Arithmetic from which. What is arithmetic? Fundamental theorem of arithmetic. Binary arithmetic. Positive real numbers

School-lyceum No. __

Essay

on the topic

“The history of arithmetic operations”

Completed: __ 5th _ grade exercises

______________
Karaganda, 2015

The Arabs did not erase numbers, but crossed them out and wrote a new number above the crossed one. It was very inconvenient. Then the Arab mathematicians, using the same method of subtraction, began to begin the action from the lowest ranks, i.e., once they worked on a new method of subtraction, similar to the modern one. To indicate subtraction in the 3rd century. BC e. in Greece they used the inverted Greek letter psi (F). Italian mathematicians used the letter M, the initial letter in the word minus, to denote subtraction. In the 16th century, the sign - began to be used to indicate subtraction. This sign probably passed into mathematics from trade. Merchants, pouring wine from barrels for sale, used a chalk line to mark the number of measures of wine sold from the barrel.

Multiplication

Multiplication is a special case adding several identical numbers. In ancient times, people learned to multiply when counting objects. So, counting the numbers 17, 18, 19, 20 in order, they were supposed to represent

20 is not only like 10+10, but also like two tens, that is, 2 10;

30 is like three tens, that is, repeat the ten term three times - 3 - 10 - and so on

People started multiplying much later than adding. The Egyptians performed multiplication by repeated addition or successive doublings. In Babylon, when multiplying numbers, they used special multiplication tables - the “ancestors” of modern ones. IN Ancient India They used a method of multiplying numbers, which was also quite close to the modern one. The Indians multiplied numbers starting from the highest ranks. At the same time, they erased those numbers that had to be replaced during subsequent actions, since they added to them the number that we now remember when multiplying. Thus, Indian mathematicians immediately wrote down the product, performing intermediate calculations in the sand or in their heads. The Indian method of multiplication was passed on to the Arabs. But the Arabs did not erase the numbers, but crossed them out and wrote a new number above the crossed out one. In Europe, for a long time, the product was called the sum of multiplication. The name "multiplier" is mentioned in works of the 6th century, and "multiplicand" in the 13th century.

In the 17th century, some mathematicians began to denote multiplication with an oblique cross - x, while others used a dot for this. In the 16th and 17th centuries, various symbols were used to indicate actions; there was no uniformity in their use. Only at the end of the 18th century did most mathematicians begin to use a dot as a multiplication sign, but they also allowed the use of an oblique cross. Multiplication signs ( , x) and the equal sign (=) became generally accepted thanks to the authority of the famous German mathematician Gottfried Wilhelm Leibniz (1646-1716).

Division

Any two natural numbers You can always add and also multiply. Subtraction from a natural number can only be performed when the subtrahend is less than the minuend. Division without a remainder is feasible only for some numbers, and it is difficult to find out whether one number is divisible by another. In addition, there are numbers that cannot be divided by any number other than one. You cannot divide by zero. These features of the action significantly complicated the path to understanding division techniques. In Ancient Egypt, the division of numbers was carried out by the method of doubling and mediation, that is, dividing by two and then adding the selected numbers. Indian mathematicians invented the "up division" method. They wrote the divisor below the dividend, and all intermediate calculations above the dividend. Moreover, those numbers that were subject to change during intermediate calculations were erased by the Indians and new ones were written in their place. Having borrowed this method, the Arabs began to cross out numbers in intermediate calculations and write others over them. This innovation made “up division” much more difficult. A method of division close to the modern one first appeared in Italy in the 15th century.

For thousands of years, the action of division was not indicated by any sign - it was simply called and written down as a word. Indian mathematicians were the first to denote division initial letter from the name of this action. The Arabs introduced a line to denote the division. The line for marking division was adopted from the Arabs in the 13th century by the Italian mathematician Fibonacci. He was the first to use the term private. The colon sign (:) to indicate division came into use in the late 17th century.

The equal sign (=) was introduced for the first time English teacher Ma themes by R. Ricorrd in the 16th century. He explained: “No two objects can be more equal to each other, like two parallel lines.” But even in Egyptian papyri there is a sign that denoted the equality of two numbers, although this sign is completely different from the = sign.

Our acquaintance with mathematics begins with arithmetic, the science of number. One of the first Russian arithmetic textbooks, written by L. F. Magnitsky in 1703, began with the words: “Arithmetic, or the numerator, is an honest, unenviable art, and conveniently understandable for everyone, most useful and most praiseworthy, from the most ancient and the newest, in different times of the greatest arithmeticians who lived, invented and expounded.” With arithmetic we enter, as M.V. Lomonosov said, into the “gates of learning” and begin our long and difficult, but fascinating path of understanding the world.

The word "arithmetic" comes from the Greek arithmos, which means "number". This science studies operations with numbers, various rules for handling them, and teaches how to solve problems that boil down to addition, subtraction, multiplication and division of numbers. Arithmetic is often imagined as some kind of first stage of mathematics, based on which one can study its more complex sections - algebra, mathematical analysis, etc. Even whole numbers - the main object of arithmetic - are related when they are considered general properties and patterns, to higher arithmetic, or number theory. This view of arithmetic, of course, has grounds - it really remains the “alphabet of counting,” but the alphabet is “most useful” and “easy to understand.”

Arithmetic and geometry are long-time companions of man. These sciences appeared when the need arose to count objects, measure plots of land, divide spoils, and keep track of time.

Arithmetic originated in the countries of the Ancient East: Babylon, China, India, Egypt. For example, the Egyptian Rind papyrus (named after its owner G. Rind) dates back to the 20th century. BC. Among other information, it contains decompositions of a fraction into a sum of fractions with a numerator equal to one, for example:

The treasures of mathematical knowledge accumulated in the countries of the Ancient East were developed and continued by scientists Ancient Greece. History has preserved many names of scientists who dealt with arithmetic in the ancient world - Anaxagoras and Zeno, Euclid (see Euclid and his Elements), Archimedes, Eratosthenes and Diophantus. The name of Pythagoras (VI century BC) sparkles here like a bright star. The Pythagoreans (students and followers of Pythagoras) worshiped numbers, believing that they contained all the harmony of the world. Individual numbers and pairs of numbers were assigned special properties. The numbers 7 and 36 were held in high esteem, and at the same time attention was paid to the so-called perfect numbers, friendly numbers, etc.

In the Middle Ages, the development of arithmetic was also associated with the East: India, the countries of the Arab world and Central Asia. From the Indians came to us the numbers we use, zero and the positional number system; from al-Kashi (XV century), who worked at the Samarkand Observatory of Ulugbek, - decimal fractions.

Thanks to the development of trade and influence oriental culture starting from the 13th century. Interest in arithmetic is also increasing in Europe. It is worth remembering the name of the Italian scientist Leonardo of Pisa (Fibonacci), whose work “The Book of Abacus” introduced Europeans to the main achievements of Eastern mathematics and was the beginning of many studies in arithmetic and algebra.

Along with the invention of printing (mid-15th century), the first printed mathematical books appeared. The first printed book on arithmetic was published in Italy in 1478. In the “Complete Arithmetic” of the German mathematician M. Stiefel (early 16th century) there are already negative numbers and even the idea of ​​logarithmization.

From about the 16th century. the development of purely arithmetic questions flowed into the mainstream of algebra - as a significant milestone, one can note the appearance of the works of the French scientist F. Vieta, in which numbers are indicated by letters. From this time on, the basic arithmetic rules are finally understood from the standpoint of algebra.

The main object of arithmetic is number. Natural numbers, i.e. the numbers 1, 2, 3, 4, ... etc., arose from counting specific objects. Many thousands of years passed before man learned that two pheasants, two hands, two people, etc. can be called by the same word “two”. An important task of arithmetic is to learn to overcome the specific meaning of the names of the objects being counted, to distract from their shape, size, color, etc. Fibonacci already has a task: “Seven old women go to Rome. Each has 7 mules, each mule carries 7 bags, each bag contains 7 loaves, each loaf contains 7 knives, each knife has 7 sheaths. How many are there?" To solve the problem, you will have to put together old women, mules, bags, and bread.

The development of the concept of number - the appearance of zero and negative numbers, ordinary and decimal fractions, ways of writing numbers (digits, notations, number systems) - all this has a rich and interesting history.

“The science of numbers refers to two sciences: practical and theoretical. Practical studies numbers insofar as we are talking about countable numbers. This science is used in market and civil affairs. The theoretical science of numbers studies numbers in the absolute sense, abstracted by the mind from bodies and everything that can be counted in them.” al-Farabi

In arithmetic, numbers are added, subtracted, multiplied and divided. The art of quickly and accurately performing these operations on any numbers has long been considered the most important task of arithmetic. Nowadays, in our heads or on a piece of paper, we make only the simplest calculations, increasingly entrusting more complex computational work to microcalculators, which are gradually replacing devices such as an abacus, an adding machine (see Computer technology), and a slide rule. However, at the core of the work of all computers- simple and complex - the simplest operation is the addition of natural numbers. It turns out that the most complex calculations can be reduced to addition, but this operation must be done many millions of times. But here we are invading another area of ​​mathematics, which originates in arithmetic - computational mathematics.

Arithmetic operations on numbers have a variety of properties. These properties can be described in words, for example: “The sum does not change by changing the places of the terms,” can be written in letters: , can be expressed in special terms.

For example, this property of addition is called a commutative or commutative law. We apply the laws of arithmetic often out of habit, without realizing it. Often students at school ask: “Why learn all these commutative and combinational laws, since it’s already clear how to add and multiply numbers?” In the 19th century mathematics took an important step - it began to systematically add and multiply not only numbers, but also vectors, functions, displacements, tables of numbers, matrices and much more, and even just letters, symbols, without really caring about their specific meaning. And here it turned out that the most important thing is what laws these operations obey. The study of operations specified on arbitrary objects (not necessarily on numbers) is already the field of algebra, although this task is based on arithmetic and its laws.

Arithmetic contains many rules for solving problems. In old books you can find problems on the “triple rule”, on “proportional division”, on the “method of scales”, on the “false rule”, etc. Most of these rules are now outdated, although the problems that were solved with their help cannot in any way be considered outdated. The famous problem about a swimming pool that is filled with several pipes is at least two thousand years old, and it is still not easy for schoolchildren. But if earlier to solve this problem it was necessary to know a special rule, then nowadays junior schoolchildren are taught to solve such a problem by entering the letter designation of the desired quantity. Thus, arithmetic problems led to the need to solve equations, and this is again an algebra problem.

PYTHAGORAS (c. 570-c. 500 BC) There are no written documents left about Pythagoras of Samos, and from later evidence it is difficult to reconstruct the true picture of his life and achievements. It is known that Pythagoras left his native island of Samos in the Aegean Sea off the coast of Asia Minor as a sign of protest against the tyranny of the ruler and already at an adult age (according to legend, at the age of 40) he appeared in the Greek city of Crotone in southern Italy. Pythagoras and his followers - the Pythagoreans - formed a secret alliance that played a significant role in life Greek colonies in Italy. The Pythagoreans recognized each other by a star-shaped pentagon - a pentagram. The Pythagoreans believed that the secret of the world was hidden in numerical patterns. The world of numbers lived a special life for the Pythagorean; numbers had their own special life meaning. Numbers, equal to the amount their divisors were perceived as perfect (6, 28, 496, 8128); Friendly were pairs of numbers, each of which was equal to the sum of the divisors of the other (for example, 220 and 284). Pythagoras was the first to divide numbers into even and odd, simple and composite, and introduced the concept of a figured number. In his school, Pythagorean triplets of natural numbers were examined in detail, in which the square of one was equal to the sum of the squares of the other two (see Fermat's last theorem). Pythagoras is credited with saying: “Everything is a number.” He wanted to reduce the whole world, and mathematics in particular, to numbers (and he meant only natural numbers). But in the school of Pythagoras itself a discovery was made that violated this harmony. It has been proven that it is not a rational number, i.e. cannot be expressed in terms of natural numbers. Naturally, Pythagoras’ geometry was subordinated to arithmetic; this was clearly manifested in the theorem that bears his name and which later became the basis for the use of numerical methods in geometry. (Later, Euclid again brought geometry to the forefront, subordinating algebra to it.) Apparently, the Pythagoreans knew the correct solids: tetrahedron, cube and dodecahedron. Pythagoras is credited with the systematic introduction of proofs into geometry, the creation of the planimetry of rectilinear figures, and the doctrine of similarity. The name of Pythagoras is associated with the doctrine of arithmetic, geometric and harmonic proportions, averages. It should be noted that Pythagoras considered the Earth to be a ball moving around the Sun. When in the 16th century The church began to fiercely persecute the teachings of Copernicus; this teaching was stubbornly called Pythagorean.

ARCHIMEDES (c. 287-212 BC) More is known about Archimedes, the great mathematician and mechanic, than about other ancient scientists. First of all, the year of his death is reliable - the year of the fall of Syracuse, when the scientist died at the hands of a Roman soldier. However, ancient historians Polybius, Livy, and Plutarch said little about his mathematical merits; from them, information about the scientist’s wonderful inventions made during his service with King Hieron II has reached our times. There is a well-known story about the king’s golden crown. Archimedes checked the purity of its composition using the law of buoyancy force he found, and his exclamation “Eureka!”, i.e. "Found!". Another legend says that Archimedes built a system of blocks with the help of which one man was able to launch huge ship"Siracosia". The words of Archimedes spoken then became winged: “Give me a fulcrum, and I will turn the Earth.” The engineering genius of Archimedes manifested itself with particular force during the siege of Syracuse, a wealthy trading city on the island of Sicily. The soldiers of the Roman consul Marcellus were detained for a long time at the walls of the city by unprecedented machines: powerful catapults targeted stone blocks, throwing machines were installed in the loopholes, throwing out hail of cannonballs, coastal cranes turned outside the walls and threw stone and lead blocks at enemy ships, hooks picked up ships and threw them down from high altitude , systems of concave mirrors (in some stories - shields) set ships on fire. In “The History of Marcellus,” Plutarch describes the horror that reigned in the ranks of the Roman soldiers: “As soon as they noticed that a rope or a log was appearing from behind the fortress wall, they fled, shouting that Archimedes had invented a new machine for their destruction.” . Archimedes' contribution to the development of mathematics was also enormous. The Archimedes spiral (see Spirals), described by a point moving in a rotating circle, stood apart from the many curves known to his contemporaries. The next kinematically defined curve - the cycloid - appeared only in the 17th century. Archimedes learned to find a tangent to his spiral (and his predecessors were able to draw tangents only to conic sections), found the area of ​​its turn, as well as the area of ​​an ellipse, the surface of a cone and a sphere, the volumes of a sphere and a spherical segment. He was especially proud of the ratio he discovered of the volume of a sphere and a cylinder circumscribed around it, which is equal to 2:3 (see Inscribed and Circumscribed Figures). Archimedes also worked a lot on the problem of squaring the circle (see Famous problems of antiquity). The scientist calculated the ratio of the circumference to the diameter (number) and found that it was between and. The method he created for calculating the circumference and area of ​​a figure was a significant step towards the creation of differential and integral calculus, which appeared only 2000 years later. Archimedes also found the sum to be infinite geometric progression A major role in the development of mathematics was played by his essay “Psammit” - “On the number of grains of sand”, in which he shows how using the existing number system one can express arbitrarily large numbers. As a basis for his reasoning, he uses the problem of counting the number of grains of sand within the visible Universe. Thus, the then existing opinion about the presence of mysterious “largest numbers” was refuted.

Among the important concepts that arithmetic introduced are proportions and percentages. Most concepts and methods of arithmetic are based on comparing various dependencies between numbers. In the history of mathematics, the process of merging arithmetic and geometry occurred over many centuries.

One can clearly trace the “geometrization” of arithmetic: complex rules and patterns expressed by formulas become clearer if they can be depicted geometrically. An important role in mathematics itself and its applications is played by the reverse process - the translation of visual, geometric information into the language of numbers (see Graphical calculations). This translation is based on the idea of ​​the French philosopher and mathematician R. Descartes about defining points on a plane by coordinates. Of course, this idea had already been used before him, for example in maritime affairs, when it was necessary to determine the location of a ship, as well as in astronomy and geodesy. But it is from Descartes and his students that the consistent use of the language of coordinates in mathematics comes. And in our time, when controlling complex processes (for example, flight spacecraft) prefer to have all information in the form of numbers, which are processed by a computer. If necessary, the machine helps a person translate the accumulated numerical information into the language of drawing.

You see that, speaking about arithmetic, we always go beyond its limits - into algebra, geometry, and other branches of mathematics.

How can we delineate the boundaries of arithmetic itself?

In what sense is this word used?

The word "arithmetic" can be understood as:

an academic subject that deals primarily with rational numbers (whole numbers and fractions), operations on them, and problems solved with the help of these operations;

part of the historical building of mathematics, which has accumulated various information about calculations;

“theoretical arithmetic” is a part of modern mathematics that deals with the construction of various numerical systems (natural, integer, rational, real, complex numbers and their generalizations);

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

“higher arithmetic”, or number theory, an independently developing part of mathematics.

What is arithmetic? When did humanity begin to use and work with numbers? Where do the roots of such everyday concepts as numbers, addition and multiplication go, which man has made an inseparable part of his life and worldview? The ancient Greek minds admired sciences such 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 forgot the elementary multiplication table? Familiar to us logical thinking, using numbers, fractions and other tools, was not easy for people and was inaccessible to our ancestors for a long time. In fact, until 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 to become acquainted with fascinating world mathematics. As M.V. Lomonosov said, arithmetic is the gate of learning, opening the path to world knowledge for us. But he’s right, can knowledge of the world be separated from 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") is of Greek origin and means "number". She studies the number and everything that can be connected with them. This is the world of numbers: various operations on numbers, numerical rules, solving problems that involve 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 entire 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 the elementary alphabet, without arithmetic you cannot solve even an elementary problem. This is why all sciences advanced only after the development of arithmetic and mathematics, having previously been just a set of assumptions.

Arithmetic is a phantom science

What is arithmetic - natural science or phantom? In fact, as they reasoned ancient greek philosophers, neither numbers nor figures exist in reality. This is just a phantom that is created in human thinking when looking at environment with its processes. In fact, nowhere around do we see anything like that that could be called a number; rather, a number is a way of the human mind to study the world. Or maybe this is a 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 position so firmly that in the modern world no one can be considered socially adapted without knowledge of its fundamentals.

How did the natural number appear?

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

But the real breakthrough came when human thought It has reached the point where the same number “two” can be used to designate 2 kilograms, 2 bricks, and 2 parts. The point 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. This is how the arithmetic of numbers was born, which further developed and expanded, occupying ever larger positions in the life of society.

Such in-depth concepts of number as zero and negative numbers, fractions, notation of numbers by digits and other methods have the richest and most interesting story development.

Arithmetic and practical Egyptians

The two most ancient companions of man in exploring the surrounding world and solving everyday problems are arithmetic and geometry.

It is believed that the history of arithmetic originates in the Ancient East: India, Egypt, Babylon and China. Thus, the Rhinda papyrus is 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 decomposition of one fraction into a 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 thinking about numbers; on the contrary, calculations were carried out only for practical purposes. That is, an 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 at the papyrus. As you can see, Egyptian progress in calculations was caused more by massive construction than by a love of science.

For this reason, the calculations found on the papyri cannot be called reflections on the topic of fractions. Most likely, this was a practical preparation that helped in the future to solve problems with fractions. The ancient Egyptians, who did not know the multiplication tables, performed rather long calculations, divided into many subproblems. Perhaps this is one of those subtasks. It is easy to see that calculations with such blanks are very labor-intensive and have little prospects. Maybe for this reason we don't see much contribution Ancient Egypt in the development of mathematics.

Ancient Greece and philosophical arithmetic

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

Deep-thinking Greeks left a significant mark on history, and their works have reached us:

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

And, of course, the Greeks, who turned everything into philosophy, and especially those who continued the work of Pythagoras, were so captivated by numbers that they considered them the sacrament of the harmony of the world. Numbers have been studied and researched so much that some of them and their pairs have been attributed special properties. For example:

• Perfect numbers are those that are equal to the sum of all their divisors except the number itself (6=1+2+3).
• Friendly numbers are those 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 pursued for 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 were left only “for beauty.”

Eastern thinkers of the Middle Ages

In the same way, in the Middle Ages, arithmetic owes its development to Eastern contemporaries. The Indians gave us numbers that we actively use, such a concept as “zero”, and a positional option 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 Abacus,” introducing Eastern innovations. It became the cornerstone of the development of algebra and arithmetic, research and scientific activity in Europe.

Russian arithmetic

And finally, arithmetic, which found its place and took root in Europe, began to spread to 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 of the first arithmetic textbook in Russia are Arabic. Although Arabic numerals were found earlier, in engravings dating back to the 17th century.

The book itself is decorated with images of Archimedes and Pythagoras, and on the first page 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 are inscribed the words “division”, “multiplication”, “addition”, etc. One can only imagine what meaning they conveyed such truths that are now considered commonplace.

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

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, it is an honest, unenvious art...” This approach to arithmetic is quite justified, because it is its widespread implementation that can be considered the beginning of the rapid development of scientific thought in Russia and general education.

Non-prime numbers

A prime number is a natural number that has only 2 positive divisors: 1 and itself. All other numbers, not counting 1, are called composite numbers. Examples of prime numbers: 2, 3, 5, 7, 11, and all others that have no divisors other than the number 1 and itself.

As for the number 1, it has a special place - there is an agreement that it should be considered neither simple nor composite. A seemingly simple number conceals a multitude unsolved mysteries inside yourself.

Euclid's theorem says that there are an infinite number of prime numbers, and Eratosthenes came up with a special arithmetic “sieve” that sifts out difficult numbers, leaving only prime ones.

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

Fundamental Theorem of Arithmetic

Among the observations about prime numbers, special mention must be made of the fundamental theorem of arithmetic.

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

The main theorem of arithmetic is quite cumbersome to prove, and its understanding is no longer similar to the simplest basics.

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 a whole universe inside it. Dedicated to prime numbers wonderful story mathematician Don Tsagir "The First Fifty Million Primes."

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 is faced with arithmetic, studying the number of legs and arms of dolls, the number of cubes, apples, etc. This is how we study arithmetic, which then develops into more complex rules.

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

Higher arithmetic is a deductive science that studies the laws of arithmetic. We know most of them, although we may not know their exact wording.

Law of addition and multiplication

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

• Commutative, which says that rearranging the terms does not change the sum, or a+b= 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 most elementary, but they are used by all sciences, not to mention Everyday life.

Any two natural numbers a and b can be expressed in the 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 the distributive or distributive law:

a(b+c)= ab+ac

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

Law of arithmetic order

The law of order is used by human logic every day, checking watches and counting bills. And, nevertheless, it also needs to be formalized 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 fundamental law that governs 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 number systems

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

Let's consider number systems from the point of view of the influence of 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 with 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, a 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 that is 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 a particular digit is located, for example, the digit 8 in the second position has the value 80. This is typical for the decimal system; there are other positional systems, for example 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 not 10, but 2 times greater. Binary numbers have 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. The second digit 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. 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 the decimal system by 10. Such a system has one drawback: it is too large an increase in the digits that are necessary to write numbers. Examples of representing decimal numbers as binary numbers can be seen in the following table.

Decimal numbers in binary form are shown below.

Both octal and hexadecimal number systems are also used.

This mysterious arithmetic

What is arithmetic, “twice two” or the unknown secrets of numbers? As we see, arithmetic may seem simple at first glance, but its non-obvious ease is deceptive. Children can study it together with Auntie Owl from the cartoon “Baby Arithmetic,” or they can immerse themselves in deeply scientific research of an almost philosophical order. In history, she went from counting objects to worshiping the beauty of numbers. One thing is certain: with the establishment of the basic postulates of arithmetic, all science can rest on its strong shoulder.

Since ancient times, work with numbers has been divided into two different areas: one directly concerned the properties of numbers, the other was associated with counting techniques. By "arithmetic" in many countries this latter field is usually meant, which is undoubtedly the oldest branch of mathematics.

Apparently, the greatest difficulty for ancient calculators was working with fractions. This can be seen from the Ahmes Papyrus (also called the Rhind Papyrus), an ancient Egyptian work on mathematics dating back to around 1650 BC. All fractions mentioned in the papyrus, with the exception of 2/3, have numerators equal to 1. The difficulty of handling fractions is also noticeable when studying ancient Babylonian cuneiform tablets. Both the ancient Egyptians and Babylonians apparently performed calculations using some form of abacus. The science of numbers received significant development among the ancient Greeks starting with Pythagoras, around 530 BC. As for the technology of calculation itself, much less was done in this area by the Greeks.

The later Romans, on the contrary, made virtually no contribution to the science of numbers, but based on the needs of rapidly developing production and trade, they improved the abacus as a counting device. Very little is known about the origins of Indian arithmetic. Only a few later works on the theory and practice of number operations have come down to us, written after the Indian positional system was improved by including zero in it. Exactly when this happened we do not know for sure, but it was then that the foundations for our most common arithmetic algorithms were laid.

The Indian number system and the first arithmetic algorithms were borrowed by the Arabs. The earliest extant Arabic arithmetic textbook was written by al-Khwarizmi around 825. It makes extensive use and explanation of Indian numerals. This textbook was later translated into Latin and had a significant influence on Western Europe. A distorted version of the name al-Khwarizmi has come down to us in the word “algorism”, which, when further mixed with the Greek word arrhythmos became the term "algorithm".

Indo-Arabic arithmetic became known in Western Europe mainly thanks to the work of L. Fibonacci Book of abacus (Liber abaci, 1202). The Abacist method offered simplifications similar to the use of our positional system, at least for addition and multiplication. The Abacists were replaced by algorithms that used zero and the Arabic method of division and extraction square root. One of the first arithmetic textbooks, the author of which is unknown to us, was published in Treviso (Italy) in 1478. It dealt with calculations when making trade transactions. This textbook became the predecessor of many arithmetic textbooks that appeared subsequently. Until the beginning of the 17th century. More than three hundred such textbooks were published in Europe. Arithmetic algorithms have been significantly improved during this time. In the 16th–17th centuries. Symbols for arithmetic operations appeared, such as =, +, -, ґ, ё and .

Mechanization of arithmetic calculations.

As society developed, so did the need for faster and more accurate calculations. This need gave rise to four remarkable inventions: Indo-Arabic numerals, decimals, logarithms and modern computing machines.

In fact, the simplest calculating devices existed before the advent of modern arithmetic, because in ancient times elementary arithmetic operations were performed on the abacus (in Russia, abacuses were used for this purpose). The simplest modern computing device can be considered a slide rule, which consists of two logarithmic scales sliding one along the other, which allows multiplication and division by summing and subtracting segments of the scales. B. Pascal (1642) is considered to be the inventor of the first mechanical adding machine. Later in the same century, G. Leibniz (1671) in Germany and S. Moreland (1673) in England invented machines for performing multiplication. These machines became the predecessors of desktop computing devices (arithmometers) of the 20th century, which made it possible to quickly and accurately perform addition, subtraction, multiplication and division operations.

In 1812, the English mathematician C. Babbage began to create a design for a machine for calculating mathematical tables. Although work on the project continued for many years, it remained unfinished. Nevertheless, Babbage’s project served as an incentive for the creation of modern electronic computers, the first examples of which appeared around 1944. The speed of these machines was amazing: with their help, in minutes or hours it was possible to solve problems that previously required many years of continuous calculations, even with the use of adding machines.

Positive integers.

Let A And B– two finite sets that do not have common elements, let it go A contains n elements, and B contains m elements. Then many S, consisting of all elements of the sets A And B, taken together, is a finite set containing, say, s elements. For example, if A consists of elements ( a, b, c), a bunch of IN– from elements ( x, y), then the set S=A+B and consists of elements ( a, b, c, x, y). Number s called amount numbers n And m, and we write it like this: s = n + m. In this entry the numbers n And m are called terms, the operation of finding the sum – addition. The operation symbol "+" is read as "plus". A bunch of P, consisting of all ordered pairs in which the first element is chosen from the set A, and the second one is from the set B, is a finite set containing, say, p elements. For example, if, as before, A = {a, b, c}, B = {x, y), That P=AґB = {(a,x), (a,y), (b,x), (b,y), (c,x), (c,y)). Number p called work numbers a And b, and we write it like this: p = aґb or p = a×b. Numbers a And b in the work are called multipliers, the operation of finding the product – multiplication. The operation symbol ґ is read as “multiplied by.”

It can be shown that from these definitions the following fundamental laws of addition and multiplication of integers follow:

– the law of commutative addition: a + b = b + a;

– law of associative addition: a + (b + c) = (a + b) + c;

– the law of commutative multiplication: aґb = bґa;

– law of associativity of multiplication: aґ(bґc) = (aґb)ґc;

– law of distributivity: aґ(b + c)= (aґb) + (aґc).

If a And b– two positive integers and if there is a positive integer c, such that a = b + c, then we say that a more b(this is written like this: a>b), or what b less a(this is written like this: b). For any two numbers a And b one of three relationships holds: either a = b, or a>b, or a.

The first two fundamental laws say that the sum of two or more terms does not depend on how they are grouped and in what order they are arranged. Similarly, from the third and fourth laws it follows that the product of two or more factors does not depend on how the factors are grouped or what their order is. These facts are known as the "generalized laws of commutativity and associativity" of addition and multiplication. It follows from them that when writing the sum of several terms or the product of several factors, the order of the terms and factors is unimportant and the parentheses can be omitted.

In particular, the repeated amount a + a + ... + a from n terms is equal to nґa. Repeated work aґaґ ... ґa from n We agreed to denote the factors a n; number a called basis, and the number n – repeated product indicator, the repeated work itself – nth power numbers a. These definitions allow us to establish the following fundamental laws for exponents:

Another important consequence of the definitions: aґ1 = a for any integer a, and 1 is the only integer that has this property. The number 1 is called unit.

Divisors of integers.

If a, b, c– integers and aґb = c, That a And b are divisors of a number c. Because aґ1 = a for any integer a, we conclude that 1 is a divisor of any integer and that any integer is a divisor of itself. Any integer divisor a, different from 1 or a, got the name proper divisor numbers a.

Any integer other than 1 and not having its own divisors is called prime number. (An example of a prime number is the number 7.) A whole number that has its own divisors is called composite number. (For example, the number 6 is composite, since 2 divides 6.) From the above it follows that the set of all integers is divided into three classes: one, prime numbers and composite numbers.

There is a very important theorem in number theory that states that “any integer can be represented as a product of prime numbers, and up to the order of the factors, such a representation is unique.” This theorem is known as the "fundamental theorem of arithmetic". It shows that prime numbers serve as the “building blocks” from which all integers other than one can be constructed using multiplication.

If a certain set of integers is given, then the largest integer that is a divisor of each number included in this set is called greatest common divisor given set of numbers; the smallest integer whose divisor is each number from a given set is called least common multiple given set of numbers. Yes, the largest common divisor numbers 12, 18 and 30 is 6. The least common multiple of the same numbers is 180. If the greatest common divisor of two integers a And b is equal to 1, then the numbers a And b are called mutually prime. For example, the numbers 8 and 9 are relatively prime, although neither of them is prime.

Positive rational numbers.

As we have seen, integers are abstractions that arise from the process of counting finite sets of objects. However, for the needs of everyday life, integers are not enough. For example, when measuring the length of a table top, the adopted unit of measurement may be too large and not fit a whole number of times into the measured length. To cope with such a difficulty, with the help of the so-called. fractional(i.e., literally, “broken”) numbers, a smaller unit of length is introduced. If d– some integer, then the fractional unit 1/ d determined by the property dґ1/d= 1, and if n is an integer, then nґ1/d we simply write it as n/d. These new numbers are called “ordinary” or “simple” fractions. Integer n called numerator fractions and numbers d – denominator. The denominator shows how many equal shares the unit was divided into, and the numerator shows how many such shares were taken. If n d, the fraction is called proper; if n = d or n>d, then it is incorrect. Integers are treated as fractions with a denominator of 1; for example, 2 = 2/1.

Since the fraction n/d can be interpreted as the result of division n units per d equal parts and taking one of those parts, a fraction can be thought of as the "quotient" or "ratio" of two whole numbers n And d, and understand the fraction line as a division sign. Therefore, fractions (including whole numbers as special case fractions) are usually called rational numbers (from the Latin ratio - ratio).

Two fractions n/d And ( kґn)/(kґd), Where k– an integer, can be considered equal; for example, 4/6 = 2/3. (Here n = 2, d= 3 and k= 2.) This is known as the “fundamental property of a fraction”: the value of any fraction will not change if the numerator and denominator of the fraction are multiplied (or divided) by the same number. It follows that any fraction can be written as the ratio of two relatively prime numbers.

From the interpretation of the fraction proposed above it also follows that as the sum of two fractions n/d And m/d having the same denominator, you should take the fraction ( n + m)/d. When adding fractions with different denominators, you must first convert them, using the basic property of a fraction, into equivalent fractions with the same (common) denominator. For example, n 1 /d 1 = (n 1 H d 2)/(d 1 H d 2) and n 2 /d 2 = (n 2 H d 1)/(d 1 H d 2), from where

One could do it differently and first find the least common multiple, say m, denominators d 1 and d 2. Then there are integers k 1 and k 2 , such that m = k 1 H d 1 = k 2 H d 2 and we get:

With this method the number m usually called lowest common denominator two fractions. These two results are equivalent by the definition of equality of fractions.

Product of two fractions n 1 /d 1 and n 2 /d 2 is taken equal to the fraction ( n 1 H n 2)/(d 1 H d 2).

The eight fundamental laws given above for integers are also valid if, under a, b, c understand arbitrary positive rational numbers. Also, if given two positive rational numbers n 1 /d 1 and n 2 /d 2, then we say that n 1 /d 1 > n 2 /d 2 if and only if n 1 H d 2 > n 2 H d 1 .

Positive real numbers.

The use of numbers to measure the lengths of line segments suggests that for any two given line segments AB And CD there must be some segment UV, perhaps very small, which could be postponed an integer number of times in each of the segments AB And CD. If so common unit length measurements UV exists, then the segments AB And CD are called commensurate. Already in ancient times, the Pythagoreans knew about the existence of incommensurable straight segments. A classic example is the side of a square and its diagonal. If we take the side of a square as a unit of length, then there is no rational number that could be a measure of the diagonal of this square. You can verify this by arguing by contradiction. Indeed, suppose that the rational number n/d is the measure of the diagonal. But then segment 1/ d could be postponed n once diagonally and d times on the side of the square, despite the fact that the diagonal and the side of the square are incommensurable. Consequently, regardless of the choice of unit of length, not all line segments have lengths that can be expressed in rational numbers. In order for all line segments to be measured by some unit of length, the number system must be expanded to include numbers representing the results of measuring the lengths of line segments that are incommensurate with the chosen unit of length. These new numbers are called positive irrational numbers. The latter, together with positive rational numbers, form a wider set of numbers, the elements of which are called positive valid numbers.

If OR– horizontal half-line emanating from a point O, U– point on OR, different from the origin O, And OU is chosen as a unit segment, then each point P on a half-line OR can be associated with a single positive real number p, expressing the length of the segment OP. In this way we establish a one-to-one correspondence between positive real numbers and points other than O, on a half-line OR. If p And q– two positive real numbers corresponding to points P And Q on OR, then we write p>q,p = q or p depending on the location of the point P to the right of the point Q on OR, coincides with Q or located to the left of Q.

The introduction of positive irrational numbers significantly expanded the scope of applicability of arithmetic. For example, if a– any positive real number and n is any integer, then there is only one positive real number b, such that bn=a. This number b called a root n th degree of a and is written as, where the symbol in its outline resembles a Latin letter r, with which the Latin word begins radix(root) and is called radical. It can be shown that

These relationships are known as the basic properties of radicals.

From a practical point of view, it is very important that any positive irrational number can be approximated as accurately as desired by a positive rational number. This means that if r is a positive irrational number and e is an arbitrarily small positive rational number, then we can find positive rational numbers a And b, such that a and b. For example, a number is irrational. If you select e= 0.01, then ; if you choose e= 0.001, then .

Indo-Arabic number system.

Algorithms, or calculation schemes, of arithmetic depend on the number system used. It is quite obvious, for example, that the calculation methods invented for the Roman number system may differ from the algorithms invented for the current Indo-Arabic system. Moreover, some number systems may be completely unsuitable for constructing arithmetic algorithms. Historical data shows that before the adoption of the Indo-Arabic number notation system, there were no algorithms at all that made it easy enough to add, subtract, multiply and divide numbers using “pencil and paper.” Over the long years of the existence of the Indo-Arabic system, numerous algorithmic procedures specially adapted to it were developed, so that our modern algorithms are the product of an entire era of development and improvement.

In the Hindu-Arabic number system, each entry representing a number is a set of ten basic symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, called numerals. For example, the Hindu-Arabic notation for the number four hundred and twenty-three takes the form of the sequence of digits 423. The meaning of a digit in the Hindu-Arabic notation of a number is determined by its place, or position, in the sequence of digits that form this notation. In the example we gave, the number 4 means four hundreds, the number 2 means two tens and the number 3 means three ones. The number 0 (zero), used to fill empty positions, plays a very important role; for example, the entry 403 means the number four hundred and three, i.e. tens are missing. If a, b, c, d, e mean individual numbers, then in the Indo-Arabic system abcde means an abbreviation for an integer

Since every integer admits a unique representation in the form

Where n is an integer, and a 0 , a 1 ,..., a n- numbers, we conclude that in a given number system, each integer can be represented in a unique way.

The Indo-Arabic number system allows you to concisely write not only integers, but also any positive real numbers. Let us introduce the notation 10 - n for 1/10 n, Where n– an arbitrary positive integer. Then, as can be shown, any positive real number can be represented, and uniquely, in the form

This record can be compressed by writing it as a sequence of numbers

where is the sign, called the decimal point, between a 0 and b 1 indicates where they begin negative powers the number 10 (in some countries a dot is used for this purpose). This method of writing a positive real number is called decimal expansion, and a fraction presented in the form of its decimal expansion is decimal.

It can be shown that for a positive rational number, the decimal expansion after the decimal point either breaks off (for example, 7/4 = 1.75) or repeats (for example, 6577/1980 = 3.32171717...). If a number is irrational, then its decimal expansion does not break off and does not repeat itself. If decimal expansion irrational number cut off at some decimal place, we get its rational approximation. The farther to the right of the decimal point the sign at which we terminate the decimal expansion is located, the better the rational approximation (the smaller the error).

In the Hindu-Arabic system, a number is written using ten basic digits, the meaning of which depends on their place, or position, in the notation of the number (the value of a digit is equal to the product of the digit and some power of 10). Therefore, such a system is called the decimal positional system. Positional number systems are very convenient for constructing arithmetic algorithms, and this is precisely why the Indo-Arabic number system is so widespread in the modern world, although in different countries Different symbols may be used to represent individual numbers.

Names of numbers.

The names of numbers in the Indo-Arabic system follow certain rules. The most common way of naming numbers is that the number is first divided into groups of three digits from right to left. These groups are called "periods". The first period is called the period of "units", the second - the period of "thousands", the third - the period of "millions", etc., as shown in the following example:

Each period is read as if it were a three-digit number. For example, the period 962 is read as "nine hundred sixty two". To read a number consisting of several periods, the group of digits in each period is read, starting with the leftmost one and then proceeding in order from left to right; Each group is followed by the name of the period. For example, the number above reads "seventy-three trillion eight hundred forty-two billion nine hundred sixty-two million five hundred thirty-two thousand seven hundred ninety-eight." Note that when reading and writing integers, the conjunction “and” is not usually used. The name of the unit category is omitted. Trillions are followed by quadrillions, quintillions, sextillions, septillions, octillions, nonallions, and decillions. Each period has a value 1000 times greater than the previous one.

In the Hindu-Arabic system, it is customary to follow the following procedure for reading the numbers to the right of the decimal point. Here the positions are called (in order from left to right): “tenths”, “hundredths”, “thousandths”, “ten-thousandths”, etc. A proper decimal is read as if the digits after the decimal point form a whole number, followed by the name of the position of the last digit to the right. For example, 0.752 is read as "seven hundred fifty-two thousandths." A mixed decimal is read by combining the rule for naming whole numbers with the rule for naming proper decimals. For example, 632.752 reads "six hundred thirty-two point seven hundred and fifty-two thousandths." Notice the word "integers" before the decimal point. IN last years decimal numbers are increasingly read more simply, for example 3.782 as "three point seven hundred and eighty two".

Addition.

Now we are ready to analyze the arithmetic algorithms that are introduced in primary school. These algorithms deal with operations on positive real numbers written as decimal expansions. We assume that elementary addition and multiplication tables have been learned by heart.

Consider the addition problem: calculate 279.8 + 5.632 + 27.54:

First, we sum up the same powers of the number 10. The number 19Х10 –1 is divided according to the distributive law into 9Х10 –1 and 10Х10 –1 = 1. We move the unit to the left and add it to 21, which gives 22. In turn, we split the number 22 into 2 and 20 = 2H10. We move the number 2H10 to the left and add it to 9H10, which gives 11H10. Finally, we divide 11H10 into 1H10 and 10H10 = 1H10 2, move 1H10 2 to the left and add it to 2H10 2, which gives 3H10 2. The final total turns out to be 312.972.

It is clear that the calculations performed can be presented in a more concise form, at the same time using it as an example of the addition algorithm that is taught in school. To do this, we write all three numbers one below the other so that the decimal points are on the same vertical:

Starting from the right, we find that the sum of the coefficients at 10 –3 is equal to 2, which we write in the corresponding column under the line. The sum of the coefficients at 10 –2 is equal to 7, which is also written in the corresponding column under the line. The sum of the coefficients for 10 –1 is 19. We write the number 9 under the line, and move 1 to the previous column, where there are ones. Taking into account this unit, the sum of the coefficient in this column turns out to be equal to 22. We write one two under the line, and move the other to the previous column, where the tens are. Taking into account the transferred two, the sum of the coefficients in this column is equal to 11. We write one unit under the line, and transfer the other to the previous column, where there are hundreds. The sum of the coefficients in this column turns out to be equal to 3, which we write below the line. The required amount is 312.972.

Subtraction.

Subtraction is the inverse of addition. If three positive real numbers a, b, c interconnected so that a+b=c, then we write a = c – b, where the symbol “-” is read as “minus”. Finding a number a according to known numbers b And c called "subtraction". Number c called minuend, number b– “subtractable”, and the number a- "difference". Since we are dealing with positive real numbers, the condition must be satisfied c > b.

Let's look at an example of subtraction: calculate 453.87 – 82.94.

First of all, borrowing a unit from the left if necessary, we transform the expansion of the minuend so that its coefficient for any power of 10 is greater than the coefficient of the subtrahend for the same power. From 4H10 2 we borrow 1H10 2 = 10H10, adding the last number to the next term in the expansion, which gives 15H10; similarly, we borrow 1Х10 0, or 10Ч10 –1, and add this number to the penultimate term of the expansion. After this we get the opportunity to subtract the coefficients at equal degrees numbers 10 and we can easily find the difference 370.93.

The recording of subtraction operations can be presented in a more compressed form and you can get an example of a subtraction algorithm studied in school. We write the subtrahend under the minuend so that their decimal points are on the same vertical. Starting from the right, we find that the difference in coefficients at 10 –2 is equal to 3, and we write this number in the same column under the line. Since in the next column on the left we cannot subtract 9 from 8, we change the three in the units position of the minuend to two and treat the number 8 in the tenths position as 18. After subtracting 9 from 18 we get 9, etc., i.e. .

Multiplication.

Let's first consider the so-called “short” multiplication is multiplication of a positive real number by one of the single-digit numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, for example, 32.67ґ4. Using the law of distributivity, as well as the laws of associativity and commutativity of multiplication, we get the opportunity to break factors into parts and arrange them in a more convenient way. For example,

These calculations can be written more compactly as follows:

The compression process can be continued. We write the factor 4 under the multiplicand 32.67, as indicated:

Since 4ґ7 = 28, we write the number 8 under the line, and place 2 above the number 6 of the multiplicand. Next, 4ґ6 = 24, which, taking into account what is transferred from the column on the right, gives 26. We write the number 6 below the line, and write 2 above the number 2 of the multiplicand. Then we get 4ґ2 = 8, which in combination with the transferred two gives 10. We sign the number 0 under the line, and the one above the number 3 of the multiplicand. Finally, 4ґ3 = 12, which, taking into account the transferred unit, gives 13; The number 13 is written below the line. Putting a decimal point, we get the answer: the product is equal to 130.68.

A "long" multiplication is simply a "short" multiplication repeated over and over again. Consider, for example, multiplying the number 32.67 by the number 72.4. Let's place the multiplier under the multiplicand, as indicated:

Doing short multiplication from right to left, we get the first quotient of 13.068, the second of 65.34, and the third of 2286.9. According to the law of distributivity, the product that needs to be found is the sum of these partial products, or 2365.308. In written notation, the decimal point in partial products is omitted, but they must be correctly arranged in “steps” in order to then be summed up to obtain the complete product. The number of decimal places in the product is equal to the sum of the number of decimal places in the multiplicand and the multiplier.

Division.

Division is the inverse operation of multiplication; just as multiplication replaces repeated addition, division replaces repeated subtraction. Consider, for example, the question: how many times is 3 contained in 14? Repeating the operation of subtracting 3 from 14, we find that 3 “enters” 14 four times, and the number 2 “remains”, i.e.

The number 14 is called divisible, number 3 – divider, number 4 – private and number 2 – the remainder. The resulting relationship can be expressed in words as follows:

dividend = (divisor ґ quotient) + remainder,

0 Ј remainder

Finding the quotient and remainder of 1400 divided by 3 by repeatedly subtracting 3 would require a lot of time and effort. The procedure could be significantly speeded up if we first subtract 300 from 1400, then 30 from the remainder, and finally 3. After subtracting 300 four times, we would get a remainder of 200; after subtracting 30 from 200 six times, the remainder would be 20; finally, after subtracting 3 from 20 six times, we get the remainder 2. Therefore,

The quotient and remainder to be found are 466 and 2, respectively. The calculations can be organized and then sequentially compressed as follows:

The above reasoning applies if the dividend and divisor are any positive real numbers expressed in the decimal system. Let's illustrate this with the example of 817.65е23.7.

First, the divisor must be converted to an integer using a decimal point shift. In this case, the decimal point of the dividend is shifted by the same number of decimal places. The divisor and dividend are arranged as shown below:

Let's determine how many times the divisor is contained in three digit number 817, the first part of the dividend that we divide by the divisor. Since it is estimated to be contained three times, we multiply 237 by 3 and subtract the product of 711 from 817. The difference of 106 is less than the divisor. This means that the number 237 appears in the trial dividend no more than three times. The number 3, written under the number 2 divisor below the horizontal line, is the first digit of the quotient that needs to be found. After we move down the next digit of the dividend, we get the next trial dividend 1066, and we need to determine how many times the divisor 237 fits into the number 1066; Let's say 4 times. We multiply the divisor by 4 and get the product 948, which we subtract from 1066; the difference turns out to be 118, which means that the next digit of the quotient is 4. We then subtract the next digit of the dividend and repeat the entire procedure described above. This time it turns out that the trial dividend 1185 is exactly (without a remainder) divisible by 237 (the remainder of the division finally turns out to be 0). By separating with a decimal point in the quotient the same number of digits as they are separated in the dividend (remember that we previously moved the decimal point), we get the answer: the quotient is equal to 34.5.

Fractions.

Calculations with fractions include addition, subtraction, multiplication and division, as well as simplifying complex fractions.

Adding fractions with the same denominator is done by adding the numerators, for example,

1/16 + 5/16 + 7/16 = (1 + 5 + 7)/16 = 13/16.

If fractions have different denominators, then they must first be reduced to a common denominator, i.e. convert to fractions with the same denominators. To do this, we find the least common denominator (the smallest multiple of each of the given denominators). For example, when adding 2/3, 1/6 and 3/5, the lowest common denominator is 30:

Summing up, we get

20/30 + 5/30 + 18/30 = 43/30.

Subtracting fractions is done in the same way as adding them. If the denominators are the same, then the subtraction comes down to subtracting the numerators: 10/13 – 2/13 = 8/13; If fractions have different denominators, then you must first bring them to a common denominator:

7/8 – 3/4 = 7/8 – 6/8 = (7 – 6)/8 = 1/8.

When multiplying fractions, their numerators and denominators are multiplied separately. For example,

5/6ґ4/9 = 20/54 = 10/27.

To divide one fraction by another, you need to multiply the first fraction (dividend) by the reciprocal fraction of the second (divisor) (to get the reciprocal fraction, you need to swap the numerator and denominator of the original fraction), i.e. ( n 1 /d 1)е( n 2 /d 2) = (n 1 H d 2)/(d 1 H n 2). For example,

3/4е7/8 = 3/4ґ8/7 = 24/28 = 6/7.

A mixed number is the sum (or difference) of a whole number and a fraction, such as 4 + 2/3 or 10 – 1/8. Since a whole number can be thought of as a fraction with a denominator of 1, a mixed number is nothing more than the sum (or difference) of two fractions. For example,

4 + 2/3 = 4/1 + 2/3 = 12/3 + 2/3 = 14/3.

A complex fraction is one that has a fraction in either the numerator, the denominator, or the numerator and denominator. This fraction can be converted into a simple one:

Square root.

If n r, such that r 2 = n. Number r called square root from n and is designated . At school they teach you to extract square roots in two ways.

The first method is more popular because it is simpler and easier to apply; calculations using this method are easily implemented on a desktop calculator and generalize to the case of cube roots and higher roots. The method is based on the fact that if r 1 – approaching the root, then r 2 = (1/2)(r 1 + n/r 1) – more accurate approximation of the root.

Let's illustrate the procedure by calculating the square root of some number between 1 and 100, say the number 40. Since 6 2 = 36 and 7 2 = 49, we conclude that 6 is the best approximation to in whole numbers. A more accurate approximation to is obtained from 6 as follows. Dividing 40 by 6 gives 6.6 (rounded to the first decimal place) even numbers of tenths). To get a second approximation to , we average the two numbers 6 and 6.6 and get 6.3. Repeating the procedure, we obtain an even better approximation. Dividing 40 by 6.3, we find the number 6.350, and the third approximation turns out to be (1/2)(6.3 + 6.350) = 6.325. Another repetition gives 40е6.325 = 6.3241106, and the fourth approximation turns out to be (1/2)(6.325 + 6.3241106) = 6.3245553. The process can continue as long as desired. In general, each subsequent approximation can contain twice as many digits as the previous one. So, in our example, since the first approximation, the integer 6, contains only one digit, we can keep two digits in the second approximation, four in the third, and eight in the fourth.

If the number n does not lie between 1 and 100, then you must first divide (or multiply) n to some power of 100, say, to k-th so that the product is in the range from 1 to 100. Then the square root of the product will be in the range from 1 to 10, and after it is extracted, we multiply (or divide) the resulting number by 10 k, find the required square root. For example, if n= 400000, then we first divide 400000 by 100 2 and we get the number 40, which lies in the range from 1 to 100. As shown above, it is approximately equal to 6.3245553. Multiplying this number by 10 2, we get 632.45553 as an approximate value for, and the number 0.63245553 serves as an approximate value for.

The second of the procedures mentioned above is based on the algebraic identity ( a + b) 2 = a 2 + (2a + b)b. At each step, the already obtained part of the square root is taken as a, and the part that still needs to be determined is for b.

Cube root.

To extract the cube root of a positive real number, there are algorithms similar to those for extracting the square root. For example, to find the cube root of a number n, first we approximate the root by some number r 1 . Then we build a more accurate approximation r 2 = (1/3)(2r 1 + n/r 1 2), which in turn gives way to an even more accurate approximation r 3 = (1/3)(2r 2 + n/r 2 2), etc. The procedure for constructing increasingly accurate approximations of the root can continue indefinitely.

Consider, for example, calculating the cube root of a number between 1 and 1000, say the number 200. Since 5 3 = 125 and 6 3 = 216, we conclude that 6 is the closest integer to the cube root of 200. Therefore, we choose r 1 = 6 and sequentially calculate r 2 = 5,9, r 3 = 5,85, r 4 = 5.8480. In each approximation, starting from the third, it is allowed to maintain a number of characters that is one less than twice the number of characters in the previous approximation. If the number from which you want to extract the cube root is not between 1 and 1000, then you must first divide (or multiply) it by some, say, k th, power of the number 1000 and thereby bring it into the desired range of numbers. The cube root of the newly obtained number lies in the range from 1 to 10. After it is calculated, it must be multiplied (or divided) by 10 k to get the cube root of the original number.

The second, more complex, algorithm for finding the cube root of a positive real number is based on the use of the algebraic identity ( a + b) 3 = a 3 + (3a 2 + 3ab + b 2)b. Currently, algorithms for extracting cube roots, as well as roots of higher degrees, in high school are not studied because they are easier to find using logarithms or algebraic methods.

Euclid's algorithm.

This algorithm was presented in Beginnings Euclid (c. 300 BC). It is used to calculate the greatest common divisor of two integers. For the occasion positive numbers it is formulated as a procedural rule: “Divide the larger of the two given numbers by the smaller. Then divide the divisor by the remainder and continue in this manner until the last divisor is evenly divided by the last remainder. The last of the divisors will be the greatest common divisor of the two given numbers.”

As numerical example Let's consider two integers 3132 and 7200. The algorithm in this case comes down to the following actions:

The greatest common divisor is the same as the last divisor - the number 36. The explanation is simple. In our example, we see from the last line that the number 36 divides the number 288. From the penultimate line it follows that the number 36 divides 324. So, moving up from line to line, we are convinced that the number 36 divides 936, 3132 and 7200 . We now claim that the number 36 is a common divisor of the numbers 3132 and 7200. Let g is the greatest common divisor of the numbers 3132 and 7200. Since g divides 3132 and 7200, from the first line it follows that g divides 936. From the second line we conclude that g divides 324. So, going down from line to line, we are convinced that g divides 288 and 36. And since 36 is a common divisor of the numbers 3132 and 7200 and is divided by their greatest common divisor, we conclude that 36 is this greatest common divisor.

Examination.

Arithmetic calculations require constant attention and are therefore prone to errors. Therefore, it is very important to check the calculation results.

1. The addition of a column of numbers can be checked by adding the numbers in the column first from top to bottom and then from bottom to top. The justification for this method of verification is the generalized law of commutativity and associativity of addition.

2. Subtraction is checked by adding the difference with the subtrahend - the minuend should be obtained. The rationale for this verification method is the definition of the subtraction operation.

3. Multiplication can be checked by rearranging the multiplicand and the multiplier. The justification for this method of verification is the law of commutative multiplication. You can check multiplication by breaking the factor (or multiplicand) into two terms, performing two separate multiplication operations and adding the resulting products - you should get the original product.

4. To check division, you need to multiply the quotient by the divisor and add the remainder to the product. It should be the dividend. The rationale for this verification method is the definition of the division operation.

5. Checking the correctness of extracting a square (or cubic) root consists of raising the resulting number by squaring (or cube) - the original number should be obtained.

A particularly simple and very reliable way to check the addition or multiplication of integers is a technique that represents a transition to the so-called. "comparisons modulo 9". Let us call “excess” the remainder of the sum of the digits used when dividing by 9 given number. Then, regarding “excesses,” two theorems can be formulated: “the excess of the sum of integers is equal to the excess of the sum of the excesses of terms,” and “the excess of the product of two integers is equal to the excess of the product of their excesses.” Below are examples of checks based on this theorem:

The method of moving to comparisons modulo 9 can also be used when testing other arithmetic algorithms. Of course, such a check is not infallible, since working with “excesses” is also subject to errors, but such a situation is unlikely.

Interest.

A percentage is a fraction whose denominator is 100; Interest can be written in three ways: as common fraction, as a decimal fraction or using the special percentage notation %. For example, 7 percent can be written as 7/100, as 0.07, or as 7%.

An example of the most common type of percentage problem is the following: “Find 17% of 82.” To solve this problem, you need to calculate the product 0.17ґ82 = 13.94. In products of this kind, 0.17 is called the rate, 82 is the base, and 13.94 is the share, expressed as a percentage. The three mentioned quantities are related to each other by the relation

Rate ґ base = percentage share.

If any two quantities are known, the third can be determined from this relationship. Accordingly, we get three types of problems “using percentages”.

Example 1. The number of students enrolled in this school increased from 351 to 396. By what percentage did this number increase?

The increase was 396 – 351 = 45 people. Writing the fraction 45/351 as a percentage, we get 45/351 = 0.128 = 12.8%.

Example 2. An ad in the store during a sale says “25% off all items.” What is the sale price for an item that normally sells for $3.60?

A 25% decrease in price of $3.60 means a decrease of 0.25-3.60 = $0.90; therefore, the price of the item during the sale will be $3.60 – $0.90 = $2.70.

Example 3. Money deposited in the bank at 5% per annum brought a profit of $40 per year. What amount was deposited into the bank?

Since 5% of the amount is $40, i.e. 5/100 ґ amount = $40, or 1/100 ґ amount = 8 dollars, the total amount is 800 dollars.

Arithmetic of approximate numbers.

Many numbers used in calculations arise either from measurements or estimates and can therefore only be considered approximations. It is obvious that the result of calculations performed with approximate numbers can only be an approximate number. For example, suppose that measurements of the counter surface yielded the following results (rounded to the nearest tenth of a meter): width 1.2 m, length 3.1 m; one could say that the area of ​​the counter is 1.2ґ3.1 = 3.72 m2. However, in reality the information is far from being so certain. Since the value 1.2 m only indicates that the width measurement is between 1.15 and 1.25 m, and 3.1 indicates that the length measurement is between 3.05 and 3.15 m, about the counter area we can only say that it should be greater than 1.15ґ3.05 = 3.5075, but less than 1.25ґ3.15 = 3.9375. Therefore, the only reasonable answer to the question about the area of ​​the counter is to say that it is approximately 3.7 m 2 .

Let us next consider the problem of adding the results of approximate measurements of 3.73 m, 52.1 m and 0.282 m. The simple sum is 56.112 m. But, as in the previous problem, all that can be said with certainty is that the true sum must be greater than 3.725 + 52.05 + 0.2815 = 56.0565 m and less than 3.735 + 52.15 + 0.2825 = 56.1765 m. Thus, the only reasonable answer to the question is to say that the sum is approximately equal to 56.1 m.

The two examples above illustrate some rules that are useful when working with approximate numbers. Exist various ways rounding numbers. One of them is to discard the lower digits of the number. Moreover, if the first digit to be discarded is more than five, then the last remaining digit must be increased by one, if less, then the last digit of the remaining part remains unchanged.

If the first digit to be discarded is exactly five, then the last digit to be retained is increased by one if it is odd and remains unchanged if it is even. For example, when rounding to the nearest hundredth the number 3.14159;17.7682; 28,999; 0.00234; 7.235 and 7.325 become 3.14; 17.77; 29.00; 0.00; 7.24 and 7.32.

Another method of rounding is associated with the concept of significant figures and is used when writing a number by machine. The significant digits of an approximate number are the digits in its decimal notation in order from left to right, starting with the first non-zero digit and ending with the digit that stands in place of the decimal place corresponding to the error. For example, the significant digits of the approximate number 12.1 are the numbers 1, 2, 1; approximate number 0.072 – numbers 7, 2; the approximate number 82000, written to the nearest hundred, is 8, 2, 0.

Now we will formulate the two rules for operating with approximate numbers mentioned above.

When adding and subtracting approximate numbers, each number should be rounded to the digit next in number to the last digit. exact number, and round the resulting sum and difference to the same number of decimal places as the least accurate number. When multiplying and dividing approximate numbers, each number should be rounded to the sign following the last significant digit of the least significant number, and the product and quotient should be rounded with the same accuracy as the least accurate number is known.

Returning to the previously considered problems, we get:

1.2ґ3.1 = 3.72 m 2 » 3.7 m 2

3.73 + 52.1 + 0.28 = 56.11 m 2 "56.1 m,

where the sign " means "approximately equal".

Some arithmetic textbooks provide algorithms for working with approximate numbers, allowing you to avoid unnecessary signs when calculating. In addition, they use the so-called. recording approximate numbers, i.e. any number is represented in the form (a number in the range from 1 to 10) ґ (power of 10), where the first factor contains only the significant digits of the number. For example, 82000 km, rounded to the nearest hundred km, would be written as 8.20ґ10 4 km, and 0.00702 cm would be written as 7.02ґ10 –3 cm.

Numbers in mathematical tables, trigonometric or logarithm tables are approximate, written with a certain number of signs. When working with such tables, you should follow the rules for calculations with approximate numbers.

Logarithms.

By the beginning of the 17th century. The complexity of applied computing problems has increased so much that it was not possible to cope with them “manually” due to too much labor and time. Fortunately, invented in time by J. Napier at the beginning of the 17th century. logarithms made it possible to cope with the problem that arose. Since the theory and applications of logarithms are described in detail in a special article LOGARITHM, we will limit ourselves to only the most necessary information.

It can be shown that if n is a positive real number, then there is a unique positive real number x, such that 10 x = n. Number x called (regular or decimal) logarithm numbers n; conventionally it is written like this: x=log n. Thus, the logarithm is an exponent, and from the laws of operations with exponents it follows that

It is these properties of logarithms that explain their widespread use in arithmetic. The first and second properties allow us to reduce any multiplication and division problem to a simpler addition and subtraction problem. The third and fourth properties make it possible to reduce exponentiation and root extraction to much simpler operations: multiplication and division.

For ease of use of logarithms, their tables have been compiled. To compile a table of decimal logarithms, it is enough to include only logarithms of numbers from 1 to 10. For example, since 247.6 = 10 2 ґ2.476, we have: log247.6 = log10 2 + log2.476 = 2 + log2.476, and since 0.02476 = 10 –2 ґ2.476, then log0.02476 = log10 –2 + log2.476 = –2 + log2.476. Note that the decimal logarithm of a number between 1 and 10 lies between 0 and 1 and can be written as a decimal. It follows that the decimal logarithm of any number is the sum of an integer, called the characteristic of the logarithm, and a decimal fraction, called the mantissa of the logarithm. The characteristic of the logarithm of any number can be found “in the mind”; The mantissa should be found using tables of logarithms. For example, from the tables we find that log2.476 = 0.39375, hence log247.63 = 2.39375. If the characteristic of the logarithm is negative (when the number is less than one), then it is convenient to represent it as the difference of two positive integers, for example, log0.02476 = –2 + 0.39375 = 8.39375 – 10. The following examples explain this technique.

Literature:

History of mathematics from ancient times to early XIX V., vol. 1–3. M., 1970–1972.
Serre J.-P. Arithmetic course. M., 1972
Nechaev V.I. Numerical systems. M., 1975
Daan-Dalmedico A., Peiffer J . Paths and labyrinths. Essays on the history of mathematics. M., 1986
Engler E. Elementary Mathematics. M., 1987



Arithmetic is the most basic, basic section of mathematics. It originated from people's needs for counting.

Mental arithmetic

What is called mental arithmetic? Mental arithmetic is a learning method quick counting who came from antiquity.

Currently, unlike the previous one, teachers are trying not only to teach children how to count, but also trying to develop their thinking.

The learning process itself is based on the use and development of both hemispheres of the brain. The main thing is to be able to use them together, because they complement each other.

Indeed, the left hemisphere is responsible for logic, speech and rationality, and the right hemisphere is responsible for imagination.

The training program includes training in operation and use of tools such as abacus.

The abacus is the main tool in learning mental arithmetic, because students learn to work with them, move the dominoes and understand the essence of the calculation. Over time, the abacus becomes your imagination, and students imagine them, build on this knowledge and solve examples.

Reviews about these teaching methods are very positive. There is one drawback - the training is paid, and not everyone can afford it. Therefore, the path of a genius depends on one’s financial situation.

Mathematics and arithmetic

Mathematics and arithmetic are closely related concepts, or rather arithmetic is a branch of mathematics that works with numbers and calculations (operations with numbers).

Arithmetic is the main section, and therefore the basis of mathematics. Basic Mathematics – the most important concepts and operations that form the basis on which all subsequent knowledge is built. The main operations include: addition, subtraction, multiplication, division.

Arithmetic is usually studied at school from the very beginning of education, that is. from first grade. Children master basic mathematics.

Addition is an arithmetic operation during which two numbers are added, and their result is a new one - the third.

a+b=c.

Subtraction is an arithmetic operation in which the second number is subtracted from the first number, and the result is the third.

The addition formula is expressed as follows: a - b = c.

Multiplication is an action that results in the sum of identical terms.

The formula for this action is: a1+a2+…+an=n*a.

Division- This is the division of a number or variable into equal parts.

Sign up for the course "Speed ​​up mental arithmetic, NOT mental arithmetic"to learn how to quickly and correctly add, subtract, multiply, divide, square numbers and even take roots. In 30 days you will learn how to use easy techniques to simplify arithmetic operations. Each lesson contains new techniques, clear examples and useful tasks.

Teaching arithmetic

Arithmetic is taught within the school walls. From the first grade, children begin to study the basic and main section of mathematics - arithmetic.

Adding numbers

Rules of arithmetic

The order of operations in an expression is very important!

If the example looks like 2+3-4, then the order in it can be whatever you want. Because addition and subtraction operations have the same priority. If we do the addition first, we get: 5-4=1, and if we do the subtraction first, then: 2-1=1. As you can see, the result is the same.

Similarly with the expression for multiplication and division. Multiplication and division operations have the same priority. For example, 2 8:4. Let's do the multiplication first: 16:4=4, and if division: 2 2=4.

Order makes sense when an expression mixes addition or subtraction operations with multiplication or division operations. For example:

2+22. The first action is to perform ALL operations of multiplication and division, and only then addition and subtraction. That is, the expression 2+2 2 = 2+4=6.

But there are parentheses in the expressions. Parentheses tend to change the order of operations. Let's consider the previous example, only with brackets: (2+2)*2. In this case, the operations inside the brackets are performed first, and then outside the brackets in the order: 1. Multiplication and division 2. Addition and subtraction.

So, (2+2) 2=4 2=8.

As you can see from the examples, parentheses have a role. And the order of operations is the same.

Arithmetic lessons

Arithmetic lessons - school lessons, up to the sixth grade. Then mathematics opens its sections: geometry and algebra, and later trigonometry.

Arithmetic 5th grade

In the fifth grade, students begin to study topics such as: fractions, mixed numbers. You can find information about operations with these numbers in our articles on the relevant operations.

A fractional number is the ratio of two numbers to each other or the numerator to the denominator. A fractional number can be replaced by division. For example, ¼ = 1:4.

Mixed number– this is a fractional number, only with the integer part highlighted. The integer part is allocated provided that the numerator is greater than the denominator. For example, there was a fraction: 5/4, it can be transformed by highlighting the whole part: 1 whole and ¼.

Examples for training:

Task No. 1:

Task No. 2:

Arithmetic 6th grade

In 6th grade, the topic of converting fractions to lowercase notation appears. What does it mean? For example, given the fraction ½, it will be equal to 0.5. ¼ = 0.25.

Examples can be compiled in the following style: 0.25+0.73+12/31.

Examples for training:

Task No. 1:

Task No. 2:

Games for developing mental arithmetic and counting speed

There are great games that promote numeracy, help develop math skills and mathematical thinking, mental counting and counting speed! You can play and develop! You are interested? Read short articles about games and be sure to try yourself.

Game "Quick Count"

The "quick count" game will help you speed up your mental counting. The essence of the game is that in the picture presented to you, you will need to choose a yes or no answer to the question “Are there 5 identical fruits?” Follow your goal, and this game will help you with this.

Game "Mathematical Comparisons"

The Math Comparison game will require you to compare two numbers against the clock. That is, you have to choose one of two numbers as quickly as possible. Remember that time is limited, and the more you answer correctly, the better your math skills will develop! Shall we try?

Game "Quick addition"

The game “Quick Addition” is an excellent quick counting simulator. The essence of the game: a 4x4 field is given, that is. There are 16 numbers, and above the field is the seventeenth number. Your goal: using sixteen numbers, make 17 using the addition operation. For example, above the field you have the number 28 written, then in the field you need to find 2 such numbers that in total will give the number 28. Are you ready to try your hand? Then go ahead and train!

Development of phenomenal mental arithmetic

We have looked at only the tip of the iceberg, to understand mathematics better - sign up for our course: Accelerating mental arithmetic - NOT mental arithmetic.

From the course you will not only learn dozens of techniques for simplified and quick multiplication, addition, multiplication, division, and calculating percentages, but you will also practice them in special tasks and educational games! Mental arithmetic also requires a lot of attention and concentration, which are actively trained when solving interesting problems.

Speed ​​reading in 30 days

Increase your reading speed by 2-3 times in 30 days. From 150-200 to 300-600 words per minute or from 400 to 800-1200 words per minute. The course uses traditional exercises for the development of speed reading, techniques that speed up brain function, methods for progressively increasing reading speed, the psychology of speed reading and questions from course participants. Suitable for children and adults reading up to 5000 words per minute.

Development of memory and attention in a child 5-10 years old

The purpose of the course: to develop the child’s memory and attention so that it is easier for him to study at school, so that he can remember better.

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