Two's Complement

wiki method of representing signed integers. Greates value as sign (Most Significan Bit)

It ensures the addition and subtraction works same, 在计算机内存角度不用考虑负号； range $-2^{n-1} - 2{n-1} - 1$.

Steps: Encode

1. start from positive version 3 => 0011
2. flip all bits 1100
3. add 1 1101 = -3

Steps: Decode

1. check sign 1 means negative
2. Flip the bit 0010
3. add 1 0011 means 3

Ones's Complement

Wiki

Formula

$-k = 2^n - k$ 公式的核心是Mod运算，-k+k=0是mod(2^n)下的0，也就是2^n; -8是特殊的，16-8=8，所以8跟-8的位表示一样；

$-k = 2^{n+1} - k$两个公式等价；

4bits数

usign => [0,15]; sign => [-8, 7]; 其中0-7跟unsigned二进制一致，-1~-8用15-8表示； 负数的表示是通过保证与整数相加得到0来获取的；

为什么刚好能对应

Why Decimal value is called magnitude