With the help of this online calculator You can convert whole and fractional numbers from one number system to another. A detailed solution with explanations is given. To translate, enter the original number, set the base of the number system of the original number, set the base of the number system to which you want to convert the number and click the "Translate" button. See the theoretical part and numerical examples below.
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Translation of integer and fractional numbers from one number system to any other - theory, examples and solutions
There are positional and non-positional number systems. The Arabic number system that we use in everyday life is positional, while the Roman one is not. In positional number systems, the position of a number uniquely determines the magnitude of the number. Consider this using the example of the number 6372 in the decimal number system. Let's number this number from right to left starting from zero:
Then the number 6372 can be represented as follows:
6372=6000+300+70+2 =6 10 3 +3 10 2 +7 10 1 +2 10 0 .
The number 10 defines the number system (in this case it is 10). The values of the position of the given number are taken as degrees.
Consider the real decimal number 1287.923. We number it starting from the zero position of the number from the decimal point to the left and to the right:
Then the number 1287.923 can be represented as:
1287.923 =1000+200+80 +7+0.9+0.02+0.003 = 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 an integer in position n, D -k - fractional number in position (-k), s- number system.
A few words about number systems. A number in the decimal number system consists of a set of digits (0,1,2,3,4,5,6,7,8,9), in the octal number system it consists of a set of digits (0,1, 2,3,4,5,6,7), in the binary system - from the set of digits (0.1), in the hexadecimal number system - from the set of digits (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 numbers 10,11,12,13,14,15. In Table 1 presented numbers in different systems reckoning.
| 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 |
Converting numbers from one number system to another
To translate numbers from one number system to another, the easiest way is to first convert the number to the decimal number system, and then, from the decimal number system, translate it into the required number system.
Converting numbers from any number system to decimal number system
Using formula (1), you can convert numbers from any number system to the decimal number system.
Example 1. Convert the number 1011101.001 from binary number system (SS) to decimal SS. Solution:
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 =64+16+8+4+1+1/8=93.125
Example2. Convert the number 1011101.001 from octal number system (SS) to decimal SS. Solution:
Example 3 . Convert the number AB572.CDF from hexadecimal to decimal SS. Solution:
Here A-replaced by 10, B- at 11, C- at 12, F- at 15.
Converting numbers from a decimal number system to another number system
To translate numbers from the decimal number system to another number system, you need to translate the integer part of the number separately and fractional part numbers.
The integer part of the number is translated from the decimal SS to another number system - by successive division of the integer part of the number by the base of the number system (for binary SS - by 2, for 8-digit SS - by 8, for 16-digit - by 16, etc. ) to obtain a whole remainder, less than the base of the SS.
Example 4 . Let's translate the number 159 from decimal SS to 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, when divided by 2, gives the quotient 79 and the remainder is 1. Further, the number 79, when divided by 2, gives the quotient 39 and the remainder is 1, and so on. As a result, by constructing a number from the remainder of the division (from right to left), we get a number in binary SS: 10011111 . Therefore, we can write:
159 10 =10011111 2 .
Example 5 . Let's convert the number 615 from decimal SS to octal SS.
| 615 | 8 | ||
| 608 | 76 | 8 | |
| 7 | 72 | 9 | 8 |
| 4 | 8 | 1 | |
| 1 |
When converting a number from decimal SS to octal SS, you need to sequentially divide the number by 8 until you get an integer remainder less than 8. As a result, building a number from the remainder of the division (from right to left) we get a number in octal SS: 1147 (see Fig. 2). Therefore, we can write:
615 10 =1147 8 .
Example 6 . Let's translate 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 Figure 3, by successively dividing the number 19673 by 16, we got the remainders 4, 12, 13, 9. In the hexadecimal number system, the number 12 corresponds to C, the number 13 - D. Therefore, our hexadecimal number is 4CD9.
To convert correct decimals ( real number with zero integer part) into a number system with base s, it is necessary to multiply this number by s in succession until the fractional part is net zero, or we will not get the required number of digits. If the multiplication results in a number with an integer part other than zero, then this integer part is not taken into account (they are sequentially added to the result).
Let's look at the above with examples.
Example 7 . Let's translate 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 successively multiplied by 2. If the result of multiplication is a number with an integer part other than zero, then the integer part is written separately (to the left of the number), and the number is written with a zero integer part. If, when multiplied, a number with a zero integer part is obtained, then zero is written to the left of it. The multiplication process continues until a pure zero is obtained in the fractional part or the required number of digits is obtained. Writing bold numbers (Fig. 4) from top to bottom, we get the required number in the binary system: 0. 0011011 .
Therefore, we can write:
0.214 10 =0.0011011 2 .
Example 8 . Let's translate the number 0.125 from the decimal number system to the binary SS.
| 0.125 | ||
| x | 2 | |
| 0 | 0.25 | |
| x | 2 | |
| 0 | 0.5 | |
| x | 2 | |
| 1 | 0.0 |
To convert the number 0.125 from decimal SS to binary, this number is successively multiplied by 2. In the third stage, 0 was obtained. Therefore, the following result was obtained:
0.125 10 =0.001 2 .
Example 9 . Let's 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 get the numbers 3, 6, 12, 8, 11, 4. But in hexadecimal SS, the numbers C and B correspond to the numbers 12 and 11. Therefore, we have:
0.214 10 =0.36C8B4 16 .
Example 10 . Let's translate the number 0.512 from the decimal number system to 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 |
Got:
0.512 10 =0.406111 8 .
Example 11 . Let's translate the number 159.125 from the decimal number system to binary SS. To do this, we translate separately the integer part of the number (Example 4) and the fractional part of the number (Example 8). Combining these results, we get:
159.125 10 =10011111.001 2 .
Example 12 . Let's translate the number 19673.214 from the decimal number system to hexadecimal SS. To do this, we translate separately the integer part of the number (Example 6) and the fractional part of the number (Example 9). Further combining these results we get.
The calculator allows you to convert whole and fractional numbers from one number system to another. The base of the number system cannot be less than 2 and more than 36 (10 digits and 26 Latin letters, after all). Numbers must not exceed 30 characters. To enter fractional numbers, use the symbol. or, . To convert a number from one system to another, enter the original number in the first field, the base of the original number system in the second, and the base of the number system to which you want to convert the number in the third field, then click the "Get Entry" button.
original 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 -th number system.
I want to get a record of a 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 -th number system.
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Translations completed: 3446071
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Number systems
Number systems are divided into two types: positional and not positional. We use the Arabic system, it is positional, and there is also the Roman one - it is just not positional. In positional systems, the position of a digit in a number uniquely determines the value of that number. This is easy to understand by looking at the example of some number.
Example 1. Let's take the number 5921 in the decimal number system. We number the number from right to left starting from zero:
The number 5921 can be written in the following form: 5921 = 5000+900+20+1 = 5 10 3 +9 10 2 +2 10 1 +1 10 0 . The number 10 is a characteristic that defines the number system. The values of the position of the given number are taken as degrees.
Example 2. Consider the real decimal number 1234.567. We number it starting from the zero position of the number from the decimal point to the left and to the right:
The number 1234.567 can be written as follows: 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 .
Converting numbers from one number system to another
Most in a simple way transferring a number from one number system to another is the translation of the number first into the decimal number system, and then, the result obtained into the required number system.
Converting numbers from any number system to decimal number system
To convert a number from any number system to decimal, it is enough to number its digits, starting from zero (the digit to the left of the decimal point) similarly to examples 1 or 2. Let's find the sum of the products of the digits of the number by the base of the number system to the power of the position of this digit:
1.
Convert number 1001101.1101 2 to decimal number system.
Solution: 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
Answer: 10011.1101 2 = 19.8125 10
2.
Convert number E8F.2D 16 to decimal number system.
Solution: 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
Answer: E8F.2D 16 = 3727.17578125 10
Converting numbers from a decimal number system to another number system
To convert numbers from a decimal number system to another number system, the integer and fractional parts of the number must be translated separately.
Converting the integer part of a number from a decimal number system to another number system
The integer part is converted from the decimal number system to another number system by successively dividing the integer part of the number by the base of the number system until an integer remainder is obtained, which is less than the base of the number system. The result of the transfer will be a record from the remains, starting with the last one.
3.
Convert number 273 10 to octal number system.
Solution: 273 / 8 = 34 and remainder 1, 34 / 8 = 4 and remainder 2, 4 is less than 8, so the calculation is complete. The record from the remnants will look like this: 421
Examination: 4 8 2 +2 8 1 +1 8 0 = 256+16+1 = 273 = 273 , the result is the same. So the translation is correct.
Answer: 273 10 = 421 8
Let's consider the translation of correct decimal fractions into various number systems.
Converting the fractional part of a number from a decimal number system to another number system
Recall that a proper decimal fraction is real number with zero integer part. To translate such a number into a number system with base N, you need to consistently multiply the number by N until the fractional part is zeroed or the required number of digits is obtained. If during multiplication a number with an integer part other than zero is obtained, then the integer part is not taken into account further, since it is sequentially entered into the result.
4.
Convert number 0.125 10 to binary number system.
Solution: 0.125 2 = 0.25 (0 is the integer part, which will be the first digit of the result), 0.25 2 = 0.5 (0 is the second digit of the result), 0.5 2 = 1.0 (1 is the third digit of the result, and since the fractional part is zero , the translation is complete).
Answer: 0.125 10 = 0.001 2
You can enter either whole numbers, such as 34 , or fractional numbers, such as 637.333 . For fractional numbers, the accuracy of the translation after the decimal point is indicated.
The following are also used with this calculator:
Ways to represent numbers
Binary (binary) numbers - each digit means the value of one bit (0 or 1), the most significant bit is always written on the left, the letter “b” is placed after the number. For ease of perception, notebooks can be separated by spaces. For example, 1010 0101b.Hexadecimal (hexadecimal) numbers - each tetrad is represented by one character 0 ... 9, A, B, ..., F. Such a representation can be denoted in different ways, only the character “h” is used here after the last hexadecimal digit. For example, A5h. In program texts, the same number can be denoted both as 0xA5 and 0A5h, depending on the syntax of the programming language. A non-significant zero (0) is added to the left of the most significant hexadecimal digit represented by a letter to distinguish between numbers and symbolic names.
Decimals (decimal) numbers - each byte (word, double word) is represented by an ordinary number, and the sign of the decimal representation (letter "d") is usually omitted. The byte from the previous examples has a decimal value of 165. Unlike binary and hexadecimal notation, decimal is difficult to mentally determine the value of each bit, which sometimes has to be done.
Octal (octal) numbers - each triple of bits (separation starts from the least significant) is written as a number 0-7, at the end the sign "o" is put. The same number would be written as 245o. The octal system is inconvenient in that the byte cannot be divided equally.
Algorithm for converting numbers from one number system to another
Whole translation decimal numbers to any other number system is carried out by dividing the number by the base new system numbering until the remainder remains a number less than the base of the new number system. The new number is written as the remainder of the division, starting with the last one.The conversion of the correct decimal fraction to another PSS is carried out by multiplying only the fractional part of the number by the base of the new number system until all zeros remain in the fractional part or until the specified translation accuracy is reached. As a result of each multiplication operation, one digit of the new number is formed, starting from the highest.
The translation of an improper fraction is carried out according to the 1st and 2nd rules. The integer and fractional parts are written together, separated by a comma.
Example #1. 
Translation from 2 to 8 to 16 number system.
These systems are multiples of two, therefore, the translation is carried out using the correspondence table (see below).
To convert a number from a binary number system to an octal (hexadecimal) number, it is necessary to divide the binary number into groups of three (four for hexadecimal) digits from a comma to the right and left, complementing the extreme groups with zeros if necessary. Each group is replaced by the corresponding octal or hexadecimal digit.
Example #2. 1010111010.1011 = 1.010.111.010.101.1 = 1272.51 8
here 001=1; 010=2; 111=7; 010=2; 101=5; 001=1
When converting to hexadecimal, you must divide the number into parts, four digits each, following the same rules.
Example #3. 1010111010.1011 = 10.1011.1010.1011 = 2B12.13 HEX
here 0010=2; 1011=B; 1010=12; 1011=13
The conversion of numbers from 2, 8 and 16 to the decimal system is carried out by breaking the number into separate ones and multiplying it by the base of the system (from which the number is translated) raised to the power corresponding to it serial number in the translated number. In this case, the numbers are numbered to the left of the decimal point (the first number has the number 0) with increasing, and to the right with decreasing (ie with a negative sign). The results obtained are added up.
Example #4.
Example of converting from binary to decimal number system.
1010010.101 2 = 1 2 6 +0 2 5 +1 2 4 +0 2 3 +0 2 2 +1 2 1 +0 2 0 + 1 2 -1 +0 2 - 2 +1 2 -3 =
= 64+0+16+0+0+2+0+0.5+0+0.125 = 82.625 10 Example of conversion from octal to decimal number system. 108.5 8 = 1* 8 2 +0 8 1 +8 8 0 + 5 8 -1 = 64+0+8+0.625 = 72.625 10 An example of converting from hexadecimal to decimal number system. 108.5 16 = 1 16 2 +0 16 1 +8 16 0 + 5 16 -1 = 256+0+8+0.3125 = 264.3125 10
Once again, we repeat the algorithm for translating numbers from one number system to another PSS
- From the decimal number system:
- divide the number by the base of the number system being translated;
- find the remainder after dividing the integer part of the number;
- write down all remainders from division in reverse order;
- From the binary system
- To convert to the decimal number system, you need to find the sum of the products of base 2 by the corresponding degree of discharge;
- To convert a number to octal, you need to break the number into triads.
For example, 1000110 = 1000 110 = 106 8 - To convert a number from binary to hexadecimal, you need to divide the number into groups of 4 digits.
For example, 1000110 = 100 0110 = 46 16
Table of correspondence of number systems:
| Binary SS | Hexadecimal SS |
| 0000 | 0 |
| 0001 | 1 |
| 0010 | 2 |
| 0011 | 3 |
| 0100 | 4 |
| 0101 | 5 |
| 0110 | 6 |
| 0111 | 7 |
| 1000 | 8 |
| 1001 | 9 |
| 1010 | A |
| 1011 | B |
| 1100 | C |
| 1101 | D |
| 1110 | E |
| 1111 | F |
Table to convert to octal system reckoning
Example #2. Convert the number 100.12 from decimal to octal and vice versa. Explain the reasons for the discrepancies.
Solution.
Stage 1. .
The remainder of the division is written in reverse order. We get the number in the 8th number system: 144
100 = 144 8
To translate the fractional part of a number, we successively multiply the fractional part by base 8. As a result, each time we write down the integer part of the product.
0.12*8 = 0.96 (whole part 0
)
0.96*8 = 7.68 (whole part 7
)
0.68*8 = 5.44 (whole part 5
)
0.44*8 = 3.52 (whole part 3
)
We get the number in the 8th number system: 0753.
0.12 = 0.753 8
100,12 10 = 144,0753 8
Stage 2. Converting a number from decimal to octal.
Reverse conversion from octal to decimal.
To translate the integer part, it is necessary to multiply the digit of the number by the corresponding degree of digit.
144 = 8 2 *1 + 8 1 *4 + 8 0 *4 = 64 + 32 + 4 = 100
To translate the fractional part, it is necessary to divide the digit of the number by the corresponding degree of digit
0753 = 8 -1 *0 + 8 -2 *7 + 8 -3 *5 + 8 -4 *3 = 0.119873046875 = 0.1199
144,0753 8 = 100,96 10
The difference of 0.0001 (100.12 - 100.1199) is due to a rounding error when converting to octal. This error can be reduced if we take a larger number of digits (for example, not 4, but 8).