Those taking the Unified State Exam and more...

It is strange that in computer science lessons in schools they usually show students the most complex and inconvenient way to convert numbers from one system to another. This method consists of sequentially dividing the original number by the base and collecting the remainders from the division in reverse order.

For example, you need to convert the number 810 10 to binary:

We write the result in reverse order from bottom to top. It turns out 81010 = 11001010102

If you need to convert to the binary system, quite big numbers, then the division ladder takes on the size of a multi-story building. And how can you collect all the ones and zeros and not miss a single one?

The Unified State Exam program in computer science includes several tasks related to converting numbers from one system to another. Typically, this is a conversion between octal and hexadecimal systems and binary. These are sections A1, B11. But there are also problems with other number systems, such as in section B7.

To begin with, let us recall two tables that would be good to know by heart for those who choose computer science as their future profession.

Table of powers of number 2:

2 1	2 2	2 3	2 4	2 5	2 6	2 7	2 8	2 9	2 10
2	4	8	16	32	64	128	256	512	1024

It is easily obtained by multiplying the previous number by 2. So, if you do not remember all of these numbers, the rest are not difficult to obtain in your mind from those that you remember.

Table of binary numbers from 0 to 15 with hexadecimal representation:

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111
0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F

The missing values ​​are also easy to calculate by adding 1 to the known values.

Integer conversion

So, let's start by converting directly to the binary system. Let's take the same number 810 10. We need to decompose this number into terms equal to powers of two.

1. We are looking for the power of two closest to 810 and not exceeding it. This is 2 9 = 512.
2. Subtract 512 from 810, we get 298.
3. Repeat steps 1 and 2 until there are no 1s or 0s left.
4. We got it like this: 810 = 512 + 256 + 32 + 8 + 2 = 2 9 + 2 8 + 2 5 + 2 3 + 2 1.

Then there are two methods, you can use any of them. How easy it is to see that in any number system its base is always 10. The square of the base will always be 100, the cube 1000. That is, the degree of the base of the number system is 1 (one), and there are as many zeros behind it as the degree is.

Method 1: Arrange 1 according to the ranks of the indicators of the terms. In our example, these are 9, 8, 5, 3 and 1. The remaining places will contain zeros. So, we got the binary representation of the number 810 10 = 1100101010 2. Units are placed in 9th, 8th, 5th, 3rd and 1st places, counting from right to left from zero.

Method 2: Let's write the terms as powers of two under each other, starting with the largest.

810 =

Now let's add these steps together, like folding a fan: 1100101010.

That's all. At the same time, the problem “how many units are in the binary notation of the number 810?” is also simply solved.

The answer is as many as there are terms (powers of two) in this representation. 810 has 5 of them.

Now the example is simpler.

Let's convert the number 63 to the 5-ary number system. The closest power of 5 to 63 is 25 (square 5). A cube (125) will already be a lot. That is, 63 lies between the square of 5 and the cube. Then we will select the coefficient for 5 2. This is 2.

We get 63 10 = 50 + 13 = 50 + 10 + 3 = 2 * 5 2 + 2 * 5 + 3 = 223 5.

And, finally, very easy translations between 8 and hexadecimal systems. Since their base is a power of two, the translation is done automatically, simply by replacing the numbers with their binary representation. For the octal system, each digit is replaced by three binary digits, and for the hexadecimal system, four. In this case, all leading zeros are required, except for the most significant digit.

Let's convert the number 547 8 to binary.

547 8 =	101	100	111
	5	4	7

One more, for example 7D6A 16.

7D6A 16 =	(0)111	1101	0110	1010
	7	D	6	A

Let's convert the number 7368 to the hexadecimal system. First, write the numbers in triplets, and then divide them into quadruples from the end: 736 8 = 111 011 110 = 1 1101 1110 = 1DE 16. Let's convert the number C25 16 to the octal system. First, we write the numbers in fours, and then divide them into threes from the end: C25 16 = 1100 0010 0101 = 110 000 100 101 = 6045 8. Now let's look at converting back to decimal. It is not difficult, the main thing is not to make mistakes in the calculations. We expand the number into a polynomial with powers of the base and coefficients for them. Then we multiply and add everything. E68 16 = 14 * 16 2 + 6 * 16 + 8 = 3688. 732 8 = 7 * 8 2 + 3*8 + 2 = 474 .

Converting Negative Numbers

Here you need to take into account that the number will be presented in two's complement code. To convert a number into additional code, you need to know the final size of the number, that is, what we want to fit it into - in a byte, in two bytes, in four. The most significant digit of a number means the sign. If there is 0, then the number is positive, if 1, then it is negative. On the left, the number is supplemented with a sign digit. We do not consider unsigned numbers; they are always positive, and the most significant bit in them is used as information.

To convert a negative number to binary's complement code, you need to convert positive number into the binary system, then change zeros to ones and ones to zeros. Then add 1 to the result.

So, let's convert the number -79 to the binary system. The number will take us one byte.

We convert 79 to the binary system, 79 = 1001111. We add zeros on the left to the size of the byte, 8 bits, we get 01001111. We change 1 to 0 and 0 to 1. We get 10110000. We add 1 to the result, we get the answer 10110001. Along the way we answer Unified State Exam question“how many ones are in the binary representation of the number -79?” The answer is 4.

Adding 1 to the inverse of a number eliminates the difference between the representations +0 = 00000000 and -0 = 11111111. In two's complement code they will be written the same as 00000000.

Converting fractional numbers

Fractional numbers are converted in the reverse way of dividing whole numbers by the base, which we looked at at the very beginning. That is, using sequential multiplication by a new base with the collection of whole parts. The integer parts obtained during multiplication are collected, but do not participate in the following operations. Only fractions are multiplied. If the original number is greater than 1, then the integer and fractional parts are translated separately and then glued together.

Let's convert the number 0.6752 to the binary system.

0	,6752
	*2
1	,3504
	*2
0	,7008
	*2
1	,4016
	*2
0	,8032
	*2
1	,6064
	*2
1	,2128

The process can be continued for a long time until we get all the zeros in the fractional part or the required accuracy is achieved. Let's stop at the 6th sign for now.

It turns out 0.6752 = 0.101011.

If the number was 5.6752, then in binary it will be 101.101011.

Using this online calculator you can convert whole and fractional numbers from one number system to another. A detailed solution with explanations is given. To translate, enter the original number, set the base of the number system of the source number, set the base of the number system into which you want to convert the number and click on the "Translate" button. See the theoretical part and numerical examples below.

The result has already been received!

Converting integers and fractions from one number system to any other - theory, examples and solutions

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 defines the number system (in in this case this 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 number system (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 zero whole 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 obtain 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.

Methods for converting numbers from one number system to another.

Converting numbers from one positional number system to another: converting integers.

To convert an integer from one number system with base d1 to another with base d2, you must sequentially divide this number and the resulting quotients by base d2 of the new system until you get a quotient less than base d2. The last quotient is the highest digit of the number in new system numbers with base d2, and the numbers following it are remainders from division, written in the reverse order of their receipt. Perform arithmetic operations in the number system in which the number being translated is written.

Example 1. Convert the number 11(10) to the binary number system.

Answer: 11(10)=1011(2).

Example 2. Convert the number 122(10) to the octal number system.

Answer: 122(10)=172(8).

Example 3. Convert the number 500(10) to hexadecimal number system.

Answer: 500(10)=1F4(16).

Converting numbers from one positional number system to another: converting proper fractions.

To convert a proper fraction from a number system with base d1 to a system with base d2, it is necessary to sequentially multiply the original fraction and the fractional parts of the resulting products by the base of the new number system d2. The correct fraction of a number in the new number system with base d2 is formed in the form of integer parts of the resulting products, starting from the first.
If the translation results in a fraction in the form of an infinite or divergent series, the process can be completed when the required accuracy is achieved.

When translating mixed numbers, it is necessary to separately translate the integer and fractional parts into a new system according to the rules for translating integers and proper fractions, and then combine both results into one mixed number in the new number system.

Example 1. Convert the number 0.625(10) to the binary number system.

Answer: 0.625(10)=0.101(2).

Example 2. Convert the number 0.6(10) to the octal number system.

Answer: 0.6(10)=0.463(8).

Example 2. Convert the number 0.7(10) to hexadecimal number system.

Answer: 0.7(10)=0.B333(16).

Convert binary, octal and hexadecimal numbers to decimal number system.

To convert a number from the P-ary system to a decimal one, you must use the following expansion formula:
аnan-1…а1а0=аnPn+ аn-1Pn-1+…+ а1P+a0 .

Example 1. Convert the number 101.11(2) to the decimal number system.

Answer: 101.11(2)= 5.75(10) .

Example 2. Convert the number 57.24(8) to the decimal number system.

Answer: 57.24(8) = 47.3125(10) .

Example 3. Convert the number 7A,84(16) to the decimal number system.

Answer: 7A.84(16)= 122.515625(10) .

Converting octal and hexadecimal numbers to the binary number system and vice versa.

To convert a number from the octal number system to binary, each digit of this number must be written as a three-digit binary number (triad).

Example: write the number 16.24(8) in the binary number system.

Answer: 16.24(8)= 1110.0101(2) .

To convert a binary number back into the octal number system, you need to divide the original number into triads to the left and right of the decimal point and represent each group with a digit in the octal number system. Extreme incomplete triads are supplemented with zeros.

Example: write the number 1110.0101(2) in the octal number system.

Answer: 1110.0101(2)= 16.24(8) .

To convert a number from the hexadecimal number system to the binary system, you need to write each digit of this number as a four-digit binary number (tetrad).

Example: write the number 7A,7E(16) in the binary number system.


Answer: 7A,7E(16)= 1111010.0111111(2) .

Note: leading zeros on the left for integers and on the right for fractions are not written.

To convert a binary number back into the hexadecimal number system, you need to divide the original number into tetrads to the left and right of the decimal point and represent each group with a digit in the hexadecimal number system. Extreme incomplete triads are supplemented with zeros.

Example: write the number 1111010.0111111(2) in hexadecimal number system.

Let's look at ways to convert numbers from one number system to another.

a) Converting a binary number to decimal.

It is necessary to add twos in powers corresponding to the positions where ones stand in binary. For example:

Let's take the number 20. In the binary system it has the following form: 10100.

So (we count from left to right, counting from 4 to 0; a number to the zero power is always equal to one)

10100 = 1*2 4 + 0*2 3 + 1*2 2 + 0*2 1 + 0*2 0 = 20

16+0+4+0+0 = 20.

b) Converting a decimal number to binary.

You need to divide it by two, writing the remainder from right to left:

20/2 = 10, remainder 0

10/2=5, remainder 0

5/2=2, remainder 1

2/2=1, remainder 0

1/2=0, remainder 1

As a result we get: 10100 = 20

c) Converting a hexadecimal number to decimal.

In the hexadecimal system, the position number of a digit in a number corresponds to the power to which the number 16 must be raised:

8A = 8*16 + 10 (0A) = 138

Finally, we present the algorithm for converting to and from the binary system, proposed by L. Radyuk.

Let A(cd) be an integer decimal number. Let's write it as a sum of powers of base 2 with binary coefficients. His entries in expanded form will not include negative powers bases (numbers 2):

A(td) = a(n-1) * 2^(n-1) + a(n-2) * 2^(n-2) + … + a(1) * 2^1 + a(0) * 2^0.

At the first step, we divide the number A(tsd) by the base of the binary system, that is, by 2. The quotient of the division will be equal to:

a(n-1) * 2^(n-2) + a(n-2) * 2^(n-3) + … + a(1), and the remainder is a(0).

At the second step, we again divide the integer quotient by 2, the remainder of the division will now be equal to a(1).

If we continue this division process, then after the nth step we get the sequence of residues:

a(0), a(1),…, a(n-1).

It is easy to notice that their sequence coincides with the reverse sequence of digits of an integer binary number written in collapsed form:

A(2) = a(n-1)...a(1)a(0).

Thus, it is enough to write the remainders in reverse order to obtain the desired binary number.

Then the algorithm itself will be as follows:

1. Consistently divide the original integer decimal number and the resulting integer quotients by the base of the system (by 2) until you get a quotient that is less than the divisor, that is, less than 2.

2. Write down the resulting remainders in reverse order, and add the last quotient to the left.

To convert numbers from octal and hexadecimal number systems to binary, you need to convert the digits of the number into groups of binary digits. To convert from the octal system to binary, each digit of the number must be converted into a group of three binary digits, a triad, and when converting a hexadecimal number into a group of four digits, a tetrad.

CONCLUSION

Summing up the results of the work, we can draw the following conclusions.

The positional number system consists of using a limited number of digits, but the position of each digit in the number provides the significance (weight) of this digit. The position of a digit in a number is called digit in mathematical language.

The base of a positional number system is the number of different signs or symbols (digits) used to display numbers in a given system.

In order to binary numbers, differing in quite a significant length, were easier to perceive and display, they are compressed into octal and hexadecimal number systems.

IN computer technologies all types of information are encoded only in numbers or, more precisely, numbers that are represented in the binary number system, a method of representing any numbers using two signs (digits) according to the positional principle.

The calculator allows you to convert whole 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 must not exceed 30 characters. To enter fractional numbers, use the symbol. or, . To convert a number from one system to another, enter the original number in the first field, the base of the original number system in the second, and the base of the number system to which you want to convert the number in the third field, then click the "Get Record" button.

Original number written 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 -th number system.

I want to get a number written 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 -th number system.

Get entry

Translations completed: 3336969

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Number systems

Number systems are divided into two types: positional And not positional. We use the Arabic system, it is positional, but there is also the Roman system - it is not positional. In positional systems, the position of a digit in a number uniquely determines the value of that number. This is easy to understand by looking at some number as an example.

Example 1. Let's take the number 5921 in the decimal number system. Let's number the number from right to left starting from zero:

The number 5921 can be written in the following form: 5921 = 5000+900+20+1 = 5·10 3 +9·10 2 +2·10 1 +1·10 0 . The number 10 is a characteristic that defines the number system. The values ​​of the position of a given number are taken as powers.

Example 2. Consider the real decimal number 1234.567. Let's 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 = 1000+200+30+4+0.5+0.06+0.007 = 1·10 3 +2·10 2 +3·10 1 +4·10 0 +5·10 -1 + 6·10 -2 +7·10 -3 .

Converting numbers from one number system to another

Most in a simple way converting a number from one number system to another is to first convert the number into a decimal number system, and then the resulting result into the required number system.

Converting numbers from any number system to the decimal number system

To convert a number from any number system to decimal, it is enough to number its digits, starting with zero (the digit to the left of the decimal point) similarly to examples 1 or 2. Let's find the sum of the products of the digits of the number by the base of the number system to the power of the position of this digit:

1. Convert the number 1001101.1101 2 to the decimal number system.
Solution: 10011.1101 2 = 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 = 16+2+1+0.5+0.25+0.0625 = 19.8125 10
Answer: 10011.1101 2 = 19.8125 10

2. Convert the number E8F.2D 16 to the decimal number system.
Solution: E8F.2D 16 = 14·16 2 +8·16 1 +15·16 0 +2·16 -1 +13·16 -2 = 3584+128+15+0.125+0.05078125 = 3727.17578125 10
Answer: E8F.2D 16 = 3727.17578125 10

Converting numbers from the decimal number system to another number system

To convert numbers from the decimal number system to another number system, the integer and fractional parts of the number must be converted separately.

Converting an integer part of a number from a decimal number system to another number system

An integer part is converted from a decimal number system to another number system by sequentially dividing the integer part of a number by the base of the number system until a whole remainder is obtained that is less than the base of the number system. The result of the translation will be a record of the remainder, starting with the last one.

3. Convert the number 273 10 to the octal number system.
Solution: 273 / 8 = 34 and remainder 1. 34 / 8 = 4 and remainder 2. 4 is less than 8, so the calculation is complete. The record from the balances will look like this: 421
Examination: 4·8 2 +2·8 1 +1·8 0 = 256+16+1 = 273 = 273, the result is the same. This means the translation was done correctly.
Answer: 273 10 = 421 8

Let's consider the translation of regular decimal fractions into various number systems.

Converting the fractional part of a number from the decimal number system to another number system

Let us remind you that the correct decimal called real number with zero integer part. To convert such a number to a number system with base N, you need to sequentially multiply the number by N until the fractional part goes to zero or the required number of digits is obtained. If, during multiplication, a number with an integer part other than zero is obtained, then the integer part is not taken into account further, since it is sequentially entered into the result.

4. Convert the number 0.125 10 to the binary number system.
Solution: 0.125·2 = 0.25 (0 is the integer part, which will become the first digit of the result), 0.25·2 = 0.5 (0 is the second digit of the result), 0.5·2 = 1.0 (1 is 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