Educational complex Tue. Transfer from one number system to another Rules for transfer from 10 to 2
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). 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.
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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
The simplest way to convert a number from one number system to another is to first convert the number to the decimal number system, and then the resulting result to 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
Let's consider the translation of regular decimal fractions into various number systems.
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 goes to zero 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 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
Converting numbers from one number system to another is an important part of machine arithmetic. Let's consider the basic rules of translation.
1. To convert a binary number to a decimal one, it is necessary to write it in the form of a polynomial, consisting of the products of the digits of the number and the corresponding power of 2, and calculate it according to the rules of decimal arithmetic:
When translating, it is convenient to use the table of powers of two:
Table 4. Powers of number 2
|
n (degree) |
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Example.
2. To convert an octal number to a decimal one, it is necessary to write it down as a polynomial consisting of the products of the digits of the number and the corresponding power of the number 8, and calculate it according to the rules of decimal arithmetic:
When translating, it is convenient to use the table of powers of eight:
Table 5. Powers of the number 8
|
n (degree) |
|||||||
Example. Convert the number to the decimal number system.
3. To convert a hexadecimal number to a decimal one, it is necessary to write it in the form of a polynomial, consisting of the products of the digits of the number and the corresponding power of the number 16, and calculate it according to the rules of decimal arithmetic:
When translating, it is convenient to use blitz of powers of number 16:
Table 6. Powers of the number 16
|
n (degree) |
|||||||
Example. Convert the number to the decimal number system.
4. To convert a decimal number to the binary system, it must be sequentially divided by 2 until a remainder less than or equal to 1 remains. A number in the binary system is written as a sequence of the last division result and the remainders from the division in reverse order.
Example. Convert the number to the binary number system.
5. To convert a decimal number to the octal system, it must be sequentially divided by 8 until a remainder less than or equal to 7 remains. A number in the octal system is written as a sequence of digits of the last division result and the remainder of the division in reverse order.
Example. Convert the number to the octal number system.
6. To convert a decimal number to the hexadecimal system, it must be sequentially divided by 16 until there is a remainder less than or equal to 15. A number in the hexadecimal system is written as a sequence of digits of the last division result and the remainders from the division in reverse order.
Example. Convert the number to hexadecimal number system.
From 16 or 8 to 2
| Translation octal And hexadecimal numbers to binary system very simple: just replace each digit with its binary equivalent triad(three digits) or notebook(four digits) (see table). | |||||||
| Binary (Radise 2) | Octal (Base 8) | Decimal (Base 10) | Hexadecimal (Base 16) | ||||
| triads | tetrads | ||||||
| 0 1 | 0 1 2 3 4 5 6 7 | 000 001 010 011 100 101 110 111 | 0 1 2 3 4 5 6 7 8 9 | 0 1 2 3 4 5 6 7 8 9 A B C D E F | 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 |
For example:
a) Translate 305.4 8 "2" s.s.
b) Translate 7B2.E 16 "2" s.s.
16A 16 =1 0110 1010 2 345 8 =11 100 101 2
From 2 to 16 or 8
For example:
a) Translate 1101111001.1101 2 "8" s.s.
b) Translate 11111111011.100111 2 "16" s.s.
1000101010010101 2 =1000 1010 1001 0101=8A95 16 = 1 000 101 010 010 101=105225 8
From 16 to 8 and back
Conversion from octal to hexadecimal and back is carried out through the binary system using triads and tetrads.
For example:
Translate 175.24 8 "16" s.s.
Result: 175.24 8 = 7D.5 16.
From 10 to any s.s.
For example:
a) Translate 181 10 "8" s.s.
Result: 181 10 = 265 8
b) Translate 622 10 "16" s.s.
Result: 622 10 = 26E 16
Translation of proper fractions
To convert a regular decimal fraction to another system, this fraction must be sequentially multiplied by the base of the system into which it is converted. In this case, only fractional parts are multiplied. Fractions in the new system are written in the form of whole parts of products, starting from the first.
For example:
Convert 0.3125 10 "8" s.s.
Result: 0.3125 10 = 0.24 8
Comment. A final decimal fraction in another number system may correspond to an infinite (sometimes periodic) fraction. In this case, the number of characters in the representation of a fraction in the new system is taken depending on the required accuracy.
For example:
Convert 0.65 10 "2" s.s. Accuracy 6 digits.
Result: 0.65 10 0.10(1001) 2
To convert an improper decimal fraction to a number system with a non-decimal base It is necessary to translate the whole part and the fractional part separately.
For example:
Translate 23.125 10 "2" s.s.
Thus: 23 10 = 10111 2 ; 0.125 10 = 0.001 2.
Result: 23.125 10 = 10111.001 2.
It should be noted that integers remain integers, and proper fractions remain fractions in any number system.
From 2, 8 or 16 to 10
For example:
a)10101101.101 2 = 1 2 7 + 0 2 6 + 1 2 5 + 0 2 4 + 1 2 3 + 1 2 2 + 0 2 1 + 1 2 0 + 1 2 -1 + 0 2 -2 + 1 2 - 3 = 173.625 10
b) Translate 703.04 8 "10" s.s.
703.04 8 = 7 8 2 + 0 8 1 + 3 8 0 + 0 8 -1 + 4 8 -2 = 451.0625 10
c) Translate B2E.4 16 "10" s.s.
B2E.4 16 = 11 16 2 + 2 16 1 + 14 16 0 + 4 16 -1 = 2862.25 10
Scheme for converting numbers from one number system to another
Arithmetic operations in positional number systems
Let's look at the basic arithmetic operations: addition, subtraction, multiplication and division. The rules for performing these operations in the decimal system are well known - these are addition, subtraction, multiplication by a column and division by an angle. These rules apply to all other positional number systems. Only addition and multiplication tables must be used specific for each system.
Addition
When adding, the numbers are summed up by digits, and if there is an excess, it is transferred to the left
When adding binary numbers in each digit, the digits of the terms are added and transferred from the adjacent low-order digit, if any. It is necessary to take into account that 1+1 gives a zero in a given digit and a carry unit to the next one.
For example:
Perform addition of binary numbers:
a) X=1101, Y=101;
Result 1101+101=10010.
b) X=1101, Y=101, Z=111;
Result 1101+101+111=11001.
Addition table in the 8th number system
| 2+2=4 | 3+2=5 | 4+2=6 | 5+2=7 | 6+2=10 | 7+2=11 |
| 2+3=5 | 3+3=6 | 4+3=7 | 5+3=10 | 6+3=11 | 7+3=12 |
| 2+4=6 | 3+4=7 | 4+4=10 | 5+4=11 | 6+4=12 | 7+4=13 |
| 2+5=7 | 3+5=10 | 4+5=11 | 5+5=12 | 6+5=13 | 7+5=14 |
| 2+6=10 | 3+6=11 | 4+6=12 | 5+6=13 | 6+6=14 | 7+6=15 |
| 2+7=11 | 3+7=12 | 4+7=13 | 5+7=14 | 6+7=15 | 7+7=16 |
Addition table in the 16th number system
| + | A | B | C | D | E | F | ||||||||||
| A | B | C | D | E | F | |||||||||||
| A | B | C | D | E | F | |||||||||||
| A | B | C | D | E | F | |||||||||||
| A | B | C | D | E | F | |||||||||||
| A | B | C | D | E | F | |||||||||||
| A | B | C | D | E | F | |||||||||||
| A | B | C | D | E | F | |||||||||||
| A | B | C | D | E | F | |||||||||||
| A | B | C | D | E | F | |||||||||||
| A | B | C | D | E | F | |||||||||||
| A | A | B | C | D | E | F | ||||||||||
| B | B | C | D | E | F | 1A | ||||||||||
| C | C | D | E | F | 1A | 1B | ||||||||||
| D | D | E | F | 1A | 1B | 1C | ||||||||||
| E | E | F | 1A | 1B | 1C | 1D | ||||||||||
| F | F | 1A | 1B | 1C | 1D | 1E |
Using 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 source number, set 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, which we use in everyday life, is positional, but the Roman number system 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 number systems.
| 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 convert the integer part of the number and the fractional part of the number separately.
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 a decimal SS to an 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 successively multiply this number by s until the fractional part contains a pure zero, or we obtain the required number of digits. If, during multiplication, a number with an integer part other than zero is obtained, 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 |
Got:
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.
Those taking the Unified State Exam and more...
It is strange that in computer science lessons in schools they usually show students the most complex and inconvenient way to convert numbers from one system to another. This method consists of sequentially dividing the original number by the base and collecting the remainders from the division in reverse order.
For example, you need to convert the number 810 10 to binary:
We write the result in reverse order from bottom to top. It turns out 81010 = 11001010102
If you need to convert fairly large numbers into the binary system, then the division ladder takes on the size of a multi-story building. And how can you collect all the ones and zeros and not miss a single one?
The Unified State Exam program in computer science includes several tasks related to converting numbers from one system to another. Typically, this is a conversion between octal and hexadecimal systems and binary. These are sections A1, B11. But there are also problems with other number systems, such as in section B7.
To begin with, let us recall two tables that would be good to know by heart for those who choose computer science as their future profession.
Table of powers of number 2:
| 2 1 | 2 2 | 2 3 | 2 4 | 2 5 | 2 6 | 2 7 | 2 8 | 2 9 | 2 10 |
| 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 | 512 | 1024 |
It is easily obtained by multiplying the previous number by 2. So, if you do not remember all of these numbers, the rest are not difficult to obtain in your mind from those that you remember.
Table of binary numbers from 0 to 15 with hexadecimal representation:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
| 0000 | 0001 | 0010 | 0011 | 0100 | 0101 | 0110 | 0111 | 1000 | 1001 | 1010 | 1011 | 1100 | 1101 | 1110 | 1111 |
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | A | B | C | D | E | F |
The missing values are also easy to calculate by adding 1 to the known values.
Integer conversion
So, let's start by converting directly to the binary system. Let's take the same number 810 10. We need to decompose this number into terms equal to powers of two.
- We are looking for the power of two closest to 810 and not exceeding it. This is 2 9 = 512.
- Subtract 512 from 810, we get 298.
- Repeat steps 1 and 2 until there are no 1s or 0s left.
- We got it like this: 810 = 512 + 256 + 32 + 8 + 2 = 2 9 + 2 8 + 2 5 + 2 3 + 2 1.
Method 1: Arrange 1 according to the ranks of the indicators of the terms. In our example, these are 9, 8, 5, 3 and 1. The remaining places will contain zeros. So, we got the binary representation of the number 810 10 = 1100101010 2. Units are placed in 9th, 8th, 5th, 3rd and 1st places, counting from right to left from zero.
Method 2: Let's write the terms as powers of two under each other, starting with the largest.
810 =
Now let's add these steps together, like folding a fan: 1100101010.
That's all. At the same time, the problem “how many units are in the binary notation of the number 810?” is also simply solved.
The answer is as many as there are terms (powers of two) in this representation. 810 has 5 of them.
Now the example is simpler.
Let's convert the number 63 to the 5-ary number system. The closest power of 5 to 63 is 25 (square 5). A cube (125) will already be a lot. That is, 63 lies between the square of 5 and the cube. Then we will select the coefficient for 5 2. This is 2.
We get 63 10 = 50 + 13 = 50 + 10 + 3 = 2 * 5 2 + 2 * 5 + 3 = 223 5.
And, finally, very easy translations between 8 and hexadecimal systems. Since their base is a power of two, the translation is done automatically, simply by replacing the numbers with their binary representation. For the octal system, each digit is replaced by three binary digits, and for the hexadecimal system, four. In this case, all leading zeros are required, except for the most significant digit.
Let's convert the number 547 8 to binary.
| 547 8 = | 101 | 100 | 111 |
| 5 | 4 | 7 |
One more, for example 7D6A 16.
| 7D6A 16 = | (0)111 | 1101 | 0110 | 1010 |
| 7 | D | 6 | A |
Let's convert the number 7368 to the hexadecimal system. First, write the numbers in triplets, and then divide them into quadruples from the end: 736 8 = 111 011 110 = 1 1101 1110 = 1DE 16. Let's convert the number C25 16 to the octal system. First, we write the numbers in fours, and then divide them into threes from the end: C25 16 = 1100 0010 0101 = 110 000 100 101 = 6045 8. Now let's look at converting back to decimal. It is not difficult, the main thing is not to make mistakes in the calculations. We expand the number into a polynomial with powers of the base and coefficients for them. Then we multiply and add everything. E68 16 = 14 * 16 2 + 6 * 16 + 8 = 3688. 732 8 = 7 * 8 2 + 3*8 + 2 = 474 .
Converting Negative Numbers
Here you need to take into account that the number will be presented in two's complement code. To convert a number into additional code, you need to know the final size of the number, that is, what we want to fit it into - in a byte, in two bytes, in four. The most significant digit of a number means the sign. If there is 0, then the number is positive, if 1, then it is negative. On the left, the number is supplemented with a sign digit. We do not consider unsigned numbers; they are always positive, and the most significant bit in them is used as information.
To convert a negative number to binary's complement, you need to convert a positive number to binary, then change the zeros to ones and the ones to zeros. Then add 1 to the result.
So, let's convert the number -79 to the binary system. The number will take us one byte.
We convert 79 to the binary system, 79 = 1001111. We add zeros on the left to the size of the byte, 8 bits, we get 01001111. We change 1 to 0 and 0 to 1. We get 10110000. We add 1 to the result, we get the answer 10110001. Along the way, we answer the Unified State Exam question “how many units are in the binary representation of the number -79?” The answer is 4.
Adding 1 to the inverse of a number eliminates the difference between the representations +0 = 00000000 and -0 = 11111111. In two's complement code they will be written the same as 00000000.
Converting fractional numbers
Fractional numbers are converted in the reverse way of dividing whole numbers by the base, which we looked at at the very beginning. That is, using sequential multiplication by a new base with the collection of whole parts. The integer parts obtained during multiplication are collected, but do not participate in the following operations. Only fractions are multiplied. If the original number is greater than 1, then the integer and fractional parts are translated separately and then glued together.
Let's convert the number 0.6752 to the binary system.
| 0 | ,6752 |
| *2 | |
| 1 | ,3504 |
| *2 | |
| 0 | ,7008 |
| *2 | |
| 1 | ,4016 |
| *2 | |
| 0 | ,8032 |
| *2 | |
| 1 | ,6064 |
| *2 | |
| 1 | ,2128 |
The process can be continued for a long time until we get all the zeros in the fractional part or the required accuracy is achieved. Let's stop at the 6th sign for now.
It turns out 0.6752 = 0.101011.
If the number was 5.6752, then in binary it will be 101.101011.
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