Number division number system online calculator. Arithmetic operations with numbers in positional number systems

When adding floating point numbers, the alignment of orders is performed towards a higher order:

Floating point addition algorithm:

1. Alignment of orders;
2. Addition of mantissas in the additional modified code;
3. Normalization of the result.

Example #4.
A=0.1011*2 10 , B=0.0001*2 11
1. Alignment of orders;
A=0.01011*2 11 , B=0.0001*2 11
2. Addition of mantissas in the additional modified code;
MA add. mod. =00.01011
MB additional mod. =00.0001
00,01011
+ 00,00010
=
00,01101
A+B=0.01101*2 11
3. Normalization of the result.
A+B=0.1101*2 10

Example #3. Write a decimal number in binary-decimal notation and add two numbers in binary notation.

Consider the basic arithmetic operations: addition, subtraction, multiplication and division. The rules for performing these operations in the decimal system are well known - this is addition, subtraction, multiplication by a column and division by an angle. These rules apply to all other positional number systems. You just need to use special addition and multiplication tables for each system.

1. Addition

Addition tables are easy to create using counting rules.

When adding, the numbers are summed up by digits, and if an excess occurs, then it is transferred to the left.

Example 1 Let's add the numbers 15 and 6 in different number systems.

Example 2 Let's add the numbers 15, 7 and 3.

Example 3 Let's add the numbers 141.5 and 59.75.

Answer: 141.5 + 59.75 = 201.25 10 = 11001001.01 2 = 311.2 8 = C9.4 16

Examination. Let's convert the received amounts to decimal form:

11001001,01 2 = 2 7 + 2 6 + 2 3 + 2 0 + 2 -2 = 201,25

311,2 8 = 3 . 8 2 + 1 . 8 1 + 1 . 8 0 + 2 . 8 -1 = 201,25

C9.4 16 = 12 . 16 1 + 9 . 16 0 + 4 . 16 -1 = 201,25

2. Subtraction

Loanhigh order units

Example 4 Subtract one from the numbers 10 2 , 10 8 and 10 16

Example 5 Subtract one from the numbers 100 2 , 100 8 and 100 16 .

Example 6 Subtract the number 59.75 from the number 201.25.

Answer: 201.25 10 - 59.75 10 = 141.5 10 = 10001101.1 2 = 215.4 8 = 8D.8 16.

Examination. Let's convert the resulting differences to decimal form:

10001101,1 2 = 2 7 + 2 3 + 2 2 + 2 0 + 2 -1 = 141,5;

215,4 8 = 2 . 8 2 + 1 . 8 1 + 5 . 8 0 + 4 . 8 -1 = 141,5;

8D,8 16 = 8 . 16 1+D . 16 0 + 8 . 16 -1 = 141,5.

How do we add in decimal notation?

Let's remember how we add numbers in the way we are already used to, in decimal.

The most important thing is to understand the ranks. Remember the alphabet of each SS and then it will become easier for you.

Addition in binary is no different from addition in decimal. The main thing to remember is that the alphabet contains only two numbers: 0 and 1. Therefore, when we add 1 + 1, we get 0, and increase the number by another 1 digit. Look at the example above:

1. We begin to fold as we used to from right to left. 0 + 0 = 0, so we write 0. Go to the next bit.
2. We add 1 + 1 and get 2, but 2 is not in the binary number system, which means we write 0, and add 1 to the next bit.
3. We get three units in this category, we add 1 + 1 + 1 = 3, this number also cannot be. So 3 - 2 = 1. And 1 is added to the next digit.
4. We again get 1 + 1 = 2. We already know that 2 cannot be, so we write 0, and add 1 to the next bit.
5. There is nothing more to add, so in the answer we get: 10100.

We analyzed one example, solve the second one yourself:

Just like in any other number systems, you need to remember the Alphabet. Let's try to add the expression.

1. Everything is as usual, we begin to fold from right to left. 4 + 3 = 7.
2. 5 + 4 = 9. There can't be nine, so we subtract 8 from 9, we get 1. And add 1 more to the next digit.
3. 3 + 7 + 1 = 11. Subtract 8 from 11, we get 3. And add one to the next digit.
4. 6 + 1 = 7.
5. There is nothing more to add. Answer: 7317.

Now do the addition yourself:

1. We perform actions already familiar to us and do not forget about the alphabet. 2 + 1 = 3.
2. 5 + 9 = 14. Remember the Alphabet: 14 = E.
3. C \u003d 12. 12 + 8 \u003d 20. Twenty is not in the hexadecimal number system. So we subtract 16 from 20 and get 4. And we add one to the next digit.
4. 1 + 1 = 2.
5. There is nothing more to add. Answer: 24E3.

Subtraction in number systems

Let's remember how we do it in the decimal number system.

1. We start from left to right, from the smallest category to the largest. 2 - 1 = 1.
2. 1 – 0 = 1.
3. 3 – 9 = ? Three is less than nine, so let's borrow a one from the highest order. 13 - 9 = 4.
4. From the last digit, we took a unit for the previous action, so 4 - 1 = 3.
5. Answer: 3411.

1. We start as usual. 1 - 1 = 0.
2. 1 – 0 = 1.
3. You cannot subtract one from 0. Therefore, we will take one category from the elder. 2 - 1 = 1.
4. Answer: 110.

Now decide for yourself:

1. Nothing new, the main thing is to remember the alphabet. 4 - 3 = 1.
2. 5 – 0 = 5.
3. We cannot immediately subtract 7 from 3, for this we need to borrow a unit from a higher order. 11 - 7 = 4.
4. Remember that we borrowed one earlier, 6 - 1 = 5.
5. Answer: 5451.

Let's take the previous example and see what the result is in hexadecimal. Same or different?

1. 4 – 3 = 1.
2. 5 – 0 = 5.
3. We cannot immediately subtract 7 from 3, for this we need to borrow a unit from a higher order. 19 - 7 \u003d 12. In hexadecimal, 12 \u003d C.
4. Remember that we borrowed one earlier, 6 - 1 = 5
5. Answer: 5S51

Example for self solution:

Multiplication in number systems

Let's remember once and for all that multiplication in any number system by one will always give the same number.

1. We multiply each digit by one, as usual from right to left, and we get the number 6748;
2. We multiply 6748 by 8 and get the number 53984;
3. We perform the operation of multiplying 6748 by 3. We get the number 20244;
4. We add all 3 numbers, according to the rules. We get 2570988;
5. Answer: 2570988.

In binary, multiplication is very easy. We always multiply either by 0 or by one. The main thing is to fold carefully. Let's try.

1. 1101 we multiply by one, as usual from right to left, and we get the number 1101;
2. We perform this operation 2 more times;
3. We add all 3 numbers carefully, remember the alphabet, not forgetting the ladder;
4. Answer: 1011011.

Example for self solution:

1. 5 x 4 \u003d 20. And 20 \u003d 2 x 8 + 4. We write the remainder of the division into a number - it will be 4, and we keep 2 in mind. We do this procedure from right to left and get the number 40234;
2. When multiplied by 0, we get four 0;
3. When multiplied by 7, we get the number 55164;
4. Now we add the numbers and get - 5556634;
5. Answer: 5556634.

Example for self solution:

Everything is as usual, the main thing is to remember the alphabet. For convenience, translate alphabetic numbers into the number system that is familiar to you, as you multiply, translate back into an alphabetic value.

For clarity, let's analyze the multiplication by 5 of the number 20A4.

1. 5 x 4 \u003d 20. And 20 \u003d 16 + 4. We write the remainder of the division into a number - it will be 4, and we keep 1 in mind.
2. A x 5 + 1 \u003d 10 x 5 + 1 \u003d 51. 51 \u003d 16 x 3 + 3. We write the remainder of the division into a number - it will be 3, and we keep 3 in mind.
3. When multiplied by 0, we get 0 + 3 = 3;
4. 2 x 5 = 10 = A; As a result, we get the A334; We do this procedure with two other numbers;
5. Remember the rule of multiplication by 1;
6. When multiplied by B, we get the number 1670C;
7. Now we add the numbers and get - 169B974;
8. Answer: 169B974.

An example for an independent solution.

With 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 original number, set the base of the number system to which you want to convert the number and click the "Translate" button. See the theoretical part and numerical examples below.

The result has already been received!

Translation of integer and fractional numbers 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, while the Roman one is not. In positional number systems, the position of a number uniquely determines the magnitude of the number. 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 this case it is 10). The values ​​of the position of the given number are taken as degrees.

Consider the real decimal number 1287.923. We number it starting from the zero position of the number from the decimal point to the left and to the 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 a set of digits (0,1,2,3,4,5,6,7,8,9), in the octal number system it consists of a set of digits (0,1, 2,3,4,5,6,7), in the binary system - from the set of digits (0.1), in the hexadecimal number system - from the 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 numbers 10,11,12,13,14,15. In Table 1 numbers are represented in different number systems.

Converting numbers from one number system to another

To translate numbers from one number system to another, the easiest way is to first convert the number to the decimal number system, and then, from the decimal number system, translate it into the required number system.

Converting numbers from any number system to 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 to decimal SS. Solution:

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

Converting numbers from a decimal number system to another number system

To convert numbers from a decimal number system to another number system, you need to translate the integer part of the number and the fractional part of the number separately.

The integer part of the number is translated from the decimal SS to another number system - by successive division of the integer part of the number by the base of the number system (for binary SS - by 2, for 8-digit SS - by 8, for 16-digit - by 16, etc. ) to obtain a whole remainder, less than the base of the SS.

Example 4 . Let's translate the number 159 from decimal SS to binary SS:

As can be seen from Fig. 1, the number 159, when divided by 2, gives the quotient 79 and the remainder is 1. Further, the number 79, when divided by 2, gives the quotient 39 and the remainder is 1, and so on. As a result, by constructing a number from the remainder of the division (from right to left), we get 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.

When converting a number from decimal SS to octal SS, you need to sequentially divide the number by 8 until you get an integer remainder less than 8. As a result, building a number from the remainder of the division (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 translate the number 19673 from the decimal number system to hexadecimal SS.

As can be seen from Figure 3, by successively dividing the number 19673 by 16, we got the remainders 4, 12, 13, 9. In the hexadecimal number system, the number 12 corresponds to C, the number 13 - D. Therefore, our hexadecimal number is 4CD9.

To convert correct decimal fractions (a real number with a zero integer part) into a number system with base s, this number must be successively multiplied by s until the fractional part is pure zero, or we get the required number of digits. If the multiplication results in a number with an integer part other than zero, 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 translate the number 0.214 from the decimal number system to binary SS.

As can be seen from Fig.4, the number 0.214 is successively 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 during multiplication a number with a zero integer part is obtained, then zero is written to the left of it. The multiplication process continues until a pure zero is obtained in the fractional part or the required number of digits is obtained. Writing bold numbers (Fig. 4) from top to bottom, we get the required number in the binary system: 0. 0011011 .

Therefore, we can write:

0.214 10 =0.0011011 2 .

Example 8 . Let's translate the number 0.125 from the decimal number system to the binary SS.

To convert the number 0.125 from decimal SS to binary, this number is successively multiplied by 2. In the third stage, 0 was obtained. Therefore, the following result was obtained:

0.125 10 =0.001 2 .

Example 9 . Let's translate the number 0.214 from the decimal number system to hexadecimal SS.

Following examples 4 and 5, we get the numbers 3, 6, 12, 8, 11, 4. But in hexadecimal SS, the numbers C and B correspond to the numbers 12 and 11. Therefore, we have:

0.214 10 =0.36C8B4 16 .

Example 10 . Let's translate the number 0.512 from the decimal number system to the octal SS.

Received:

0.512 10 =0.406111 8 .

Example 11 . Let's translate 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). Combining these results, we get:

159.125 10 =10011111.001 2 .

Example 12 . Let's translate 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 get.