Numbering history. Multi-digit subtraction

In the elementary course in mathematics numbering we will understand the totality of methods for designating and naming natural numbers.

Natural numbers are studied by concentrations. Concentration is the region of the considered numbers, united by common features. In the initial course, the following concentrations are distinguished: ten, one hundred (2 stages - from 11 to 20; from 21 to 100); thousand, multiple digits.

The ultimate goal of studying numbering is the assimilation of a number of general principles underlying the decimal number system, oral and written numbering, leading students to systematic generalizations, the ability to highlight and emphasize the general that is found in a new area of numbers, and consideration of the new on the basis and in comparison with previously learned.

The main educational tasks of studying numbering can be called:

1. Form a knowledge system:

On the natural number and the number "0";

On natural succession;

About oral and written numbering.

2. To acquaint with computational techniques based on knowledge of numbering.

When studying this topic, students should develop the following skills:

Indicate the number in writing;

Compare any numbers in different ways;

Replace the number with the sum of bit terms;

Describe any number.

Consider the method of familiarization with the basic mathematical concepts studied in this topic.

The concept of a natural number is given at the empirical level.

The number is indicated in the order of establishing a one-to-one correspondence between the objects of a given set and words - numerals.

In primary school:

Number is a quantitative characteristic of a class of equivalent sets.

A number is an element of an ordered set, a member of a natural sequence.

When studying actions, the number acts as an object on which an arithmetic operation is performed.

Students need to develop the following knowledge and skills:

Select a number from other concepts;

Correctly name the number;

Know how to form a number (as a result of counting; as a result of measurement; as a result of performing arithmetic operations);

Know how to designate numbers using numbers; a digit is a sign for a number;

Know the various functions of a number (quantity function, order function, measuring function).

Number and number "0".

Zero is considered as a quantitative characteristic of the class of empty sets (2-2, 4-4), i.e. set containing no elements.

Zero is considered as a number indicating the beginning of measurement (measurement) on the ruler.

Zero is considered as a component of actions of I and II steps (5+0, 05).

4. The number zero is used if there are no units of any digit (but there is no digit).

For example, in the number 300 there are no units of I and II category, i.e. units and tens, we denote the number of units and tens by zeros.

Natural sequence of numbers.

According to the traditional program, the natural sequence is entered as a series of numbers, according to which the score is kept.

Properties of a segment of the natural series:

The natural series of numbers starts with one.

Each number has its place. Each next number is one more than the previous one; each previous one is less than the next one.

All numbers before the selected number are less than it; standing after - more than the studied number.

Infinity of the natural series of numbers.

In the natural series of numbers, students should be able to identify finite sequences: single-digit, double-digit, n-digit numbers.

9, 99, 999, 9999… - the largest one-digit, two-digit, three-digit, four-digit, n-digit numbers.

Why? If we add 1 to each of them, we get the smallest number of the next sequence.

10, 100, 1000, 10000 ... - the smallest two-digit, three-digit, n-digit number, because when subtracting from each unit, we get the most more previous sequence.

Distinguish between oral and written numbering.

Oral numbering is a set of rules that make it possible, with the help of a few words, to make names for many numbers. In the course of studying oral numbering, it is necessary to reveal the rules for counting, reading, and the formation of numbers; know the numbers from 0 to 9, the words-numerals - forty, ninety, one hundred, thousand, million, billion. Account rules:

When counting, the final number refers to the entire set.

Rules for the formation of names and reading numbers.

1. The names of numbers from 10 to 20 are formed using the names adopted for the first ten numbers, but it has its own peculiarity - when reading, the lower digit is first called, then the rest (one-on-twenty; two-on-twenty).

2. The remaining names of numbers are formed according to the principle of bits; reading numbers begins with units of the highest digit.

3. When forming and reading multi-digit numbers, the principle of reading by class is observed.

Written numbering is a set of rules that make it possible to designate any number with the help of a few characters.

In the course of studying written numbering, the concept of "numbers" is introduced.

A digit is a symbol for a number. Purposeful systematic work is being carried out to distinguish between the concepts of "number" and "number".

Signs (numbers) are entered to indicate the first nine numbers. All other numbers are written using the same ten digits (from 0 to 9), but using two or more digits, the value of which depends on the place occupied by the digit in the number entry (i.e. the local value of the digit or the positional principle of writing numbers ).

Oral and written numbering of numbers is based on knowledge of the decimal number system. In mathematics, the number system is a set of signs, rules of operations and the order in which these signs are written when forming a number. There are two types of number systems:

A non-positional system, which is characterized by the fact that each character, regardless of the form of writing a number, is assigned one well-defined value (for example, Roman numeration).

Positional system (for example, decimal number system), which is characterized by the following properties:

Each digit takes on different meanings depending on its position in the notation of the number (positional notation principle).

Each digit, depending on its position, is called a bit unit; bit units are as follows: units, tens, hundreds, etc.

10 units of one digit make up one unit of the next digit, i.e. the ratio of bit units is ten (10 units = 1 dec; 10 dec = 1 hundred, etc.).

Starting from right to left and in a row, every 3 bit units form bit classes (units, thousands, millions, etc.).

Adding one more unit of a given category to nine units gives a unit of the next, higher (senior) category.

It is necessary to highlight the basic concepts of the decimal number system:

The unit of account is what we take as the basis of the account. Each next counting unit is 10 times larger than the previous one.

A digit is the place of a digit in a number entry.

3. Units of I, II, III categories, etc. - units standing on the first (units), second (tens), third (hundreds) place in the number record, counting from right to left.

4. Digit number - a number consisting of units of one digit.

5. Non-digit number - a number consisting of units of different digits.

6. Class - a union of units of three categories according to certain criteria. Each unit of the next class is more than a thousand times the previous one. (Thus, the first unit of the class of units is 1000 times less than the first unit of the class of thousands, etc.)

The order of studying the numbering can be reflected in the table:

The technique for studying the enumeration of non-negative integers suggests the possibility of various approaches.

In the methodology of primary education, it is traditional to study the numbering by concentrations. This approach is reflected in the textbooks of mathematics developed by Bantova M.A., Beltyukova G.V. and etc.

The gradual expansion of the numerical area creates good conditions for the formation of knowledge, skills in numbering: knowledge about numbers and how to designate them is gradually enriched; practical actions with numbers become more complicated (formation, name, recording, comparison, transformation, etc.).

There are three main stages in the study of numbering: preparatory, familiarization with new material, consolidation of knowledge and skills.

At the preparatory stage, it is necessary to form in students a psychological attitude to the study of numbering, to activate their previous experience and existing knowledge, to arouse interest in new numbers. To this end, it is proposed to include in advance exercises to repeat the main issues of numbering the numbers of the previous concentration: the ratio of the studied counting units, the decimal composition of numbers, the natural sequence, the rules for writing and ways to compare numbers; addition and subtraction techniques based on knowledge of numbering. Also, exercises have been developed in counting objects or in naming numbers in a natural sequence with access to a new concentration, this helps students understand that there are numbers outside the studied concentration and that they are somewhat similar to numbers already familiar to children.

When getting acquainted with the numbering, the exercises help students to highlight the essential features of the concepts being formed, to master the methods of the studied actions.

The selection of questions was carried out and the order of study in each concentre was determined:

first, the formation of a counting unit is considered, the count of objects is kept using this counting unit;

on the basis of the account, new bit numbers are introduced, their formation and names are revealed;

on the basis of the account with the help of all known counting units, the formation and oral designation of non-digit numbers is shown; their composition from bit;

exercises are included in counting objects using new numbers; the natural sequence of numbers is assimilated;

on the basis of knowledge of the decimal composition and the local meaning of numbers, the written numbering of numbers is revealed;

in all concentres, along with the account, the measurement of such quantities as length, mass, cost is considered; the units of measurement of these quantities and their ratio are studied in comparison with the corresponding counting units and help to assimilate them (for example, 1 dm \u003d 10 cm; 1 r. \u003d 100 k.; 1 kg \u003d 1000 g, etc.);

methods for comparing numbers are introduced based on:

the principle of formation of a natural sequence;

establishing a one-to-one correspondence between elements of sets;

knowledge of the bit composition of numbers;

knowledge of the class composition;

in each concentre, computational techniques are introduced based on knowledge of numbering:

a) the principle of formation of a natural sequence, cases of the form a + 1, where a is any natural number;

b) bit composition of numbers (exercises in adding bit numbers and reverse exercises in replacing non-bit numbers with the sum of bit numbers, as well as subtracting individual bit numbers from non-bit numbers) for example:

400+70+3=473; 506=500+6; 842-40=802;

842-800=42; 842-2=840.

When familiarizing with the numbering, it is necessary to rely on the subject actions of students. To do this, it is proposed to use various teaching aids: counting material, on which it is easy to illustrate the decimal grouping of objects when counting (sticks, bunches of sticks, squares, stripes of squares, triangles with 10 circles); visual aids that form ideas about the natural sequence of numbers (rulers, tape measures, ribbons with highlighted centimeters, decimeters, meters); visual aids that help to understand the positional principle of writing numbers (numbering tables of categories and classes, abacus).

After the introduction, purposeful work is carried out to consolidate knowledge and develop skills. Training exercises are combined with creative exercises.

Tasks are given to analyze typical errors, to compare, classify, generalize, to characterize any number. The scheme (plan) for parsing numbers, starting with single-valued to multi-valued, will gradually expand, deepen, and be enriched with new theoretical material. At the initial stage, it can be compiled on the basis of a generalization of the formulated answers of students and include the following questions:

Reading a number.

The place of a number in counting.

Decimal composition.

Write a number using numbers.

When studying the numbering of multi-digit numbers, the parsing scheme will include more tasks.

This work will allow to generalize and systematize the knowledge of students on the numbering of non-negative integers.

Another approach to the study of numbering is possible, which is reflected in the program and textbooks developed by Istomina N.B.

In connection with the thematic structure of the course, it does not distinguish concentres, but themes: “Single-digit numbers”, “Two-digit numbers”, “Three-digit numbers”, “Four-digit numbers”, “Five-digit and six-digit numbers”, in the process of studying which children form conscious reading and writing skills.

Highlighting topics whose names are oriented to the number of characters in a number helps children understand the differences between a number and a number.

At the first stage, in the topic “Single-digit numbers”, students form ideas about quantitative and ordinal numbers, counting skills; they get acquainted with the notation of numbers and with a segment of the natural series of single-digit numbers. Then they learn the meaning of addition and subtraction and the composition of single-digit numbers. The work of assimilation of numbering begins with the realization that a two-digit number consists of tens and ones.

Subsequent work, aimed at mastering the decimal number system and at developing the ability to read and write two-digit numbers, is associated with establishing a correspondence between the object model of a number and its symbolic notation. A visual aid in the form of a triangle with 10 circles is used as a ten object model.

Jobs offered:

To identify signs of similarity and difference between two-digit and three-digit numbers;

To write numbers in certain numbers;

To compare numbers;

To identify the rules (patterns) for constructing a series of numbers.

These types of tasks are also used in the study of other topics.

Exercise: Compare the exercises in the process of implementation, in which students learn the oral and written numbering of numbers in various textbooks of mathematics for elementary grades. What are the features of these exercises in each textbook?

The purpose of any numbering is to depict any natural number using a small number of individual signs. This could be achieved with a single sign - 1 (one). Each natural number would then be written by repeating the unit symbol as many times as there are units in this number. Addition would be reduced to simply ascribing units, and subtracting to deleting (erasing) them. The idea underlying such a system is simple, but this system is very inconvenient. It is practically not suitable for writing large numbers, and it is used only by peoples whose the account does not go beyond one or two tens.

With the development of human society, people's knowledge increases and the need to count and record the results of counting fairly large sets, measuring large quantities becomes more and more.

Primitive people did not have a written language, there were no letters or numbers, every thing, every action was depicted with a picture. These were real drawings showing this or that quantity. Gradually they became simpler, became more and more convenient for writing. We are talking about writing numbers in hieroglyphs. numbers. However, in order to further improve the account, it was necessary to switch to a more convenient notation that would allow numbers to be denoted by special, more convenient signs (numbers). The origin of numbers for each people is different.

The first figures are found more than 2 thousand years BC in Babylon. The Babylonians wrote with sticks on soft clay slabs and then dried their records. The writing of the ancient Babylonians was called cuneiform. The wedges were placed both horizontally and vertically, depending on their value. The vertical wedges denoted units, and the horizontal, so-called tens, units of the second digit.

Some cultures used letters to write numbers. Instead of numbers, they wrote the initial letters of numeral words. Such a numbering, for example, was among the ancient Greeks. By the name of the scientist who proposed it, she entered the history of culture under the name gerodian numbering. So, in this numbering, the number "five" was called "pinta" and denoted by the letter "P", and the number ten was called "deka" and denoted by the letter "D". Currently, no one uses this numbering. In contrast to it Roman numbering has been preserved and has come down to our days. Although now Roman numerals are not so common: on watch dials, to indicate chapters in books, centuries, on old buildings, etc. There are seven key signs in Roman numeration: I, V, X, L, C, D, M.

You can guess how these signs appeared. The sign (1) - one - is a hieroglyph that depicts the finger (kama), the sign V is the image of the hand (the wrist with the thumb extended), and for the number 10, the image of two fives (X) together. To write down the numbers II, III, IV, use the same signs, displaying actions with them. So, the numbers II and III repeat the unit the corresponding number of times. To write the number IV, I is placed before five. In this notation, the unit placed before the five is subtracted from V, and the units placed after V are

are added to it. And in the same way, the unit written before ten (X) is subtracted from ten, and the one on the right is added to it. The number 40 is denoted by XL. In this case, 10 is subtracted from 50. To write the number 90, 10 is subtracted from 100 and XC is written.

Roman numeration is very convenient for writing numbers, but almost unsuitable for calculations. It is almost impossible to do any actions in writing (calculations with “columns” and other calculation methods) with Roman numerals. This is a very big drawback of Roman numbering.

For some peoples, numbers were recorded using the letters of the alphabet, which were used in grammar. This record took place among the Slavs, Jews, Arabs, and Georgians.

alphabetical the numbering system was first used in Greece. The oldest record made according to this system is attributed to the middle of the 5th century. BC. In all alphabetic systems, numbers from 1 to 9 were designated by individual characters using the corresponding letters of the alphabet. In Greek and Slavic numbering, a dash “titlo” (~) was placed above the letters that denoted numbers to distinguish numbers from ordinary words. For example, a, b,<Г иТ -Д-Все числа от 1 до999 записывали на основе принципа прибавления из 27 индивидуальных знаков для цифр. Пробызаписать в этой системе числа больше тысячи привели к обозначениям,которые можно рассматривать как зародышипозиционной системы. Так,для обозначения единиц тысячиспользовались те же буквы,что и для единиц,но с черточкой слева внизу,например, @ , q; etc.

Traces of the alphabetic system have survived to our time. Thus, we often number the paragraphs of reports, resolutions, etc. with letters. However, we have retained the alphabetic numbering method only to designate ordinal numbers. We never designate cardinal numbers with letters, much less we never operate with numbers written in the alphabetical system.

The old Russian numbering was also alphabetic. The Slavic alphabetic designation of numbers arose in the 10th century.

Now exists Indian system number entries. It was brought to Europe by the Arabs, which is why it got the name Arabic numbering. Arabic numbering has spread throughout the world, displacing all other number entries. In this numbering, 10 icons are used to write numbers, which are called numbers. Nine of them represent numbers from 1 to 9.

2 Order1391

The tenth icon - zero (0) - means the absence of a certain digit of numbers. With the help of these ten characters, you can write any large numbers you like. Until the 18th century. in Russia, written signs, except for zero, were called signs.

So, the peoples of different countries had different written numbering: hieroglyphic - among the Egyptians; cuneiform - among the Babylonians; Herodian - among the ancient Greeks, Phoenicians; alphabetic - among the Greeks and Slavs; Roman - in the western countries of Europe; Arabic - in the Middle East. It should be said that Arabic numbering is now used almost everywhere.

Analyzing the systems of writing numbers (numbering) that took place in the history of the cultures of different peoples, we can conclude that all writing systems are divided into two large groups: positional and non-positional number systems.

Non-positional number systems include: writing numbers in hieroglyphs, alphabetic, Roman and some other systems. A non-positional number system is such a system of writing numbers when the content of each character does not depend on the place in which it is written. These characters are, as it were, nodal numbers, and algorithmic numbers are combined from these characters. For example, the number 33 in non-positional Roman numeration is written as follows: XXXIII. Here, the signs X (ten) and I (one) are used in the notation of the number three times each. Moreover, each time this sign denotes the same value: X is ten units, I is one, regardless of the place where they stand in a row of other signs.

In positional systems, each sign has a different meaning depending on where it stands in the number entry. For example, in the number 222, the number “2” is repeated three times, but the first digit on the right indicates two units, the second - two tens, and the third - two hundred. In this case we mean decimal number system. Along with the decimal number system in the history of the development of mathematics, there were binary, five-fold, two-decimal, etc.

Positional number systems are convenient in that they make it possible to write large numbers using a relatively small number of characters. An important advantage of positional systems is the simplicity and ease of performing arithmetic operations on numbers written in these systems.

The emergence of positional systems for designating numbers was one of the major milestones in the history of culture. It should be said that this did not happen by chance, but as a natural step in the cultural development of peoples. This is confirmed by the independent emergence of positional systems at different peoples: among the Babylonians - more than 2 thousand years BC; among the Mayan tribes (Central America) - at the beginning of a new era; among the Indians - in the IV-VI century AD.

The origin of the positional principle should first of all be explained by the appearance of a multiplicative form of notation. So, in the multiplicative notation, the number 154 can be written: 1xYu 2 + 5x10 + 4. As you can see, this record displays the fact that when counting some numbers of units of the first digit, in this case ten units, are taken for one unit of the next digit, a certain number of units of the second digit is taken, in turn, as the unit of the third digit, and so on. This allows you to use the same numerical symbols to display the number of units of different digits. The same notation is possible when counting any elements of finite sets.

In the five-fold system, counting is carried out by "heels" - five each. So, African blacks count on pebbles or nuts and put them in piles of five items each. They combine five such heaps into a new heap, and so on. At the same time, pebbles are first counted, then heaps, then large heaps. With this method of counting, the fact is emphasized that the same operations should be performed with piles of pebbles as with individual pebbles. The Russian traveler Miklukho-Maclay illustrates the technique of counting according to this system. Thus, characterizing the process of counting goods by the natives of New Guinea, he writes, that in order to count the number of strips of paper, which indicated the number of days before the return of the Vityaz corvette, the Papuans did the following: ten, the second repeated the same word, but at the same time he bent his fingers, first on one, then on the other hand. Having counted to ten and bending the fingers of both hands, the Papuan lowered both fists to his knees, pronouncing "iben kare" - two hands. The third Papuan at the same time bent one finger on his hand. With another ten, it was

the same thing was done, with the third Papuan bending the second finger, and for the third ten, the third finger, etc. A similar account also took place among other nations. For such an account, at least three people were needed. One counted units, the other - tens, the third - hundreds. If we replace the fingers of those who counted with pebbles placed in different recesses of a clay board or strung on twigs, then the simplest calculating device would turn out.

Over time, the names of the digits began to be skipped when writing numbers. However, to complete the positional system, the last step was missing - the introduction of zero. With a relatively small counting basis, which was the number 10, and operating with relatively large numbers, especially after the names of bit units began to be skipped, the introduction of zero became simply necessary. the place of the missed digit. One way or another, however, the introduction of zero was an absolutely inevitable stage in the natural development process, which led to the creation of a modern positional system.

The number system can be based on any number except 1 (one) and 0 (zero). In Babylon, for example, there was the number 60. If the number system is based on big number, then the notation of the number will be very short, but the execution of arithmetic operations will be more difficult. If, on the contrary, take the number 2 or 3, then the arithmetic operations are performed very easily, but the notation itself will become cumbersome. It would be possible to replace the decimal system with a more convenient one, but the transition it would be associated with great difficulties: first of all, it would be necessary to reprint all scientific books anew, to remake all calculating instruments and machines. It is unlikely that such a replacement would be appropriate. The decimal system has become familiar, and therefore convenient.

Exercises for self-examination

A sequential series of numbers is determined

faded gradually. The main role in the creation of ... numbers was played by ... addition. In addition, ... was used, as well as multiplication.

algorithmic

operation

subtraction

signs

cuneiform hieroglyphs alphabetic

To write numbers, different peoples invented different .... So, before our

days, the following types of records have arrived:,

Gerodianov, ..., Roman, etc.

And now people sometimes
use alphabetical and .., numbering, Roman

most often when denoting ordinal numbers.

In today's society, most
peoples uses Arabic (...) numbers- Hindu

Written numbering (systems) de
fall into two large groups: position
nye and ... number systems. non-positional

§ 6. Calculating instruments

The most ancient devices for facilitating counting and calculations were the human hand and pebbles. Thanks to the counting on the fingers, five-digit and decimal (decimal) number systems arose. It was correctly noted by the scientist mathematician N.N. we had not ten fingers on our hands, but eight, then humanity would use the octal system.

In practical activities, when counting objects, people used pebbles, tags with notches, ropes with knots, etc. The first and more advanced device specifically designed for computing was a simple abacus, from which the development of computer technology began. Accounting with the help of an abacus, already known in China, Ancient Egypt and Ancient Greece long before our era, existed for many millennia when written calculations replaced the abacus. It should be noted that the abacus served not so much to facilitate the actual calculations, but to remember the intermediate results .

Several varieties of abacus are known: Greek, which was made in the form of a clay tablet, on which lines were drawn with a solid object and pebbles were placed in the resulting recesses (grooves). Even simpler was the Roman abacus, on which the pebbles could move not along the grooves, but simply along the lines drawn on the board.

In China, an abacus-like device was called a suan-pan, and in Japan, a soroban. The basis for these devices were balls

ki strung on twigs; counting tables, consisting of horizontal lines corresponding to units, tens, hundreds, etc., and vertical lines intended for individual terms and factors. Tokens were laid out on these lines - up to four.

Our ancestors also had abacus - Russian abacus. They appeared in the 16th-17th centuries, they are still used today. The main merit of the inventors of the abacus is the use of a positional number system.

The next important step in the development of computer technology was the creation of adding machines and adding machines. Such machines were designed independently by different inventors.

In the manuscripts of the Italian scientist Leonardo da Vinci (1452-1519) there is a sketch of a 13-bit adding device. A 6-bit sketch was developed by the German scientist V. Schickard (1592-1636), and the machine itself was built around 1623. It should be noted that these inventions became known only in the middle of the 20th century, so they had no effect on the development of computer technology. It was believed that the first adding machine (8-bit) was designed in 1641, and built in 1645 by B. Pascal. Therefore the project was launched their serial production. Several copies of these machines have survived to this day. Their advantage was that they allowed you to perform all four arithmetic operations: addition, subtraction, multiplication and division.

The term "computer technology" is understood as a set of technical systems, i.e. computers, mathematical tools, methods and techniques used to facilitate and speed up the solution of labor-intensive tasks related to information processing (calculations), as well as the branch of technology involved in the development and operation of computers. The main functional elements of modern computers, or computers, are made on electronic devices, therefore they are called electronic computers - computers. According to the method of presenting information, computers are divided into three groups;

Analogue computers (AVM), in which information is presented in the form of continuously changing variables, expressed by some physical quantities;

digital computers (DCM), in which
information is presented in the form of discrete values
belt (numbers) expressed as a combination of discrete values
values of any physical quantity (numbers);
hybrid computers (HVM)
ryh, both ways of presenting information are used.

The first analog computing device appeared in the 17th century. It was a slide rule.

In the XVIII-XIX centuries. continued improvement of mechanical arithmometers with an electric drive. This improvement was purely mechanical in nature and lost its significance with the transition to electronics. The only exceptions are the machines of the English scientist Ch. Be-bidzha: difference (1822) and analytical (1830).

The difference machine was intended for tabulating polynomials and, from a modern point of view, was a specialized computer with a fixed (hard) program. The machine had a "memory" - several registers for storing numbers. When a given number of calculation steps was performed, the counter of the number of operations was triggered - a bell was heard. The results were printed out by a printing device. Moreover, in time this operation was combined with calculations.

While working on the difference engine, Bebidge came up with the idea of creating a digital computer for performing various scientific and technical calculations. Working automatically, this machine performed a given program. The author called this machine analytical. This machine is a prototype of modern computers. Bebidzh's analytical engine included the following devices:

for storing digital information (now called
stored by a storage device);
to perform operations on numbers (now this
arithmetic device);
device for which Babyj did not come up with a name
and which controlled the sequence of actions of the ma
tires (now this is a control device);
for input and output of information.

As information carriers for input and output, Bebidge intended to use perforated cards (punched cards) of the type used in the control of a loom. Bebidge provided for the input of function value tables with control into the machine.

which made it possible, if necessary, to re-enter it into the car.

Thus, Bebidzh's analytical engine was the world's first program-controlled computer. For this machine, the world's first programs were also compiled. The first programmer was the daughter of the English poet Byron, Augusta Ada Lovelace (1815-1852). In her honor, one of the modern programming languages is called "Ada".

The first electronic computer is considered to be a machine developed at the University of Pennsylvania in the USA. This ENIAC machine was built in 1945, had automatic program control. The disadvantage of this machine was the lack of a memory device for storing commands.

The first computer with all the components of modern machines was the English EDSAK machine, built in 1949 at Cambridge University. The memory device of this machine contains numbers (written in binary code) and the program itself. Thanks to the numerical form of writing program commands, the machine can perform various operations.

Under the leadership of S.A. Lebedev (1902-1974), the first domestic computer (electronic computer) was developed. MESM performed only 12 commands, the nominal speed of actions was 50 operations per second. The MESM RAM could store 31 seventeen-bit binary numbers and 64 twenty-bit commands. In addition, there were external storage devices. In 1966, under the guidance of the same designer, a large electronic calculating machine (BESM) was developed.

Electronic computers use various programming languages - this is a notation system for describing information data and programs (algorithms).

The program in machine language has the form of a table of numbers, each line corresponds to one operator-machine command. At the same time, in the command, for example, the first few digits are the operation code, i.e. they tell the machine what to do (add, multiply, etc.), and the rest of the numbers indicate exactly where the necessary numbers are located in the machine’s memory (terms, factors) and where you should remember the result of operations (the sum of products, etc.).

A programming language is defined by three components: alphabet, syntax, and semantics.

Most of the programming languages (BASIC, FORTRAN, PASCAL, ADA, COBOL, LISP) developed to date are sequential. The programs written in them are a sequence of orders (instructions). They are sequentially, one after the other, processed by the machine when help of so-called translators.

The performance of computers will increase due to the parallel (simultaneous) execution of operations, while most of the existing programming languages are designed for sequential execution of operations. Therefore, the future, apparently, belongs to such programming languages that will allow describing the problem being solved, and not the sequence of execution of operators.

Self Test Exercises

The development ... of instruments in the history of mathematics counting
matics happened gradually. From is
use of parts of one's own body - fingers
... - to the use of various special abacus
alno created devices: ... linearly logarithmic
ka, abacus, ... , analytical engine and computing
electronic... car.

Programs for ... machines are electronic computing

tables of numbers. telny

Components of programming languages
niya are the alphabet, ... and semantics. syntax

§ 7. Formation, current state and prospects

developed methodology for teaching children the elements of mathematics

preschool age

The issues of mathematical development of preschool children are rooted in classical and folk pedagogy. Various counting rhymes, proverbs, sayings, riddles, nursery rhymes were good material in teaching children to count, allowed the child to form concepts about numbers, shape, size, space and time. For example,

The white-sided magpie cooked porridge, fed the children.

I gave this, I gave this And I gave this, But I did not give this:

You didn't carry water, You didn't chop firewood, You didn't cook porridge - There's nothing for you.

The first printed textbook by I. Fedorov "Primer" (1574) included thoughts about the need to teach children to count in the process of various exercises. pedagogical works of Ya.A. Comenius, M.G. Pestalozzi, K.D. Ushinsky, F. Frebel, L.N. Tolstoy and others.

So, Y.A. Komensky (1592-1670) in the book "Mother's School" recommends even before school to teach a child to count within twenty, the ability to distinguish between big-small, even-odd numbers, compare objects by size, recognize and name some geometric figures, use in practice units of measurement: inch, span, step, pound, etc.

The classical systems of sensory learning by F. Frebel (1782-1852) and M. Montessori (1870-1952) present a methodology for introducing children to geometric shapes, sizes, measurement and counting. The "gifts" created by Froebel are still used as didactic material to acquaint children with number, shape, size and spatial relationships.

KD Ushinsky (1824-1871) repeatedly wrote about the importance of teaching children to count before school. He considered it important to teach the child to count individual objects and their groups, to perform addition and subtraction, to form the concept of ten as a unit of account. However, all this was just wishes that had no scientific justification.

Of particular importance are the issues of the methodology of mathematical development in the pedagogical literature of elementary school at the turn of the 19th-20th centuries. The authors of the methodological recommendations at that time were advanced teachers and methodologists. The experience of practical workers was not always scientifically substantiated.

nym, but it was tested in practice. Over time, it improved, stronger and more fully, progressive pedagogical thought came to light in it. At the end of the 19th - at the beginning of the 20th century, methodologists needed to develop a scientific foundation for the methodology of arithmetic. A significant contribution to the development of the methodology was made by advanced Russian teachers and methodologists P.S. Guryev, A. I.Goldenberg, D.F.Egorov, VAEvtushevsky, D.D.Galanin and others.

The first teaching aids on the methodology of teaching preschoolers to count, as a rule, were addressed simultaneously to teachers, parents and educators. Based on the experience of practical work with children, V.A. conversations, games, practical exercises are offered by methods of working with children. The author considers it necessary to acquaint children with such concepts as: one, many, several, pair, more, less, the same, equally, equal, the same and others. The main task is to study numbers from 1 to 10, with each number considered separately. At the same time, children learn actions on these numbers. Visual material is widely used.

In the course of conversations and classes, children gain knowledge about form, space and time, about dividing the whole into parts, about quantities and their measurement.

Questions about the methods, content of teaching children to count and mathematical development in general, which could become the basis for their successful further education at school, have been especially heatedly debated in preschool pedagogy since the creation of a wide network of public preschool education.

The most extreme position was to prohibit any purposeful teaching of mathematics. It is most clearly reflected in the works of K. Flebedintsev. children on the basis of distinguishing groups of objects, the perception of sets. And further, beyond these small aggregates, the main role in the formation of the concept of number belongs to the account, which displaces the simultaneous (holistic) perception of sets. At the same time, he considered it desirable that the child acquire knowledge during this period “imperceptibly”, independently. K.F. Lebedintsev came to this conclusion based on observations of children learning the first numerical representations and mastering them

In fact, very early children begin to isolate some small groups of homogeneous objects and, imitating adults, call it a number. But this knowledge is still shallow, not sufficiently conscious. The ability of children to name numbers is not always an objective indicator of mathematical abilities. And yet, in the 20s, many methodologists, educators adopted the point of view of K.F. Lebedintsev. In their opinion, numerical representations arise in a child mainly due to the holistic perception of small groups of homogeneous objects located in the environment table, car wheels, etc.). On this basis, it was considered optional to teach children to count.

However, the leading teachers - "preschoolers" in the 20-30s (E.I. Tikheeva, L.K. Shleger and others) noted that the process of forming numerical representations in children is very complex, and therefore it is necessary to purposefully teach them to count. Play was recognized as the main way of teaching children to count. So, the authors of the book “Living Numbers, Living Thoughts and Hands at Work” (Kyiv, 1920) E. Gorbunov-Pasadov and I. Tsunzer wrote that the child is trying to introduce into his activity-game what is interesting to him at the moment. Therefore, familiarization with elements of mathematics should be based on the active activity of the child. It was believed that when playing, children learn the account better, get better acquainted with numbers and actions on them.

Most teachers of the 1920s and 1930s had a negative attitude towards the need to create programs for kindergarten, towards goal-oriented learning. In particular, L.K. Schleger argued that children should freely choose their own activities, at their own request, i.e. everyone can do what he has in mind, choose the appropriate material, set goals for himself and achieve them. This program, in her opinion, should be based on the natural inclinations and aspirations of children. The role of the educator would be only to create conditions conducive to the self-education of children. L.K. Schleger believed that the account should be connected with various activities of the child, and the educator should use various moments from the life of children to exercise them in the account.

In the works of E. I. Tikheeva, M. Ya. Morozova and others, it was emphasized that the child must learn knowledge about the first ten numbers even before school and at the same time learn them “without any systematic classes and special teaching methods.

different nature." In the work "Modern Kindergarten, Its Significance and Equipment" (Petersburg, 1920), the authors noted that the very life of the kindergarten, the activities of children, the game provide a huge number of moments that can be used for children to learn the account within the limits available their age, and assimilation is completely at ease. The foundation of mathematical thinking is easily laid in the soul of a child, which is so necessary for both the student and the teacher if the school (kindergarten) strives for scientific and systematic education.

E. I. Tikheeva clearly imagined the content of familiarizing preschool children with numbers and counting and repeatedly emphasized that the modern methodology seeks to lead children to the assimilation of knowledge on their own, creating conditions for the child that provide him with an independent search for cognitive material and the use his. She wrote that children should not be taught calculations, but the child must learn the first ten, of course, before school. All numerical representations available to children of this age, they must take from the life in which they take an active part. And the participation of the child in life under normal conditions should be expressed only in one thing - work, play, i. That is, while playing, working, living, the child will certainly learn to count on his own, if adults at the same time are inconspicuous assistants and leaders for him.

In the work “Account in the Life of Young Children” (1920), E. I. Tikheeva also opposed “oppression and violence” in the mathematical development of the child. moments, but also objected to the spontaneous upbringing of the child. Quite rightly, she considered sensory perception as the main source of mathematical knowledge. The concept of number should enter the child's life only in "an inseparable unity with the objects" that are around the child. In this regard, the author draws attention to the availability of the necessary visual material in kindergarten and at home. After certain numerical representations have been received by the child, you can use game-lessons. The author recommends special games-lessons with didactic materials to familiarize and consolidate these ideas, deepen the necessary skills in counting.

Realizing that the spontaneous mastery of numerical representations cannot have the proper sequence, consistency, E. I. Tikheeva offered special sets of didactic material as a means of systematizing knowledge. She recommended using natural material as a counting material: pebbles, leaves, beans, cones, etc. She created didactic material such as paired pictures and lotto, developed tasks for consolidating quantitative and spatial representations.

The content of mathematical knowledge E. I. Tikheeva represented quite widely. This is an acquaintance with the value, measurement, numbers, even fractions. E. I. Tikheeva assigned a significant place in the content of teaching mathematics to the formation of children's ideas about magnitude and measure. She considered it important to reveal to children the functional relationship between the measurement result and the magnitude of the measure. All types of measurements should be appropriate, associated with practical tasks, for example, playing in a store ("shop").

Unfortunately, E. I. Tikheeva did not at all appreciate the role of collective activities, considering them to be imposed on the child from the outside. She assumed that in kindergarten the knowledge of children would be different; the degree of their development is not the same, but this “should not frighten the educator.” Although the author does not give specific recommendations anywhere on how to work with children of different levels of development.

E. I. Tikheeva made a certain contribution to the development of methods for teaching children to count, having determined the amount of knowledge available to “preschoolers.” She paid much attention to familiarizing children with the relationship between objects of different sizes: more-less, wider-narrower, shorter-longer and others. An excellent master practitioner who knew the child deeply, she felt the need for training, the gradual complication of educational material, although she basically recognized only individual training. In fact, E. I. Tikheeva did not develop and theoretically substantiate the methodology for teaching counting, did not show the main ways for children to master the initial mathematical knowledge, however, the didactic material and didactic games created by her are also used in modern pedagogical practice.

At the end of the 1930s, there was a move away from unorganized education in kindergarten, and from that moment on, problems arise related to determining the content and methods of teaching children of different age groups in kindergarten.

A significant stage in the development of methods for the development of mathematical representations was the work of F.N. Bleher. Being an innovator-practitioner of her time in the field of preschool education, she developed, tested and offered teachers a wide program for teaching preschoolers elementary knowledge in mathematics. size, quantity, space, time, and measurement. While Learning to Count is generally designed for individual use, there is plenty of material to bring children together. To make it easier for the teacher to distribute the material, the entire content of the manual is divided into lessons (81 lessons) - this is how the author calls the classes.

The image of any natural number is possible with the help of a small number of individual characters. This could be achieved with a single sign - 1 (one). Each natural number would then be written by repeating the unit symbol as many times as there are units in this number. Addition would be reduced to a simple attribution of units, and subtraction - to the deletion (erasing) of them. The idea behind such a system is simple, but this system is very inconvenient. It is practically unsuitable for recording large numbers, and it is used only by peoples whose count does not go beyond one or two dozen.

With the development of human society, people's knowledge increases and the need to count and record the results of counting fairly large sets, measuring large quantities becomes more and more.

Primitive people did not have a written language, there were no letters or numbers, every thing, every action was depicted with a picture. These were real drawings showing this or that quantity. Gradually they were simplified, became more and more convenient for recording. We are talking about writing numbers in hieroglyphs. The hieroglyphs of the ancient Egyptians testify that the art of counting was highly developed among them, with the help of hieroglyphs large numbers were depicted. However, in order to further improve the account, it was necessary to switch to a more convenient notation that would allow numbers to be denoted by special, more convenient signs (numbers). The origin of numbers for each nation is different.

The first figures are found more than 2 thousand years BC. in Babylon. The Babylonians wrote with sticks on soft clay slabs and then dried their notes. The ancient Babylonian alphabet was called cuneiform. The wedges were placed both horizontally and vertically, depending on their value. Vertical wedges denoted units, and horizontal, so-called tens, units of the second category.

Some cultures used letters to write numbers. Instead of numbers, they wrote the initial letters of words-numerals. Such a numbering, for example, was among the ancient Greeks. By the name of the scientist who proposed it, she entered the history of culture under the name gerodian numbering. So, in this numbering, the number "five" was called "pinta" and denoted by the letter "P", and the number ten was called "deka" and denoted by the letter "D". Currently, no one uses this numbering. Unlike her Roman numbering has been preserved and has come down to our days. Although now Roman numerals are not so common: on watch dials, to indicate chapters in books, centuries, on old buildings, etc. There are seven key signs in Roman numeration: I, V, X, L, C, D, M.

You can guess how these signs appeared. The sign (1) - the unit - is a hieroglyph that depicts the I finger (kama), the sign V is the image of the hand (the wrist with the thumb extended), and for the number 10 - the image of two fives (X) together. To write down the numbers II, III, IV, use the same signs, displaying actions with them. So, the numbers II and III repeat the unit the corresponding number of times. To write the number IV, five is preceded by I. In this notation, the unit placed before the five is subtracted from V, and the units placed after it are added to it. And in the same way, the unit written before ten (X) is subtracted from ten, and the one on the right is added to it. The number 40 is denoted by XL. In this case, 10 is subtracted from 50. To write the number 90, 10 is subtracted from 100 and XC is written.

Roman numeration is very convenient for writing numbers, but almost unsuitable for calculations. It is almost impossible to do any actions in writing (calculations by "columns" and other calculation methods) with Roman numerals. This is a very big drawback of Roman numeration.

For some peoples, numbers were recorded using the letters of the alphabet, which were used in grammar. This record took place among the Slavs, Jews, Arabs, Georgians.

alphabetical the numbering system was first used in Greece. The oldest record made according to this system is attributed to the middle of the 5th century BC. BC. In all alphabetic systems, numbers from 1 to 9 were designated by individual characters using the corresponding letters of the alphabet. In Greek and Slavic numerations, above the letters that denoted numbers, in order to distinguish numbers from ordinary words, a dash “titlo” (~) was placed. For example, a B C etc. All numbers from 1 to 999 were written on the basis of the principle of adding 27 individual characters for numbers.

Traces of the alphabetic system have survived to our time. So, we often number the paragraphs of reports, resolutions, etc. with letters. However, the alphabetical numbering method has been preserved with us only to denote ordinal numbers. We never designate quantitative numbers with letters, especially since we never operate with numbers written in the alphabetical system.

The old Russian numbering was also alphabetical. The Slavic alphabetic designation of numbers arose in the 10th century.

The tenth icon - zero (0) - means the absence of a certain digit of numbers. With the help of these ten characters, you can write any large numbers you like. Until the 18th century in Russia, written signs, except for zero, were called signs.

So, the peoples of different countries had different written numbering: hieroglyphic - among the Egyptians; cuneiform - among the Babylonians; gerodian - among the ancient Greeks, Phoenicians; alphabetical - among the Greeks and Slavs; Roman - in the Western countries of Europe; Arabic - in the Middle East. It should be said that now Arabic numbering is used almost everywhere.

Non-positional number systems include: writing numbers in hieroglyphs, alphabetic, Roman and some other systems. A non-positional number system is such a system of writing numbers when the content of each character does not depend on the place in which it is written. These symbols are like nodal numbers, and algorithmic numbers are combined from these symbols. For example, the number 33 in non-positional Roman numeration is written like this: XXXIII. Here, the signs X (ten) and I (one) are used three times each in writing a number. Moreover, each time this sign denotes the same value: X - ten units, I - one, regardless of the place where they stand in a number of other signs.

In positional systems, each sign has a different meaning, depending on where it stands in the notation of the number. For example, in the number 222, the number "2" is repeated three times, but the first number on the right indicates two units, the second two tens, and the third two hundreds. In this case, we mean the decimal number system. Along with the decimal number system in the history of the development of mathematics, there were binary, five-fold, twenty-decimal, etc.

The emergence of positional systems for designating numbers was one of the major milestones in the history of culture. It should be said that this did not happen by chance, but as a natural step in the cultural development of peoples. This is confirmed by the independent emergence of positional systems among different peoples: among the Babylonians - more than 2 thousand years BC; among the Mayan tribes (Central America) - at the beginning of a new era; among the Hindus - in the IV-VI centuries. AD

The origin of the positional principle, first of all, should be explained by the appearance of a multiplicative notation. Multiplicative notation is notation using multiplication. By the way, this record appeared simultaneously with the invention of the first counting device, which the Slavs called the abacus. So, in a multiplicative notation, the number 154 can be written: 1 x 104 - 5 x 10 + 4. As you can see, this notation reflects the fact that when counting, some numbers of units of the first digit, in this case ten units, are taken as one unit of the next rank, a certain number of units of the second rank is taken, in turn, as a unit of the third rank, and so on. This allows you to use the same numerical symbols to display the number of units of different digits. The same notation is possible when counting any elements of finite sets.

The technique of counting according to this system is illustrated by the Russian traveler Miklukho-Maclay. Thus, characterizing the process of counting goods by the natives of New Guinea, he writes that in order to count the number of strips of paper that indicated the number of days before the return of the Vityaz corvette, the Papuans did the following: (one), “square” (two) and so on up to ten, the second repeated the same word, but at the same time bent his fingers first on one, then on the other hand. Having counted to ten and bending the fingers of both hands, the Papuan lowered both fists to his knees, pronouncing "iben kare" - two hands. The third Papuan at the same time bent one finger on his hand. The same was done with the other ten, with the third Papuan bending the second finger, and for the third ten - the third finger, and so on. A similar account took place among other peoples. For such an account, at least three people were needed. One counted units, the other counted tens, the third counted hundreds. If we replace the fingers of those who counted with pebbles placed in different recesses of a clay board or strung on twigs, then we would get the simplest counting device.

Over time, the names of the digits began to be skipped when writing numbers. However, to complete the positional system, the last step was missing - the introduction of zero. With a relatively small counting basis, which was the number 10, and operating with relatively large numbers, especially after the names of bit units began to be skipped, the introduction of zero became simply necessary. The zero symbol could first be an image of an empty abacus token or a modified simple dot that could be put in place of a missed discharge. One way or another, however, the introduction of zero was an absolutely inevitable stage in the natural development process, which led to the creation of a modern positional system.

The number system can be based on any number except 1 (one) and 0 (zero). In Babylon, for example, there was the number 60. If a large number is taken as the basis of the number system, then the record of the number will be very short, but the performance of arithmetic operations will be more difficult. If, on the contrary, we take the number 2 or 3, then arithmetic operations are performed very easily, but the notation itself will become cumbersome. It would be possible to replace the decimal system with a more convenient one, but the transition to it would be associated with great difficulties: first of all, all scientific books would have to be reprinted, all calculating instruments and machines would be remade. It is unlikely that such a replacement would be appropriate. The decimal system has become familiar, and therefore convenient.

With the development of human society, people's knowledge increases and the need to count and record the results of counting fairly large sets, measuring large quantities becomes more and more.

Primitive people did not have a written language, there were no letters or numbers, every thing, every action was depicted with a picture. These were real drawings showing this or that quantity. Gradually they were simplified, became more and more convenient for recording. We are talking about writing numbers in hieroglyphs. However, in order to further improve the account, it was necessary to switch to a more convenient notation that would allow numbers to be denoted by special, more convenient signs (numbers). The origin of numbers for each nation is different.

The first figures are found more than 2 thousand years BC. in Babylon. The Babylonians wrote with sticks on soft clay slabs and then dried their notes.

Some cultures used letters to write numbers. Instead of numbers, they wrote the initial letters of words-numerals. Such a numbering, for example, was among the ancient Greeks. So, in this numbering, the number "five" was called "pinta" and was denoted by the letter "P". Currently, no one uses this numbering. Unlike her Roman numbering has been preserved and has come down to our days. Although now Roman numerals are not so common: on watch dials, to indicate chapters in books, centuries, on old buildings, etc. There are seven key signs in Roman numeration: I, V, X, L, C, D, M.

For some peoples, numbers were recorded using the letters of the alphabet, which were used in grammar. This record took place among the Slavs, Jews, Arabs, Georgians.

alphabetical the numbering system was first used in Greece. For example, a B C etc.

The old Russian numbering was also alphabetical. The Slavic alphabetic designation of numbers arose in the 10th century.

Over time, the names of the digits began to be skipped when writing numbers. However, to complete the positional system, the last step was missing - the introduction of zero. With a relatively small counting basis, which was the number 10, and operating with relatively large numbers, especially after the names of bit units began to be skipped, the introduction of zero became simply necessary. The zero symbol could first be an image of an empty abacus token or a modified simple dot that could be put in place of a missed discharge. One way or another, however, the introduction of zero was an absolutely inevitable stage in the natural development process, which led to the creation of a modern positional system.

With the development of human society, people's knowledge increases and the need to count and record the results of counting fairly large sets, measuring large quantities becomes more and more.

For some peoples, numbers were recorded using the letters of the alphabet, which were used in grammar. This record took place among the Slavs, Jews, Arabs, and Georgians.

The old Russian numbering was also alphabetic. The Slavic alphabetic designation of numbers arose in the 10th century.

2 Order1391

The number system can be based on any number except 1 (one) and 0 (zero). In Babylon, for example, there was the number 60. If a large number is taken as the basis of the number system, then the record of the number will be very short, but the execution of arithmetic operations will be more difficult. If, on the contrary, take the number 2 or 3, then arithmetic operations are performed very easily, but the record itself will become cumbersome. It would be possible to replace the decimal system with a more convenient one, but the transition to it would be associated with great difficulties: first of all, all scientific books would have to be reprinted, all counting instruments and machines would be redone. It is unlikely that such a replacement would be appropriate .The decimal system has become familiar, and therefore convenient.

Exercises for self-examination

A sequential series of numbers is determined

faded gradually. The main role in the creation of ... numbers was played by ... addition. In addition, ... was used, as well as multiplication.

algorithmic

operation

subtraction

signs

cuneiform hieroglyphs alphabetic

To write numbers, different peoples invented different .... So, before our

days, the following types of records have arrived:,

Gerodianov, ..., Roman, etc.

And nowadays people sometimes use alphabetic and .., numbering, Roman

most often when denoting ordinal numbers.

In modern society, most peoples use Arabic (...) numbers- Hindu

Written numberings (systems) are divided into two large groups: positional and ... number systems. non-positional

Portal for the student. Self-training

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