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, 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 $1$. 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 binary system Reckoning.

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 octal system Reckoning.

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 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. 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.

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 fraction the product will not equal zero or the required accuracy of number representation will be 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 arbitrary number, written in the number system with the base q=2 n, converted to 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

To quickly convert numbers from the decimal number system to the binary system, you need to have a good knowledge of the numbers “2 to the power”. For example, 2 10 =1024, etc. This will allow you to solve some translation examples literally in seconds. One of these tasks is Problem A1 from the USE demo 2012. You can, of course, take a long and tedious time to divide a number by “2”. But it’s better to decide differently, saving valuable time on the exam.

The method is very simple. Its gist is this: If the number that needs to be converted from the decimal system is equal to the number "2 to the power", then this number in the binary system contains a number of zeros equal to the power. We add a “1” in front of these zeros.

• Let's convert the number 2 from the decimal system. 2=2 1 . Therefore, in the binary system, a number contains 1 zero. We put “1” in front and get 10 2.
• Let's convert 4 from the decimal system. 4=2 2 . Therefore, in the binary system, a number contains 2 zeros. We put “1” in front and get 100 2.
• Let's convert 8 from the decimal system. 8=2 3 . Therefore, in the binary system, a number contains 3 zeros. We put “1” in front and get 1000 2.

Similarly for other numbers "2 to the power".

If the number that needs to be converted is less than the number “2 to the power” by 1, then in the binary system this number consists only of units, the number of which is equal to the power.

• Let's convert 3 from the decimal system. 3=2 2 -1. Therefore, in the binary system, a number contains 2 ones. We get 11 2.
• Let's convert 7 from the decimal system. 7=2 3 -1. Therefore, in the binary system, a number contains 3 ones. We get 111 2.

In the figure, the squares indicate the binary representation of the number, and the decimal representation in pink on the left.

The translation is similar for other numbers “2 to the power-1”.

It is clear that the translation of numbers from 0 to 8 can be done quickly or by division, or simply know by heart their representation in the binary system. I gave these examples so that you understand the principle this method and used it to translate more "impressive numbers", for example, to translate the numbers 127,128, 255, 256, 511, 512, etc.

You can encounter problems where you need to convert a number that is not equal to the number “2 to the power”, but close to it. It may be greater or less than 2 to the power. The difference between the translated number and the number "2 to the power" should be small. For example, up to 3. The representation of numbers from 0 to 3 in the binary system just needs to be known without translation.

If the number is greater than , then we solve it like this:

First we convert the number “2 to the power” into the binary system. And then we add to it the difference between the number “2 to the power” and the number being translated.

For example, let's convert 19 from the decimal system. It is greater than the number "2 to the power" by 3.

16=2 4 . 16 10 =10000 2 .

3 10 =11 2 .

19 10 =10000 2 +11 2 =10011 2 .

If the number is less than the number "2 to the power", then it is more convenient to use the number "2 to the power-1". We solve it like this:

First we convert the number “2 to the power-1” into the binary system. And then we subtract from it the difference between the number “2 to the power of 1” and the number being translated.

For example, let's convert 29 from the decimal system. It is greater than the number “2 to the power-1” by 2. 29=31-2.

31 10 =11111 2 .

2 10 =10 2 .

29 10 =11111 2 -10 2 =11101 2

If the difference between the number being translated and the number "2 to the power" is more than three, then you can break the number into its components, convert each part into the binary system and add.

For example, convert the number 528 from the decimal system. 528=512+16. We translate 512 and 16 separately.
512=2 9 . 512 10 =1000000000 2 .
16=2 4 . 16 10 =10000 2 .
Now let's add it in a column:

Let's look at one of the most important topics in computer science -. IN school curriculum it is revealed rather “modestly,” most likely due to the lack of hours allocated to it. Knowledge on this topic, especially on translation of number systems, are prerequisite for successfully passing the Unified State Exam and admission to universities at the relevant faculties. Below in detail concepts such as positional and non-positional number systems, examples of these number systems are given, rules for converting integers are presented decimal numbers, regular decimal fractions and mixed decimal numbers to any other number system, converting numbers from any number system to decimal, converting from octal and hexadecimal number systems to binary number system. On exams in large quantities There are problems on this topic. The ability to solve them is one of the requirements for applicants. Coming soon: For each topic of the section, in addition to detailed theoretical material, almost all possible options tasks For self-study. In addition, you will have the opportunity to download ready-made ones from a file hosting service completely free of charge. detailed solutions to these tasks, illustrating various ways getting the correct answer.

positional number systems.

Non-positional number systems- number systems in which the quantitative value of a digit does not depend on its location in the number.

Non-positional number systems include, for example, Roman, where instead of numbers there are Latin letters.

I	1 (one)
V	5 (five)
X	10 (ten)
L	50 (fifty)
C	100 (one hundred)
D	500 (five hundred)
M	1000 (thousand)

Here the letter V stands for 5 regardless of its location. However, it is worth mentioning that although the Roman number system is classic example non-positional number system is not completely non-positional, because The smaller number in front of the larger one is subtracted from it:

IL	49 (50-1=49)
VI	6 (5+1=6)
XXI	21 (10+10+1=21)
MI	1001 (1000+1=1001)

positional number systems.

Positional number systems- number systems in which the quantitative value of a digit depends on its location in the number.

For example, if we talk about the decimal number system, then in the number 700 the number 7 means “seven hundred”, but the same number in the number 71 means “seven tens”, and in the number 7020 - “seven thousand”.

Each positional number system has its own base. A natural number greater than or equal to two is chosen as the base. It is equal to the number of digits used in a given number system.

For example:
• Binary- positional number system with base 2.
• Quaternary- positional number system with base 4.
• Five-fold- positional number system with base 5.
• Octal- positional number system with base 8.
• Hexadecimal- positional number system with base 16.

To successfully solve problems on the topic “Number systems”, the student must know by heart the correspondence of binary, decimal, octal and hexadecimal numbers up to 16 10:

10 s/s	2 s/s	8 s/s	16 s/s
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
16	10000	20	10

It is useful to know how numbers are obtained in these number systems. You can guess that in octal, hexadecimal, ternary and others positional number systems everything happens in the same way as the decimal system we are used to:

One is added to the number and a new number is obtained. If the units place becomes equal to the base of the number system, we increase the number of tens by 1, etc.

This “transition of one” is what frightens most students. In fact, everything is quite simple. The transition occurs if the units digit becomes equal to number base, we increase the number of tens by 1. Many, remembering the good old decimal system, are instantly confused about the digits in this transition, because decimal and, for example, binary tens are different things.

Hence, resourceful students develop “their own methods” (surprisingly... working) when filling out, for example, truth tables, the first columns (variable values) of which are, in fact, filled with binary numbers in ascending order.

For example, let's look at getting numbers in octal system: We add 1 to the first number (0), we get 1. Then we add 1 to 1, we get 2, etc. to 7. If we add one to 7, we get a number equal to the base of the number system, i.e. 8. Then you need to increase the tens place by one (we get the octal ten - 10). Next, obviously, are the numbers 11, 12, 13, 14, 15, 16, 17, 20, ..., 27, 30, ..., 77, 100, 101...

Rules for converting from one number system to another.

1 Converting integer decimal numbers to any other number system.

The number must be divided by new number system base. The first remainder of the division is the first minor digit of the new number. If the quotient of the division is less than or equal to the new base, then it (the quotient) must be divided again by the new base. The division must be continued until we get a quotient less than the new base. This is the highest digit of the new number (you need to remember that, for example, in the hexadecimal system, after 9 there are letters, i.e. if the remainder is 11, you need to write it as B).

Example ("division by corner"): Let's convert the number 173 10 to the octal number system.

Thus, 173 10 =255 8

2 Converting regular decimal fractions to any other number system.

The number must be multiplied by the new number system base. The digit that has become the integer part is the highest digit of the fractional part of the new number. to obtain the next digit, the fractional part of the resulting product must again be multiplied by a new base of the number system until the transition to the whole part occurs. We continue multiplication until the fractional part equals zero, or until we reach the accuracy specified in the problem (“... calculate with an accuracy of, for example, two decimal places”).

Example: Let's convert the number 0.65625 10 to the octal number system.