Any natural number can be represented using a small number of individual signs. This could be achieved with a single sign - 1 (units). Each natural number then it would be written by repeating the unit symbol as many times as there are units in this number. Addition would be reduced to simply adding units, and subtraction would be to crossing out (wiping out) them. The idea behind such a system is simple, but the system is very inconvenient. It is practically unsuitable for recording large numbers, and it is used only by peoples whose counting does not go beyond one or two tens.
With development human society People's knowledge is increasing and there is an increasing need to count and record the results of counting quite large sets and measuring large quantities.
U primitive people there was no writing, there were no letters or numbers, every thing, every action was depicted with a picture. These were real drawings depicting one or another quantity. Gradually they were simplified and became more and more convenient for recording. It's about about writing numbers in hieroglyphs. However, to further improve the counting, it was necessary to move to a more convenient notation, which would allow numbers to be designated with special, more convenient signs (numbers). The origin of numbers is different for each nation.
The first figures are found more than 2 thousand years BC. in Babylon. The Babylonians wrote with sticks on slabs of soft clay and then dried their notes.
Some peoples used letters to write numbers. Instead of numbers they wrote initial letters numeral words. Such numbering, for example, was used by 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 The numbering has been preserved and has survived to this day. Although now Roman numerals are not found so often: on watch dials, to indicate chapters in books, centuries, on old buildings, etc. There are seven node signs in Roman numbering: I, V, X, L, C, D, M.
Among some peoples, numbers were written using letters of the alphabet, which were used in grammar. This recording took place among the Slavs, Jews, Arabs, and Georgians.
Alphabetical The numbering system was first used in Greece. For example, a B C etc.
Traces of the alphabetic system have survived to this day. Thus, we often use letters to number paragraphs of reports, resolutions, etc. However, we have retained the alphabetical method of numbering only for designating ordinal numbers. Quantitative numbers We never designate with letters, much less we never operate with numbers written in the alphabetical system.
Ancient Russian numbering was also alphabetical. The Slavic alphabetic notation for numbers arose in the 10th century.
So, among the peoples different countries there were different written numberings: hieroglyphic - among the Egyptians; cuneiform - among the Babylonians; Herodian - among the ancient Greeks, Phoenicians; alphabetical - among the Greeks and Slavs; Roman - in Western countries Europe; Arabic - in the Middle East. It should be said that Arabic numbering is now used almost everywhere.
Positional number systems are convenient because they make it possible to write big numbers using a relatively small number of characters. An important advantage of positional systems is simplicity and ease of implementation arithmetic operations over the numbers written in these systems.
The origin of the positional principle should first of all be explained by the appearance of the multiplicative form of notation. Multiplicative notation is notation using multiplication. By the way, this entry appeared simultaneously with the invention of the first calculating device, which the Slavs called the abacus. So, in multiplicative notation, the number 154 can be written: 1 x 104 – 5 x 10 + 4.
In the five-digit system, counting is done in heels - five at a time. Thus, African blacks count on pebbles or nuts and put them in piles of five items each. They combine five such piles into a new pile, and so on. At the same time, first they count the pebbles, then the heaps, then the 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.
Over time, the names of the digits began to be omitted when writing numbers. However, to complete the positional system, the last step was missing - introducing a zero. With a relatively small counting base, such as the number 10, and operating with relatively large numbers, especially after the names of digit units began to be omitted, 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, which could be placed in the place of the missing digit. One way or another, however, the introduction of zero was a completely inevitable stage in the natural process of development, which led to the creation of the 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 basis of the number system is taken big number, then writing the number will be very short, but performing arithmetic operations will be more complex. If, on the contrary, we take the number 2 or 3, then arithmetic operations are very easy to do, but the recording 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, it would be necessary to retype everything again science books, remodel all calculating instruments and machines. It is unlikely that such a replacement would be advisable. Decimal system has become familiar, and therefore convenient.
The purpose of any numbering is to represent any natural number using 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 that number. Addition would be reduced to simply adding units, and subtraction would be to crossing out (wiping out) 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 who the count does not go beyond one or two tens.
With the development of human society, people's knowledge increases and there is an increasing need to count and record the results of counting quite large sets and measuring large quantities.
Primitive people had no writing, no letters, no numbers, every thing, every action was depicted with a picture. These were real drawings depicting this or that quantity. Gradually they were simplified and became more and more convenient for writing. We are talking about writing numbers in hieroglyphs. The hieroglyphs of the ancient Egyptians indicate that the art of counting was quite highly developed among them; large numbers were depicted with the help of hieroglyphs numbers. However, to further improve the counting, it was necessary to move to a more convenient notation, which would allow numbers to be designated with special, more convenient signs (numbers). The origin of numbers is different for each nation.
The first numbers are found more than 2 thousand years BC in Babylon. The Babylonians wrote with sticks on slabs of soft clay and then dried their notes. The writing of the ancient Babylonians 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 peoples used letters to write numbers. Instead of numbers, they wrote the initial letters of numeral words. Such numbering, for example, was used by the ancient Greeks. After the name of the scientist who proposed it, it entered the history of culture under the name Herodian 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 it Roman the numbering has been preserved and has survived to this day. Although now Roman numerals are not found so often: on watch dials, to indicate chapters in books, centuries, on old buildings, etc. There are seven node signs in Roman numbering: I, V, X, L, C, D, M.
One can guess how these signs appeared. The sign (1) - unit is a hieroglyph that depicts the I finger (kama), the sign V is the image of a hand (the wrist with the thumb extended), and for the number 10 - an image of two fives (X) together. To write down the numbers II, III, IV, use the same signs, displaying actions with them. So, numbers II and III repeat one corresponding number once. To write the number IV, I is placed before five. In this notation, the one placed before the five is subtracted from V, and the ones placed after V are subtracted.
are added to it. And in the same way, the one written before ten (X) is subtracted from ten, and the one on the right is added to it. The number 40 is designated XL. In this case, 10 is subtracted from 50. To write the number 90, 10 is subtracted from 100 and HS is written.
Roman numbering is very convenient for writing numbers, but almost unsuitable for carrying out calculations. It is almost impossible to do any written actions (calculations in “columns” and other calculation methods) with Roman numerals. This is a very big drawback of Roman numbering.
Some peoples recorded numbers using letters of the alphabet that were used in grammar. This recording took place among the Slavs, Jews, Arabs, and Georgians.
Alphabetical The numbering system was first used in Greece. The oldest record made using this system dates back to the middle of the 5th century. BC. In all alphabetic systems, numbers from 1 to 9 were designated by individual symbols using the corresponding letters of the alphabet. In Greek and Slavic numbering, a dash “title” (~) was placed above the letters that denoted numbers in order 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 this day. Thus, we often use letters to number paragraphs of reports, resolutions, etc. However, we have retained the alphabetical method of numbering only to designate ordinal numbers. We never denote cardinal numbers with letters, much less we never operate with numbers written in the alphabetical system.
Ancient Russian numbering was also alphabetical. The Slavic alphabetic designation of numbers arose in the 10th century.
Now exists Indian system recording numbers. 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 records of numbers. In this numbering, 10 icons called numerals are used to record numbers. Nine of them represent numbers from 1 to 9.
2 Order1391
The tenth symbol - zero (0) - means the absence of a certain category of numbers. Using these ten symbols, you can write any large numbers. Until the 18th century. in Rus', written signs other than 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; alphabetical - among the Greeks and Slavs; Roman - in Western European countries; Arabic - in the Middle East. It should be said that Arabic numbering is now used almost everywhere.
Analyzing the systems of recording numbers (numbering) that took place in the history of cultures of different peoples, we can conclude that all written 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 a system for writing numbers when the content of each symbol 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 in writing the number three times each. Moreover, each time this sign denotes the same value: X - ten units, I - one, regardless of the place in which 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 record. For example, in the number 222, the digit “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-digit, twenty-digit, etc.
Positional number systems are convenient because 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 notating numbers was one of the main 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 the new era; among the Hindus - in the 4th-6th centuries AD.
The origin of the positional principle should first of all be explained by the appearance of the multiplicative form of notation. Multiplicative notation is notation using multiplication. By the way, this notation appeared simultaneously with the invention of the first calculating device, which the Slavs called the abacus. So, in multiplicative notation, the number 154 can be written: 1 x 10 2 + 5 x 10 + 4. As you can see, this notation reflects the fact that when counting, certain quantities of units of the first digit, in this case ten units, are taken as one unit of the next digit, a certain number of units of the second digit is taken, in turn, as a unit of the third category, etc. This allows you to use the same numeric symbols to depict the number of units of different digits. The same notation is possible when counting any elements of finite sets.
In the five-digit system, counting is done in heels - fives at a time. Thus, African blacks count on pebbles or nuts and put them in piles of five items each. They combine five such piles into a new pile, and so on. In this case, 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 counting technique using this system is illustrated by the Russian traveler Miklouho-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 until the return of the corvette “Vityaz”, the Papuans did the following: the first, laying strips of paper on his knees, with each laying aside, repeated “square” (one), “square” (two) and so on until ten, the second repeated the same word, but at the same time bent his fingers first on one hand, 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 bent one finger on his hand. With the other ten there was
the same thing was done, and the third Papuan bent the second finger, and for the third ten - the third finger, etc. Similar counting took place among other peoples. For such a count, at least three people were needed. One counted units, another - 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 we would get the simplest calculating device.
Over time, the names of digits began to be omitted when writing numbers. However, to complete the positional system, the last step was missing - introducing a zero. With a relatively small counting base, such as the number 10, and dealing with relatively large numbers, especially after the names of digit units began to be omitted, the introduction of zero became simply necessary. The zero symbol could initially be an image of an empty abacus token or a modified simple dot that could be placed on place of the missing discharge. One way or another, however, the introduction of zero was a completely inevitable stage in the natural process of development, which led to the creation of the 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 for the number system, then writing the number will be very short, but performing arithmetic operations will be more difficult. If, on the contrary, you take the number 2 or 3, then arithmetic operations are performed very easily, but the recording itself would 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, it would be necessary to reprint all scientific books, remake all calculating instruments and machines. It is unlikely that such a replacement would be advisable .The decimal system has become familiar, and therefore convenient.
Self-test exercises
A sequential series of numbers determines
fell gradually. The main role in the creation of... numbers was played by... addition. In addition,..., as well as multiplication were used.
algorithmic
operation
subtraction
signs
cuneiform hieroglyphs alphabetic
To record numbers, different peoples invented different ones.... So, until our
days the following types of records have reached: ,
Herodianova, ..., Roman, etc.
And nowadays people sometimes use alphabetical and..., numbering, Roman
most often when denoting ordinal numbers.
In modern society, most peoples use Arabic (...) numbers - Hindu
Written numbering systems (systems) are divided into two large groups: positional and... number systems. non-positional
Topic: Studying the numbering of numbers.
Plan :
1. The purpose and educational objectives of studying numbering.
2. The sequence of studying the numbering of non-negative integers.
3. Methodology for studying numbering.
Basic theoretical provisions of this section.
In the initial mathematics course numbering is understood as a set of techniques for designating and naming natural numbers .
There are oral and written numbering.
Verbal numbering- 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, words - numerals - forty, ninety, one hundred, thousand, million, billion.
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. (eleven twelve).
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- this is a set of rules that make it possible to designate any numbers with the help of a few symbols. In the course of studying written numbering, the concept of “numbers” is introduced. 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 that the digit occupies in the number record (i.e. the place value of the digit or the positional principle of writing numbers).
Oral and written numbering numbers relies on knowledge of the decimal number system.
Basic concepts of the decimal number system:
1. The counting unit is what we take as the basis for the count. Each subsequent counting unit is 10 times larger than the previous one (one ten is 10 times more than one unit; one hundred is 10 times more than one ten, etc.).
2. Place – the place of a digit in the notation of a number.
3. Units of the 1st, 2nd, 3rd category, 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 one digit, for example: 10,20,30,40,50,60... - numbers consisting only of tens (round tens); 100, 200, 300, ... - numbers consisting only of hundreds (round hundreds); 1000, 2000, 3000 - numbers consisting only of units of thousands (round units of thousands), etc.
5. Non-digit number – a number consisting of units of different digits, for example, numbers consisting of tens and units (11,22,35,47,89); numbers consisting of hundreds and units (208, 406); consisting of hundreds and tens (240, 560); consisting of hundreds, tens and units (346, 683), etc.
6. Complete numbers - numbers in which there are units of all digits, for example, a complete three-digit number 134, a four-digit number 5674
7. Incomplete numbers - numbers in which there are no units of one category or another (in this case, a zero is written in their place), for example: incomplete three-digit numbers 560, 404, incomplete four-digit numbers 1002, 1020, 1200, 1220, etc.
8. 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, 1 unit of the units class is 1000 times less than 1 unit of the thousand class, etc.)
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:
1. 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).
2. A positional system (for example, a decimal number system), which is characterized by the following properties:
Each digit takes on 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 decade; 10 decades = 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.
Properties of a segment of a natural series:
1. The natural series of numbers begins with one.
2. 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.
3. All numbers preceding the highlighted number are less than it; everything after is greater than the number studied.
4. Infinity of the natural series of numbers.
The purpose and educational objectives of studying numbering
The purpose of studying numbering is to master the general principles underlying the decimal number system, oral and written numbering.
Basic educational objectives studying numbering:
1.Form a knowledge system:
About the natural number and the number “0”;
On the natural sequence of numbers;
About oral and written numbering;
2.Introduce computational techniques based on knowledge of numbering.
When studying this topic, students should develop the following skills :
2. indicate a number in writing;
3. compare any numbers in different ways;
4. replace the number with the sum of the bit terms;
5. characterize any number.
Students must develop the following knowledge and skills:
1. Select number from other concepts.
2. Name the number correctly.
3. Know the ways of forming a number (as a result of counting; as a result of measurement; as a result of performing arithmetic operations).
4. Know how to designate numbers using numbers.
5. Know the various functions of number. (Quantitative function, order function, measurement function.)
The purpose of any numbering is to represent any natural number using a small number of individual characters. This could be achieved with a single sign - 1 (units). Each natural number would then be written by repeating the unit symbol as many times as there are units in that number. Addition would be reduced to simply adding units, and subtraction would be to crossing out (wiping out) them. The idea behind such a system is simple, but the 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 tens.
With the development of human society, people's knowledge increases and there is an increasing need to count and record the results of counting quite large sets and measuring large quantities.
Primitive people did not have writing, there were no letters or numbers; every thing, every action was depicted with a picture. These were real drawings depicting one or another quantity. Gradually they were simplified and became more and more convenient for recording. We are talking about writing numbers in hieroglyphs. The hieroglyphs of the ancient Egyptians indicate that the art of counting was quite highly developed among them; large numbers were depicted with the help of hieroglyphs. However, to further improve the counting, it was necessary to move to a more convenient notation, which would allow numbers to be designated with special, more convenient signs (numbers). The origin of numbers is different for each nation.
The first figures are found more than 2 thousand years BC. in Babylon. The Babylonians wrote with sticks on slabs of soft clay and then dried their notes. The writing of the ancient Babylonians 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 peoples used letters to write numbers. Instead of numbers, the initial letters of numeral words were written. Such numbering, for example, was used by the ancient Greeks. After the name of the scientist who proposed it, it entered cultural history under the name Herodian 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 The numbering has been preserved and has survived to this day. Although now Roman numerals are not found so often: on watch dials, to indicate chapters in books, centuries, on old buildings, etc. There are seven node signs in Roman numbering: I, V, X, L, C, D, M.
One can guess how these signs appeared. The sign (1) - unit - is a hieroglyph that depicts the first finger (kama), the sign V is the image of a hand (the wrist with the thumb extended), and for the number 10 - an image of two fives (X) together. To write numbers II, III, IV, use the same signs, displaying actions with them. Thus, numbers II and III repeat one the corresponding number of times. To write the number IV, an I is placed before five. In this notation, the one placed before the five is subtracted from V, and the ones placed after V are subtracted.
are added to it. And in the same way, the one written before ten (X) is subtracted from ten, and the one on the right is added to it. The number 40 is designated XL. In this case, 10 is subtracted from 50. To write the number 90, 10 is subtracted from 100 and HS is written.
Roman numbering is very convenient for writing numbers, but almost unsuitable for carrying out calculations. It is almost impossible to do any written actions (calculations in “columns” and other calculation methods) with Roman numerals. This is a very big drawback of Roman numbering.
Among some peoples, numbers were written using letters of the alphabet, which were used in grammar. This recording took place among the Slavs, Jews, Arabs, and Georgians.
Alphabetical The numbering system was first used in Greece. The oldest record made using this system dates back to the middle of the 5th century. BC. In all alphabetic systems, the numbers 1 to 9 were represented by individual symbols using the corresponding letters of the alphabet. In Greek and Slavic numbering, above the letters that denoted numbers, in order to distinguish numbers from ordinary words, a dash “titlo” (~) was placed. For example, a, b,<Г и Т -Д-Все числа от 1 до 999 записывали на основе принципа прибавления из 27 индивидуальных знаков для цифр. Пробы записать в этой системе числа больше тысячи привели к обозначениям, которые можно рассматривать как зародыши позиционной системы. Так, для обозначения единиц тысяч использовались те же буквы, что и для единиц, но с черточкой слева внизу, например, @ , q; etc.
Traces of the alphabetic system have survived to this day. Thus, we often use letters to number paragraphs of reports, resolutions, etc. However, we have retained the alphabetical method of numbering only for designating ordinal numbers. We never denote cardinal numbers with letters, much less we never operate with numbers written in the alphabetical system.
Ancient Russian numbering was also alphabetical. The Slavic alphabetic notation for numbers arose in the 10th century.
Now exists Indian system recording numbers. It was brought to Europe by the Arabs, which is why it got the name Arabic numbering. Arabic numbering spread throughout the world, displacing all other records of numbers. This numbering uses 10 symbols called numerals to write numbers. Nine of them represent numbers from 1 to 9.
2 Order 1391
The tenth icon - zero (0) - means the absence of a certain digit of numbers. Using these ten characters you can write any large numbers you like. Until the 18th century in Rus', written signs other than 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; alphabetical - among the Greeks and Slavs; Roman - in Western European countries; Arabic - in the Middle East. It should be said that Arabic numbering is now used almost everywhere.
Analyzing the systems of recording numbers (numbering) that took place in the history of cultures of different peoples, we can conclude that all written 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 a system for writing numbers where the content of each symbol 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 as follows: XXXIII. Here the signs X (ten) and I (one) are used three times each in writing the number. Moreover, each time this sign denotes the same value: X - ten units, I - one, regardless of the place in which they stand in a row of other signs.
In positional systems, each sign has a different meaning depending on where in the number it appears. For example, in the number 222, the digit “2” is repeated three times, but the first digit on the right stands for two ones, the second for two tens, and the third for two hundreds. 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-digit, twenty-digit, etc.
Positional number systems are convenient because 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 notating numbers was one of the main 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 the new era; among the Hindus - in the 4th-6th centuries AD.
The origin of the positional principle should first of all be explained by the appearance of the multiplicative form of notation. Multiplicative notation is notation using multiplication. By the way, this entry appeared simultaneously with the invention of the first calculating device, which the Slavs called the abacus. So, in multiplicative notation, the number 154 can be written: 1x10 2 + 5x10 + 4. As we can see, this record reflects the fact that when counting, certain quantities of units of the first digit, in this case ten units, are taken as one unit of the next digit, a certain number of units of the second digit are taken, in turn, as a unit of the third digit, etc. d. This allows you to use the same numeric symbols to depict the number of units of different digits. The same notation is possible when counting any elements of finite sets.
In the five-digit system, counting is done in heels - five at a time. Thus, African blacks count on pebbles or nuts and put them in piles of five items each. They combine five such piles into a new pile, and so on. In this case, 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 technique of counting using this system is illustrated by the Russian traveler Miklouho-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 until the return of the corvette “Vityaz”, the Papuans did the following: the first, laying strips of paper on his knees, repeated “square” with each delay. (one), “square” (two) and so on until ten, the second repeated the same word, but at the same time bent the 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 bent one finger on his hand. It was with another ten
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 took place among other nations. For such a count, at least three people were needed. One counted units, another 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 omitted when writing numbers. However, to complete the positional system, the last step was missing - introducing a zero. With a relatively small counting base, such as the number 10, and operating with relatively large numbers, especially after the names of digit units began to be omitted, 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, which could be placed in the place of the missing digit. One way or another, however, the introduction of zero was a completely inevitable stage in the natural process of development, which led to the creation of the 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 for the number system, then writing the number will be very short, but performing arithmetic operations will be more complex. If, on the contrary, you take the number 2 or 3, then the arithmetic operations are performed very easily, but the recording 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, it would be necessary to reprint all scientific books, redo all calculating instruments and machines. It is unlikely that such a replacement would be advisable. The decimal system has become familiar, and therefore convenient.
Self-test exercises
A sequential series of numbers determines
algorithmic
operation
subtraction
To record numbers, different peoples invented different.... So, before our
days the following types of records have arrived: ....... ,
Herodianova, ..., Roman, etc.
And nowadays people sometimes
use alphabetical and..., numbering, Roman
most often when denoting ordinal numbers.
In modern society the majority
peoples use Arabic (...) numbers - Hindu
Written numbering systems (systems)
fall into two large groups: position
nal and... number systems. non-positional
§ 6. Counting devices
The most ancient devices for facilitating counting and calculations were the human hand and pebbles. Thanks to finger counting, pentary and decimal (decimal) number systems emerged. The scientist mathematician N.N. Luzin correctly noted that “the advantages of the decimal system are not mathematical, but zoological. If we had eight fingers instead of ten on our hands, 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 calculations was a simple abacus, from which the development of computing technology began. Counting with the 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 calculations themselves, but rather to remember 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 hard object and pebbles were placed in the resulting recesses (grooves). Even simpler was the Roman abacus, on which pebbles could move not along grooves, but simply along lines marked on the board.
In China, a device similar to an abacus was called suan-pan, and in Japan - soroban. The basis for these devices were ball-
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 an abacus - a Russian abacus. They appeared in the 16th-17th centuries and are still used today. The main merit of the inventors of the abacus is the use of a positional number system.
The next important stage in the development of computer technology was the creation of adding machines and adding machines. Such machines were designed independently of each other 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. The German scientist W. Schickard (1592-1636) developed a 6-digit sketch, 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 did not have any impact 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. Based on this project, their mass production was launched. 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 problems associated with information processing (computing), as well as the branch of technology involved in the development and operation of computers. The main functional elements of modern computing machines, or computers, are made on electronic devices, which is why they are called electronic computers - computers. According to the method of presenting information, computers are divided into three groups;
Analog 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
variable (numbers) expressed by a combination of discrete values
values of any physical quantity (number);
Hybrid computers (HCM), which include
In many cases, both methods 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. The improvement of electrically driven mechanical adding machines continued. 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. Bebij: difference (1822) and analytical (1830).
The difference engine was intended for tabulating polynomials and, from a modern point of view, was a specialized computer with a fixed (rigid) program. The machine had “memory” - several registers for storing numbers. When a specified number of calculation steps were completed, the counter for the number of operations was triggered and a bell rang. The results were printed out on a printing device. Moreover, this operation was combined with calculations in time.
While working on the difference machine, Babidge came up with the idea of creating a digital computer to perform a variety of scientific and technical calculations. Working automatically, this machine carried out a given program. The author called this machine analytical. This machine is a prototype of modern computers. The Babidzh Analytical Engine included the following devices:
For storing digital information (now called
stored by a storage device);
To perform operations on numbers (now this is
arithmetic device);
A device for which Babidge did not come up with a name
tion and which controlled the sequence of actions of the ma
buses (now this is a control device);
For input and output of information.
As information carriers for input and output, Babidge proposed to use perforated cards (punched cards) of the type that are used to control a weaving loom. Babidzh provided for entering into the machine tables of function values with control. The output information could be printed and also punched out on punched cards,
which made it possible to reintroduce it into the machine if necessary.
Thus, Bebage's Analytical Engine was the world's first software-controlled computer. The world's first programs were compiled for this machine. 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 the one developed at the University of Pennsylvania in the USA. This ENIAC machine was built in 1945 and had automatic program control. The disadvantage of this machine was the lack of a memory device to store commands.
The first computer with all the components of modern machines was the English EDSAC machine, built in 1949 at the University of Cambridge. The memory of this machine contains numbers (written in binary code) and the program itself. Thanks to the numerical form of recording 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 executed 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 leadership 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 data information and programs (algorithms).
A profamma in machine language looks like a table of numbers, each line corresponds to one operator - a machine command. In this case, in a command, for example, the first few digits are the operation code, i.e. tell the machine what to do (add, multiply, etc.), and the remaining numbers indicate exactly where in the machine’s memory the required numbers (additions, factors) are located and where the result of the operations (sum of products, etc.) should be remembered. .
A programming language is defined by three components: alphabet, syntax and semantics.
Most programming languages (BASIC, FORTRAN, PASCAL, ADA, COBOL, LISP) developed so far are sequential. The programs written on them represent a sequence of orders (instructions). They are sequentially, one after another, processed on the machine using so-called translators.
The performance of computers will be increased due to parallel (simultaneous) execution of operations, while most existing programming languages are designed for sequential execution of operations. Therefore, the future, apparently, lies in programming languages that will make it possible to describe the problem being solved itself, and not the sequence of execution of operators.
Self-test exercises
Development... of instruments in the history of mathematics counting
matics happened gradually. From is
using parts of one's own body - fingers
...
- to the use of various special- abacus
ally created devices: ... linear- logarithmic
ka, abacus, ..., analytical engine and computing
electronic ... machine.
Programs for... machines are electronic-computing
tables of numbers. body
Components of programming languages
tions are the alphabet, ... and semantics. syntax
§ 7. Formation, current state and prospects
developed methods of teaching the elements of mathematics to children
preschool age
Issues of mathematical development of preschool children have their roots in classical and folk pedagogy. Various counting rhymes, proverbs, sayings, riddles, and nursery rhymes were good material for teaching children to count and allowed the child to form concepts about numbers, shape, size, space and time. For example,
Gave to this, Gave to this, And gave to this, But did not give to this:
You didn’t carry water, you didn’t chop wood, you didn’t cook porridge - you have nothing.
The first printed educational book by I. Fedorov, “The Primer” (1574), included thoughts on the need to teach children to count through various exercises. Questions about the content of methods of teaching mathematics to preschool children and the formation of their knowledge about size, measurement, time and space can be found in the pedagogical works of Ya.A. Comenius, M.G. Pestalozzi, K.D. Ushinsky, F. Frebel, L.N. Tolstoy and others.
Thus, J. A. Komensky (1592-1670) in the book “Mother’s School” recommends that even before school, teach the child to count within twenty, the ability to distinguish between large and small numbers, even and odd, to compare objects by size, to recognize and name some geometric figures, use units of measurement in practical activities: inch, span, step, pound, etc.
The classical systems of sensory learning by F. Frebel (1782-1852) and M. Montessori (1870-1952) present a method for introducing children to geometric shapes, quantities, measurement and counting. The “gifts” created by Froebel are still used as didactic material to familiarize children with number, shape, size and spatial relationships.
K.D. Ushinsky (1824-1871) repeatedly wrote about the importance of teaching children to count before school. He considered it important to teach a child to count individual objects and their groups, perform addition and subtraction operations, and form the concept of ten as a unit of counting. However, all this was just wishes without any scientific basis.
Issues of methods of mathematical development acquire particular importance in the pedagogical literature of primary schools at the turn of the 19th-20th centuries. The authors of the methodological recommendations were then advanced teachers and methodologists. The experience of practical workers was not always scientifically substantiated.
nom, but it was tested in practice. Over time, he improved, and progressive pedagogical thought emerged stronger and more fully in him. At the end of the 19th and beginning of the 20th centuries, methodologists had a need to develop a scientific foundation for arithmetic methods. 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 methods 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. Kemnitz published a methodological manual “Mathematics in Kindergarten” (Kiev, 1912), where the main methods of working with children are conversations, games, and practical exercises. The author considers it necessary to introduce children to such concepts as: one, many, several, pair, more, less, the same, equally, equal, the same etc. The main task is to study numbers from 1 to 10, with each number being considered separately. At the same time, children learn operations on these numbers. Visual material is widely used.
During conversations and activities, children gain knowledge about shape, space and time, about dividing a whole into parts, about quantities and their measurement.
Questions about the methods and 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 hotly 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. In the book “The Development of Numerical Concepts in Early Childhood” (Kiev, 1923), the author came to the conclusion that the first ideas about numbers within 5 arise in children on the basis of distinguishing groups of objects and the perception of sets. And then, beyond these small aggregates, the main role in the formation of the concept of number belongs to counting, which displaces the simultaneous (holistic) perception of sets. At the same time, he considered it desirable for the child to acquire knowledge during this period “invisibly,” on his own. K.F. Lebedintsev came to this conclusion based on observations of children’s assimilation of the first numerical concepts and mastery of them
account. Children actually very early begin to identify some small groups of homogeneous objects and, imitating adults, call this a number. But this knowledge is still shallow and not sufficiently conscious. Children's ability to name numbers is not always an objective indicator of mathematical ability. And yet, in the 20s, many methodologists and educators accepted the point of view of K.F. Lebedintsev. In their opinion, numerical ideas arise in a child mainly due to the holistic perception of small groups of homogeneous objects located in the environment (arms, legs, table legs, car wheels, etc.). On this basis, it was considered unnecessary to teach children to count.
However, advanced “preschool” teachers in the 20-30s (E.I. Tikheeva, L.K. Shleger, etc.) noted that the process of forming numerical concepts in children is very complex, and therefore it is necessary to purposefully teach them to count. Play was recognized as the main way to teach children to count. Thus, the authors of the book “Living Numbers, Living Thoughts and Hands at Work” (Kyiv, 1920) E. Gorbunov-Pasadov and I. Tsunzer wrote that a child tries to introduce into his activity - play - what is interesting to him at the moment. Therefore, familiarization with the elements of mathematics should be based on the active activity of the child. It was believed that by playing, children better master counting and become better acquainted with numbers and operations with them.
Most teachers of the 20s and 30s had a negative attitude towards the need to create programs for kindergarten and targeted teaching. In particular, L.K. Shleger argued that children should freely choose their own activities according to their own wishes, i.e. everyone can do what he intended, 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 children's self-learning. L.K. Shleger believed that counting should be combined with various types of child activities, and the teacher should use various moments from the children’s lives to practice counting.
Olga Perkova
Types of written numbering (presentation)
Types of written numbering.
The development of counting began at a time when forms of production such as hunting and fishing became familiar to man. It became necessary to make tools to master these industries. And having moved to cold countries, people began to make tools that could easily be used to process durable leather.
Finger counting.
Counting began to develop faster from the time when people figured out to use their fingers. It is they who have become so simple and at the same time unique "apparatus", which laid the foundation for further development written numbering.
There was, of course, verbal counting, but it only became active after agriculture developed.
Over time, many peoples began to come up with various words for names, which were assigned to numbers. For example, if it was necessary to designate the number one, then it was designated as "nose". "mouth", "head" (what a person has in one quantity). Accordingly, the number two is associated with the words "eyes", "hands", "legs" etc.
Finger counting gradually led to the fact that the counting began to be ordered, and the person, accordingly, verbally simplified the numbers. Let's say the expression that corresponded to the number 13 - "ten toes on both feet and three fingers on one hand"- simplified in "finger on hand"; to express the number 26, instead of the words “ten toes on both feet, ten fingers on both hands and three toes on the other person’s foot” it was said otherwise: "three fingers of another person".
The emergence of number systems
The transition of man to finger counting led to the creation of several different number systems.
The most ancient of the finger number systems is considered to be fivefold. This system originated and spread in America.
The further development of number systems followed two paths. The tribes that did not stop at counting on the fingers of one hand moved on to counting on the fingers of the second hand and then on the toes.
The natural unit of the highest rank during the emergence of the decimal system was "Human" as the owner of 20 fingers. In this system, 40 is expressed as "two people", 80 – "four people" and so on.
Thus, humanity gradually created its own methods of calculation and reached the point when the method that modern mathematics uses appeared.
Numbering in Rus'.
The first Russian monument of mathematical content to this day is considered to be a handwritten work by the Novgorod monk Kirik, written by him in 1136.
By the 16th century refers to the invention of a remarkable calculating device, which later received the name "Russian abacus"
Written The number system has undergone many changes
with the development and formation of human society, with a smooth transition from ancient man to modern personality.