"Simple arithmetic" Solve some unusual mathematical problems. 1) I go to the pool 1 time in 3 days; Seryozha - 1 time in 4 days, and Kolya - 1 time in 5 days. Last Monday we all met at the pool. When will we meet again and what day of the week will it be? (Across

How the dog "Let us drive" appeared on the buses. Greyhound, the oldest and most popular bus company in the country, began in Hibbing, Minnesota. In this city, many years later, Bob Dylan was born, who, unlike

Arithmetic is the most basic, basic section of mathematics. It owes its appearance to the needs of people in the account.

mental arithmetic

What is mental arithmetic? Mental arithmetic is a method of learning to count quickly, which came from antiquity.

At present, unlike the previous one, teachers are trying not only to teach children the speed of counting, but also try to develop 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, while the right hemisphere is responsible for imagination.

The training program includes training in operation and the use of a tool such as abacus.

Abacus is the main tool in the study of mental arithmetic, because students learn to work with them, sort out the bones and realize the essence of counting. Over time, the abacus becomes your imagination, and the students imagine them, build on this knowledge and solve examples.

Feedback on these teaching methods is very positive. There is one minus - training is paid, and not everyone can afford it. Therefore, the path of a genius depends on the financial situation.

Mathematics and arithmetic

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

Arithmetic is the main section, and therefore the basis of mathematics. The basis of mathematics is 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, as a rule, is studied at school from the very beginning of training, that is. from first grade. Children learn the basics of mathematics.

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

a+b=c.

Subtraction- this is an arithmetic operation, during which the second number is subtracted from the first number, and the third number will be the result.

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

Multiplication is an action, as a result of which the sum of identical terms is found.

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

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

Sign up for the course "Speed ​​up mental counting, 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 tricks to simplify arithmetic operations. Each lesson contains new techniques, clear examples and useful tasks.

Learning arithmetic

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

Number addition

Arithmetic rules

The order in which operations are performed in an expression is very important!

If the example has the form 2+3-4, then the order in it can be anything. Because addition and subtraction operations have the same precedence. If we perform addition first, we get: 5-4=1, and if we subtract first, then: 2-1=1. As you can see the result is the same.

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

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

2+22. The first action is performed 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 brackets in the expressions. Parentheses tend to change the order of operations. 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 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. Further, mathematics opens its sections: geometry and algebra, and later trigonometry.

Arithmetic Grade 5

In the fifth grade, the student begins to study such topics as: fractional numbers, mixed numbers. You can find information about operations with these numbers in our articles on the corresponding 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 the division operation. For example, ¼ = 1:4.

mixed number is a fractional number, only with a highlighted integer part. 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 converted by highlighting the whole part: 1 whole and ¼.

Examples for training:

Task number 1:

Task number 2:

Arithmetic Grade 6

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

Examples can be written in this style: 0.25+0.73+12/31.

Examples for training:

Task number 1:

Task number 2:

Games for the development of mental counting and counting speed

There are wonderful games that help develop counting, 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 Score"

The "quick counting" 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 "Fast Addition"

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

Development of phenomenal mental arithmetic

We have considered only the tip of the iceberg, in order to understand mathematics better - sign up for our course: Speed ​​up mental arithmetic - NOT mental arithmetic.

From the course, you will not only learn dozens of tricks for simplified and fast multiplication, addition, multiplication, division, calculating percentages, but also work them out in special tasks and educational games! Mental counting also requires a lot of attention and concentration, which are actively trained in 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 wpm or from 400 to 800-1200 wpm. The course uses traditional exercises for the development of speed reading, techniques that speed up the work of the brain, a method for progressively increasing the speed of reading, understands the psychology of speed reading and the questions of course participants. Suitable for children and adults reading up to 5,000 words per minute.

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

The purpose of the course is 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.

After completing the course, the child will be able to:

1. 2-5 times better to remember texts, faces, numbers, words
2. Learn to remember for longer
3. The speed of remembering the necessary information will increase

With arithmetic, the science of numbers, our acquaintance with mathematics begins. One of the first Russian arithmetic textbooks, written by L. F. Magnitsky in 1703, began with the words: “Arithmetic or numerator, is an art that is honest, unenviable, and conveniently understandable to everyone, most useful and most praised, from the oldest and newest, who lived at different times of the finest arithmeticians, 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 knowing the world.

The word "arithmetic" comes from the Greek arithmos, which means "number". This science studies operations on numbers, various rules handling them, teaches you to solve problems that boil down to addition, subtraction, multiplication and division of numbers. Arithmetic is often imagined as some first step in mathematics, based on which it is possible to study its more complex sections - algebra, mathematical analysis, etc. Even integers - the main object of arithmetic - are referred, when considering their general properties and patterns, to higher arithmetic, or number theory. Such a view of arithmetic, of course, has grounds - it really remains the "alphabet of counting", but the alphabet is "most useful" and "comfortable".

Arithmetic and geometry are old companions of man. These sciences appeared when it became necessary to count objects, measure land, divide booty, keep track of time.

Arithmetic originated in countries ancient east: Babylon, China, India, Egypt. For example, the Egyptian papyrus Rinda (named after its owner G. Rinda) dates back to the 20th century. BC. Among other information, it contains expansions of a fraction into the 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. Many names of scientists involved in arithmetic in the ancient world have been preserved for us by history - Anaxagoras and Zeno, Euclid (see Euclid and his "Beginnings"), Archimedes, Eratosthenes and Diophantus. The name of Pythagoras (VI century BC) sparkles here as a bright star. The Pythagoreans (disciples 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 in high esteem, at the same time attention was paid to the so-called perfect numbers, friendly numbers, etc.

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

Thanks to the development of trade and the influence of oriental culture since the XIII century. increasing interest in arithmetic in Europe. One should remember the name of the Italian scientist Leonardo of Pisa (Fibonacci), whose work "The Book of the Abacus" introduced Europeans to the main achievements of the mathematics of the East and was the beginning of many studies in arithmetic and algebra.

Together 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. The Complete Arithmetic by the German mathematician M. Stiefel (early 16th century) already contains negative numbers and even the idea of ​​taking a logarithm.

Around 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. Since that time, the basic arithmetic rules have been fully understood from the standpoint of algebra.

The basic object of arithmetic is the number. Natural numbers, i.e. the numbers 1, 2, 3, 4, ... etc., arose from counting specific items. Many millennia passed before man learned that two pheasants, two hands, two people, etc. can be called the same word "two". An important task of arithmetic is to learn to overcome the specific meaning of the names of counted objects, to abstract from their shape, size, color, etc. Fibonacci already has a task: “Seven old women are going to Rome. Each has 7 mules, each mule carries 7 bags, each bag has 7 loaves, each loaf has 7 knives, each knife has 7 sheaths. How many? To solve the problem, you will have to put together old women, and mules, and 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 (numbers, symbols, number systems) - all this has a rich and interesting history.

“The science of numbers means 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. Now, in our minds or on a piece of paper, we do only the simplest calculations, more and more often entrusting more complex computational work to microcalculators, which are gradually replacing such devices as abacus, adding machine (see Computing), slide rule. However, the operation of all computers - simple and complex - is based on the simplest operation - the addition of natural numbers. It turns out that the most complex calculations can be reduced to addition, only this operation must be done many millions of times. But here we are invading another area of ​​mathematics that 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 from a change in 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 displacement and combination laws, because it’s so 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 given on arbitrary objects (not necessarily on numbers) is already the domain 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 for the "triple rule", for "proportional division", for the "method of weights", for the "false rule", etc. Most of these rules are now obsolete, although the tasks that were solved with their help can by no means be considered obsolete. The famous problem about a 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 today even younger students are taught to solve such a problem by entering the letter designation of the desired value. Thus, arithmetic problems led to the need to solve equations, and this is again the task of algebra.

Among the important concepts introduced by arithmetic, proportions and percentages should be noted. Most of the concepts and methods of arithmetic are based on comparing various relationships between numbers. In the history of mathematics, the process of merging arithmetic and geometry took place over many centuries.

One can clearly trace the "geometrization" of arithmetic: complex rules and patterns expressed by formulas become clearer if one can represent them geometrically. plays an important role in mathematics itself and its applications. reverse process- 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 on the definition of points on the 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 the ship, as well as in astronomy and geodesy. But it is precisely from Descartes and his students that the consistent use of the language of coordinates in mathematics comes. And in our time, when managing complex processes (for example, the flight of a spacecraft), they prefer to have all the information in the form of numbers, which are processed by a computer. If necessary, the machine helps a person to translate the accumulated numerical information into the language of the drawing.

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

How to delineate the boundaries of arithmetic itself?

In what sense is this word used?

The word "arithmetic" can be understood as:

an academic subject dealing 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" - 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" - 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.

• Arithmetic (ancient Greek ἀριθμητική; from ἀριθμός - number) is a branch of mathematics that studies numbers, their relationships and properties. The subject of arithmetic is the concept of number in the development of ideas about it (natural, integer and rational, real, complex numbers) and its properties. In arithmetic, measurements, computational operations (addition, subtraction, multiplication, division) and calculation methods are considered. Higher arithmetic, or number theory, deals with the study of the properties of individual integers. Theoretical arithmetic pays attention to the definition and analysis of the concept of number, while formal arithmetic operates with logical constructions of predicates and axioms. Arithmetic is the oldest and one of the main mathematical sciences; it is closely related to algebra, geometry and number theory.

  The reason for the emergence of arithmetic was the practical need for an account, and calculations related to the tasks of accounting during the centralization of agriculture. Science has evolved along with the increasing complexity of the problems that need to be solved. A great contribution to the development of arithmetic was made by Greek mathematicians, in particular the Pythagorean philosophers, who tried to comprehend and describe all the laws of the world with the help of numbers.

  In the Middle Ages, arithmetic, following the Neoplatonists, was included among the so-called seven liberal arts. Main areas practical application arithmetic then were trade, navigation, construction. In this regard, approximate calculations of irrational numbers, which are necessary primarily for geometric constructions, have received special importance. Arithmetic developed especially rapidly in India and the countries of Islam, from where the latest achievements of mathematical thought penetrated into Western Europe; Russia got acquainted with mathematical knowledge "both from the Greeks and from the Latins."

  With the advent of the New Age, nautical astronomy, mechanics, and more complex commercial calculations set new demands on the technique of computing and gave impetus to further development arithmetic. At the beginning of the 17th century, Napier invented logarithms, and then Fermat singled out number theory as an independent section of arithmetic. By the end of the century, an idea was formed of an irrational number as a sequence of rational approximations, and over the next century, thanks to the works of Lambert, Euler, Gauss, arithmetic included operations with complex quantities, acquiring a modern look.

  The subsequent history of arithmetic was marked by a critical revision of its foundations, attempts at its deductive justification. The theoretical substantiation of the idea of ​​a number is associated primarily with the strict definition of the natural number and Peano's axioms, formulated in 1889. The consistency of the formal construction of arithmetic was shown by Gentzen in 1936.

  The basics of arithmetic have long and invariably been given great attention in primary school education.

On the one hand, this is a very simple question. On the other hand, schoolchildren, and many adults, often confuse arithmetic and mathematics and do not really know what is the difference between these two subjects. Mathematics is the most extensive concept that includes any operations with numbers. Arithmetic is just one branch of mathematics. Arithmetic includes familiarity with numbers, simple counting, and operations with numbers. Previously, in schools, lessons were called precisely arithmetic, and only over time they began to bear the name of mathematics, which smoothly flows into algebra. In fact, algebra begins when unknown numbers appear in the examples and letters are used instead. That is, in a simple way, operations with x And y.

Term "arithmetic" derived from the Greek word "arithmos" which means "number". In the 14-15 centuries, this term was translated in England not quite correctly - “the metric art”, which essentially meant “metric art”, more suitable for geometry than simple counting and simple operations with numbers.

One of the reasons why the concept of “arithmetic” is not used in schools is that even in elementary school lessons, in addition to numbers, they also study geometric shapes and units of measurement (centimeter, meter, etc.), and this is already beyond regular account. Nevertheless, learning mental arithmetic occurs in a child's life to some extent by itself, in the process of getting to know the outside world. Term "mental arithmetic" means the ability to count in the mind. Agree, each of us at some point in our lives learns this and not only thanks to school lessons.

Today there are whole methods for developing high-speed mental counting skills in children. For example, the ancient Abacus training is especially popular, which is based on the ability to count on special accounts (different from the usual ones with tens). Abacus translated from English and is "accounts", therefore the name of the technique sounds the same. The Japanese call this technique Soroban training, because. in their language, "abacus" is called "soroban".

There are four elementary operations in arithmetic: addition, subtraction, multiplication, and division. And it does not matter whether integers are used in the example or decimals and fractions. You can introduce a child to numbers from early childhood, and do it at ease and in the game. Not only imagination will help parents in this, but also a lot of special educational materials that can be found in any store.

By modern requirements by the first grade, the child should already count at least within ten (and preferably up to 20), and also carry out basic operations with familiar numbers - add and subtract them. It is also important that the child can compare which of the numbers is greater, which is less, and which numbers are equal. Thus, we can say that it is arithmetic that a child should know even before entering school.

Such requirements are presented not only in Russia, but throughout the world, because. the pace of life is accelerating, and the amount of knowledge is increasing daily. What was enough to know school curriculum 20-30 years ago, today it occupies no more than 50% of the information taught by teachers. Be that as it may, arithmetic will always remain the basis for the study of numbers and counting, as well as the initial level of mathematics, without which it is impossible to learn more complex tasks and skills.