Terms of arithmetic. What is arithmetic and how does it differ from mathematics? The law of folding and multiplying

Arithmetic is the most important, basic branch of mathematics. Viniknennu won't go to the needs of people at the rakhunku.

Mental arithmetic

What is called mental arithmetic? Mental arithmetic is a way of learning the Swedish language, which has been around for a long time.

Nina, first of all, the authors are trying to teach children the strength of their children, and they are trying to develop their minds.

The process itself will begin on the brain and the development of both brains. Golovne - take them all out at once, because the stench will be stronger than one.

In truth, the left side stands for logic, reasoning and rationality, and the right side stands for manifestation.

Before the program begins to enter the beginning of the robot and the use of such a tool as abacus.

The abacus is the main tool in learned mental arithmetic, so scientists begin to work with them, sort out the brushes and understand the essence of the structure. Then the abacus becomes your symbol, and the scientists represent them, build on their knowledge and use their butts.

The messages about these methods of learning are very positive. And one downside is that it costs money, and not everyone can afford it. Therefore, the way for genius to lie away from the material form.

Mathematics and arithmetic

Mathematics and arithmetic are closely related concepts, and more precisely, arithmetic is a branch of mathematics that deals with numbers and calculations (actions with numbers).

Arithmetic is the main branch, and also the basis of mathematics. The basis of mathematics is the most important concepts and operations, the basis for all future knowledge. The main operations include: addition, removal, multiplication, division.

Arithmetic, as a rule, is taught at school as a beginning, then. in first grade. Children learn the basics of mathematics.

Dodavannya– this is an arithmetic operation, in which two numbers are added up, which results in a new third.

a+b=c.

Vіdnіmannya– this is an arithmetic operation, from each number of the first number comes another number, and the result is the third.

The folding formula looks like this: a - b = c.

multiplication- This action, as a result of which there is a sum of new donations.

The formula looks like this: a1+a2+…+an=n*a.

Podil- this is divided into equal parts, no matter how large or small.

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Learning arithmetic

The beginning of arithmetic is carried out at the walls of the school. From the first grade, children begin to learn the basic and main branch of mathematics - arithmetic.

Adding numbers

Arithmetic 5th grade

In the fifth grade, students begin learning the following: shot numbers, mixed numbers. You can find information about operations with these numbers in our articles on similar operations.

Drobov's number- This is the relationship of two numbers, one to one, or the number to the sign. Shot grit can be replaced with a half operation. For example, ¼ = 1:4.

Mixed number- This is a shot number, only with the whole part seen. The whole part seems to be that the number is greater than the sign. For example, the word is: 5/4, which can be converted by the way of seeing the whole part: 1 whole and ¼.

Butt for training:

Zavdannya No. 1:

Zavdannya No. 2:

Arithmetic 6th grade

In the 6th grade, the topic of converting fractions in small notation appears. What does this mean? For example, given drib ½, won additionally 0.5. ¼ = 0.25.

The buttstock can be folded in this style: 0.25+0.73+12/31.

Butt for training:

Zavdannya No. 1:

Zavdannya No. 2:

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Material from Uncyclopedia

Our knowledge of mathematics begins with arithmetic, the science of number. One of the first Russian handbooks of arithmetic, written by L. F. Magnitsky in 1703, beginning with the words: “Arithmetic, or numbers, is a mysticism, honestly, immediately, and in everything clearly understandable, richly colorful and richly praised, From the most recent and the newest, among the most common arithmetic , vinaydene ta vikladene.” With arithmetic we enter, as M.V. Lomonosov said, at the “bram of eternity” and we begin our long and difficult, if not spluttering, path to knowledge of the world.

The word "arithmetic" comes from the Greek arithmos, which means "number". This science deals with numbers, different rules for dealing with them, learning to understand the problem that comes down to adding, adding, multiplying and subdividing numbers. Arithmetic is often considered to be the first stage of mathematics, based on which it can be combined with its branches - algebra, mathematical analysis, etc. bud. , or number theory. Such a view of arithmetic, of course, may be subversive - it is effectively deprived of the “alphabet of the alphabet”, and the alphabet is “highest” and “hands-on”.

Arithmetic and geometry are people’s long-time companions. These sciences began to grow when there was a need to save objects, to destroy land plots, to divide crops, and to keep order.

Vinyl arithmetic in the ancient lands: Babylonia, China, India, Egypt. For example, the Egyptian papyrus of Rinda (titles in the name of the ruler G. Rinda) dates back to the XX century. BC Among other statements, you should place the divided fraction into the sum of fractions with a number equal to one, for example:

2/73 = 1/60 + 1/219 + 1/292 + 1/365.

The treasures of mathematical knowledge accumulated in the ancient lands were dissolved and continued by ancient Greece. History has preserved for us many names of scientists who studied arithmetic in the ancient world - Anaxagoras and Zeno, Euclid, Archimedes, Eratosthenes and Diophantus. The name of Pythagoras (VI century BC) shines here like a bright mirror. The Pythagoreans (scholars and followers of Pythagoras) admired numbers, respecting that they contained all the harmony of the world. Let's call numbers and pairs of numbers ascribed special powers. In great honor there were numbers 7 and 36, then there was a great deal of respect for the rank of numbers, friendly numbers, and so on.

In the middle century, the development of arithmetic was also related to: India, the edges of the Arab world and Central Asia. From the Indians came to us the numbers by which we use the zero positional numerical system; from al-Kashi (XV century), who worked at the Samarkand Observatory of Ulugbek, – tens of fractions.

The development of trade and the influx of similar culture began in the 13th century. Interest in arithmetic is growing in Europe. The discovery of the name of the Italian scientist Leonardo of Pisa (Fibonacci), whose “Book of Abacus” introduced Europeans to the main achievements of mathematics immediately became the beginning of rich research in arithmetic and algebra .

At the same time, the first mathematical books appeared as a result of the friendship (the middle of the 15th century). The first book on arithmetic was published in Italy in 1478. In “Recent Arithmetic” by the German mathematician M. Stiefel (early 16th century), there are already negative numbers and the idea of ​​logarithm.

Around the 16th century. The development of arithmetic fundamentals flowed into the mainstream of algebra - as a milestone can be called the appearance of the work of the French scientist F. Viet, in which numbers are indicated by letters. Beginning from this hour, the basic arithmetic rules are already fully understood from the standpoint of algebra.

The main object of arithmetic is number. natural numbers, then. numbers 1, 2, 3, 4, ... etc. We went to the store for specific items. More than a thousand years have passed, the first one has been taken over by two pheasants, two hands, two people, etc. can be called one and with that very word “two”. An important task of arithmetic is to learn to calculate the specific names of objects that appear, depending on their shape, size, color, etc. Fibonacci also has a task: “This woman will go to Rome . A leather bag has 7 mules, a leather bag has 7 bags, a leather bag has 7 loaves, a leather bag has 7 knives, a leather bag has 7 knives. How many are there? For the best task, you will have to put together old things, mules, sacks, and bread.

The development of the concept of number - the appearance of zero and negative numbers, prime and tenth fractions, ways of writing numbers (digits, place values, number systems) - all has a rich and rich history.

In arithmetic, numbers are added, subtracted, multiplied and divided. The mystery of working quickly and without mercy on any numbers has long been respected by the most important tasks of arithmetic. It is in our minds and on our minds that we are working on the simplest calculations, more and more often we rely on microcalculators, which are gradually replacing such devices as calculators, arithmometers (div. calculating equipment), logarithmic ruler. Protely, the operation of all calculating machines - simple and folding - is based on the simplest operation - the addition of natural numbers. It turns out that the most complex structures can be reduced to folding, only this operation requires many millions of times. But here we are invading another area of ​​mathematics, like taking a cob in arithmetic - calculus mathematics.

Arithmetic operations over numbers loom across different powers. This power can be described in words, for example: “When you change the place, the sum of the dodanks does not change,” can be written in letters: a + b = b + a can be expressed in special terms.

For example, the assigned power of addition is called a permutative or commutative law. We often stagnate the laws of arithmetic without anyone knowing. Schoolchildren often ask: “It’s important to read all the moving and adding laws, even if it’s so clear how to add and multiply numbers?” In the 19th century Mathematics was grunted by the important Krok - Vaughn became systematically wandered around the number of numbers, ale like vector, function, change, numbers, matrix, so bagato, just a liper, symbols, it is not enough to cross the specific zero. And the point here was that the most important ones are those which laws govern these operations. The implementation of operations on additional objects (not necessarily on numbers) is also a field of algebra, although it is based on arithmetic and laws.

Arithmetic has a lot of rules for solving problems. In old books you can focus on the “triple rule”, the “proportional division”, the “vag method”, the “false rule”, etc. You can’t respect the old ones. The famous problem about a swimming pool, which is filled with a lot of pipes, has been around for at least two thousand years, and is still not easy for schoolchildren. If earlier for the perfect task it was necessary to know a special rule, today even young schoolchildren begin to follow such a task by entering the letters of the designated value. Therefore, the knowledge of arithmetic has made it necessary to develop a new knowledge of algebra.

It is important for those who are interested to understand how arithmetic has gone wrong, then calculate the proportions and hundreds. It is best to understand that the methods of arithmetic are based on an equal number of different deposits between numbers. In the history of mathematics, the process of developing arithmetic and geometry lasted for centuries.

You can clearly see the “geometrization” of arithmetic: complex rules and patterns expressed by formulas become intelligible when it is possible to represent them geometrically. The great role of mathematics itself and its additions is played by the reverse process of transferring visual, geometric information into the language of numbers (div. Graphic calculations). This translation is based on the idea of ​​the French philosopher and mathematician R. Descartes about assigning coordinates to points on a plane. It is clear that until now this idea had already been explored, for example, in maritime law, when it was necessary to determine the location of a ship, as well as in astronomy and geodesy. Just like Descartes and his scientists go to the consistent definition of coordinates in mathematics. And in our time, when carrying out complex processes (for example, the flow of a spacecraft), we value, for the most part, all the information in the form of numbers, which is processed by a calculating machine. If necessary, the machine helps people transfer accumulated numerical information to their little ones.

You see that, speaking about arithmetic, we gradually cross the borders - into algebra, geometry, and other branches of mathematics.

How to christen the borders of arithmetic itself?

Which sense experiences this word?

Under the word “arithmetic” you can understand:

a basic subject that deals with rational numbers (whole numbers and fractions), operations on them, and tasks that follow these actions;

part of historical modern mathematics, which has piled up a variety of information about calculation;

“theoretical arithmetic” is a part of everyday mathematics that deals with the construction of various numerical systems (natural, integral, rational, functional, complex numbers and their intelligibility);

“formal arithmetic” is a part of mathematical logic (div. Mathematical logic), which deals with the analysis of the axiomatic theory of arithmetic;

“I’m looking for arithmetic,” or number theory, a part of mathematics that develops independently.

What is arithmetic? When has mankind begun to vitiate numbers and make use of them? Where do the roots of such everyday things go to understand the numbers, additions and multiplications that people have made with the invisible part of their life and the world? Long ago, Greek minds were filled with such sciences as geometry, as with the miraculous symphonies of human logic.

Perhaps arithmetic is not as deep as other sciences, but what would happen to them, forget about the elementary multiplication table? It is more logical for us to understand that calculating numbers, fractions and other tools was not easy for people and was inaccessible to our ancestors at such a difficult time. In fact, before the development of arithmetic, the thirst for human knowledge was based on scientific knowledge.

Arithmetic - the beginning of mathematics

Arithmetic is the science of numbers, with which every person begins to become familiar with the burning light of mathematics. As M.V. Lomonosov said, arithmetic is the temple of eternity, which guides us on our way to the world. If you’re right, how can knowledge of the world be enhanced by knowledge of numbers and letters, mathematics and language? It is possible, if not in the current world, that the rapid development of science and technology dictates its own laws.

The word "arithmetic" (Greek "Arithmos") means "number" in Greek. She adds the number and everything that can be connected with it. This is the world of numbers: various operations on numbers, numerical rules, the solution of problems related to multiplication, and so on.

Basic object of arithmetic

The basis of arithmetic is the whole number, the power and patterns of which are seen in most arithmetic, but in essence, because of how accurate the approach of taking from the consideration of such a small block as a natural number, lies the value of everything were mathematicians.

Therefore, in regards to what arithmetic is, we can simply say: it is the science of numbers. So, about the first seven, the nine and all this different variety. And just like before, you can’t write good and most mediocre ideas without elementary alphabet, without arithmetic you can’t write elementary knowledge. Why did all the sciences emerge only after the development of arithmetic and mathematics, being a set aside?

Arithmetic is a phantom science

What is arithmetic - a natural science and a phantom? In truth, as the ancient Greek philosophers faded, there are no numbers, no articles. This is just a phantom that appears in the human mind when looking at the extreme middle ground of its processes. In truth, we have never seen anything similar that could be called a number, which stands for everything; number is a way for the human mind to embrace the world. Or maybe we should see ourselves in the middle? Philosophers have been arguing about this for a long time, but we cannot give any definitive proof. So otherwise, arithmetic has become so important to take its position that in the world no one can be concerned with social adaptation without knowledge of its fundamentals.

As a natural number appeared

Naturally, the main object with which arithmetic operates is a natural number, such as 1, 2, 3, 4, …, 152... etc. The arithmetic of natural numbers is the result of the arrangement of basic objects, for example, cores on a pocket. Still, the difference between “a lot” and “a little” has ceased to control people, and it has become possible to develop more thoroughly the technology of the market.

But a new breakthrough has arisen when human thought has reached the point where it is possible to use the same number “two” to mean 2 kilograms, 2 goals, and 2 details. On the right is that it is necessary to abstract from the forms, powers and places of objects, then it is possible to work with these objects in the form of natural numbers. This is how the arithmetic of numbers was born, as it further developed and expanded, occupying a larger position in the life of a marriage.

Such lost concepts of numbers, such as zero and the same number, fractions, assigning numbers with digits and other methods, may have the richest and most historical development.

Arithmetic and practical Egyptians

Two of the most ancient companions of people in the world and the highest daily tasks are arithmetic and geometry.

It is important that the history of arithmetic began at the Ancient Gathering: in India, Egypt, Babylon and China. Thus, the papyrus of Rhinda of the Egyptian campaign (names so, fragments belong to the same ruler), dating from the 20th century. That is, in addition to other valuable data, place the distribution of one fraction among the sum of fractions with different signifiers and a number equal to one.

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

Why do you have such a folding layout? On the right is that the Egyptian approach does not tolerate abstract thoughts about numbers, however, the calculations were carried out in a practical manner. So the Egyptian would take up such work as the destruction of the tomb, including in order to create a tomb, for example. It was necessary to collect the full length of the rib of the sporudi, and there was no doubt about killing a man for papyrus. As you can see, Egyptian progress has been made in the wake of revolutions, leading to everything from mass, everyday life, from love to science.

Therefore, the findings found on papyrus cannot be called thoughts on the topic of shotguns. It was better for everything, this practical preparation, which helped to solve problems with fractions. The ancient Egyptians, who did not know the multiplication table, carried out long calculations and calculations on a daily basis. Possibly, one of these days. It is important to note that development of such preparations is very difficult and unpromising. Possibly, for this reason we do not need to make a great contribution to Ancient Egypt in the developments of mathematics.

Ancient Greece and philosophical arithmetic

The rich knowledge of the Ancient One was successfully mastered by the ancient Greeks, known as amateurs of abstract, abstract and philosophical thoughts. Their practice has proven no less, but it is difficult to know the greatest theorists and thinkers. This has come to the ruin of science, and the fragments are inevitably buried in arithmetic without being separated from reality. Of course, you can multiply 10 cows and 100 liters of milk, but you won’t be able to get far.

The ancient Greeks lost a significant trace in history, and their works have come down to us:

• Euclid and "Cobs".
• Pythagoras.
• Archimedes
• Eratosthenes.
• Zeno.
• Anaxagoras.

And, of course, the Greeks, who turn everything into philosophy, and especially the lasting works of Pythagoras, the tables were buried with numbers that respected them for the mystery of harmony of the world. The numbers of the table were based on research that certain of them and their pairs were attributed special power. For example:

• The complete numbers are those that equal the sum of all their partners, except the number itself (6=1+2+3).
• Friendly numbers are the same numbers, one of which is the ancient sum of all the partners of another, and incidentally (the Pythagoreans knew only one such pair: 220 and 284).

The Greeks, who respected the need to love science, and who did not profit from it, achieved great success by searching, studying and adding numbers. It should be noted that not all of these investigations were widely rejected, and some of them were deprived of “for beauty’s sake.”

Similar thinkers of the Middle East

So, in the Middle Ages, arithmetic, with its development, follows similar branches. The Indians gave us numbers that we are actively vikorystvo, such as the concept of “zero” and the positional variant of the immediate crisis. Seeing Al-Kashi, who worked in Samarkand in the 15th century, we fell asleep without any importance in understanding daily arithmetic.

Much of what makes Europe aware of its achievements immediately became possible thanks to the Italian scientist Leonardo Fibonacci, who wrote the book “The Book of Abacus,” who became aware of similar innovations. It became the cornerstone of the development of algebra and arithmetic, pre-Slednitsa and scientific activity in Europe.

Russian arithmetic

And, they say, arithmetic, which found its place and took root in Europe, expanded on Russian lands. The first Russian arithmetic was published in 1703 - there was a book about arithmetic by Leonty Magnitsky. For a long time I was deprived of my only teacher in mathematics. Vaughn to take into account the basic aspects of algebra and geometry. Numbers, like the one used in the butts of Russia's first assistant in arithmetic, are Arabic. However, Arabic numerals were denominated even earlier, on engravings that date back to the 17th century.

The book itself is decorated with images of Archimedes and Pythagoras, and on the first arkush there is an image of arithmetic in the form of a woman. There sits on the throne, under it is written in Hebrew a word that means the name of God, and at the gatherings that lead to the throne, the words “field”, “multiply”, “folded”, etc. are written. what meaning was given to such truths, which today are respected as an essential phenomenon.

The handbook with 600 pages describes both the basics of the basic table of addition and multiplication, as well as additions to navigational sciences.

It’s not surprising that the author chose images of Greek mystels for his book, even though he himself was filled with the beauty of arithmetic, saying: “Arithmetic is the science of numbers, it’s a mysticism, honestly, not too late...”. Such an approach to the arithmetic of entire calculations, and even the same everywhere, can be used as a starting point for the vigorous development of scientific thought in Russia and abroad.

Not easy simple numbers

A simple number is also a natural number because it has only 2 positive counterparts: 1 and itself. All other numbers, except 1, are called warehouse numbers. Apply prime numbers: 2, 3, 5, 7, 11 and all others, as there are no other debtors except number 1 and yourself.

As for the number 1, there is a special understanding - there is an agreement that we will neither forgive nor forget to respect any trace. Simple at first glance, a simple number conceals a secret mystery within itself.

Euclid's theorem states that there are no prime numbers, and Eratosthenes invented a special arithmetic “sieve” that distinguishes non-prime numbers that are not prime.

The point is to emphasize the first uncorrected number, and then to emphasize those that are multiples of it. We repeat this procedure completely and extract the table of prime numbers.

Fundamental theorem of arithmetic

To be careful about prime numbers, it is necessary to use a special method to guess the fundamental theorem of arithmetic.

The main theorem of arithmetic states that if there is an integer number greater than 1, then it is forgivable, but it can be decomposed into a number of prime numbers, up to the order of repetition of the multiples, and in a single order.

The basic theorem of arithmetic is cumbersome, and its understanding is not similar on the simplest basis.

At first glance, simple numbers are an elementary concept, but they are not. Physics, even if it regarded the atom as elementary, did not yet find the whole universe in the middle. Forgive numbers are dedicated to the miraculous revelation of mathematician Don Tsagir “The First Fifty Million Prime Numbers.”

From the “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 struggles with arithmetic, the number of arms and legs of dolls, the number of cubes, the number of apples, etc. This is how we learn arithmetic, so we can proceed with the complex rules.

All our life we ​​have learned from the rules of arithmetic, which have become for the common people the most useful thing that science gives. The study of numbers is “baby arithmetic”, which is how to familiarize people with the light of numbers and the appearance of numbers even in early childhood.

In general, arithmetic is a deductive science, as it learns the laws of arithmetic. Most of them are familiar to us, although we may not know their exact formulations.

The law of folding and multiplying

Two natural numbers a and b can be expressed in the form of a sum a+b, which is also a natural number. The following laws have been formed:

• Commutative, as it seems that by rearranging the dodanks the sum changes, so that a+b= b+a.
• Associative, Who can say that the sum cannot be stored in the way of grouping the add-ons by places, because a + (b + c) = (a + b) + c.

The rules of arithmetic, such as addition, are one of the elementary ones, and they are also used by all sciences, not even though they are about everyday life.

Two natural numbers a and b can be expressed in the creation a*b or a*b, which is also a natural number. Before the creation of stagnation, the most commutative and associative laws, as before the formation:

• a * b = b * a;
• a * (b * c) = (a * b) * c.

It is important that there is a law, which is given by the addition and multiplication of titles, also as a separate or distributive law:

a(b+c)= ab+ac

This law is actually taught to us to practice with the arches, unscrewing them, so we ourselves can practice with more complex formulas. These are the same laws that will guide us through the chimerical and unforgivable light of algebra.

Law of arithmetic order

The law of order, human logic, vikoryst’s daily life, blazing years and flamboyant bills. And, therefore, it is necessary to formalize it as specific formulas.

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

• a is superior 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 may be fair. The fundamental law that keeps order is as follows: yakscho a< b и b < c, то a< c.

There are also laws that bind order to the actions of multiplication and addition: yakscho a< b, то a + c < b+c и ac< bc.

The laws of arithmetic lead us to practice with numbers, signs and arcs, transforming everything into a symphony of numbers.

Positional and non-positional calculation systems

We can say that numbers are not a mathematical language, and there is a lot of knowledge in them. There are a number of numerical systems, which, like the alphabets of different languages, differ from each other.

Let's take a look at the numerical system at a glance at the flow of position on the scale of the value of the number of that position. So, for example, the Roman system is non-positional, where each number is encoded with a different set of special characters: I/V/X/L/C/D/M. They are equal to the numbers 1/5/10/50/100/500/1000. In such a system, the number does not change its value depending on what position the position is on: first, second, etc. about subtracting other numbers, you need to fold the base ones. For example:

• DCC = 700.
• CCM = 800.

The most important system for us is the number system of Arabic numerals, which is positional. In such a system, the digit of a number indicates the number of digits, for example, three-digit numbers: 333, 567, etc. It is important for any rank to lie in the position where the same number is located, for example, the number 8 in another position has the value 80. This is typical for the tens system, there are other positional systems, for example the two.

Dviykov arithmetic

Two-fold arithmetic works with a two-fold alphabet, which consists of everything from 0 to 1. And the corresponding alphabet is called a two-fold numbering system.

The importance of double arithmetic compared to tens arithmetic means that the importance of the evil position is ten times greater, and 2 times. Two numbers look like 111, 1001 or so. bud. How to understand such numbers? Well, let’s look at the number 1100:

1. The first digit is 1*8=8, keeping in mind that the fourth digit also needs to be multiplied by 2, so we remove the position 8.
2. Another number is 1 * 4 = 4 (position 4).
3. The third digit is 0 * 2 = 0 (position 2).
4. The fourth digit is 0 * 1 = 0 (position 1).
5. Well, our number is 1100 = 8 +4 +0 +0 = 12.

So, when switching to a new rank of evil, its value in the two-fold system is multiplied by 2, and in the tenth – by 10. This system has one minus: it does not have a very large increase in the ranks that are necessary for recording numbers. The application of tens numbers in the following table can be seen in the following table.

The tens numbers in the double view are shown below.

Vysimkova and sixteenth calculation systems are also used.

Tsya mysterious arithmetic

What is arithmetic, “two and two” and the unknown secrets of numbers? As a matter of fact, arithmetic may seem simple at first glance, but its ease is not obvious. You can teach your children about little Owl from the cartoon “Baby Arithmetic”, or you can learn from deep scientific research and non-philosophical research. In history, there has been a path from the classification of objects to the worship of the beauty of numbers. One thing is clear: from the establishment of the basic postulates of arithmetic, the whole science can rest on its shoulder.

• Arithmetic (Ing.-Greek: ἀριθμητική; from ἀριθμός - number) - a branch of mathematics that calculates numbers, hundreds of hundreds and powers. The subject of arithmetic is the concept of numbers in development of a new (natural, goals and rational, functional, complex numbers) and power. In arithmetic, calculating operations (addition, subtraction, multiplication, subdivision) and calculation methods are considered. The study of powers around whole numbers is mainly dealt with in arithmetic and number theory. Theoretical arithmetic pays attention to the importance and analysis of the concept of number, just as formal arithmetic operates with logical elements of predicates and axioms. Arithmetic is one of the most ancient and fundamental mathematical sciences; It is closely related to algebra, geometry and number theory.

  The reason for the use of arithmetic was the practical need for the economy and the calculations associated with the debts imposed during the centralization of the rural state. Science developed at the same time from the complexity of tasks that required excellence. A great contribution to the development of arithmetic was made by the Greek mathematicians, together with the Pythagorean philosophers, who tried to use numbers to comprehend and describe all the laws of the world.

  In the Middle Ages, arithmetic was elevated, after the Neoplatonists, to the so-called seven great mysteries. The main areas of practical arithmetic were trade, navigation, and everyday life. In connection with this, special significance has arisen in the near calculation of irrational numbers, which are necessary for us first for geometric purposes. Arithmetic developed especially vigorously in India and the lands of Islam; the signs of new advances in mathematical thought penetrated to Western Europe; Russia was familiar with mathematical knowledge “both from the Greeks and from the Latins.”

  With the advent of the New Hour, nautical astronomy, mechanics, and commercial developments have become more complex, new challenges have been introduced to calculation techniques, and arithmetic has been further developed. At the beginning of the 17th century, Napier introduced logarithms, and then Fermat saw the theory of numbers in an independent branch of arithmetic. Until the end of the century, the concept of the irrational number was formulated as about the sequence of rational approaches, and during the new century, thanks to Lambert, Euler, and Gaus, arithmetic included operations with complex quantities, which appeared .

  The subsequent history of arithmetic is marked by a critical review of its foundations and attempts at deductive reasoning. The theoretical grounding of the number is related to us in advance of the values ​​of the natural number and Peano's axioms, formulated in 1889. The inconsistency of formal everyday arithmetic was shown by Gentzen in 1936.

  The fundamentals of arithmetic have a long history and are invariably given great importance in early school education.

What is "arithmetic"? How to spell this word correctly. I understand that interpretation.

arithmetic It is a mystery to calculate what vibrates with positive active numbers. A short history of arithmetic. Since ancient times, work with numbers has been divided into two different areas: one dealt with the power of numbers, the other was associated with the technology of the market. Under "arithmetic" in many countries, the remaining field itself is respected, which, of course, is the oldest branch of mathematics. Obviously, the robot with fractions was the one with the greatest complexity among long-time calculators. This can be judged from the Papyrus of Ahmes (also called the Papyrus of Rhinda), an ancient Egyptian work of mathematics dating back to approximately 1650 BC. All the fractions that can be guessed in the papyrus, behind the sign 2/3, are numerals, equal to 1. It is important to use fractions and note them when inscribing old-Vylonian cuneiform tablets. Both the ancient Egyptians and the Babylonians, perhaps, paid for this type of abacus. The science of numbers took away from the ancient Greeks the development of sutta, starting with Pythagoras, around 530 BC. Due to the lack of technology of calculation, in this Galusia the Greeks collected much less. The Romans, who lived later, however, practically did not make any contribution to the science of quantity, but in response to the needs of manufacturing and trade, which quickly developed, they improved the abacus as a shell device. Little is known about the origin of Indian arithmetic. Only a few more recent works have come down to us about the theory and practice of operations with numbers, written after the Indian positional system was perfected to include zero. When exactly this happened, we are completely unaware of whether the foundation was laid for our most extensive arithmetic algorithms (including NUMBERS and MEDIATION SYSTEMS). The Indian numerical system and the first arithmetic algorithms were introduced by the Arabs. The oldest of the Arabic arithmetic textbooks that have come down to us, the writings of al-Khwarizm are close to 825. He widely interprets and explains Indian numbers. This handbook was later translated into Latin and added to Western Europe. A variant of the name of al-Khorezm came before us from the word “algorism”, which, when further mixed with the Greek word aritmos, was transformed into the term “algorithm”. Indo-Arabic arithmetic became popular in Western Europe, especially due to the work of L. Fibonacci, The Book of Abacus (Liber abaci, 1202). The Abacist method is based on simplicity, similar to our positional system, used for addition and multiplication. Abacists were replaced by algorithms that vikorized the zero and Arabic method for the subtraction of the square root. One of the first assistants of arithmetic, the author of whom is unknown to us, was published in Treviso (Italy) in 1478. He learned about the breakdown of the current trading areas. This assistant became the successor of the rich assistants in arithmetic who showed up this year. On the cob of the 17th century. More than three hundred such handbooks were published in Europe. Arithmetic algorithms were completely thorough in this hour. At 16-17 st. symbols of arithmetic operations appeared, such as =, +, -, *, “root” and /. It is customary to note that tens of fractions of Vinays have 1585 rubles. S. Stevin, logarithms - J. Napier in 1614 rub., slide rule - W. Outred in 1622 rub. Current analogue and digital computing devices were found in the mid-20th century. also MATHEMATICS; MATHEMATICS HISTORY; NUMBER THEORY; ROWS. Mechanization of arithmetic calculations. With the development of marriage, the demand for more money and accurate payments grew. This need has called forth through life several miraculous discoveries: Indo-Arabic numerical notations, tens fractions, logarithms and daily calculating machines. In fact, the simplest rahunkov devices were in existence before the advent of modern arithmetic, since in ancient times elementary arithmetic operations were carried out on the abacus (in Russia, rahunkas were used in this way). The simplest daily calculating device is to use a logarithmic rule, which consists of two logarithmic scales that fit one into each other, which allows you to multiply and subdivide. and the visible sections of the scales. The inventor of the first mechanical pouch machine is considered to be B. Pascal (1642). Later in the same century, G. Leibnitz (1671) from Germany and S. Moreland (1673) in England produced the most advanced machines for wagon multiplication. These machines became the predecessors of desktop calculating devices (arithmometers) of 20 degrees, which allowed the operations of folding, lifting, multiplying and hemming to be carried out quickly and accurately. In 1812 The English mathematician C. Babbage began designing a machine for calculating mathematical tables. Although she had been working on the project for many years, she ended up left unfinished. Babbage's proto-project served as a stimulus for the creation of modern electronic computing machines, the first visions of which appeared around 1944. The speed of these machines was contrary to reality: from them, it was possible to figure out the problems that had previously generated a lot of uninterrupted calculations due to the stagnation of arithmometers. The essence of the reference can be explained with the help of a specific arithmetic task, for example, calculating the number p (from the bottom of the stake to its diameter). The first systematic attempts at calculation appear in Archimedes (around 240). Vikoristov, who has not yet fully completed the numerical system, can, after many works of sums, calculate p with an accuracy equivalent to our current numerical system in two digits after the coma. Using the Vikorist method of Archimedes, L. van Zeylen (1540-1610), who dedicated a significant part of life to this, sums were calculated with an accuracy of 35 digits after the coma. In 1873, after fifteen years, the work of W. Shanks cut off the values ​​of 707 characters, but later it became clear that starting from the 528th character, amends crept into the calculation. In 1958, an IBM computer calculated 707 digits of the number p i in 40 seconds, and continued further calculations, subtracting 10,000 digits in 100 digits. also COMP'YUTER; NUMBER PI. The whole number is positive. The basis of our understanding of numbers is the intuitive understanding of multiplicities, similarities between multipliers and the endless sequence of symbols and sounds. We all know the sequence of symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... and nothing else, as there is an endless sequence of markings and an endless sequence There are significant sounds (number of words) “one”, “two”, “three”, “chotiri”, “five”, “six”, “sim”, “visim”, “nine”, “ten”, “eleven”, “twelve” ", . .., similar to the song symbols. Any multiplicity, all the elements of which can be put in a one-to-one relationship with the elements of any cob segment of our unbroken sequence of symbols, is called a terminal multiplier. In this case, the number of elements of impersonality indicates the remaining symbol of the segment. For example, there are no objects that can be placed in a one-to-one relationship with the cob segment 1, 2, 3, 4, 5, 6, 7, 8, and the end multiplier, which accommodates 8 (“weight”) elements. The symbol 8 indicates the “number” of objects in the output multiplier. This number is a symbol or a label that is assigned to this multiplicity. This label is assigned to all these and more than these multipliers that can be put in a one-to-one relationship with this multiplier. The uniquely designated label for any given terminal multiplicity is called the “reinterpretation” of the elements of the given multiplicity, and the labels themselves are named after the natural or whole positive numbers (div. also NUMBER; MULTIPLICITY THEORY). Let A and B be the two terminal multiples, so that they don’t mix up extra elements, and don’t mix up A n elements, and let B mix up m elements. Then there is no element S, which is the sum of all the elements of the multiplicities A and B, taken at once, and the end multiplier, which contains, say, the elements s. For example, if A is composed of elements (a, b, c), impersonal is composed of elements (x, y), then impersonal S = A + B is composed of elements (a, b, c, x, y). The number s is called the sum of numbers n and m, and it is written like this: s = n + m. In this case, the records of numbers n and m are called dodanki, the operation of finding sum is called dodavannyam. The operation symbol "+" is read as "plus". The multiplicity P, which is formed from all ordered pairs, in which the first element is from the multiplier A, and the other from the multiplicity B, is the final multiplicity, which contains, say, p elements. For example, as before, A = (a, b, c), B = (x, y), then P = AґB = ((a, x), (a, y), (b, x), (b, y), (c, x), (c, y)). The number p is called the complement of the numbers a and b, and we write it like this: p = a*b or p = a*b. The numbers a and b in the creation are called multipliers, the operation of the creation is called multipliers. The symbol for the operation ґ is read as “multiplied by”. It can be shown that these values ​​follow the fundamental laws of addition, which are imposed below, and the multiplication of whole numbers: - The law of commutativity of addition: a + b = b + a; - Law of association: a + (b + c) = (a + b) + c; - Law of commutative multiplication: a * b = b * a; - Law of associative multiplication: a * (b * c) = (a * b) * c; - Law of distributivity: aґ(b + c)= (a*b) + (a*c). Since a and b are two positive integers and since the number c is a positive integer, so that a = b + c, then we say that a is greater than b (this is written like this: a > b), or b is less than a (this is written like this: b b, or a