Designation of numbers on writing in ancient Rus'. Numeral system of ancient Rus'

Lesson – excursion

in mathematics on the topic: “Old Russian number system”

Lesson objectives:

Educational:

To familiarize students with historical information about the ancient Russian number system;

Illustrate to students the ancient Russian number system;

Educational:

Development of cognitive interest and mathematical speech in schoolchildren;

Development of skills to systematize and generalize this material;

Educators:

Foster a spirit of competition;

Develop work discipline;

Formation of self-organization skills.

Progress of the lesson:

Organizing time

Hello guys. Today we will get acquainted with the ancient Russian number system, consider its features and disadvantages, and at the end of the event we will write a test to test your knowledge on this topic, so listen to me carefully, I will dwell on the main points.

1. Historical background:

Number system (Lat. numbering) numeratio ) - a method of denoting numbers using signs - numbers, or words. A notation system based on numbers is written numbering. A notation system based on words is verbal numbering.

Our ancient ancestors also had their own ancient Russian alphabetic number system.Our ancestors used 27 Cyrillic letters as numbers. , only above them, to distinguish them, they put a special sign - TITLO.

And the number 10000 was denoted by the same letter as 1, only without the title, it was circled and the number was called “DARKNESS”.

The largest of the quantities was called “DECK” and it was equal to 1050, it was believed that “THE HUMAN MIND CAN UNDERSTAND MORE THAN THIS.”

Old Russian numbering

Cyrillic number system

Cyrillic number system - the number system of Ancient Rus', based on the alphabetical notation of numbers using the Cyrillic or Glagolitic alphabet.

In its main features it repeats the Greek number system.

It was used in Russia until the beginning of the 18th century, when it was replaced by a number system based on Arabic numerals.

Currently used in books in Church Slavonic.

Clock using Cyrillic alphabet

Most of the letters of the Old Russian alphabet had a numerical correspondence. So, the letter “Az” meant “one”, “Vedi” - “two”... Some letters did not have numerical correspondences. Numbers were written and pronounced from left to right, with the exception of numbers from 11 to 19 (for example, 17 - seventeen).

The Glagolitic number system was built using the same principle, in which Glagolitic letters were used.

At the beginning of the 18th century, a mixed system of notating numbers was sometimes used, consisting of both Cyrillic and Arabic numerals. For example, some copper kopecks have the date 17K1 (1721) minted on them.

Features of the Cyrillic number system

Lowercase letters were used almost exclusively to write numbers.

The numerical value 5 was originally carried by the ordinary letter “e”, but later its so-called “long” version began to be used, from which the Ukrainian letter “є” subsequently developed.

For the numerical value 6 in ancient times, both the usual letter “zelo” (S) and a mirror inverted one were used.

The letter “i” in numerical use does not have dots.

For the numerical value 60, it is usually not the usual letter “o” that is used, but its so-called “wide” version (in Unicode, due to a misunderstanding, called “round omega”).

The meaning of 90 in the most ancient Cyrillic texts was expressed not by the letter “ch”, but by the sign “koppa” borrowed from Greek ( ҁ ).

The meaning of 400 in ancient times was expressed by the letter “Izhitsa ( ѵ )», later the so-called “ik” is a y-shaped sign, used only as a numeric sign and as part of the digraph “uk” (“ou”). Use in numerical value“ika” is typical for Russian publications, and “izhitsy” is typical for early printed Ukrainian, later South Slavic and Romanian ones.

At a value of 800 it could be used as a “naked omega (ѡ )", and (more often) the compound sign "from (ѿ )"; For more details, see the article “Omega (Cyrillic)”.

The value of 900 in ancient times was expressed by “small yus” (ѧ ), somewhat similar to the corresponding Greek letter "disigma" (Ϡ ); later the letter “ts” began to be used in this meaning.

Old Russian numbering

Thousands

To indicate thousands, to the left of the corresponding letter-number, a small diagonal was written down to the left and on it two small lines -҂ (U+0482).

Examples:

- 1706;

- 7118 year according to chronology “from the creation of the world” (1610 from the Nativity of Christ).

Tens and hundreds of thousands, millions

Large numbers (tens and hundreds of thousands, millions and billions) could be expressed not through the sign “҂ ”, and a specially circled letter used to designate units. However, for large numbers these notations were quite unstable.

Dark

To indicate darkness, the letter was surrounded by a solid circle.

Small count - ten thousand (104) or one hundred thousand (105);

The great count is a million (106, great darkness).

Darkness of topics:

The great count is a million millions (1012, great darkness).

In small counting, the number served as the last limit of natural (correlated with any activity) counting. The darkness is overwhelming - an infinite number, an innumerable multitude.

From the word darkness comes the military rank temnik - a major military leader. Temnik was, for example, Mamai.

Similar names are tumen and miriada.

Legion (ignorant)

To indicate legion (ignorance), the letter was circled with dots.

Small account - one hundred thousand (105);

The great count is a million millions (1012).

Leodre

To designate a leodr, the letter was circled with dashes.

Small account - million (106);

The great count is a legion of legions (1024).

Raven (raven)

To designate a corvid (raven), the letter was circled with crosses or commas.

Small account - ten million (107);

The great count is leodr leodrov (1048).

Deck

The largest number is the deck. The letter was square brackets, but not to the right and left, as with ordinary letters, but above and below. Plus two diamonds were placed on the right and left.

Small account - one hundred million (108);

The great count is ten ravens (1049).

Arrangement in order Example

Test work

Instructions for performing test work:

From the 15 proposed tasks below, choose only one correct answer and circle the correct answer. Enter all answers into the table:

Number

tasks

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Answers

Evaluation criteria:

For each correctly completed task, 1 point is given.

The mark “5” is given if 14-15 points are completed correctly

The mark “4” is given if 12-13 points are completed correctly.

The mark “3” is given if 10-11 points are completed correctly

The mark “2” is given if correctly performed from 9 points and below

Number

tasks

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Answers

Which letter in numerical use does not have dots:

A) "i”;

b) "k”;

V) "o”?

2. The number system is the designation of numbers using signs:

a) numbers;

b) words;

c) numbers or words.

3. How many letters in the Cyrillic alphabet were used by our ancestors as numbers:

a) 26;

b) 37;

c) 27?

4. What is a “titlo”:

a) a special sign to distinguish letters from numbers;

b) a special sign to distinguish numbers from letters;

c) a special sign to distinguish digits from numbers?

5. What was the name of the largest value:

a) darkness;

b) deck;

c) legion?

6. What was the name of the number system of Ancient Rus':

a) Cyrillic;

b) Ionian;

c) Indo-Arab?

7. Which letter from the modern Russian alphabet is missing in the Old Russian numbering:

a) A;

b) B;

c) B?

8. The initial numerical value “5” was carried by which letter:

a) “e”;

b) “”;

V) "s».

9. “Izhitsa (v)” is the meaning of the number:

a) 800;

b) 600;

c) 400.

10. What symbol is used to indicate “leodr”:

A) ;

b) ;

V) ?

11. Translate the number 539 into Old Russian numbering:

a) FLO;

b) FLO;

c) FLO.

12. Which of the following numbering arrangements is ascending:

a) darkness, legion, leodr, deck, thousand, raven;

b) thousand, darkness, leodr, raven, deck, legion;

c) thousand, darkness, legion, leodr, raven, deck?

13. Which symbol from Old Russian numbering means “ignorant”:

a) darkness;

b) legion

c) deck?

14. “Raven” in Old Russian numbering is designated as:

a) corvid;

b) crow;

c) a liar?

15. The meaning of what number is used by the Greek sign “kopa”:

a) 80;

b) 90;

c) 100?

Summarizing:

You worked well today, met the goals set for you, and also showed good knowledge on the topic “Old Russian number system”. For your work in the lesson you receive the following grades (the grades of each student for the work in the lesson are announced).

Thanks everyone for Good work. Well done!

GBPOU "Pedagogical College No. 4 St. Petersburg" GBPOU "PEDAGOGICAL COLLEGE No. 4 St. Petersburg"
Mathematics.
"Ancient number system
Rus'"
Presentation
carried out
students 12
gr.
Zamyatina Karina and Shablovskaya
Anastasia

10.

We have written monuments of the mathematical knowledge of the Russian people starting from about the thousandth year of our chronology. This knowledge is the result of a long previous development and is based on the practical needs of man.

Interest in science arose early in Russia among broad sections of the population. Information has been preserved about schools under Vladimir Svyatoslavovich (978-1015), under Yaroslav the Wise (1036-1054). In a very early era there were “number lovers” who were interested in mathematics not only to the extent that it was needed directly for practical activities.

An example of such “number lovers” was the Novgorod monk of the early twelfth century, Kirik.

Speaking about the interest of the Russian people in mathematics in those centuries separated from our time, we must not forget that we are talking here about the advanced layers of the people who strived for knowledge, building a national culture that flourished magnificently in subsequent centuries.

Next to these progressive elements there were significant circles of clergy and exploiters who were hostile to knowledge in general and mathematics in particular. We find evidence of a hostile attitude towards knowledge back in the 17th and 18th centuries.

The main prerequisite for all mathematical knowledge is numbering, which had different forms among different ancient peoples.

Apparently, all nations initially marked numbers with notches on sticks, which the Russians called tags. This method of recording debt obligations or taxes was used by the illiterate population of different countries. The stick had cuts corresponding to the amount of debt or tax.

The stick was split in half; one half was left with the debtor or payer, the other was kept with the lender or in the treasury.

When paying, both halves were checked by folding. In England, this method of recording taxes existed until the end of the seventeenth century. During the liquidation of the old tax obligations of the peasants, a bonfire was made from the accumulated tags in the courtyard of the London Treasury. This fire turned out to be so large that the treasury building itself burned down, and along with it, a sample of the English length measure embedded in the wall perished, so that since then the English have not known the exact length of their foot.

The Greeks in the sixth century before our chronology began to denote numbers with letters equipped with a special symbol.

Our ancestors wrote numbers in the same way using letters of the Slavic alphabet, above which a special icon was placed - a title. The table below shows which letters denoted which number in Slavic numbering. The influence of this numbering explains some terms in the Russian language. In old grammar textbooks, the letter “i” was called “and octal”, the letter “i” was called “and decimal”. These names are explained by the fact that in Slavic numbering the letter “i” stood for 8, the letter “i” for 10.

The needs of economic life in the distant past were satisfied with relatively small numbers - the so-called “small account” of our ancestors. He reached the number 10,000, which in the oldest monuments is called “darkness,” that is, a dark number that cannot be clearly imagined.

Subsequently, the limit of small counting was pushed back to 108, to the number of “darkness of topics.” An ancient manuscript on this occasion states that “more than this number the human mind cannot comprehend.” But along with this “small number”, “if a great count and list occurred,” a second system was used, called the “great number or count” or “great Slovenian number.” It used higher categories: darkness - 10 6, legion - 10 12, leodr - 10 24, raven - 10 48; sometimes there is also a deck - ten ravens - 10 49 (although it was necessary to take 10 96 as a deck, following the system). The author of the manuscript again states that “there is no greater number.”

To denote these large numbers, our ancestors used original way, not found among any of the peoples known to us: the number of units of any of the listed higher ranks was denoted by the same letter as simple units, but framed for each number by a corresponding border.

Greek mathematicians did not think of this method of writing, even among their brilliant representatives.

Such large numbers were not required at that time, and are not required now, by any practical problem. Archimedes, the greatest Greek mathematician, calculated that the number of grains of sand in the entire cosmic space, as understood at that time, does not exceed 10 63.

A Slavic “number lover” would say that this number of grains of sand is no more than “a thousand legions of ravens” (10 68 = 10 3 · 10 12 · 10 48). The number of grains of sand in the entire cosmic space could indeed seem to a person of that time to be the greatest imaginable number, which justifies the statements of the authors of the manuscripts that “more than this is not given to man to understand.” Examination of the large Slavic account repeatedly in Russian mathematical manuscripts indicates that “number lovers” were quite numerous in ancient Rus'.

In the first printed Russian mathematics textbook, in “Arithmetic” by L. F. Magnitsky * (1703), international terms for large numbers are given (million, billion, trillion, quadrillion). Reaching 1024 (quadrillion), the author states:

* (In 1699 in Amsterdam, I. F. Kopievsky published “Guide to Arithmetic, that is, at all counts,” in which the basics of notation are outlined on sixteen small pages. The book was not distributed in Russia.)

“The number is infinite, Our minds are not enough, And no one knows the end... ........................ ........ .......idlely search for multiple numbers and write more of this excellent table * ........................ And who else needs to calculate what inside the sky, the number of this is sufficient for the things of all the world."

* (That is, reaching up to 1024.)

With the adoption of Hindu numbering, Slavic numbering lost all practical meaning.

A characteristic “number lover” of ancient Rus' was the monk Kirik, who was already mentioned by us, and who wrote in 1134 the book “Kirik - the deacon of the Novgorod Anthony Monastery, the teaching, he told a person the numbers of all years.”

He excitedly counts how many months, how many days, how many hours he has lived, and then counts in years, months, weeks and days the time that has passed from the creation of the world to the year 1134 (6644 from the “creation of the world”), calculates the day of Easter in the future. When calculating time, Kirik uses “fractional hours,” meaning by them fifths, twenty-fifths, one hundred and twenty-fifths (and so on) of a twelve-hour day. Having reached the seventh fractional hour in this count, of which there are 937,500 in the day, he declares: “this does not happen anymore,” which apparently means that smaller divisions of the day were not used.

In "Russian Truth", the famous legal monument of ancient Rus', the compilation of which dates back to the period of time between the eleventh and fifteenth centuries, there are articles devoted to the calculation of the offspring of a certain initial number of sheep, goats and pigs. The calculator assumes that the available number of sheep doubles per year, and then, for example, from twenty-two sheep in 12 years there will be a herd of 22 · 2 12 = 90112 sheep, which is the result given in "Russkaya Pravda".

This is a problem that appears at about the same time in the arithmetic manuals of equal nations, either about the offspring of rabbits, or in the form of a problem about rewarding the inventor of the chess game. These calculations, apparently, were the creation of such “number lovers” as the already mentioned Kirik of Novgorod.

It is natural to compare with what has been said about the mathematical culture of our ancestors the state of mathematical knowledge among the peoples inhabiting Western Europe. Arithmetic operations there they are made using a counting board (abacus), on which pebbles (beans) or circles with dashes are placed *. Our abacus is one of the abacus pitchforks.

* (It is believed that the proverb “left on the beans” originates from here. The man who lost all his money was left with his counting board and the beans - "left with the beans.")

Numbers are written using cumbersome Roman numeration, in which even small numbers require a large number of characters (for example, 878 is written like this: DCCCLXXVIII), and writing large numbers is much more difficult than in the great Slavic counting. Our modern numbers in Western Europe do not appear in books until the thirteenth century, meeting with strong opposition from supporters of the old method of counting on the abacus or using Roman numbering.

Petrikova Anna

This report was offered as homework. The children were interested in searching for material on this topic and then listening to information from their friends. A mini-conference was held as a test for homework.

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Report on the topic “The Emergence of Number”

Completed by 5B class student Anna Petrikova

1.1. The origin of counting in ancient times

Our initial ideas about number and shape date back to the very distant era of the ancient Stone Age - the Paleolithic. For hundreds of thousands of years of this period, people lived in caves, in conditions little different from the life of animals, and their energy was spent mainly on obtaining food in the simplest way - collecting it wherever possible. People made hunting and fishing tools, developed a language to communicate with each other, and in the late Paleolithic era they decorated their existence by creating works of art, figurines and drawings.

Until there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, people made little progress in understanding numerical quantities and spatial relationships. Only with the onset of this fundamental turning point, revolution, when man’s passive attitude towards nature was replaced by an active one, do we enter a new Stone Age, the Neolithic.

The most difficult stage that humanity has gone through in developing the concept of number is considered to be its separation of the concept of one from the concept of “many”. It happened, in all likelihood, even when humanity was at the lowest stage of development. V.V. Bobynin explains this selection by the fact that a person usually grabs one object with his hand, and this, in his opinion, distinguishes one from the many. Thus, Bobynin thinks of the beginning of notation as the creation of a system consisting of two representations: one and an indefinite set. .

For example, the Botocud tribe, who lived in Brazil, expressed numbers only with the words “one” and “many.” The appearance of the element “two” is explained by the identification of the possibility of taking one object in each hand. At the initial stage of counting, a person associated this concept with the concept of both hands, which contain one object in each. When expressing the concept of “three,” a difficulty was encountered: a person does not have a third hand; this difficulty was overcome when the man thought of placing a third object at his feet. Thus, "three" was characterized by raising both hands and pointing to the feet. This is where the separation and concept of “four” came from, on the one hand, because on the one hand, this was prompted by the juxtaposition of two arms and two legs, and on the other, by the possibility of placing one object at each leg. At the first stage of development of counting, a person did not yet use the name of numbers, but expressed them either at his feet, or with appropriate body movements or gestures.

The further development of counting probably dates back to the era when humanity became familiar with certain forms of production - hunting and fishing. Man had to make the simplest tools to master these industries. In addition, the advancement of man into cold countries forced him to make clothes and create tools for processing leather.

Little by little, a primitive communist society emerged with an appropriate distribution of food, clothing and tools. All these circumstances forced a person, in one way or another, to keep track of the common property, the forces of the enemy, with whom he had to fight to conquer new territories. The counting process could no longer stop at four and had to develop further and further.

At this stage of development, a person no longer needs to take the objects being counted in his hand or place them at his feet. Mathematics includes the first abstraction, which consists in the fact that the objects being counted are replaced by some other homogeneous objects or signs: pebbles, knots, branches, notches. The operation is carried out according to the principle of one-to-one correspondence: each item being counted is associated with one of the items selected as a counting instrument (that is, one pebble, one knot on a rope, etc.). Traces of this kind of counting have been preserved among many peoples to this day. Sometimes such primitive counting instruments (pebbles, shells, bones) were strung on a cord or stick so as not to be lost. This subsequently led to the creation of more advanced calculating instruments, which have retained their significance to this day: Russian abacus and the Chinese suan-pan, which is similar to them.

1.2. Finger counting.

The development of counting went much faster when a person decided to turn to the closest to him, the most natural counting apparatus - his fingers. Perhaps the first act of counting on fingers was placing the object with the index finger; here the finger played the role of a unit. The participation of fingers in counting helped a person move beyond the number four, since when all the fingers on one hand began to be considered equal units, this immediately made it possible to bring the count to five. Further development of counting required the complication of the counting apparatus, and man found a way out by first involving the fingers of the second hand in counting, and then extending his technique to his toes: for tribes that did not wear shoes, the use of toes was quite natural. Moreover, such an expansion of the counting stages obviously occurred as a result of the possibility of bringing fingers and toes into a unique correspondence, which is noted among some peoples.

Thus, to express the number “twenty,” South American Indians contrast their fingers with their toes.

In the era described, people’s economic calculations were limited to the fact that after the distribution of food and clothing captured as a result of a skirmish with the enemy, there was no longer a need to remember the numbers that arose during the calculations, and therefore counting did not need names for numbers, but was carried out mainly through appropriate gestures.

For example, the native inhabitants of the Andoman Islands, located in the Bay of Bengal of the Indian Ocean, did not have words to express numbers and used certain gestures when counting. From this it is clear that gesticulation when counting as a relic remained for a long time among many peoples who did not develop verbal numbering.

Verbal counting began to develop only when agriculture became the leading form of production. At that time, private property gradually arose, the objects of which were fields, vegetable gardens, and herds. Owners of fields and domestic animals, being tightly connected to them, were forced not only to count the objects belonging to them, but also to remember their number, and this pushed man towards the creation of named numbers. At first, memorization was carried out in a very cumbersome and clumsy way: by recalling external signs memorized items. For example, the owner of a herd of oxen remembered the number of animals he owned based on the characteristics that one ox was gray, the other black, etc. Of course, this method of memorization could not be suitable when the number of memorized objects was large.

The next step in the development of the naming of numbers must be recognized as the appearance of descriptive expressions for a collection of several units. For example, instead of the name of the number expressing two objects, the phrase “as many as my hands” was used, the name four was conveyed by the phrase: “as many as the legs of the animal.” So, verbal expressions of several objects were mainly parts of the human and animal body.

Subsequently, these descriptions of the expression among many peoples were replaced by the names of the corresponding words, and thus these names were assigned to numbers. Thus, the number two began to be expressed in words denoting “ears”, “hands”, “wings”, four - “ostrich leg” (four-fingered), etc.

Finger counting gradually led to the ordering of counting, and people spontaneously came to simplify the verbal expression of numbers. So, for example, the expression that should correspond to the number 11 - “ten fingers on both hands and one toe on one foot” - was simplified into “toe”; to express the number 23, instead of saying “ten fingers on both hands, ten fingers on both feet and three fingers on another person’s hand,” it was simply said: “three fingers of another person.”

Reductions of this kind at the same time seemed to lead to the separation of units from the highest category. In fact, such names as “hand” - to designate five, “two hands” - to designate ten, “leg” - to designate fifteen, “people” - to designate twenty, etc., served to designate units of higher category than the finger, and the fingers played the role of units of the lowest category.

In this sense, the expression "one on the other hand" meaning "six" can be considered as "one from the second heel" or as "five and one", i.e. “hand” is a unit of the highest category. Likewise, the name “two on the foot,” meaning “twelve,” indicated that two units were taken from the second ten; this could also be conveyed by the following phrase: “two hands and two fingers,” where “two hands” plays the role of a higher-order unit in relation to the fingers.

For example, some tribes from the Torres Strait Islands have only one - “urapoon” and two - “okaza”. These numbers are used to count. In their language, three is expressed as “Okaza Urapun”, four is “Okaza Okaza”, five is “Okaza Okaza Urapun”, six is ​​“Okaza Okaza Okaza”, etc. Here are examples of the counting of some Australian tribes: Murray River Tribe: 1 – “enea”, 2 – “petcheval”, 3 – “petcheval enea”, four – “petcheval petcheval”.

1.3. The emergence of number systems

The transition of man to finger counting led to the creation of several different number systems.

The most ancient of the finger number systems is considered to be fivefold. This system is believed to have originated and become most widespread in America. Its creation dates back to this era, when a person counted on the fingers of one hand. Obviously, with this method of counting, some kind of counting was done every time the counting of all the fingers of one hand was completed. Until recently, some tribes retained the fivefold system in its pure form (for example, among the inhabitants of Polynesia and Melanesia).

The further development of number systems followed two paths. The tribes that did not stop at counting on the fingers of one hand moved on to counting on the fingers of the second hand and then on the toes. At the same time, some of the tribes settled on counting fingers only on their hands and this laid the basis for the decimal number system, while another part of the tribes, probably a large number, extended counting to the toes and thereby created the preconditions for the foundation of a system with a base of 20. This system became widespread mainly way among a significant part of the Indian tribes of North America and the indigenous inhabitants of Central and South America, as well as in the northern part of Siberia and Africa.

The decimal number system is predominant among the peoples of Europe. However, this does not mean that in Europe this system was always the only one: some peoples switched to the decimal system in later times, and earlier ones used a different system.

The natural unit of the highest rank when the 20-digit system emerged was “man” as the owner of 20 fingers. In this system, 40 is expressed as “two people”, 60 as “three people”, etc. The decimal system has big drawback: to express it verbally, you need to have 20 different names for the main numbers. Therefore, when some tribes developed a decimal number system, many other tribes that used the decimal number system gradually moved away from it, adopting the decimal number system. It is believed that the transition from the decimal system to the decimal system was facilitated by the fact that since people began to wear shoes that covered their toes, the ability to directly count in two tens was lost. The decimal system in our time is not observed among any people in pure water; it is usually combined with decimal or pental. However, traces of this system were preserved in the naming of some peoples, even those who achieved high cultural development.

So, for example, among the French the number 80 is expressed by the word quatre-vingts (four times twenty), and 90 - by the word quatre-vingt-dix (four times twenty and ten), among the Georgians the numbers 40, 60 and 80 are called ormatsy, somatsy and otkhmatsy, i.e. .e. 2x20, 3x20 and 4x20 (where “otsy” means 20, “ori” means 2, “sami” means 3, and “othi” means 4). The numbers 30, 50, 70 and 90 are called otsdaati, ormotsdaati, tsamotsdaati and otkhmotsdaati, i.e. 20+10, 2x20+10, 3x20+10 and 4x20+10.

Some tribes used not the fingers themselves, but their joints as a counting device. In this case, counting sometimes also developed quite productively and was formalized into coherent systems. Here the counting process proceeded in this way: the thumb of one hand is the counter of the joints of the remaining fingers of this hand; because each of the other four fingers of this hand contains three joints, then the unit next to the joint above was the number 12, which served as the duodecimal number system. This process sometimes did not stop at twelve, but continued further, with each finger of the other hand serving as a unit of the highest category, i.e. represented 12, and after counting all the fingers on the second hand, a new unit of the highest category 12x5 was created, i.e. 60. It is possible that this kind of counting contributed to the creation of the sexagesimal number system, which was widespread in ancient Babylon and later passed on to many other peoples.

Traces of duodecimal and hexadecimal number systems have survived to this day. It is worth remembering at least the counting of hours in a day, the measurement of angles in degrees, minutes and seconds.

So gradually, under the influence of economic needs, humanity created its own methods of calculation and finally achieved a harmonious method, which was further consciously improved and simplified until it turned into the method that modern mathematics uses.

1.4. Written numbering among ancient peoples.

If the development of labor processes and the emergence of property forced man to invent numbers and their names, then the further growth of people's economic needs led them along the path of greater and greater expansion and deepening of the concept of number. Particularly significant changes in this sense occurred when states emerged with a more or less complex state apparatus that required accounting for property and the creation of a tax system, and when commodity exchange moved into the stage of development of trade using a monetary system. On the one hand, this led to the emergence of written numbering, and on the other, counting operations began to develop, i.e. operations on numbers appeared.

A kind of recording of numbers was carried out even in those distant eras of human life: all these knots, notches strung on a shell cord were nothing more than the embryo of a recorded number. Then they began to denote the number 1 with one dash, 2 with two, 3 with three, etc.

The development of numerical notation has always accompanied the general rise in the cultural level of peoples, and therefore proceeded most intensively in those countries that quickly followed the path of statehood development.

Among the peoples of the globe in the most favorable conditions for the development of their economic and political life were those who lived at the junction of three continents: Europe, Africa and Asia, as well as the peoples who occupied the territories of the Hindustan Peninsula and modern China. The natural conditions in these places were extremely diverse. This diversity and extreme differentiation were observed in the development of the productive forces and, accordingly, social life.

The states located in these territories were the first states in the history of mankind, where we find the embryo of modern sciences and mathematics in particular.

Numbering of states of the Ancient East and Rome.

The ancient Babylonian state was located in that part of Mesopotamia where the beds of the Tigris and Euphrates rivers come closest. The main city of this state, Babylon, was located on the banks of the Euphrates.

The heyday of the Babylonian state dates back to the second half of the 18th century. BC. Agricultural products (grain, fruits, livestock) were exported to neighboring countries. Trade was favored by Babylon's central position on the banks of navigable rivers. The flourishing of trade led to the development of a monetary system of measures. In Babylon, a system of measures similar to our metric one was created, only it was based not on the number 10, but on the number 60. This system was fully maintained by the Babylonians for measuring time and angles, and we inherited from them the division of hours and degrees into 60 minutes, and minutes for 60 seconds.

Researchers explain in different ways the appearance of the sexagesimal number system among the Babylonians. Most likely, the base 60 was taken into account here, which is a multiple of 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60, which greatly simplifies all calculations.

Numerical notation among the Babylonians arose in a very distant era. It is believed that the Babylonians borrowed it from the peoples who lived on the territory of the Babylonian state even before its formation. This recording, like Babylonian writing, was made on clay tablets by pressing triangular wedges onto them, with a triangular block serving as the recording tool. This kind of cuneiform consisted mainly of three positions of the blade: vertical with the tip down, horizontal with the tip to the left and horizontal with the tip to the right. In this case, the ▼ sign meant one, 3 - ten. With the help of these signs, using also the method of addition, it was possible to express multi-digit numbers. For example, the sign ▼▼▼ represented 5, the sign 33 ▼▼▼ – number 23, etc. ▼▼

The origins of Egyptian culture date back to 4000 BC. It is believed that Egyptian writing was created during this era. Initially it was hieroglyphic in nature, i.e. Each concept was depicted as a separate picture. But gradually the hieroglyphic records took on a slightly different form, calledhieroglyphic notation.

The same method was used to record numbers. When writing hieroglyphically, numbers were already expressed in the decimal system, and there were special signs for place numbers: units, tens, hundreds, etc. The unit was represented by the sign |, ten, hundred, thousand, ten thousand, one hundred thousand, million, ten million. Moreover, if a unit of some category was contained in a number several times, then it was repeated the same number of times in the record, i.e. the law of addition was observed. For example, the number 5 was expressed like this: . The number 122 looked like: .

The Egyptians used only unit fractions, i.e. those that express only one fraction in our notation have one in the numerator (we call such fractions aliquot ). The exception was the fraction 2/3, for which there was a special sign: ; ½ also had a special sign, and all the rest were expressed using the symbol “ro”, which had the form. To represent a fraction, they drew this symbol and put a number under it that represented the denominator. For example, one seventh was written like this: .

Recordings were made mainly with paints on papyrus. Sometimes the recording materials were stone, wood, leather, or canvas. The text was written in lines predominantly from right to left and in columns from top to bottom.

The initial concepts of mathematics, which originated in Ancient China, served to develop the mathematical culture of neighboring peoples who occupied the territory of modern Korea, Indochina and, especially, Japan.

In China, information of a mathematical nature began to accumulate early and the recording of numbers appeared. Moreover, the Chinese hieroglyphic numbers were even more complex in writing than the Egyptian ones. (Fig. in app.).

But, in addition to these hieroglyphic numbers, simpler digital signs were also widespread in China, used in trade transactions.

They looked like this: |=1; ||=2; |||=3; ||||=4; |||||=5;| =6; ||=7; |||=8;||||=9; 0=0. Numbers were written in columns from top to bottom. A great advantage of the Chinese notation of numbers was the introduction of zero to express missing digits. It is believed that zero was borrowed from India in the 12th century.

Since ancient times, a saun-pan calculating device has come into use in China, its design reminiscent of modern Russian abacus (Fig. in appendix). Its main difference from Russian abacus is that our abacus is based on the decimal number system, while saun-pan has a mixed quinary and binary system. In a saun-pan, each wire is divided into two parts: in the lower part there are 5 bones strung, and in the upper part - 2. When all five bones are counted out from the lower part of the wire, they are replaced by one in the upper part; where the bones in the upper part are replaced by one bone of the highest category.

At the dawn of human culture, China was far ahead of Babylon and Egypt in the development of mathematics.

The method of writing numbers from the Romans was borrowed from the ancient Etruscans, one of the tribes of Ancient Italy. In this record, traces of the five-fold number system were preserved, and numbers were expressed using letters, namely the numbers 1, 5, 10, 50, 100, 500 and 1000 were designated by the actual letters I, V, X, L, C, D and M. For For larger numbers (10000, 100000, 1000000) there were special signs. There was no sign to indicate zero. In their notes, they adhered to the principle of addition and subtraction: numbers written on the right were added, and numbers written on the left were subtracted from the number written next to it. Thus, IX, XII, XC and CXXX meant 9, 12, 90 and 130, respectively. The Roman notation of numbers is used in our time in cases where it is necessary to write down some strictly fixed number, on which you do not have to make any arithmetic operations, for example, the date of construction of a monument or building, century, chapter in a book, etc.

Due to the difficulty of calculations, the Romans resorted to using finger counting or the abacus. (rice).

This abacus is a metal board with grooves along which tokens can be passed. There are nine longitudinal grooves, and seven of them make it possible to count units, tens, hundreds, thousands, tens of thousands, hundreds of thousands and millions. The digits of the units become larger when moving from the right grooves to the left ones (as can be seen in the figure). The two rightmost grooves make it possible to count fractional parts. The grooves for integers are divided into two parts: one token is placed in the upper one, and four in the lower one. The top token replaces the bottom five. The second groove on the right is also divided into two parts and makes it possible to count twelfths, with the upper part containing one token, and the lower part five. The rightmost groove is divided into three parts, of which the upper one accounts for 24 lobes, the middle 48 lobes and the lower one – 72 lobes. The right drawing shows a report equal to 84,071+2|12+1|72.

Numbers in India.

Indians made particularly valuable contributions to arithmetic. In this regard, mathematics owes to the Indians the ordering of numerical notation by introducing numbers for the decimal number system and establishing the principle of place value of numbers. In addition, in India, the use of zero to indicate the corresponding digit units has become widespread, which also played a big role in improving numerical records and facilitating operations on numbers.

The digital signs of India do not coincide in outline with modern numbers, but still have a great resemblance to them in some cases. For example, the Indian signs depicting one, seven and zero were very similar to modern numbers. The remaining signs have changed greatly over the many centuries separating us from the time of their origin.

The introduction of zero, numbers and the principle of their place value facilitated computational operations on numbers, and therefore arithmetic calculations received significant development in India. The main advantage of the Indians' introduction of number writing methods was that they greatly reduced the number of digits, applied the positional system to decimal counting, and introduced the zero sign. While the Greeks, Jews, Syrians, etc. to write numbers, up to 27 different digital signs were used; among the Indians, the number of such digital signs decreased to 10, including the designation of zero. As for the positional system, its beginnings were still among the Babylonians, but there this system was used for sexagesimal counting, and the Indians introduced it for decimal counting. Finally, the use of a sign for zero in the positional system gave a great advantage over the recording of numbers by the Babylonians. So, for example, among the Babylonians the ▼ symbol could denote one and 1/60, and in general any number of the form 60 n , and in Indian writing, the sign 1 could only denote one, since to denote a ten, a hundred, and so on, the corresponding number of zeros was written after the unit.

The process of writing numbers and performing arithmetic operations on them was done by Indians on a white board covered with red sand. The recording instrument was a stick. Thus, when writing, white marks appeared on the red surface, drawn with a stick.

Numbers of peoples of Central Asia.

Since the 7th century. In the history of the peoples that make up the states of Central Asia and the Middle East, the Arab state begins to play a significant role. From the small Arab states that entirely fit on the Arabian Peninsula in the 7th-8th centuries, the Arab Caliphate was created - a state occupying a vast territory. It included, in addition to the main territory of the Arabs, Palestine, Syria, Mesopotamia, Persia, Transcaucasia, Central Asia, Northern India, Egypt, North Africa and the Iberian Peninsula. The capital of the caliphate was first Damascus, and then in the 8th century. A new city was built near the former Babylon - Baghdad, where the capital was moved.

Since many of the representatives of the peoples who entered the caliphate wrote in Arabic, bourgeois historians incorrectly include the works of scientists of these peoples among the works of the Arabs.

The first major mathematician among the peoples that were part of the caliphate was the great Uzbek (Khorezmian) mathematician and astrologer of the 9th century. Muhammad ben Mussa al-Khwarizmi (2nd half of the 8th century - between 830-840).

Al-Khwarizmi's work on arithmetic has reached our time only in translation into Latin. It played a significant role in the development of European mathematics, since it was in it that Europeans became acquainted with Indian methods of writing numbers, that is, with the system of Indian numerals, with the use of zero and with the mixed meaning of digits. Due to the fact that this information was obtained by Europeans from a book whose author lived in an Arab state and wrote in Arabic, Indian numerals decimal system began to be incorrectly called "Arabic numerals".

Numbering in Rus'.

East Slavic tribes, the ancient ancestors of the Russian, Ukrainian and Belarusian peoples, began to form around 2-3 thousand years BC. In the 7th and 8th centuries. The Slavs had their first cities. The first big cities of Rus' were Kyiv and Novgorod.

In the 10th century, during the reign of Vladimir Svyatoslavovich (?-1015), the ancient Russian state (Kievan Rus) reached its greatest prosperity and power. In terms of cultural development, it occupied one of the prominent places among European states. In Rus' during this era, in parallel with the general development of culture, there was a relatively rapid dissemination of information from mathematics.

True, no monuments of mathematical literature have survived to our time that would give us the opportunity to judge the development of mathematics in Rus' in the 9th-10th centuries, but documents of a different nature allow us to draw some conclusions in this regard. The first Russian monument of mathematical content to this day is considered to be a handwritten work by a Novgorod monk Kirika, written by him in 1136 and bearing the title “Criticism of the deacon and domestic of the Novgorod Anthony Monastery, the teaching of how to tell a person the number of all years.”

In this work, Kirik revealed himself to be a very skillful counter and a great lover of numbers. The main tasks that Kirik solves are of chronological order: calculating the time that has elapsed between any event. When making calculations, Kirik used a numbering system called the small list and expressed by the following names: 10,000 - darkness, 100,000 - legion, or ignorant, 1,000,000 - leodr.

In addition to the small list, in Ancient Rus' there was an even larger list, which made it possible to operate with very large numbers. In the list system, the main digit units had the same names as in the small one, but the relationships between these units were different, namely:

A thousand thousand is darkness;

The darkness of those is legion, or pevedia;

Legion of legions - leodr;

Leodr leodrov - raven;

10 ravens - deck.

In the last of these numbers, i.e. about the deck, it was said: “And more than this cannot be understood by the human mind.”

Units, tens and hundreds were depicted in Slavic letters with a sign above them, called a title, to distinguish numbers from letters. Thousands were depicted with the same letters, but the sign So was placed in front of them, depicting one, - twenty-two, - six thousand, etc.

Darkness, legion and leodr were depicted with the same letters, but to distinguish them from units, tens, hundreds and thousands they were circled. So, it depicted three darknesses; - three legions, and - three leodres.

By the 16th century refers to the invention of a remarkable calculating device, which later received the name “Russian abacus” (Fig.). It is believed that the idea of ​​​​creating this device belonged to the Russian merchants Strogonov. Fractions in Ancient Rus' were called shares, later “broken numbers”. In old manuals we find the following names of fractions in Rus':

Half, half,- a third, - a quarter, - a half a third, - half a third, - half a third, - half a third, - half and half a third (small third),- half-half-half,- five, - seven, - tithe.

Slavic numbering was used in Russia until the 16th century; only in this century did the decimal positional number system gradually begin to penetrate into our country. It finally supplanted the Slavic numbering under Peter I.