What non-procurement number systems. Notation. Translation of decimal numbers to other number systems
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Positioning and Non-Purpose Systems
A variety of numbers that existed earlier and are used in our time, can be divided into non-procurement and positional. The signs used when recording numbers are called numbers.
In the non-phase surgery systems from the position of the number in the record of the number does not depend on the value that it denotes. An example of a non-sample number system is a Roman system in which Latin letters are used as numbers.
In the positional viewing systems, the value indicated by the number in the number of numbers depends on its position. The number of numbers used is called the base of the number system. The place of each digit is among the position. The first system known to us, based on the positional principle - the sixteen Babylonian. The numbers in it were two species, one of whom was designated units, others - dozens.
Currently, the positional numbering systems are more widespread than non-procurement. This is explained by the fact that they allow you to record big numbers With the help of a relatively small number of characters. An even more important advantage of positional systems is the simplicity and ease of performing arithmetic operations over the numbers recorded in these systems.
The Indo-Arab decimal system was most common. Indians were the first to use zero to indicate the positional significance of the values \u200b\u200bin the number of numbers. This system was named decimal, as it is ten digits.
The difference between the positional and non-phase surge systems is easiest to understand the example of a comparison of two numbers. In a positional number system, a comparison of two numbers occurs in the following way: In the numbers under consideration, the numbers standing in the same positions are compared to the left right. The bungling digit corresponds to the value of the number. For example, for numbers 123 and 234, 1 less than 2, therefore, the number 234 is larger than the number 123. In the non-phase number system, this rule does not work. An example of this can be a comparison of two numbers IX and VI. Despite the fact that i is less than V, the number IX is larger than the number Vi.
The base of the number system in which the number is recorded is usually indicated by the lower index. For example, 555 7 - the number recorded in the seminar number system. If the number is recorded in the decimal system, the base is usually not specified. The base of the system is also a number, and it is indicated in the usual decimal system. Any integer in the position system can be written in the form of a polynomial:
X s \u003d (a n a n-1 a n-2 ... a 2 a 1) s \u003d a n · s n-1 + a n-1 · s n-2 + a n-2 · s n- 3 + ... + a 2 · s 1 + a 1 · s 0
where S is the base of the number system, and n is the numbers recorded in this number system, n is the number of digits of the number.
So, for example, the number 6293 10 will be recorded in the form of polynomial as follows:
6293 10 \u003d 6 · 10 3 + 2 · 10 2 + 9 · 10 1 + 3 · 10 0
Examples of positional positioning systems:
· Binary (or Number System 2) This is a positive integer positional (local) number system that allows you to present various numerical values \u200b\u200bwith two characters. Most often it is 0 and 1.
· Octal - positional integer count system with base 8. To represent numbers, it uses figures 0 to 7. The octal system is often used in the areas associated with digital devices. Earlier was widely used in programming and computer documentation, however, hexadecimal is almost completely displaced.
· Decimal number system is a positional surgery system for an integer base 10. The most common system of surgery in the world. To write numbers, the most commonly used characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, called Arabic numbers.
· Twelve (widely used in antiquity, in some private areas used now) - a positional surgery system with integer base 12. Figures are used 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, B. Some peoples of Nigeria and Tibet still use a twelve number of the number system, but the echoes can be found almost in any culture. In Russian, there is a word "dozen", in English "Dozen", in some places the word twelve use instead of "ten", like a round number, for example, wait 12 minutes.
· Hexadecimal (most common in programming, as well as in fonts) - a positional numbering system for integer base 16. Usually, decimal numbers from 0 to 9 and Latin letters from a to f to designate numbers from 10 to 15 are used as hexadecimal digits. Widely used in low-level programming and in general in computer documentation, since modern computers The minimum unit of memory is the 8-bit byte, the values \u200b\u200bof which is convenient to record two hexadecimal numbers.
· Sixties (measurement of angles and, in particular, longitude and latitude) - a positional surgery system for an integer base 60. Used in ancient times in the Middle East. The consequences of this number system is the division of angular and arc degrees (as well as an hour) for 60 minutes and a minute to 60 seconds.
The greatest interest when working on a computer represents the number systems with bases 2, 8 and 16. These number systems are usually enough for full-fledged work as a person and a computing machine, but sometimes due to different circumstances still have to refer to other number systems, for example to a ternary, seminary or level system based on the base 32.
To operate with numbers recorded in such unconventional systems, it should be borne in mind that they do not differ in principle from the usual decimal. Addition, subtraction, multiplication in them is carried out on the same scheme.
Other number systems are not used mainly because everyday life People are used to using a decimal number system, and no other is required. In the computational machines, a binary number system is used, as it is quite simple to operate with numbers recorded in binary form.
Often, a hexadecimal system is used in computer science, since the recording of numbers in it is much shorter than the recording of numbers in binary system. A question may arise: why not use to write very large numbers a number system, for example, for the basis of 50? For such a number system, 10 conventional numbers plus 40 characters are required, which would correspond to the numbers from 10 to 49 and it is unlikely that someone will like to work with these forty signs. Therefore, in the real life of the surge system based on the base, more than 16, practically not used.
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For the first time, the positional number system appeared in ancient Babylon. In India, the system works in the form of a positional decimal numbering using zero, in Hindus this system The numbers borrowed the Arab nation, they, in turn ...
The number system is a recording method (image) of numbers. Different number systems that existed earlier and which are currently being used are divided into two groups: · positional, · non-procurement ...
Notation. Recording actions over numbers
A variety of numbers that existed earlier and are used in our time, can be divided into non-procurement and positional. Signs used when recording numbers are called numbers ...
Notation. Recording actions over numbers
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Non-sample number systems
People learned to consider a long time ago. Subsequently, there was a need for numbers. The number of items was depicted by applying dashes, serifs on some solid surface. For two people, two people could accurately keep some numeric information, they took a wooden tag, they did the desired number of scubons on it, and then they split the tag in half. Everyone hurt her half and kept her. This technique allowed to avoid controversial situations. Archaeologists found such entries during excavations. They refer to 10-11 millennium BC.
Scientists called such numbers recording such single (unary)Since any number in it is formed by repetition of one sign, symbolizing the unit.
Later, these badges began to combine in groups of 3, 5 and 10 sticks. Therefore, more convenient numbering systems occurred.
Around the third millennium BC, Egyptians came up with their numerical system in which special icons were used to designate key numbers - hieroglyphs. Each such hieroglyph could be repeated not more than 9 times. The required system of number is called Ancient Egyptian decimal non-phase surgery
An example of an unexpected survey system that has survived to this day can be a numbering system that used more than two and a half thousand years ago in ancient Rome. It is calledroman number system.
The basis of signs I (1), V (5), x (10), L (50), C (10), D (50), C (100), D (500), M (1000) are based.
Roman numbers enjoyed very long, today they are used mainly to name significant dates, volumes, sections and chapters in books.
To write a number, the Romans used not only addition, but also subtraction.
Rules for compiling numbers in the Roman number system:
- Going in a row several identical numbers fold (a group of the first type).
- If the left of the greater digit is less, then the value of the smaller figure (the second type group) is greater.
- The values \u200b\u200bof groups and numbers that are not included in the first and second type groups are folded.
In the old days, the surge systems resembling Roman were widely used in Russia. They were called yasacha. With their help, the collectors of the filters filled the receipts about the payment of Podachi (Yasaka) and made entries in the applied notebook.
"Russian Book of Finners"
Non-aid numbering systems have a number of significant drawbacks:
- There is a constant need to introduce new signs to record large numbers.
- It is impossible to represent fractional and negative numbers.
- It is difficult to perform arithmetic operations, since there are no algorithms for their execution. In particular, in all nations, along with the number systems, there were ways of a finger account, and the Greeks had a countable board of Abaca - something like our accounts.
But we still use elements of the non-sacrification system in everyday speech, in particular, we are talking to a hundred, and not ten dozen, a thousand, a million, billion, trillion.
Laboratory work №16
Number systems
Theoretical part
IN base
<10 используют n первых арабских цифр, а при n>
| Base | Name | Alphabet |
| N \u003d 2. | binary | 0 1 |
| N \u003d 3. | Tropic | 0 1 2 |
| N \u003d 4. | Ferry | 0 1 2 3 |
| N \u003d 5. | PAT. | 0 1 2 3 4 |
| N \u003d 6. | Sherteric | 0 1 2 3 4 5 |
| N \u003d 7. | semal | 0 1 2 3 4 5 6 |
| N \u003d 8. | octal | 0 1 2 3 4 5 6 7 |
| N \u003d 10. | decimal | 0 1 2 3 4 5 6 7 8 9 |
| N \u003d 16. | Hexadecimal |
| Radix | ||||||
| IV \u003d 5 - 1 \u003d 4 | XL \u003d 50 - 10 \u003d 40 |
Consider the numbers:
Translation from a decimal number system to other
Example: We transfer the number 75 of the decimal system to binary, octal and hexadecimal:
Answer: 75 10 \u003d 1 001 011 2 \u003d 113 8 \u003d 4B 16.
Translation into a decimal number system
Transfer of integers from the number system with the base Q (non-followed system) into a decimal number system is performed according to the rule: if all the terms in the unfolded form are submitted in the decimal system and calculate the resulting expression according to the rules of decimal arithmetic, it turns out the number in the decimal system equal to This. Consider examples:
112 3 \u003d 1 · 3 2 + 1 · 3 1 + 2 · 3 0 \u003d 9 + 3 + 2 \u003d 14 10
101101 2 \u003d 1 · 2 5 + 0 · 2 4 + 1 · 2 3 + 1 · 2 2 + 0 · 2 1 + 1 · 2 0 \u003d 32 + 0 + 8 + 4 + 1 \u003d 45 10
15fs 16. \u003d 1 · 16 3 + 5 · 16 2 + 15 (f) · 16 1 + 12 (C) · 16 0 \u003d 4096 + 1280 + 240 + 12 \u003d 5628 10
Deployed form of numbers
The detailed form of the number of numbers - This is a record in the form of discharge terms recorded by the degree of appropriate discharge and the foundation of the degree.
Consider examples:
32478 10 \u003d 3 · 10,000 + 2 · 1000 + 4 · 100 + 7 · 10 + 8 \u003d
3 · 10 4 + 2 · 10 3 + 4 · 10 2 + 7 · 10 1 + 8 · 10 0
112 3 \u003d 1 · 3 2 + 1 · 3 1 + 2 · 3 0
101101 2 \u003d 1 · 2 5 + 0 · 2 4 + 1 · 2 3 + 1 · 2 2 + 0 · 2 1 + 1 · 2 0
15fc. 16 \u003d 1 · 16 3 + 5 · 16 2 + 15 · 16 1 + 12 · 16 0
With l o f n and e
Addition tables are easy to compile using the account rule.
In h and t and n and e
Example 4.
Subscribe a unit from numbers 10 2, 10 8 and 10 16
Example 5.
Submount the unit from numbers 100 2, 100 8 and 100 16. 
Example 6. Pull out the number 59.75 from among 201.25.
Answer: 201.25 10 - 59.75 10 \u003d 141.5 10 \u003d 10001101.1 2 \u003d 215.4 8 \u003d 8D, 8 16.
Check. We transform the obtained differences to the decimal form:
10001101,1 2 = 2 7 + 2 3 + 2 2 + 2 0 + 2 -1 = 141,5;
215,4 8 = 2 . 8 2 + 1 . 8 1 + 5 . 8 0 + 4 . 8 -1 = 141,5;
8D, 8 16 \u003d 8 . 16 1 + D . 16 0 + 8 . 16 -1 = 141,5.
U m n o zh e n and e
By performing multiplication of multivalued numbers in various positional positioning systems, it is possible to use the usual multiplication algorithm in the column, but the results of multiplication and addition of unambiguous numbers must be boring from the corresponding multiplication and addition table system.
D E L EN AND E
The division in any positioning system is made according to the same rules as an angle division in the decimal system. In the binary system, the division is especially simply performed, because the next digit of the private can only be zero or one.
Example 9.
We divide the number 30 to the number 6.
Answer: 30: 6 = 5 10 = 101 2 = 5 8 .
Example 10. We divide the number 5865 by the number 115.
Octal: 13351 8:163 8
Answer: 5865: 115 = 51 10 = 110011 2 = 63 8 .
Check.
110011 2 = 2 5 + 2 4 + 2 1 + 2 0 = 51; 63 8 = 6 .
8 1 + 3 .
8 0 = 51.
Example 11. We divide the number 35 to the number 14.
Octal: 43 8: 16 8
Answer: 35: 14 = 2,5 10 = 10,1 2 = 2,4 8 .
Check. We transform the obtained private to the decimal form:
10,1 2 = 2 1 + 2 -1 = 2,5;
2,4 8 = 2 . 8 0 + 4 .
Octal and hexadecimal
A binary system, convenient for computers, is uncomfortable for a person because of its cumbersomeness and unusual recording.
The translation of the numbers from the decimal system to binary and on the contrary performs the machine. However, to professionally use the computer, you should learn to understand the word car. For this, both the octal and hexadecimal systems are developed.
The numbers in these systems are read almost as easily as decimal, they require three (octal) and four (hexadecimal) times less discharges, respectively than in the binary system (after all, the number 8 and 16 - respectively, the third and fourth degrees of the number 2) .
For example: 
For example,
How to move a privilence decimal on any other positional number system?
| To transfer the right decimal dpobi F. to the level of the base q. necessary F. Multiply by q. , recorded in the same decimal system, then fractional part of the resulting work again multiplied by q, etc., until the dual part of the next clicking will not be an inquisitive zero, or the required accuracy of the number of the number will not be achieved F. in q.- Official system. Representation of the fractional part of the number F. in new system There will be a sequence of integers of the obtained works recorded in the order of their receipt and depicted one q.- friend. If the required translation accuracy F. make up k. After the semicolons, the limit absolute error is equal to q - (K + 1) / 2. |
Example. We translate the number 0.36 of the decimal system to binary, octal and hexadecimal:
Practical work.
1. Translate a given number from a decimal number system to a binary, octal and hexadecimal number system.
c) 712.25 (10);
d) 670.25 (10);
2. Translate a given number to a decimal number system.
a) 1001110011 (2);
b) 1001000 (2);
c) 1111100111.01 (2);
d) 1010001100,101101 (2);
e) 413.41 (8);
e) 118.8c (16).
3. Fold numbers.
a) 1100001100 (2) +1100011001 (2);
b) 110010001 (2) +1001101 (2);
c) 111111111,001 (2) +1111111110,0101 (2);
d) 1443.1 (8) +242.44 (8);
e) 2B4, C (16) + EA, 4 (16).
Laboratory work number 16.
Number systems
Theoretical part
Positional Number Systems
IN positional viewing systems The value indicated by the number in the number of numbers depends on its position. The number of numbers used is called base Positional number system.
The number system used in modern mathematics is positional decimal system. Its base is 10, because The recording of the numbers is made using 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
The positional nature of this system is easy to understand by the example of any multi-valued number. For example, among the 333 first 3 means 3 hundred, the second is 3 dozen, the third - 3 units (the value of each digit depends on the place that this figure takes).
To write numbers in the position system with the base n, you need to have an alphabet from N numbers. Usually for this with n<10 используют n первых арабских цифр, а при n>10 to ten Arabic figures add letters. Here are examples of alphabets of several systems:
| Base | Name | Alphabet |
| N \u003d 2. | binary | 0 1 |
| N \u003d 3. | Tropic | 0 1 2 |
| N \u003d 4. | Ferry | 0 1 2 3 |
| N \u003d 5. | PAT. | 0 1 2 3 4 |
| N \u003d 6. | Sherteric | 0 1 2 3 4 5 |
| N \u003d 7. | semal | 0 1 2 3 4 5 6 |
| N \u003d 8. | octal | 0 1 2 3 4 5 6 7 |
| N \u003d 10. | decimal | 0 1 2 3 4 5 6 7 8 9 |
| N \u003d 16. | Hexadecimal | 0 1 2 3 4 5 6 7 8 9 A B C D E F |
If you want to specify the base of the system to which the number belongs, then it is attributed to the lower index to this number: 101101 2, 3671 8, 3B8F 16
We write the first 17 numbers in binary and octal systems Note:
| Radix | ||||||
Non-sample number systems
In addition to positional, there are also other - non-phase surgery systems built on other principles.
In the non-phase surgery systems from the position of the number in the record of the number does not depend on the value that it denotes. The well-known example of such a system is the Roman system (Roman numbers). In the Roman system, Latin letters are used as numbers:
If a smaller figure is recorded on the left, and the right is large, then their values \u200b\u200bare subtracted:
| IV \u003d 5 - 1 \u003d 4 | XL \u003d 50 - 10 \u003d 40 |
Consider the numbers:
a) lxxxvii \u003d (50 + 30) + (5 + 2) \u003d 87. In this example Digit X, participating 3 times, each time means the same value - 10 units.
b) mcmxcvi \u003d 1000 + (1000 - 100) + (100 - 10) + (5 + 1) \u003d 1996
Roman numbers we often meet and now, for example, on Hours dials, in books with chapter numbering, in the designation of centuries. However, in mathematical practice they do not apply. Positional systems are convenient because allowing you to record large numbers with a relatively small number of characters. An even more important advantage of positional systems is the simplicity and ease of performing arithmetic operations over numbers. Try multiplying two three digits for comparison by writing them with Roman numbers.
Introduction
The topic of abstract at the rate of "Informatics-1" - "Number Systems".
The purpose of writing an abstract: get acquainted with the concept of a number system and classification; Translation of numbers from one number system to another.
The concept of a number system. Positioning and Non-Purpose Systems
whole number algebraic binary
The number system is called a system of receptions and rules that allow you to establish a mutually unambiguous match between any number and its representation in the form of a set of a finite number of characters. Many characters used for such a representation are called numbers.
Notation:
gives representations of a set of numbers (integers and / or real);
gives each number a unique representation (or at least a standard presentation);
reflects the algebraic and arithmetic structure of numbers.
Numbers are divided into positional and non-procurement. In non-phase systems, any number is defined as some function from the numerical values \u200b\u200bof the set of numbers representing this number. Figures in non-phase surgery systems correspond to some fixed numbers. An example of a non-sacrification system is a Roman number system.
Historically, non-phase systems were the first number systems. One of the main deficiencies is the difficulty of recording large numbers. The record of large numbers in such systems is very cumbersome and the alphabet of the system is extremely large.
IN computing technology Non-phase systems do not apply. 3.
The number system is called the positional, if the same figure can take different numerical values \u200b\u200bdepending on the number of the discharge number in the set of numbers representing a specified number. An example of such a system is an Arab decimal number system.
The basis of the positioning system defines its name. Computing equipment used binary, octal, decimal and hexadecimal systems.
Currently, the positional numbering systems are more widespread than non-procurement. This is explained by the fact that they allow you to record large numbers with a relatively small number of characters. An even more important advantage of positional systems is the simplicity and ease of performing arithmetic operations over the numbers recorded in these systems.
We give examples where you can meet the use of positional numbering systems:
binary in discrete mathematics, computer science, programming;
decimal - used everywhere;
twelve - a dozen account;
hexadecimal - used in programming, computer science;
sixties - time measurement units, measurement of corners and, in particular, coordinates, longitude and latitude.
Single number system
The need for the number of numbers began to occur in people still in antiquity after they learned to count. Certificate of this is archaeological finds in the places of camp of primitive people who belong to the Paleolithic period ($ 10 $ - $ 11 $ thousand. BC). Initially, the number of items depicted using certain signs: dashes, notches, mugs, applied to stones, wood or clay, as well as nodes on the ropes.
Picture 1.
Scientists this number of recording numbers are called single (unary)Since the number in it is formed by repetition of one mark, which symbolizes the unit.
Disadvantages of the system:
when writing a large number, it is necessary to use a large number of chopsticks;
it is possible to be easily mistaken when applying sticks.
Later, to facilitate the bill, these signs people began to unite.
Example 1.
Examples of using a single number system can be found in our lives. For example, small children are trying to portray how many years they are on their fingers, or counting sticks are used to teach a bill in the first grade.
Single system Not quite convenient, as the records look very long and their application is quite tedious, so more practical in the use of the number system began to appear.
Here are some examples.
Ancient Egyptian decimal non-phase surgery
This number system appeared about 3000 years BC. As a result of the fact that the inhabitants of the ancient Egypt came up with their numerical system, in which, with the designation of key numbers $ 1 $, $ 10 $, $ 100 $, etc. Hieroglyphs were used, which was convenient when writing on clay plates, which replaced the paper. Other numbers were made up of them by addition. At first, the number of higher order was recorded, and then the lower one. Moved and divided the Egyptians, twisting the numbers consistently. Each digit could repeat to $ 9 $ 9 times. Examples of the numbers of this system are shown below.
Figure 2.
Roman number system
This system is not fundamentally much different from the previous one and has survived to this day. It is based on signs:
$ I $ (one finger) for a number $ 1 $;
$ V $ (opened palm) for a number $ 5 $;
$ X $ (two folded palms) for $ 10 $;
for the designation of numbers $ 100 $, $ 500 $ and $ 1000 $, the first letters of the corresponding Latin words were used ( Sentum - one hundred, Demimille - Half thousand Mille - one thousand).
In compiling the number of Romans used the following rules:
The number is equal to the sum of the values \u200b\u200bof the in a row of several identical "digits" forming a group of the first type.
The number is equal to the difference between the values \u200b\u200bof the two "digits", if the left is more than less than the smaller. In this case, the significance is greater, the significance is less. Together they form a second type group. At the same time, the left "figure" may be less than the right maximum on $ 1 $ Order: before $ l (50) $ and $ C ($ 100) from the "younger" can stand only $ ($ 10), before $ D ($ 500 ) and $ M ($ 1000) - only $ C ($ 100), before $ V (5) - I (1) $.
The number is equal to the sum of the values \u200b\u200bof groups and "digits", which are not included in the group $ 1 $ or $ 2 $ of type.
Figure 3.
Roman numbers enjoy ancient times: they are denoted dates, volumes of volumes, partitions, chapters. Previously, it believed that ordinary Arabic numbers can be easily faked.
Alphabetic numbering systems
The number system data is more perfect. These include Greek, Slavic, Phoenician, Jewish and others. In these systems, the number from $ 1 $ to $ 9 $, as well as the number of dozens (from $ 10 $ to $ 90 $), hundreds (from $ 100 $ to $ 900 $) were denoted by the letters of the alphabet.
In an ancient Greek alphabetic number of numbers of $ 1, 2, ..., $ 9 was designated the first nine letters of the Greek alphabet, etc. For the designation of numbers $ 10, 20, ..., 90 $ the following $ 9 $ of letters were used to designate numbers $ 100, 200, ..., $ 900 - the last $ 9 $ letters.
In the Slavic peoples, the numerical values \u200b\u200bof the letters were installed in accordance with the order of the Slavic alphabet, which used the verbolitz initially, and then Cyrillic.
Figure 4.
Note 1.
The alphabetical system was used in ancient Russia. Until the end of $ XVII $ century, Cyrillic letters were used as numbers.
Non-aid numbering systems have a number of significant drawbacks:
There is a constant need to introduce new signs to record large numbers.
It is impossible to represent fractional and negative numbers.
It is difficult to perform arithmetic operations, since there are no algorithms for their execution.