Instructions

Video on the topic

In the counting system that we use every day, there are ten digits - from zero to nine. That's why it's called decimal. However, in technical calculations, especially those related to computers, other systems, specifically binary and hexadecimal. Therefore you need to be able to translate numbers from one systems counting to another.

You will need

• - a piece of paper;
• - pencil or pen;
• - calculator.

Instructions

The binary system is the simplest. It has only two digits - zero and one. Each digit of binary numbers, starting from the end, corresponds to a power of two. Two in equals one, in the first - two, in the second - four, in the third - eight, and so on.

Suppose you are given the binary number 1010110. The units in it are in second, third, fifth and seventh places. Therefore, in the decimal system this number is 2^1 + 2^2 + 2^4 + 2^6 = 2 + 4 + 16 + 64 = 86.

Inverse problem - decimal numbers system. Let's say you have the number 57. To get it, you must sequentially divide the number by 2 and write the remainder. The binary number will be built from end to beginning.
The first step will give you the last digit: 57/2 = 28 (remainder 1).
Then you get the second one from the end: 28/2 = 14 (remainder 0).
Further steps: 14/2 = 7 (remainder 0);
7/2 = 3 (remainder 1);
3/2 = 1 (remainder 1);
1/2 = 0 (remainder 1).
This last step, because the result of division is zero. As a result, you got the binary number 111001.
Check your answer: 111001 = 2^0 + 2^3 + 2^4 + 2^5 = 1 + 8 + 16 + 32 = 57.

The second, used in computer matters, is hexadecimal. It has not ten, but sixteen digits. So as not to be new symbols, first ten digits of hexadecimal systems are designated by ordinary numbers, and the remaining six are with Latin letters: A, B, C, D, E, F. They correspond to decimal notation numbers m from 10 to 15. To avoid confusion, the number written in hexadecimal is preceded by the # sign or the symbols 0x.

To make a number from hexadecimal systems, you need to multiply each of its digits by the corresponding power of sixteen and add the results. For example, the number #11A in decimal notation is 10*(16^0) + 1*(16^1) + 1*(16^2) = 10 + 16 + 256 = 282.

Reverse conversion from decimal systems to hexadecimal is done using the same method of remainders as to binary. For example, take the number 10000. Consistently dividing it by 16 and writing down the remainders, you get:
10000/16 = 625 (remainder 0).
625/16 = 39 (remainder 1).
39/16 = 2 (remainder 7).
2/16 = 0 (remainder 2).
The result of the calculations will be hexadecimal number #2710.
Check your answer: #2710 = 1*(16^1) + 7*(16^2) + 2*(16^3) = 16 + 1792 + 8192 = 10000.

Transfer numbers from hexadecimal systems It's much easier to convert to binary. The number 16 is a two: 16 = 2^4. Therefore, each hexadecimal digit can be written as a four-digit binary number. If you have less than four digits in a binary number, add leading zeros.
For example, #1F7E = (0001)(1111)(0111)(1110) = 1111101111110.
Check the answer: both numbers in decimal notation they are equal to 8062.

To translate, you need to break the binary number into groups of four digits, starting from the end, and replace each such group with a hexadecimal digit.
For example, 11000110101001 becomes (0011)(0001)(1010)(1001), which in hexadecimal notation equals #31A9. The correctness of the answer is confirmed by conversion to decimal notation: both numbers are equal to 12713.

Tip 5: How to convert a number to binary system calculus

Due to its limited use of symbols, the binary system is most convenient for use in computers and other digital devices. There are only two symbols: 1 and 0, so this system used in the operation of registers.

Instructions

Binary is positional, i.e. The position of each digit in a number corresponds to a certain digit, which is equal to two to the appropriate power. The degree starts at zero and increases as you move from right to left. For example, number 101 is equal to 1*2^0 + 0*2^1 + 1*2^2 = 5.

Octal, hexadecimal and decimal systems are also widely used among positional systems. And if for the first two the second method is more applicable, then for translation from both are applicable.

Consider a decimal number to binary system by sequential division by 2. To convert a decimal number 25 V

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 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 $1$. Represent a number in the binary system 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 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 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.

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 next date. 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 connection between the two various sets 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.

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 digits 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 this 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 got four binary numbers, which now need to be added 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.