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 decimal system number system, and only then convert from decimal 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, it is required 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 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 decimal system When calculating into binary, it must be sequentially divided by $2$ until there is a remainder less than or equal to $1$. Number in binary system represent as a sequence last result division and 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 $15$. Number in hexadecimal system present as a sequence of digits of the last division result and the remainders from 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 the decimal number system to the non-decimal number system, it is necessary fractional part the number being converted is sequentially multiplied by the base of the system into which it needs to be converted. Fraction in new system will be presented in the form of entire parts of works, 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 a finite decimal fraction can 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.

Hello, site visitor! We continue to study the protocol network layer IP, or to be more precise, its version IPv4. At first glance the topic binary numbers and binary number system has nothing to do with the IP protocol, but if we remember that computers work with zeros and ones, then it turns out that the binary system and its understanding is the basis of the fundamentals, we need learn to convert numbers from binary to decimal and vice versa: decimal to binary. This will help us better understand the IP protocol, as well as the principle of operation of variable-length network masks. Let's get started!

If the topic of computer networks is interesting to you, you can read other course recordings.

4.4.1 Introduction

Before we begin, it’s worth explaining why a network engineer needs this topic. Although you could be convinced of its necessity when we spoke, you can say that there are IP calculators that greatly facilitate the task of allocating IP addresses, calculating the necessary subnet/network masks and determining the network number and host number in the IP address. That’s right, but the IP calculator is not always at hand, this is the reason number one. Reason number two is that in the Cisco exams they won't give you an IP calculator and that's it. you will have to do the conversion of IP addresses from decimal to binary on a piece of paper, and there are not so few questions where this is required in the exam/exams for obtaining the CCNA certificate, it would be a shame if the exam was failed because of such a trifle. And finally, understanding the binary number system leads to better understanding operating principle.

At all network engineer You don’t have to be able to convert numbers from binary to decimal and vice versa in your head. Moreover, rarely does anyone know how to do this mentally; teachers of various courses in computer networks, since they are faced with this constantly day after day. But with a piece of paper and a pen, you should learn how to translate.

4.4.2 Decimal digits and numbers, digits in numbers

Let's start simple and talk about binary digits and numbers, you know that numbers and numbers are two different things. The number is special character to designate, and number is an abstract notation meaning quantity. For example, to write down that we have five fingers on our hand, we can use Roman and Arabic numerals: V and 5. In this case, five is both a number and a digit. And, for example, to write the number 20 we use two digits: 2 and 0.

In total, in the decimal number system we have ten numbers or ten symbols (0,1,2,3,4,5,6,7,8,9), by combining which we can write different numbers. What principle are we guided by when using the decimal number system? Yes, everything is very simple, we raise ten to one power or another, for example, let’s take the number 321. How can it be written differently, like this: 3*10 2 +2*10 1 +1*10 0 . Thus, it turns out that the number 321 represents three digits:

1. The number 3 means the most significant place or in this case it is the hundreds place, otherwise their number.
2. The number 2 is in the tens place, we have two tens.
3. The number one refers to the least significant digit.

That is, in this entry a two is not just a two, but two tens or two times ten. And three is not just three, but three times a hundred. The following dependence is obtained: the unit of each next digit is ten times greater than the unit of the previous one, because what 300 is is three times a hundred. A digression regarding the decimal number system was necessary to make it easier to understand the binary system.

4.4.3 Binary digits and numbers, as well as their recording

There are only two digits in the binary number system: 0 and 1. Therefore, writing a number in the binary system is often much larger than in the decimal system. With the exception of the numbers 0 and 1, zero in the binary number system is equal to zero in the decimal number system, and the same is true for one. Sometimes, in order not to confuse which number system the number is written in, sub-indices are used: 267 10, 10100 12, 4712 8. The number in the sub-index indicates the number system.

The symbols 0b and &(ampersand) can be used to write binary numbers: 0b10111, &111. If in the decimal number system, to pronounce the number 245 we use this construction: two hundred and forty-five, then in the binary number system, to name the number, we need to pronounce a digit from each digit, for example, the number 1100 in the binary number system should not be pronounced as a thousand one hundred, but like one, one, zero, zero. Let's look at writing the numbers from 0 to 10 in the binary number system:

I think the logic should be clear by now. If in the decimal number system for each digit we had ten options available (from 0 to 9 inclusive), then in the binary number system in each of the digits of a binary number we have only two options: 0 or 1.

To work with IP addresses and subnet masks, we only need natural numbers in the binary number system, although the binary system allows us to write fractional and negative numbers, but we don’t need it.

4.4.4 Converting numbers from decimal to binary

Let's take a better look at this how to convert a number from decimal to binary. And here everything is actually very, very simple, although it’s difficult to explain in words, so I’ll give it right away example of converting numbers from decimal to binary. Let's take the number 61, to convert to the binary system, we need to divide this number by two and see what is the remainder of the division. And the result of division is again divided by two. In this case, 61 is the dividend, we will always have two as a divisor, and we divide the quotient (the result of division) by two again, continue dividing until the quotient contains 1, this last unit will be the leftmost digit . The picture below demonstrates this.

Please note that the number 61 is not 101111, but 111101, that is, we write the result from the end. In the latter particular, there is no sense in dividing one by two, since in this case integer division is used, and with this approach it turns out as in Figure 4.4.2.

This is not the most quick way converting a number from binary to decimal. We have several accelerators. For example, the number 7 in binary is written as 111, the number 3 as 11, and the number 255 as 11111111. All these cases are incredibly simple. The fact is that the numbers 8, 4, and 256 are powers of two, and the numbers 7, 3, and 255 are one less than these numbers. So, for numbers that are one less than a number equal to a power of two, a simple rule applies: in the binary system, such a decimal number is written as a number of units equal to a power of two. So, for example, the number 256 is two to the eighth power, therefore, 255 is written as 11111111, and the number 8 is two to the third power, and this tells us that 7 in the binary number system will be written as 111. Well, understand, how to write 256, 4 and 8 in the binary number system is also not difficult, just add one: 256 = 11111111 + 1 = 100000000; 8 = 111 + 1 = 1000; 4 = 11 + 1 = 100.
You can check any of your results on a calculator and it’s better to do so at first.

As you can see, we have not yet forgotten how to divide. And now we can move on.

4.4.5 Converting numbers from binary to decimal

Converting numbers from binary is much easier than converting from decimal to binary. As an example of translation, we will use the number 11110. Pay attention to the table below, it shows the power to which you need to raise two in order to eventually get a decimal number.

To get a decimal number from this binary number, you need to multiply each number in the digit by two to the power, and then add the results of the multiplication, it’s easier to show:

1*2 4 +1*2 3 +1*2 2 +1*2 1 +0*2 0 = 16+8+4+2+0=30

Let's open the calculator and make sure that 30 in the decimal number system is 11110 in binary.

We see that everything was done correctly. From the example it is clear that Converting a number from binary to decimal is much easier than converting it back. To work with confidence you just need to remember powers of two up to 2 8. For clarity, I will provide a table.

We don’t need more, since the maximum possible number that can be written in one byte (8 bits or eight binary values) is 255, that is, in each octet of the IP address or IPv4 subnet mask, the maximum possible value is 255. There are fields , in which there are values ​​greater than 255, but we do not need to calculate them.

4.4.6 Addition, subtraction, multiplication of binary numbers and other operations with binary numbers

Let's now look at operations that can be performed on binary numbers. Let's start with simple arithmetic operations and then move on to Boolean algebra operations.

Adding binary numbers

Adding binary numbers is not that difficult: 1+0 =1; 1+1=0 (I’ll give an explanation later); 0+0=0. These were simple examples where only one digit was used, let's look at examples where the number of digits is more than one.
101+1101 in the decimal system is 5 + 13 = 18. Let's count in a column.

The result is highlighted in orange, the calculator says that we calculated correctly, you can check it. Now let's see why this happened, because at first I wrote that 1+1=0, but this is for the case when we have only one digit, for cases when there are more than one digits, 1+1=10 (or two in decimal), which is logical.

Then look what happens, we perform additions by digits from right to left:

1. 1+1=10, write zero, and one goes to the next digit.

2. In the next digit we get 0+0+1=1 (this unit came to us from the result of addition in step 1).

4. Here we have a unit only in the second number, but it has also been transferred here, so 0+1+1 = 10.

5. Glue everything together: 10|0|1|0.

If you’re lazy in a column, then let’s count like this: 101011+11011 or 43 + 27 = 70. What can we do here, but let’s look, because no one forbids us to make transformations, and changing the places of the terms does not change the sum, for the binary number system this rule is also relevant.

1. 101011 = 101000 + 11 = 101000 + 10 + 1 = 100000 + 1000 + 10 + 1.
2. 11011 = 11000 + 10 + 1 = 10000 + 1000 + 10 + 1.
3. 100000 + 10000 + (1000 +1000) + (10+10) + (1+1).
4. 100000 + (10000 + 10000) + 100 + 10.
5. 100000 + 100000 +110
6. 1000000 + 110.
7. 1000110.

You can check with a calculator, 1000110 in binary is 70 in decimal.

Subtracting Binary Numbers

Immediately an example for subtracting single-digit numbers in the binary number system, we didn’t talk about negative numbers, so we don’t take 0-1 into account: 1 – 0 = 1; 0 – 0 = 0; 1 – 1 = 0. If there is more than one digit, then everything is also simple, you don’t even need any columns or tricks: 110111 – 1000, this is the same as 55 – 8. As a result, we get 101111. And the heart stopped beating , where does the unit in the third digit come from (numbering from left to right and starting from zero)? It's simple! In the second digit of the number 110111 there is 0, and in the first digit there is 1 (if we assume that the numbering of digits starts from 0 and goes from left to right), but the unit of the fourth digit is obtained by adding two units of the third digit (you get a kind of virtual two) and from this For twos, we subtract one, which is in the zero digit of the number 1000, and 2 - 1 = 1, and 1 is a valid digit in the binary number system.

Multiplying binary numbers

It remains for us to consider the multiplication of binary numbers, which is implemented by shifting one bit to the left. But first, let's look at the results of single-digit multiplication: 1*1 = 1; 1*0=0 0*0=0. Actually, everything is simple, now let's look at something more complex. Let's take the numbers 101001 (41) and 1100 (12). We will multiply by column.

If it is not clear from the table how this happened, then I will try to explain in words:

1. It is convenient to multiply binary numbers in a column, so we write out the second factor under the first; if the numbers have different numbers of digits, it will be more convenient if the larger number is on top.
2. The next step is to multiply all the digits of the first number by the lowest digit of the second number. We write the result of the multiplication below; we need to write it so that under each corresponding digit the result of the multiplication is written.
3. Now we need to multiply all the digits of the first number by the next digit of the second number and write the result one more line below, but this result needs to be shifted one digit to the left; if you look at the table, this is the second sequence of zeros from the top.
4. The same must be done for subsequent digits, each time moving one digit to the left, and if you look at the table, you can say that one cell to the left.
5. We have four binary numbers that we now need to add and get the result. We recently looked at addition, there shouldn't be any problems.

In general, the multiplication operation is not that difficult, you just need a little practice.

Boolean algebra operations

There are two very important concepts in Boolean algebra: true and false, the equivalent of which is zero and one in the binary number system. Boolean algebra operators expand the number of available operators over these values, let's take a look at them.

Logical AND or AND operation

The Logical AND or AND operation is equivalent to multiplying single-digit binary numbers.

1 AND 1 = 1; 1 AND 0 = 1; 0 AND 0 = 0; 0 AND 1 = 0.

1 AND 1 = 1 ; 1 AND 0 = 1 ; 0 AND 0 = 0 ; 0 AND 1 = 0.

The result of “Logical AND” will be one only if both values ​​are equal to one; in all other cases it will be zero.

Operation "Logical OR" or OR

The operation “Logical OR” or OR works on the following principle: if at least one value is equal to one, then the result will be one.

1 OR 1 = 1; 1 OR 0 = 1; 0 OR 1 = 1; 0 OR 0 = 0.

1 OR 1 = 1 ; 1 OR 0 = 1 ; 0 OR 1 = 1 ; 0 OR 0 = 0.

Exclusive OR or XOR operation

The operation "Exclusive OR" or XOR will give us a result of one only if one of the operands equal to one, and the second is equal to zero. If both operands are equal to zero, the result will be zero and even if both operands are equal to one, the result will be zero.

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. Express the base of the new number system using the numbers of the original number system and carry out all subsequent actions in original system Reckoning.

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 down 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 translate hexadecimal number 4AC35 16 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 with the same thing in each row a fractional number must be written in different number systems.

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 computing technology that uses machine arithmetic, the conversion of numbers from one number system to another plays an important role. 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 a finite decimal fraction can 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.

The result has already been received!

Number systems

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 system notation (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 proper decimal fractions ( real number with a zero integer part) into a number system with base s is necessary given number successively multiply by s until the fractional part is pure zero, or we get 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.