The calculator allows you to convert whole and fractional numbers from one number systems 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). The length of 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 Record" button.
Original number written 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 number written 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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Number systems
Number systems are divided into two types: positional And not positional. We use the Arabic system, it is positional, but there is also the Roman system - it is 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 some number as an example.
Example 1. Let's take the number 5921 in the decimal number system. Let's 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 a given number are taken as powers.
Example 2. Consider the real decimal number 1234.567. Let's number it starting from the zero position of the number from the decimal point to the left and right:
The number 1234.567 can be written in the following form: 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 converting a number from one number system to another is converting the number first to decimal system number, and then, the resulting result into the required number system.
Converting numbers from any number system to the decimal number system
To convert a number from any number system to decimal, it is enough to number its digits, starting with 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 the number 1001101.1101 2 to the 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 the number E8F.2D 16 to the 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 the decimal number system to another number system
To convert numbers from the decimal number system to another number system, the integer and fractional parts of the number must be converted separately.
Converting an integer part of a number from a decimal number system to another number system
An integer part is converted from a decimal number system to another number system by sequentially dividing the integer part of a number by the base of the number system until a whole remainder is obtained that is less than the base of the number system. The result of the translation will be a record of the remainder, starting with the last one.
3.
Convert the number 273 10 to the 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 balances 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. This means the translation was done correctly.
Answer: 273 10 = 421 8
Consider the translation of proper decimal fractions into various systems Reckoning.
Converting the fractional part of a number from the decimal number system to another number system
Recall that a proper decimal fraction is called real number with zero integer part. To convert such a number to a number system with base N, you need to sequentially multiply the number by N until the fractional part is zeroed or the required number of digits is obtained. If multiplication results in a number with an integer part other than zero, then whole part is not taken into account further, since it is sequentially entered into the result.
4.
Convert the number 0.125 10 to the binary number system.
Solution: 0.125·2 = 0.25 (0 is the integer part, which will become 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 , then the translation is completed).
Answer: 0.125 10 = 0.001 2
Purpose of the work. Studying methods and developing skills for converting numbers from one positional number system to another.
The number of different digits used in a positional system determines the name of the number system and is called basis
th number system.
Any number N in a positional number system with a base 
can be represented as a polynomial from the base 
:
Where 
- number, 
- digits of the number (coefficients at powers 
),
- base of the number system ( 
>1).
Numbers are written as a sequence of numbers:
.
, a point in the sequence separates the integer part of the number from the fractional part (coefficients for non-negative powers, from coefficients for negative powers). The dot is omitted if the number is an integer (no negative powers).
Used in computer systems positioning systems Numbers with non-decimal base: binary, octal, hexadecimal.
The computer hardware is based on two-position elements that can only be in two states; one of which is designated 0, and the other - 1. Therefore, the arithmetic-logical main computer is the binary number system.
Binary number system. Two digits are used: 0 and 1. In the binary system, any number can be represented as: 
.
, Where 
either 0 or 1.
This entry corresponds to the sum of the powers of 2 taken with the indicated coefficients:
Octal number system. Eight digits are used: 0, 1, 2, 3, 4, 5, 6, 7. Used in a computer as auxiliary for recording information in abbreviated form. To represent a single digit octal system three binary digits (triad) are used (see Table 1).
Hexadecimal number system. 16 digits are used to represent numbers. The first ten digits of this system are designated by numbers from 0 to 9, and the upper six digits by Latin letters: A (10), B (11), C (12), D (13), E (14), F (15). The hexadecimal system, like the octal system, is used to record information in abbreviated form. To represent one digit of the hexadecimal number system, four binary digits (tetrad) are used (see Table 1).
Table 1.
Alphabets of positional number systems (ss)
|
Binary ss (Base 2) |
Octal ss (Base 8) |
Decimal ss (Base 10) |
Hexadecimal ss (Base 16) |
||
|
Binary |
Binary tetrads |
||||
Task 1. Convert numbers from given systems numbers in the decimal system.
Methodical instructions.
Converting numbers to the decimal system is carried out by compiling the sum of a power series with the base of the system from which the number is being converted. The value of this amount is then calculated.
Examples.
a) Translate s.s.
.
b) Translate 
s.s.
c) Translate 
s.s.
Task 2. Convert integers from decimal to octal, hexadecimal and binary.
Methodical instructions.
Conversion of integer decimal numbers into octal, hexadecimal and binary systems is carried out by sequentially dividing the decimal number by the base of the system into which it is converted until the quotient equals zero. The number in the new system is written as division remainders, starting from the last one.
Examples.
a) Translate 
s.s.
181: 8 = 22 (remainder 5)
22: 8 = 2 (remainder 6)
2: 8 = 0 (remainder 2)
Answer: 
.
b) Translate 
s.s.
The table shows the division:
622: 16 = 38 (remainder 14 10 = E 16)
38: 16 = 2 (remainder 6)
2: 16 = 0 (remainder 2)
Answer: 
.
Task 3. Convert regular decimals from decimal to octal, hexadecimal and binary.
With 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, specify the base of the number system of the original number, specify the base of the number system into which you want to convert the number and click on the "Translate" button. See the theoretical part and numerical examples below.
The result has already been received!
Converting integers and fractions 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, but Roman is not. In positional number systems, the position of a number uniquely determines the magnitude of the number. Let's 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 determines the number system (in this case it is 10). The values of the position of a given number are taken as powers.
Consider the real decimal number 1287.923. Let's number it starting from zero, position of the number from the decimal point to the left and 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 many digits (0,1,2,3,4,5,6,7,8,9), in the octal number system it consists of many digits (0,1, 2,3,4,5,6,7), in the binary number system - from a set of digits (0,1), in the hexadecimal number system - from a 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 the numbers 10,11,12,13,14,15. In the table Tab.1 numbers are presented 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 convert numbers from one number system to another, the easiest way is to first convert the number to the decimal number system, and then convert from the decimal number system to the required number system.
Converting numbers from any number system to the 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 number system to decimal SS. Solution:
Here A-replaced by 10, B- at 11, C- at 12, F- by 15.
Converting numbers from the decimal number system to another number system
To convert numbers from the decimal number system to another number system, you need to separately convert the whole part of the number and fractional part numbers.
The integer part of a number is converted from decimal SS to another number system by sequentially dividing the integer part of the number by the base of the number system (for binary SS - by 2, for 8-ary SS - by 8, for 16-ary SS - by 16, etc. ) until a whole residue is obtained, less than the base CC.
Example 4 . Let's convert 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 remainder 1. Further, the number 79 when divided by 2 gives the quotient 39 and remainder 1, etc. As a result, constructing a number from division remainders (from right to left), we obtain 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, constructing a number from division remainders (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 convert 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, the remainders are 4, 12, 13, 9. In the hexadecimal number system, the number 12 corresponds to C, the number 13 to D. Therefore, our hexadecimal number is 4CD9.
To convert regular decimal fractions (a real number with a zero integer part) into a number system with base s, it is necessary to sequentially multiply this number by s until the fractional part is pure zero, or we obtain 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 included in the result).
Let's look at the above with examples.
Example 7 . Let's convert 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 sequentially 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 the multiplication results in a number with a zero integer part, then a zero is written to the left of it. The multiplication process continues until the fractional part reaches a pure zero or we obtain the required number of digits. Writing bold numbers (Fig. 4) from top to bottom we get the required number in the binary number system: 0. 0011011 .
Therefore we can write:
0.214 10 =0.0011011 2 .
Example 8 . Let's convert 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 convert the number 0.125 from decimal SS to binary, this number is sequentially multiplied by 2. In the third stage, the result is 0. Consequently, the following result is obtained:
0.125 10 =0.001 2 .
Example 9 . Let's convert 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 12 and 11 correspond to the numbers C and B. Therefore, we have:
0.214 10 =0.36C8B4 16 .
Example 10 . Let's convert the number 0.512 from the decimal number system to 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 . Let's convert 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). Further combining these results we get:
159.125 10 =10011111.001 2 .
Example 12 . Let's convert 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 obtain.
All positional number systems are equal, but depending on the problems that a person solves using numbers, he can use number systems with different bases.
The most commonly used number system is the decimal number system, i.e. a number system whose alphabet consists of ten digits (0,1,2,3,4,5,6,7,8,9) and, accordingly, the base is equal to ten. The widespread use of this number system is easily explained. Firstly, writing a number in the decimal number system is quite compact, and secondly, the decimal number system has been used by humanity for several centuries. During this time, people have become accustomed to numbers, to writing numbers, and to pronouncing numbers in the decimal number system, for example, the entry “15” is understandable to any person and he will read it as fifteen, but the same number written in the binary number system “1111” causes at least slight bewilderment as to how to read this number.
And yet it is unequivocal to state that the decimal number system is optimal choice humanity cannot work with numbers. Let's prove this with several examples.
You all remember the multiplication table and, of course, you remember how much effort you had to put in to learn this table. We will not give here the multiplication table in the decimal number system, but for comparison we give the multiplication table in the binary number system:
As you can see, the multiplication table in the binary number system looks much simpler than in the decimal number system.
The compactness of writing numbers in the decimal number system is also not the highest; in all number systems with a base greater than ten, numbers will be written more compactly, for example, the same number “15” will be written as “F” in the hexadecimal number system.
As already mentioned in paragraph 5, the binary number system is adopted for recording numbers in a computer. In this paragraph we must understand how numbers are represented in computer memory; for this it will be enough to understand the rules for converting decimal numbers into the binary number system.
In practice, to convert numbers from a base ten number system to a base two number system, use the following rule:
1. A number written in a number system with base ten is divided with a remainder by two (base new system notation), written in digits of the base ten number system ( old system number), until the quotient turns out to be 0.
2. The remainders obtained from division, written in reverse order, form a number in the new number system with base two.
This rule is more convenient to use for converting numbers from the decimal number system. To convert back to the decimal number system, it is more convenient to use the so-called Horner scheme.
1.Number the positions in the number, from right to left, starting from zero;
2. Compose a series representing the sum of the products of the digits of a number by the base of the old number system, written in the digits of the new number system, raised to a power equal to the position number of the digit in the number;
3. Find the sum of the series.
Let's look at these rules using specific examples.
Example 1: Write the decimal number 121 in the binary number system.
121 | 2 121 D =1111001 B
120 60 | 2
1 60 30 | 2
0 30 15 | 2
0 14 7 | 2
1 6 3 | 2
