Let denote the number of bits. Then the value of a two's
complement integer fixed-point number can be expressed in terms of its
bits
as
(G.1)
We visualize the binary word containing these bits as
Each bit is of course either 0 or 1. Check that the case
in Table G.3 is computed correctly using this formula. As an
example, the number 3 is expressed as
while the number -3 is expressed as
and so on.
The most-significant bit in the word, , can be interpreted as the
``sign bit''. If is ``on'', the number is negative. If it is
``off'', the number is either zero or positive.
The least-significant bit is . ``Turning on'' that bit adds 1 to
the number, and there are no fractions allowed.
The largest positive number is when all bits are on except , in
which case
. The largest (in magnitude) negative number is
, i.e., and for all . Table G.4 shows
some of the most common cases.
Table G.4:
Numerical range limits in -bit two's-complement.