Why is the integer type 128 to byte -128

 public static void main(String[] args) {
        short a = 128;
        byte b = (byte) a;
        System.out.println(b);
    }

I believe many people know that the result of the above code is -128 , But this is why

Everybody knows short The length of is 16 individual bit position So in int in 128 The binary representation of   0000 0000 1000 0000
java During forced rotation If the long bit is converted to the short bit, the binary value at the end of the long bit that is the same as the short bit is retained , therefore b The binary representation of 1000 0000 but 1000 0000 Convert to decimal to 128, Why is it byte It will be expressed as -128 Well ？

Students who have studied computer know that digital binary in computer The highest bit is the sign bit ,1 Represents a negative value ,0 Represents a positive value . And with complement （ The complement of a positive number is the same as the original code , The complement of negative numbers is based on the original code, and then you add 1.） Is stored in the form of . but 1000 0000 The original code of is still 1000 0000.

Let's understand from two perspectives -128 Why is the complement of 1000 0000.

Understand from the meaning of complement  

because ：256-128=256+(-128) Complement --（ There is only addition in the machine ）. Subtraction becomes the addition of complement .

and      256-128=128 
therefore    256+（-128） Complement =128 
therefore           （-128） Complement =256-128 
                 （-128） Complement  =128 （1000 0000）

0000 0000   0000 0000     0
1000 0001   1111 1111    -1
1000 0010   1111 1110    -2
1000 0011   1111 1101    -3
    ...          .. Complement constantly -1...
1111 1111   1000 0001 -127
Unable to express      1000 0000 -128

So the rules are 8 In bit binary 1000 0000 Representation bit -128  So when other integers 128 Turn into byte When is -128