Our acquaintance with mathematics begins with arithmetic, the science of number. One of the first Russian arithmetic textbooks, written by L. F. Magnitsky in 1703, began with the words: “Arithmetic, or the numerator, is an honest, unenviable art, and conveniently understandable for everyone, most useful and much praised, from the most ancient and the newest, who lived at different times of the fairest 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 understanding the world.
The word "arithmetic" comes from the Greek arithmos, which means "number". This science studies operations with numbers, various rules for handling them, and teaches how to solve problems that boil down to addition, subtraction, multiplication and division of numbers. Arithmetic is often imagined as some kind of first stage of mathematics, based on which one can study its more complex sections - algebra, mathematical analysis, etc. Even integers - the main object of arithmetic - are referred, when their general properties and patterns are considered, to higher arithmetic, or number theory. This view of arithmetic, of course, has grounds - it really remains the “alphabet of counting,” but the alphabet is “most useful” and “easy to understand.”
Arithmetic and geometry are long-time companions of man. These sciences appeared when the need arose to count objects, measure plots of land, divide spoils, and keep track of time.
Arithmetic originated in the countries of the Ancient East: Babylon, China, India, Egypt. For example, the Egyptian Rind papyrus (named after its owner G. Rind) dates back to the 20th century. BC. Among other information, it contains decompositions of a fraction into a sum of fractions with a numerator equal to one, for example:
The treasures of mathematical knowledge accumulated in the countries of the Ancient East were developed and continued by the scientists of Ancient Greece. History has preserved many names of scientists who dealt with arithmetic in the ancient world - Anaxagoras and Zeno, Euclid (see Euclid and his Elements), Archimedes, Eratosthenes and Diophantus. The name of Pythagoras (VI century BC) sparkles here like a bright star. The Pythagoreans (students and followers of Pythagoras) worshiped numbers, believing that they contained all the harmony of the world. Individual numbers and pairs of numbers were assigned special properties. The numbers 7 and 36 were held in high esteem, and at the same time attention was paid to the so-called perfect numbers, friendly numbers, etc.
In the Middle Ages, the development of arithmetic was also associated with the East: India, the countries of the Arab world and Central Asia. From the Indians came to us the numbers we use, zero and the positional number system; from al-Kashi (XV century), who worked at the Samarkand Observatory of Ulugbek, - decimal fractions.
Thanks to the development of trade and the influence of oriental culture since the 13th century. Interest in arithmetic is also increasing in Europe. It is worth remembering the name of the Italian scientist Leonardo of Pisa (Fibonacci), whose work “The Book of Abacus” introduced Europeans to the main achievements of Eastern mathematics and was the beginning of many studies in arithmetic and algebra.
Along with the invention of printing (mid-15th century), the first printed mathematical books appeared. The first printed book on arithmetic was published in Italy in 1478. In the “Complete Arithmetic” of the German mathematician M. Stiefel (early 16th century) there are already negative numbers and even the idea of logarithmization.
From about the 16th century. the development of purely arithmetic questions flowed into the mainstream of algebra - as a significant milestone, one can note the appearance of the works of the French scientist F. Vieta, in which numbers are indicated by letters. From this time on, the basic arithmetic rules are finally understood from the standpoint of algebra.
The main object of arithmetic is number. Natural numbers, i.e. the numbers 1, 2, 3, 4, ... etc., arose from counting specific objects. Many thousands of years passed before man learned that two pheasants, two hands, two people, etc. can be called by the same word “two”. An important task of arithmetic is to learn to overcome the specific meaning of the names of the objects being counted, to distract from their shape, size, color, etc. Fibonacci already has a task: “Seven old women go to Rome. Each has 7 mules, each mule carries 7 bags, each bag contains 7 loaves, each loaf contains 7 knives, each knife has 7 sheaths. How many are there?" To solve the problem, you will have to put together old women, mules, bags, and bread.
The development of the concept of number - the appearance of zero and negative numbers, ordinary and decimal fractions, ways of writing numbers (digits, notations, number systems) - all this has a rich and interesting history.
“The science of numbers refers to two sciences: practical and theoretical. Practical studies numbers insofar as we are talking about countable numbers. This science is used in market and civil affairs. The theoretical science of numbers studies numbers in the absolute sense, abstracted by the mind from bodies and everything that can be counted in them.” al-Farabi
In arithmetic, numbers are added, subtracted, multiplied and divided. The art of quickly and accurately performing these operations on any numbers has long been considered the most important task of arithmetic. Nowadays, in our heads or on a piece of paper, we make only the simplest calculations, increasingly entrusting more complex computational work to microcalculators, which are gradually replacing devices such as an abacus, an adding machine (see Computer technology), and a slide rule. However, 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, but this operation must be done many millions of times. But here we are invading another area of mathematics, which originates in arithmetic - computational mathematics.
Arithmetic operations on numbers have a variety of properties. These properties can be described in words, for example: “The sum does not change by changing the places of the terms,” can be written in letters: , can be expressed in special terms.
For example, this property of addition is called a commutative or commutative law. We apply the laws of arithmetic often out of habit, without realizing it. Often students at school ask: “Why learn all these commutative and combinational laws, since it’s already clear how to add and multiply numbers?” In the 19th century mathematics took an important step - it began to systematically add and multiply not only numbers, but also vectors, functions, displacements, tables of numbers, matrices and much more, and even just letters, symbols, without really caring about their specific meaning. And here it turned out that the most important thing is what laws these operations obey. The study of operations specified on arbitrary objects (not necessarily on numbers) is already the field of algebra, although this task is based on arithmetic and its laws.
Arithmetic contains many rules for solving problems. In old books you can find problems on the “triple rule”, on “proportional division”, on the “method of scales”, on the “false rule”, etc. Most of these rules are now outdated, although the problems that were solved with their help cannot in any way be considered outdated. The famous problem about a swimming pool that is filled with several pipes is at least two thousand years old, and it is still not easy for schoolchildren. But if earlier in order to solve this problem it was necessary to know a special rule, today younger schoolchildren are taught to solve such a problem by entering the letter designation of the desired quantity. Thus, arithmetic problems led to the need to solve equations, and this is again an algebra problem.
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PYTHAGORAS
There are no written documents left about Pythagoras of Samos, and from later evidence it is difficult to reconstruct the true picture of his life and achievements. It is known that Pythagoras left his native island of Samos in the Aegean Sea off the coast of Asia Minor as a sign of protest against the tyranny of the ruler and already at an adult age (according to legend, at the age of 40) he appeared in the Greek city of Crotone in southern Italy. Pythagoras and his followers - the Pythagoreans - formed a secret alliance that played a significant role in the life of the Greek colonies in Italy. The Pythagoreans recognized each other by a star-shaped pentagon - a pentagram. The teachings of Pythagoras were greatly influenced by the philosophy and religion of the East. He traveled a lot in the countries of the East: he was in Egypt and Babylon. There Pythagoras also became acquainted with eastern mathematics. Mathematics became part of his teaching, and the most important part. The Pythagoreans believed that the secret of the world was hidden in numerical patterns. The world of numbers lived a special life for the Pythagorean; numbers had their own special life meaning. Numbers equal to the sum of their divisors were perceived as perfect (6, 28, 496, 8128); Friendly were pairs of numbers, each of which was equal to the sum of the divisors of the other (for example, 220 and 284). Pythagoras was the first to divide numbers into even and odd, simple and composite, and introduced the concept of a figured number. In his school, Pythagorean triplets of natural numbers were examined in detail, in which the square of one was equal to the sum of the squares of the other two (see Fermat's last theorem). Pythagoras is credited with saying: “Everything is a number.” He wanted to reduce the whole world, and mathematics in particular, to numbers (and he meant only natural numbers). But in the school of Pythagoras itself a discovery was made that violated this harmony. It has been proven that it is not a rational number, i.e. cannot be expressed in terms of natural numbers. Naturally, Pythagoras’ geometry was subordinated to arithmetic; this was clearly manifested in the theorem that bears his name and which later became the basis for the use of numerical methods in geometry. (Later, Euclid again brought geometry to the forefront, subordinating algebra to it.) Apparently, the Pythagoreans knew the correct solids: tetrahedron, cube and dodecahedron. Pythagoras is credited with the systematic introduction of proofs into geometry, the creation of the planimetry of rectilinear figures, and the doctrine of similarity. The name of Pythagoras is associated with the doctrine of arithmetic, geometric and harmonic proportions, averages. It should be noted that Pythagoras considered the Earth to be a ball moving around the Sun. When in the 16th century The church began to fiercely persecute the teachings of Copernicus; this teaching was stubbornly called Pythagorean. |
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ARCHIMEDES
More is known about Archimedes, the great mathematician and mechanic, than about other ancient scientists. First of all, the year of his death is reliable - the year of the fall of Syracuse, when the scientist died at the hands of a Roman soldier. However, ancient historians Polybius, Livy, and Plutarch said little about his mathematical merits; from them, information about the scientist’s wonderful inventions made during his service with King Hieron II has reached our times. There is a well-known story about the king’s golden crown. Archimedes checked the purity of its composition using the law of buoyancy force he found, and his exclamation “Eureka!”, i.e. "Found!". Another legend says that Archimedes built a system of blocks with the help of which one man was able to launch the huge ship Syracosia. The words of Archimedes spoken then became winged: “Give me a fulcrum, and I will turn the Earth.” The engineering genius of Archimedes manifested itself with particular force during the siege of Syracuse, a wealthy trading city on the island of Sicily. The soldiers of the Roman consul Marcellus were detained for a long time at the walls of the city by unprecedented machines: powerful catapults targeted stone blocks, throwing machines were installed in the loopholes, throwing out hail of cannonballs, coastal cranes turned outside the walls and threw stone and lead blocks at enemy ships, hooks picked up ships and They threw them down from a great height, systems of concave mirrors (in some stories - shields) set the ships on fire. In “The History of Marcellus,” Plutarch describes the horror that reigned in the ranks of the Roman soldiers: “As soon as they noticed that a rope or a log was appearing from behind the fortress wall, they fled, shouting that Archimedes had invented a new machine for their destruction.” . Archimedes' contribution to the development of mathematics was also enormous. The Archimedes spiral (see Spirals), described by a point moving in a rotating circle, stood apart from the many curves known to his contemporaries. The next kinematically defined curve - the cycloid - appeared only in the 17th century. Archimedes learned to find a tangent to his spiral (and his predecessors were able to draw tangents only to conic sections), found the area of its turn, as well as the area of an ellipse, the surface of a cone and a sphere, the volumes of a sphere and a spherical segment. He was especially proud of the ratio he discovered of the volume of a sphere and a cylinder circumscribed around it, which is equal to 2:3 (see Inscribed and Circumscribed Figures). Archimedes also worked a lot on the problem of squaring the circle (see Famous problems of antiquity). The scientist calculated the ratio of the circumference to the diameter (number) and found that it was between and. The method he created for calculating the circumference and area of a figure was a significant step towards the creation of differential and integral calculus, which appeared only 2000 years later. Archimedes also found the sum of an infinite geometric progression with denominator . In mathematics, this was the first example of an infinite series. A major role in the development of mathematics was played by his essay “Psammit” - “On the number of grains of sand”, in which he shows how using the existing number system one can express arbitrarily large numbers. As a basis for his reasoning, he uses the problem of counting the number of grains of sand within the visible Universe. Thus, the then existing opinion about the presence of mysterious “largest numbers” was refuted. |
Among the important concepts that arithmetic introduced are proportions and percentages. Most concepts and methods of arithmetic are based on comparing various dependencies between numbers. In the history of mathematics, the process of merging arithmetic and geometry occurred over many centuries.
One can clearly trace the “geometrization” of arithmetic: complex rules and patterns expressed by formulas become clearer if they can be depicted geometrically. An important role in mathematics itself and its applications is played by the reverse process - the translation of visual, geometric information into the language of numbers (see Graphical calculations). This translation is based on the idea of the French philosopher and mathematician R. Descartes about defining points on a plane by coordinates. Of course, this idea had already been used before him, for example in maritime affairs, when it was necessary to determine the location of a ship, as well as in astronomy and geodesy. But it is from Descartes and his students that the consistent use of the language of coordinates in mathematics comes. And in our time, when controlling complex processes (for example, 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 translate the accumulated numerical information into the language of drawing.
You see that, speaking about arithmetic, we always go beyond its limits - into algebra, geometry, and other branches of mathematics.
How can we delineate the boundaries of arithmetic itself?
In what sense is this word used?
The word "arithmetic" can be understood as:
an academic subject that deals primarily with rational numbers (whole numbers and fractions), operations on them, and problems solved with the help of these operations;
part of the historical building of mathematics, which has accumulated various information about calculations;
“theoretical arithmetic” is a part of modern mathematics that deals with the construction of various numerical systems (natural, integer, rational, real, complex numbers and their generalizations);
“formal arithmetic” is a part of mathematical logic (see Mathematical logic), which deals with the analysis of the axiomatic theory of arithmetic;
“higher arithmetic”, or number theory, an independently developing part of mathematics.
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 the difference is between these two subjects. Mathematics is the most extensive concept that includes any operations with numbers. Arithmetic is just one of the branches of mathematics. Arithmetic includes introduction to numbers, simple counting, and number operations. Previously, lessons in schools were called arithmetic, and only over time they began to bear the name mathematics, which smoothly flows into algebra. Essentially, algebra begins when unknown numbers appear in examples and letters are used instead. That is, in a simple way, operations with x And y.
Term "arithmetic" comes from the Greek word "arithmos", which means "number". In the 14th-15th centuries, this term was translated in England not entirely correctly - “the metric art”, which essentially meant “metric art”, suitable more 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 primary school lessons, in addition to numbers, they also study geometric shapes and units of measurement (centimeter, meter, etc.), and this goes beyond regular account. However, learning mental arithmetic occurs to some extent naturally in a child’s life, in the process of getting to know the world around him. Term "mental arithmetic" means the ability to do mental math. Agree, each of us learns this at some point in our lives, and not only through school lessons.
Today there are entire methods for developing children’s speed mental arithmetic skills. For example, especially popular is ancient Abacus training, which is based on the ability to count on special abacuses (different from ordinary ones with tens). Abacus translated from English is "abacus", that’s why the name of the technique sounds the same. The Japanese call this technique Soroban training, because... in their language, “abacus” is called “soroban”.
Arithmetic uses four elementary operations - addition, subtraction, multiplication and division. It doesn’t matter whether integers are used in the example or decimals and fractions. You can introduce your child to numbers from early childhood, and do it at ease and through play. Parents will be helped with this not only by their imagination, but also by a variety of special educational materials that can be found in any store.
According to modern requirements for the first grade, a child should already count at least up to ten (and preferably up to 20), and also carry out basic operations with familiar numbers - adding and subtracting them. It is also important that the child can compare which numbers are larger, which are smaller, 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 accelerates, and the volume of knowledge increases daily. What was enough to know in the school curriculum 20-30 years ago today takes up no more than 50% of the information taught by teachers. Be that as it may, arithmetic will always remain the basis for learning numbers and counting, as well as the initial level of mathematics, without which it is impossible to learn more complex tasks and skills.
Arithmetic
Arithmetic and.
1.
A branch of mathematics that studies the simplest properties of numbers, ways of writing them and operations on them.
2.
An academic subject containing the basics of this section of mathematics.
3. decomposition
A textbook setting out the content of a given academic subject.
Explanatory Dictionary by Efremova. T. F. Efremova. 2000.
Synonyms:
See what “Arithmetic” is in other dictionaries:
- (from the Greek arithmos number, and toche art). A science that deals with numbers. 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
Female, Greek the doctrine of counting, the science of notation; the basis of all mathematics (the science of quantities, of the measurable); old counting or numerical wisdom; counting, reckoning, numerical calculation, calculation. Arithmetic, arithmetical, relating to it. Arithmetician... ... Dahl's Explanatory Dictionary
Digital business, digital science, digital, counting Dictionary of Russian synonyms. arithmetic tsifir (obsolete) Dictionary of synonyms of the Russian language. Practical guide. M.: Russian language. Z. E. Alexandrova. 2011… Synonym dictionary
- (from the Greek words ariJmoV number and tecnh art) part of mathematics that deals with the study of the properties of certain specific quantities; in a closer sense, arithmetic is the science of numbers expressed in numbers, and deals with operations on numbers. Can i… … Encyclopedia of Brockhaus and Efron
Modern encyclopedia
- (from Greek arithmos number) part of mathematics; studies the simplest properties of numbers, primarily natural (positive integers) and fractions, and operations on them. The development of arithmetic led to the separation from it of algebra and number theory... Big Encyclopedic Dictionary
ARITHMETICS, a method of calculation using addition, subtraction, multiplication and division. The formal axiomatic basis for these operations was provided by Giuseppe Peano at the end of the 19th century. Based on some postulates, for example, that there is only one... ... Scientific and technical encyclopedic dictionary
ARITHMETICS, arithmetic, many. no, female (Greek arithmetike). The study of numbers expressed by digits and operations on them. Ushakov's explanatory dictionary. D.N. Ushakov. 1935 1940 ... Ushakov's Explanatory Dictionary
ARITHMETICS, and, female. 1. A branch of mathematics that studies the simplest properties of numbers expressed by digits and operations on them. 2. transfer Same as counting (in 2 digits) (colloquial). We checked the expenses and it turned out disappointing. | adj. arithmetic, ah, ... ... Ozhegov's Explanatory Dictionary
arithmetic- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Topics of energy in general EN arithmetics ... Technical Translator's Guide
Arithmetic- (from the Greek arithmos number), part of mathematics that studies the simplest properties of integers and fractions and operations on them. It arose in ancient times from the practical needs of counting, measuring distances, time, etc. Improvement... ... Illustrated Encyclopedic Dictionary
Books
- Arithmetic, Kiselev Andrey Petrovich. 2017 marks the 165th anniversary of the birth of A.P. Kiselyov. His first school textbook on arithmetic was published in 1884. In 1938, it was approved as an arithmetic textbook for 5-6...
Arithmetic is the branch of mathematics whose subject of study is 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 calculations and the simplest properties of numbers. In that section called number theory (or higher arithmetic), the properties of individual integers are studied.
Arithmetic is most closely related to number theory, algebra and geometry, and is one of the main mathematical sciences, as well as the most ancient of them.
The main subjects of arithmetic are operations on numbers, their properties, as well as numerical sets. In addition, arithmetic studies such issues as the origin and development of the concept of numbers, measurement and counting techniques.
The number operations that are the subject of arithmetic are addition, subtraction, division and multiplication. These also include operations such as root extraction, exponentiation and solving various numerical equations.
In addition, historically it has developed that arithmetic operations include, in addition to multiplication, doubling; in addition to division, division with a remainder and by two; check; calculating the sum of geometric and arithmetic progressions. Moreover, all arithmetic operations have their own hierarchy, in which the highest level is occupied by the extraction of roots and exponentiation, the lower level by multiplication and division, and then by addition and subtraction.
It should be noted that those measurements and mathematical calculations that find wide practical application (for example, percentages, proportions, etc.) belong to the so-called lower arithmetic, and the concept of number and its logical analysis belong to theoretical arithmetic.
Arithmetic is in a very close connection with algebra, the main subject of study of which is various operations with numbers that do not take into account their properties and features. At the same time, extracting roots and exponentiation are the technical part of algebra.
Because in everyday life arithmetic is used almost everywhere, then absolutely everyone needs certain knowledge in this science. Throughout life, operations such as counting, calculating volumes, areas, speeds, time intervals and lengths have to be performed very often.
To master any profession, you must have basic arithmetic knowledge, and this is especially true for those specialties related to economics, technology and the natural sciences.
Arithmetic (Greek arithmetika, from arithmys - number)
the science of numbers, primarily about natural (positive integers) numbers and (rational) fractions, and operations on them. Possession of a sufficiently developed concept of natural numbers and the ability to perform operations with numbers are necessary for the practical and cultural activities of a person. Therefore, A. is an element of preschool education of children and a compulsory subject of the school curriculum. Many mathematical concepts are constructed using natural numbers (for example, the basic concept of mathematical analysis is a real number). In this regard, mathematics is one of the main 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, operations on numbers are studied without taking into account their individual properties. The individual properties of integers form the subject of number theory (See Number theory).
Historical reference. Having arisen in ancient times from the practical needs of counting and simple measurements, arithmetic developed in connection with the increasing complexity of economic activity and social relations, monetary calculations, problems of measuring distances, time, areas, and the requirements that other sciences placed on 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 cultural peoples and observed during the acquisition of these concepts by children. These data indicate that the development of those elements of mental activity that underlie the counting process goes through a number of intermediate stages. These include: the ability to recognize the same object and distinguish objects in a set of objects to be counted; the ability to establish an exhaustive decomposition of this totality 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, first directly, and then by comparing them with the elements of a once and for all ordered collection of objects, that is, a collection of objects located 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 a variety of 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 aggregates of a relatively small number of objects, beyond which quantitative differences are vaguely realized and are characterized by words that are synonymous with the word “many”; in this case, the counting tools are notches on the tree (“tag” count), counting pebbles, rosary beads, fingers, etc., as well as sets containing a constant number of elements, for example: “eyes” - as a synonym for the numeral “two”, hand (“metacarpus”) - 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 (sequential pronunciation of the names of fingers, parts of the hands) in some cases can be traced directly, is further associated with counting 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), 40, 12 (“dozen”), 60 and even 11 (New Zealand). In the era of developed trade relations, numbering methods (both oral and written) naturally showed a tendency towards uniformity among tribes and nationalities communicating with each other; this circumstance played a decisive role in the establishment and dissemination of the system used in the present day. time of the numbering system (notation (See Notation)), the principle of place (bitwise) meaning of numbers and methods of performing arithmetic operations. Apparently, similar reasons explain the well-known similarity of numeral names in different languages: for example, two - dva (Sanskrit), δυο (Greek), duo (Latin), two (English). The source of the first reliable information about the state of arithmetic knowledge in the era of ancient civilizations are 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 operating on integers and fractions with auxiliary tables, without any theoretical explanations. Some of the problems in this collection are solved essentially by setting up and solving equations; Arithmetic and geometric progressions are also found. About the rather high level of arithmetic culture of the Babylonians for 2-3 thousand years BC. e. allow judging Cuneiform mathematical texts. The written numbering of the Babylonians in cuneiform texts is a peculiar combination of the decimal system (for numbers less than 60) with the sexagesimal system, with digit units 60, 60 2, etc. The most significant indicator of a high level of arithmetic is the use of sexagesimal fractions with the same numbering system applied to them, similar to modern decimal fractions. The Babylonians' arithmetic technique, which was theoretically similar to conventional techniques 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 teaching aids, there are also corresponding tables of reciprocal numbers (two-digit and three-digit, i.e., with an accuracy of 1/60 2 and 1/60 3), which were used in division. Among the ancient Greeks, the practical side of architecture did not receive further development; the system of written numbering they used using letters of the alphabet was much less suitable for complex calculations than the Babylonian one (it is significant, in particular, that ancient Greek astronomers preferred to use the sexagesimal system). On the other hand, ancient Greek mathematicians laid the foundation for the theoretical development of arithmetic in terms of the doctrine of natural numbers, the theory of proportions, the measurement of quantities and, in an implicit form, also the theory of irrational numbers. In Euclid’s Elements (3rd century BC) there are proofs of the infinity of the number of prime numbers, basic theorems on divisibility, and algorithms for finding the common measure of two segments and the common greatest divisor of two numbers, which have retained their significance and are still significant (see. Euclid's algorithm), a proof of the non-existence of a rational number whose square is 2 (the irrationality of the number √2), and a theory of proportions expressed in geometric form. 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 isolating prime numbers (Eratosthenes' sieve) and solving a number of indeterminate equations of 2nd and higher degrees (Diophantus). A significant role in the formation of the concept of an infinite natural series of numbers was played by the “Psammit” of Archimedes (3rd century BC), which proves the possibility of naming and denoting arbitrarily large numbers. The works of Archimedes indicate a fairly high skill in obtaining approximate values of the desired quantities: extracting the root of multi-digit numbers, finding rational approximations for irrational numbers, for example The Romans did not advance the technology of calculations; however, they left behind a numbering system that has survived to this day (Roman numerals), which is poorly suited for operations and is now used almost exclusively to designate ordinal numbers. It is difficult to trace continuity in the development of mathematics in relation to previous, more ancient cultures; however, extremely important stages in the development of Africa 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 achievement 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. Scientists of the medieval East not only preserved the heritage 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, primarily into Italy and Spain. The relatively slow progress of architecture in the Middle Ages gives way to the beginning of the 17th century. rapid improvement of calculation methods in connection with increased practical demands on computing technology (problems of nautical astronomy, mechanics, increasingly complex commercial calculations, etc.). Fractions with a denominator of 10, which were 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 sine, tangent, etc. with radius taken as 10 5). For the first time (1427), al-Kashi described in detail the system of decimal fractions and the rules for operating with them.
The notation of decimal fractions, which essentially coincides with the modern one, is found in the works of S. Stevin in 1585 and from that time has become widespread. The invention of logarithms at the beginning of the 17th century dates back to the same era. J. Napier om.
At the beginning of the 18th century. techniques for performing and recording calculations are taking on a modern form. In Russia until the beginning of the 17th century. numbering similar to Greek was used; The oral numbering system was well and uniquely developed, reaching the 50th digit. From Russian arithmetic manuals of the early 18th century. Of greatest importance was L. F. Magnitsky's Arithmetic, highly appreciated by M. V. Lomonosov (See Magnitsky) (1703). It contains the following definition of A.: “Arithmetic, or numerator, is an honest, unenviable art, and easy to understand for everyone, most useful, and most praiseworthy, invented and expounded by the most ancient and modern arithmeticians who lived at different times.” Along with numbering questions, a presentation of calculation techniques with integers and fractions (including decimals) and related problems, this manual also contains elements of algebra, geometry and trigonometry, as well as a number of practical information related to commercial calculations and navigation problems . A.'s presentation takes on a more or less modern form from L. Euler and his students. Theoretical questions of arithmetic. The theoretical development of questions relating to 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 associated with moments that equally determined the development of algebra, geometry and analysis. The most important should be considered the creation of a general doctrine of Quantities, a corresponding abstract doctrine of number (See Number) (integer, rational and irrational) and the alphabetic apparatus of algebra. The fundamental importance of arithmetic 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 arithmetic of the concept of an irrational number, defined 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).
Grassmann's construction was further completed by the work of G. Peano,
in which a system of basic (not defined through other concepts) concepts is clearly highlighted, namely: the concept of a natural number, the concept of one number immediately following another in a natural series, and the concept of the initial member of a 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 a natural number b and beyond the natural number With, That b And With are identical; 5) if any proposition has been 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 sentence is true for all natural numbers. This axiom - the axiom of complete induction - makes it possible to further use Grassmann's definitions of actions and prove the general properties of natural numbers. These constructions, which provide a solution to the problem of substantiating formal statements of arithmetic, leave aside the question of the logical structure of arithmetic of natural numbers in the broader sense of the word, including those operations that define the applications of arithmetic both within mathematics itself and in practical applications. life. Analysis of this side of the issue, having clarified the content of the concept of cardinal number, at the same time showed that the question of justification of arithmetic is closely related to more general fundamental problems of methodological analysis of mathematical disciplines. If the simplest propositions of mathematics, relating to the elementary counting of objects and being a generalization of the centuries-old experience of mankind, naturally fit into the simplest logical scheme, then mathematics, as a mathematical discipline that studies the infinite collection of natural numbers, requires a study of the consistency of the corresponding system of axioms and a more detailed analysis of the meaning of the resulting from its general proposals. Lit.: Klein F., Elementary mathematics from a higher point of view, trans. with him. vol. 3 ed., vol. 1, M.-L., 1935; Arnold I.V., Theoretical arithmetic, 2nd ed., M., 1939; Bellustin V.K., How people gradually reached real arithmetic, M., 1940; Grebencha M.K., Arithmetic, 2nd ed., M., 1952; Berman G.N., Number and the science of it, 3rd ed., M., 1960; Deptyaan I. Ya., History of arithmetic, 2nd ed., M., 1965; Vygodsky M. Ya., Arithmetic and algebra in the Ancient World, 2nd ed., M., 1967. I. V. Arnold.
Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
Synonyms:See what “Arithmetic” is in other dictionaries:
- (from the Greek arithmos number, and toche art). A science that deals with numbers. 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
