Dual number system. Binary numbers: deuce number system deuce number system with butts

A number system is a set of methods and rules for naming and assigning numbers. Mental signs that are used to assign numbers are called numbers.

All numerical systems should be divided into two classes: non-positional and positional.

In positional numerical systems, the number of skin digits varies depending on the position (position) of the sequence of digits that represent the number. For example, in the number 757.7, the first semka means 7 hundreds, the other - 7 units, and the third - 7 tenths of a unit.

And the writing of the number 757.7 itself means a shorthand notation of virazu:

In non-positional number systems, the number of digits (or the input that must be made for the value of the number) does not lie in its position in the number record. So, in the Roman system of numbers in the number XXXII (thirty-two), the number X in any position is equal to just ten.

Historically, the first numerical systems were the most non-positional systems. One of the main shortcomings is the difficulty of writing large numbers. The recording of large numbers in such systems is either very cumbersome, or the alphabet of the system is extremely large. The use of a non-positional number system, which is widely stagnant at this time, may be the so-called Roman numbering.

Dual number system, then. The system is based on a “minimal” system, in which the principle of positionality in the digital form of recording numbers is implemented. In the two-dimensional numerical system, the value of the skin digit “behind the place” increases twofold during the transition from the younger category to the older one.

The history of the development of the two-digit number system is one of the brightest stories in the history of arithmetic. The official “nativity” of double arithmetic is associated with the names of G.V. Leibniz, who published a paper in which the rules for establishing all arithmetic operations on double numbers were examined.

Leibniz, however, did not recommend two-digit arithmetic for practical calculations instead of the tens system, but emphasized that “calculation using additional twos, such as 0 and 1, is fundamental for science and gives rise to new , which appear in brown weather, in the practice of numbers, and especially in geometry: the reason for this is the fact that when numbers are reduced to the simplest beginnings, such as 0 and 1, a miraculous order appears here.

Leibniz introduced the two-system system in a simple, manual and beautiful way. Vin said that “the calculation of two... is fundamental for science and gives rise to new ideas... When numbers are reduced to the simplest cobs, which are 0 and 1, a miraculous order appears.”

In honor of the “dyadic system” – that’s how the dyadic system was then called – a medal was knocked out. It displayed a table with numbers and simple actions with them. Along the edge of the medal there was a line with the inscription: “To bring everything out of nothing, just one is enough.”

Then they forgot about the dual system. For nearly 200 years, no work has been seen on this topic. They turned to it only in 1931, when the feasibility of a practical stagnation of a double number was demonstrated.

Leibniz's brilliant transference arose only two and a half centuries later, when the famous American scientist, physicist and mathematician John von Neumann, introduced the two-dimensional number system as a universal way of encoding information in electronic computers ("John von Neumann's principles").

2. It is widely implemented in digital electronics and is used in most current computing devices, including computers, mobile phones and various types of sensors. We can safely say that the technologies of our time are based on binary numbers.

Writing numbers

Any number, no matter how large, is written in the two-digit system using two additional symbols: 0 and 1. For example, the number 5 of all known tens systems in the two-digit system will be represented as 101. Binary numbers can be you are designated by the prefix 0b or ampersand (&), for example: &101.
In all number systems, except tens, the symbols are read one by one, so if you take 101 into account, it is read as “one zero one.”

Transfers from one system to another

Programs that regularly work with the two-digit number system can convert binary numbers to tens on the fly. This can be done effectively without any formulas, especially as people know how the smallest part of the computer “brain” works - bit.

The number zero also means 0, and the number one in the two-digit system will still be one, but what else can we do if we run out of numbers? The ten system “prompted” to introduce the term “ten” at this time, and in the binary system it would be called “two”.

As 0 is &0 (ampersand is the symbol of the two-system), 1 = &1, 2 will be symbolized as &10. A three can also be written in two digits, as you can see &11, then one two and one one. Possible combinations are drawn up, and in the tens system at this stage hundreds are obtained, and in the twos system - “fours”. Chotiri – tse &100, five – &101, six – &110, sem – &111. It’s coming, more than one rakhunku is the same.

You can note the peculiarity: while in the tens system the digits are multiplied by ten (1, 10, 100, 1000 and so on), then in the double system they are multiplied by two: 2, 4, 8, 16, 32. This corresponds to the size of flash cards other accumulated people who are victorious in computers and other devices.

What is binary code?

Numbers presented in the binary numeral system are called binary, but in this view you can also see non-numeric meanings (letters and symbols). In this way, words and texts can be encoded in numbers, although the appearance of the word is not so laconic, and even to write just one letter you need a number of zeros and ones.

How are computers able to read such a large amount of information? In reality, everything is simpler, as it seems. People who lived before the tenth number system usually convert double numbers into larger ones, and then work with them in some kind of manipulation, and the basis of computer logic initially lies in the binary system of numbers. One in the equipment indicates high voltage, and zero - low, and for one there is voltage, and for zero the day is off.

Binary numbers in culture

We appreciate the mercy, which is the merit of modern mathematicians. Although binary numbers are fundamental in the technologies of our time, they have been used for a long time, and in different parts of the planet. A long line (one) and a level (zero) are used to encode all the symbols that mean all the elements: sky, earth, grim, water, mountains, wind, fire and water (mass of water). This analogue of 3-bit numbers was described in the classic text of the book Change. Trigrams became 64 hexagrams (6-bit digits), the order of which in the Book of Changes is expanded to double digits from 0 to 63.

This order of additions was made in the eleventh century by the Chinese scientist Shao Yun, although there is no evidence that he effectively understood the two-fold number system as a whole.

In India, even before our era, binary numbers were also used in the mathematical basis for describing poetry, compiled by the mathematician Pingala.

The Ink script (Kipu) is considered to be the prototype of modern databases. They themselves first stagnated not only the binary code of the number, but not the numeric records in the two-system. Stos is characterized not only by primary and additional keys, but also by substitution of positional numbers, coding for additional colors and series of repetitions of data (cycles). The Incas first established a method of maintaining an accounting system, which is called a permanent entry.

First from programmers

The double number system is based on the numbers 0 and 1, described by the famous physicist and mathematician Gottfried Wilhelm Leibniz. We buried the ancient Chinese culture and the traditional texts of the book of Changes, noting the similarity of hexagrams to binary numbers from 0 to 111111. We buried evidence of similar achievements Philosophy and mathematics at that time. Leibniz can be called the first of the programmers and information theorists. Having discovered that if you write groups of double numbers vertically (one below the other), then in the vertical stacks of numbers that come out, zeros and ones are regularly repeated. This prompted me to admit that it was possible to create completely new mathematical laws.

Leibniz understands that binary numbers are optimal for stagnation in mechanics, which is based on the change of passive and active cycles. It was the 17th century, and this great teaching of the Vinaishas on the paper of the calculating machine, which worked on the basis of new discoveries, was evidently clear that civilization had not yet reached such technological development, and at this hour The creation of such a machine would be impossible.

Lesson plan

Here you find out:

♦ how to work with numbers;
♦ what is an electronic table;
♦ how calculations are carried out;
♦ for additional electronic tables;
♦ how can you vikorist electronic tables for information modeling.

Dual number system

Main topics of the paragraph:

♦ ten and two number system;
♦ the form for writing the number is opened;
♦ conversion of two numbers to the tens system;
♦ converting tens numbers to the two system;
♦ arithmetic of two numbers.

Whose section has information about the organization to calculate the computer. The payments are related to savings and the calculation of numbers.

The computer works with numbers using a two-digit number system.

This idea belongs to John von Neumann, who formulated the principles for organizing the EOM in 1946. It is clear that this is a numerical system.

Tens and twos number systems

The number system, or shortened version of SS, is a system of recording numbers that involves dialing numbers.

You learned about the history of various numerical systems when you studied the 7th section of the handbook. And today with you we express our respect for such numerical systems as the two tens of the SS.

As you already know from the material learned earlier, one of the most common stagnant number systems is ten SS. And this system is called so because at the heart of this wording is the number 10. That’s why the number system itself is called ten.

You already know that in this system there are ten numbers such as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Ten digits are the basis of this numerical system.

And the axis of the two-digit number system involves only two digits, such as 0 and 1, and the basis of the system is the number 2.

Now let's try to figure out how to figure out the value with just two numbers.

Opened form for writing numbers

Let's go back to our memory and guess what is the principle of recording numbers in the tenth SS. It will no longer be a secret to you that in such a SS record of the number lies in the place where the numbers are expanded, so that their position is visible.

So, for example, a number, which is the extreme right-hander, tells us about the number of ones of the number that follows this number, as a rule, indicates the number of twos, etc.

If we, for example, take a number like 333, then it is important that the rightmost number represents three ones, then three tens and then three hundreds.

Now it is imaginable to look at such zeal:

Here there is a lot of jealousy, in whom the right-handed hand has been expanded into one sign, given as a kindled form of recording this rich-valued number.

Let's look at another example of a rich tenth number, which is also represented by the flared form:

Translation of two numbers into the tens system

Now let’s take, for example, such a rich double number as:

This rich-valued number has a two on the bottom right side, which indicates the basis of the number system. Then we realized that we have a number two in front of us and it is no longer possible to confuse it with tens.

І the value of the cutaneous figure in the double number increases twice with the cutaneous cut from the right hand to the left. Now let's marvel at how the form of writing the double number appears:

How can we convert the number two into the tens system?

Now let's look at some examples of converting double numbers to the tenth number system:

This example shows those that have a two-digit tenth number, at times, it suggests a six-digit two-digit number. The double system is characterized by an increase in the number of digits as the value of the number increases.

Now let’s marvel at how we see the beginning of the natural series of numbers in the tenth (A10) and twofold (A2) SS:

Converting tens numbers to the two system

Having looked at the butt further, I now understand that you are about to convert the double number to the exact tenth number. Well, now let’s try to make a turnaround. I wonder what we need to earn for this. For such a translation, we need to try to divide the tenth number into additional ones, which are the steps of two. Let's look at this example:

Yak bachimo, it’s not so easy to do it. Let’s try to look at another, more simple method of converting from tenths SS to twos. This method is based on the fact that the number in the ten is usually divisible by two, and the excess is removed and will be the youngest digit of the number being found. Once again, the number is removed and again divided by two and the offensive rank of the number being found is removed. We will continue this process until the privacy of the system becomes less than two. The axis is also kept private and will be the highest digit of the number, as we joked about.

Let's now look at the method of writing half by two. For example, let’s take the number 37 and try to convert it to the two-digit system.

On the butts of the two, a5, a4, a3, a2, a1, a0 are the designated digits of the double number, which occur in order from left to right. As a result, we take away from you:

Arithmetic of double numbers

If we go by the rules in arithmetic, it is easy to note that in the two-digit system the number of notes is much simpler, lower than tens.

Now let's guess the variants of folding and multiplying single-digit double numbers.

Due to its simplicity, which is easy to use with the bit structure of computer memory, the two-digit number system won the respect of computer creators.

Pay attention to how the butt of adding two rich-valued double numbers is calculated for the help of a counter:

And the axis in front of you is a multiplication of richly-valued two-digit numbers in a stack:

You noted how easy it is to simply remove such butts.

Briefly about smut

Number system - the rules for recording numbers and related to these rules are the ways of counting.

The basis of the numerical system is the number of its digits.

Double numbers are numbers in the double number system. This entry has two numbers: 0 and 1.

The form for writing a two-digit number is opened - this is the number given in the form of the sum of the steps of the two, multiplied by 0 or 1.

The number of double numbers in a computer is related to the bit structure of the computer memory and the simplicity of double arithmetic.

Advantages of the double number system

Now let’s take a look at the advantages of the two-tier calculation system:

First of all, the advantage of the two-dimensional number system is that it helps to simply carry out the processes of saving, transferring and processing information on a computer.
In other words, for this witchcraft, not ten elements are sufficient, but only two;
Thirdly, the display of information requires only two stages, which is more reliable and stable to a large extent;
Fourthly, there is the possibility of using the logic of algebra to create logical transformations;
By the way, two's arithmetic is still simpler than ten's arithmetic, which is more manual.

Few parts of the two-digit number system

The two-digit system of numbers is less simple, because people are more inclined to use the tens system, since it is much shorter. And axis, the double system has a large number of discharges, which is also a small part.

Why is the two-count number system so wide?

The two-digit number system is popular because it uses numerical techniques, where the skin number may be represented on the physical nose.

Even if it’s simpler than two stages when a physical element is prepared, you can see the device, in which there may be ten different stages. Wait, it would be much more complex.

In fact, this is one of the main reasons for the popularity of the two-digit number system.

History of the twin number system

The history of the creation of the two-dimensional number system in arithmetic, dosit yaskra i strimok. The founder of this system is considered to be the famous German mathematician G. V. Leibniz. He published a paper in which he described the rules by which various arithmetic operations on double numbers could be performed.

Unfortunately, until the beginning of the 20th century, the deuce number system was of little use in applied mathematics. And after simple healing mechanical devices began to appear, they began to actively apply more and more attention to the two-digit numerical system and began to actively develop them, the fragments for calculation devices were created manually and without noyu. This is a minimal system, with the help of which it is possible to fully implement the principle of positionality in the digital form of recording numbers.

Food and food

1. Name the advantages and shortcomings of the two-digit number system, which is equal to the tenth.
2. Which double numbers represent the next tens numbers:
128; 256; 512; 1024?
3. Why are these two numbers equal in the tens system:
1000001; 10000001; 100000001; 1000000001?
4. Convert the following two-digit numbers to the tens system:
101; 11101; 101010; 100011; 10110111011.
5. Convert the numeral system from the two to the next tenth number:
2; 7; 17; 68; 315; 765; 2047.
6. Enter the additions to the two-fold number system:
11 + 1; 111 + 1; 1111 + 1; 11111 + 1.
7. Calculate the multiplication for the double number system:
111 10; 111 11; 1101 101; 1101 · 1000.

I. Semakin, L. Zalogova, S. Rusakov, L. Shestakova, Computer Science, 9th grade
Submitted by readers from Internet sites

Dual number system Today, vikoryst is available on almost all digital devices. Computers, controllers and other computing devices generate calculations in the dual system. Digital devices for recording and producing sound, photos and videos save and process signals in a two-digit number system. The transmission of information via digital channels also uses the vikoryst model of the two-dimensional number system.

The system is called this because the basis of the system is the number two ( 2 ) or in a dual system 10 2 - this means that for the display of numbers, only two digits “0” and “1” are used. The two is written at the bottom right of the number, and here is the significant basis of the number system. For the tenth system, do not indicate the basis.

Zero - 0 ;
One - 1 ;

What else are you going to do? All numbers have expired. How to represent the number two? In the tens system, in a similar situation (when the digits have run out), we introduced the concept of ten, immediately we are tempted to introduce the concept of “two” and, say, two - not one two and zero one. But you can also write it down as “10 2”.

Otje, Two - 10 2 (one two, zero one)
Three - 11 2 (one two, one one)

Chotiri - 100 2 (one four, zero twos, zero one)
Five - 101 2 (one four, zero two, one one)
Six - 110 2 (one four, one two, zero one)
Sim - 111 2 (one four, one two, one one)

The possibilities of three ranks have been exhausted, we introduce a larger one - the scale (we are mastering a new rank).

Everything - 1000 2 (one weight, zero four, zero two, zero one)
Nine - 1001 2 (one weight, zero four, zero two, one one)
Ten - 1010 2 (one weight, zero four, one two, zero one)
...
and so on...
...

Next, if the possibilities of the discharge discharges, to display the date of the day, are exhausted, we introduce more units of the scale, then. affects the offensive discharge.

Let's look at the number 1011 2 is written in the two-digit number system. You can say about him what to avenge: one weight, zero four, one two and one one. And you can deduce this value through the numbers, so you can enter it in the next step.

1011 2 = 1 *8+0 *4+1 *2+1 *1, here the sign * (star) means multiplication.

The series of numbers 8, 4, 2, 1 is nothing more than the whole stage of the number two (substitutes of the number system) and we can write this:

1011 2 = 1 *2 3 +0 *2 2 +2 *2 1 +2 *2 0

A similar rank for a double fraction (fractional number), for example: 0.101 2 (five-eight), you can say about him what to take revenge on: one for a friend, zero quarters and one eighth. This value can be calculated as follows:

0.101 2 = 1 *(1/2) + 0 *(1/4) + 1 *(1/8)

I here is a series of numbers 1/2; 1/4 and 1/8 is nothing more than two steps and we can also write:

0.101 2 = 1 *2 -1 + 0 *2 -2 + 1 *2 -3

For the mixed number 110.101 we can similarly write:

110.101 = 1 *2 2 +1 *2 1 +0 *2 0 +1 *2 -1 +0 *2 -2 +1 *2 -3

Let's number the digits of the whole part of the double number, right hand to left, like 0,1,2...n (numbering starts from zero!). And the shotgun discharges, left-handed, are -1, -2, -3...-m. Then the value of a double number can be calculated using the following formula:

N = d n 2 n +d n-1 2 n-1 +…+d 1 2 1 +d 0 2 0 +d -1 2 -1 +d -2 2 -2 +…+d -(m-1) 2 -(m-1) +d -m 2 -m

De: n- the number of discharges for the whole part of the number minus one;
m- number of discharges in the shot part of the number
d i- the number is what it costs i-th category

This formula is called layout formula double number, then. number recorded in the double number system. If this formula has the number two, replace it with an abstract number q, then we remove the decomposition formula for the number written in qth number systems:

N = d n q n +d n-1 q n-1 +…+d 1 q 1 +d 0 q 0 +d -1 q -1 +d -2 q -2 +…+d -(m-1) q - (m-1) +d-m q-m

With the help of this formula, you can calculate the values of a double number and a number written in any other positional number system. We recommend reading the following articles about other numerical systems.

It’s easy to send your money to the robot to the base. Vikorist the form below

Students, post-graduate students, young people, who have a strong knowledge base in their new job, will be even more grateful to you.

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In the face of computer theory, computer programs sometimes forget about the role that numerical systems played in the history of computers. Even the first medical devices (abacus and arithmometers), prototypes of modern computers, began to be created long before the algebra of logic, the theory of algorithms - and the number systems themselves played a leading role in their creation. About this memory track, forecasting the development of computer technology.

1. Similar history of the development of numerical systems

In the early days, people were unable to grasp the development of marriage. The first peoples do not have an advanced numerical system. Back in the 19th century, the rich tribes of Australia and Polynesia had only two numbers: one and two; The numbers they added were: 3 – two – one, 4 – two – two, 5 – two – two – one and 6 – two – two – two. They said “richly” about all the numbers, great 6, without individualizing them. This is not yet a rakhunok, but rather a germ.

Over the years, the idea of small groups interacting with each other has developed; Vinikli words poznachen understand “chotiri”, “five”, “six”, “sim”. The remaining word trivaliy hour also meant an unmistakably great quantity. Our arrivals have preserved the memory of this era (“seven times you see - once you see”, “seven nannies have a child without an eye”, “this trouble - one answer”, etc.).

During the reign of the Maury and Guptian dynasties (IV - II centuries BC - VIII century AD), the Indians created a dozen numerical systems, constantly depicting numbers (later the names had a slightly changed appearance in Arabic).

One of the oldest number systems is the Egyptian hieroglyphic numbering, which dates back to 2500 - 3000 BC. e. There was a tenth non-positional number system, in which the principle of addition (numbers expressed in order by digits, standing, adding up) was stagnated for writing numbers. There were special signs for one, ten, one hundred and other tens digits up to.

With the development of the husband-and-government life, there was a need for the creation of numerical systems that would make it possible to carry out exchanges over larger boundaries and to designate ever larger sets of objects. For this person, she carved objects that were used for him, like tools: making notches on clubs and trees, tying bundles on skeins, putting fireplaces in shopping bags, etc. This type of calculation is called a unary number system, then. A number system in which any number record has at least one type of symbol. This is true, since the number of signs is immediately visually indicated and is associated with the number of objects that these signs represent. We all went to first grade and waited there, on healing sticks - all the way back to that distant era. Before the speech, from the shell behind the stones, you carry out your cob of carved, sophisticated tools, such as, for example, Russian shells, Chinese shells ("swan-pan"), the ancient Egyptian "abacus" (a board divided into smugs, where the tokens were placed). Similar instruments were used in many nations. However, in the Latin language the concept of “rakhunok” is expressed by the word “calculatio” (from our word “calculation”); and it’s similar to the word “calculus”, which means “little stone”.

Particularly important is the role of man’s natural tool – his fingers. This tool could not preserve the result of the crash for long, otherwise it would be “at hand” and suffer great fragility. The language of the first people was poor; The gestures symbolized the marriage of the words, and the numbers, which were also named, “appeared” on the fingers.

From now on, the supply of numbers was sufficiently expanded. At first, people became infected with a few dozen and then reached a hundred. For many rich peoples, the number 40 has long been a boundary of great importance. In Russian language the word “centipede” has the sense “richness”; The expression “forty forty” meant, for the old hours, the number that will exceed in the future.

At the beginning of the year, the rocket reaches a new boundary: ten tens, and a name is created for the number 100. Suddenly the word “hundred” evokes the sense of an unidentifiably great number. The same sensation is then evoked successively by the numbers one thousand, ten thousand (in the old days this number was called “darkness”), and one million.

At the cordon stage, the term “infinity” is used, which means a specific number.

2. History of the twin number system

A number system is a set of methods and rules for naming and assigning numbers. Mental signs that are used to assign numbers are called numbers.

All numerical systems should be divided into two classes: non-positional and positional.

And the writing of the number 757.7 itself means a shorthand notation of virazu:

3. Recording a number in the double system

The fewer symbols - digits in one digit for recording in the double system, the more digits are needed to indicate that number. Let’s take, for example, the number 8. The two-digit system requires four digits: 1000.

Now let’s take another entry from the dual system – 1111. The most important, the remaining number will still be one. Ale is already in the highest category - the greater for it is twice the greater and means 2, the third is again twice the greater - 4, the fourth is equal - 8.

Let's try to write down some number, for example 1017, in a double system. For this, as in the tens system, we break it down into ranks, but the ranks here look different. Of course, from the bottom, from 7. The fragments in the double skin system have a discharge that is twice as large for the attack, the number 7 will be written as the sum of three double discharges: 7 = 4+2+ 1 (1 for 2 times less than 2; 2 for 2 times less than 4) . Middle 7 one four, one two, one one: 7=4+2+ 1. This entry can be made in another way: 1*22+ 1*21 + 1. Also, for each of these categories we put 1-111.

Then comes the number 10. It is made up of one digit and one two: 10 = 8 +2 = 1 * 23 + 0 * 22 + 1 * 21 + 0 * 20. Please note, there are not many digits of one and four, so there are them we put zeros and write the number like this: 1010.

All forward discharges can also be distributed. Then the whole number 1017 will be written as 512 + 256 + 128 + 64 + 32 + 16 + 8 + 1 = 1 * 29 + 1 * 28 + 1 * 27 + 1 * 26 + 1 * 25 + 1 * 24 + 1 * 23 + 0*22 + 0*21 + 1*20 c. We write down the discharges and subtract 1,111,111,001.

We know the basics of the two-system system, which is unimportant through the tradition of operating first and through the tenth system. The double system is used only by calculating machines. The machine overinsurances zeros and ones with very high speed.

The advantages of the two-dimensional numerical system lie in the simplicity of implementing the processes of saving, transferring and processing information on a computer:

1. For its implementation, the required elements are from two possible countries, and not from ten.

2. Providing information appears to be reliable and reliable in two ways.

3. The possibility of stagnating the algebra of logic before concluding logical transformations.

4. Two's arithmetic is simpler than ten's arithmetic.

Few parts of the two-digit number system.

Also, the code of the number written in the double number system is a sequence of 0 and 1. Large numbers take up a large number of digits.

The rapid increase in the number of ranks is the largest shortcoming of the two-tier numerical system.

Visnovok

Dual computer coding

People have been using the ten system, perhaps, since ancient times they have been using their fingers, but even before computer technology and EOM, the two number system has a number of advantages over other systems, because for its implementation, the required technical devices require only two stable conditions (e strum - no strum, magnetic - non-magnetic, etc.), and not, for example, with ten, as with ten; Submission of information appears to be reliable and reliable; the possibility of using the Boolean algebra apparatus for the creation of logical transformations of information; Two's arithmetic is simpler than ten. Prote, the shortcomings of the double system - the increase in the number of discharges, the necessary recording of numbers.

Today, the double number system itself is being used to encode and encrypt information. Of all existing numerical systems, the two-fold numerical system is the most manual and essential in computer technology and EOM.

List of Wikilists

1. Bobinin V.V. “Lectures on the history of mathematics” (“Physical and Mathematical Sciences”, vol. IX and X, lectures 2-6);

2. Bobinin V.V. "Research on the history of mathematics" (Vip. II, M., 1896).

3. Vigodsky M.Ya. Adviser of Elementary Mathematics, M.: State University of Technical and Theoretical Literature, 1956.

4. Rolich Ch.M. - View 2 to 16, Minsk, “Vishcha School”, 1981

5. Fomin S.V. Systems of numbers, M.: Nauka, 1987.

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