Basic understanding of numerical systems

The numerical system is a set of rules and when you write numbers behind an additional set of digital signs. The number of digits required to record a number in a system is called by the system of numbers. When the system is entered, it will be written in the right numbers in the lower index :; ; etc.

There are two types of numerical systems:

position, if the value of the skin digit of the number is indicated by the position in the record of the number;

non-position, if the value of the digit in the number does not lie in the record of the number.

The application of a non-positional number system є Roman: numbers IX, IV, XV, etc. By the application of the positional system, the number is ten system, as it is victorious all over the place.

If only the number in the position system can be written in the polynomial form:

de S - the basis of the numerical system;

Digits of the number written in the data of the numerical system;

n - number of rows of the number.

Butt. number be written in the polynomial form with the next rank:

See the numerical systems

Roman numeral system is a non-positional system. In order to write down numbers, use the Latin alphabet. At the same time, the letter I always means one, the letter - V five, X - ten, L - fifty, C - one hundred, D - five hundred, M - one thousand, etc. For example, the number 264 will be recorded at the CCLXIV viglyad. When writing numbers in the Roman system, the numerical values ​​of the number є are algebraic sum of numbers, which should be included. With a number of digits in the recording of the number, it follows, as a rule, in the order of decreasing the value, and it is not allowed to write down the order of more than three of the same digits. At the same time, if the number with the great values ​​is followed by the number for the smallest, the contribution to the value of the number in the whole is negative. Typical butts, how to create the basic rules for writing numbers in the Roman numeral system, guided in the tables.

Table 2. Writing numbers in the Roman numeral system

Not much of the Roman system є the presence of formal rules for writing numbers і, apparently, arithmetic diy with large-valued numbers. Through the lack of handiness and great foldability in the Danish hour, the Roman system of numerical vikorisovyutsya there, pretty efficiently by hand: in the literature (numbering of chapters), in the issued documents (gray passports, valuable papers in the row and in.), In decorative plates vipadk_v.

Desyatkov's system of numbers - in Danish hour of the most vidom and vicoristovuvan. The winners of dozens of systems and numbers are brought to the attention of the heads of the people. Without it, it’s unlikely that I could have been able to understand, but the technology is more likely to fail. The reason why the ten system of numbers has become widely accepted is not mathematical. People sounded vvazhati in dozens of numerical systems, so they have 10 fingers on their hands.

The ancient image of dozens of digits (Fig. 1) is not vague: a skin number means the number for a number of kutiv in niy. Napryklad, 0 - kutіv nemaє, 1 - one kut, 2 - two kuti, etc. The writing of dozens of numbers has become aware of hundreds of changes. The form, like mi koristuєmosya, arose in the XVI century.

The Desyatkova system first appeared in India approximately in the 6th century of the new era. The indian numbering vikorisovat nine numerical symbols and zero for the meaning empty position. In the early Indian manuscripts, which came to us, the numbers were written down in a ringing order - the most significant number was put on the right. Ale nezabarom became the rule of roztashovuvat taku figure from the left side. Especially the meaning was given to the zero symbol, which was introduced for the positional system of meaning. Indіyska numbering, including zero, went until our hour. In Europe and India, dozens of arithmetic have been expanded on the ear of the XIII century. clerks to robots of the Italian mathematician Leonardo Pizansky (Fibonacci). The Europeans posed the Indian system of numbers among the Arabians, calling it the “Arab”. It’s historically incorrect to call it “flatten” and “pony”.

The desyatkov system is vikoristy ten digits - 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, as well as the symbols "+" and "-" for the sign of the number and to whom or a speck for the subset of the whole and fractional parts numbers.

In the numbering machines there is a double number system, the base is number 2. To write numbers in the whole system, only two digits - 0 and 1 appear computers, shche in the XVII - XIX centuries. The first publication of the discussion of the two systems and numbers is due to the Spanish priest Juan Karamuel Lobkovits (1670). The statute of the Nimetsian mathematician Gotfrid Wilhelm Leibnitz, published in 1703 r. Leibnits did not recommend a vicaristic system for practical calculations, but rather important for theoretical educators. With the help of a two-way system of numerical growth, we will be good at home and neglect development.

Vibir of two systems for storing in the numerical technology, it will be explained how the electronic elements are triggers, from which EOM microcircuits are stored, can be transferred to two working mills.

For the additional help of the two-way system of coding, it is possible to register both the knowledge and the knowledge. It’s easy to see if you can guess the principle of coding and transmission of information for additional Morse code. Telegraphist, vikoristovuchi only two symbols of the abetka - dots and dashes, you can transmit practically any text.

The dviykova system is handy for a computer, but not handy for people: the numbers to go around are very important and important to record and memorize. Surely, you can translate a number into a dozen system and write it down in such a view, and then, if you need to translate it back, albeit all the shifting work. To this, the system of numbers becomes stagnant, the dispute is two-fold - eightfold and shistnytsyatkov. To write numbers in cix systems, 8 and 16 digits are required. The 16-ternary first has 10 zagalny digits, and the distance is great Latin letters. The sixteen-digit A is the decimal number 10, the hexadecimal B is the tenth number 11, etc. D. Vikoristannya of the systems will be explained by the way that you go to write the number in any of the number of two systems. Below is a table of the type of numbers recorded in the latest systems.

Table 3. Variety of numbers recorded in different numerical systems

Rules for the transfer of numbers from the same system and number in іnshu

The transfer of numbers from one system of numerical to one becomes an important part of machine arithmetic. The basic rules of the transfer are clear.

1. For the conversion of a two-digit number in tens, it is necessary to write down the polynomial at the top of the view, to add the number of digits and the corresponding step of the number 2, and calculate it according to the rules of tens of arithmetic:

When shifting manually, follow the table of steps of two steps:

Table 4. Levels of number 2

Butt. Convert the number to the tens numerical system.

2. For the transfer of a large number in dozens, it is necessary to write down the polynomial at the top of the view, so that it can be added up from the creation of the digits of the number and of the given step of the number 8, and calculate it according to the rules of dozens of arithmetic:

When shifting, manually adjust the table of steps:

Table 5. Steps of number 8

To translate numbers from one system of numbers into one, it is necessary to provide basic information about the systems of numbers and the form of submitting numbers in them.

number s small numbers, scho get used to the system of numbers, are called as a base, or the base of the system of numbers. The zahal has a positive number X in a positional system with a base s can be presented at the viglyadі polіnoma:

de s- the base of the numerical system, - the numbers, admissible in the given numerical systems. After the end of the part X, And the last is a fraction of a shot X.

In the numerical technology, most of the definition was the binary (BIN - binary), and two-coded systems: octal (OCT - octal), sixteen (HEX - hexadecimal) and two-coded - ten (BCDded)

Nadal for the designated vicorist system, the numerical number will fall into the bows, and in the index, the system is assigned. number X by submission s will be recognized.

Dviykova numerical system

The number 2 ( s= 2) and to write down the numbers, only two digits must be entered: 0 and 1. If you want to see a double number, you can finish a physical element with two clearly different stuns, one of which is 1, and only 0.

First of all, take care of translating from any system of numbers to two, it is necessary to respectfully change the butt of recording the number in two systems of numbers:

If you don’t need to go into the theory, but if you don’t need to correct the result, then speed up Online calculator Conversion of whole numbers from tens of systems and numerical in інші systems .

Visimkova and Shistnachyatkova systems and numbers

The number of systems and numbers are reduced to two-coded ones, in which the basis of the number system is the purpose of the two steps: - for the large and - for the sixteen.

In a viscous system of numerical strength ( s= 8) 8 digits are used: 0, 1, 2, 3, 4, 5, 6, 7.

First, you need to take care of the transfer from any system and the number to visimkov, it is necessary to respectfully change the butt of the number recording in the visimkov system:

In sixteen systems of numerical strength ( s= 16) 16 digits should be selected: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.

Butt will write numbers in sixteen systems:

The wide range of octal and sixteen numerical systems is matched by two factors.

Accordingly, the systems allow replacing the recording of a double number with larger compact representations (the recording of the number in the large and sixteen systems will be displayed in 3 or 4 times and is shorter than the double number). In another way, instead of reconfiguring the numbers between the two-sided system from one side and the top and the sixteen-letter system, it is simple to work out from one side to the other. Surely, fragments for the octal number of leather rows are represented as a group of three two rows (triads), and for a sixteen-row number - as a group of two two rows (zoshiti), then for converting a two-number number, add 3 groups of 3 , see through from the Komi to the right and left. At the same time, if necessary, add zero evil from the whole part і / or the right-handed person from the fraction of the part and the skin to such a group - a triad or a tetrad - to replace the evalent octal or sixteen-digit tabular (div.

If you don’t need to go into the theory, but if you don’t need to correct the result, then speed up Online calculator Conversion of whole numbers from tens of systems and numerical in інші systems .

For a vocal shift of the skin OCT or HEX, the digit will be replaced by a triad or a notebook of predominant numbers, and insignificant zeros of evil and on the right are displayed.

For those who have seen it earlier, use the next rank:

If you don’t need to go into the theory, but if you don’t need to correct the result, then speed up Online calculator Conversion of whole numbers from tens of systems and numerical in інші systems .

Dviykoviy-tenyatkov numerical system

In a two-tenth system of a vag, a skin discharge is of a level 10, as in dozens of systems, and a ten-figure skin is coded in two-digit numbers. To record the tenth number in the BCD-system, replace the skin with a tenth digit of the equivalent four-digit two-digit combination:

Whether a dozen number can be represented in a two-tenth record, ale to the memory, but not a double equivalent of a number. Tse can be seen from this butt:

Transferring numbers from one system and number to one

hey X- number in the system of numerical base s, Yake, you can supply power to the system with the base h... Manually resolved two vipads.

At the first v h you can vikoristovuvati the arithmetic of the whole system. The method of converting the field into the given numbers in the view of the polynomial behind the steps s, And also in the numerical polynomial according to the rules of arithmetic of the system and numerical base h... So, for example, it is easy to switch to two or two systems and numbers up to a dozen. Descriptions of how to use the following tools:

.

.

In both types of arithmetic, follow the rules of the system and numerical base 10.

Another vipad () handily use arithmetic according to the present s... Here it is a matter of vrahovuvati, where the transfer of whole numbers and correct fractions is carried out according to different rules. When shifting the small fractions of the whole and the fraction of the fraction, they are translated skinnically according to their own rules, if the number is rejected, it is written through a coma.

Translation of whole numbers

The rules for the transfer of whole numbers have become clear from the alphanumeric formulas for recording a number in the correct position of the system. Don't worry about the number of the outgoing systems of numbers s maє viglyad. It is necessary to take away the record of the number in the numerical system with the base h:

.

For a known value, the value is divided by a polynomial by h:

.

The yak is visible, the youngest rozryad, tobto, dorіvnyuє to the first surplus. The upcoming significant class of visas starts on the private h:

.

Reshta is also counted by the way of private dots, until it becomes equal to zero.

To transfer an integer from the s-number system and the number to the h-number, it is necessary to change the number and the private number to h (according to the rules of the number system with the base h) to quiet pir, until the private does not become equal to zero. The older digit in the record of the number with the base h is the remaining surplus, and the advancing figures behind it are the surplus of the foregrounds, so that they are written in the last, zolotniy and rejected.

The calculator allows you to translate numbers and fractions from one system and number into one. The number of the system cannot be less than 2 or more than 36 (10 digits and 26 Latin letters after all). Dovzhina numbers are not guilty of changing 30 symbols. Use the symbol to enter fractional numbers. abo ,. To translate a number from one system into one, enter the number in the first field, after the current system and number in another and the number of systems, in which it is necessary to translate the number, in the third field, to write down the button.

cob number recorded in 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 -y systems of numbers.

I want to correct the writing of the number in 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 -y systems of numbers.

edit the record

Viconano shifts: 3036712

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numerical systems

The numerical systems are divided into two types: positionі NOT position... Mi koristuєmosya in the Arabic system, won positsіynoї, and є still roman - won’t yakraz not positsіyno. In positional systems, the position of the digit in the number is uniquely signified by the value of the number. It is easy to see, looking at the butt of a certain number.

butt 1... The number is 5921 in dozens of numerical systems. Numbered the number from the right to the left, correcting from zero:

The number 5921 can be written in the offensive view: 5921 = 5000 + 900 + 20 + 1 = 5 · 10 3 + 9 · 10 2 + 2 · 10 1 + 1 · 10 0. The number 10 is a characteristic, which is the basis of the numerical system. The number of steps taken is the value of the position of the given number.

butt 2... Dozens of speech numbers are visible 1234.567. Numbered from the zero position of the decimal number to the left and to the right:

The number 1234.567 can be written in the offensive view: 1234.567 = 1000 + 200 + 30 + 4 + 0.5 + 0.06 + 0.007 = 1 · 10 3 + 2 · 10 2 + 3 · 10 1 + 4 · 10 0 + 5 · 10 -1 + 6 · 10 -2 + 7 · 10 -3.

Transferring numbers from one system and number to one

There is a simple way to transfer the number from one number system to the number, є transfer the number from the number to a dozen number system, and then, the given result into the required number system.

Transfer of numbers from any number system and number to tens number system

To transfer the number from either system to a dozen, complete the number of the th order, correcting from the zero (dot code) similarly to butts 1 or 2. We know how to create the digits of the number on the display of the system and the number in the step of the position of the digit:

1. Convert the number 1001101.1101 2 to the tens numerical system.
Decision: 10011.1101 2 = 1 2 4 + 0 2 3 +0 2 2 + 1 2 1 + 1 2 0 + 1 2 -1 + 1 2 -2 + 0 2 -3 + 1 2 - 4 = 16 + 2 + 1 + 0.5 + 0.25 + 0.0625 = 19.8125 10
as follows: 10011.1101 2 = 19.8125 10

2. Convert the number E8F.2D 16 to the tens numerical system.
Decision: E8F.2D 16 = 14 16 2 + 8 16 -1 + 15 16 0 + 2 16 -1 + 13 16 -2 = 3584 + 128 + 15 + 0.125 + 0.05078125 = 3727.17578125 10
as follows: E8F.2D 16 = 3727.17578125 10

Transfer of numbers from tens of systems and numbers to the system of numbers

To translate numbers from tens of systems and numbers into the system of numbers and fractions, it is necessary to translate okremo.

The transfer of the whole part of the number from the tens of the number system to the number system

The whole part is transferred from dozens of numerical systems to the whole numerical system after the addition of the last distribution of the whole part of the number on the basis of the system and the number to reject the whole surplus, the lesser submission of the system and the number. The result of the transfer will be a record of the surplus, correcting the rest.

3. Convert the number 273 10 into the visimkov system of numbers.
Decision: 273/8 = 34 і surplus 1, 34/8 = 4 і surplus 2, 4 less than 8, the calculation is completed. Record from surplus in matime of such viglyad: 421
Perevirka: 4 8 2 + 2 8 1 + 1 8 0 = 256 + 16 + 1 = 273 = 273, the result is scored. It means that the transfer of the viconings is correct.
as follows: 273 10 = 421 8

The transfer of the correct tens of fractions in the numerical systems is easy to understand.

Shifting the fractional part of the number from tens of systems and the number in the system of numbers

Nagadaєmo, it is correct to be called a dozen fractions make a number with a zero whole part... To translate such a number into the numerical system with the base N, it is necessary to multiply the number by N until the fractional part is zeroed out, or if there is no need for a number of rows. If, when multiple numbers are entered, the whole part is shown as zero, then the whole part is far from being lost, so it is last entered into the result.

4. Convert the number 0.125 10 into the two-sided numerical system.
Decision: 0.125 2 = 0.25 (0 is the whole part, which will become the first digit in the result), 0.25 2 = 0.5 (0 is the other digit in the result), 0.5 , then transfer of completions).
as follows: 0.125 10 = 0.001 2

People didn’t immediately come across rahuvati. The first suspension is applied to a small number of objects - one or two. Everything, more, for the change of the new "bagato". Itself tse vvazhaya ear of the current system of calculation.

A short history

In the process of developing civilization in people, it became necessary to distribute small items, united by special signs. We celebrated the winners of the witnesses: "three", "chotiri" and so far to "seven". However, the tse buv zakrytiy, obmezheniya a number of, staying understood in yaku prodovzhuvalo carry the message of navantazhennya more early "bagato". We will use the butt of a whole є folklore, which moves before us from the primordial viglyad (for example, adding "Sem raziv vidmiryai - once vidrіzh").

Winning of folding rakhunka methods

In the last hour of life and all the processes of human activity have been accelerated. Tse summoned, in his own heart, to the determination of the larger folding system of calculation. At the same time, people were victorious for the purpose of turning the simplest tools of the rakhunka. They knew the stench of them about themselves: the stench of sticks on the walls of the stove with hand-held handlers, hammered notches, wicked numbers for them from clubs and stones - the axis deprived of a small list of current todi of development. On the other hand, with the help of this type, the bulo is assigned the unique name "unary system of calculation". The essence of the polarity is in the recording of the number from the stasis of the same type of signs. This is the best hand-held system that allows you to visually separate a number of objects and signs. The most widespread won in the cob classes of schools (rakhunkovy sticks). With the fall of the "Kamintsev rakhunka", you can boldly respect the modern equipment in the latest modifications. Cicavo and vyniknennya of the bitter word "calculus", the root of which is found in Latin calculus, but it does not change as "kamenchik".

Rahunka on fingers

In the minds of the wretched vocabulary stock of the first people, they often served important updates to the information that was transmitted. Perevaga fingers were in full versatility and in permanent transfer with an object, which would like to transmit information. However, here there are some shortcomings: interconnection and short transmission times are meant. That is why all the rakhunok of people, who used the "finger method", intermingled with numbers, multiples of the number of fingers: 5 - the number of fingers on one hand; 10 - on both hands; 20 - the backbone on the hands and feet. The managers, according to the development of the numerical stock, were given the system to complete the hourly progress.

pershі vivid

With the development of the system of calculating and expanding the possibilities and needs of the people, the maximum victorious number in the cultures of the Bagatian peoples has become 40. Because of them, there is also a sense of non-meanings (so it’s not a matter of rakhunku) number. In Russia, a wide spread nabulo viraz "forty forties". Yogo zm_st is made to a number of objects, as it is uncomfortable to fuck. The upward slope of the development is the time of the number 100. The distance has gone up to dozens. By the year, the numbers of 1000, 10,000 and so far, the skin of those who carried a message of nurturing, analogous to seven and forty, began to appear. At the current light, the cordon of the kintsev rakhunka is not worth it. On the current day, the univer- sal concept of "lack of inconsistency" has been introduced.

Cycles and fractions of numbers

The current system of calculation for the smallest number of items is taken as one. A large number of won are of an unimaginable size. However, with more exact wins, it can also be crushed. The same s tsim are tied to the singing stage of the development of the figure of the fractional number. For example, the Babylonian system of pennies (wagons) became 60 mins, and 1 Talanov were added. At your own house, 1 min was added to 60 shekels. Itself on the basis of the Babylonian mathematician widely used the shistdecimal fragmentation. The fractions came to us widely from the ancient Greeks and Indians. At the same time, the records are identical to those of India. The visibility of the remaining shot rice is insignificant. The Greeks prescribed the number at the top, and the standard at the bottom. The Indian version of the writing of fractions, having made a wide development in Asia and Europe, two students: Muhammad Khorezmskaya and Leonardo Fibonacchi. The Roman numeral system was based on 12 units, called ounces, to the whole (1 ac), apparently, in the basis of all the numbers were twelve fractions. At the same time, from those who were taken away from the reception, they often got stuck and special dates. So, for example, until the 17th century astronomers used to call the sixty fractions so called, as they got dozens of years old (introduced to the household of Simon Stevin, a scientist-engineer). As a result of the further progress of the people, there is a need for an even larger expansion of the number row. This is how the negative, erratic and knowledge appeared to be zero, having appeared quite recently. Win pochav zastosovuvatisya when introduced into the current system of calculating negative numbers.

Vikoristannya non-positivnykh alphabet

Which is such an alphabet? For this system of calculation it is characteristic that the value of the numbers does not change from the arrangement. Non-positional alphabet powerfully manifestation of non-interconnected number of elements. The principle of additivity lies in the basis of systems, which will be based on a given type of alphabet. In other words, the initial value of the number is summed up with the sum of all numbers, including the notation. Winning non-positional systems became earlier than positional ones. Falling in the way of rakhunka, the number of values ​​is beginning as a difference, or the sum of all digits, which is included in the warehouse of the number.

Find the shortcomings of such systems. Among the main tracks of the video:

• the introduction of new numbers in the form of a great number;
• the inability to portray negative and fractional numbers;
• the foldability of arithmetic operations.

In the history of people, the development of systems and numbers has been stagnant. Naybilsh vidomimy vvazhayutsya: walnut, roman, alphabetic, unarn, davnoєgipetska, vavilonska.

One of the most advanced rachunka methods

It has been saved up to our days practically in an unassuming viglyad, one of the best ones. For the help of it, there are known dates, juvileyni in that number. Also, she knew a wider stasis in literature, science and spheres of life. In the Roman system of numerical strength, all seven letters of the skin are similar to the number: I = 1; V = 5; X = 10; L = 50; C = 100; D = 500; M = 1000.

viniknennya

The very similarity of Roman numerals is not sound, the history has not spared the exact tributes that have appeared. At the same time it’s insane є fact: the meaningful infusion on the Roman numbering was broken by the p'yatirkova system of numbering. However, in Latin Movies, there are riddles about her. At the time of the presentation of the Vinicla, the hypothesis about the suspicion of their system by the ancient Romans in the nation (imovirno, among the Etruscans).

Features

The recording of all whole numbers (up to 5000) is carried out with the help of repeating the description of the numbers. The key feature is the spreading of signs:

• folding to see for this wash, how to stand more before the younger (XI = 11);
• vidnimannya vidbuvaєtsya, as the mensha figure stands in front of a larger one (IX = 9);
• one and the same sign can not stand for more than three times (for example, 90 is recorded XC instead of LXXXX).

Inadvertently ї the lack of success in displaying arithmetic works. At the same time, she was annoyed to finish it and she stopped being vicious in Europe in the main system and was calculated on a regular basis - in the 16th century.

The Roman numerical system does not flaunt absolutely non-positional. It is tied with a ts that, in the row of vipadks, there is one more figure in the row (for example, IX = 9).

Sposib rakhunki in Ancient Egypt

The third thousand years before ours is the moment of accrual of the system in the Ancient Egypt. The essence of the її was in the recording with special signs of the numbers 1, 10, 102, 104, 105, 106, 107. All the numbers were recorded in the form of the combination of the given signs. At the same time, it was obmezhennya - the dermal figure was to blame for the bullet being repeated no more than nine times. The basic method of rakhunka, which is called "non-positional ten system of numbers", is based on a simple principle. The sense of yogo pole is in the fact that the written number added up to the sum of all the numbers that were added up.

Unary way of rakhunka

The numerical system, in every hour I will write down the numbers of vikoristany one sign - I - is called unary. The number of skin starts coming in as a result of the addition of the news to the foregoing. At the same time, there are a number of such I, according to the number recorded behind them.

Visimkov system of calculation

There is a central positioning method based on the number 8. For displaying numbers, a digital row from 0 to 7 is displayed. The main її experience is an easy transfer of numbers. Їx can be converted to і backward. The date of the manipulation is to come back to replace the numbers. From the octal system, the stink is translated into two triplets (for example, 28 = 0102, 68 = 1102). Denmark has a way of expanding the area of ​​computer software and software.

Shistnachyatkov system of calculation

It’s an hour to stay in the computer spheres of Denmark. At the root of the given system, the number of letters is 16. The system of calculation, which is based on new, transfers the numbers from 0 to 9 and a number of letters of the Latin alphabet (from A to F), which are stagnant for a certain interval of 10 from 1010 to 15 as it is already meant, vikorystovuyutsya at the creation of software security and documentation, connected with computers and warehouses. It is based on the power of a modern computer, the main unit of which is 8-bit memory. Її manually re-create and write down two sixteen digits behind the scenes. The IBM / 360 system became the founder of such a process. Documentation for it was for the first time shifted in a different way. Unicode standard transmissionє to record any character in a sixteen-point view with at least 4 digits.

how to write

The mathematically formalized method of rakhunku primed on the designated th in the lower index in dozens of systems. Butt, number 1444 will be recorded at viglyad 144410. Moving programs for writing sixteen systems may be different syntax:

visnovok

Yak vyvchayutsya Informatics is the main discipline, within the framework of how to build up the accumulation of tribute, the process of forming it into a form for living. From the storage of special tools, the registration and transfer of all available information to the program is updated. Win nadal vikoristovuyutsya at the establishment of software security and computer documentation. Vivchayuchi development systems and calculations, informatics transmission, vikorystannya, as it has already been said vishche, new tools. There are a lot of them with a quick replay of numbers. One of such "tools" is a table of numerical systems. Koristuvatisya her finish by hand. For the help of this table, you can, for example, quickly translate the number from the hexadecimal system into two, not volodyucha with special scientific knowledge. The current power of digital transformation є is practical for a skin-tight person in all people, some of the necessary tools are needed to be used by people on the basis of critical resources. In addition, browse and program online translation. Quite simply, I will forgive the task of recreating the numbers and the quickest hour of operations.

If you are engaged in the settings of a mesh of a large scale and the skin day stays with the numbers, then such a kind of cheat sheet is not necessary to start, everything is so timid on a mad reflex. Ale if the fancywork is even more rare, then do not expect a mask in ten form there for the prefix 21, or else the address of the fenestration when there is a prefix. At the connection with the cipher and the virus, write a number of small articles-cheat sheets on the transfer of numbers in the different systems of numbers, hedgehog addresses, masks, etc. In the whole part of the talk about translating numbers into different numerical systems.

1. Numerical systems

If you are busy tying the knitting with the computer fences and IT, you will see whether you will connect with the cymbals. І as a tlumachny IT-shnik you will need to get to the point where you want to get a little better if you will be stuck in practice.
The transfer of skin digits from the IP address is clearly visible 98.251.16.138 in the offensive system of numbers:

• Dviykova
• visimkova
• desyatkova
• shisstnachyatkova

1.1 Desyatkova

So, as the numbers are written in tens, shifting from tens to tens is permissible 🙂

1.1.1 Desyatkova → Binary

It’s known that there’s a dviykova system of numerical strengths that is practically used in all modern computers and bagaton’s numerical annexes. The system is even simpler - we have only 0 and 1.
To convert the number with a decimal into a two-letter form, it is necessary to vikoristovuvati rozpodil modulo 2 (that is, an integer number by 2), as a result, we will probably go to the surplus either 1, or 0. At the same time, the result is written from right to left. Put everything on your butt:

Malyunok 1.1 - Transfer of numbers from tens to two system

Malyunok 1.2 - Transfer of numbers from tens to two system

I will describe the rise of the number 98. The difference is 98 by 2, the result is a maximum of 49 and an excess of 0. to a friend. Let's write down the result from right to left.

1.1.2 Desyatkova → Visimkova

Vіsіmkovo system is a whole integer system of numerical base 8. So all numbers in them are represented by the range 0 - 7, for transferring from dozens of systems and it is necessary to choose the number 8 modulo 8.

Malunok 1.3 - Transferring numbers from tens to the visimkov system

Rozpodil is similar to the 2-part system.

1.1.3 Desyatkova → Shistnachyatkova

The Shistnachyatkov system may have increased the octal system. At her, from 16 to 16, there are dozens of numbers from 0 to 9 + Latin letters from A (number 10) to F (number 15). Check it for skin once, if you change the settings of the bridging adapter - the MAC address. So it is, if you are using IPv6.

Malunok 1.4 - Transfer of numbers from tens to sixteen system

1.2 Binary

In front of the front, all tens of numbers were translated into the same numerical systems, one in binary. Now the dermal number is translated from two forms.

1.2.1 Binary → Desyatkova

To translate numbers from two forms into tens, two nuances are required for the nobility. The first one - at the skin zero and odnichki є a multiplier of 2 in the n-th step, with which n it goes from right to left just by one. The other is to multiply all the numbers that are required in the form of a decimal number. At once we will have a formula like this:

D = (a n × p n-1) + (a n-1 × p n-2) + (a n-2 × p n-3) + ..., (1.2.1)

de,
D - tse number in the tenth form, yake mi shukaєmo;
n- number of symbols in pre-numbers;
a - a number in two forms at the n-th position (to be the first character, another, etc.);
p - efficiency, for road transport 2.8 or 16 in a step n(In the abundance of the systems and numbers)

For example, the number is 110102. We are dazzled by the formula and write:

• The number is stored in 5 characters ( n=5)

a 5 = 1, a 4 = 1, a 3 = 0, a 2 = 1, a 1 = 0

• p = 2 (so yak is translated from two into tens)

At the pidsumku maєmo:

D = (1 × 2 5-1) + (1 × 2 5-2) + (0 × 2 5-3) + (1 × 2 5-4) + (0 × 2 5-5) = 16 + 8 + 0 + 2 + 0 = 26 10

Hto write the sound on the right on the left, the form will look like this:

D = (0 × 2 5-5) + (1 × 2 5-4) + (0 × 2 5-3) + (1 × 2 5-2) + (1 × 2 5-1) = 0 + 2 + 0 + 8 + 16 = 26 10

Ale, as I know, because of the rearrangement of the sum does not change. Let's now translate our numbers into tens.

Malunok 1.5 - Converting numbers from two to ten systems

1.2.2 Binary → Visimkova

When shifting, we need two number of split into groups of three symbols from right to left. As the rest of the group does not consist of three symbols, it is simply that it is displayed in all numbers with zeros. For example:

10101001 = 0 10 101 001

1011100 = 00 1 011 100

The skin group of bites is one of the largest numbers. As a matter of fact, I need to write a formula 1.2.1 for the skin group of beats. As a result, we are otrimaєmo.

Malunok 1.6 - Number shifting from two to one system

1.2.3 Binary → Shistnachyatkova

Here we need two numbers to break into groups according to the number of symbols from the right to the next steps in the daily battles in groups with zeros, like writing a game. If the rest of the group is stored from zeros, then it is necessary to ignore it.

110101011 = 000 1 1010 1011

1011100 = 0 101 1100

001010000 = 00 0101 0000 = 0101 0000

The skin group of beats is one of sixteen numbers. Vikoristovuєmo formula 1.2.1 for skin groups of beats.

Malunok 1.7 - Transfer of numbers from two to sixteen system

1.3 Visimkova

In the whole system, we can only fold the folds when shifting to the 16-point system, the pieces of the grid will pass smoothly.

1.3.1 Visimkova → Binary

Skin number in a viscous system is a whole group of three beats in a two-bit system, as it was written in the word. For the transfer, we need to speed up the cheat sheet:

Malunok 1.8 - Spur for translating numbers from the viscous system

Vikoristovuchi tsyu plate is translated by our numbers in the dviykovu system.

Malunok 1.9 - Transfer of numbers from visimkov to two system

I will describe the triplets of visnovok. The first number we have is 142, which means there will be three groups of three bits per skin. We use the spur і bachimo, so the number 1 tse 001, the number 4 tse 100 і the number 2 tse 010. The result is the number 001100010.

1.3.2 Visimkova → Desyatkova

Here, my victorious formula 1.2.1 is only with coefficient 8 (tobto p = 8). As a result, maєmo

Malyunok 1.10 - Transfer of numbers from vіsіmkovіy to tens system

• The number is stored in 3 characters ( n=3)

a 3 = 1, a 2 = 4, a 1 = 2

• p = 8 (so yak is translated from visimkov to tens)

As a result, maєmo:

D = (1 × 8 3-1) + (4 × 8 3-2) + (2 × 8 3-3) = 64 + 32 + 2 = 98 10

1.3.3 Visimkova → Shistnachyatkova

Yak wrote earlier, for the translation, we need to translate the numbers into a two-way system, then from two into a shistnadtsyatkov, added to groups of 4 bytes. I can vikoristovuvati step on the spur.

Malunok 1.11 - Spur for translating numbers from the sixteen system

It is also possible to translate a plate from two into a sixteen-point system. Now our numbers are translated.

Malunok 1.12 - Transfer of numbers from visimkov to the sixteen-point system

1.4 Shistnachyatkova

The whole system has the same problem when transferring to visimkov. Ale about the tse piznishe.

1.4.1 Shistnachyatkova → Binary

The skin number in the sixteen system is a whole group of the number of bits in the two systems, as it was written in the word. For the transfer, we can speed up with a cheat sheet, yak known as a vische. As a result:

Malunok 1.13 - Transfer of numbers from sixteen into two system

At the same time, the number is 62. The vicoristovuchi plate (Fig. 1.11) is m bachimo, scho 6 tse 0110, 2 tse 0010, as a result, the number is 01100010.

1.4.2 Shistnachyatkova → Desyatkova

Here, my victorious formula 1.2.1 is only with a coefficient of 16 (tobto p = 16). As a result, maєmo

Malunok 1.14 - Conversion of numbers from sixteen to ten system

The first is the number. From formula 1.2.1:

• The number of stored in 2 characters ( n=2)

a 2 = 6, a 1 = 2

• p = 16 (so yak translates from sixteen to tens)

The result is maєmo.

D = (6 × 16 2-1) + (2 × 16 2-2) = 96 + 2 = 98 10

1.4.3 Shisstnachyatkova → Visimkova

For transferring to the visimkov system, it is necessary to translate the copy into two, then split into groups according to the 3rd bit and speed up with a plate (Fig. 1.8). As a result:

Malyunok 1.15 - Transfer of numbers from sixteen into a system

Let's talk about IP-addresses, masks and hemlines.