What is "arithmetic"? What is the correct spelling of this word. Concept and interpretation.
arithmetic the art of computing with positive real numbers. A Brief History of Arithmetic. Since ancient times, work with numbers has been divided into two different areas: one directly concerned the properties of numbers, the other was related to the technique of counting. By "arithmetic" in many countries it is usually meant this last branch, which is undoubtedly the oldest branch of mathematics. Apparently, the greatest difficulty for the ancient calculators was caused by working with fractions. This can be inferred from the Ahmes Papyrus (also called the Rhinda Papyrus), an ancient Egyptian work on mathematics dating from around 1650 BC. All fractions mentioned in the papyrus, with the exception of 2/3, have numerators equal to 1. The difficulty of dealing with fractions is also noticeable when studying ancient Babylonian cuneiform tablets. Both the ancient Egyptians and the Babylonians seem to have calculated with some form of abacus. The science of numbers has been significantly developed by the ancient Greeks since Pythagoras, around 530 BC. As for the technique of calculation itself, the Greeks did much less in this area. The Romans who lived later, on the contrary, made practically no contribution to the science of number, 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 operations with numbers have come down to us, written after the Indian positional system was improved by including zero in it. We do not know exactly when this happened, but it was then that the foundations for our most common arithmetic algorithms were laid (see also NUMBERS AND NUMBER SYSTEMS). The Indian number system and the first arithmetic algorithms were borrowed by the Arabs. The earliest surviving 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 impact 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 aritmos, turned into the term "algorithm". Indo-Arabic arithmetic became known in Western Europe mainly due to the work of L. Fibonacci The Book of the Abacus (Liber abaci, 1202). The Abacist method offered simplifications similar to the use of our positional system, at least for addition and multiplication. Abatsistov changed 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 settlements in commercial transactions. This textbook became the forerunner of many arithmetic textbooks that appeared later. Until the beginning of the 17th century. more than three hundred such textbooks have been published in Europe. Arithmetic algorithms have been significantly improved during this time. In the 16-17 centuries. symbols for arithmetic operations appeared, such as =, +, -, *, "root" and /. It is generally accepted that decimal fractions were invented in 1585 by S. Stevin, logarithms by J. Napier in 1614, and the slide rule by W. Outred in 1622. Modern analog and digital computing devices were invented in the middle of the 20th century. See also MATHEMATICS; MATHEMATICS HISTORY; NUMBERS THEORY; ROWS. Mechanization of arithmetic calculations. With the development of society, the need for faster and more accurate calculations grew. This need gave birth to four remarkable inventions: Hindu-Arabic numerical designations, decimal fractions, logarithms, and modern computers. In fact, the simplest counting devices existed before the advent of modern arithmetic, because in ancient times elementary arithmetic operations were performed on an abacus (in Russia, abacus was used for this purpose). The simplest modern computing device can be considered a slide rule, which is two sliding one along the other logarithmic scales, which allows multiplication and division, summing and subtracting scale segments. 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. Morland (1673) in England invented machines for performing multiplication. These machines became the forerunners of the 20th century desktop computing devices (arithmometers), which made it possible to perform addition, subtraction, multiplication, and division operations quickly and accurately. In 1812, the English mathematician C. Babbage set about creating a project 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 impetus for the creation of modern electronic computers, the first samples of which appeared around 1944. The speed of these machines amazed the imagination: 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. The essence of the matter can be explained by the example of a specific arithmetic problem, for example, calculating the number p (the ratio of the circumference of a circle to its diameter). The first systematic attempts to calculate p are found in Archimedes (c. 240 BC). Using a very imperfect number system, he, after much effort, managed to calculate p with an accuracy equivalent in our modern number system to two decimal places. Using the method of Archimedes, L. van Zeulen (1540-1610), having devoted a significant part of his life to this, was able to calculate p with an accuracy of 35 decimal places. In 1873, after fifteen years of work, W. Shanks obtained the value of p with 707 digits, but later it turned out that, starting from the 528th digit, errors crept into his calculations. In 1958, an IBM computer calculated 707 digits of the number p in 40 seconds and, continuing further calculations, received 10,000 digits in 100 minutes. See also COMPUTER; PI. Integer positive numbers. The basis of our ideas about numbers is the intuitive concepts of a set, the correspondence between sets, and an endless sequence of distinguishable signs or sounds. The sequence of symbols familiar to all of us 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... 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 characters. Any set, all of whose elements can be put in one-to-one correspondence with the elements of some initial segment of our infinite sequence of symbols, is called a finite set. The number of elements in the set is indicated by the last symbol of the segment. For example, the set of items that can be put in a one-to-one correspondence with the initial segment 1, 2, 3, 4, 5, 6, 7, 8 is a final set containing 8 ("eight") elements. Symbol 8 indicates the "number" of items in the original set. This number is the symbol or label assigned to the given set. The same label is assigned to all those and only those sets that can be put in one-to-one correspondence with the given set. The unambiguous definition of a label for any given finite set is called "recalculating" the elements of the given set, and the labels themselves are called natural or positive integers (see also NUMBER; SET THEORY). Let A and B be two finite sets with no common elements, and let A contain n elements and B contain m elements. Then the set S, consisting of all elements of sets A and B taken together, is a finite set containing, say, s elements. For example, if A consists of elements (a, b, c), set B consists of elements (x, y), then set S = A + B and consists of elements (a, b, c, x, y). The number s is called the sum of the numbers n and m, and we write it like this: s = n + m. In this notation, the numbers n and m are called terms, the operation of finding the sum is called addition. The operator symbol "+" is read as "plus". The set P, consisting of all ordered pairs in which the first element is chosen from set A and the second from set B, is a finite set containing, say, p elements. For example, if, as before, A = (a, b, c), B = (x, y), then P = AґB = ((a, x), (a, y), (b, x), (b, y), (c, x), (c, y)). The number p is called the product of the numbers a and b, and we write it like this: p = a*b or p = a*b. The numbers a and b in the product are called factors, the operation of finding the product is called multiplication. The operation symbol ґ is read as "multiplied by". It can be shown that the following fundamental laws of addition and multiplication of integers follow from these definitions: - the law of commutativity of addition: a + b = b + a; - the law of associativity of addition: a + (b + c) = (a + b) + c; - law of commutativity of multiplication: a*b = b*a; - law of multiplication associativity: a*(b*c) = (a*b)*c; - distributive law: aґ(b + c)= (a*b) + (a*c). If a and b are two positive integers, and if there is a positive integer c such that a = b + c, then we say that a is greater than b (this is written as a > b), or that b is less than a ( it is written like this: b b, or a
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 the classroom primary school in addition to numbers, they also study geometric shapes and units of measurement (centimeter, meter, etc.), and this already goes beyond the usual calculation. 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 the limit of 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.
Arithmetic is the branch of mathematics that deals with numbers, their properties and relationships.
Its name is of Greek origin: in the language of ancient Hellas, the word " arrhythmos"(it is also pronounced as" arithmos") means " number».
Arithmetic studies the rules of calculation and the simplest properties of numbers. In its section, which is called number theory (or higher arithmetic), the properties of individual integers are studied.
Arithmetic most closely related to number theory, algebra and geometry, and is one of the main mathematical sciences, as well as the oldest of them.
The main subjects of arithmetic are operations on numbers, their properties, and number sets. In addition, arithmetic studies such issues as the origin and development of the concept of numbers, measurements and counting techniques.
Operations on numbers, which are the subject of the study of arithmetic, are addition, subtraction, division and multiplication. They also include operations such as extracting a root, raising to a power, and solving various numerical equations.
In addition, historically it turned out that arithmetic operations include, in addition to multiplication, doubling; in addition to division, division with a remainder and by two; check; calculation of the sum of geometric and arithmetic progressions. At the same time, all arithmetic operations have their own hierarchy, in which the highest level is occupied by extracting roots and raising to a power, the lower one is multiplication and division, and then addition and subtraction.
It should be noted that those measurements and mathematical calculations that are widely practical use(for example, percentages, proportions, etc.) belong to the so-called lower arithmetic, and the concept of a number and its logical analysis belong to theoretical arithmetic.
Arithmetic is in a very close relationship with algebra, the main subject of study of which are various operations with numbers that do not take into account their properties and features. At the same time, extracting roots and raising to a power are the technical part of algebra.
Because in Everyday life arithmetic is used almost everywhere, then certain knowledge in this science is necessary for absolutely everyone. Throughout life, operations such as counting, calculating volumes, areas, speeds, time intervals and lengths have to be performed very often.
To master any profession, it is necessary to have basic knowledge of arithmetic, and this especially applies to those specialties that are related to economics, technology and the natural sciences.
Arithmetic (Greek arithmetika, from arithmys - number)
the science of numbers, primarily natural (positive integer) numbers and (rational) fractions, and operations on them. Possession of a sufficiently developed concept of a natural number and the ability to perform operations with numbers are necessary for practical and cultural activities person. Therefore A. is an element preschool education children and a compulsory subject of the school curriculum. Via natural numbers many mathematical concepts(for example, the basic concept of mathematical analysis is a real number). In this regard, A. is one of the basic mathematical sciences. When emphasis is placed on the logical analysis of the concept of number (See Number), the term theoretical arithmetic is sometimes used. Algebra is closely related to algebra (see Algebra), in which, in particular, actions on numbers are studied without taking into account their individual properties. The individual properties of integers are the subject of number theory (See number theory).
History reference. Originating in ancient times from the practical needs of counting and the simplest measurements, arithmetic developed in connection with the increasing complexity economic activity and social relations, monetary calculations, the tasks of measuring distances, time, areas and the requirements that other sciences made of it. The emergence of counting and the initial stages of the formation of arithmetic concepts are usually judged by observations relating to the process of counting among primitive peoples, and, indirectly, by studying traces of similar stages preserved in the languages of civilized peoples and observed during the assimilation of these concepts by children. These data indicate that the development of those elements of mental activity that underlie the process of counting goes through a number of intermediate stages. These include: the ability to recognize one and the same object and to distinguish objects in a set of objects to be counted; the ability to establish an exhaustive decomposition of this set into elements that are distinguishable from each other and at the same time equal in counting (using a named “unit” of counting); the ability to establish a correspondence between the elements of two sets, at first directly, and then by comparing them with the elements of a once-for-all ordered set of objects, that is, a set of objects arranged in a certain sequence. The elements of such a standard ordered set are words (numerals) used in counting objects of any qualitative nature and corresponding to the formation of the abstract concept of number. Under the most diverse conditions, one can observe similar features of the gradual emergence and improvement of the listed skills and the arithmetic concepts corresponding to them. At first, counting turns out to be possible only for collections of a relatively small number of objects, beyond which quantitative differences are vaguely recognized and are characterized by words that are synonymous with the word "many"; at the same time, notches on a tree (“tag” account), counting pebbles, rosaries, fingers, etc., as well as sets containing a constant number of elements, for example: “eyes” - as a synonym for the numeral “two”, hand ("pascarpus") - as a synonym and the actual basis of the numeral "five", etc. Verbal ordinal counting (one, two, three, etc.), the direct dependence of which on finger counting (consistent pronunciation of the names of fingers, parts of hands) in some cases can be traced directly, is further associated with the counting of groups containing a certain number of objects. This number forms the base of the corresponding number system, usually as a result of counting on the fingers of two hands, equal to 10. However, there are also groupings of 5, 20 (French 80 "quatre-vingt" = 4 × 20), 12 ("dozen"), 60 and even 11 each ( New Zealand). In the era of developed trade relations, the methods of numbering (both oral and written) naturally showed a tendency towards uniformity among the tribes and nationalities that communicated with each other; this circumstance played a decisive role in the establishment and dissemination of the applied in present. the time of the numbering system (calculus (See Reckoning)), the principle of the local (bitwise) value of digits, and the methods for performing arithmetic operations. Apparently, similar reasons explain the well-known similarity of numerals in different languages: for example, two - dva (Skt.), δυο (Greek), duo (Lat.), two (English). The source of the first reliable information about the state of arithmetic knowledge in the era of ancient civilizations is the written documents of Dr. Egypt (Mathematical Papyri), written approximately 2 thousand years BC. e. These are collections of problems indicating their solutions, rules for operations on integers and fractions with auxiliary tables, without any explanation of a theoretical nature. The solution of some problems in this collection is made, in essence, with the help of compiling and solving equations; there are also arithmetic and geometric progressions. About a rather high level of arithmetic culture of the Babylonians for 2-3 thousand years BC. e. allow to judge cuneiform mathematical texts. The written numbering of the Babylonians in cuneiform texts is a kind of combination of the decimal system (for numbers less than 60) with sexagesimal, with bit units 60, 60 2, etc. The most significant indicator high level A. is the use of sexagesimal fractions with the distribution of the same numbering system to them, similar to modern decimal fractions. The technique of performing arithmetic among the Babylonians, theoretically similar to the usual methods in the decimal system, was complicated by the need to resort to extensive multiplication tables (for numbers from 1 to 59). In the surviving cuneiform materials, which, apparently, were study guides, there are, in addition, the corresponding tables of reciprocals (two-digit and three-digit, that is, with an accuracy of 1/60 2 and 1/60 3), used in division. Among the ancient Greeks, the practical side of A. did not receive further development; the system of written numbering they used using the letters of the alphabet was much less suitable for complex calculations than the Babylonian one (it is significant, in particular, that the ancient Greek astronomers preferred to use the sexagesimal system). On the other hand, the ancient Greek mathematicians laid the foundation for the theoretical development of A. in terms of the doctrine of natural numbers, the theory of proportions, the measurement of quantities, and - implicitly - also the theory of irrational numbers. In the "Elements" of Euclid (3rd century BC) there are proofs of the infinity of the number of prime numbers that have retained their significance and still exist, basic theorems on divisibility, algorithms for finding the common measure of two segments and the common greatest divisor of two numbers (see. Euclidean algorithm), proof of the non-existence of a rational number whose square is 2 (the irrationality of the number √2), and set out in geometric shape proportion theory. The number-theoretic problems considered include problems on perfect numbers (See Perfect numbers) (Euclid), on Pythagorean numbers (See Pythagorean numbers),
and also - already in a later era - an algorithm for extracting prime numbers (Eratosthenes sieve) and solving a number of indefinite equations of the 2nd and higher degrees (Diophantus). A significant role in the formation of the concept of an infinite natural series of numbers was played by Archimedes' Psammit (3rd century BC), in which the possibility of naming and denoting arbitrarily large numbers is proved. The writings of Archimedes testify to a rather high art in obtaining approximate values of the desired quantities: extracting a root from multi-valued numbers, finding rational approximations for irrational numbers, for example The Romans did not advance the technique of computing, leaving, however, the numbering system (Roman numerals) that has come down to our time, which is little adapted to the production of actions and is currently used almost exclusively to designate ordinal numbers. It is difficult to trace the continuity in the development of mathematics in relation to previous, more ancient cultures; however, extremely important stages in the development of A. are associated with the culture of India, which influenced both the countries of Western Asia and Europe, and the countries of the East. Asia (China, Japan). In addition to the application of algebra to solving problems of arithmetic content, the most significant merit of the Indians was the introduction of a positional number system (using ten digits, including zero to indicate the absence of units in any of the digits), which made it possible to develop relatively simple rules for performing basic arithmetic operations. The scientists of the medieval East not only preserved the legacy of ancient Greek mathematicians in translations, but also contributed to the dissemination and further development of the achievements of the Indians. Methods for performing arithmetic operations, largely still far from modern, but already using the advantages of the positional number system, from the 10th century. n. e. began to gradually penetrate into Europe, especially in Italy and Spain. By the beginning of the 17th century, the comparatively slow progress of architecture in the Middle Ages was replaced by the rapid improvement of calculation methods in connection with the increased practical demands for computing technology (problems of nautical astronomy, mechanics, more complicated commercial calculations, etc.). Fractions with a denominator of 10, which were still used by the Indians (when extracting square roots) and repeatedly attracted the attention of European scientists, were first used in an implicit form in trigonometric tables (in the form of integers expressing the lengths of the lines of the sine, tangent, etc. with a radius taken as 10 5). For the first time (1427) he described in detail the system of decimal fractions and the rules of action on them al-Kashi.
The notation of decimal fractions, essentially coinciding with the modern one, is found in the writings of S. Stevin in 1585 and since that time has become widespread. The invention of logarithms at the beginning of the 17th century belongs to the same era. J. Napier om.
At the beginning of the 18th century methods of performing and recording calculations acquire a modern form. In Russia before the beginning of the 17th century. a numbering similar to Greek was used; a system of oral numbering was developed well and in a peculiar way, reaching up to the 50th category. From Russian arithmetic manuals of the early 18th century. The Arithmetic of L. F. Magnitsky (See Magnitsky) (1703), highly appreciated by M. V. Lomonosov, was of the greatest importance. It contains the following definition of A.: “Arithmetic, or numerator, is an art that is honest, unenviable, and understandable to everyone, most useful, and most praised, from the most ancient and the latest, who lived at different times, the fairest arithmeticians, invented, and expounded.” Along with numbering issues, a presentation of the technique of calculating with integers and fractions (including decimals) and related tasks, this manual also contains elements of algebra, geometry and trigonometry, as well as a number of practical information related to commercial calculations and navigation tasks . A.'s presentation is already becoming more or less modern look by L. Euler and his students. Theoretical questions of arithmetic. The theoretical development of questions concerning the doctrine of number and the doctrine of the measurement of quantities cannot be divorced from the development of mathematics as a whole: its decisive stages are connected with moments that equally determined the development of algebra, geometry and analysis. The creation of a general doctrine of the quantity x, a corresponding abstract doctrine of the number (whole, rational, and irrational) and the literal apparatus of algebra must be considered most important. The fundamental importance of mathematics as a science sufficient for the study of continuous quantities of various kinds was realized only towards the end of the 17th century. in connection with the inclusion in A. of the concept of an irrational number determined by a sequence of rational approximations. An important role in this was played by the apparatus of decimal fractions and the use of logarithms, which expanded the range of operations carried out with the required accuracy on real numbers (irrational as well as rational).
The construction of Grassmann was further completed by the work of J. Peano,
in which the system of basic (not defined through other concepts) concepts is clearly distinguished, namely: the concept of a natural number, the concept of the succession of one number directly after another in the natural series, and the concept of the initial member of the natural series (which can be taken as 0 or 1). These concepts are interconnected by five axioms, which can be considered as an axiomatic definition of these basic concepts. Peano's axioms: 1) 1 is a natural number; 2) the next natural number is a natural number; 3) 1 does not follow any natural number; 4) if a natural number a follows natural number b and for a natural number with, then b and with identical; 5) if any proposition is proven for 1 and if from the assumption that it is true for a natural number n, it follows that it is true for the following P natural number, then this proposition is true for all natural numbers. This axiom - the axiom of complete induction - makes it possible in the future to use Grasmann's definitions of actions and prove general properties natural numbers. These constructions, which provide a solution to the problem of substantiating the formal provisions of A., leave aside the question of the logical structure of A. natural numbers in the broader sense of the word, with the inclusion of those operations that determine the applications of A. both within the framework of mathematics itself and in practical life. An analysis of this side of the question, having clarified the content of the concept of a quantitative number, at the same time showed that the question of substantiating an A. is closely connected with more general fundamental problems of the methodological analysis of mathematical disciplines. If the simplest sentences of A., related to the elementary counting of objects and being a generalization of the centuries-old experience of mankind, naturally fit into the simplest logical schemes, then A., as a mathematical discipline that studies an infinite set of natural numbers, requires an investigation of the consistency of the corresponding system of axioms and a more detailed analysis of the meaning of the ensuing from it general proposals. Lit.: Klein F., Elementary mathematics from the point of view of higher, trans. with him. vol. 3 ed., vol. 1, M.-L., 1935; Arnold I. V., Theoretical arithmetic, 2nd ed., M., 1939; Belyustin V.K., How people gradually came to real arithmetic, M., 1940; Grebencha M.K., Arithmetika, 2nd ed., M., 1952; Berman G.N., Number and science about it, 3rd ed., M., 1960; Deptyan I. Ya., History of arithmetic, 2nd ed., M., 1965; Vygodsky M. Ya., Arithmetic and algebra in ancient world, 2nd ed., M., 1967. I. V. Arnold.
Big soviet encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
Synonyms:See what "Arithmetic" is in other dictionaries:
- (from Greek arithmos number, and toche art). A science that has numbers as its subject. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. ARITHMETICS from Greek. arithmos, number, and techne, art. The science of numbers. ... ... Dictionary of foreign words of the Russian language
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 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.
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"
A game " Quick addition"- 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 counting - 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.