When you are setting up networks of various sizes and deal with calculations every day, you don’t need to create this kind of cheat sheet, everything is done on an unconditional reflex. But when you poke around in networks very rarely, you don’t always remember what the mask is in decimal form for the prefix 21 or what the network address is for the same prefix. In this regard, I decided to write several small articles-cheat sheets on converting numbers to various number systems, network addresses, masks, etc. In this part we will talk about converting numbers into different number systems.
1. Number systems
When you do anything related to computer networks and IT, you will come across this concept anyway. And as a smart IT guy, you need to understand this at least a little, even if in practice you will use it very rarely.
Let's look at the translation of each digit from an IP address 98.251.16.138
in the following number systems:
- Binary
- Octal
- Decimal
- Hexadecimal
1.1 Decimal
Since the numbers are written in decimal, we will skip the conversion from decimal to decimal :)
1.1.1 Decimal → Binary
As we know, the binary number system is used in almost all modern computers and many other computing devices. The system is very simple - we only have 0 and 1.
To convert a number with a tithe into binary form, you need to use division modulo 2 (i.e. integer division by 2), as a result of which we will always have a remainder of either 1 or 0. In this case, the result is written from right to left. An example will put everything in its place:
Figure 1.1 – Converting numbers from decimal to binary system
Figure 1.2 – Converting numbers from decimal to binary system
I will describe the division of the number 98. We divide 98 by 2, as a result we have 49 and the remainder is 0. Next we continue division and divide 49 by 2, as a result we have 24 with a remainder of 1. And in the same way we get to 1 or 0 in divisible. Then we write the result from right to left.
1.1.2 Decimal → Octal
The octal system is an integer number system with base 8. I.e. all numbers in it are represented in the range 0 – 7 and to convert from the decimal system you need to use division modulo 8.
Figure 1.3 – Converting numbers from decimal to octal system
The division is similar to the 2-point system.
1.1.3 Decimal → Hexadecimal
The hexadecimal system has almost completely replaced the octal system. It has a base of 16, but uses decimal digits from 0 to 9 + Latin letters from A (number 10) to F (number 15). You encounter it every time you check your network adapter settings - this is the MAC address. Same when IPv6 is used.
Figure 1.4 – Converting numbers from decimal to hexadecimal
1.2 Binary
In the previous example, we converted all decimal numbers to other number systems, one of which is binary. Now let's convert each number from binary form.
1.2.1 Binary → Decimal
To convert numbers from binary to decimal, you need to know two nuances. The first is that each zero and one have a multiplier of 2 in nth degree, in which n increases from right to left by exactly one. The second is that after multiplying, all the numbers need to be added and we get the number in decimal form. As a result, we will have a formula like this:
D = (a n × p n-1) + (a n-1 × p n-2) + (a n-2 × p n-3) +…, (1.2.1)
Where,
D is the decimal number we are looking for;
n– the number of characters in a binary number;
a – number in binary form on nth position(i.e. first character, second, etc.);
p – coefficient equal to 2.8 or 16 to the power n(depending on the number system)
For example, let’s take the number 110102. We look at the formula and write:
- The number consists of 5 characters ( n=5)
- p = 2 (since we are converting from binary to decimal)
a 5 = 1, a 4 = 1, a 3 = 0, a 2 = 1, a 1 = 0
As a result we have:
D = (1 × 2 5-1) + (1 × 2 5-2) + (0 × 2 5-3) + (1 × 2 5-4) + (0 × 2 5-5) = 16 + 8 + 0 + 2 + 0 = 26 10
For those who are used to writing from right to left, the form will look like this:
D = (0 × 2 5-5) + (1 × 2 5-4) + (0 × 2 5-3) + (1 × 2 5-2) + (1 × 2 5-1) = 0 + 2 + 0 + 8 + 16 = 26 10
But, as we know, rearranging the terms does not change the sum. Let's now convert our numbers to decimal form.
Figure 1.5 – Converting numbers from binary to decimal system
1.2.2 Binary → Octal
When translating, we need to divide the binary number into groups of three characters from right to left. If the last group does not consist of three characters, then we simply replace the missing bits with zeros. Eg:
10101001 = 0 10 101 001
1011100 = 00 1 011 100
Each group of bits is one of the octal numbers. To find out which one, you need to use the formula 1.2.1 written above for each group of bits. As a result we get.
Figure 1.6 – Converting numbers from binary to octal system
1.2.3 Binary → Hexadecimal
Here we need to split the binary number into groups of four characters from right to left, followed by adding zeros to the missing bits of the group, as described above. If the last group consists of zeros, then they should be ignored.
110101011 = 000 1 1010 1011
1011100 = 0 101 1100
001010000 = 00 0101 0000 = 0101 0000
Each group of bits is one of the hexadecimal numbers. We use formula 1.2.1 for each group of bits.
Figure 1.7 – Converting numbers from binary to hexadecimal
1.3 Octal
In this system, we may have difficulties only when converting to hexadecimal, since the rest of the translation goes smoothly.
1.3.1 Octal → Binary
Each number in the octal system is a group of three bits in the binary system, as described above. To translate, we need to use a cheat sheet:
Figure 1.8 – Spur for converting numbers from the octal system
Using this tablet, we will convert our numbers to the binary system.
Figure 1.9 – Converting numbers from octal to binary
I'll describe the conclusion a little. Our first number is 142, which means there will be three groups of three bits each. We use the spur and see that number 1 is 001, number 4 is 100 and number 2 is 010. As a result, we have the number 001100010.
1.3.2 Octal → Decimal
Here we use formula 1.2.1 only with a coefficient of 8 (i.e. p=8). As a result we have
Figure 1.10 – Converting numbers from octal to decimal system
- The number consists of 3 characters ( n=3)
- p = 8 (since we are converting from octal to decimal)
a 3 = 1, a 2 = 4, a 1 = 2
As a result we have:
D = (1 × 8 3-1) + (4 × 8 3-2) + (2 × 8 3-3) = 64 + 32 + 2 = 98 10
1.3.3 Octal → Hexadecimal
As was written earlier, to translate, we first need to convert the numbers into the binary system, then from binary to hexadecimal, dividing them into groups of 4 bits. You can use the following spur.
Figure 1.11 – Spur for converting numbers from the hexadecimal system
This table will help you convert from binary to hexadecimal. Now let's convert our numbers.
Figure 1.12 – Converting numbers from octal to hexadecimal
1.4 Hexadecimal
This system has the same problem when converting to octal. But more on that later.
1.4.1 Hex → Binary
Each number in hexadecimal is a group of four bits in binary, as described above. To translate, we can use the cheat sheet located above. As a result:
Figure 1.13 – Converting numbers from hexadecimal to binary
Let's take the first number - 62. Using the table (Fig. 1.11) we see that 6 is 0110, 2 is 0010, as a result we have the number 01100010.
1.4.2 Hex → Decimal
Here we use formula 1.2.1 only with a coefficient of 16 (i.e. p=16). As a result we have
Figure 1.14 – Converting numbers from hexadecimal to decimal
Let's take the first number. Based on formula 1.2.1:
- The number consists of 2 characters ( n=2)
- p = 16 (since we are converting from hexadecimal to decimal)
a 2 = 6, a 1 = 2
As a result we have.
D = (6 × 16 2-1) + (2 × 16 2-2) = 96 + 2 = 98 10
1.4.3 Hex → Octal
To convert to the octal system, you must first convert to binary, then divide it into groups of 3 bits and use the table (Fig. 1.8). As a result:
Figure 1.15 – Converting numbers from hexadecimal to octal
We will talk about IP addresses, masks and networks.
2.3. Converting numbers from one number system to another
2.3.1. Converting integers from one number system to another
It is possible to formulate an algorithm for converting integers from a radix system p into a system with a base q :
1. Base new system express numbers using numbers from the original number system and carry out all subsequent actions in the original number system.
2. Consistently divide the given number and the resulting integer quotients by the base of the new number system until we obtain a quotient that is smaller than the divisor.
3. The resulting remainders, which are digits of the number in the new number system, are brought into accordance with the alphabet of the new number system.
4. Compose a number in the new number system, writing it starting from the last remainder.
Example 2.12. Convert the decimal number 173 10 to octal number system:
We get: 173 10 =255 8
Example 2.13. Convert the decimal number 173 10 to hexadecimal number system:
We get: 173 10 =AD 16.
Example 2.14. Convert the decimal number 11 10 to the binary number system. It is more convenient to depict the sequence of actions discussed above (translation algorithm) as follows:
We get: 11 10 =1011 2.
Example 2.15. Sometimes it is more convenient to write down the translation algorithm in table form. Let's convert the decimal number 363 10 to a binary number.
|
Divider |
|||||||||
We get: 363 10 =101101011 2
2.3.2. Converting fractional numbers from one number system to another
It is possible to formulate an algorithm for converting a proper fraction with a base p into a fraction with a base q:
1. Express the base of the new number system with numbers from the original number system and carry out all subsequent actions in the original number system.
2. Consistently multiply the given numbers and the resulting fractional parts of the products by the base of the new system until the fractional part of the product becomes equal to zero or the required accuracy of number representation is achieved.
3. The resulting integer parts of the products, which are digits of the number in the new number system, should be brought into conformity with the alphabet of the new number system.
4. Compose the fractional part of a number in the new number system, starting from the integer part of the first product.
Example 2.17. Convert the number 0.65625 10 to the octal number system.
We get: 0.65625 10 =0.52 8
Example 2.17. Convert the number 0.65625 10 to hexadecimal number system.
|
x 16 |
|
We get: 0.65625 10 =0.A8 1
Example 2.18. Convert the decimal fraction 0.5625 10 to the binary number system.
|
x 2 |
|
|
x 2 |
|
|
x 2 |
|
|
x 2 |
|
We get: 0.5625 10 =0.1001 2
Example 2.19. Convert the decimal fraction 0.7 10 to the binary number system.
Obviously, this process can continue indefinitely, giving more and more new signs in the image of the binary equivalent of the number 0.7 10. So, in four steps we get the number 0.1011 2, and in seven steps the number 0.1011001 2, which is a more accurate representation of the number 0.7 10 in binary number system, and etc. Such an endless process is terminated at a certain step, when it is considered that the required accuracy of number representation has been obtained.
2.3.3. Translation of arbitrary numbers
Translation of arbitrary numbers, i.e. numbers containing an integer and a fractional part are carried out in two stages. The integer part is translated separately, and the fractional part separately. In the final recording of the resulting number, the integer part is separated from the fractional part by a comma (dot).
Example 2.20. Convert the number 17.25 10 to the binary number system.
We get: 17.25 10 =1001.01 2
Example 2.21. Convert the number 124.25 10 to octal system.
We get: 124.25 10 =174.2 8
2.3.4. Converting numbers from base 2 to base 2 n and vice versa
Translation of integers. If the base of the q-ary number system is a power of 2, then the conversion of numbers from the q-ary number system to the 2-ary number system and back can be carried out using more simple rules. In order to write an integer binary number in the number system with base q=2 n, you need:
1. Divide the binary number from right to left into groups of n digits each.
2. If the last left group has less than n digits, then it must be supplemented on the left with zeros to the required number of digits.
Example 2.22. The number 101100001000110010 2 will be converted to the octal number system.
We divide the number from right to left into triads and under each of them write the corresponding octal digit:
We get the octal representation of the original number: 541062 8 .
Example 2.23. The number 1000000000111110000111 2 will be converted to the hexadecimal number system.
We divide the number from right to left into tetrads and under each of them write the corresponding hexadecimal digit:
We get the hexadecimal representation of the original number: 200F87 16.
Converting fractional numbers. In order to write a fractional binary number in a number system with base q=2 n, you need:
1. Divide the binary number from left to right into groups of n digits each.
2. If the last right group has less than n digits, then it must be supplemented on the right with zeros to the required number of digits.
3. Consider each group as an n-bit binary number and write it with the corresponding digit in the number system with the base q=2 n.
Example 2.24. The number 0.10110001 2 will be converted to the octal number system.
We divide the number from left to right into triads and under each of them we write the corresponding octal digit:
We get the octal representation of the original number: 0.542 8 .
Example 2.25. The number 0.100000000011 2 will be converted to the hexadecimal number system. We divide the number from left to right into tetrads and under each of them write the corresponding hexadecimal digit:
We get the hexadecimal representation of the original number: 0.803 16
Translation of arbitrary numbers. In order to write an arbitrary binary number in the number system with the base q=2 n, you need:
1. Divide the integer part of a given binary number from right to left, and the fractional part from left to right into groups of n digits each.
2. If the last left and/or right groups have less than n digits, then they must be supplemented on the left and/or right with zeros to the required number of digits;
3. Consider each group as an n-bit binary number and write it with the corresponding digit in the number system with the base q = 2 n
Example 2.26. Let's convert the number 111100101.0111 2 to the octal number system.
We divide the integer and fractional parts of the number into triads and under each of them write the corresponding octal digit:
We get the octal representation of the original number: 745.34 8 .
Example 2.27. The number 11101001000,11010010 2 will be converted to the hexadecimal number system.
We divide the integer and fractional parts of the number into notebooks and under each of them write the corresponding hexadecimal digit:
We get the hexadecimal representation of the original number: 748,D2 16.
Converting numbers from number systems with base q=2n to binary. In order to convert an arbitrary number written in the number system with the base q=2 n into the binary number system, you need to replace each digit of this number with its n-digit equivalent in the binary number system.
Example 2.28.Let's convert the hexadecimal number 4AC35 16 to the binary number system.
According to the algorithm:
We get: 1001010110000110101 2 .
Tasks for independent completion (Answers)
2.38. Fill out the table, in each row of which the same integer must be written in various systems Reckoning.
|
Binary |
Octal |
Decimal |
Hexadecimal |
2.39. Fill out the table, in each row of which the same fractional number must be written in different number systems.
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Binary |
Octal |
Decimal |
Hexadecimal |
2.40. Fill out the table, in each row of which the same arbitrary number (the number can contain both an integer and a fractional part) must be written in different number systems.
|
Binary |
Octal |
Decimal |
Hexadecimal |
|
59.B |
Note 1
If you want to convert a number from one number system to another, then it is more convenient to first convert it to the decimal number system, and only then convert it from the decimal number system to any other number system.
Rules for converting numbers from any number system to decimal
IN computer technology, using machine arithmetic, an important role is played by the conversion of numbers from one number system to another. Below we give the basic rules for such transformations (translations).
When converting a binary number to a decimal, you need to represent the binary number as a polynomial, each element of which is represented as the product of a digit of the number and the corresponding power of the base number, in this case $2$, and then you need to calculate the polynomial using the rules of decimal arithmetic:
$X_2=A_n \cdot 2^(n-1) + A_(n-1) \cdot 2^(n-2) + A_(n-2) \cdot 2^(n-3) + ... + A_2 \cdot 2^1 + A_1 \cdot 2^0$
Figure 1. Table 1
Example 1
Convert the number $11110101_2$ to the decimal number system.
Solution. Using the given table of $1$ powers of the base $2$, we represent the number as a polynomial:
$11110101_2 = 1 \cdot 27 + 1 \cdot 26 + 1 \cdot 25 + 1 \cdot 24 + 0 \cdot 23 + 1 \cdot 22 + 0 \cdot 21 + 1 \cdot 20 = 128 + 64 + 32 + 16 + 0 + 4 + 0 + 1 = 245_(10)$
To convert a number from the octal number system to the decimal number system, you need to represent it as a polynomial, each element of which is represented as the product of a digit of the number and the corresponding power of the base number, in this case $8$, and then you need to calculate the polynomial according to the rules of decimal arithmetic:
$X_8 = A_n \cdot 8^(n-1) + A_(n-1) \cdot 8^(n-2) + A_(n-2) \cdot 8^(n-3) + ... + A_2 \cdot 8^1 + A_1 \cdot 8^0$
Figure 2. Table 2
Example 2
Convert the number $75013_8$ to the decimal number system.
Solution. Using the given table of $2$ powers of the base $8$, we represent the number as a polynomial:
$75013_8 = 7\cdot 8^4 + 5 \cdot 8^3 + 0 \cdot 8^2 + 1 \cdot 8^1 + 3 \cdot 8^0 = 31243_(10)$
To convert a number from hexadecimal to decimal, you need to represent it as a polynomial, each element of which is represented as the product of a digit of the number and the corresponding power of the base number, in this case $16$, and then you need to calculate the polynomial according to the rules of decimal arithmetic:
$X_(16) = A_n \cdot 16^(n-1) + A_(n-1) \cdot 16^(n-2) + A_(n-2) \cdot 16^(n-3) + . .. + A_2 \cdot 16^1 + A_1 \cdot 16^0$
Figure 3. Table 3
Example 3
Convert the number $FFA2_(16)$ to the decimal number system.
Solution. Using the given table of $3$ powers of the base $8$, we represent the number as a polynomial:
$FFA2_(16) = 15 \cdot 16^3 + 15 \cdot 16^2 + 10 \cdot 16^1 + 2 \cdot 16^0 =61440 + 3840 + 160 + 2 = 65442_(10)$
Rules for converting numbers from the decimal number system to another
- To convert a number from the decimal number system to the binary system, it must be sequentially divided by $2$ until there is a remainder less than or equal to $1$. A number in the binary system is represented as a sequence of the last result of division and the remainders from division in reverse order.
Example 4
Convert the number $22_(10)$ to the binary number system.
Solution:
Figure 4.
$22_{10} = 10110_2$
- To convert a number from the decimal number system to octal, it must be sequentially divided by $8$ until there is a remainder less than or equal to $7$. A number in the octal number system is represented as a sequence of digits of the last division result and the remainders from the division in reverse order.
Example 5
Convert the number $571_(10)$ to the octal number system.
Solution:
Figure 5.
$571_{10} = 1073_8$
- To convert a number from the decimal number system to the hexadecimal system, it must be successively divided by $16$ until there is a remainder less than or equal to $15$. A number in the hexadecimal system is represented as a sequence of digits of the last division result and the remainder of the division in reverse order.
Example 6
Convert the number $7467_(10)$ to hexadecimal number system.
Solution:
Figure 6.
$7467_(10) = 1D2B_(16)$
In order to convert a proper fraction from a decimal number system to a non-decimal number system, it is necessary to sequentially multiply the fractional part of the number being converted by the base of the system to which it needs to be converted. Fractions in the new system will be represented as whole parts of products, starting with the first.
For example: $0.3125_((10))$ in octal number system will look like $0.24_((8))$.
In this case, you may encounter a problem when the final decimal may correspond to an infinite (periodic) fraction in the non-decimal number system. In this case, the number of digits in the fraction represented in the new system will depend on the required accuracy. It should also be noted that integers remain integers, and proper fractions remain fractions in any number system.
Rules for converting numbers from a binary number system to another
- To convert a number from the binary number system to octal, it must be divided into triads (triples of digits), starting with the least significant digit, if necessary, adding zeros to the leading triad, then replace each triad with the corresponding octal digit according to Table 4.
Figure 7. Table 4
Example 7
Convert the number $1001011_2$ to the octal number system.
Solution. Using Table 4, we convert the number from the binary number system to octal:
$001 001 011_2 = 113_8$
- To convert a number from the binary number system to hexadecimal, it should be divided into tetrads (four digits), starting with the least significant digit, if necessary, adding zeros to the most significant tetrad, then replace each tetrad with the corresponding octal digit according to Table 4.
1. Ordinal counting in various number systems.
In modern life, we use positional number systems, that is, systems in which the number denoted by a digit depends on the position of the digit in the notation of the number. Therefore, in the future we will talk only about them, omitting the term “positional”.
In order to learn how to convert numbers from one system to another, we will understand how sequential recording of numbers occurs using the example of the decimal system.
Since we have a decimal number system, we have 10 symbols (digits) to construct numbers. We start counting: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The numbers are over. We increase the bit depth of the number and reset the low-order digit: 10. Then we increase the low-order digit again until all the digits are gone: 11, 12, 13, 14, 15, 16, 17, 18, 19. We increase the high-order digit by 1 and reset the low-order digit: 20. When we use all the digits for both digits (we get the number 99), we again increase the digit capacity of the number and reset the existing digits: 100. And so on.
Let's try to do the same in the 2nd, 3rd and 5th systems (we introduce the notation for the 2nd system, for the 3rd, etc.):
| 0 | 0 | 0 | 0 |
| 1 | 1 | 1 | 1 |
| 2 | 10 | 2 | 2 |
| 3 | 11 | 10 | 3 |
| 4 | 100 | 11 | 4 |
| 5 | 101 | 12 | 10 |
| 6 | 110 | 20 | 11 |
| 7 | 111 | 21 | 12 |
| 8 | 1000 | 22 | 13 |
| 9 | 1001 | 100 | 14 |
| 10 | 1010 | 101 | 20 |
| 11 | 1011 | 102 | 21 |
| 12 | 1100 | 110 | 22 |
| 13 | 1101 | 111 | 23 |
| 14 | 1110 | 112 | 24 |
| 15 | 1111 | 120 | 30 |
If the number system has a base greater than 10, then we will have to enter additional characters; it is customary to enter letters of the Latin alphabet. For example, for the 12-digit system, in addition to ten digits, we need two letters ( and ):
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |
| 3 | 3 |
| 4 | 4 |
| 5 | 5 |
| 6 | 6 |
| 7 | 7 |
| 8 | 8 |
| 9 | 9 |
| 10 | |
| 11 | |
| 12 | 10 |
| 13 | 11 |
| 14 | 12 |
| 15 | 13 |
2. Conversion from the decimal number system to any other.
To convert a positive integer decimal number to a number system with a different base, you need to divide this number by the base. Divide the resulting quotient by the base again, and further until the quotient is less than the base. As a result, write down in one line the last quotient and all remainders, starting from the last.
Example 1. Let's convert the decimal number 46 to the binary number system.
Example 2. Let's convert the decimal number 672 to the octal number system.
Example 3. Let's convert the decimal number 934 to the hexadecimal number system.
3. Conversion from any number system to decimal.
In order to learn how to convert numbers from any other system to decimal, let’s analyze the notation we are familiar with decimal number.
For example, the decimal number 325 is 5 units, 2 tens and 3 hundreds, i.e.
The situation is exactly the same in other number systems, only we will multiply not by 10, 100, etc., but by the powers of the base of the number system. For example, let's take the number 1201 in the ternary number system. Let's number the digits from right to left starting from zero and imagine our number as the sum of the products of a digit and three to the power of the digit of the number:
This is the decimal notation of our number, i.e.
Example 4. Let's convert the octal number 511 to the decimal number system.
Example 5. Let's convert the hexadecimal number 1151 to the decimal number system.
4. Conversion from the binary system to the system with the base “power of two” (4, 8, 16, etc.).
To convert a binary number into a number with a power of two base, it is necessary to divide the binary sequence into groups according to the number of digits equal to the power from right to left and replace each group with the corresponding digit of the new number system.
For example, Let's convert the binary number 1100001111010110 to the octal system. To do this, we will divide it into groups of 3 characters starting from the right (since ), and then use the correspondence table and replace each group with a new number:
We learned how to build a correspondence table in step 1.
| 0 | 0 |
| 1 | 1 |
| 10 | 2 |
| 11 | 3 |
| 100 | 4 |
| 101 | 5 |
| 110 | 6 |
| 111 | 7 |
Those.
Example 6. Let's convert the binary number 1100001111010110 to hexadecimal.
| 0 | 0 |
| 1 | 1 |
| 10 | 2 |
| 11 | 3 |
| 100 | 4 |
| 101 | 5 |
| 110 | 6 |
| 111 | 7 |
| 1000 | 8 |
| 1001 | 9 |
| 1010 | A |
| 1011 | B |
| 1100 | C |
| 1101 | D |
| 1110 | E |
| 1111 | F |
5. Conversion from a system with the base “power of two” (4, 8, 16, etc.) to binary.
This translation is similar to the previous one, done in the opposite direction: we replace each digit with a group of digits in the binary system from the correspondence table.
Example 7. Let's convert the hexadecimal number C3A6 to the binary number system.
To do this, replace each digit of the number with a group of 4 digits (since ) from the correspondence table, supplementing the group with zeros at the beginning if necessary:
Write the number in the binary number system, and the powers of two from right to left. For example, we want to convert the binary number 10011011 2 to decimal. Let's write it down first. Then we write the powers of two from right to left. Let's start with 2 0, which is equal to "1". We increase the degree by one for each subsequent number. We stop when the number of elements in the list is equal to the number of digits in the binary number. Our example number, 10011011, has eight digits, so a list of eight elements would look like this: 128, 64, 32, 16, 8, 4, 2, 1
Write the digits of the binary number under the corresponding powers of two. Now simply write 10011011 under the numbers 128, 64, 32, 16, 8, 4, 2, and 1, so that each binary digit corresponds to a different power of two. The rightmost "1" of the binary number must correspond to the rightmost "1" of the powers of two, and so on. If you prefer, you can write the binary number above powers of two. The most important thing is that they match each other.
Match the digits in a binary number with the corresponding powers of two. Draw lines (from right to left) that connect each successive digit of the binary number to the power of two above it. Start drawing lines by connecting the first digit of a binary number to the first power of two above it. Then draw a line from the second digit of the binary number to the second power of two. Continue connecting each number to the corresponding power of two. This will help you visually see the relationship between two different sets of numbers.
Write down the final value of each power of two. Go through each digit of a binary number. If the number is 1, write the corresponding power of two under the number. If this number is 0, write 0 under the number.
- Since "1" matches "1", it remains "1". Since "2" matches "1", it remains "2". Since "4" corresponds to "0", it becomes "0". Since "8" matches "1", it becomes "8", and since "16" matches "1" it becomes "16". "32" matches "0" and becomes "0", "64" matches "0" and therefore becomes "0", while "128" matches "1" and therefore becomes 128.
Add up the resulting values. Now add the resulting numbers under the line. Here's what you have to do: 128 + 0 + 0 + 16 + 8 + 0 + 2 + 1 = 155. This is the decimal equivalent of the binary number 10011011.
Write the answer along with a subscript equal to the number system. Now all you have to do is write 155 10 to show that you are working with a decimal answer, which deals with powers of ten. The more you convert binary numbers to decimals, the easier it will be for you to remember powers of two, and the faster you will be able to complete the task.
Use this method to convert a binary number with a decimal point to decimal form. You can use this method even if you want to convert a binary number such as 1.1 2 to decimal. All you need to know is that the number on the left side of the decimal is a regular number, and the number on the right side of the decimal is the "halve" number, or 1 x (1/2).
- "1" to the left of the decimal number corresponds to 2 0, or 1. 1 to the right of the decimal number corresponds to 2 -1, or.5. Add 1 and .5 and you get 1.5, which is the decimal equivalent of 1.1 2.