With the help of this online calculator You can translate integers and fractional numbers from one number system to another. A detailed solution is given with explanations. To translate, enter the original number, set the source number system base, set the base of the number system to which you want to translate the number and click on the "Translate" button. Theoretical part and numerical examples see below.
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Translation of whole and fractional numbers from one number system to any other - theory, examples and solutions
There are positional and not positional number systems. Arabic number system, which we use in everyday life is a positional, and Roman - no. In positional surgery systems, the position of the number uniquely determines the value of the number. Consider this on the example of the number 6372 in a decimal number system. Number this number on the right left since scratch:
Then the number 6372 can be represented as follows:
6372 \u003d 6000 + 300 + 70 + 2 \u003d 6 · 10 3 + 3 · 10 2 + 7 · 10 1 + 2 · 10 0.
The number 10 defines the number system (in this case This is 10). As degrees, the positions of the number of this number are taken.
Consider a real decimal number 1287.923. Number it starting from scratch the position of the number from the decimal point to the left and right:
Then the number 1287.923 can be represented as:
1287.923 \u003d 1000 + 200 + 80 + 7 + 0.9 + 0.02 + 0.003 \u003d 1 · 10 3 + 2 · 10 2 + 8 · 10 1 + 7 · 10 0 + 9 · 10 -1 + 2 · 10 -2 + 3 · 10 -3.
In general, the formula can be represented as follows:
C n · s. N + C N-1 · s. N-1 + ... + C 1 · s. 1 + C 0 · s 0 + d -1 · s -1 + d -2 · s -2 + ... + d -k · s -k
where c n is a number in position n., D -k - fractional number in position (-K), s. - Number system.
A few words about the number systems. The number in the decimal number system consists of a plurality of numbers (0.1,2,3,4,5,6,7,8,9), in an octaous number system - from a plurality of numbers (0.1, 2,3,4,5,6,7), in a binary number system - from a plurality of numbers (0.1), in a hexadecimal number system - from a plurality of numbers (0,1,2,3,4,5,6, 7,8,9, A, B, C, D, E, F), where A, B, C, D, E, F correspond to the number 10,11,12,13,14,15. In Table Table.1 Presented numbers B. different systems Note.
| Table 1 | |||
|---|---|---|---|
| Notation | |||
| 10 | 2 | 8 | 16 |
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 2 | 10 | 2 | 2 |
| 3 | 11 | 3 | 3 |
| 4 | 100 | 4 | 4 |
| 5 | 101 | 5 | 5 |
| 6 | 110 | 6 | 6 |
| 7 | 111 | 7 | 7 |
| 8 | 1000 | 10 | 8 |
| 9 | 1001 | 11 | 9 |
| 10 | 1010 | 12 | A. |
| 11 | 1011 | 13 | B. |
| 12 | 1100 | 14 | C. |
| 13 | 1101 | 15 | D. |
| 14 | 1110 | 16 | E. | 15 | 1111 | 17 | F. |
Translation of numbers from one number system to another
To transfer numbers from one number system to another, the easiest way to first translate the number in decimal system Number, and then, from a decimal number system to translate to the desired number system.
Translation of numbers from any number system in a decimal number system
Using formula (1), you can translate numbers from any number system to a decimal number system.
Example 1. Translate the number 1011101.001 from the binary number system (SS) in a decimal SS. Decision:
1 · 2 6 +0 · 2 5 + 1 · 2 4 + 1 · 2 3 + 1 · 2 2 + 0 · 2 1 + 1 · 2 0 + 0 · 2 -1 + 0 · 2 -2 + 1 · 2 -3 \u003d 64 + 16 + 8 + 4 + 1 + 1/8 \u003d 93.125
Example2. Translate the number 1011101.001 from the octaous number system (SS) in a decimal SS. Decision:
Example 3 . Translate the number AB572.CDF from a hexadecimal number system in a decimal SS. Decision:
Here A. - per 10, B. - by 11, C.- at 12, F. - by 15.
Translation of numbers from a decimal number system to another number system
To transfer numbers from a decimal numbering system to another number system, it is necessary to translate separately by the integer part of the number and fractional part of the number.
An integer part of the number is translated from a decimal SS to another number system - a sequential division of a whole part of the number on the base of the number system (for a binary CC - by 2, for an 8-character SS - by 8, for 16-smoke-16, etc. ) Before getting a whole residue, less than the base of the SS.
Example 4 . We translate the number 159 of the decimal SS into the binary SS:
| 159 | 2 | ||||||
| 158 | 79 | 2 | |||||
| 1 | 78 | 39 | 2 | ||||
| 1 | 38 | 19 | 2 | ||||
| 1 | 18 | 9 | 2 | ||||
| 1 | 8 | 4 | 2 | ||||
| 1 | 4 | 2 | 2 | ||||
| 0 | 2 | 1 | |||||
| 0 |
As can be seen from fig. 1, the number 159 during division by 2 gives the private 79 and the residue 1. Next, the number 79 during division by 2 gives Private 39 and the residue 1, etc. As a result, by building a number from the balances of divisions (right to left) we get a number in binary ss: 10011111 . Consequently, you can write:
159 10 =10011111 2 .
Example 5 . We translate the number 615 of the decimal SS into the octal SS.
| 615 | 8 | ||
| 608 | 76 | 8 | |
| 7 | 72 | 9 | 8 |
| 4 | 8 | 1 | |
| 1 |
When the number from the decimal SS in the octal SS, it is necessary to sequentially divide the number on 8 until the whole residue is less than 8. As a result, building a number from the balances of division (right to left), we get a number in the octane SS: 1147 (See Fig. 2). Consequently, you can write:
615 10 =1147 8 .
Example 6 . We transfer the number 19673 from the decimal number system to hexadecimal SS.
| 19673 | 16 | ||
| 19664 | 1229 | 16 | |
| 9 | 1216 | 76 | 16 |
| 13 | 64 | 4 | |
| 12 |
As can be seen from Fig.
To transfer the right decimal fractions (real number with a zero integer) to the level of the n base system this number Consistently multiplied by S until the fractional part does not get pure zero, or we will not get the required number of discharges. If you get a number with a whole part, different from zero, then this whole part does not take into account (they are consistently enrolled in the result).
Consider the foregoing on the examples.
Example 7 . We transfer the number 0.214 from the decimal number system to binary SS.
| 0.214 | ||
| x. | 2 | |
| 0 | 0.428 | |
| x. | 2 | |
| 0 | 0.856 | |
| x. | 2 | |
| 1 | 0.712 | |
| x. | 2 | |
| 1 | 0.424 | |
| x. | 2 | |
| 0 | 0.848 | |
| x. | 2 | |
| 1 | 0.696 | |
| x. | 2 | |
| 1 | 0.392 |
As can be seen from Fig. 4, the number 0.214 is multiplied by 2. If the multiplication is obtained with a whole part, different from zero, then the integer part is written separately (to the left of the number), and the number is written to the zero integer. If, when multiplying, a number with a zero integer is obtained, then zero is written to the left. The multiplication process continues until the fractional part does not get pure zero or do not get the required number of discharges. Recording fatty numbers (Fig. 4) from top to bottom We obtain the desired number in the binary number system: 0. 0011011 .
Consequently, you can write:
0.214 10 =0.0011011 2 .
Example 8 . We translate the number 0.125 from the decimal number system to binary SS.
| 0.125 | ||
| x. | 2 | |
| 0 | 0.25 | |
| x. | 2 | |
| 0 | 0.5 | |
| x. | 2 | |
| 1 | 0.0 |
To bring the number of 0.125 of the decimal SS into a binary, this number is multiplied by 2. In the third stage it turned out 0. Therefore, the following result turned out:
0.125 10 =0.001 2 .
Example 9 . We translate the number 0.214 from the decimal number system to hexadecimal SS.
| 0.214 | ||
| x. | 16 | |
| 3 | 0.424 | |
| x. | 16 | |
| 6 | 0.784 | |
| x. | 16 | |
| 12 | 0.544 | |
| x. | 16 | |
| 8 | 0.704 | |
| x. | 16 | |
| 11 | 0.264 | |
| x. | 16 | |
| 4 | 0.224 |
Following examples 4 and 5, we obtain numbers 3, 6, 12, 8, 11, 4. But in hexadecimal CC, the numbers 12 and 11 correspond to the number C and B. Therefore, we have:
0.214 10 \u003d 0.36C8B4 16.
Example 10 . We translate the number 0.512 from a decimal number system in the octal SS.
| 0.512 | ||
| x. | 8 | |
| 4 | 0.096 | |
| x. | 8 | |
| 0 | 0.768 | |
| x. | 8 | |
| 6 | 0.144 | |
| x. | 8 | |
| 1 | 0.152 | |
| x. | 8 | |
| 1 | 0.216 | |
| x. | 8 | |
| 1 | 0.728 |
Received:
0.512 10 =0.406111 8 .
Example 11 . We translate the number 159.125 from a decimal number system to binary SS. To do this, we translate separately an integer part of the number (Example 4) and the fractional part of the number (Example 8). Next, we get the merging of these results:
159.125 10 =10011111.001 2 .
Example 12 . We transfer the number 19673.214 from a decimal number system to hexadecimal. To do this, we translate separately an integer part of the number (Example 6) and the fractional part of the number (example 9). Next, we get the combining results.
Translation of binary SS numbers in 8-richene and 16-richene and back
1. Transfer from binary number system to hexadecimal:
the initial number is broken down on the notebooks (i.e. 4 digits), starting on the right for integers and the left for fractional. If the number of digits of the source binary number is not multiple 4, it is supplemented on the left with zeros to 4 for integers and on the right for fractional;
each tetrad is replaced with a hexadecimal digit in accordance with the table.
1. 10011 2 = 0001 0011 2 = 13 16
2. 0.1101 2 \u003d 0, D 16.
2. From the hexadecimal number system in binary:
each digit of the hexadecimal number is replaced by a binary digit notebook in accordance with the table. If a binary number is less than 4 digits, it is complemented to the left with zeros to 4;
1. 13 16 = 0001 0011 2 = 10011 2
2. 0,2A 16 \u003d 0.0010 1010 2 \u003d 0.0010101 2.
3. From the binary number system in the octal
the initial number is divided into triads (i.e. 3 digits), starting on the right for integers and the left for fractional. If the number of digits of the original binary number is not multiple 3, it is supplemented on the left with zeros to 3 for integers and on the right for fractional;
each triad is replaced by the octal digit in accordance with the table
1. 1101111001.1101 2 =001 101 111 001.110 100 2 = 1571,64
2. 11001111.1101 2 = 011 001 111.110 100 2 = 317, 64 8
4. To translate the octal number to the binary number system
each digit of the octal number is replaced by triad binary digits in accordance with the table. If the binary number table has less than 3 digits, it is complemented to the left by zeros to 3 for integers and to the right up to 3 for fractional;
miscellaneous zeros are discarded in the result.
1. 305,4 8 = 011 000 101 , 100 2 = 11000101,1 2
2. 2516,1 8 = 010 101 001 110 , 001 2 = = 10101001110,001 2
5. Transfer from octal to hexadecimal system and back It is carried out through the binary system with the help of triad and tetrad.
1. 175.24 8 \u003d 001 111 101, 010 100 2 \u003d 0111 1101, 0101 2 \u003d 7D, 5 16
2. 426,574 8 \u003d 100 010 110, 101 111 100 2 \u003d 0001 0001 0110, 1011 1110 2 \u003d 116, BE
3. 0.0010101 2 \u003d 0.0010 1010 2 \u003d 0,2A 16.
4. 7b2, E 16 \u003d 0111 1011 0010, 1110 2 \u003d 11110110010,111 2
5. 11111111011111111 2 \u003d 0111 1111 1011,1001 1100 2 \u003d 7FB, 9C 16
6. 11000110111 2 \u003d 0011 0001,1011 1000 2 \u003d 31, B8 16
Only one thing is important for computer chip. Either there is a signal (1), or it is not (0). But it is not easy to record programs in binary code. On paper, very long combinations of zeros and units are obtained. It is hard for a person.The use of the usual decimal system in computer documentation and programming is very uncomfortable. Transformations from binary to decimal systems and back - very laborious processes.
The origin of the octal system, as well as decimal, is associated with the score on the fingers. But not fingers need to be considered, but the gaps between them. They are just eight.
The solution to the problem was the octal. At least at the dawn computer equipment. When the prohibition of processors was small. The octal system made it possible to translate as binary numbers In the octal and vice versa.
The octal number system is a surcharge system with a base 8. To represent numbers, numbers are used in it from 0 to 7.
Conversion
In order to translate the number into binary, it is necessary to replace each figure of the octal number on the top three of the binary digits. It is only important to remember which binary combination corresponds to numbers numbers. They are quite a bit. Total eight!In all number systems, except for decimal, signs are read one by one. For example, in the octal system, the number 610 is pronounced "six, one, zero".
If you know the number system well, you can not remember the correspondence of some numbers to others.
The binary system is no different from any other position system. Each category number has. As soon as the limit is achieved, the current category is reset, and a new one appears in front of it. Only one remark. This limit is very small and equal to one!
Everything is very simple! Zero will appear by a group of three zeros - 000, 1 will be wrapped by a sequence 001, 2 will turn into 010, etc.
As an example, try converting the octal number 361 to binary.
The answer is 011 110 001. Or, if you drop incognent zero, then 11110001.
The translation from the binary system in the octal is similar to the above. Only begin the breakdown on the top three from the end of the number.
Author Eternal AUM. asked a question in the section Other languages \u200b\u200band technologies
translation of numbers in binary, octaous number system and received the best answer
Answer from Emil Ivanov [Guru]
// See the answer of the user Gennady!
// Task: 100 (10) \u003d? (2).
(* "Translate 100 (out of 10 s) in a 2-item number system!",
i accidentally heard when I passed past the street table of the "Markrit" cafe,
(At the angle of the street "Patriarch Evtimiy" and "Prince Boris" in Sofia) June 05, 2009. *)
The decision (which I spoke out loud, because I had to wait a lot of cars passing along the boulevard):
І Method - the number 100 is divided into 2 (until 1) is not available, and the remnants of division form a number from the bottom-up (from left to right).
100: 2 \u003d 50 i 0
50: 2 \u003d 25 i 0
25: 2 \u003d 12 i 1
12: 2 \u003d 6 i 0
6: 2 \u003d 3 І 0
3: 2 \u003d 1 i 1
1: 2 \u003d 1 i 1
100 (10) = 1100100 (2)
II method - a number decomposes in the degrees of Number 2, starting at a maximum number of 100 degrees (Numbers 2).
(If the extent 2 is not known in advance, you can count:
2 to 7 degrees 128
2 by 6 degrees 64
2 to 5 degrees 32
2 by 4 degrees 16
2 by 3 degrees 8
2 2 degrees 4
2 per 1 degree 2
2 at 0 degrees 1).
1. 64 <100 является первым слагаемым,
64 + 32 <100, (32 второе слагаемое)
64 + 32 + 16\u003e 100 (hence and 16 are not the term)
...
64 + 32 + 4 \u003d 100 (4 is the third term - the number 100 is obtained).
2. On the discharge ** of each terms (from q. 1) to write the number 1,
on the rest of the discharges ** write 0.
** The discharge of the number corresponds to the degree of number 2.
** For example, 2 digit corresponds to the 2nd degree of number 2,
where should be 1, since the number 4 (2nd degrees of the number 2) is the foundation.)
100 (10) = 64 +32 +4 = 1100100 (2)
// Since 2 by 3 degrees 8,
for the rapid transformation of the number:
1. Of the 2-% in the 8-character number system,
can:
- group numbers of a 2-digit number in three;
- record the resulting 8-character number in each of the top three.
100 (10) = 1 100 100 (2) = 144 (8)
2. Of the 8-% in the 2nd number of the severity,
you can record each 8-southern number of 3 digits of a 2-% number system.
100 (10) = 144 (8) = 1 100 100 (2)
Answer from Kitty[newcomer]
use the calculator on the computer and all problems))))
Answer from Alexander Radko[active]
At the Calculator in Windows Change the view of the engineering))
then point the phone model, try something from this link,
Answer from Gennady[guru]
Good day.
Remember the simple algorithm.
While the number is greater than zero, divide it on the base of the system and record the remnants of the right on the left. Everything!
Example. Translate 13 to the binary system. After the sign is equal to the private and residue.
13: 2 = 6 1
6: 2 = 3 0
3: 2 = 1 1
1: 2 = 0 1
TOTAL 13 (10) \u003d 1101 (2)
Similarly, with other grounds.
Reverse transfer is performed by multiplying each discharge to the corresponding degree of the base of the system followed by summation.
1101 -> 1*2^2 + 1*2^2 + 0*2^1 + 1*2^0 = 1*8 + 1*4 + 0*2 + 1*1 = 8 + 4 + 0 + 1 = 13
Translation from, admitting, the octal system in a five-year one must be done through a decimal on these rules.
If you are aware of this, you will not need a mobile on the exam.
Good luck!
