Together with this calculator also use the following:
Translation of numbers into binary, hexadecimal, decimal, octaous number systems
Multiplication of binary numbers
Format representation of floating semicolons
Example number 1. Present the number 133.54 in the form of a floating point number.
Decision. Imagine the number 133.54 in a normalized exponential form:
1.3354 * 10 2 \u003d 1.3354 * EXP 10 2
The number 1.3354 * EXP 10 2 consists of two parts: Mantissa M \u003d 1.3354 and Exhibitors Exp 10 \u003d 2
If the mantissa is in the range of 1 ≤ m The representation of the number in denormalized exponential form.
If the mantissa is in the range of 0.1 ≤ M present the number in the denormalized exponential form: 0.13354 * exp 10 3
Example number 2. Submit a binary number 101.10 2 in a normalized form, write in a 32-bit IEEE754 standard.
Tank truth
Calculation of limits
Arithmetic in a binary number system
Arithmetic action B. binary system Perform the same as in decimal. But, if in the decimal system for the transfer and the loan is carried out at ten units, then in binary - two units. The table presents the rules for addition and subtraction in the binary number system.- When adding in the binary system, the number of two units in this discharge will be 0 and the transfer of units to the eldest discharge will appear.
- When subtracting from zero, the units are made by a unit of a senior discharge, where there are 1. The unit occupied in this discharge gives two units in the discharge, where the action is calculated, as well as one by one, in all intermediate discharges.
The addition of numbers, taking into account their signs on the machine, is a sequence of the following actions:
- transformation of source numbers to the specified code;
- discontinuous addition of codes;
- analysis of the result.
When performing an operation in an additional (modified additional) code, if a unit of transfer occurs as a result of addition in the iconic discharge, it is discarded.
Operation of subtraction in the computer is performed through the addition by rule: X-y \u003d x + (- y). Further actions are performed as well as for the addition operation.
Example number 1.
Danched: x \u003d 0.110001; Y \u003d -0.001001, folded in the reverse modified code.
Danched: x \u003d 0,101001; Y \u003d -0.001101, folded in an additional modified code. 
Example number 2. Solve examples on the subtraction of binary numbers using the addition method to 1 and cyclic transfer.
a) 11 - 10.
Decision.
Imagine the number 11 2 and -10 2 in the reverse code.
Binary number 0000011 has a reverse code 0.0000011
Moving the number 00000011 and 11111101
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | |||||||
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 |
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | ||||||
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 |
In the 2nd discharge, an overflow arose (1 + 1 \u003d 10). Therefore, write 0, and 1 transfer to the 3rd category.
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | 1 | |||||
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 0 |
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | 1 | 1 | ||||
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 |
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 | |||
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 |
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | ||
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
As a result, we get:
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
There was a transfer from a sign discharge. Add it (i.e. 1) to the resulting number (thereby carrying out the procedure of cyclic transfer).
As a result, we get:
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
Result of addition: 00000001. We translate into the decimal representation. To transfer the whole part, you must multiply the discharge of the number to the corresponding degree of discharge.
00000001 = 2 7 *0 + 2 6 *0 + 2 5 *0 + 2 4 *0 + 2 3 *0 + 2 2 *0 + 2 1 *0 + 2 0 *1 = 0 + 0 + 0 + 0 + 0 + 0 + 0 + 1 = 1
Result of addition (in decimal representation): 1
b) 111-010 Imagine the number 111 2 and -010 2 in the reverse code.
The reverse code for a positive number coincides with the direct code. For a negative number, all numbers are replaced with the opposite (1 to 0, 0 to 1), and a unit is entered into the iconic discharge.
Binary number 0000111 has a reverse code 0.0000111
Binary number 0000010 has a reverse code 1.1111101
Moving the number 00000111 and 11111101
In the 0th discharge, an overflow arose (1 + 1 \u003d 10). Therefore, write 0, and 1 transfer to the 1st category.
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | |||||||
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 |
In the 1st discharge, an overflow occurred (1 + 1 \u003d 10). Therefore, we write 0, and 1 transfer to the 2nd category.
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | ||||||
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 |
In the 2nd discharge, an overflow arose (1 + 1 + 1 \u003d 11). Therefore, write 1, and 1 transfer to the 3rd category.
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | 1 | |||||
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 1 | 0 | 0 |
In the 3rd discharge, an overflow occurred (1 + 1 \u003d 10). Therefore, write 0, and 1 transfer to the 4th category.
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | 1 | 1 | ||||
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 1 | 0 | 0 |
In the 4th discharge, an overflow arose (1 + 1 \u003d 10). Therefore, write 0, and 1 transfer to the 5th category.
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 | |||
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
In the 5th discharge, an overflow occurred (1 + 1 \u003d 10). Therefore, write 0, and 1 transfer to the 6th category.
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | ||
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
In the 6th category there was an overflow (1 + 1 \u003d 10). Therefore, write 0, and 1 transfer to the 7th category.
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
In the 7th discharge, an overflow occurred (1 + 1 \u003d 10). Therefore, write 0, and 1 transfer to the 8th category.
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
As a result, we get:
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 |
| 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
There was a transfer from a sign discharge. Add it (i.e. 1) to the resulting number (thereby carrying out the procedure of cyclic transfer).
As a result, we get:
| 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 |
Result of the addition: 00000101
The number 00000101 was obtained. To translate the whole part, you must multiply the discharge of the number to the corresponding degree of discharge.
00000101 = 2 7 *0 + 2 6 *0 + 2 5 *0 + 2 4 *0 + 2 3 *0 + 2 2 *1 + 2 1 *0 + 2 0 *1 = 0 + 0 + 0 + 0 + 0 + 4 + 0 + 1 = 5
Result of addition (in decimal representation): 5
Addition of binary real-sized material numbers
In a computer, any number can be represented in a floating point format. Floating point format is shown in Figure:
For example, the number 10101 in a floating point format can be written as follows:
In computers, the normalized form of the number of the number is used, in which the position of the comma is always set before the meaning of the mantissa, i.e. Condition is satisfied:
B -1 ≤ | M | Normalized number - This is a number that has a significant digit after the comma (ie 1 in a binary number system). An example of normalization:
0,00101*2 100 =0,101*2 10
111,1001*2 10 =0,111001*2 101
0,01101*2 -11 =0,1101*2 -100
11,1011*2 -101 =0,11011*2 -11
When the floating point numbers are addition, alignment of orders are performed towards more of the order: 
Algorithm for addition of floating point numbers:
- Alignment of orders;
- Addition of mantiss in an additional modified code;
- Normalization of the result.
Example number 4.
A \u003d 0,1011 * 2 10, b \u003d 0.0001 * 2 11
1. Alignment of orders;
A \u003d 0,01011 * 2 11, b \u003d 0.0001 * 2 11
2. Addition of mantiss in an additional modified code;
MA extra.mode. \u003d 00,01011
MB extra.mode. \u003d 00.0001
00,01011
+ 00,00010
=
00,01101
A + b \u003d 0,01101 * 2 11
3. Normalization of the result.
A + b \u003d 0,1101 * 2 10
Example number 3. Record a decimal number in a binary-decimal number system and fold two numbers in a binary number system.
Note:
You can only perform actions in one number system if you are given different number systems, first transfer all the numbers into one number system
If you work with a number system, the base of which is more than 10 and in your example met the letter, mentally replace it with the number in the decimal system, draw the necessary operations and translate the result back to the source number system
Addition:
Everyone remembers how in elementary school we were taught to fold the column, the discharge with the discharge. If, when adding in the discharge, a number was obtained more than 9, we subtracted from it 10, the result was recorded in response, and 1 was added to the next discharge. From this you can formulate a rule:
- Fold more convenient to "column"
- Folding downwardly, if the figure is discharged\u003e more the largest digit of the alphabet of this number system, we subtract from this number the base of the number system.
- The result is recorded in the desired discharge
- Add a unit to the next discharge
Fold 1001001110 and 100111101 in a binary number system
1001001110 |
100111101 |
1110001011 |
Answer: 1110001011
Fasten F3B and 5A in a hexadecimal number system
Fe0. |
Answer: Fe0.
Subtraction: Everyone remembers how in elementary school we were taught to deduct the column, the discharge from the category. If, when subtracting in the discharge, there was a number less than 0, we "occupied" a unit from the older discharge and added to the desired figure 10, from the new number it was subtracted. From this you can formulate a rule:
Example:
Subscribe from 1001001110 Number 100111101 in a binary number system
1001001110 |
100111101 |
100010001 |
Answer: 100010001
Release from F3B number 5a in a hexadecimal number system
D9.6 |
Answer: D96.
Most importantly, do not forget about the fact that you have only the numbers of this number system, do not forget about the transitions between the discharge terms.
Multiplication:
Multiplication in other number systems occurs just as we used to multiply.
- Multiply more convenient by the "Stage"
- Multiplication in any number system occurs according to the same rules as in decimal. But we can only use the alphabet, this system Note
Multiply 10111 by number 1101 in a binary number system
10111 |
1101 |
10111 |
10111 |
10111 |
100101011 |
Answer: 100101011
Multiply F3B by number A in a hexadecimal number system
F3B. |
984E. |
Answer: 984E.
Answer: 984E.
Most importantly, do not forget about the fact that you have only the numbers of this number system, do not forget about the transitions between the discharge terms.Division:
Division in other survey systems occurs just as we used to share.
- Sharing more convenient to "column"
- Division in any number system occurs according to the same rules as in decimal. But we can only use alphabet, this number system
Example:
Divided 1011011 to the number 1101 in the binary number system
Split F 3. B for number 8 in a hexadecimal number system
Most importantly, do not forget about the fact that you have only the numbers of this number system, do not forget about the transitions between the discharge terms.
Non-aposition
Non-sample number systems
Non-sample number systems appeared historically first. In these systems, the value of each digital symbol is constantly independent of its position. The simplest case of the non-sacrification system is a single, for which the single symbol is used to designate numbers, as a rule, it is a feature, sometimes a point that the number corresponds to the indicated number is always installed:
- 1 - |
- 2 - ||
- 3 - |||, etc.
Thus, this single symbol is important. unitsFrom which sequential addition obtained the required number:
||||| = 1+1+1+1+1 = 5.
Modification of a single system is a system with a base in which there are characters not only to designate a unit, but also for the degrees of the base. For example, if the base is taken number 5, then there will be additional characters for the notation 5, 25, 125, and so on.
An example of such a system with the base 10 is the ancient Egyptian, which arose in the second half of the third millennium to the new era. This system had the following hieroglyphs:
- six - units,
- arc - dozens,
- palm sheet - hundreds,
- lotus flower - thousands.
The numbers were obtained by simply addiction, the order of the following could be any. So, for the designation, for example, the number 3815, the three lotus flower painted, eight palm leaves, one arc and five poles. More complex systems with additional signs - old Greek, Roman. Roman also uses an element of the positioning system - a large figure that stands in front of the smaller, is added, smaller before - it is subtracted: IV \u003d 4, but Vi \u003d 6, this method, however, is used exclusively to designate numbers 4, 9, 40, 90, 400 , 900, 4000, and their additions.
The newly Russian systems used as numbers 27 letters of the alphabet, where they were designated every number from 1 to 9, as well as tens and hundreds. This approach provided the ability to record numbers from 1 to 999 without repetitions.
In the old-circuit system, special framing around numbers were used to designate large numbers.
As a verbal system, the number is still almost everywhere inspiration. The verbal numbering systems are strongly tied in the language, and their general elements are mainly related to the general principles and names of large numbers (trillion and higher). General principles based on modern verbal numbering damage to the formation of the designation by adding and multiplying the values \u200b\u200bof unique names.
Arithmetic operations in the binary number system
The rules for performing arithmetic actions over binary numbers are set by tables of addition, subtraction and multiplication.
The execution rule of the addition operation is equally for all number systems: if the amount of the figures folded is greater than or equal to the base of the number system, the unit is transferred to the next left discharge. When subtracting, if necessary, make a loan.
Similarly, arithmetic action in octal, hexadecimal and other surcharge systems are performed. In this case, it is necessary to take into account that the value of the transfer in the next discharge when adding and a loan from the older discharge, when subtracting, determines the value of the base of the surcharge system.
Arithmetic operations in the octal number system
To represent numbers in an octal number system, eight digits (0, 1, 2, 3, 4, 5, 6, 7, 7) are used, since the base of the octal number system is 8. All operations are manufactured by these eight digits. Operations of addition and multiplication in the octal number system are manufactured using the following tables:
Tables of addition and multiplication in the octaous number system
|
Example 5.. Interest octal numbers 5153- 1671I2426,63- 1706.71 |
Example 6.Mimal octal numbers51 16I16.6 3.2 |
Arithmetic operations in a hexadecimal number system
To represent numbers in a hexadecimal number system, sixteen digits are used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, 9. in the hexadecimal system numbering in the hexadecimal system. The execution of arithmetic operations in the hexadecimal system is performed as in a decaderal system, but when performing arithmetic operations over large numbers, it is necessary to use the formation tables and multiplication of numbers in a hexadecimal number system.
Table of addition in a hexadecimal number system
Multiplication table in a hexadecimal number system 
|
Example 7. Squeeze hexadecimal numbers |
Adjustment and subtraction of numbers in any positioning system is performed. To find the amount, there are units of the same discharge, starting with the units of the first discharge (right). If the sum of the units of the folded discharge exceeds the number equal to the base of the system, then the unit of the senior discharge is distinguished from this amount, which is added to the adjacent category on the left. Therefore, addition can be done directly, as in the decimal system, in the "Column" using the table of addition of unambiguous numbers.
For example, in a surge system with a base 4, the addition table has this kind:
Yet simply table Additions in the binary number system:
| 0 + 0 = 0 | 0 + 1 = 1 | 1 + 1 = 10. |
|
Example:
|
Subtraction We carry out the same way as in the decimal system: we subscribe subtractable under the reduced and produce the subtraction of numbers in the discharges, starting from the first. If the subtraction of units in the category is impossible, "occupy" a unit in the highest discharge and transform it into units of the neighboring right discharge.
|
Example: 2311 4 - 1223 4 .
|
With the help of this online calculator, you can translate entire 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.
The result is already received!
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 systems Number positioning uniquely determines the number of numbers. 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 to another to another, the easiest way to first translate the number to a decimal number system, and then, from the 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.- by 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. hexadecimal - This is 4CD9.
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.
