What is arithmetic? When did humanity start using numbers and working with them? Where do the roots of such everyday concepts as numbers, addition and multiplication, which a person has made an inseparable part of his life and worldview, go? Ancient Greek minds admired such sciences as geometry as the most beautiful symphonies of human logic.

Perhaps arithmetic is not as deep as other sciences, but what would happen to them if a person forgets the elementary multiplication table? The logical thinking habitual to us, using numbers, fractions and other tools, was not easy for people and long time was inaccessible to our ancestors. In fact, before the development of arithmetic, no area of ​​human knowledge was truly scientific.

Arithmetic is the ABC of mathematics

Arithmetic is the science of numbers, with which any person begins acquaintance with the fascinating world of mathematics. As M. V. Lomonosov said, arithmetic is the gate of learning, opening the way to world knowledge for us. But he is right, can the knowledge of the world be separated from the knowledge of numbers and letters, mathematics and speech? Perhaps in the old days, but not in the modern world, where the rapid development of science and technology dictates its own laws.

The word "arithmetic" (Greek "arithmos") of Greek origin, means "number". She studies numbers and everything that can be connected with them. This is the world of numbers: various operations on numbers, numerical rules, solving problems that are related to multiplication, subtraction, etc.

Basic object of arithmetic

The basis of arithmetic is an integer, the properties and patterns of which are considered in higher arithmetic or In fact, the strength of the whole building - mathematics depends on how correct the approach is taken in considering such a small block as a natural number.

Therefore, the question of what arithmetic is can be answered simply: it is the science of numbers. Yes, about the usual seven, nine and all this diverse community. And just as you cannot write good or even the most mediocre poetry without an elementary alphabet, you cannot solve even an elementary problem without arithmetic. That is why all the sciences advanced only after the development of arithmetic and mathematics, having before that been just a set of assumptions.

Arithmetic is a phantom science

What is arithmetic - natural science or phantom? In fact, as the ancient Greek philosophers argued, neither numbers nor figures exist in reality. This is just a phantom that is created in human thinking when considering environment with its processes. In fact, nowhere around we see anything like that, which could be called a number, rather, the number is a way of the human mind to study the world. Or maybe it is the study of ourselves from the inside? Philosophers have been arguing about this for many centuries in a row, so we do not undertake to give an exhaustive answer. One way or another, arithmetic has managed to take its positions so firmly that in the modern world no one can be considered socially adapted without knowing its foundations.

How did the natural number come about?

Of course, the main object that arithmetic operates on is a natural number, such as 1, 2, 3, 4, ..., 152 ... etc. The arithmetic of natural numbers is the result of counting ordinary objects, such as cows in a meadow. Still, the definition of "a lot" or "little" once ceased to suit people, and they had to invent more advanced counting techniques.

But a real breakthrough happened when human thought reached the point that it is possible to designate 2 kilograms, and 2 bricks, and 2 details with the same number “two”. The fact is that you need to abstract from the forms, properties and meaning of objects, then you can perform some actions with these objects in the form of natural numbers. Thus was born the arithmetic of numbers, which further developed and expanded, occupying ever greater positions in the life of society.

Such profound concepts of number as zero and negative number, fractions, designations of numbers by numbers and in other ways, have a rich and interesting history of development.

Arithmetic and Practical Egyptians

The two oldest human companions in the study of the world around us and in solving everyday problems are arithmetic and geometry.

It is believed that the history of arithmetic originates in the Ancient East: in India, Egypt, Babylon and China. Thus, the Rinda papyrus of Egyptian origin (so named because it belonged to the owner of the same name), dating back to the 20th century. BC, in addition to other valuable data, contains the expansion of one fraction into the sum of fractions with different denominators and a numerator equal to one.

For example: 2/73=1/60+1/219+1/292+1/365.

But what is the meaning of such a complex decomposition? The fact is that the Egyptian approach did not tolerate abstract thoughts about numbers, on the contrary, calculations were made only for practical purposes. That is, the Egyptian will engage in such a thing as calculations, solely in order to build a tomb, for example. It was necessary to calculate the length of the edge of the structure, and this forced a person to sit down behind the papyrus. As you can see, the Egyptian progress in the calculations was caused, rather, by mass construction, rather than by love for science.

For this reason, the calculations found on the papyri cannot be called reflections on the theme of fractions. Most likely, this is a practical preparation, which helped to solve problems with fractions in the future. The ancient Egyptians, who did not know the multiplication tables, made rather long calculations, decomposed into many subtasks. Perhaps this is one of those subtasks. It is easy to see that calculations with such workpieces are very laborious and unpromising. Perhaps for this reason we do not see the great contribution of Ancient Egypt to the development of mathematics.

Ancient Greece and philosophical arithmetic

Many knowledge of the Ancient East was successfully mastered by the ancient Greeks, known lovers of abstract, abstract and philosophical reflections. They were no less interested in practice, but it is difficult to find the best theorists and thinkers. This has benefited science, since it is impossible to delve into arithmetic without breaking it off from reality. Of course, you can multiply 10 cows and 100 liters of milk, but you won't get very far.

The deep-thinking Greeks left a significant mark on history, and their writings have come down to us:

• Euclid and the Elements.
• Pythagoras.
• Archimedes.
• Eratosthenes.
• Zeno.
• Anaxagoras.

And, of course, the Greeks, who turned everything into philosophy, and especially the successors of the work of Pythagoras, were so fascinated by numbers that they considered them the mystery of the harmony of the world. Numbers have been studied and researched to such an extent that some of them and their pairs have been assigned special properties. For example:

• Perfect numbers are those that are equal to the sum of all their divisors, except for the number itself (6=1+2+3).
• Friendly numbers are such numbers, one of which is equal to the sum of all divisors of the second, and vice versa (the Pythagoreans knew only one such pair: 220 and 284).

The Greeks, who believed that science should be loved, and not be with it for the sake of profit, achieved great success by exploring, playing and adding numbers. It should be noted that not all of their research was widely used, some of them remained only "for beauty".

Eastern thinkers of the Middle Ages

In the same way, in the Middle Ages, arithmetic owed its development to Eastern contemporaries. The Indians gave us the numbers that we actively use, such a concept as "zero", and a positional variant familiar to modern perception. From Al-kashi, who worked in Samarkand in the 15th century, we inherited without which it is difficult to imagine modern arithmetic.

In many ways, Europe's acquaintance with the achievements of the East became possible thanks to the work of the Italian scientist Leonardo Fibonacci, who wrote the work "The Book of the Abacus", introducing Eastern innovations. It became the cornerstone of the development of algebra and arithmetic, research and scientific activities in Europe.

Russian arithmetic

And, finally, arithmetic, which found its place and took root in Europe, began to spread to the Russian lands. The first Russian arithmetic was published in 1703 - it was a book about arithmetic by Leonty Magnitsky. For a long time it remained the only textbook in mathematics. It contains the initial moments of algebra and geometry. The numbers used in the examples by the first arithmetic textbook in Russia are Arabic. Although Arabic numerals were found earlier, on engravings dating back to the 17th century.

The book itself is decorated with images of Archimedes and Pythagoras, and on the first sheet there is an image of arithmetic in the form of a woman. She sits on a throne, under her is written in Hebrew a word denoting the name of God, and on the steps that lead to the throne, the words “division”, “multiplication”, “addition”, etc. are inscribed. truths that are now considered commonplace.

The 600-page textbook covers both basics like the addition and multiplication tables and applications to navigational sciences.

It is not surprising that the author chose images of Greek thinkers for his book, because he himself was captivated by the beauty of arithmetic, saying: “Arithmetic is a numerator, there is honest, unenviable art ...”. This approach to arithmetic is quite justified, because it is its widespread introduction that can be considered the beginning of the rapid development of scientific thought in Russia and general education.

Non-prime primes

A prime number is a natural number that has only 2 positive divisors: 1 and itself. All other numbers, except for 1, are called composite. Examples of prime numbers: 2, 3, 5, 7, 11, and all others that have no other divisors other than 1 and itself.

As for the number 1, it is on a special account - there is an agreement that it should be considered neither simple nor composite. Simple at first glance, a simple number hides many unsolved mysteries within itself.

Euclid's theorem says that there are an infinite number of prime numbers, and Eratosthenes came up with a special arithmetic "sieve" that eliminates non-prime numbers, leaving only simple ones.

Its essence is to underline the first non-crossed out number, and subsequently to cross out those that are multiples of it. We repeat this procedure many times - and we get a table of prime numbers.

Fundamental theorem of arithmetic

Among the observations about prime numbers, the fundamental theorem of arithmetic must be mentioned in a special way.

The fundamental theorem of arithmetic says that any integer greater than 1 is either prime or can be decomposed into a product of primes up to the order of the factors, and in a unique way.

The basic theorem of arithmetic is proved rather cumbersome, and its understanding is no longer similar to the simplest foundations.

At first glance, prime numbers are an elementary concept, but they are not. Physics also once considered the atom to be elementary, until it found the whole universe inside it. Prime numbers are the subject of a wonderful story by mathematician Don Tzagir "The First Fifty Million Primes".

From "three apples" to deductive laws

What can truly be called the reinforced foundation of all science is the laws of arithmetic. Even in childhood, everyone encounters arithmetic, studying the number of legs and arms of dolls, the number of cubes, apples, etc. This is how we study arithmetic, which then turns into more complex rules.

Our whole life acquaints us with the rules of arithmetic, which have become for the common man the most useful of all that science gives. The study of numbers is "arithmetic-baby", which introduces a person to the world of numbers in the form of numbers in early childhood.

Higher arithmetic is a deductive science that studies the laws of arithmetic. Most of them are known to us, although we may not know their exact wording.

Law of addition and multiplication

Any two natural numbers a and b can be expressed as a sum a + b, which will also be a natural number. The following laws apply to addition:

• commutative, which says that the sum does not change from the rearrangement of terms, or a + b \u003d b + a.
• Associative, which says that the sum does not depend on the way the terms are grouped in places, or a+(b+c)= (a+ b)+ c.

The rules of arithmetic, such as addition, are among the elementary ones, but they are used by all sciences, not to mention everyday life.

Any two natural numbers a and b can be expressed as a product a*b or a*b, which is also a natural number. The same commutative and associative laws apply to the product as to addition:

• a*b=b*a;
• a*(b*c)= (a* b)* c.

Interestingly, there is a law that combines addition and multiplication, also called distributive, or distributive law:

a(b+c)=ab+ac

This law actually teaches us to work with brackets, opening them, thus we can work with more complex formulas. These are precisely the laws that will guide us through the bizarre and complex world of algebra.

Law of arithmetic order

The law of order is used by human logic every day, comparing watches and counting banknotes. And, nevertheless, and it needs to be issued in the form of specific formulations.

If we have two natural numbers a and b, then the following options are possible:

• a is equal to b, or a=b;
• a is less than b, or a< b;
• a is greater than b, or a > b.

Of the three options, only one can be fair. The basic law that governs the order says: if a< b и b < c, то a< c.

There are also laws relating order to the operations of multiplication and addition: if a< b, то a + c < b+c и ac< bc.

The laws of arithmetic teach us to work with numbers, signs and brackets, turning everything into a harmonious symphony of numbers.

Positional and non-positional calculus systems

We can say that numbers are a mathematical language, on the convenience of which much depends. There are many number systems that, like alphabets, different languages, differ from each other.

Consider the number systems from the point of view of the influence of the position on the quantitative value of the digit at this position. So, for example, the Roman system is non-positional, where each number is encoded by a certain set of special characters: I/ V/ X/L/ C/ D/ M. They are equal, respectively, to the numbers 1/ 5/10/50/100/500/ 1000. In such a system, the number does not change its quantitative definition depending on what position it is in: first, second, etc. To get other numbers, you need to add the base ones. For example:

• DCC=700.
• CCM=800.

The number system more familiar to us using Arabic numerals is positional. In such a system, the digit of a number determines the number of digits, for example, three-digit numbers: 333, 567, etc. The weight of any digit depends on the position in which this or that digit is located, for example, the number 8 in the second position has the value 80. This is typical for the decimal system, there are other positional systems, such as binary.

Binary arithmetic

Binary arithmetic works with the binary alphabet, which consists of only 0 and 1. And the use of this alphabet is called the binary number system.

The difference between binary arithmetic and decimal arithmetic is that the significance of the position on the left is no longer 10, but 2 times. Binary numbers are of the form 111, 1001, etc. How to understand such numbers? So, consider the number 1100:

1. The first digit on the left is 1 * 8 = 8, remembering that the fourth digit, which means it needs to be multiplied by 2, we get position 8.
2. Second digit 1*4=4 (position 4).
3. Third digit 0*2=0 (position 2).
4. Fourth digit 0*1=0 (position 1).
5. So, our number is 1100=8+4+0+0=12.

That is, when moving to a new digit on the left, its significance in the binary system is multiplied by 2, and in decimal - by 10. Such a system has one drawback: this is too large an increase in the digits that are needed to write numbers. Examples of representing decimal numbers as binary numbers can be found in the following table.

Decimal numbers in binary form are shown below.

Both octal and hexadecimal systems are also used.

This mysterious arithmetic

What is arithmetic, "twice two" or the unknown secrets of numbers? As you can see, arithmetic may seem simple at first glance, but its unobvious ease is deceptive. It can also be studied by children along with Aunt Owl from the cartoon “Baby Arithmetic”, or you can immerse yourself in deeply scientific research of an almost philosophical order. In history, she has gone from counting objects to worshiping the beauty of numbers. Only one thing is known for sure: with the establishment of the basic postulates of arithmetic, all science can rely on its strong shoulder.

Of the more than 500 thousand clay tablets found by archaeologists during excavations in ancient Mesopotamia, about 400 contain mathematical information. Most of them have been deciphered and allow one to get a fairly clear idea of ​​the amazing algebraic and geometric achievements of the Babylonian scientists.

Opinions differ about the time and place of the birth of mathematics. Numerous researchers of this issue attribute its creation to various peoples and date it to different eras. The ancient Greeks did not yet have a single point of view on this matter, among whom the version was especially widespread that the Egyptians came up with geometry, and Phoenician merchants who needed such knowledge for trading calculations, and arithmetic. Herodotus in "History" and Strabo in "Geography" gave priority to the Phoenicians. Plato and Diogenes Laertius considered Egypt to be the birthplace of both arithmetic and geometry. This is also the opinion of Aristotle, who believed that mathematics was born due to the presence of leisure among the local priests.

This remark follows the passage that in every civilization the practical crafts are born first, then the pleasure arts, and only then the sciences aimed at knowledge. Eudemus, a student of Aristotle, like most of his predecessors, also considered Egypt to be the birthplace of geometry, and the reason for its appearance was the practical needs of land surveying. According to Evdem, geometry goes through three stages in its improvement: the emergence of practical skills in land surveying, the emergence of a practically oriented applied discipline and its transformation into a theoretical science. To all appearances, Eudemus attributed the first two stages to Egypt, and the third to Greek mathematics. True, he nevertheless admitted that the theory of calculating areas arose from the solution of quadratic equations, which were of Babylonian origin.

Small clay plaques found in Iran were supposedly used to record grain measurements from 8000 BC. Norwegian Institute of Palaeography and History,
Oslo.

The historian Joseph Flavius ​​("Ancient Judea", book 1, ch. 8) has his own opinion. Although he calls the Egyptians the first, he is sure that they were taught arithmetic and astronomy by the forefather of the Jews, Abraham, who fled to Egypt during the famine that befell the land of Canaan. Well, the Egyptian influence in Greece was strong enough to impose on the Greeks a similar opinion, which, with their light hand, is still in circulation in historical literature. Well-preserved clay tablets covered with cuneiform texts found in Mesopotamia and dated from 2000 BC. and before 300 AD, testify both to a somewhat different state of affairs, and to what mathematics was like in ancient Babylon. It was a rather complex alloy of arithmetic, algebra, geometry, and even the rudiments of trigonometry.

Mathematics was taught in scribe schools, and each graduate had a fairly serious amount of knowledge for that time. Apparently, this is exactly what Ashurbanipal, the king of Assyria in the 7th century, is talking about. BC, in one of his inscriptions, saying that he learned to find "complex reciprocals and multiply." To resort to calculations, life forced the Babylonians at every turn. Arithmetic and simple algebra were needed in housekeeping, when exchanging money and paying for goods, calculating simple and compound interest, taxes, and the share of the crop handed over to the state, temple or landowner. Mathematical calculations, and rather complex ones, were required for large-scale architectural projects, engineering work during the construction of an irrigation system, ballistics, astronomy, and astrology.

An important task of mathematics was to determine the timing of agricultural work, religious holidays, and other calendar needs. How high achievements were in the ancient city-states between the Tigris and Euphrates in what the Greeks would later call so surprisingly accurately mathema ("knowledge"), let us judge the deciphering of Mesopotamian clay cuneiforms. By the way, among the Greeks, the term mathema at first denoted a list of four sciences: arithmetic, geometry, astronomy and harmonics, it began to denote mathematics itself much later. In Mesopotamia, archaeologists have already found and continue to find cuneiform tablets with records of a mathematical nature, partly in Akkadian, partly in Sumerian, as well as mathematical reference tables. The latter greatly facilitated the calculations that had to be done on a daily basis, so a number of deciphered texts quite often contain the calculation of interest.

The names of the arithmetic operations of the earlier, Sumerian period of Mesopotamian history have been preserved. So, the operation of addition was called "accumulation" or "addition", when subtracting, the verb "pull out" was used, and the term for multiplication meant "eat." It is interesting that in Babylon they used a more extensive multiplication table - from 1 to 180,000 than the one that we had to learn at school, i.e. calculated on numbers from 1 to 100. In ancient Mesopotamia, uniform rules for arithmetic operations were created not only with integers, but also with fractions, in the art of operating with which the Babylonians were significantly superior to the Egyptians. In Egypt, for example, operations with fractions continued to remain primitive for a long time, since they only knew aliquot fractions (i.e., fractions with a numerator equal to 1). Since the time of the Sumerians in Mesopotamia, the main counting unit in all economic affairs was the number 60, although the decimal number system was also known, which was used by the Akkadians.

The most famous of the mathematical tablets of the Old Babylonian period, stored in the library of Columbia University (USA). Contains a list of right triangles with rational sides, that is, triples of Pythagorean numbers x2 + y2 = z2 and indicates that the Pythagorean theorem was known to the Babylonians at least a thousand years before the birth of its author. 1900 - 1600 BC.

Babylonian mathematicians widely used the sexagesimal positional (!) counting system. On its basis, various calculation tables were compiled. In addition to multiplication tables and tables of reciprocals, with the help of which division was carried out, there were tables of square roots and cubic numbers. Cuneiform texts devoted to solving algebraic and geometric problems indicate that Babylonian mathematicians were able to solve some special problems, including up to ten equations with ten unknowns, as well as certain varieties of cubic equations and equations of the fourth degree. At first, quadratic equations served mainly purely practical purposes - the measurement of areas and volumes, which was reflected in the terminology. For example, when solving equations with two unknowns, one was called "length" and the other was called "width." The product of the unknowns was called the "area". Just like now!

In tasks leading to a cubic equation, there was a third unknown quantity - "depth", and the product of three unknowns was called "volume". Later, with the development of algebraic thinking, the unknowns began to be understood more abstractly. Sometimes, as an illustration of algebraic relations in Babylon, geometric drawings were used. Later, in Ancient Greece they became the main element of algebra, while for the Babylonians, who thought primarily algebraically, drawings were only a means of visualization, and the terms “line” and “area” most often understood dimensionless numbers. That is why there were solutions to problems where the "area" was added to the "side" or subtracted from the "volume", etc. Of particular importance in ancient times was the accurate measurement of fields, gardens, buildings - the annual floods of the rivers brought a large amount of silt that covered the fields and destroyed the boundaries between them, and after the decline in water, land surveyors, by order of their owners, often had to re-measure allotments. In the cuneiform archives, many such land surveying maps, compiled over 4 thousand years ago, have been preserved.

Initially, the units of measurement were not very accurate, because the length was measured with fingers, palms, elbows, which are different for different people. The situation was better with large quantities, for the measurement of which they used a reed and a rope of certain sizes. But here, too, the measurement results often differed from each other, depending on who measured and where. Therefore, different measures of length were adopted in different cities of Babylonia. For example, in the city of Lagash, the "cubit" was 400 mm, and in Nippur and Babylon itself - 518 mm. Many surviving cuneiform materials were textbooks for Babylonian schoolchildren, which provided solutions to various simple problems that were often encountered in practical life. It is not clear, however, whether the student solved them in his mind or did preliminary calculations with a twig on the ground - only the conditions of mathematical problems and their solution are written on the tablets.

Geometric problems with drawings of trapezoids and triangles and the solution of the Pythagorean theorem. Plate dimensions: 21.0x8.2. 19th century BC. British museum

The main part of the mathematics course at school was occupied by the solution of arithmetic, algebraic and geometric problems, in the formulation of which it was customary to operate with specific objects, areas and volumes. On one of the cuneiform tablets, the following problem was preserved: “In how many days can a piece of fabric of a certain length be made if we know that so many cubits (a measure of length) of this fabric are made daily?” The other shows tasks related to construction work. For example, "How much earth will be needed for an embankment, the dimensions of which are known, and how much earth must each worker move, if their total number is known?" or “How much clay should each worker prepare to build a wall of a certain size?”

The student also had to be able to calculate coefficients, calculate totals, solve problems on measuring angles, calculating areas and volumes of rectilinear figures - this was a common set for elementary geometry. The names of geometric figures preserved from Sumerian times are interesting. The triangle was called the “wedge”, the trapezoid was called the “forehead of the bull”, the circle was called the “hoop”, the container was denoted by the term “water”, the volume was “earth, sand”, the area was called the “field”. One of the cuneiform texts contains 16 problems with solutions that relate to dams, ramparts, wells, water clocks and earthworks. One task is provided with a drawing relating to a circular shaft, another considers a truncated cone, determining its volume by multiplying the height by half the sum of the areas of the upper and lower bases.

Babylonian mathematicians also solved planimetric problems using the properties of right triangles, subsequently formulated by Pythagoras in the form of a theorem on the equality in a right triangle of the square of the hypotenuse to the sum of the squares of the legs. In other words, the famous Pythagorean theorem was known to the Babylonians at least a thousand years before Pythagoras. In addition to planimetric problems, they also solved stereometric problems related to determining the volume of various kinds of spaces, bodies, and widely practiced drawing plans for fields, areas, individual buildings, but usually not to scale. The most significant achievement of mathematics was the discovery of the fact that the ratio of the diagonal and the side of a square cannot be expressed as a whole number or a simple fraction. Thus, the concept of irrationality was introduced into mathematics.

It is believed that the discovery of one of the most important irrational numbers - the number π, expressing the ratio of the circumference of a circle to its diameter and equal to an infinite fraction ≈ 3.14 ..., belongs to Pythagoras. According to another version, for the number π, the value 3.14 was first proposed by Archimedes 300 years later, in the 3rd century BC. BC. According to another, Omar Khayyam was the first to calculate it, this is generally the 11th - 12th century. AD It is only known for certain that the Greek letter π first denoted this ratio in 1706 by the English mathematician William Jones, and only after the Swiss mathematician Leonhard Euler borrowed this designation in 1737 did it become generally accepted. The number π is the oldest mathematical riddle, this discovery should also be sought in ancient Mesopotamia.

Babylonian mathematicians were well aware of the most important irrational numbers, and the solution to the problem of calculating the area of ​​a circle can also be found in the decoding of cuneiform clay tablets of mathematical content. According to these data, π was taken equal to 3, which, however, was quite sufficient for practical land surveying purposes. Researchers believe that the sexagesimal system was chosen in ancient Babylon for metrological reasons: the number 60 has many divisors. Hexadecimal notation of integers did not become widespread outside of Mesopotamia, but in Europe until the 17th century. both sexagesimal fractions and the usual division of the circle into 360 degrees were widely used. The hour and minutes, divided into 60 parts, also originate in Babylon.

The ingenious idea of ​​the Babylonians to use the minimum number of digital characters to write numbers is remarkable. The Romans, for example, did not even think that the same number can denote different quantities! To do this, they used the letters of their alphabet. As a result, a four-digit number, for example, 2737 contained as many as eleven letters: MMDCCXXXVII. And although in our time there are extreme mathematicians who will be able to divide LXXVIII into a column by CLXVI or multiply CLIX by LXXIV, one can only feel sorry for those residents of the Eternal City who had to perform complex calendar and astronomical calculations with the help of such mathematical balancing act or calculated large-scale architectural projects and various engineering objects.

The Greek number system was also based on the use of the letters of the alphabet. Initially, the Attic system was adopted in Greece, which used a vertical line to designate a unit, and for the numbers 5, 10, 100, 1000, 10,000 (essentially it was a decimal system) - the initial letters of their Greek names. Later, around the 3rd c. BC, the Ionic number system became widespread, in which 24 letters of the Greek alphabet and three archaic letters were used to denote numbers. And to distinguish numbers from words, the Greeks placed a horizontal line over the corresponding letter. In this sense, Babylonian mathematical science stood above the later Greek or Roman, since it is she who owns one of the most outstanding achievements in the development of number notation systems - the principle of positionality, according to which the same numerical sign (symbol) has different meanings depending on whether the place where it is located. By the way, the Egyptian number system was inferior to the Babylonian and the modern Egyptian number system.

The Egyptians used a non-positional decimal system, in which the numbers from 1 to 9 were denoted by the corresponding number of vertical lines, and individual hieroglyphic symbols were introduced for successive powers of 10. For small numbers, the Babylonian number system in general terms resembled the Egyptian one. One vertical wedge-shaped line (in the early Sumerian tablets - a small semicircle) meant a unit; repeated the required number of times, this sign served to write numbers less than ten; to designate the number 10, the Babylonians, like the Egyptians, introduced a new symbol - a wide wedge-shaped sign with a point directed to the left, resembling an angle bracket in shape (in early Sumerian texts - a small circle). Repeated an appropriate number of times, this sign served to designate the numbers 20, 30, 40 and 50. Most modern historians believe that ancient scientific knowledge was purely empirical in nature.

With regard to physics, chemistry, natural philosophy, which were based on observations, it seems to be true. But the concept of sensory experience as the source of knowledge faces an insoluble question when it comes to such an abstract science as mathematics operating with symbols. Especially significant were the achievements of Babylonian mathematical astronomy. But whether the sudden leap raised Mesopotamian mathematicians from the level of utilitarian practice to a vast knowledge, allowing them to apply mathematical methods to predict the positions of the Sun, Moon and planets, eclipses and other celestial phenomena, or whether the development proceeded gradually, we unfortunately do not know. The history of mathematical knowledge in general looks strange.

We know how our ancestors learned to count on their fingers and toes, making primitive numerical records in the form of notches on a stick, knots on a rope, or pebbles laid out in a row. And then - without any transitional link - suddenly information about the mathematical achievements of the Babylonians, Egyptians, Chinese, Hindus and other ancient scientists, so solid that their mathematical methods withstood the test of time until the middle of the recently ended II millennium, i.e. for more than than three thousand years...

What is hidden between these links? Why did the ancient sages, in addition to practical significance, revere mathematics as sacred knowledge, and gave the names of gods to numbers and geometric figures? Is it just behind this a reverent attitude towards Knowledge as such? Perhaps the time will come when archaeologists will find answers to these questions. In the meantime, let's not forget what Oxfordian Thomas Bradwardine said 700 years ago: "He who has the shamelessness to deny mathematics should have known from the very beginning that he would never enter the gates of wisdom."

Municipal Autonomous educational institution

secondary school No. 211 named after L.I. Sidorenko

Novosibirsk

Research:

Does mental arithmetic develop the mental abilities of a child?

Section "Mathematics"

The project was completed by:

Klimova Ruslana

3rd "B" class student

MAOU secondary school No. 211

named after L.I. Sidorenko

Project Manager:

Vasilyeva Elena Mikhailovna

Novosibirsk 2017

Introduction 3

2. Theoretical part

2.1 History of arithmetic 3

2.2 First counting devices 4

2.3 Abacus 4

2.4 What is mental arithmetic? five

3. Practical part

3.1 Classes at the school of mental arithmetic 6

3.2 Lesson summary 6

4. Conclusions on the project 7.8

5. List of used literature 9

1. INTRODUCTION

Last summer, my grandmother and mother and I watched the program “Let them talk”, where a 9-year-old boy, Daniyar Kurmanbaev from Astana, counted in his mind (mentally) faster than a calculator, while doing manipulations with the fingers of both hands. And in the program they talked about an interesting method for developing mental abilities - about mental arithmetic.

It struck me and my mother and I became interested in this technique.

It turned out that in our city there are 4 schools where they teach mental counting tasks and examples of any complexity. These are Abacus, AmaKids, Pythagoras, Menard. Classes in schools are not cheap. My parents and I chose a school so that it was close to home, classes were not very expensive, so that there were real reviews about the teaching program, as well as certified teachers. In all respects, the Menard school was suitable.

I asked my mother to enroll me in this school, because I really wanted to learn how to count quickly, improve my performance at school and discover something new.

The technique of mental arithmetic is more than five hundred years old. This technique is a system of oral counting. Training in mental arithmetic is carried out in many countries of the world - in Japan, the USA and Germany, Kazakhstan. In Russia, they are just beginning to master it.

Objective of the project: to find out:

Does mental arithmetic develop the mental abilities of a child?

Project object: student 3 "B" class MAOU secondary school No. 211 Klimova Ruslana.

Subject of study: mental arithmetic - a system of mental counting.

Research objectives:

Learn how mental arithmetic is taught;

Understand whether mental arithmetic develops the mental abilities of a child?

Find out if it is possible to learn mental arithmetic on your own at home?

2.1 HISTORY OF ARITHMETIC

In every case, you need to know the history of its development.

Arithmetic originated in the countries of the Ancient East: Babylon, China, India, Egypt.

Arithmetic studies numbers and operations on numbers, various rules for handling them, teaches how to solve problems that reduce to addition, subtraction, multiplication and division of numbers.

The name "arithmetic" comes from the Greek word (arithmos) - number.

The emergence of arithmetic is associated with the labor activity of people and with the development of society.

The importance of mathematics in everyday life is great. Without counting, without the ability to correctly add, subtract, multiply and divide numbers, the development of human society is unthinkable. We study the four arithmetic operations, the rules of oral and written calculations, starting from elementary grades. All these rules were not invented or discovered by any one person. Arithmetic originated from the daily life of people.

Ancient people obtained their food mainly by hunting. The whole tribe had to hunt for a large animal - a bison or an elk: you cannot cope with it alone. In order for the prey not to leave, it had to be surrounded, well, at least like this: five people on the right, seven behind, four on the left. Here you can't do without an account! And the leader of the primitive tribe coped with this task. Even in those days when a person did not know such words as "five" or "seven", he could show the numbers on his fingers.

The basic object of arithmetic is the number.

2.2 FIRST COUNTING DEVICES

People have long tried to ease their account with the help of various means and devices. The first, most ancient "calculating machine" were the fingers and toes. This simple device was quite enough - for example, to count the mammoths killed by the entire tribe.

Then there was trade. And ancient merchants (Babylonian and other cities) made calculations using grains, pebbles and shells, which they began to lay out on a special board called an abacus.

The analogue of the abacus in ancient China was the counting device "su-anpan", in ancient China - the Japanese abacus, called "soroban".

Russian abacus first appeared in Russia in the 16th century. They were a board with parallel lines drawn on it. Later, instead of the board, they began to use a frame with wires and bones.

2.3 ABACUS

Word "abacus" (abacus) means scoreboard.

Let's look at the modern abacus...

To learn how to use accounts, you need to know what they are.

Accounts consist of:

• dividing line;

  upper bones;

  lower bones.

There is a center point in the middle. The top bones represent fives, and the bottom bones represent ones. Each vertical strip of bones, starting from right to left, denotes one of the digits of the numbers:

• tens of thousands, etc.

For example, to postpone the example: 9 - 4=5, you need to move the top bone on the first line on the right (it means five) and raise the 4 lower bones. Then lower the 4 lower bones. So we get the required number 5.

The mental abilities of children develop through the ability to count in the mind. To train both hemispheres, you need to constantly engage in solving arithmetic problems. In a short time, the child will already be solving complex problems without using a calculator.

2.4 WHAT IS MENTAL ARITHMETIC?

mental arithmetic- This is a method for developing the mental abilities of children from 4 to 14 years old. The basis of mental arithmetic is the abacus score. The child counts on the abacus with both hands, making calculations twice as fast. On the abacus, children not only add and subtract, but also learn to multiply and divide.

mentality - it is the mental capacity of man.

During mathematics lessons, only the left hemisphere of the brain, which is responsible for logical thinking, develops, while the right hemisphere develops such subjects as literature, music, and drawing. There are special training techniques that are aimed at developing both hemispheres. Scientists say that those people who have fully developed both hemispheres of the brain achieve success. Many people have a more developed left hemisphere and a less developed right.

There is an assumption that mental arithmetic allows you to use both hemispheres, performing calculations of varying complexity.
The use of an abacus makes the left hemisphere work - it develops fine motor skills and allows the child to visually see the counting process.
Skills are trained gradually with the transition from simple to complex. As a result, by the end of the program, the child can mentally add, subtract, multiply and divide three- and four-digit numbers.

Therefore, I decided to go to classes in the school of mental arithmetic. Since I really wanted to learn how to quickly learn poetry, develop my logic, develop determination, and also develop some qualities of my personality.

3. 1 LESSONS AT THE SCHOOL OF MENTAL ARITHMETICS

My mental arithmetic lessons took place in classrooms equipped with computers, a television, a magnetic whiteboard, and a large teacher's abacus. Diplomas of teacher education and certificates of a teacher, as well as patents for the use of international methods of mental arithmetic, hang on the wall near the classrooms.

At a trial lesson, the teacher showed me and my mother an abacus abacus, briefly told how to use them and the very principle of counting.

The training is structured as follows: once a week for 2 hours I studied in a group of 6 people. In the lessons we used the abacus (accounts). By moving the bones on the abacus with their fingers (fine motor skills), they learned to perform arithmetic operations physically.

There was a mental warm-up at the lesson. And there were always breaks where we could have a little snack, drink water or play games. At home, we were always given sheets with examples, for independent work Houses.

In 1 month of training I:

met with accounts. I learned how to use my hands correctly when counting: with the thumb of both hands we raise the knuckles on the abacus, with the index fingers we lower the knuckles.

During the 2nd month of training I:

learned to count two-stage examples with tens. Tens are located on the second needle from the far right. When counting with tens, we already use the thumb and forefinger of the left hand. Here the technique is the same as with the right hand: we raise it with a large one, we lower it with our index one.

In the 3rd month of training I:

solved on the abacus examples of subtraction and addition with units and tens - three-stage.

Solve examples of subtraction and addition with thousandths - two-stage

In the 4th month of study:

Get to know the mind map. Looking at the card, I had to mentally move the knuckles and see the answer.

Also, in the classes on mental arithmetic, she trained to work on a computer. There is a program installed where the number of numbers for the account is set. The frequency of their display is 2 seconds, I watch, remember and count. While counting on the accounts. Give 3, 4 and 5 numbers. The numbers are still single digits.

In mental arithmetic, more than 20 formulas are used for calculations (close relatives, help from a brother, help from a friend, etc.) that need to be remembered.

3.2 LESSON CONCLUSIONS

I worked out 2 hours a week and 5-10 minutes a day on my own for 4 months.

First month of training	fourth month
1. I count on the abacus 1 sheet (30 examples)
2. Mentally count 1 sheet (10 examples)
3. I am learning a poem (3rd quatrains)
	20-30 minutes
4. Doing homework (mathematics: one task, 10 examples)
40-50 minutes

4. CONCLUSIONS ON THE PROJECT

1) I was interested in logic puzzles, puzzles, crossword puzzles, games to find differences. I became more diligent, attentive and collected. My memory has improved.

2) The purpose of mental mathematics is to develop the child's brain. While doing mental arithmetic, we develop our skills:

We develop logic and imagination by performing mathematical operations first on a real abacus, and then imagining the abacus in the mind. As well as solving logical problems in the classroom.

We improve concentration by performing arithmetic counting of a huge number of numbers on imaginary abacuses.

Memory improves. After all, all pictures with numbers, after performing mathematical operations, are stored in memory.

The speed of thought. All "mental" mathematical operations are performed at a speed that is comfortable for children, which is gradually increased and the brain "accelerates".

3) At the lessons in the center, teachers create a special playful atmosphere, and sometimes children, even against their will, are included in this fascinating environment.

Unfortunately, such interest in studies cannot be realized when studying independently.

There are many video courses on the Internet and on the YouTube channel with which you can understand how to count on an abacus.

You can learn this technique on your own, but it will be very difficult! First, it is necessary that mom or dad understand the essence of mental arithmetic - they learn to add, subtract, multiply and divide themselves. Books and videos can help them with this. The instructional video of the lessons demonstrates at a slow pace how to work with the abacus. Of course, videos are preferable to books, as everything is clearly shown on it. And then they explained it to the child. But adults are very busy, so this is not an option.

It's hard without a teacher-instructor! After all, the teacher in the classroom monitors the correct operation of both hands, corrects, if necessary. Another extremely important thing is the correct setting of the counting technique, as well as the timely correction of incorrect skills.

The 10-level program is designed for 2-3 years, it all depends on the child. All children are different, some are given quickly, while others need a little more time to master the program.

Our school now also has classes in mental arithmetic - this is the Formula Aikyu center at the Moscow Autonomous Educational Institution Secondary School No. L.I. Sidorenko. The method of mental arithmetic in this center was developed by Novosibirsk teachers and programmers, with the support of the Department of Education Novosibirsk region! And I started attending classes at school, as it is generally convenient for me.

For me, this technique is like an interesting way to improve my memory, increase concentration and develop my personality traits. And I will continue to do mental arithmetic!

And maybe my work will attract other children to mental arithmetic classes, which will affect their academic performance.

Literature:

Ivan Yakovlevich Depman. History of arithmetic. A guide for teachers. Second edition, corrected. M., Education, 1965 - 416 p.

Depman I. World of Numbers M.1966.

A. Benjamin. Secrets of mental mathematics. 2014. - 247 p. - ISBN: N/A.

"Mental arithmetic. Addition and subtraction "Part 1. Tutorial for children 4-6 years old.

G.I. Glaser. History of mathematics, Moscow: Education, 1982. - 240 p.

Karpushina N.M. Liber abaci by Leonardo Fibonacci. Journal "Mathematics at School" No. 4, 2008. Popular Science Department.

M. Kutorgi “On the accounts of the ancient Greeks” (“Russian Bulletin”, vol. SP, p. 901 et seq.)

Vygodsky M.L. "Arithmetic and algebra in the ancient world" M. 1967.

ABACUSxle - seminars on mental arithmetic.

UCMAS-ASTANA- articles.

Internet resources.

Popova L.A. 1

Koshkin I.A. 1

1 Municipal Budgetary Educational Institution "Education Center - Gymnasium No. 1"

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Introduction

Relevance. Mental arithmetic is now gaining great popularity. Thanks to new teaching methods, children quickly absorb new information, develop their creativity, learn to solve complex mathematical problems in their minds, without using a calculator.

Mental arithmetic is a unique method for developing the mental abilities of children from 4 to 16 years old, based on a system of mental counting. Learning by this technique, the child can solve any arithmetic problem in a few seconds (addition, subtraction, multiplication, division, calculating the square root of a number) in his mind faster than using a calculator.

Objective:

Learn the history of mental arithmetic

Show how you can use the abacus when solving mathematical problems

To analyze what other alternative methods of calculation are that simplify the calculation and make it entertaining

Hypothesis:

Let's assume that arithmetic can be fun and easy, you can calculate much faster and more productively using mental arithmetic methods and various tricks.

Classes with Chinese accounts have a positive effect on memory, which is reflected in the assimilation educational material. This applies to memorizing poetry and prose, theorems, various mathematical rules, foreign words, that is, a large amount of information.

Research methods: search on the Internet, study of literature, practical work on mastering the abacus, solving examples using the abacus,

Study execution plan:

To study the literature of the history of arithmetic from the very beginning

Outline the principles of computing on the abacus

To analyze how mental arithmetic classes go, and draw conclusions from my classes

Find out the benefits and analyze the possible difficulties in the mental account

Show what other ways to calculate in arithmetic

Chapter 1. The history of the development of arithmetic

Arithmetic originated in the countries of the Ancient East: Babylon, China, India, Egypt. The name "arithmetic" comes from the Greek word "arithmos" - number.

Arithmetic studies numbers and operations on numbers, various rules for handling them, teaches you how to solve problems that reduce to addition, subtraction, multiplication and division of numbers.

The emergence of arithmetic is associated with the labor activity of people and with the development of society.

The importance of mathematics in everyday life is great. Without counting, without the ability to correctly add, subtract, multiply and divide numbers, the development of human society is unthinkable. We study the four arithmetic operations, the rules of oral and written calculations, starting from elementary grades. All these rules were not invented or discovered by any one person. Arithmetic originated from the daily life of people.

1.1 First counting devices

People have long tried to ease their account with the help of various means and devices. The first, most ancient "calculating machine" were the fingers and toes. This simple device was quite enough - for example, to count the mammoths killed by the entire tribe.

Then there was trade. And ancient merchants (Babylonian and other cities) made calculations using grains, pebbles and shells, which they began to lay out on a special board called an abacus.

The analogue of the abacus in ancient China was the Su-anpan counting device. It is a small elongated box, divided along the length into unequal parts by partitions. Across the box are twigs on which balls are strung.

The Japanese did not lag behind the Chinese and, using their example, in the 16th century created their own counting device - the Soroban. It differed from the Chinese one in that there was one ball each in the upper compartment of the device, while in the Chinese version there were two.

Russian abacus first appeared in Russia in the 16th century. They were a board with parallel lines drawn on it. Later, instead of the board, they began to use a frame with wires and bones.

1.2 Abacus

Around the fourth century BC, the first counting device was invented. Its creator is the scientist Abacus, and the device was named after him. It looked like this: a clay plate with grooves into which stones were placed, denoting numbers. One groove was for units, and the other for tens.

Word "abacus" (abacus) means scoreboard.

Let's look at the modern abacus...

To learn how to use accounts, you need to know what they are.

Accounts consist of:

dividing line;

upper bones;

lower bones.

There is a center point in the middle. The top bones represent fives, and the bottom ones represent ones. Each vertical strip of bones, starting from right to left, denotes one of the digits of the numbers:

tens of thousands, etc.

For example, to postpone the example: 9 - 4=5, you need to move the top bone on the first line on the right (it means five) and raise the 4 lower bones. Then lower the 4 lower bones. So we get the required number 5.

Chapter 2. What is mental arithmetic?

mental arithmetic is a method of developing the mental abilities of children from 4 to 14 years old. The basis of mental arithmetic is the abacus score. It originated in ancient Japan over 2000 years ago. The child counts on the abacus with both hands, making calculations twice as fast. On the accounts, not only add and subtract, but also learn to multiply and divide.

mentality - it is the mental capacity of man.

During mathematics lessons, only the left hemisphere of the brain, which is responsible for logical thinking, develops, while the right hemisphere develops such subjects as literature, music, and drawing. There are special training techniques that are aimed at developing both hemispheres. Scientists say that those people who have fully developed both hemispheres of the brain achieve success. Many people have a more developed left hemisphere and a less developed right.

There is an assumption that mental arithmetic allows you to use both hemispheres, performing calculations of varying complexity.
The use of an abacus makes the left hemisphere work - it develops fine motor skills and allows the child to visually see the counting process.
Skills are trained gradually with the transition from simple to complex. As a result, by the end of the program, the child can mentally add, subtract, multiply and divide three- and four-digit numbers.

In addition to solving examples without using notes and drafts, doing mental arithmetic allows you to:

improve academic performance in various subjects at school;

diversify from mathematics to music;

learn foreign languages ​​faster;

become more proactive and independent;

develop leadership qualities;

be confident.

imagination: in the future, the link to the accounts is weakened, which allows you to make calculations in your mind, work with imaginary accounts;

the representation of the number is perceived not objectively, but figuratively, the image of the number is formed in the form of an image of combinations of bones;

observation;

hearing, the method of active listening improves auditory skills;

concentration of attention, as well as the distribution of attention increases: simultaneous involvement in several types of thought processes.

Practicing mental arithmetic is not a direct training of mathematical skills. Quick counting is only a means and indicator of the speed of thinking, but not an end in itself. The purpose of mental arithmetic is the development of intellectual and creativity, and this will be useful to future mathematicians and humanities. However, one must be prepared for the fact that at the very beginning of training it will be necessary to put in enough effort, diligence, perseverance and attentiveness. There may be errors in the calculations - so do not rush.

Chapter 3. Classes in the school of mental arithmetic.

The entire program for the development of oral counting is built on the successive passage of two stages.

At the first of them, acquaintance and mastery of the technique of performing arithmetic operations using bones takes place, during which two hands are involved simultaneously. In his work, the child uses an abacus. This subject allows him to absolutely freely subtract and multiply, add and divide, calculate the square and cube roots.

During the passage of the second stage, students are taught mental counting, which is performed in the mind. The child ceases to be constantly attached to the abacus, which also stimulates his imagination. The left hemispheres of children perceive numbers, and the right hemispheres perceive the image of knuckles. This is the basis of the method of mental counting. The brain begins to work with an imaginary abacus, while perceiving numbers in the form of pictures. The performance of the mathematical calculation is associated with the movement of the bones.

In mental arithmetic, more than 20 formulas are used for calculations (close relatives, help from a brother, help from a friend, etc.) that need to be remembered.

For example, Brothers in mental arithmetic are two numbers, the addition of which gives five.

There are 5 brothers in total.

1+4 = 5 Brother 1 - 4 4+1 = 5 Brother 4 - 1

2+3 = 5 Brother 2 - 3 5+0 = 5 Brother 5 - 0

3+2 = 5 Brother 3 - 2

Friends in mental arithmetic are two numbers that add up to ten.

Only 10 friends.

1+9 = 10 Friend 1 - 9 6+4 = 10 Friend 4 - 6

2+8 = 10 Friend 2 - 8 7+3 = 10 Friend 7 - 3

3+7 = 10 Friend 3 - 7 8+2 = 10 Friend 8 - 2

4+6 = 10 Friend 4 - 6 9-1 = 10 Friend 9 -1

5+5 = 10 Friend 5 - 5

Chapter 4. My studies in mental arithmetic.

At the trial lesson, the teacher showed us the abacus abacus, briefly told us how to use them and the very principle of counting.

There was a mental warm-up at the lesson. And there were always breaks where we could have a little snack, drink water or play games. At home, we were always given sheets with examples for independent work at home. I also trained in a special program where examples were launched - they flashed on the monitor at different speeds.

At the very beginning of my training, I:

Get familiar with accounts. I learned how to use my hands correctly when counting: with the thumb of both hands we raise the knuckles on the abacus, with the index fingers we lower the knuckles.

Over time I:

I learned to count two-stage examples with tens. Tens are located on the second needle from the far right. When counting with tens, we already use the thumb and forefinger of the left hand. Here the technique is the same as with the right hand: we raise it with a large one, we lower it with our index one.

In the 3rd month of study:

I used the abacus to solve examples of subtraction and addition with units and tens - three-stage.

Solve examples of subtraction and addition with thousandths - two-stage

Further:

Get to know the mind map. Looking at the card, I had to mentally move the knuckles and see the answer.

I worked out 2 hours a week and 5-10 minutes a day on my own for 4 months.

First month of training	fourth month
1. I count on the abacus 1 sheet (30 examples of 3 terms)
2. I mentally count 30 examples (5-7 terms each)
3. I am learning a poem (3rd quatrains)
4. Doing homework (mathematics: one task, 10 examples)
Send your good work in the knowledge base is simple. Use the form below Students, graduate students, young scientists who use the knowledge base in their studies and work will be very grateful to you. Posted on http://www.allbest.ru/ Introduction 1. The beginning of mathematics in primitive society 2. The origin of mathematics in the ancient East 2.1 Egypt 2.2 Babylon Conclusion Bibliography Introduction Mathematics (Greek - knowledge, science) - the science of quantitative relations and spatial forms of the real world. A clear understanding of the independent position of mathematics as a special science, having its own subject and method, became possible only after the accumulation of sufficiently large factual material and arose for the first time in Dr. Greece in the 6th-5th centuries. BC. The development of mathematics up to this time is naturally attributed to the period of the birth of mathematicians and, and to the 6th-5th centuries. BC. date the beginning of the period of elementary mathematics, which lasted until the 16th century. During these first two periods, mathematical research deals mainly with a very limited stock of basic concepts that arose even at very early stages of historical development in connection with the simplest demands of economic life, reduced to counting objects, measuring the amount of products, areas of land, determining the size individual parts of architectural structures, time measurement, commercial calculations, navigation, etc. The first problems of mechanics and physics, with the exception of individual studies by Archimedes (3rd century BC), which already required the beginnings of infinitesimal calculus, could still be satisfied with the same stock of basic mathematical concepts. The only science that, long before the widespread development of the mathematical study of natural phenomena in the 17-18 centuries. systematically presented its special and very high demands to mathematics, there was astronomy, which completely determined, for example, early development trigonometry. In the 17th century new demands of natural science and technology are forcing mathematicians to focus their attention on creating methods that allow them to mathematically study the movement, the processes of changing quantities, the transformation of geometric shapes (during design, etc.). With the use of variables in the analytical geometry of R. Descartes and the creation of differential and integral calculus, the period of mathematics of variables begins. Further expansion of the range of quantitative relations and spatial forms studied by mathematics led at the beginning of the 19th century. to the need to treat the process of expanding the subject of mathematical research consciously, setting ourselves the task of systematically studying possible types of quantitative relations and spatial forms from a fairly general point of view. Creation of N.I. Lobachevsky of his "imaginary geometry", which later received quite real applications, was the first significant step in this direction. The development of this kind of research introduced such important features into the structure of mathematics that mathematics in the 19th and 20th centuries. naturally attributed to special period modern mathematics. 1. The beginning of mathematics in primitive society Our initial ideas about number and form belong to a very distant era of the ancient Stone Age - the Paleolithic. For hundreds of thousands of years of this period, people lived in caves, in conditions not much different from the life of animals, and their energy was spent mainly on obtaining food in the simplest way - collecting it, wherever possible. People made tools for hunting and fishing, developed a language for communicating with each other, and in the Late Paleolithic era, they decorated their existence by creating works of art, figurines and drawings. Perhaps the drawings in the caves of France and Spain (about 15 thousand years ago) had a ritual significance, but undoubtedly a wonderful sense of form is found in them. Until there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, people made little progress in understanding numerical values ​​​​and spatial relationships. Only with the onset of this fundamental change, a revolution, when the passive attitude of man to nature was replaced by an active one, do we enter a new stone age, the Neolithic. This great event in the history of mankind took place about ten thousand years ago, when the ice sheet in Europe and Asia began to melt and give way to forests and deserts. Nomadic wanderings in search of food gradually ceased. Fishermen and hunters were more and more forced out by primitive farmers. Such farmers, staying in one place as long as the soil remained fertile, built dwellings designed for longer periods. Villages began to spring up to protect them from bad weather and from predatory enemies. Many such Neolithic settlements have been excavated. Their remains show how such simple crafts as pottery, weaving and carpentry gradually developed. There were granaries so that the population could, by producing surpluses, store food for the winter and in case of crop failure. Bread was baked, beer was brewed, and copper and bronze were smelted and processed in the Late Neolithic. Discoveries were made, the potter's wheel and the cart wheel were invented, boats and dwellings were improved. All these remarkable innovations arose only within one zone or another and did not always spread outside it. For example, the American Indians learned about the existence of the cart wheel only after the arrival of the whites. Nevertheless, the pace of technological progress has accelerated tremendously compared to the ancient Stone Age. The villages conducted significant trade among themselves, which developed so much that it is possible to trace the existence of trade relations between areas hundreds of kilometers away from each other. This commercial activity was strongly stimulated by the discovery of the technique of smelting copper and bronze and the manufacture of first copper and then bronze tools and weapons. This, in turn, contributed to the further formation of languages. The words of these languages ​​expressed very concrete things and very few abstract concepts, but the languages ​​already had a certain vocabulary for simple numerical terms and for some spatial images. Many tribes in Australia, America and Africa were at this level when they first met white people, and some tribes still live in such conditions, so it is possible to study their customs and ways of expressing thoughts. Numerical terms expressing some of the "most abstract concepts that the human mind can create," as Adam Smith D.Ya. Stroyk said. Brief essay on the history of mathematics. - M, 1984 .- P.23. , slowly came into use. For the first time they appear as qualitative rather than quantitative terms, expressing the difference between only one (or rather "some" - "some" rather than "one person") and two and many. The ancient qualitative origin of numerical concepts is still revealed in those special binary terms that exist in some languages, such as, for example, Greek and Celtic. With the expansion of the concept of number, large numbers were first formed by addition: 3 by adding 2 and 1, 4 by adding 2 and 2, 5 by adding 2 and 3. Here are examples of counting some Australian tribes: Murray River Tribe: 1 = enea, 2 = petcheval, 3 = petcheval-enea, 4 = petcheval-petcheval. Kamilaroi: 1 = small, 2 = bulan, 3 = guliba, 4 = bulan-bulan, 5 = bulan-guliba, 6 = guliba-guliba. The development of crafts and trade contributed to the crystallization of the concept of number. Numbers were grouped and combined into larger units, usually using the fingers of one hand or both hands, a common technique in trading. This led to counting first to base five, then to base ten, which was completed by addition and sometimes subtraction, so that twelve was perceived as 10 + 2, and nine as 10 - I2). Sometimes 20 was taken as the basis - the number of fingers and toes. Of the 307 primitive American peoples studied by Eales, 146 were decimal, 106 were five and five-decimal, and the rest were twenty and five-twenty. In its most characteristic form, the base twenty system existed among the Maya in Mexico and among the Celts in Europe. Numerical records were made with the help of bundles, notches on sticks, knots on ropes, pebbles or shells stacked in piles of five, techniques very similar to those used in ancient times by the owner of the inn, who used tags. To move from such tricks to special characters for 5, 10, 20, etc. only one step had to be taken, and it is precisely such symbols that we find in use at the beginning of recorded history, at the so-called dawn of civilization. The oldest example of the use of tags dates back to the Paleolithic era. This is a radius of a young wolf, discovered in 1937 in Vestonice (Moravia), about 17 centimeters long with 55 deep notches. The first twenty-five notches are placed in groups of five, followed by a double-length notch ending this row, and then a new row of notches begins with a new double-length notch). So, it is obvious that the old statement, which we find in Jacob Grimm and which was often repeated, that counting arose as counting on fingers, is wrong. Finger counting, that is, counting with heels and tens, arose only at a certain stage of social development. But since it came to this, it became possible to express numbers in the number system, which made it possible to form large numbers. So a primitive kind of arithmetic arose. Fourteen was expressed as 10 + 4, sometimes as 15--1. Multiplication originated when 20 was expressed not as 10 + 10, but as 2 x 10. Similar binary operations were performed for millennia, representing a cross between addition and multiplication, in particular in Egypt and in the pre-Aryan culture of Mohenjo-Daro on the Indus. The division began with the fact that 10 began to be expressed as "half of the body", although the conscious use of fractions remained extremely rare. For example, among the North American tribes, only a few cases of the use of fractions are known, and almost always it is only a fraction, although sometimes It is curious that they were very fond of big numbers, which, perhaps, was prompted by the universal human desire to exaggerate the number of herds or killed enemies; vestiges of this bias are visible in the Bible and other religious books. There was also a need to measure the length and capacity of objects. Units of measurement were crude, and often based on the size of the human body. We are reminded of this by such units as a finger, a foot (that is, a foot), an elbow. When they began to build houses such as those of the farmers of India or the inhabitants of the piled buildings of Central Europe, rules began to be worked out how to build in straight lines and at right angles. English word"straight" (straight) is related to the verb "stretch" (stretch), which indicates the use of a rope). The English word "line" (line) is cognate with the word "linen" (cloth), which indicates the connection between the weaving craft and the birth of geometry. This was one of the ways along which the development of mathematical interests proceeded. Neolithic man also had a keen sense of geometric form. The firing and coloring of clay vessels, the manufacture of reed mats, baskets and fabrics, and later metalworking developed an idea of ​​planar and spatial relationships. Dance figures also had to play their part. Neolithic ornaments were pleasing to the eye, revealing the equality, symmetry and similarity of the figures. Numerical ratios can also appear in these figures, as in some prehistoric ornaments depicting triangular numbers; in other ornaments we find "sacred" numbers. Such ornaments remained in use in historical times. We see fine examples on dipylon vases of the Minoan and early Greek period, later in Byzantine and Arabic mosaics, in Persian and Chinese carpets. Initially, early ornaments may have had a religious or magical significance, but gradually their aesthetic purpose became predominant. In Stone Age religion we can catch the first attempts to come to grips with the forces of nature. Religious rites were thoroughly permeated with magic, the magical element was part of the then existing numerical and geometric representations, also manifesting itself in sculpture, music, and drawing. There were magical numbers such as 3, 4, 7, and magical figures, such as the five-pointed star and the swastika; some authors even believe that this side of mathematics was a decisive factor in the development1), but although the social roots of mathematics in modern times may have become less noticeable, they are quite obvious in the early period of human history. Modern "numerology" is a remnant of magical rites dating back to the Neolithic, and perhaps even to the Paleolithic era. Even among the most backward tribes we find some measure of time and, consequently, some information about the movement of the sun, moon and stars. Information of this kind first acquired a more scientific character when agriculture and trade began to develop. The use of the lunar calendar dates back to a very ancient era in the history of mankind, since the change in the course of plant growth was associated with the phases of the moon. Primitive peoples paid attention to both the solstice and the rising of the Pleiades at dusk. The most ancient civilized peoples attributed astronomical information to the most remote, prehistoric period of their existence. Other primitive peoples used the constellations as landmarks when sailing. This astronomy gave some information about the properties of the sphere, circles, and angles. This brief information from the era of mathematics in primitive society shows that science in its development does not necessarily go through all the stages that now form its teaching. Only recently have scientists paid due attention to some of the oldest geometric shapes known to mankind, such as knots or ornaments. On the other hand, some of the more elementary branches of our mathematics, like graphing or elementary statics, are of comparatively recent origin. A. Speiser remarked with a certain causticity: “The late origin of elementary mathematics is at least evidenced by the fact that it is clearly inclined to be boring, a property that is apparently inherent in it, while a creative mathematician will always prefer to deal with interesting and beautiful problems” Kolmogorov A.N. Mathematics //Large Russian Encyclopedia/ Ed. B.A. Vvedensky.- M, 1998.- S.447. . 2. The origin of mathematics in the ancient East 2.1 Egypt The counting of objects at the earliest stages of the development of culture led to the creation of the simplest concepts of the arithmetic of natural numbers. Only on the basis of the developed system of oral numeration do written numeral systems arise and methods of performing over natural numbers four arithmetic operations (of which only division presented great difficulties for a long time). The needs of measurement (the amount of grain, the length of the road, etc.) lead to the appearance of names and symbols for the simplest fractional numbers and to the development of methods for performing arithmetic operations on fractions. In this way, material was accumulated that gradually formed into the most ancient mathematical science - arithmetic. Measurement of areas and volumes, the needs of building technology, and a little later - astronomy, cause the development of the rudiments of geometry. These processes went on among many peoples to a large extent independently and in parallel. Of particular importance for the further development of science was the accumulation of arithmetic and geometric knowledge in Dr. Egypt and Babylon. In Babylon, on the basis of the developed technique of arithmetic calculations, the rudiments of algebra also appeared, and in connection with the demands of astronomy, the rudiments of trigonometry. The oldest surviving mathematical texts by Dr. Egypt, related to the beginning of the 2nd millennium BC. e., consist mainly of examples for solving individual problems and, at best, recipes for solving them, which can sometimes be understood only by analyzing the numerical examples given in the texts; these decisions are often followed by a check of the answer. We should talk about recipes for solving certain types of problems, because mathematical theory in the sense of a system of interconnected and, generally speaking, one way or another proved general theorems, apparently did not exist at all. This is evidenced, for example, by the fact that the exact solutions were used without any difference from the approximate ones. Nevertheless, the very stock of established mathematical facts was, in accordance with high construction technology, the complexity of land relations, the need for an accurate calendar, etc., quite large. According to papyri 1st floor. 2nd millennium BC The state of Egyptian mathematics at that time can be characterized in the following terms. Having overcome the difficulties of operations with integers based on a non-positional decimal number system, clear from the example. The Egyptians created a peculiar and rather complex apparatus for dealing with fractions, which required special auxiliary tables. The main role in this was played by the operations of doubling and splitting integers, as well as the representation of fractions as sums of fractions of one and, in addition, fractions 2/3. Doubling and bifurcation, as a special kind of action, through a number of intermediate links reached Europe of the Middle Ages. Problems were systematically solved to find unknown numbers, which would now be written as an equation with one unknown. Geometry was reduced to the rules for calculating areas and volumes. The areas of a triangle and a trapezoid, the volumes of a parallelepiped and a pyramid with a square base were correctly calculated. The highest known achievement of the Egyptians in this direction was the discovery of a method for calculating the volume of a truncated pyramid with a square base, corresponding to the formula The rules for calculating the area of ​​a circle and the volumes of a cylinder and a cone correspond sometimes to a roughly approximate value of the number p = 3, sometimes to a much more accurate one. The presence of a rule for calculating the volume of a truncated pyramid, instructions on how to calculate, for example, the area of ​​an isosceles trapezoid by converting it into an equal rectangle, and a number of other circumstances indicate that the formation of mathematical deductive thinking was already planned in Egyptian mathematics. The ancient papyri themselves had an educational purpose and did not fully reflect the amount of knowledge and methods of Egyptian mathematicians. math fraction 2.2 Babylon There are incomparably more mathematical texts that allow one to judge mathematics in Babylon than Egyptian ones. Babylonian cuneiform mathematical texts cover the period from the beginning of the 2nd millennium BC. e. (the era of the Hammurabi dynasty and the Kassites) before the emergence and development of Greek mathematics. However, even the first of these texts belong to the heyday of Babylonian mathematics, further texts, despite the presence of some new points, testify, on the whole, rather to its stagnation. The Babylonians of the Hammurabi dynasty received from the Sumerian period a developed mixed decimal-hexadecimal numbering system, which already included a positional principle with signs for 1 and 60, as well as 10 (the same signs denote the same number of units of different sexagesimal digits) . For example: Sexagesimal fractions were also designated similarly. This made it possible to perform actions with integers and with sexagesimal fractions according to uniform rules. At a later time, a special sign also appears to indicate the absence of intermediate digits in a given number. Division using tables of reciprocals was reduced to multiplication (this technique is sometimes found in Egyptian texts). In later texts, the calculation of reciprocals other than 2 a , 3 b , 5 g , i.e. not expressed by a final sexagesimal fraction, sometimes brought to the eighth sexagesimal sign; it is possible that in this case the periodicity of such fractions was discovered; for example, in the case of 1 / 7 . In addition to tables of reciprocals, there are tables of products, squares, cubes, etc. A large number of economic records proves the widespread use of all these means in complex economic palace and temple activities. The calculation of interest on debts has also been widely developed. There are also a number of texts from the Hammurabi dynasty devoted to solving problems that, from a modern point of view, are reduced to equations of the first, second, and even third degrees. Problems on quadratic equations arose, probably, by reversing purely practical geometric problems, which in many cases indicate a significant development of abstract mathematical thought. Such, for example, is the problem of determining the side of a rectangle by its area and perimeter. However, this problem was not reduced to a three-term quadratic equation, but was apparently solved using a transformation that we would write (x+y)2=(xy)2+4xy, which leads almost immediately to a system of two linear equations with two unknowns. Another problem related to the so-called Pythagorean theorem, known in Babylon since ancient times, to determine the legs from the given hypotenuse and area, was represented by a three-term equation with a single positive root. The tasks are selected so that the roots are always positive integers and for the most part the same. This shows that the surviving clay tablets -- study exercises; the teaching was apparently oral. But the Babylonians also knew the methods of approximate calculation of the square root, for example, the length of the diagonal of a square with a given side. Thus, the algebraic component of Babylonian mathematics was significant and reached a high level. Along with this, the Babylonians knew how to sum arithmetic progressions, at least the simplest finite geometric progressions, and even knew the rule for summing successive square numbers, starting from 1. There is an assumption that such more abstract scientific interests, not limited to the recipe directly necessary in practice, but leading to the creation of general algebraic methods for solving problems, arose in the “schools of scribes”, where students prepared for counting and economic activities. Texts of this kind later disappear. But then the technique of computing with multi-digit numbers develops further in connection with the development in the 1st millennium BC. e. more accurate methods in astronomy. On the basis of astronomy, the first extensive tables of empirically found dependencies arise, in which one can see the prototype of the idea of ​​a function. The Babylonian cuneiform mathematical tradition continues in Assyria, the Persian state, and even into the Hellenistic era up to the 1st century BC. BC. Of the achievements of Babylonian mathematics in the field of geometry, which went beyond the knowledge of the Egyptians, it should be noted the developed measurement of angles and some rudiments of trigonometry, obviously associated with the development of astronomy; later, some regular polygons appear in cuneiform texts inscribed in a circle. If we compare the mathematical sciences of Egypt and Babylon in terms of the way of thinking, then it will not be difficult to establish their commonality in terms of such characteristics as authoritarianism, uncriticality, following tradition, and the extremely slow evolution of knowledge. These same features are found in the philosophy, mythology, religion of the East. As E. Kolman wrote about this, “in this place, where the will of the despot was considered law, there was no place for thinking, searching for the causes and justifications of phenomena, much less for free discussion” Kolmogorov A.N. Mathematics // Great Russian Encyclopedia / Ed. B.A. Vvedensky.- M, 1998.- S.447. . Conclusion As already mentioned, mathematics is the science of spatial forms (geometric aspect) and quantitative ratios (numerical aspect) of the objects under study. At the same time, it abstracts from the qualitative certainty of objects, so the mathematical results are universal, applicable to any objects and any scientific problems. The number "20" can mean the number of basic amino acids (biochemistry); age of the Universe, billions of years (cosmology); duration of the geological epoch, millions of years (geology); human age, years (anthropology); the number of employees of the company (management); the number of neurons in the human brain; billions (physiology); percentage of profitability of production (economics), etc. It is precisely because of the universality of its application, and also in connection with the study of the most important quantitative aspects of any processes, that the role of mathematics in the progress of all sciences is extremely high. This has long been obvious to eminent scientists. That is why the level of development of any known science can be established primarily by the degree of use of mathematics in it. At the same time, we are talking not just about the use of numbers (then history could be considered the most developed science), but about the level of mathematization of specific scientific achievements. Domestic methodologists (Akchurin A.I.) distinguish three levels of knowledge mathematization: 1. The first (lowest) level is the use of mathematics in processing the results of quantitative experiments. 2. The second (middle) level is the development of theoretical and mathematical models. 3. The third (highest) level is the creation of a mathematical theory of the objects under study. Different sciences, both natural and humanitarian, and even sections of individual sciences have a different level of mathematization: 1. The lowest level is typical for such sciences as jurisprudence, linguistics (excluding mathematical linguistics), historiography, pedagogy, psychology, sociology and some others. 2. The average level is typical for such sciences as biophysics, genetics, ecology, military sciences, economics, management, geology, chemistry, etc. 3. The highest level is typical for such sciences as astronomy, geodesy, physics (especially mechanics, acoustics, hydrodynamics, electrodynamics, optics), etc. The sciences that currently have the highest level of mathematization are called exact. Of course, mathematics itself is also an exact science. Thus, mathematical modeling is an effective method of cognition, but it is not applicable in all sciences and their sections, but only in those where the use of mathematics has advanced sufficiently. Bibliography 1. Besov K. History of science and technology from ancient times to the end of the twentieth century.- M: UNITI, 1997.- P.14-16. 2. Kolmogorov A.N. Mathematics // Great Russian Encyclopedia / Ed. B.A. Vvedensky.- M: TSB, 1998 .- S.446-449. 3. The concept of modern natural science /Ed. S.I. Samygina.- Rostov-on-Don: Phoenix, 1997 .- P.8-12. 4. Lipovko P.O. The concept of modern natural science. - Rostov n / D: Phoenix, 2004 .- P.41-45. 5. Polikarpov V.S. History of science and technology. - Rostov-on-Don: Phoenix, 1999 .- P.56-59. 6. Stroyk D.Ya. A Brief Essay on the History of Mathematics.- M: Main Editorial Board of Physics and Mathematics, 1984 .- P.21-53. Hosted on Allbest.ru Similar Documents The study of the historical development of mathematics in Russian Empire in the period of the 18th-19th centuries as a science of quantitative relations and spatial forms of the real world. Analysis of the level of mathematical education and its development by Russian scientists. abstract, added 01/26/2012 Background of the origin of mathematics in ancient Egypt. Tasks for calculating "aha". Science of the Ancient Egyptians. Problem from the Rhind Papyrus. Geometry in Ancient Egypt. Sayings of great scientists about the importance of mathematics. 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