Binary original code, inverse code, complement code

Binary source code 、 Inverse code 、 Complement code

I used to see binary source code in College 、 Inverse code 、 When complementing these, it is almost a glance , Because I think it has nothing to do with upper level development , Now, because there is a project to do the encapsulation of audio stream , You need to operate on binary , Let's review .

Any data in the computer is stored in binary , For a number , The computer should use a certain encoding method for storage , Original code 、 Inverse code 、 Complement is the way that a machine stores a specific number .

Original code

The source code is the easiest way for human brain to understand and calculate

Is the absolute value of the sign bit plus the value , The first bit represents the sign bit , The rest of the bits represent the value . Comparison 8 Bit binary ：

+1=0000 0001

-1=1000 0001

Because the first sign bit is occupied , therefore 8 The binary value range of bits is ：

[1111 1111 , 0111 1111]

namely

[-127 , 127]

Inverse code

Complement representation the human brain cannot intuitively see its value . It is also necessary to convert the original code to calculate its value .

The inverse of a positive number is itself

The inverse of a negative number is based on its original code , The sign bits remain the same , The rest of the bits are reversed .

+1= 0000 0001 primary = 0000 0001 back

-1= 1000 0001 primary = 1111 1110 back

Complement code

Complement representation the human brain cannot intuitively see its value . It is also necessary to convert the original code to calculate its value .

The complement of a positive number is itself

The complement of a negative number is based on its inverse , Last +1.

+1= 0000 0001 primary = 0000 0001 back = 0000 0001 repair

-1= 1000 0001 primary = 1111 1110 back = 1111 1111 repair

Meaning of existence

Since the original code is best for human brain to read directly , Then why are there inverse codes and complements , Because when it comes to calculation , When the addition and subtraction of the original code symbol bits are involved, it will make the addition and subtraction extremely complex .

for example ：1-1=1+(-1)=0000 0001 + 1000 0001 =1000 0010 = -2 This is obviously wrong

• Inverse code calculation

  When using the original code for calculation, subtraction will lead to an error in the calculation result , People began to study how to involve symbolic bits in operations . That is, the inverse code .

  1-1=1+(-1)=0000 0001 + 1111 1110 = 1111 1111 back = 1000 0000 primary =-0=0

  In this way, the problem that the original code cannot be subtracted has been solved , But there is a drawback , Namely 0 When , In human understanding ,+0 and -0 It's the same , But it may appear in the complement 0000 0000 and 1000 0000 Two codes represent 0 The situation of , So the complement appears .

• Complement calculation

  1-1=1+(-1)=0000 0001 + 1111 1111 = 0000 0000 repair = 0000 0000 primary =0

  Use complement , It's not just fixed 0 There are two coding problems , It can also represent a minimum number . That's why 8 Bit binary , The range represented by the original code or the inverse code is [-127, +127], And the range represented by complement is [-128, 127].

  Because the machine uses complement , So for the programming commonly used in 32 position int type , The range can be expressed as : [-231, 231-1] Because the first bit represents the sign bit . When using the complement representation, you can save another minimum value .

Be careful

stay java Chinese binary is encoded by complement , So we can often see such a saying ：

Java in Byte The value range of the type is -128～127

Many people, including me, were confused when I studied this field , Why not symmetrical , Generally speaking, I don't explain it too carefully, and it's easy to forget , Knowing the principle of complement coding here is much easier to understand .

8 The value range of bit source code and inverse code is -127~127, It's a number less than the complement , Why is that , Because just now I mentioned that there are +0 and -0 Divided into , Therefore, the value range is just symmetrical , And in the complement , There is only 0 This number . So you can empty Give one digit to indicate other numbers .

So here comes the question , Why does it happen to be -128 Well , It can be deduced as follows .

primary 1111 1101= back 1000 0010= repair 1000 0011=-125

primary 1111 1110= back 1000 0001= repair 1000 0010=-126

primary 1111 1111= back 1000 0000= repair 1000 0001=-127

Just look at whether the complement has found any rules

Then the value of our complement is ：1000 0000 How much should it be

According to the above rules, we can see repair 1000 0000=-128

summary

Finally, I have solved the problem that has been bothering me for a long time , Now let's understand a lot about byte The calculation of binary correlation is much clearer .