Common arithmetic. Value of the Word & LaquaReet

What is "arithmetic"? How the word is written correctly. Concept and interpretation.

arithmetic The art of calculations produced with positive valid numbers. Brief history of arithmetic. With deep ancient times, work with numbers was divided into two various areas: One concerned directly properties of numbers, the other was associated with the account technique. Under the "arithmetic" in many countries, it is usually in mind that this is the last region, which is undoubtedly the oldest industry of mathematics. Apparently, the greatest difficulty in the ancient calculations caused work with fractions. This can be judged by the papyrus Akhmes (also called Papyrus Rinda), the ancient Egyptian composition of mathematics, dating from about 1650 BC. All the fractions mentioned in the papyrus, with the exception of 2/3, have numerals equal to 1. The difficulty of circulation of fractions is noticeable and when studying the ancient-wheeled clinical tablets. And the ancient Egyptians, and Babylonians, apparently, produced calculations with a certain variety of abaca. Science of numbers received substantial development from Pythagora from the ancient Greeks, about 530 BC. As for the technique of calculations directly, in this field, the Greeks were made much less. Later later, the Romans, on the contrary, practically did not make any contribution to the science of the number, but on the basis of the needs of rapidly developed production and trade, the abacus was improved as a countable device. Very little is known about the origin of Indian arithmetic. Only some later works about the theory and practice of operations with numbers, written after the Indian positioning system, have reached us after the Indian positioning system has been enhanced by inclusion in it zero. When it happened exactly, it is unknown to us, but it was then that the foundations for our most common arithmetic algorithms were laid (see also numbers and numbers). The Indian number system and the first arithmetic algorithms were borrowed by the Arabs. The earliest of the arrithmetic desired to us the Arab textbooks was written by Al-Khorezmi around 825. It is widely used and explained by Indian figures. Later, this textbook was translated into Latin and had a significant impact on Western Europe. The distorted version of Al-Khorezmi has reached us in the word "algorithus", which, with further mixing with the Greek word, ADROSOS turned into the term "algorithm". Indo-Arabic arithmetic became known in Western Europe Basically, thanks to the composition of L. Fibonacci, Abaca Book (Liber Abaci, 1202). The abacy method proposed simplified, similar to the use of our positional system, in any case for addition and multiplication. Abacists changed algorithms that used zero and Arabic method of dividing and extracting square root. One of the first arithmetic textbooks, the author of which we unknown, came out in Treviso (Italy) in 1478. It was about calculations when making trade transactions. This textbook has become the predecessor of many arithmetic textbooks that have subsequently. Until 17 V. In Europe, more than three hundred such textbooks were published. Arithmetic algorithms during this time were significantly improved. In 16-17 centuries. There are symbols of arithmetic operations, such as \u003d, +, -, *, "root" and /. It is believed that decimal fractions invented in 1585 S. Sustin, Logarithmia - J. Never in 1614, the logarithmic line - W. Outred in 1622. Modern analog and digital computing devices were invented in the middle of 20 V. See also mathematics; Mathematics history; Numbers theory; Rows. Mechanization of arithmetic calculations. With the development of society, the need for faster and accurate calculations grew. This need caused to live four wonderful inventions: Indo-Arabic numerical notation, decimal fractions, logarithms and modern computing machines. In fact, the simplest countable devices existed before the appearance of modern arithmetic, for in ancient times, elementary arithmetic operations were made on abacus (in Russia, scores were used for this purpose). The simplest modern computing device can be considered a logarithmic line of two moving one along another logarithmic scales, which allows you to produce multiplication and division, summing up and cleaned the segments of the scale. The inventor of the first mechanical summing machine is considered by B. Pascal (1642). Later in the same century of Leibniz (1671) in Germany and S. Morland (1673) in England, machines invented multiplication machines. These machines have become predecessors of the desktop computing devices (arithmometers) of the 20 V., Allowed the operation quickly and accurately produce operations, subtracting, multiplying and divisions. In 1812, English Mathematician C. Babbage began to create a project for calculating mathematical tables. Although the work on the project lasted for many years, it remained unfinished. Nevertheless, the project of Babbide served as an incentive to create modern electronic computing machines, the first samples of which appeared around 1944. The speed of these machines amazed imagination: with their help for a minute or hours it was possible to solve problems previously demanding many years of continuous computing even with the use of arithmometers. The essence of the case can be explained by an example of a specific arithmetic problem, for example, calculating the number P (the ratio of the circumference length to its diameter). The first systematic attempts of calculating P are found at Archimedes (approx. 240 BC). Using a very imperfect numbering system, after long work managed to calculate P with an accuracy equivalent to our modern system Number of two silent signs. Using the Method of Archimedes, L. W. Zeilen (1540-1610), byienting this considerable part of life, managed to calculate P with an accuracy of 35 decimal plates. In 1873, after fifteen years of work, U.Shenks received the value of P with 707 signs, but later it turned out that since the 528th sign, errors were embryo in its calculation. In 1958, IBM computer calculated 707 characters in the number of P and, continuing further computation, received 10,000 characters for 100 minutes. See also Computer; PI. Whole positive numbers. The basis of our ideas about numbers are the intuitive concepts of the set, the correspondence between the sets and the infinite sequence of distinguishable signs or sounds. Familiar to all of us sequence of symbols 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... there is nothing but an infinite sequence of distinguishable signs and an infinite sequence of distinguishable sounds (or words ) "One", "Two", "Three", "Four", "Five", "Six", "Seven", "Eight", "Nine", "Ten", "Eleven", "Twelve" ,. .. corresponding to certain symbols. Any set, all the elements of which can be put in a mutually unique correspondence with the elements of a certain initial segment of our infinite sequence of symbols, is called the final set. At the same time, the number of elements of the set indicates the last symbol of the segment. For example, a plurality of items that can be put into a mutually unique correspondence with the initial segment 1, 2, 3, 4, 5, 6, 7, 8, is a final set containing 8 ("eight") elements. The symbol 8 indicates the "number" of objects in the source set. This number is a character, or a shortcut attributed to this set. The same label is attributed to all those and only those sets that can be put into a mutually unique correspondence with this set. The unambiguous definition of a label for any given finite set is called "recalculation" of the elements of this set, and the labels themselves obtained the name of natural or integer positive numbers (see also the number; the theory sets). Let A and B be two final sets that do not have common elementsAnd let a be contains n elements, and B contains m elements. Then the set S, consisting of all elements of sets a and b, taken together, is a final set containing, say, s elements. For example, if a consists of elements (A, B, C), a set of B - from the elements (x, y), then the set S \u003d A + B and consists of elements (A, B, C, X, X, Y). The number S is called the sum of N and M numbers, and we write it as follows: S \u003d n + m. In this record, the number N and M is called the terms, the surgery of the amount - addition. The symbol of the operation "+" is read as "plus". The set P consisting of all ordered steam, in which the first element is selected from the set A, and the second - from the set B, is a final set containing, say, p elements. For example, if, as before, a \u003d (a, b, c), b \u003d (x, y), then p \u003d aґb \u003d ((a, x), (a, y), (b, x), (b, y), (C, X), (C, Y)). The number P is called the product of the numbers a and b, and we record it like this: p \u003d a * b or p \u003d a * b. The numbers a and b in the work are called multipliers, the operation of finding a product - multiplication. The operation symbol ґ is read as "multiplied by". It can be shown that from these definitions follow the fundamental laws of addition and multiplication of integers: - the law of commutation of addition: A + B \u003d B + A; - the law of associative addition: A + (B + C) \u003d (A + B) + C; - the law of commutation of multiplication: A * B \u003d b * A; - the law of the associative multiplication: A * (B * C) \u003d (A * B) * C; - Distribution law: Aґ (B + C) \u003d (A * B) + (A * C). If a and b are two positive integers and if there is a positive integer C, such that A \u003d B + C, then we say that a larger b (this is written as follows: a\u003e b), or that b is less than A ( This is written as: bb, or a

The arithmetic is called the section of mathematics, the subject of studying the numbers, their properties and relationships.

Her name has a Greek origin: in the language of the ancient Ellala word " aRITOS."(It is still pronounced as" ariff") Means" number».

Arithmetic Learns the rules of calculations and the simplest properties of numbers. In its section, which is called the theory of numbers (or the highest arithmetic), the properties of individual integers are studied.

Arithmetic The most closely is associated with the theory of numbers, algebra and geometry, and is one of the main mathematical sciences, as well as the most ancient of them.

The main objects of arithmetic are actions on the numbers, their properties, as well as numeric sets. In addition, issues such as the origin and development of the concept of numbers, measurement and techniques of the account are studied in arithmetic.

Actions on the numbers that are subject to learning arithmetic is addition, subtraction, division and multiplication. These operations such as the extraction of the root, the construction of various numerical equations can also be attributed.

In addition, historically developed in such a way that the arithmetic action includes, in addition to multiplying, doubling; In addition to division, division with the residue and two; score; calculation of the amount of geometric and arithmetic progress. At the same time, all arithmetic actions have their own hierarchy, in which the highest level occupies the extraction of the roots and the construction of the degree, lower - multiplication and division, and further - addition and subtraction.

It should be noted that those measurements and mathematical calculations that are widely found practical use (for example, interest, proportions, etc.) refer to the so-called lower arithmetic, and the concept of the number and its logical analysis is to the theoretical arithmetic.

Arithmetic Located in a very close connection with the algebra, the main subject of which are various operations with numbers that do not take into account their properties and features. At the same time, the extraction of the roots and the construction of the degree is the technical part of the algebra.

Since B. everyday life arithmetic It is used almost everywhere, then certain knowledge in this science is needed absolutely to all. Throughout life, operations such as an account, calculation of volumes, areas, speeds, time intervals and lengths have to be performed very often.

To master any profession, it is necessary to own the main arithmetic knowledge, and especially these are the specialties that are related to the economy, appliances and natural sciences.

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

Mental arithmetic

What is called mental arithmetic? Mental arithmetic is a learning method. rapid accountwho came from antiquity.

Currently, in contrast to the previous, teachers try not only to train children's speed children, 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, and the right - for imagination.

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

Abacus is the main tool in the study of mental arithmetic, because students learn to work with them, squeeze the knuckles and realize the essence of the account. Over time, Abacus becomes your imagination, and the learners represent them, based on these knowledge and decide examples.

Reviews of these learning methods are very positive. There is one minus - the training is paid, and not everyone can afford it. Therefore, the path of genius depends on the material situation.

Mathematics and arithmetic

Mathematics and arithmetic are closely related concepts, and rather arithmetic - a section of mathematics, working with numbers and calculations (actions with numbers).

Arithmetic is the main partition, and therefore the basis of mathematics. The basis of mathematics is the most important concepts and operations that make up the base on which all subsequent knowledge is being built. The main operations include: addition, subtraction, multiplication, division.

Arithmetic, as a rule, is studied at school from the very beginning of learning, that is. From the first class. Children are mastering the mathematics base.

Addition - This is an arithmetic effect, in the process of which two numbers are folded, and their result will be the third one.

a + B \u003d C.

Subtraction - This is an arithmetic effect, in the process of which the second number is subtracted from the first number, and the result will be the third.

The formula for addition is expressed as: a - b \u003d c.

Multiplication - This is an action, as a result of which the amount of the same terms is located.

The formula of this action is: a1 + a2 + ... + an \u003d n * a.

Division- This is a breaking on equal parts of any number or variable.

Sign up for the course "Accelerate the oral account, not mental arithmetic" to learn how to quickly and correctly fold, deduct, multiply, divide, erect numbers into a square and even extract the roots. For 30 days, you will learn how to use easy techniques to simplify arithmetic operations. In each lesson, new techniques, understandable examples and useful tasks.

Training arithmetic

Training arithmetic is made in the school walls. From the first class, children begin to study the basic and main section of mathematics - arithmetic.

Addition of numbers

Arithmetic class 5.

In the fifth grade, the schoolboy begins to study such how: fractional numbers, mixed numbers. Information about the operations with these numbers you can find in our articles according to the relevant operations.

Fractional number - This is the ratio of two numbers to each other or the numerator to the denominator. Fractional number can be replaced by division. For example, ¼ \u003d 1: 4.

Mixed number - this is a fractional number, only with the dedicated whole part. The whole part is allocated under the condition that the numerator is greater than the denominator. For example, it was a fraction: 5/4, it can be converted by allocating the whole part: 1 comes and ¼.

Examples for training:

Task number 1:

Task number 2.:

Arithmetic grade 6.

In the 6th grade, the topic of conversion of fractions in the line record appears. What does it mean? For example, the fraction of ½ is given, it will be 0.5. ¼ \u003d 0.25.

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

Examples for training:

Task number 1:

Task number 2.:

Games for the development of an oral account and account speed

There are excellent games that contribute to the development of an account to help develop mathematical abilities and mathematical thinking, oral account and account speed! You can play and develop! You are interested? Read brief articles About games and be sure to try yourself.

"Figure" game

The game "Figure" will help you speed up an oral account. The essence of the game is that on the picture presented to you, you will need to choose the answer yes or not to the question "Are there 5 identical fruits?". Go for their purpose, and this game will help you.

Mathematical comparison game

The "mathematical comparison" game will require a comparison of two numbers for a while. 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 mathematical abilities will develop! Let's try?

Game "Quick Addition"

The game "Quick Addition" is an excellent quick account simulator. The essence of the game: the 4x4 field is given, that is. 16 numbers, and over the field seventeenth. Your goal: with the help of sixteen numbers to make 17, using the operation of addition. For example, on the field you have written number 28, then in the field you need to find 2 such numbers that will give number 28 in the amount. Are you ready to try your strength? Then forward, train!

The development of a phenomenal oral account

We reviewed only the top of the iceberg to understand the mathematics better - sign up for our course: accelerate the oral account is not a mental arithmetic.

From the course you will not just recognize dozens of techniques for simplified and fast multiplication, addition, multiplication, divisions, calculating interest, but also work them in special tasks and educational games! The oral account also requires a lot of attention and concentrations that are actively trained in solving interesting tasks.

30 days

Increase reading speed 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 the traditional exercises for the development of speeds are used, techniques accelerating the brain, the method of progressive increase in reading speed, the psychology of the steps and the questions of the course participants are dealing. Suitable for children and adults, reading up to 5,000 words per minute.

Development of memory and attention from a child 5-10 years

The goal of the course: to develop memory and attention from the child so that it is easier for him to learn at school so that it can be better remembered.

What is arithmetic? When does humanity begin to use numbers and work with them? Where are the roots of such ordinary concepts, how are the numbers, addition and multiplication, which a person did an inseparable part of his life and the worldview? The ancient Greek minds admired such sciences, as well as geometry, as the most beautiful symphones of human logic.

Perhaps the arithmetic is not as deep as other sciences, but what would happen to them, forget the person an elementary multiplication table? We are familiar to us logical thinkingUsing numbers, fractions and other tools, it was not easy for people and was not available for our ancestors for a long time. In fact, to the development of arithmetic, no area of \u200b\u200bhuman knowledge was truly scientific.

Arithmetic is an alphabet of mathematics

Arithmetic is a science of numbers with which anyone starts familiarizing with fascinating peace mathematics. As M. V. Lomonosov said, the arithmetic is the gates of the scholars, opening our way to the world-cycle. But he is right, can the knowledge of the world be separated from the knowledge of numbers and letters, mathematics and speech? Perhaps in the old days, but not in modern worldwhere the rapid development of science and technology dictates their laws.

The word "arithmetic" (Greek. "Arifimo") greek origin, denotes a "number". It studies the number and everything that can be connected with them. This is the world of numbers: various actions on numbers, numeric rules, solving problems that are associated with multiplication, subtraction, etc.

The main object of arithmetic

The basis of arithmetic is an integer, properties and patterns of which are considered in the highest arithmetic or in fact, on how much the right approach is taken to consider such a small block, as a natural number, the strength of the entire building - mathematics depends.

Therefore, to the question of what arithmetic is, you can simply answer: this is a science of numbers. Yes, about the familiar seven, nine and all this diverse community. And just like the good, and the most mediocre verses do not write without an elementary alphabet, without arithmetic not solve even the elementary task. That is why all sciences have advanced only after the development of arithmetic and mathematics, being before this is just a set of assumptions.

Arithmetic - Phantom Science

What is arithmetic - Natural Science or Phantom? In fact, as ancient Greek philosophers argued, nor numbers nor the figures in reality exist. This is just a phantom, which is created in human thinking when viewed ambient With its processes. In fact, anywhere around we do not see anything like that, which could be called a number, rather, the number is the way of the human mind to study the world. Or maybe this is the study of ourselves from the inside? Philosophers are arguing about this many centuries in a row, so we do not take an exhaustive answer. Anyway, arithmetic managed so firmly to take its positions that in the modern world no one can be considered socially adapted without knowledge of its foundations.

How did a natural number appeared

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

But the real breakthrough happened when the human thought reached the fact that you can designate 2 kilograms with the same number of two kilograms, and 2 bricks, and 2 details. The fact is that you need to abstract from the forms, properties and meaning of objects, then you can make some actions with these objects in the form of natural numbers. So the arithmetic numbers were born, which further developed and wound, occupying all the big position in the life of society.

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

Arithmetic and practical Egyptians

The two oldest human satellites in the study of the surrounding world and solving household problems are arithmetic and geometry.

It is believed that the history of arithmetic takes its origin in the ancient East: in India, Egypt, Babylon and China. So, Papyrus Rinda of Egyptian origin (named as it belonged to the owner of the same name), dated XX century. BC, besides other valuable data, it contains a decomposition of one fraction for the amount of fractions with different denominators and a numerator equal to one.

For example: 2/73 \u003d 1/60 + 1/219 + 1/2 292 + 1/365.

But what is the meaning of such a complex decomposition? The fact is that egyptian approach I did not tolerate abstracted reflections about numbers, on the contrary, the calculations were made only with a practical purpose. That is, the Egyptian will be engaged in such a matter as calculations, exclusively in order to build a tomb, for example. It was necessary to calculate the length of the rib of the structures, and it forced a man for a papyrus. As can be seen, Egyptian progress in the calculations was caused, rather mass, construction, rather than love for science.

For this reason, the calculations found on papyrus cannot be called reflections on the topic of fractions. Most likely, this is a practical billet that helped in the future to solve problems with fractions. The ancient Egyptians who did not know the multiplication tables produced rather long calculations laid out on the set of subtasks. Perhaps this is one of those subtasks. It is easy to see that the calculations with such billets are very laborious and lowered. Maybe for this reason we don't see a big contribution Ancient Egypt in the development of mathematics.

Ancient Greece and Philosophical Arithmetic

Many of the knowledge of the Ancient East were successfully mastered by the ancient Greeks, famous lovers of distracted, abstract and philosophical reflections. Practice they were interested in no less, but the best theorists and thinkers find it difficult. It went to the benefit of science, because in arithmetic it is impossible to deepen, without breaking it with reality. Of course, you can multiply 10 cows and 100 liters of milk, but it will not be possible to move far.

Thinking deep Greeks left a significant mark in history, and their works reached us:

• Euclidean and "Beginning".
• Pythagoras.
• Archimedes.
• Eratosthene.
• Zenon.
• Anaxagor.

And, of course, turning everything into the Greeks philosophy, and especially the continuers of Pythagore, were so passionate about the numbers that they considered them the sacrament of the harmony of the world. The numbers were so studied and investigated that some of them were attributed to some of them and their pairs. For example:

• Perfect numbers are those that are equal to the sum of all their divisors, except for the number itself (6 \u003d 1 + 2 + 3).
• Friendly numbers are such numbers, one of which is equal to the sum of all divisors of the second, and on the contrary (the Pythagoreans knew only one such a pair: 220 and 284).

Greeks, who considered that science need to love, and not be with her for the benefit, they have achieved great success, exploring, playing and folding numbers. It should be noted that not all of their research was widely used, some of them remained only "for beauty."

Eastern Thinkers Middle Ages

In the same way and in the Middle Ages, arithmetic is obliged to easily contemporaries. Indians handed us the numbers that we actively use, such a concept as "zero", and the positional option is familiar with modern perception. From Al-Kashi, who in the 15th century he worked in Samarkand, we inherited without which it is difficult to present modern arithmetic.

In many respects, the acquaintance of Europe with the achievements of the East has become possible due to the work of the Italian scientist Leonardo Fibonacci, who wrote the work of the "Book of Abaka", acquainted with the eastern innovations. It has become the cornerstone of the development of algebra and arithmetic, research and scientific activity in Europe.

Russian arithmetic

And finally, the arithmetic, who found his place and rooted in Europe, began to spread to Russian lands. The first Russian arithmetic came out in 1703 - it was a book about the arithmetic of Leonty Magnitsky. For a long time, she remained the only educational guidance on mathematics. It contains the initial moments of algebra and geometry. Figures that used in the examples in Russia in Russia, the textbook of arithmetic, Arabic. Although the Arabic numbers met earlier, on engravings dating in the 17th century.

The book itself is decorated with images of Archimedes and Pythagore, and on the first sheet - the image of arithmetic in the form of a woman. She sits on the throne, it is written under it on Hebrew the word denoting the name of God, and on the steps that lead to the throne, the words "division", "multiplication", "addition", etc. can only be represented as important Such truths that are now considered ordinary phenomenon.

The textbook out of 600 pages describes both the bases such as the table of addition and multiplication and applications to navigation sciences.

It is not surprising that the author chose the image of Greek thinkers for his book, because he himself was captive by the beauty of arithmetic, saying: "Arithmetic is a numerator, there is an honest artistic, unkind ...". This approach to arithmetic is quite substantiated, because it is its widespread introduction that can be considered the beginning of the rapid development of scientific thought in Russia and general education.

Not easy simple numbers

A simple number is a natural number that has only 2 positive divisors: 1 and itself. All other numbers, not counting 1, are called composite. Examples of prime numbers: 2, 3, 5, 7, 11, and all others that do not have other divisors, in addition to the number 1 and themselves.

As for the number 1, it is on a special account - there is a conspiracy that it needs to be considered a simple or composite. Simple at first glance, a simple number of tatt unsolved secrets inside yourself.

The Euclid Theorem says that simple numbers are an infinite set, and Eratosthene came up with a special arithmetic "sieve", which sifts difficult numbers, leaving only simple.

Its essence is to emphasize the first unsecured number, and subsequently cross out those that he is painted. We repeated many times this procedure - and we obtain a table of prime numbers.

The main theorem of arithmetic

Among the observations about the simple numbers, it is necessary to specifically mention the main theorem of arithmetic.

The main arithmetic theorem states that any integer, greater than 1, or is simple, or it can be decomposed on the work of prime numbers with an accuracy of the order of the factory, and the only way.

The main theorem of arithmetic is proved quite cumbersome, and an understanding of it is no longer like the simplest basics.

At first glance, simple numbers are an elementary concept, but it is not. Physics also once considered an elementary atom until found a whole universe within it. Simple numbers The beautiful story of Mathematics Don Troagira "The first fifty million prime numbers is devoted."

From "three apples" to deductive laws

What truly can be called a reinforced foundation of all science - these are the laws of arithmetic. As a child, everyone is faced with arithmetic, studying the number of legs and pens in dolls, the number of cubes, apples, etc. So we study the arithmetic, which goes further into more complex rules.

All our life introduces us with the rules of arithmetic, which have become for a simple person most useful from everything that gives science. The study of numbers is "arithmetic-baby", which introduces a person with the world of numbers in the form of numbers in early childhood.

Higher arithmetic - deductive science, which studies the laws of arithmetic. Most of them are known, although, perhaps, we do not know their exact wording.

The law of addition and multiplication

Two any natural numbers A and B can be expressed as the sum of A + B, which will also be a number of natural. Regarding addition, the following laws are valid:

• Commutative, which says that the amount does not change from the permutation of the allegations, or a + b \u003d b + a.
• Associativewhich says that the amount does not depend on the method of grouping terms of places, or a + (B + C) \u003d (A + B) + C.

The rules of arithmetic, such as addition, are one of the elementary, but they use them all sciences, not to mention everyday life.

Two any natural numbers a and b can be expressed in the product A * B or A * B, which is also a number of natural. The same commutative and associative laws are applicable to the work as to addition:

• a * b \u003d b * a;
• a * (B * C) \u003d (A * B) * C.

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

a (B + C) \u003d AB + AC

This law actually teaches us to work with brackets, revealing them, thereby we can work with more complex formulas. These are precisely those laws that will lead us on the bizarre and difficult to the world of algebra.

Law of arithmetic order

The law of the order of human logic uses every day, checking hours and counting bills. And, nevertheless, it needs to be issued as a specific wording.

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

• a equal to b, or a \u003d b;
• a less b, or a< b;
• a More b, or a\u003e b.

Of the three options, only one can be fair. The main law that manages the procedure, says: if A.< b и b < c, то a< c.

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

Arithmetic laws teach us to work with numbers, signs and brackets, turning everything into a slim symphony of numbers.

Positional and non-phase calculation systems

It can be said that numbers are a mathematical language, which depends on the convenience of which. There are many calculus systems that, like the alphabets different languagesDifferent with each other.

Consider the number systems from the point of view of the impact of the position on the quantitative value of the numbers in this position. Thus, for example, the Roman system is non-sacrificial, where each number is encoded by a specific set of special characters: I / V / X / L / C / D / M. They are equal, respectively, Numbers 1/5/10 / 50/100 / 500 / 1000. In such a system, the figure does not change its quantitative determination depending on which it is worth it: the first, second, etc. To get other numbers, you need to fold the basic. For example:

• DCC \u003d 700.
• CCM \u003d 800.

A more familiar to us system for the number using Arabic numbers is positional. In such a system, the discharge of the number determines the number of numbers, for example, three-digit numbers: 333, 567, etc. The weight of any category depends on the position on which one or another digit is located, for example, the figure 8 in the second position is 80. This is characteristic of the decimal system, there are other positional systems, for example binary.

Binary arithmetic

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

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

1. The first digit on the left is 1 * 8 \u003d 8, remembering that the fourth digit, which means it must be multiplied by 2, we get position 8.
2. The second digit 1 * 4 \u003d 4 (position 4).
3. Third digit 0 * 2 \u003d 0 (position 2).
4. Fourth digit 0 * 1 \u003d 0 (position 1).
5. So, our number is 1100 \u003d 8 + 4 + 0 + 0 \u003d 12.

That is, when switching to a new bit to the left, its significance in the binary system is multiplied by 2, and in decimal - by 10. Such a system has one minus: this is too much growth in the discharges that are necessary for recording numbers. Examples of representation decimal numbers In the form of a dummy can be viewed in the following table.

Decimal numbers in binary form are shown below.

Eight and hexadecimal calculus systems are also used.

This mysterious arithmetic

What is arithmetic, "twice two" or unknown secrets of numbers? As you can see, the arithmetic, may, it seems at first glance, simple, but its non-obvious lightness is deceptive. It can be studied with children together with the aunt owl from the cartoon "Arithmetic-Baby", and you can plunge into deeply scientific research almost a philosophical order. In history, she passed the way from the counting items before worshiping the beauty of the numbers. One thing is just known: with the establishment of basic arithmetic postulates, all science can rely on her strong shoulder.

On the one hand, this is a very simple question. On the other hand, schoolchildren, and many adults, are often confused by arithmetic and mathematics and really do not know what the difference between these two items. Mathematics is the most extensive concept that includes any actions with numbers. The arithmetic is just one of the sections of mathematics. Arithmetic includes familiarity with numbers, a simple account and operations with numbers. Previously, in schools, the lessons were called precisely arithmetic and only with time they began to wear a mathematics name, which smoothly flows into the algebra. In essence, the algebra begins when unknown numbers appear in the examples and letters are used instead. That is, in a simple operation with x. and y..

Term "arithmetic" happened from the Greek word "Arithmos"What does "Number" means. In the 14th and 15th centuries, this term was translated into England not quite right - "The Metric Art", which essentially meant "metric art", suitable more for geometry, rather than a simple account and simple actions with numbers.

One of the reasons why in schools does not use the concept of "arithmetic" is that even in lessons in primary grades In addition to figures are also studying geometric forms and units of measure (centimeter, meter, etc.), and this is already out of the usual account. However, learning mental arithmetic takes place in a child's life to some extent, in the process of acquaintance with the outside world. Term "Mental arithmetic" Means the ability to read in the mind. Agree, each of us at some point in life is learning this and not only thanks to school lessons.

Today there are whole techniques for the development of children's skills in the mind. For example, the ancient abusus learning is particularly popular, which is based on the ability to count on special accounts (differ from ordinary with dozens). Abacus. Translated from English "Scores"Therefore, the name of the technique sounds the same. The Japanese is called Soroban training, because In their language "scores" are called "Soroban".

The arithmetic uses four elementary operations - addition, subtraction, multiplication and division. Moreover, no matter the integers are used in the example or decimal and fractions. You can get acquainted with numbers from early childhood, and it is easy to do it in the game. In this parents will help not only imagination, but also many special developing materials, which can be found in any store.

According to modern requirements for the first class, the child must already consider the minimum of ten (and better to 20), as well as to carry out the basic figures of the basic operations - to put them on and subtract them. It is also important that the child can compare which of the numbers is greater, how less, and what numbers are equal. Thus, it can be said that it is precisely an arithmetic child to know even before entering school.

Such requirements are presented not only in Russia, but also around the world, because The pace of life is accelerated, and the volume of knowledge increases daily. What was enough to know in school Program Another 20-30 years ago, today it takes no more than 50% taught teachers of information. Whatever it was, the arithmetic will always remain the basis of the foundations for studying the numbers and account, as well as the initial level of mathematics, without which it is impossible to study more complex tasks and skills.