The two's complement representation has several properties, familiar to designers of digital logic, that are responsible for its near-universal use in general purpose computers. For convenience we now introduce some formal notation for binary representations and review the relevant properties.
The unsigned interpretation of a vector of n bits is given by
Rather than give a bit-by-bit definition of the representation function it will suffice to define it as the inverse of the interpretation function:
for non-negative x.
The two's complement (signed) representation can be defined by concatenating a sign bit onto an unsigned bit vector:
The interpretation function for the two's complement representation was given in equation (2); it can also be defined in terms of the unsigned interpretation function: