What are the three ways to express the number of computer machines?

The three representation methods of computer machine numbers are original code, complement code and complement code. When the " " sign in the true value form of a number is represented by "0" and the "-" sign is represented by "1", it is called the original code form of the number. In order to overcome the shortcomings of the original code operation, the machine number's complement and complement representation are used; for positive numbers, the complement code has the same form as the original code, and for negative numbers, the complement code is the numerical part of the original code. opposite. The complement code is introduced based on the concept of congruence. In computer systems, values ​​are always represented and stored in the complement code.

The operating environment of this tutorial: Windows 7 system, Dell G3 computer.

The three representation methods of computer machine numbers are original code, inverse code and complement code.

Introduction to machine numbers

Machine numbers are numbers that "digitize" symbols and are the binary representation of numbers in the computer . Machine numbers have two characteristics: one is the digitization of symbols, and the other is that the size of the number is limited by the machine word length.

Machine numbers have two basic characteristics:

1. Symbol digitization of numbers. Practical data includes positive numbers and negative numbers. Since the hardware inside the computer can only represent two physical states (represented by 0 and 1), the positive sign " " or negative sign "-" of the practical data is used in the machine. Bit binary 0 or 1 to distinguish. Usually this symbol is placed in the highest bit of the binary number, which is called the sign bit. 0 represents the symbol " ", and 1 represents the symbol "-". Because the sign occupies one bit, the formal value of the number is not equal to the real value. The value corresponding to the machine number with the sign bit is called the true value of the machine number. For example, the binary truth number -011011 has a machine number of 1011011.

2. The number of binary digits is limited by the machine equipment. The number of binary digits that the machine's internal equipment can represent at one time is called the machine's word length. The word length of a machine is fixed. A word length of 8 bits is called a byte. Machine word lengths are generally integer multiples of bytes, such as word lengths of 8 bits, 16 bits, 32 bits, and 64 bits.

Form of machine number

1. Original code

Use "0" to represent the " " sign in the true value form of the number, and "- "When the number is represented by "1", it is called the original code form of the number, or the original code for short. If the word length is n bits, the original code can generally be expressed as:

When X is a positive number, [X] is the same as X, that is, [X] is = X. When X is negative. Since X itself is a negative number, in fact, the sign bit before the absolute value of the numerical part of |

The original code representation is relatively intuitive. Its numerical part is the absolute value of the number, and the conversion with true value and decimal number is very convenient. But its addition and subtraction operations are more complicated. When adding two numbers, the machine must first determine whether the signs of the two numbers are the same. If they are the same, the two numbers are added. If the signs are different, the two numbers are subtracted. Before doing subtraction, you need to judge the absolute value of the two numbers, then subtract the decimal from the large number, and finally determine the sign of the difference. In other words, when adding in such a direct form, the sign bit of the negative number cannot be the same as the sign of the difference. The numerical part participates in the operation together, but a separate line must be used to determine the sign bit of the sum. To realize these operations, the circuit is very complicated, which is obviously not economical and practical. In order to reduce equipment and solve the problem of the sign bit of negative numbers participating in operations in the machine, subtraction operations are always changed into addition operations, and two machine numbers, one's complement and one's complement, are introduced.

2. Inverse code

As mentioned above, in order to overcome the shortcomings of the original code operation, the inverse code and complement representation of machine numbers are used. That is, for positive numbers, the complement code has the same form as the original code; for negative numbers, the complement code is the inversion of the numerical part of the original code.

3. Complement code

Complement code is introduced based on the concept of congruence. Let’s look at an example where subtraction is implemented by addition. Assume that the current time is 6 o'clock Beijing time, but there is a watch that is 8 o'clock, which is 2 hours ahead of Beijing time. There are two calibration methods, one is to set back 2 hours, and the other is to set 10 hours forward. If it is stipulated that the reverse dial is for subtraction and the forward dial is for addition, then subtracting 2 and adding 10 are equivalent to the watch, that is to say, subtracting 2 can be achieved by adding 10. This is because 8 plus 10 equals 18. However, the watch can only indicate a maximum of 12. When it is greater than 12, 12 is naturally lost. 18 minus 12 leaves only 6. This shows that subtraction can be replaced by addition under certain conditions. Here "12" is called "modulo", and 10 is called the complement of "-2" modulo 12. Generalized to general:

A – B = A ( – B M ) = A ( – B ) complement

It can be seen that under the condition of modulus M, A minus B can be Use A plus -B's complement to achieve this. Here the module can be regarded as the capacity of the counter. For the above example of the watch, the module is 12. In the computer, its components have a fixed number of digits. If the number of digits is n, the count value is, that is, the counter capacity is. Therefore, the complement code in the computer is modulo " ", and its definition is as follows:

In short, for a positive number, its complement has the same form as the original code; and from equations (3) and (4), it can be seen that for a negative number, the complement is the last digit of its complement plus 1.

In short, the original code, complement code and complement code of positive numbers are exactly the same; the original code, complement code and complement code of negative numbers have different forms. In addition, special attention should be paid to the fact that for the one's complement and one's complement of a negative number (that is, a number with a sign bit of 1), the several digits after the sign bit do not represent the value of the number. If you want to know the size of this number, you must require its complement or complement.

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