Operations in various theory surge systems. Arithmetic operations in various binary number systems. Some number systems

Because in binary system The numbers in the number of numbers use only 2 digits - 0 and 1, it means when 1 + 1 is added in the lower discharge, 0, and 1 goes into a senior discharge.

By analogy with 10-ss: 9 + 1 (ten numbers in numbers in numbers), 0 and 1 is written in the older discharge, it turns out 10.

Examples

1) Moving in a column 10110 2 and 111011 2. Units from above designate the transfer from the previous discharge:

2) Perform addition for the following binary numbers:

3) add numbers: 10000000100 2 + 111000010 2 and check

10000000100 2 + 111000010 2 = 10111000110 2 .

We will verify the results of the calculation of the transfer to the decimal number system. To do this, we will transfer each term and amount into a decimal number system, we will perform the addition of the components in decimal system Note. The result must coincide with the amount.

10000000100 2 \u003d 1 × 2 10 + 1 × 2 2 \u003d 1024 + 4 \u003d 1028 10

111000010 2 \u003d 1 × 2 8 + 1 × 2 7 + 1 × 2 6 + 1 × 2 1 \u003d 256 + 128 + 64 + 2 \u003d 450 10

10111000110 2 \u003d 1 × 2 10 + 1 × 2 8 + 1 × 2 7 + 1 × 2 6 + 1 × 2 2 + 1 × 2 1 \u003d

1024 + 256 + 128 + 64 + 4 + 2 =1478 10

1028 10 + 450 10 =1478 10 .

The results coincide, therefore, the calculations in the binary number system are made correctly.

Octal numbers

Table of addition of octal numbers

+

When calculating B. octal system It must be remembered that the maximum figure is 7. The transfer when adding occurs when the amount in the next discharge is obtained more than 7. The loan from the older discharge is 10 8 \u003d 8, and all the "intermediate" discharges are filled with a number of the 7th number of the number system.

Example

1) In the example, the record 1⋅8 + 2 means that the amount turned out to be large 7, which is not placed in one category. The unit goes into the transfer, and the twice remains in this discharge.

2) perform addition 223,2 8 + 427,54 8 and check the result obtained.

223,2 8 + 427,54 8 = 652,74 8 .

We will verify the results of the calculation results into a decimal number system:

223.2 8 \u003d 2 × 8 2 + 2 × 8 1 + 3 × 8 0 + 2 × 8 -1 \u003d 128 + 16 + 3 + 0.25 \u003d

427,54 8 \u003d 4 × 8 2 + 2 × 8 1 + 7 × 8 0 + 5 × 8 -1 + 4 × 8 -2 \u003d

256 + 16 + 7 + 0,625 + 0,0625= 279,6875 10

652.74 8 \u003d 6 × 8 2 + 5 × 8 1 + 2 × 8 0 + 7 × 8 -1 + 4 × 8 -2  \u003d

384 + 40 + 2 + 0,875 + 0,0625 = 426,9375 10

147,25 10 + 279,6875 10 =426,9375 10

The results coincide, therefore, the calculations in the octaous number system are fulleased.

Hexadecimal numbers

Table of addition of hexadecimal numbers

1A.

1b.

1A.

1b.

1C.

1A.

1b.

1C.

1d.

1A.

1b.

1C.

1d.

1e.

When performing addition, it should be remembered that in the system with a base 16, the transfer appears when the amount in the next discharge exceeds 15. It is convenient to first rewrite the initial numbers, replacing all the letters on their numerical values.

Examples

2) Adopting 3B3,6 16 + 38B, 4 16 and check

3b3,6 16 + 38b, 4 16 \u003d 73e, a 16.

Perform check:

3b3,6 16 \u003d 3 × 16 2 + 11 × 16 1 + 3 × 16 0 + 6 × 16 -1 \u003d 768 + 176 +

3 + 0,375 = 947,375 10

38B, 4 16 \u003d 3 × 16 2 + 8 × 16 1 + 11 × 16 0 + 4 × 16 -1 \u003d 768 + 128 +

11 + 0,25 = 907,25 10

73E, a 16 \u003d 7 × 8 2 + 3 × 8 1 + 14 × 8 0 + 10 × 8 -1 \u003d 1792 + 48 + 14 + 0.625 \u003d 1854,625 10

947,375 10 + 907,25 10 = 1854,625 10 .

The results coincide, therefore, the calculations in the hexadecimal number system are executed correctly.

Subtraction

Binary numbers

Subtraction is performed almost the same as in the decimal system. Here are the basic rules:

0 – 0 = 0, 1 – 0 = 1, 1 – 1 = 0, 10 2 – 1 = 1.

In the latter case, you have to take a loan from the previous discharge.

Subtraction is made by analogy with a decimal number system.

To understand the principle, we will temporarily return to the decimal system. Submount in the column of 21 numbers 9:

Because of 1, it is impossible to subtract 9, you need to take a loan from the previous discharge, in which it costs 2. As a result, it is added to the younger discharge, and next 2 decreases to 1. Now you can perform subtraction: 1 + 10 - 9 \u003d 2. In the older discharge by subtract from the remaining units zero:

A more difficult case - a loan from the far (not the closest) discharge. I will read 9 out of 2001. In this case, it is not possible to take from the nearest discharge (there 0), so we take a loan from that discharge, where the figure is worth 2. All intermediate discharges are filled with digit 9, this is the senior digit of the decimal number system:

In the binary number system, when the loan is taken, no longer is added to the "working" discharge (base of the number system), and all the "intermediate" discharges (between the "workers" and those coming from the loan) are filled with nines , and units (senior digit number).

Examples

If required more Of the smaller, it will be deducted with a smaller one and put the "minus" sign:

3) 4)

Octal numbers

When subtracting "- 1" means that from this discharge used to be a loan (its value was decreased by 1), and "+ 8" - a loan from the following discharge.

2) Subtraction

Hexadecimal numbers

When subtracting a loan from the older discharge is 10 16 \u003d 16, and all the "intermediate" discharges are filled with the F - senior digit number.

For example,

Multiplication

Binary numbers

H.

The multiplication and division of the column in the binary system are performed almost the same as in the decimal system (but using the rules of binary addition and subtraction).

for example ,

1) 2)

Octal numbers

Octal table multiplication

´

Using an octal multiplication table, using the same rules that are used in a decimal number system, multiplication and division of octal multi-digit numbers are produced.

Example

Hexadecimal numbers

Multiplication table

1A.

1C.

1e.

1b.

1e.

2a.

2d.

1C.

2c.

3c.

1e.

2d.

3c.

4b.

1e.

2a.

3c.

4E.

1C.

2a.

3F.

4d.

5b.

1b.

2d.

3F.

6C.

7e.

1e.

3c.

6E.

8c.

2c.

4d.

6E.

8f.

9A.

A5.

3c.

6C.

9C.

A8.

B4.

1A.

4E.

5b.

8f.

9C.

A9.

B6.

C3.

1C.

2a.

7e.

8c.

9A.

A8.

B6.

C4.

D2.

1e.

2d.

3c.

4b.

A5.

B4.

C3.

D2.

Example

Divisionseparately into the decimal system, since for numbers from 0 to 7, their octal record coincides with decimal);

3) fold

Solution (across a hexadecimal system):

1) (first transferred to the binary system, then the binary recording of the number broke on the notebooks from right to left, each tetrad was transferred to the hexadecimal system; In this case, the tetrads can be translated from the binary system in decimaland then replace all the numbers, large 9, on the letters - a, b, c, d, e, f);

2) not need to translate anywhere;

3) fold

4) Transfer all the answers to the hexadecimal system:

121 8 \u003d 001 010 001 2 \u003d 0101 0001 2 \u003d 51 16 (transferred to the binary system of triads, broke on the tetrads to the right left, translated each tetrade separately In the decimal system, all numbers, large 9, replaced the letters - a, b, c, d, e, f).

171 2 = 001 111 001 2 = 0111 1001 2 = 79 16 ,

69 16, no need to translate

1000001 2 = 0100 0001 2 = 41 16 .

Number system.

Number system Call a set of characters (digits) and rules for their use to represent numbers.

There are positional and non-procurement systems.

INnon-aposition Systems weight numbers (i.e., the contribution that it contributes to the value of the number) does not depend on its positionin the number of numbers. Thus, in the Roman system of number, among the XXXI (thirty two), the weight of the numbers x in any position is simply ten.

INpositional Systems Note the weight of each digit varies depending on its position. (Positions) in the sequence of numbers depicting the number. For example, among 757.7, the first seven means 7 hundred, the second - 7 units, and the third - 7 tenths of the unit.

The same recording of the number 757.7 means abbreviated expression

700 + 50 + 7 + 0,7 = 7 10 2 + 5 10 1 + 7 10 0 + 7 10 -1 = 757,7.

Any positional system is characterized by its reason.

Foundation of the positioning system - These are the number of different characters or characters used for the image of the numbers in this system.

Perhaps countless positions: Binary, Tropic, Chetner, etc. Recording numbers in each of the surcharge systems q. means abbreviated expression

a. n-1 Q. n-1 + A. n-2 Q. n-2 + ... + a 1 Q. 1 + A. 0 Q. 0 + A. -1 Q. -1 + ... + a. - m. q. - m. , Where a. i. - digits of the number system; n. and m. - the number of integers and fractional discharges, respectively.

For example:

In addition to decimal, systems are widely used. whole the degree of number 2, namely:

binary (Figures are used 0, 1);

octal (Figures are used 0, 1, ..., 7);

hexadecimal (For the first integers from zero to nine, figures are used 0, 1, ..., 9, and for the following numbers - from ten to fifteen - symbols A, B, C, D, E, F) are used as numbers.

It is useful to remember the record in these number systems of the first two tens of integers:

From all number systems especially simple and therefore interesting for technical implementation in computers Binary number system.

Transfer octal and hexadecimal numbers in binary system very simple: Each digit is enough to replace the binary equivalent to it. tryiadi (triple numbers) or tetraje (Four digits).

For example:

To translate the number from binary Systems B. octal or hexadecimal, it needs to be broken left and right from the comma on triads (for octal) or tetradda (For hexadecimal) and each such group replaced the corresponding octal (hexadecimal) digit.

For example,

When translating the whole decimal Numbers in the system q. Its necessary sequentially share on the q. As long as the residue is less than or equal q-1. The number in the system with the base q. It is written as a sequence of balances from division recorded in the reverse order, starting from the latter.

Example: Transfer the number 75 of the decimal system to binary, octal and hexadecimal:

Answer: 75 10 \u003d 1 001 011 2 \u003d 113 8 \u003d 4B 16.

When translating the number from binary (octal, hexadecimal) system in decimal it is necessary to present this number as the amount of degrees of the base of its number system.

examples:

Lesson number 15-20.

Subject

Arithmetic operations in positional viewing systems. Multiplication and division.

The purpose of the lesson: Show methods of arithmetic operations (multiplication and division) numbers in different number systems, check the assimilation of the topic "Addition and subtraction of numbers in various number systems".

Tasks lesson:

educational

practical use

developing:

educational:

Materials and equipment to the lesson: Cards for independent work, multiplication tables.

Type of lesson:combined lesson

Form of conducting lesson: individual, frontal.

During the classes:

1. Checking homework.

Homework:

1. № 2.41 (1 and 2 columns), workshop, p. 55

Decision:

A) 11102 + 10012 \u003d 101112

B) 678 + 238 \u003d 1128

C) AF16 + 9716 \u003d 14616

D) 11102-10012 \u003d 1012

E) 678-238 \u003d 448

E) AF16-9716 \u003d 1816

2. №2.48 (p. 56)

2. Independent work "Addition and subtraction of numbers in various number systems". (20 minutes)

Independent work. Grade 10 .
11 + 1110 ; 10111+111 ; 110111+101110 3. Subtract: 10111-111; 11 - 1110. 4. Fold and subtract in the 8-riche system: 738 and 258	Option 1

Independent work. Grade 10. Binary number system: 2® 10; addition.
1. Perform a translation from a binary number system to decimal. 2. Fold two binary numbers. 1110+111 ; 111+1001 ; 1101+110001 3. Subtract: 111-1001; 1110 + 111. 4. Fold and subtract in a 16-riche system: 7316 and 2916	Option 2.

3. New material.

1. U m n o f n and e

By performing multiplication of multivalued numbers in various positional positioning systems, it is possible to use the usual multiplication algorithm in the column, but the results of multiplication and addition of unambiguous numbers must be boring from the corresponding multiplication and addition table system.

Multiplication in binary system

Multiplication in the octal system

Due to the emergency simplicity of multiplication table in the binary system, multiplication is reduced only to the shifts of the multiple and additions.

Example 1. Move the number 5 and 6 in decimal, binary, octal and hexadecimal number systems.

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Answer: 5 . 6 = 3010 = 111102 = 368.
Check.
111102 = 24 + 23 + 22 + 21 = 30;
368 = 381 + 680 = 30.

Example 2. Move the number 115 and 51 in decimal, binary, octal and hexadecimal number systems.

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Answer: 115 . 51 = 586510 = 10110111010012 = 133518.
Check. We transform the obtained works to the decimal form:
10110111010012 = 212 + 210 + 29 + 27 + 26 + 25 + 23 + 20 = 5865;
133518 = 1 . 84 + 3 . 83 + 3 . 82 + 5 . 81 + 1 . 80 = 5865.

2. D E LEN AND E

The division in any positioning system is made according to the same rules as an angle division in the decimal system. In the binary system, division is performed especially simplybecause the next digit of the private can be only zero or unit.
Example 3. We divide the number 30 to the number 6.

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Answer: 30: 6 = 510 = 1012 = 58.

Example 4. We divide the number 5865 by the number 115.

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Octal: 133518:1638

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Answer: 35: 14 = 2,510 = 10,12 = 2,48.
Check. We transform the obtained private to the decimal form:
10,12 = 21 + 2 -1 = 2,5;
2,48 = 2 . 80 + 4 . 8-1 = 2,5.

4. Homework:

1. Prepare for control work number 2 "On the topic of the number system. Translation of numbers. Arithmetic operations in number systems "

2. Workshop Ugrinovich, №2.46, 2.47, p. 56.

Literature:

1. Workshop on computer science and information technologies. Tutorial For general education institutions /,. - M.: Binom. Laboratory of Knowledge, 2002. 400 p.: Il.

2. Ugrinovich I. information Technology. Tutorial for 10-11 classes. - M.: Binin. Laboratory of Knowledge, 2003.

3. Shautsukova: Educational. Manual for 10-11 CL. general education. institutions. - M.: Enlightenment, 2003.9 - s. 97-101, 104-107.

Arithmetic operations in positional surgery systems

Arithmetic operations in all positional viewing systems are performed by the same rules well known to you.

Addition. Consider the addition of numbers in the binary number system. It is based on the table of addition of single-digit binary numbers:

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10

It is important to pay attention to the fact that when the two units are addition, the discharge overflow occurs and transferred to the senior discharge. The discharge overflow occurs when the value of the number in it becomes equal to or greater base.

The addition of multi-digit binary numbers occurs in accordance with the above-mentioned table of addition, taking into account possible transfers from younger discharges to the elders. As an example, lay in the column binary numbers 110 2 and 11 2:

We verify the correctness of the calculation by adding in the decimal number system. We translate binary numbers into a decimal number system and then fold them:

110 2 \u003d 1 × 2 2 + 1 × 2 1 + 0 × 2 0 \u003d 6 10;

11 2 \u003d 1 × 2 1 + 1 × 2 0 \u003d 3 10;

6 10 + 3 10 = 9 10 .

Now we will transfer the result of binary addition to the decimal number:

1001 2 \u003d 1 × 2 3 + 0 × 2 2 + 0 × 2 1 + 1 × 2 0 \u003d 9 10.

Compare the results - the addition is made correctly.

Subtraction. Consider the subtraction of binary numbers. It is based on a table of subtraction of single-digit binary numbers. When subtracting from a smaller number (0) more (1), a loan from the older discharge is made. In the table, the loan is indicated 1 with a feature:

Multiplication. The multiplication is based on the multiplication table of single-digit binary numbers:

Division. The division operation is performed according to an algorithm similar to the algorithm for performing a division operation in a decimal number system. As an example, we will produce a division of binary number 110 2 to 11 2:

For conducting arithmetic operations on numbers expressed in various number systems, it is necessary to pre-translate them into the same system.

Tasks

1.22. Conduct addition, subtraction, multiplication and division of binary numbers 1010 2 and 10 2 and check the correctness of the execution of arithmetic actions using an electronic calculator.

1.23. Fold the octal numbers: 5 8 and 4 8, 17 8 and 41 8.

1.24. Conduct the subtraction of hexadecimal numbers: F 16 and A 16, 41 16 and 17 16.

1.25. Fold numbers: 17 8 and 17 16, 41 8 and 41 16

Arithmetic operations in all positional viewing systems are performed according to the same rules. To carry out arithmetic operations on the numbers presented in various number systems, it is necessary to pre-convert them into one number system and take into account the fact that the transfer to the next discharge during the addition operation and loan from the older discharge during the subtraction operation is determined by the value of the base system base.

Arithmetic operations in the binary number system are based on the folding tables, subtraction and multiplication of single-digit binary numbers.

When folding two units, the discharge overflow occurs and the units are transferred to the senior discharge, when subtracting 0-1, it is made from the older discharge, in the "subtraction" table, this loan is indicated 1 with a feature over the number (Table 3).

Table 3.

Below are examples of performing arithmetic operations over the numbers presented in various surgery systems:

Arithmetic operations on the integers presented in various number systems are simply implemented using the Calculator and MS Excel programs.

1.3. Presentation of numbers in the computer

Numeric data is processed in a computer in a binary number system. The numbers are stored in the computer's memory in binary code, i.e., in the form of a sequence of zeros and units, and can be represented in a fixed or floating semicolon format.

The integers are stored in memory in a fixed semicolon format. With this format of the representation of the numbers for storing integer non-negative numbers, a memory register is assigned consisting of eight memory cells (8 bits). Each category of memory cells always corresponds to the same number of numbers, and the comma is on the right after the youngest discharge and beyond the discharge mesh. For example, the number 110011012 will be stored in the memory register as follows:

Table 4.

The maximum value of an integer non-negative number, which can be stored in the register in a fixed-plated format, can be determined from the formula: 2N - 1, where N is the number of digits of the number. The maximum number will be equal to 28 - 1 \u003d 25510 \u003d 111111112 and the minimum 010 \u003d 000000002. Thus, the range of changes in integer non-negative numbers will be from 0 to 25510.

In contrast to the decimal system in a binary number system with a computer representation of a binary number, there are no symbols that indicate the number of numbers: positive (+) or negative (-), so for the representation of integers with a sign in the binary system, two numbers representation format are used: the number of number with the sign and format of the additional code. In the first case, two memory registers (16 bits) are allocated to store integers with a sign, and the older discharge (extreme left) is used under the number: if the number is positive, then 0 if the number is negative, then - 1. For example The number 53610 \u003d 00000010000110002 will be presented in memory registers as follows:

Table 5.

and the negative number is -53610 \u003d 10000010000110002 in the form:

Table 6.

Maximum positive number or minimal negative in the format of the value of the number with a sign (taking into account the view of one discharge under the sign) is 2N-1 - 1 \u003d 216-1 - 1 \u003d 215 - 1 \u003d 3276710 \u003d 1111111111111112 and the range of numbers will be within - 3276710 to 32767.

Most often to represent integers with a familiar binary system, an additional code format is applied, which allows you to replace the arithmetic operation of subtraction in the computer with an operation of addition, which significantly simplifies the structure of the microprocessor and increases its speed.

To represent entire negative numbers in such a format, an additional code is used, which is the addition of a negative number module to zero. The transfer of a whole negative number to the additional code is carried out using the following operations:

1) the module of the number to record direct code in n (n \u003d 16) binary discharges;

2) get the reverse code of the number (invert all the discharges of the number, i.e. all units are replaced by zeros, and zeros - by units);

3) To the resulting reverse code, add a unit to the younger category.

For example, for the number -53610 in such format, the module will be equal to 00000010000110002, the reverse code - 1111110111100111, and the additional code - 1111110111101000.

It must be remembered that the additional code of a positive number is the number.

To store integers with a sign in addition to a 16-bit computer representation when used two memory registers (Such a format of the number is also called a format of short integers with a sign), the formats of medium and long integers are applied with a sign. To represent numbers in the middle number format, four registers are used (4 x 8 \u003d 32 bits), and for the presentation of numbers in the format of long numbers - eight registers (8 x 8 \u003d 64 bits). Ranges of values \u200b\u200bfor the format of medium and long numbers will be respectively equal: - (231 - 1) ... + 231 - 1 and - (263-1) ... + 263 - 1.

The computer representation of the numbers in a fixed-comma format has its advantages and disadvantages. TO benefits The simplicity of the presentation of numbers and algorithms for the implementation of arithmetic operations, to disadvantages - the final range of numbers representation, which may be insufficient to solve many practical problems (mathematical, economic, physical, etc.).

The real numbers (final and infinite decimal fractions) are processed and stored in a floating-point compute. With this format of the representation of the number, the position of the comma in the record may vary. Any real number to a floating semicolon can be represented as:

where a is mantissa numbers; h is the base of the number system; P is the order of the number.

Expression (2.7) for a decimal number system will take the form:

for binary -

for octal -

for hexadecimal -

This form representation is also called normal . With the change of order of the comma, the number is shifted, i.e., as it were, it is floating left or right. Therefore, the normal form of representation of numbers is called floating semicolon. The decimal number is 15.5, for example, in a floating semicolon format may be represented as: 0.155 · 102; 1.55 · 101; 15,5 · 100; 155.0 · 10-1; 1550.0 · 10-2, etc. This form of a decimal number of 15.5 floating semicolons is not used when writing computer programs And enter them into a computer (computer input devices perceive only linear data record). Based on this expression (2.7), to represent decimal numbers and enter them to the computer converts to the form

where p is the order of the number

i.e., instead of the foundation of the number 10, the letter E is written, instead of a comma dot, and the multiplication sign is not put. Thus, the number 15.5 in a floating semicolon format and a linear recording (computer representation) will be recorded in the form: 0.155E2; 1.55E1; 15.5e0; 155.0E-1; 1550.0E-2, etc.

Regardless of the number system, any number in a floating-semicolon can be represented by an infinite set of numbers. This form of recording is called abnormalized . For a unambiguous representation of floating point numbers use the normalized form of the number of the number, at which the Mantissa number must meet the condition

where | A | - the absolute value of the Mantissa number.

Condition (2.9) means that the mantissa should be a correct shot and have a digit after a semicolon, different from zero, or, in other words, if after the comma in the mantissa is not zero, the number is called normalized. Thus, the number 15.5 in the normalized form (normalized mantisum) in a floating point shape will look as follows: 0.155 · 102, i.e., normalized Mantius will be a \u003d 0.155 and order p \u003d 2, or in a computer representation of the number 0.155E2 .

The floating-semicolons have a fixed format and occupy four (32 bits) or eight bytes (64 bits) in the computer's memory. If the number takes 32 discharge in the memory of the computer, then this is the number of conventional accuracy, if 64 discharge, then this is the number of double accuracy. When recording a floating point, discharges are highlighted for storing the mantissa sign, order, mantissa and order sign. The number of discharges that are given to the procedure for the number of numbers determines the range of changes, and the number of discharges allocated for the storage of the mantissa is the accuracy with which the number is specified.

When performing arithmetic operations (addition and subtraction) above the numbers presented in a floating semicolon format, the following procedure is implemented (algorithm):

1) the orders of the numbers are aligned, over which arithmetic operations are performed (the order of smaller in the module of the number is increased to the value of the order of the number of the number of numbers, the mantissa decreases at the same number of times);

2) arithmetic operations are performed on the Mantissarms of the numbers;

3) Normalization of the result obtained is performed.

Operations in various theory surge systems. Arithmetic operations in various binary number systems. Some number systems

Arithmetic operations in positional surgery systems

1.3. Presentation of numbers in the computer

Last

It is necessary to know