Definition of Two's Complement?

Question

I've been reading over a couple of questions/answers here:

twos-complement-in-python

is-twos-complement-notation-of-a-positive-number-the-same-number

Someone gave some sample code to create a two's complement of a number:

def twos_comp(val, bits):
    """compute the 2's compliment of int value val"""
    if( (val&(1<<(bits-1))) != 0 ):
        val = val - (1<<bits)
    return val

In addition someone defined two's complement as thus:

Two's complement notation uses the n-bit two's complement to flip the sign. For 8-bit numbers, the number is subtracted from 2^8 to produce its negative.

These declarations went unchallenged. However, that chides with my understand of a two's complement is. I thought it's calculated by inverting the binary number and adding 1. (With the understanding that the number representation has a limited number of bits.)

Additionally, the two's complement is supposed to have the property that it is the additive inverse of the original number. But, the output from twos_comp doesn't appear to have that. In my hand calculations (and some test code I wrote) with my definition , I see that when a number and the twos complement of it are added together, a 1 is overflowed and the rest of the bits are zero, thus it has the additive inverse property.

Are there multiple definitions for twos complement, am I confused, or is that definition and function from the other posts just plain wrong?

Answer 1

Twos complement is in fact calculated by inverting the binary number and adding 1, for negative numbers. Such that abs(-1)=1=01 -> bitwise_inv(abs(-1))+abs(-1)=FE+1=FF. This is equivalent to the definition provided of subtracting the number from 2^8 (this should not be hard to see).

The sample code you provided, does not calculate twos complement in any useful way. I don't at all understand whats its trying to do, appears to quite be quite different from "subtracting the number from 2^8" as subtracts 2*8 from the number, while also failing to remember that when we refer to the value of a twos complement number what we mean is its unsigned value.

Here is a more correct implementation, using the same template. Notice that this precisely does "subtract the number from 2^8."

def twos_c(val,bits):
  if ((val&(1<<(bits-1)))!=0):
    val=(1<<bits)-abs(val)
  return val