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.
 |
 |
 |
8 |
-128 |
127 |
16 |
-32768 |
32767 |
24 |
-8,388,608 |
8,388,607 |
32 |
-2,147,483,648 |
2,147,483,647 |
|
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