take the correct equality: 2p = 200 k. Let’s square it piece by piece. We will receive: 4 rubles = 40,000 k. What is the mistake?

2) 5=6. Let's try to prove that 5=6. For this purpose, let's take a numerical identity:

35+10-45=42+12-54. Let's take the common factors of the left and right sides out of brackets. We get: 5(7+2-9)=6(7+2-9). Let's divide both sides of this equality by a common factor (enclosed in parentheses). We get 5=6. Where is the mistake?

3) . 2*2=5. Find the error in the following reasoning. We have the correct numerical equality: 4:4=5:5. Let us take its common factor out of brackets in each part. We get: 4(1:1)=5(1:1). The numbers in brackets are equal, so 4=5, or 2*2=5.

4) All numbers are equal to each other.Let m=n. Let's take the identity: m 2 -2mn+n 2 =n 2 -2mn+m 2 . We have: (m-n) 2 = (n-m) 2 . Hence m-n=n-m? or 2m=2n, which means m=n. Where is the mistake?

WE ARE LEARNING

REALIZE!

• A plane from Moscow flies to Kyiv and returns back to Moscow. In what weather will this plane make the entire journey faster: in calm weather; with the wind blowing with the same force in the direction Moscow-Kyiv?

• From a conversation on September 1: “How much longer do you have to study?” - “As much as you have already studied. And you?" - “One and a half times more.” Who went to what grade?

• In the notation KTS+KST=TSK, each letter has its own number. Find what the number TSC is equal to!

PROVE!

• The square of an odd number is an odd number.

• The square of an even number is a multiple of 4.

• The difference of the squares of two consecutive odd numbers is divisible by 8.

• The sum of the product of two consecutive natural numbers and the larger of them is equal to the square of that larger number.

• If you take some two-digit number with different digits, rearrange the digits in it and subtract the resulting number from the taken number, then the difference will be divided by 9.Will this be true for three-digit numbers (outer digits are rearranged)?

WONDERFUL CURVES

Archimedes' spiral. Imagine that a fly is crawling along the radius of a uniformly rotating disk at a constant speed. The path described by the fly is a curve called the Archimedes spiral. Draw some kind of Archimedes spiral.

Sine wave. Make a tube out of thick paper by folding it several times. Cut this tube at an angle. Look at the cut line if you unfold one of the parts of this tube. Redraw this line onto a piece of paper. You'll end up with one of those wonderful curves called a sine wave. You encounter it especially often when studying electrical and radio engineering.

Cardioid. Take two equal circles cut out of plywood (you can take two identical coins). Secure one of these circles. Attach the second one to the first one, mark point A on its edge, which is farthest from the center of the first circle. Then roll the movable circle along the stationary one without sliding and observe what line point A describes. Draw this line. It is one of Pascal's snails and is called a cardioid. In technology, this curve is often used to design cam mechanisms.

Geometric puzzles

• Fold three equal squares: 1) from 11 matches; 2) from 10 matches.

• The figure shown in the figure needs to be divided into 6 parts by drawing only 2 straight lines. How to do it?

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Rules of conduct for students

in the office

The mathematics classroom is equipped with modern equipment for conducting classes: PC, projector, screen, printing device.

This equipment does not tolerate dust and requires careful handling.

Prohibiting rules of behavior

in the office

Rules of conduct for students

at the lesson

1. When the teacher enters the classroom, students stand up. They sit down after greeting and permission from the teacher. Students also greet any adult who enters the classroom during class. When the teacher leaves the class, the students also stand up.
2. During the lesson, the teacher sets rules for behavior in the lesson.
3. During the lesson, you must not make noise, be distracted yourself or distract your comrades from their studies with conversations, games and other matters not related to the lesson.
4. If a student wants to say something, ask the teacher a question, or answer a question, he raises his hand and, after permission, speaks. The teacher may set other rules.
5. The end of lesson bell is given to the teacher. He determines the end time of the lesson and announces to the students its end.
6. If a student misses classes at school, he must present the class teacher with a medical certificate or a note from his parents. Missing or being late for lessons without good reason is not permitted.

Rules of conduct for students

at a break

1. At the end of the lesson, students are required to:

• tidy up your workplace;
• leave the class;
• obey the requirements of the teacher and students on duty.

1. During recess, students are in the hallway. There are two attendants in the classroom who:

• ventilate the classroom
• erased from the board,
• prepare chalk and rag,
• make sure that no one is in the class during breaks,
• help the teacher prepare material for the lesson,
• Allow students to enter the classroom two minutes before the bell and with the permission of the teacher.

1. During a break it is prohibited:

• run in places unsuitable for play, push each other;
• use obscene expressions and gestures, make noise, disturb others from resting or preparing for a lesson.

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Will make it through the road

going,

And mathematics -

thinking!

Did you know that the first calculating device was the abacus?

The first “computing devices” that people used in ancient times were fingers and pebbles. In Ancient Egypt and Ancient Greece, long before our era, they used an abacus - a board with stripes along which pebbles moved. This was the first device specifically designed for computing. Over time, the abacus was improved - in the Roman abacus, pebbles or balls moved along grooves. The abacus lasted until the 18th century, when it was replaced by written calculations. Russian abacus - abacus appeared in the 16th century. They are still used today. The big advantage of Russian abacus is that it is based on the decimal number system, and not on the five-digit number system, like all other abaci.

Algorithm for working on a task

1. I read the whole problem.
2. I read the condition and highlight the data.
3. I read the question and highlight what I'm looking for.
4. I determine the structure of the task (simple or complex).
5. I find the missing datum (if compound).
6. I am bringing the decision to the end.
7. I'm re-reading the question.
8. I answer it.

Comic problems

1. Firefighters are trained to put on their pants in three seconds. How many pants can a well-trained firefighter put on in 1 minute?
2. There is one hole in a bagel, and twice as many in a pretzel. How many fewer holes are there in 7 bagels than in 12 pretzels?
3. If baby Kuzya is weighed together with his grandmother, the result will be 59 kg. If you weigh grandma without Kuzya, you get 54 kg. How much does Kuzya weigh without his grandmother?
4. A boxer, a karateka, and a weightlifter chased a cyclist at a speed of 12 km/h. Will they catch up with a cyclist if he, having covered 45 km at a speed of 15 km/h, lies down to rest for an hour?.
5. Katya's height is 1 m 75 cm. Stretched out to her full height, she sleeps under a blanket whose length is 155 cm. How many centimeters does Katya stick out from under the blanket?.
6. How many holes will there be in an oilcloth if you pierce it 12 times with a 4-tooth fork during lunch?.
7. At a math lesson in the 7th group, there were students who had 56 ears, the teacher had 54 fewer ears. How many ears can you count during a math lesson?
8. The area of ​​one elephant's ear is 10,000 sq.cm. Find out in apt. m., area 2 elephant ears..
9. Let's say that you decide to jump into water from a height of 8 meters. And, having flown 5 meters, he changed his mind. How many more meters will you have to fly involuntarily?
10. Baby Kuzya screams like crazy 5 hours a day. Sleeps like the dead 16 hours a day. The rest of the time, baby Kuzya enjoys life in all ways available to him. How many hours a day does baby Kuzya enjoy life?
11. Koschey the Immortal was born in 1123, and received a passport only in 1936. How many years did he live without a passport?
12. Hungry Vasya eats it in 9 minutes. 3 bars, a well-fed Vasya spends 3 baht. 15 minutes. How much min. Is hungry Vasya faster with one candy bar?
13. Baby Kuzi has 4 more teeth, but his grandmother only has 3. How many teeth do the grandmother and grandson have?
14. Who will be heavier after dinner: the first is the cannibal, who weighed 48 kg before dinner and ate the 2nd cannibal for dinner, or the second, who weighed 52 kg and ate the first.

Rules of conduct in the mathematics classroom

1. Enter the office only with the permission of the teacher. Students must enter the office in a change of shoes and without outerwear
2. Students should enter the classroom calmly, without jostling, and maintaining order. Loud conversations and disputes over the workplace are prohibited
3. You cannot touch any device in the office without permission, open cabinets, or touch projection equipment.
4. It is prohibited to bring things into the office that are not intended for study. It is prohibited to use a cell phone
5. Chewing gum, no matter how tasty it may seem, is strictly prohibited for use in the classroom, both in class and during recess.
6. The main and most important requirement in the office is discipline. Dust raised in the classroom is harmful to both equipment and students
7. You cannot bring bread, nuts, sweets, or seeds into the office. Lunch in the dining room must be eaten at the dining room table

Thanks for following the rules!

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In the world of mathematics

PERIMETER consists of two Greek words peri (around) and metreō (measure). Compare it with the words periscope (ckopeo - look), periphery (phero - carry), pericardia (kardia - heart), period (hogjs - way, road)

CHORD (Greek chordē) translated from Greek - string. The origin of this term in geometry is associated with the manufacture of a bow, in which a tightly stretched string - a bowstring - tightens its ends.

The words SECTOR and SEGMENT , it turns out, are related, because they come from the same Latin word (like the word axe), which is translated into Russian as cut. So, the sector and the segment dissect the circle, but each in its own way.

MEDIAN , mediator, physician - cognate. They come from the word medium - intermediary, average. A mediator is an object that allows a musician to extract sound from his musical instrument; physician - a doctor with the help of whom the patient is healed.

Word RHOMBUS comes from the Greek rhombos meaning tambourine. It turns out that in ancient times, tambourines - musical instruments - were not round, as they are now, but had the shape of a quadrangle with equal sides.

In the word BISEXTER the root is sectr - (familiar truth), and the prefix "bis" - which means repeat, twice. So, by the very structure of the word “bisector” it is easy to determine its meaning, and also understand why you need to write a double consonant in this word With .

The word CATET is the same root as the words catacombs, cataract. The root kata is of Greek origin, meaning down, to fall. The word cataract (clouding of the eye lens) was previously used in the form of cataracts and had 2 meanings: a waterfall in the mountains, as well as movable barriers in the fortress gates. Catacombs – kata under; down + kumbē bowl.

The word HYPOTENUSE translated from Greek as to be opposite, i.e. the side of a triangle opposite its right angle.

Rebuses

Answers:

1. Task
2. Axiom
3. Apothem

Answers:

1. Vector
2. Cone
3. Pyramid

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Golden Ratio

Geometry has two treasures:
one of them is the Pythagorean theorem,
another is the division of a segment in average and extreme ratio.
I. Kepler

There are things that cannot be explained. So you come to an empty bench and sit down on it. Where will you sit - in the middle? Or maybe from the very edge? No, most likely, neither one nor the other. You will sit so that the ratio of one part of the bench to the other, relative to your body, will be approximately 1.62. A simple thing, absolutely instinctive... Sitting on a bench, you produced the “golden ratio”. The golden ratio was known back in ancient Egypt and Babylon, in India and China. The great Pythagoras created a secret school where the mystical essence of the “golden ratio” was studied. Euclid used it when creating his geometry, and Phidias - his immortal sculptures. Plato said that the Universe is arranged according to the “golden ratio”. And Aristotle found a correspondence between the “golden ratio” and the ethical law. The highest harmony of the “golden ratio” will be preached by Leonardo da Vinci and Michelangelo, because beauty and the “golden ratio” are one and the same thing. And Christian mystics will draw pentagrams of the “golden ratio” on the walls of their monasteries, fleeing from the Devil. At the same time, scientists - from Pacioli to Einstein - will search, but will never find its exact meaning. An endless series after the decimal point - 1.6180339887... Everything living and everything beautiful - everything obeys the divine law, whose name is the “golden ratio”.

Angel de Coitiers

Golden ratio in mathematics

In mathematics, proportion call the equality of two relations: a : b = c : d .

Line segment AB can be divided into two parts in the following ways:

• into two equal parts - AB: AC = AB: BC;
• into two unequal parts in any respect (such parts do not form proportions);
• thus, when AB: AC = AC: BC.

The latter is the golden division or division of a segment in extreme and average ratio.

The golden ratio is such a proportional division of a segment into unequal parts, in which the entire segment is related to the larger part as the larger part itself is related to the smaller one; or in other words, the smaller segment is to the larger as the larger is to the whole

a: b = b: c or c: b = b: a.

Practical acquaintance with the golden ratio begins with dividing a straight line segment in the golden proportion using a compass and ruler.

From point B a perpendicular equal to half is restored AB . Received point WITH connected by a line to a point A . A segment is plotted on the resulting line Sun ending with a dot D. Segment AD transferred to direct AB . The resulting point E divides segment AB in the golden ratio ratio.

Segments of the golden ratio are expressed as an infinite irrational fraction AE = 0.618..., if AB take as one BE = 0.382... For practical purposes, approximate values ​​of 0.62 and 0.38 are used. If the segment AB taken as 100 parts, then the larger part of the segment is 62, and the smaller part is 38.

The properties of the golden ratio are described by the equation:

x 2 – x – 1 = 0.

Solution to this equation:

Golden Triangle

To find segments of the golden proportion of the ascending and descending series, you can use pentagram.

To build a pentagram, you need to build a regular pentagon. The method of its construction was developed by the German painter and graphic artist Albrecht Durer (1471...1528). Let O – center of the circle, A – a point on a circle and E – the middle of the segment OA . Perpendicular to radius OA , restored at the point ABOUT , intersects the circle at the point D . Using a compass, plot a segment on the diameter CE = ED . The side length of a regular pentagon inscribed in a circle is DC . Lay out segments on the circle DC and we get five points to draw a regular pentagon. We connect the corners of the pentagon through one another with diagonals and get a pentagram. All diagonals of the pentagon divide each other into segments connected by the golden ratio.

Golden ratio in architecture

One of the most beautiful works of ancient Greek architecture is the Parthenon (5th century BC).

The figures show a number of patterns associated with the golden ratio. The proportions of the building can be expressed through various powers of the number Ф=0.618...

All architectural structures, temples and even dwellings from Ancient Egypt and Ancient Greece to the present day were created and are being created in the harmony of numbers - according to the rules of the “Golden Section”.

Golden ratio in sculpture

The golden proportion was used by many ancient sculptors. The golden proportion of the statue of Apollo Belvedere is known: the height of the depicted person is divided by the umbilical line in the golden section.

Back in the Renaissance, artists discovered that any picture has certain points that involuntarily attract our attention, the so-called visual centers. In this case, it does not matter what format the picture has - horizontal or vertical. There are only four such points; they divide the image size horizontally and vertically in the golden ratio, i.e. they are located at a distance of approximately 3/8 and 5/8 from the corresponding edges of the plane.

Golden ratio in fonts and household items

Golden ratio in biology

Rostock

Among the roadside herbs grows an unremarkable plant - chicory. Let's take a closer look at it. A shoot has formed from the main stem. The first leaf was located right there.

The shoot makes a strong ejection into space, stops, releases a leaf, but this time it is shorter than the first one, again makes an ejection into space, but with less force, releases a leaf of an even smaller size and is ejected again. If the first emission is taken as 100 units, then the second is equal to 62 units, the third – 38, the fourth – 24, etc. The length of the petals is also subject to the golden proportion. In growing and conquering space, the plant maintained certain proportions. The impulses of its growth gradually decreased in proportion to the golden ratio.

Golden ratio in body parts

By comparing the lengths of the phalanges of the fingers and the hand as a whole, as well as the distances between individual parts of the face, one can also find the “golden” ratios:

Sculptors claim that the waist divides the perfect human body in relation to the golden ratio. Measurements of several thousand human bodies have revealed that for adult men this ratio averages approximately 13/8 = 1.625

Preview:

5-6 grades
Warm-up

1. An orange is not lighter than a pear, and an apple is not lighter than an orange. Can a pear be heavier than an apple? Isn't it lighter than an apple?

2. A sister has four times as many brothers as sisters. And the brother has one more brothers than sisters. How many brothers and how many sisters are there in the family?

3. Two diggers dig a 2 m ditch in 2 hours. How many diggers will dig a 5 m ditch in 5 hours?

Comparison problems

Weighing problems

1. Available pan scales without weights and three coins, one of them is counterfeit- easier others. Detect a counterfeit coin with one weighing.
2. Solve the previous problem if there are 4 coins; 5; 6; 8; 9 and two weigh-ins.

Transfusion tasks

1. There are 18 liters of gasoline in a barrel. There is a scoop with a volume4 l and two buckets of 7 l, inwhich you need to pour 6 liters of gasoline. Howto carry out a spill?

Number problems

Problems on “Graphs”

1. The figure shows a diagram of bridges in the city of Königsberg. Is it possible to take a walk so that you cross each bridge exactly once?

Getting ready for the Olympics

We enter a university based on the results of olympiads

5-6 grades
Small Olympiad (autumn round)

1. Puss in Boots caught four pike and half the catch. How many pikes did Puss in Boots catch?

2. The hares sawed several logs. They made 10 cuts and got 16 logs. How many logs did they cut?

3. What do you think - even or odd - will be the sum:
a) two even numbers;
b) two odd numbers;
c) even and odd numbers;
d) odd and even numbers?

4. The guys brought a full basket of mushrooms from the forest. A total of 289 mushrooms were collected, with the same amount in each basket. How many guys were there?

5. The boy had 10 coins worth 1 ruble. and 5 rub. He counted 57 rubles. Was the boy wrong?

6. From a barrel containing at least 10 l gasoline, pour exactly 6 l, using a can with the capacity of a nine-liter bucket.

7. 7 chocolates are more expensive than 8 packs of cookies. What is more expensive - 8 chocolates or 9 packs of cookies?

9. There are less than 100 apples in the basket. They can be divided between two, three or five children, but cannot be divided equally between four children. How many apples are in the basket?

10. Rumor reached King Gorokh that, finally, someone had killed the Serpent Gorynych. The Tsar guessed that this was the work of either Ilya Muromets, or Dobrynya Nikitich, or Alyosha Popovich. He invited them to the court and began to question them. Each hero spoke three times. And they said this:

Ilya Muromets: 1) I did not kill Zmey Gorynych. 2) I went to overseas countries. 3) And Alyosha Popovich killed the Serpent Gorynych.

Nikitich:4) Snake Gorynych was killed by Alyosha Popovich. 5) But even if I had killed, I would not have confessed. 6) There is still a lot of evil spirit left.

Alesha Popovich: 7) It was not I who killed Zmey Gorynych. 8) I have been looking for a long time for some feat to accomplish. 9) Indeed, Ilya Muromets left for overseas countries.

Then King Gorokh learned that twice each hero spoke the truth, and once he was disingenuous. So who killed Zmey Gorynych?

7-8 grades
Invariant

Invariant - a term used in mathematics, physics, and also in programming, denotes something unchangeable.

All tasks, united by the conventional name “invariant”, have the following form: certain objects are given on which certain operations are allowed to be performed. As a rule, the problem asks, is it possible to obtain another from one object using these operations? If possible, then you need to give an example of how to do this. If it is not possible, you need to prove that it is impossible.

A variety of quantities can act as an invariant: parity, sum, product, remainder, etc.

Problem 1

The change machine exchanges one coin for five others. Is it possible to use it to exchange one coin for 27 coins?

Solution. After each such exchange, the number of coins increases by 4, while the remainder of the number of coins when divided by 4 remains unchanged. At first we had 1 coin, which means the remainder will always be 1. The number 27 when divided by 4 has a remainder of 3, so you cannot exchange one coin for 27 coins.

Problem 2

The bully Vasya tore the wall newspaper, and he tore each piece he came across into four parts. Could it have been 2009 pieces? What if each piece was torn into 4 or 10 pieces?

Solution. No. The number of pieces changes each time by 3 or 9, that is, the remainder when divided by 3 is invariant. Initially there was one newspaper, which means that the number of pieces must have a remainder of 1 modulo 3, and 2009 is divided by 3 with a remainder of 2.

Problem 3

The numbers 1, 2, 3,..., 100 are written in a row. You can swap any two numbers between which there is exactly one. Is it possible to get the series 100, 99, 98,..., 2, 1?

Solution. Note that during permitted operations, either only even numbers or only odd numbers are swapped. In this case, even numbers will always be in even places. This means that it is impossible to get a row in which 100 is in the first place.

Problem 4

80 tons of peaches, which contained 99% water, were transported from Astrakhan to Moscow. On the way, they dried out and began to contain 98% water. How many tons of peaches were brought to Moscow?

Solution. In this problem, the invariant is the weight of the “dry residue”, i.e. the difference between the weight of peaches and the weight of the water they contain. In Astrakhan, peaches contained 1%, i.e. 8 tons of “dry residue”, in Moscow these 8 tons already accounted for 2% of the peaches brought. Then the weight of peaches is 8:2-100 = 40t. Weight has halved!

Problem 5

You can add the sum of its digits to a number. Is it possible to get the number 20092009 from three in a few steps?

Solution . With each step, the number increases by the sum of the digits. Note that the number and the sum of its digits have the same remainder when divided by 3. Three is divisible by 3 without a remainder, which means that the numbers that can be obtained from it by such an operation will also be divisible by 3. And the number 20092009 is not a multiple of 3.

Answer: no.

Problem 6

An 8x8 table is given, in which numbers from 1 to 64 are written. 8 cells are shaded so that in each horizontal and in each vertical there is exactly one shaded cell. Prove that the sum of the numbers written in these 8 cells does not depend on the set of shaded cells.

Solution. Let us number the columns in the table from left to right with numbers from 1 to 8. Then we will represent the numbers in the first row as the sum of 0 and the column number; numbers written in the second line as 8+column no.; in the third line: 16+ No., etc. Since exactly one cell is shaded in each row and each column, then, regardless of the choice, the sum of the eight numbers in the set is equal to: (0 + 8 + 16 + ... + 56) + (1 + 2 + ... + 8) = 260.

Problem 7

Solve the equation in whole numbers x 2 +y 2 +z 2 =8k - 1.

Solution. Let's consider the remainders of perfect squares when divided by 8. The square of an even number can give remainders 0 and 4, and an odd one always gives remainder 1, since(2k + 1) 2 = 4k 2 + 4k + 1 = 4k(k + 1) + 1. The sum of the remainders of three complete squares can be either even, or 1, or 3. But 8k - 1 is divisible by 8 with a remainder of 7. This means that this equation has no solutions.

Problem 8

Given a convex quadrilateral with diagonals 10 cm and 7 cm. Prove thatthat when cutting such a quadrangle, it is impossible to pave a 6x6 cm square with the resulting pieces.

Solution. The area of ​​such a quadrilateral is 5∙7 sinα (α - angle between diagonals). Therefore, the area of ​​a figure equivalent to a given quadrilateral cannot exceed 35. The area of ​​a 6x6 square is 36.

7-8 grades
Problems to solve independently

2.1. There are 50 glasses in the dining room, 25 of them are upside down. Will the person on duty, turning over 4 glasses at a time, be able to get all the glasses standing correctly, that is, on the bottom?

2.2. The numbers 1,2,..., 2009 are written on the board. You are allowed to erase any two numbers and write the difference of these numbers instead. Is it possible to ensure that all the numbers on the board are zeros?

2.4. Ivan Tsarevich has two magic swords, one of which can cut off 21 heads of the Serpent Gorynych, and the second - 4 heads, but then the Serpent Gorynych grows 2008 heads. Note that if the Serpent Gorynych has, for example, only three heads left, then it is impossible to chop them with either one or the other sword. Can Ivan Tsarevich cut off all the heads of the Serpent Gorynych, if at the very beginning he had 100 heads?

2.5. On a chessboard, you are allowed to recolor all the cells in one row or one column in one move. Can there be exactly one white cell left after several moves?

2.7. There are two letters in the alphabet of the language of the UYU tribe: U and Y, and this language has an interesting property: if you remove the adjacent letters UY and UYUU from a word, the meaning of the word will not change. In the same way, the meaning of the word does not change when the letter combinations УУ, ыыууыы and Уыыу are added anywhere in the word. a) Is it possible to say that the words UYY and UYYY have the same meaning? In this problem, the expressions “have the same meaning” and “obtain from each other by transformation” are equivalent, b) Do the words UYY and UYY have the same meaning?

2.8. There are only two letters in the alphabet - A and Z. Combinations of letters AYA and YAYA, YA and AAYA, YAYA and AAA in any word can be replaced with each other. Is it possible to get the word YAA from the word AYA?

2.10. Numbers from 1 to 20 are written on the board. Any pair of numbers can be(x, y) replace with a number x + y + 5xy. Could it end up being 20082009?

2.17. There is a pile of 1001 stones on the table. The first move is to throw a stone out of the pile and then divide it into two. Each subsequent move consists of throwing away a stone from any pile containing more than one stone, and then one of the piles is again divided into two. Is it possible to leave only piles of three stones on the table after a few moves?

2.18. Prove that numbers of the form 2009n + 3 and 2009n + 4 cannot be represented as the sum of two cubes of natural numbers.

2.20. The entire set of dominoes was laid out according to the rules of the game. It is known that five comes first. What is the last number?

2.23. There are 100 pros and 100 cons written on the board. You can replace any 2 minuses with a plus, a plus and a minus with a minus, two pluses with a plus. Prove that the sign that remains at the end does not depend on the order of operations.

2.26. Prove that the equation 15x 2 - 7y 2 = 9 has no solutions in integers.

2.27. Prove that the equation x 2 - 7y = 10 has no integer solutions.

From the history of mathematics

the date of the

Solve number puzzles where the same letters correspond to the same numbers, and different - different.

***

find the correct answer for example

Mathematical puzzle

Questions for Chinaword. 1. 2.
1. Geometric figure. 1. Measure of area.2. Regular polygon. 2. Place occupied by a digit in a number notation 3. Number. 3. Number defining the length of the line4. An ancient measure of length. 4. 100 square meters.5. A relation connecting two numbers. 5. A segment connecting a point on a circle with its6. Part of a straight line limited by two centers dots. 6. Number.7. School team. 7. Rhombus with equal angles.8. Mathematical operation. 8. One hundred tens.9. A segment whose length is 1. 9. Part of mathematics, the science of numbers.

Pythagoras (c. 570 - c. 500 BC)

The judges of one of the first Olympics in history did not want to allow a young man with a strong physique to participate in sports competitions, since he was not tall enough. But he not only became a participant in the Olympics, but also defeated all his opponents. This is the legend... This young man was Pythagoras, the famous mathematician.
His whole life is a legend, or rather, a layering of many legends. He was born on the island of Samos, off the coast of Asia Minor. Only five kilometers of water separated this island from the mainland. Pythagoras left his homeland when he was very young. He walked along the roads of Egypt, lived for 12 years in Babylon, where he listened to the speeches of the priests who revealed to him the secrets of astronomy and astrology, then for several years in Italy. Already in adulthood, Pythagoras moved to Sicily and there, in Crotone, created an amazing school,

which will be called Pythagorean. Here are the “commandments” of the Pythagoreans:

Do only what will not upset you later and will not force you to repent.
Never do what you don’t know, but learn everything you need to know.
Don't neglect the health of your body.
Learn to live simply and without luxury.
Before you go to bed, analyze your actions for the day.

Pythagoras did not write down his teachings. It is known only in the retellings of Aristotle and Plato.

How many triangles do you see

on the image?

Hello, friends! Sometimes we have unusual weeks at school. For example, a week of literature, a week of Pushkin, a week of traffic rules or a healthy lifestyle. During these unusual subject weeks, various interesting events are held: thematic Olympiads, creative competitions, craft exhibitions and poster competitions. That’s what I want to talk about posters in this article. Namely, about posters for mathematics week at school.

If mathematics seems boring to you, if your child agrees with you, and mathematics does not attract him at all, then you just need to start making an interesting wall newspaper together.

What will this give you?

1. The child's warmed interest in the queen of sciences.
2. Teacher's gratitude.
3. Respect from classmates.
4. Well, if the work takes part in a poster competition, then it is quite possible that you will receive a certificate from the school administration, which will undoubtedly take pride of place in your little schoolchild’s portfolio.
5. Add to this the time you will spend with your beloved child.

Despite the fact that mathematics is an exact science, it is better to approach poster making creatively. You need to try to come up with something like this, and design it somehow so that during recess boys and girls do not run headlong through the corridors, but crowd around your creation.

Do you think this is impossible? It's very possible! I saw it with my own eyes. The children lined up to look at the poster and play with it for a while. Well, my daughter Sasha was proud that we made this poster.

So, here it is - our work!

We made it when Alexandra was in first grade. Still intact. Although it is hung on the wall in the elementary school recreation every year when it's math week.

When we brought the finished poster to school, the teacher was very surprised and at first did not believe that Sasha had a hand in making it. Nevertheless, we made the poster together, as a family. Sasha painted, chose pictures, came up with riddles, even tried to draw using a ruler (though it worked every time). She also cut it out, glued it, in general, a lot of her work was invested.

And now more about the poster. It consists of four zones called:

1. Cryptic math crossword puzzle.
2. Fun puzzles.
3. Inverted examples.
4. Bubble-bulb examples.

A snake crossword puzzle acts as a frame. In it, the last letter of one word is the first letter of the next. The crossword runs along the perimeter of the Whatman paper and also divides it into two parts, separating different blocks from each other.

The title of the poster is “Mathematics”. This word is also the first word in the crossword puzzle.

The crossword puzzle tasks were printed out and glued to whatman paper.

1. Queen of Sciences. (mathematics)
2. Himself - scarlet sugar. Kaftan – green velvet. (watermelon)
3. She wants to be solved. (task)
4. A month in which there is no school. (August)
5. The most popular fairy tale figure. (three)
6. Lives far away, shoots a bow, wears feathers. (Indian)
7. She's the master of mathematics. (number)
8. This word is very popular among those who still don’t know how to count. (yeah)
9. It contains problems and examples. (textbook)
10. It has four angles and all sides are equal. (square)
11. And it only has three corners. (triangle)
12. The figure drawn by a compass. (circle)
13. The hut was built without hands, without an axe. (nest)
14. A good number and the worst estimate. (one)
15. Who am i? Always with you! Whether you stand or sit, I am always ahead! (nose)
16. And for those who don’t like mathematics - shame and... (shame)

Immediately below the tasks for the crossword puzzle there was a “Fun Puzzles” block. You can come up with many tasks. The main thing is to remember for which class you are making the poster, so that the children already have enough knowledge to solve them. Moreover, problems can also be logical, and not just addition and subtraction. All the heroes of our tasks are real people, Sasha’s classmates and teacher. It’s much more interesting to decide this way!

These are the problems we got.

1. Sveta brought two apples to class. Masha brought three oranges. Then Vova came and took everything for himself. What did the girls name Vova and how much fruit did he eat?
2. At recess, three girls were jumping ropes, and five boys were playing hide and seek. Everyone was shouting loudly, so the director came. How many mothers of children were called to school if the director did not find the boys?
3. Vanya had two pistols, and Kolya had four machine guns. They started running around the classroom and shooting loudly. But then Olga Vladimirovna came in. How many weapons did she add to her arsenal?
4. Ira ate three chocolates, but then Vova came running and ate five more. What was Vova's name again? And how many chocolates were eaten?

The upper left corner is occupied by a block called “Flip Examples”. Our invention!

The examples are not just written on whatman paper. Numbers, as well as signs of mathematical operations, are drawn on round double-sided cards. They need to be turned over to get the correct result.

These are the correct examples.

This is how it turns out wrong. I should think.

Moreover, in the first row the examples are simple, only for addition.

The second row contains examples of addition and subtraction.

The third row is the most difficult. This is an inequality with addition and subtraction.

What numbers are written on the cards:

1st row (from left to right):

• card No. 1 – “4” and “3”;
• card No. 2 – “+” (we just glued it);
• card No. 3 – “5” and “4”;
• card No. 4 – “=” (also pasted);
• card No. 4 – “9” and “7”;

• card No. 1 – “8” and “2”;
• card No. 2 – “-” and “+”;
• card No. 3 – “5” and “4”;
• card No. 4 – “=” (both sides);
• card No. 5 – “3” and “6”;

• card No. 1 – “4” and “9”;
• card No. 2 – “-” and “+”;
• card No. 3 – “3” and “2”;
• card number 4 – “<» и «>»;
• card number 5 – “8” and “6”.

How do these cards stay on the poster? They hang on threads.

You need strong threads. We have white wool ones. Using a needle, attach one end of the thread to the card and tie a neat knot. And then, again, using a needle, we pierce the Whatman paper in the right place and stretch the thread to the back side of the poster. Cut it off, leaving a long tail, and tie a knot right at the surface of the Whatman paper.

In principle, the cards will already be held. But this is an unreliable mount. They'll rip it out right away. Therefore, we tie all the tails together in two or three pieces and glue them to the poster.

You can secure them with tape. But my experience shows that an ordinary adhesive plaster is better suited for these purposes. Sold in rolls. It sticks tightly to the Whatman paper and holds tightly.

In the lower left corner there is a block with a beautiful name “Bul-bul examples”.

Why "Bul-bul"? Because here the fish are in aquariums. In big and small. The fish are also placed on round double-sided cards with numbers written on the reverse sides. Essentially these are examples of addition. The task sounds like this:

“In a small aquarium, choose two fish. Add up the numbers written on the fish. Then, in a large aquarium, find the fish with the correct answer.”

We have 5 fish swimming in a small aquarium. They have numbers on them: “1”, “2”, “3”, “4”, “5”.

There are 7 fish in a large aquarium. On them: “3”, “4”, “5”, “6”, “7”, “8”, “9”.

Circles with fish are attached to whatman paper in the same way as “reversible examples”.

Alexandra was even given a certificate for this poster! Yes, it is not easy to perform, and the child, of course, will need the help of his parents. But that’s why parents exist, to help their children. The main thing is not to do absolutely everything for the child. We need to invent, invent, make together!

Here are some more math posters. I found them on the Internet. They are easier to make than ours. Each is decorated in its own style. Pay attention to unusual names.

More options for names for posters: “Jolly Archimedes”, “Children of Pythagoras”, “Cyphrus”, “Labyrinthus”, “Twice Two”, “Plus and Minus”, etc.

The poster can include:

• puzzles with numbers;
• interesting examples;
• funny logic puzzles;
• riddles, sayings, proverbs, fairy tales about numbers and figures;
• interesting stories about great mathematicians;
• math charades;
• math crosswords;
• digital labyrinths;
• math puzzles.

During Maths Week at school, you can do more than just draw posters. You can make a craft, play KVN, and take part in math relay races and olympiads!

Such events give children a lot of positive emotions and help them understand and love mathematics.

So, friends, do not deny yourself the pleasure! Create together with your little schoolchild! Very soon he will learn everything on his own and will no longer ask for your help)

We dedicated another interesting and unusual poster. Or you can see a wall newspaper on the topic “Red Book of Russia”.

With this, I say goodbye to you until we meet again on the blog!

I wish you health and inspiration!

And girls and boys have great success in their studies!

Always yours, Evgenia Klimkovich!

The family has 5 sons and each has a sister.
How many children are in this family?

The striking clock strikes once every 1 second.
How long will it take the clock to strike 12 o'clock?

Three hens will lay three eggs in three days.
How many eggs will 6 hens lay in 6 days?
4 chickens in 9 days?

Do you think that mathematics is a boring science?

Are mathematicians terrible bores?

You just don't know anything about them! Read our newspaper and your opinion will change!

Do you know that Charles Perrault, the author of "Little Red Riding Hood", wrote the fairy tale "The Love of Compasses and Ruler"?

Do you know that Napoleon Bonapartewrote mathematical works and one geometric fact is called “Napoleon's Problem”?

Do you know that L. N. Tolstoy, author of the novel “War and Peace”, wrote textbooks for primary schools and, in particular, an arithmetic textbook?

Do you know that A. S. Pushkinwrote the following lines: “Inspiration is needed in geometry, as in poetry”?

Did you know that great Euclidsaid to King Ptolemy: “There is no royal road in geometry”?

Why do houses in the east skip floors with number 4?

In China, Korea and Japan, the number 4 is considered unlucky, as it is consonant with the word “death”. In these countries, floors with numbers ending in four are almost always absent.

How do Arabs write and read numbers?

The Arabs use their own signs to write numbers, although the Arabs of Europe and North Africa use the “Arabic” numbers that are familiar to us. However, no matter what the signs of the numbers are, the Arabs write them, like letters, from right to left, but starting from the lower digits. It turns out that if we come across familiar numbers in an Arabic text and read the number in the usual way from left to right, we will not be mistaken.

Why do numbers on a calculator increase from bottom to top, but on a phone - from top to bottom?

The numbers on the calculator increase from bottom to top, and on the phone keyboard - from top to bottom. This is explained by the fact that calculators evolved from mechanical adding machines, where numbers were historically usually arranged from bottom to top. Telephones were equipped with a dial for a long time, and when it became possible to produce push-button devices with tone dialing, they decided to arrange the numbers on the buttons by analogy with the dial - in ascending order from top to bottom with a zero at the end.

In many sources, often with the purpose of encouraging poorly performing students, there is a statement that Einstein failed mathematics at school or, moreover, generally studied very poorly in all subjects. In fact, everything was not like that: Albert began to show talent in mathematics at an early age and knew it far beyond the school curriculum. Later, Einstein was unable to enter the Swiss Higher Polytechnic School of Zurich, showing the highest results in physics and mathematics, but not achieving the required number of points in other disciplines. Having mastered these subjects, a year later, at the age of 17, he became a student at this institution.

The decimal number system we use arose because humans have 10 fingers. The ability for abstract counting did not appear in people right away, and it turned out to be most convenient to use fingers for counting. The Mayan civilization and, independently of them, the Chukchi historically used the twenty-digit number system, using fingers not only on the hands, but also on the toes. The duodecimal and sexagesimal systems common in ancient Sumer and Babylon were also based on the use of hands: the phalanges of the other fingers of the palm, the number of which is 12, were counted with the thumb.

To get the opportunity to engage in science, Sofya Kovalevskaya had to enter into a fictitious marriage and leave Russia. At that time, Russian universities simply did not accept women, and in order to emigrate, a girl had to have the consent of her father or husband. Since Sophia's father was categorically against it, she married the young scientist Vladimir Kovalevsky. Although in the end their marriage became de facto, and they had a daughter.

One of the most laconic letters of recommendation from the university was received by mathematician John Nash, the prototype of the hero of the film “A Beautiful Mind.” The teacher wrote one line in it: “This man is a genius!”

The English mathematician Abraham de Moivre, in his old age, once discovered that the duration of his sleep increased by 15 minutes per day. Having made an arithmetic progression, he determined the date when it would reach 24 hours - November 27, 1754. On this day he died.

Pi has two unofficial holidays. The first is March 14, because this day in America is written as 3.14. The second is July 22, which is written in European format as 22/7, and the value of such a fraction is a fairly popular approximate value of Pi.

Accuracy - the politeness of kings

The great commander Alexander Suvorov loved precision in everything. His step on the march was equal to 1 arshin, that is, 71 cm. In the army they still say “Suvorov’s step.” In everyday life, we also measure distances in steps. The stride is the distance between the heels and toes of a walking person. Thus, the duel between Pushkin and Dantes took place at a distance of 10 steps, that is, 10 arshins, and between Lermontov and Martynov - at a distance of 15 steps.

To the memory box

Span - this is the distance between the outstretched thumb and index fingers.

Inch – means “thumb”, equal to 25mm.

Foot – this is the average length of an adult man’s foot, equal to 30cm 48mm.

Arshin – equal to 71 cm.

Elbow – this is the distance from the ends of the fingers to the elbow of the bent arm, equal to 45 cm.

Fathom – this is the distance between the thumbs of a person’s outstretched hands, equal to 2m 23cm.

Verst – measured long distances. The name comes from the verb “virtjeti”, which could mean “turning the plow”.

Yard – this is the distance from the nose to the thumb of an outstretched hand, equal to 91cm 44mm.