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 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 nations globe at most favorable conditions for the development of their economic and political life there were those who lived at the junction of three continents: Europe, Africa and Asia, as well as peoples who occupied the territories of the Hindustan Peninsula and modern China. 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 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. Main city 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. Products Agriculture(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 Ў 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 ЎЎЎ represented 5, the sign 33ЎЎЎ- number 23, etc. ЎЎ
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, called hieroglyphic 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. Units were represented by |, 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: ; Ѕ also had a special sign, and all the others were expressed using the symbol “rho”, 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 five-digit 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 rank. notation numbering fractional rational
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 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 no arithmetic operations have to be performed, 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 are placed 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 sign Ў could denote both one and 1/60, and in general any number of the form 60 n, and in the Indian record, the sign 1 could only denote one, since to denote a ten, a hundred, and so on, it was written after the unit the corresponding number of zeros.
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, middle Asia, Northern India, Egypt, North Africa and the Iberian Peninsula. The capital of the caliphate was first Damascus, and then in the 8th century. was built near the former Babylon new town- Baghdad, where the capital was moved.
So many of the representatives of the peoples who entered the caliphate wrote on Arabic, then 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 language. 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 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. First big cities Rus' had Kyiv and Novgorod.
In the 10th century, during the reign of Vladimir Svyatoslavovich (? -1015), 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' in this era, in parallel with general development 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 problems solved by Kirik are: chronological order: Calculate the time that has passed between an 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 ignoramuses, 1,000,000 - leodr.
In addition to the small list, Ancient Rus' there was an even larger list that 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 Slavic letters with a sign placed 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, - third, - fourth, - half a third, - half, - half and half a third, - half a third, - half and half a third (small third), - half and 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.
Post-printing processing is an integral and important part of the entire printing process. It is this that influences the properties and final appearance of printed products. The printing house performs such types of post-printing work as numbering, perforation, winding stitching, staple stitching, gluing into blocks, lamination, and corner rounding.
Numbering
Numbering means printing variable data on copies of printed publications, namely changing numbers assigned to them. Numbering is used on ready-made forms. Numbering makes it easier for consumers to search necessary information, and in some cases it is a mandatory procedure provided for by law. Numbering in printing houses is carried out using a numberer.
Numbering applies:
- To navigate through the text
- To prevent falsification
- To comply with legal requirements
- To control and record the relevant forms.
Types of numbering
The most common types of numbering:
- Direct continuous numbering. Each first sheet corresponds to a number X, the next X+1, etc.
- Reverse continuous numbering.
- Direct or reverse numbering with a given step.
Types of numbering can be used at the request of the customer, if this does not violate the requirements of the relevant regulatory documents (lottery tickets, strict reporting forms, etc.)
Winding stitching
With this type of stitching, the printed publication is wound onto a spring of arbitrary diameter and color, usually metal. Most often, coiling on a spring is used to make calendars.
Lamination
When laminating, printed products are covered with a special film, which protects it from mechanical damage and dirt, while maintaining an attractive appearance. appearance. We are ready to offer you single- and double-sided matte and glossy lamination of various densities.
Stitching, folding, creasing
Booklet stitching is a technology that allows you to combine a certain number of sheets into a notebook (brochure). Stitching, in which sheets are held together with metal clips, is called staple stitching.
Folding (German: fold) - drawing a fold line on thin and medium paper. Subsequently, the printed products are folded along the fold line.
Creasing is the application of straight, deep-convex lines to sheets. In the future, this makes it easier to bend the products.
Rounding corners
By rounding corners we mean giving the corners of small-format sheet products a rounded shape. These products are made from thick paper or cardboard. The rounding radius can be 10R, 6R, 3.5R.
Ticket 19
Question 1. Methodology for teaching oral and written numbering of numbers within 1000.
I. Oral numbering
Tasks:
1) Introduction of a new counting unit of hundreds;
2) Introduction of new bit numbers;
3) Introduction of non-digit three-digit numbers:
By counting 1;
By forming from hundreds, tens and units;
4) Establishing the total number of units of any category in the entire number.
Introduction of the new counting unit of hundreds:
Using sticks or models of place value units under the guidance of a teacher, children repeat known place value units, and then tie 10 tens into a bundle and listen to its name - a hundred. Next, you count in hundreds (1 hundred, 2 hundreds... 10 hundreds or a thousand). A record and drawings of digit units appear on the board
1 unit 1 cm
10 units = 1 dec. 10 cm = 1 dm
10 dec. = 1 cell 10 dm = 1 m
Next, it is useful for children to compare units of counting - place units with measures of length and introduce the thousand tape. The role of a simple unit on the tape is 1 cm, the role of a ten is 1 dm, and the role of a hundred is 1 m. You can repeat the counting of hundreds on the tape and mark the hundreds on the tape with flags or bright ribbons.
Introduction of new digit numbers (third digit numbers - round hundreds), their formation and name, introduction to new numerals: one hundred, two hundred...nine hundred, thousand.
Visibility: models of bit units (large squares) and 1000 tape.
Introduction of non-digit three-digit numbers:
a) By counting 1 to the previous one, going beyond 100: 100 and 1-101..
b) By formation from hundreds, tens and units. The inverse task is immediately performed - decomposing the numbers into digit terms, finding out the decimal composition of the number.
Tasks:
1) Designation of numbers by digits in the table of digits. Finding out the local meaning of numbers;
2) Reading and writing numbers written outside the table;
3) Consolidation of knowledge of numbering.
1.Designation of numbers by digits in the table of digits. Learning to read numbers using a numbering table. Visualization: numbering table, vertical and horizontal abacus.
As a result of observations at this stage, children are led to the conclusion that hundreds are units of the third rank, written in the number in third place, counting from right to left. Here the concept of a three-digit number is introduced and that zero denotes the absence of units of any digit.
2. Reading three-digit numbers written outside the table and writing them based on knowledge of the local meaning of the numbers.
Types of exercises:
1) Of these numbers, write down only those in which the number 7 stands for des, units, hundreds.
2) Use the numbers 3, 0, 1 to write down everything three digit numbers(numbers are not repeated)
3) What does the number 0 mean in these numbers?
3. Consolidating knowledge of numbering:
a) In the process of studying written numbering, work continues on mastering the decimal composition of numbers. For this purpose, cards with place numbers are now used. (Numbers are formed by superposition and vice versa)
b) Work is also underway to master the natural sequence, but now they also use written exercises: recording the previous and subsequent ones; add 1, subtract 1; Fill in the gap - write down the numbers from ... to ...
c) Identifying the largest and smallest among single-digit, two-digit and three-digit numbers.
Note that the smallest is written as 1 and zeros, and the largest as tens.
d) When learning numbering, children learn to identify total number units of any category in the entire number, and not just in the corresponding category.
Visualization: models of bit units.
In the initial mathematics course numbering We will understand a set of techniques for notating and naming natural numbers.
Natural numbers are studied by concentrations. Concentration is a combined common features area of numbers under consideration. In the initial course, the following concentrations are distinguished: ten, hundred (2 stages - from 11 to 20; from 21 to 100); thousand, multi-digit numbers.
The ultimate goal of studying numbering is to master 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 what is common in a new area of numbers, and considering new things based on and in comparison with previously studied.
The main educational objectives of studying numbering can be called:
1. Create a knowledge system:
About the natural number and the number “0”;
About natural sequence;
About oral and written numbering.
2. Introduce 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 a number with the sum of digit terms;
Describe any number.
Let's consider the methodology for introducing the basic mathematical concepts studied in this topic.
The concept of a natural number is given at an empirical level.
The number is designated in the order of establishing a one-to-one correspondence between the objects of a given set and words - numerals.
In primary school:
The number is quantitative characteristic class of equivalent sets.
A number is an element of an ordered set, a member of a natural sequence.
When studying operations, a number acts as an object on which an arithmetic operation is performed.
Students must develop the following knowledge and skills:
Distinguish number from other concepts;
Name the number correctly;
Know the ways of forming 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 to indicate a number;
Know the various functions of number (quantitative function, order function, measurement function).
Number and digit "0".
We consider zero as a quantitative characteristic of the class of empty sets (2-2, 4-4), i.e. a set that does not contain a single element.
We consider zero as a number indicating on the ruler the beginning of measurement (measuring).
We consider zero as a component of steps I and II (5+0, 05).
4. The number zero is used if there are no units of any digit (but not a missing digit).
For example, in the number 300 there are no units of category I and II, i.e. units and tens, let's 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, which is used to count.
Properties of a segment of a natural series:
The natural series of numbers begins with one.
Each number has its place. Each next number is one more than the previous one; each previous one is one less than the next one.
All numbers preceding the highlighted number are less than it; those coming after are greater than the studied number.
Infinity of natural 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 single-digit, two-digit, three-digit, four-digit, n-digit numbers.
Why? If we add 1 to each of them, we get the smallest number in the following sequence.
10, 100, 1000, 10000... - the smallest two-digit, three-digit, n-digit number, because when subtracting one from each, we obtain the largest number of the previous sequence.
There are oral and written numbering.
Oral numbering is a set of rules that make it possible to create names for many numbers using a few words. In the course of studying oral numbering, it is necessary to reveal the rules of counting, reading, and number formation; know numbers from 0 to 9, numeral words - forty, ninety, one hundred, thousand, million, billion. Account rules:
When counting, the final number is referred to the entire set.
Rules for forming 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 called first, then the rest (one-twenty; two-twenty).
2. The remaining names of numbers are formed according to the principle of digit order; reading numbers begins with units of the highest rank.
3. When forming and reading multi-digit numbers, the principle of reading by grade is observed.
Written numbering is a set of rules that make it possible to designate any numbers using a few characters.
In the course of studying written numbering, the concept of “numbers” is introduced.
A number is a sign to indicate a number. Purposeful systematic work is being carried out to distinguish between the concepts of “number” and “digit”.
Signs (numbers) are introduced 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 meaning of which depends on the place occupied by the digit in the number record (i.e. the place value of the digit or positional principle recording numbers).
Oral and written numbering of numbers is based on knowledge of the decimal number system. In mathematics, a 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 sign, regardless of the form in which the number is written, is assigned one very specific meaning (for example, Roman numeration).
A positional system (for example, a decimal number system), which is characterized by the following properties:
Each digit takes different meanings depending on its position in the number notation (positional notation principle).
Each digit, depending on its position, is called a digit unit; The digit units are as follows: units, tens, hundreds, etc.
10 units of one digit constitute one unit of the next digit, i.e. the ratio of digit units is equal to ten (10 units = 1 dec.; 10 dec. = 1 hundred, etc.).
Starting from right to left and in a row, every 3 digit units form digit 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.
The basic concepts of the decimal number system should be highlighted:
The counting unit is what we take as the basis for counting. Each subsequent counting unit is 10 times larger than the previous one.
Place is the place of a digit in a number.
3. Units of I, II, III categories, etc. - units standing in the first (units), second (tens), third (hundreds) place in the notation of a number, counting from right to left.
4. Place number - a number consisting of units of the same 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 characteristics. Each unit of the next class is a thousand times larger than the previous one. (So, 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 numbering can be reflected in the table:
The methodology for studying the numbering of non-negative integers suggests the possibility of different approaches.
In the methodology primary education traditionally the study of numbering by concentration. This approach is reflected in mathematics textbooks developed by M.A. Bantova, G.V. Beltyukova. and etc.
The gradual expansion of the numerical field creates good conditions for the formation of knowledge, skills, and habits in numbering: knowledge about numbers and ways of designating them is gradually enriched; Practical operations with numbers become more complicated (formation, naming, recording, comparison, transformation, etc.).
There are three main stages of studying numbering: preparatory, familiarization with new material, consolidation of knowledge and skills.
At the preparatory stage, it is necessary to form a psychological attitude in students to study numbering, to activate their previous experience and existing knowledge, and to arouse interest in new numbers. For this purpose, it is proposed to include in advance exercises to review the basic issues of numbering numbers from the previous concentration: the ratio of the studied counting units, the decimal composition of numbers, the natural sequence, writing rules and methods of comparing numbers; addition and subtraction techniques based on knowledge of numbering. Exercises have also 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 becoming familiar with numbering, exercises help students identify the essential features of the concepts being formed and master the methods of the actions being studied.
A selection of questions was carried out and the order of study in each concentration was determined:
first, the formation of a counting unit is considered, objects are counted using this counting unit;
based on the counting, new digit numbers are introduced, their formation and names are revealed;
based on counting using all known counting units, the formation and verbal designation of non-digit numbers is shown; their composition from bits;
includes exercises in counting objects using new numbers; the natural sequence of numbers is learned;
based on knowledge of the decimal composition and place value of numbers, the written numbering of numbers is revealed;
in all concentrations, along with counting, the measurement of such quantities as length, mass, cost is considered; units of measurement of these quantities and their relationship are studied in comparison with the corresponding counting units and help their assimilation (for example, 1 dm = 10 cm; 1 rub. = 100 k.; 1 kg = 1000 g, etc.);
methods for comparing numbers are introduced based on:
the principle of formation of natural sequence;
establishing a one-to-one correspondence between elements of sets;
knowledge of the digit composition of numbers;
knowledge of class composition;
In each concentration, computational techniques based on knowledge of numbering are introduced:
a) the principle of formation of a natural sequence, cases of the form a are introduced + 1, where a is any natural number;
b) digit composition of numbers (exercises in adding digit numbers and reverse exercises in replacing non-digit numbers with the sum of digit numbers, as well as subtracting individual digit numbers from non-digit numbers) for example:
400+70+3=473; 506=500+6; 842-40=802;
842-800=42; 842-2=840.
When familiarizing yourself with numbering, it is necessary to rely on the students’ objective actions. 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, strips of squares, triangles with 10 circles); visual aids that form ideas about the natural sequence of numbers (rulers, tape measures, tapes with highlighted centimeters, decimeters, meters); visual aids that help to understand the positional principle of writing numbers (numbering tables of ranks and classes, abaci).
After the introduction, targeted work is carried out to consolidate knowledge and practice skills. Training exercises are combined with exercises of a creative nature.
Analysis tasks are given typical mistakes, for comparison, classification, generalization, to characterize any number. The scheme (plan) for parsing numbers, starting from single-digit to multi-valued, will gradually expand, deepen, and be enriched with new theoretical material. At the initial stage, it can be compiled based on a generalization of the students’ formulated answers and include the following questions:
Reading a number.
Place of number in counting.
Decimal composition.
Writing a number using digits.
When studying the numbering of multi-digit numbers, the parsing scheme will include a larger number of tasks.
This work will allow us to generalize and systematize students’ knowledge of the numbering of non-negative integers.
Another approach to studying the numbering of numbers is possible, which is reflected in the program and textbooks developed by N.B. Istomina.
In connection with the thematic structure of the course, it does not highlight concentrations, but topics: “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 develop conscious number reading and writing skills.
Highlighting topics whose names are focused on the number of characters in a number helps children understand the differences between number and number.
At the first stage, in the topic “Single-digit numbers,” students develop ideas about cardinal and ordinal numbers, and counting skills; They get acquainted with the writing of numbers and with a segment of the natural series of single-digit numbers. They then learn the meaning of addition and subtraction and the composition of single-digit numbers. The work of mastering numbering begins with the understanding that a two-digit number consists of tens and units.
Subsequent work aimed at mastering the decimal number system and developing the skill of reading and writing two-digit numbers is associated with establishing a correspondence between the subject model of a number and its symbolic notation. Ten is used as a subject model visual material in the form of a triangle with 10 circles.
Suggested tasks:
To identify signs of similarities and differences between two-digit and three-digit numbers;
To write numbers in certain numbers;
To compare numbers;
To identify the rule (pattern) for constructing a series of numbers.
The listed types of tasks are also used when studying other topics.
Exercise: Compare the work-in-progress exercises that students use to learn verbal and written numbering in various elementary school mathematics textbooks. What are the features of these exercises in each textbook?
Wedge numbering. Even the Chaldeans and Babylonians had written signs to depict numbers. Their numbering is called wedge-shaped and is found on the tombs of ancient Persian kings.
Hieroglyphic numbering. The Egyptians attribute the invention of arithmetic to the mythical figure Thoth (Phot). They had decimal notation even under Fra-Sesostris. Egyptian numbering is called hieroglyphic. The Egyptians denoted one, ten, hundred and thousand with special signs, hieroglyphs. Several units, tens, hundreds and thousands were depicted simple construction these signs.
Chinese numbering. Numbering should also be considered among the oldest Chinese. According to the Chinese, they have been using it since the time of Fugue, the Chinese emperor who lived 300 years BC. In this numbering, the first nine numbers are depicted with special signs. There were also signs to indicate 10, 100, 1000. Large numbers were written in columns from top to bottom.
Phoenician numbering. Finally, numbering should also be considered among the most ancient Phoenician. The Phoenicians, compared to the Egyptians, carried out a reform in numbering in the sense that they replaced hieroglyphs with letters of their alphabet. Jews also used this numbering.
The Phoenicians and Jews represented the first nine numbers and the first nine tens with the 18 initial letters of their alphabet and wrote big numbers from right hand to the left.
In Egypt itself, hieroglyphic numbering was abandoned and first hieratic, and then demotic writing was introduced for general use (600 BC). IN hieratic The numbering of the first three numbers is similar to real numbers.
Greek, Roman and Church Slavonic numbering. The Greeks adopted from the Phoenicians a system of representing numbers with letters. Some claim that until then they represented numbers with the very signs that are known as Roman numbering, and that Roman numbering is, therefore, ancient Greek. Church Slavonic is nothing more than Greek, expressed only in Slavic letters.
The Romans used the following signs when depicting numbers:
1 – I, 5 – V, 10 – X, 50 – L, 100 – C, 500 – D, 1000 – M.
When depicting the remaining numbers, they were guided by the following rule:
If a smaller number follows a larger one, it increases the number; if a smaller number precedes a larger one, it reduces the number by its value.
In accordance with this rule, they depicted numbers as follows:
1 – I, 2 – II, 3 – III, 4 – IV, 5 – V, 6 – VI, 7 – VII, 8 – VIII, 9 – IX, 10 – X, 11 – XI, 12 – XII, 13 – XIII, 14 – XIV, 15 – XV, 16 – XVI, 17 – XVII, 18 – XVIII, 19 – XIX, 20 – XX, … 27 – XXVII, … 40 – XL, 60 – LX, 90 – XC, 100 – C, 110 – CX, 150 – CL, 400 – CD, 600 – DC, 900 – CM, 1100 – MC.
Numbers consisting of several thousand were written in the same way as numbers up to a thousand are written, with the only difference being that after the number of thousands below with right side the letter m (mille - thousand) was assigned. Thus, 505197 = DV m CXCVII.
In Slavic and Greek numerals, the first nine numbers, nine tens and nine hundreds were designated by special letters.
In Slavic numerals, a title (¯) is placed on the letter to indicate that the letter represents a number.
The following table shows parallel Greek and Slavic numbering:
To denote thousands, the sign was placed before the number of thousands in the Slavic notation, and in the Greek notation, a dash was added to the number denoting thousands.
Thus,
Origin and distribution of decimal numbering
Although it is not yet possible to draw a definitive conclusion regarding the image, introduction and spread of the decimal numbering system throughout Europe, however, the literature provides many very important indications on this issue. Some call this system Arabic. Indeed, history shows that the decimal system was borrowed from the Arabs. Thus, it is known that at the beginning of the 13th century, the Tuscan merchant Leonard introduced his compatriots to the methods of the decimal system after his travels through Syria and Egypt. Sarco-Bosco, a famous teacher of mathematics in Paris (died 1256), and Roger Bacon, with their writings, most contributed to the spread of this system throughout Europe. They already indicate that decimal numbering was borrowed by the Arabs from the Indians. It is reliably known from the monuments of Arabic literature that Abu Abdallah Mohammed Ibn Muse, originally from Koraism, traveled for a long time in India in the 9th century and, after his return, introduced Arab scholars to Indian numbering. Arab writers Avicena Aben-Ragel and Alsefadi also attribute the invention of numbering to the Indians.
Written monuments of Sanskrit, language ancient india, confirm the instructions of Arab writers.
From the work of Baskara, an Indian writer of the 12th century, it is clear that the Indians knew several centuries before Baskara the representation of numbers in ten signs, for this work sets out a coherent theory of four arithmetic operations and even the extraction square roots. Both Baskara and the more ancient writer Bramegupta consider the invention of numbering to be very ancient. In the even more ancient writer Aryabgat we find solutions to many remarkable mathematical questions.
These indications seem to make the assurances of the French geometer Chals unlikely that the decimal system is a development of the Roman method of using the calculation table (Abacus) in calculations and that one introduction of a zero was enough to obtain a real decimal system.
Arithmetic and logistics among the Greeks. The Greeks called arithmetic the doctrine of general properties numbers. The art of counting, or a set of practical techniques in calculation, the Greeks called logistics.
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