Houjun Liu

two's complement

Say we want to find the number which is the additive inverse (“negative”) of a number.

We can just flip each of the digit, and then add 1:

  • take \(0101\), invert it to get \(1010\)
  • adding these two numbers will give you \(1111\). If we just added one more \(0001\), it will flip over to be \(0000\).
  • Therefore, \(1010+0001 = 1011\) is the additive inverse of \(0101\).

The left most bit being one: still a mark of whether or not something is negative. It just works backwards:

pros and cons of twos complement

  • con: more difficult to represent and difficult to convert
  • pro: only 1 representation for 0
  • pro: the most significant bit still indicates the sign of a number
  • pro: addition works for any combination of positive/negative


  • all zeros: its always 0
  • zero plus all ones (011111…111): it always is the largest signed value and some middle value for unsigned
  • all ones: its always -1 (11111 => 00000 +1 => 1) for signed
  • one plus all zeros

mnemonic for remembering where overflows happened