Transfer the number 121 to a binary number system. Translation of numbers from one number system to another online. Translation of the fractional part of the number from the decimal number system to another number system

All positional numbering systems are equal, but depending on the tasks that the person solves the use of numbers it can use different bases with different bases.

The most commonly used decimal number system, i.e. The number system, the alphabet of which consists of ten digits (0.1,2,3,4,5,6,7,8,9) and, accordingly, the base is ten. The wide use of this number system is easily explained. First, the recording of the number in the decimal number system is quite compact, secondly, the decimal number system is used by humanity for several centuries. During this time, people are already accustomed to numbers, and to the record of numbers, and to the pronunciation of numbers in a decimal number system, for example, the "15" record is clear to any person and it will read it as fifteen, but the same number recorded in the binary number system "1111" causes at least a slight bewilderment, but how to read this number.

Nevertheless, it is clear to say that the decimal number system is the optimal choice of humanity to work with numbers it is impossible. We prove it with several examples.

All of you remember the multiplication table and of course remember how much effort you had to attach to learn this table. We will not give here a multiplication table in a decimal number system, but for comparison, we give the multiplication table in the binary number system:

As you can see, the multiplication table in the binary number system looks much easier than in decimal.

The compactness of the number of numbers in the decimal number system, the same is not the highest, in all numbering systems with a base for more than ten numbers will be recorded more compact, for example, the number "15" will also be recorded in a hexadecimal number system as "F".

As already mentioned in paragraph 5, a binary number system has been adopted to record numbers in AUM. In this paragraph, we must figure out however, the numbers in the memory of the computer will seem to understand the rules for the transfer of decimal numbers in binary system Note.

In practice, to transfer numbers from the number system with a base of ten to the number of hours with a base of two, use the following rule:

1. The 1, recorded in the number system with the base ten, is divided with the residue for two (base new system Number) recorded by numbers of the number system with the base ten (old number system), until it turns out in private 0.

2.The results obtained from divisions recorded in the reverse order form a number in a new number system with a base of two.

It is more convenient to use this rule to transfer numbers from a decimal number system. For reverse translation, in a decimal number system is more convenient to use the so-called schema Gorner.

1. Reconnect positions in the number, on the right to the left, starting with zero;

2. Create a number representing the amount of the number of numbers of numbers on the basis of the old number system recorded by the numbers of the new number system, erected to the degree equal number of position numbers among the number;

3. Init the sum of the row.

We will analyze these rules on specific examples.

Example 1.: Write a decimal number 121 in a binary number system.

121 | 2 121 d \u003d 1111001 b

120 60 | 2

1 60 30 | 2

0 30 15 | 2

0 14 7 | 2

1 6 3 | 2

Purpose of work.Studying methods and testing of the transformation skills from one positioning system for another to another.

The number of different numbers used in the positional system determines the name of the number system and is called base The number system.

Any number N in a positioning system with a base can be represented as a polynomial from the base :

where
- number, - numbers numbers (coefficients in degrees ),- base of the number system ( >1).

Numbers are recorded as a sequence of numbers:

.
The point in the sequence separates the whole part of the number from fractional (coefficients for non-negative degrees, from coefficients with negative degrees). The point is lowered if the number is integer (no negative degrees).

In computer systems, there are positional numbering systems with non-definite base: binary, octal, hexadecimal.

In hardware basis, the computer is two-position elements that can be only in two states; One of which is denoted by 0, and the other - 1. Therefore, the arithmetic and logical main computer is a binary number system.

Binary number system. Two digits are used: 0 and 1. In the binary system, any number can be represented as:
.
where either 0 or 1.

This entry corresponds to the sum of the degrees of the number 2, taken with the specified coefficients:

Octal number system. Eight digits are used: 0, 1, 2, 3, 4, 5, 6, 7. Used in computer as an auxiliary to record information in abbreviated form. To represent one digit of the octal system, three binary discharges are used (triad) (see Table 1).

Hex number system. For the image of numbers, 16 digits are used. The first ten digits of this system are denoted by numbers from 0 to 9, and the older six digits are Latin letters: A (10), in (11), C (12), D (13), E (14), F (15). Hexadecimal system, as well as octal, is used to record information in abbreviated form. To represent one digit of a hexadecimal number system, four binary discharge (tetrad) are used (see Table 1).

Table 1.

Alphabets of positioning systems (ss)

Binary SS (Base 2)	Octal (Base 8)		Decimal (Base 10)	Hexadecimal (Base 16)
		Binary			Binary Tetrads

Exercise 1.Translate numbers from specified numbering systems to a decimal system.

Methodical instructions.

The translation of the numbers into the decimal system is systematized by drawing up the amount of the power series with the base of the system, from which the number is translated. Then the value of this amount is then calculated.

Examples.

a) Translate S.S. 

.

b) Translate
s.S.

c) Translate
s.S.

Task 2.Translate entire numbers from a decimal system in an octal, hexadecimal and binary system.

Methodical instructions.

The transfer of entire decimal numbers into an octal, hexadecimal and binary system is valid for the sequential division of the decimal number on the basis of the system in which it is translated, until it turns out a private equal to zero. The number in the new system is recorded in the form of balance from division, starting with the latter.

Examples.

a) Translate
s.S.

181: 8 \u003d 22 (residue 5)

22: 8 \u003d 2 (residue 6)

2: 8 \u003d 0 (residue 2)

Answer:
.

b) Translate
s.S.

The table shows the division:

622: 16 \u003d 38 (residue 14 10 \u003d e 16)

38: 16 \u003d 2 (residue 6)

2: 16 \u003d 0 (residue 2)

Answer:
.

Task 3.Translate the right decimal fractions from the decimal system in an octal, hexadecimal and binary system.

The calculator allows you to transfer integers and fractional numbers from one number system to another. The base of the number system cannot be less than 2 and more than 36 (10 digits and 26 Latin letters after all). The length of numbers should not exceed 30 characters. To enter fractional numbers, use a symbol. or, . To translate a number from one system to another, enter the source number in the first field, the base of the source number system to the second and the base of the number system to which you want to translate the number in the third field, and then click the "Get Record" button.

Source number Recorded at 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 36 System number system.

I want to get a record of the number in 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 System number system.

Get writing

Translations: 3443470

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Number systems

Numbers are divided into two types: positional and not positional. We use the Arab system, it is a positional, and there is another Roman - it is just not a positional one. In the positional systems, the position of the numbers in the number uniquely determines the value of this number. It is easy to understand, examined on the example of some number.

Example 1.. Take the number 5921 in the decimal number system. Number the number on the right left since scratch:

The number 5921 can be written in the following form: 5921 \u003d 5000 + 900 + 20 + 1 \u003d 5 · 10 3 + 9 · 10 2 + 2 · 10 1 + 1 · 10 0. The number 10 is a characteristic that defines the number system. As degrees, the positions of the number of this number are taken.

Example 2.. Consider the real decimal number 1234.567. Number it starting from the zero position of the number from the decimal point to the left and right:

The number 1234.567 can be written in the following form: 1234.567 \u003d 1000 + 200 + 30 + 4 + 0.5 + 0.06 + 0.007 \u003d 1 · 10 3 + 2 · 10 2 + 3 · 10 1 + 4 · 10 0 + 5 · 10 -1 + 6 · 10 -2 + 7 · 10 -3.

Translation of numbers from one number system to another

Most simple way The translation of the number from one number system to another is the translation of the number first into a decimal number system, and then the result obtained in the desired number system.

Translation of numbers from any number system in a decimal number system

To transfer the number from any number system to decimal, it is enough to numbered its discharges, starting with zero (discharge from the decimal point), similar to examples 1 or 2. Find the amount of the number of numbers on the base of the number system to the degree of position of this figure:

1. Transfer the number 1001101.1101 2 to a decimal number system.
Decision: 10011.1101 2 \u003d 1 · 2 4 + 0 · 2 3 + 0 · 2 2 + 1 · 2 1 + 1 · 2 0 + 1 · 2 -1 + 1 · 2 -2 + 0 · 2 -3 + 1 · 2 - 4 \u003d 16 + 2 + 1 + 0.5 + 0.25 + 0.0625 \u003d 19.8125 10
Answer: 10011.1101 2 = 19.8125 10

2. Transfer the number E8F.2D 16 to a decimal number system.
Decision: E8F.2D 16 \u003d 14 · 16 2 + 8 · 16 1 + 15 · 16 0 + 2 · 16 -1 + 13 · 16 -2 \u003d 3584 + 128 + 15 + 0.125 + 0.05078125 \u003d 3727.17578125 10
Answer: E8F.2D 16 \u003d 3727.17578125 10

Translation of numbers from a decimal number system to another number system

To transfer numbers from a decimal number system to another number system, a whole and fractional part of the number must be translated separately.

Transfer of a whole part of the number from a decimal number system to another number system

The integer part is translated from a decimal number system to another number system using a sequential division of a whole part of the number based on the number of the number system until a whole balance is obtained, a smaller base system base. The result of the translation will be an entry from residues, starting with the latter.

3. Transfer the number 273 10 to an eight-lit count.
Decision: 273/8 \u003d 34 and residue 1, 34/8 \u003d 4 and residue 2, 4 less than 8, so the calculations are completed. Recording from residues will have the following form: 421
Check: 4 · 8 2 + 2 · 8 1 + 1 · 8 0 \u003d 256 + 16 + 1 \u003d 273 \u003d 273, the result coincided. So the translation is performed correctly.
Answer: 273 10 = 421 8

Consider the translation of the right decimal fractions in various systems Note.

Translation of the fractional part of the number from the decimal number system to another number system

Recall, the correct decimal fraction is called real number with zero integer. In order to translate such a number into the NUMBA system with the base n, you need to multiply the number on n until the fractional part is reset or the required number of discharges will not be obtained. If the multiplication is obtained with a whole part, different from zero, then the whole part is not taken into account, as it is consistently entered into the result.

4. Transfer a number 0.125 10 to a binary number system.
Decision: 0.125 · 2 \u003d 0.25 (0 - a whole part that will be the first digit of the result), 0.25 · 2 \u003d 0.5 (0 - the second digit of the result), 0.5 · 2 \u003d 1.0 (1 - the third digit of the result, and since the fractional part is zero , then the translation is completed).
Answer: 0.125 10 = 0.001 2

With the help of this online calculator You can translate integers and fractional numbers from one number system to another. A detailed solution is given with explanations. To translate, enter the original number, set the source number system base, set the base of the number system to which you want to translate the number and click on the "Translate" button. Theoretical part and numerical examples see below.

The result is already received!

Translation of whole and fractional numbers from one number system to any other - theory, examples and solutions

There are positional and not positional number systems. Arabic number system, which we use in everyday life is a positional, and Roman - no. In positional surgery systems, the position of the number uniquely determines the value of the number. Consider this on the example of the number 6372 in a decimal number system. Number this number on the right left since scratch:

Then the number 6372 can be represented as follows:

6372 \u003d 6000 + 300 + 70 + 2 \u003d 6 · 10 3 + 3 · 10 2 + 7 · 10 1 + 2 · 10 0.

The number 10 defines the number system (in this case it is 10). As degrees, the positions of the number of this number are taken.

Consider a real decimal number 1287.923. Number it starting from scratch the position of the number from the decimal point to the left and right:

Then the number 1287.923 can be represented as:

1287.923 \u003d 1000 + 200 + 80 + 7 + 0.9 + 0.02 + 0.003 \u003d 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 a number in position n., D -k - fractional number in position (-k), s. - Number system.

A few words about the number systems. The number in the decimal number system consists of a plurality of numbers (0.1,2,3,4,5,6,7,8,9), in an octaous number system - from a plurality of numbers (0.1, 2,3,4,5,6,7), in a binary number system - from a plurality of numbers (0.1), in a hexadecimal number system - from a plurality of numbers (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 number 10,11,12,13,14,15. In Table Table.1 Presented numbers B. different systems Note.

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.

Translation of numbers from one number system to another

To transfer numbers from one number to another to another, the easiest way to first translate the number to a decimal number system, and then, from the decimal number system to translate to the desired number system.

Translation of numbers from any number system in a decimal number system

Using formula (1), you can translate numbers from any number system to a decimal number system.

Example 1. Translate the number 1011101.001 from the binary number system (SS) in a decimal SS. Decision:

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 \u003d 64 + 16 + 8 + 4 + 1 + 1/8 \u003d 93.125

Example2. Translate the number 1011101.001 from the octaous number system (SS) in a decimal SS. Decision:

Example 3 . Translate the number AB572.CDF from a hexadecimal number system in a decimal SS. Decision:

Here A. - per 10, B. - by 11, C.- by 12, F. - at 15.

Translation of numbers from a decimal number system to another number system

To transfer numbers from a decimal number system to another number system, it is necessary to translate separately by the integer part of the number and fractional part numbers.

An integer part of the number is translated from a decimal SS to another number system - a sequential division of a whole part of the number on the base of the number system (for a binary CC - by 2, for an 8-character SS - by 8, for 16-smoke-16, etc. ) Before getting a whole residue, less than the base of the SS.

Example 4 . We translate the number 159 of the decimal SS into the 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 during division by 2 gives the private 79 and the residue 1. Next, the number 79 during division by 2 gives Private 39 and the residue 1, etc. As a result, by building a number from the balances of divisions (right to left) we get a number in binary ss: 10011111 . Consequently, you can write:

159 10 =10011111 2 .

Example 5 . We translate the number 615 of the decimal SS into the octal SS.

615	8
608	76	8
7	72	9	8
	4	8	1
		1

When the number from the decimal SS in the octal SS, it is necessary to sequentially divide the number on 8 until the whole residue is less than 8. As a result, building a number from the balances of division (right to left), we get a number in the octane SS: 1147 (See Fig. 2). Consequently, you can write:

615 10 =1147 8 .

Example 6 . We transfer 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 Fig.

To transfer the correct decimal fractions (real number with a zero integer) to the counting system with the base S, a given number must be multiplied to S, until a clean zero does not get in the fractional part, or we will not get the required number of discharges. If you get a number with a whole part, different from zero, then this whole part does not take into account (they are consistently enrolled in the result).

Consider the foregoing on the examples.

Example 7 . We transfer 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 multiplied by 2. If the multiplication is obtained with a whole part, different from zero, then the integer part is written separately (to the left of the number), and the number is written to the zero integer. If, when multiplying, a number with a zero integer is obtained, then zero is written to the left. The multiplication process continues until the fractional part does not get pure zero or do not get the required number of discharges. Recording fatty numbers (Fig. 4) from top to bottom We obtain the desired number in the binary number system: 0. 0011011 .

Consequently, you can write:

0.214 10 =0.0011011 2 .

Example 8 . We translate 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 bring the number of 0.125 of the decimal SS into a binary, this number is multiplied by 2. In the third stage it turned out 0. Therefore, the following result turned out:

0.125 10 =0.001 2 .

Example 9 . We translate 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 obtain numbers 3, 6, 12, 8, 11, 4. But in hexadecimal CC, the numbers 12 and 11 correspond to the number C and B. Therefore, we have:

0.214 10 \u003d 0.36C8B4 16.

Example 10 . We translate the number 0.512 from a decimal number system in the 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

Received:

0.512 10 =0.406111 8 .

Example 11 . We translate the number 159.125 from a decimal number system to binary SS. To do this, we translate separately an integer part of the number (Example 4) and the fractional part of the number (Example 8). Next, we get the merging of these results:

159.125 10 =10011111.001 2 .

Example 12 . We transfer the number 19673.214 from a decimal number system to hexadecimal. To do this, we translate separately an integer part of the number (Example 6) and the fractional part of the number (example 9). Next, we get the combining results.