Two's-Complement, Integer Fixed-Point Numbers

Let $N$ denote the (even) number of bits. Then the value of a two's complement integer fixed-point number can be expressed in terms of its bits $\{b_i\}_{i=0}^{N-1}$ as

$$x = - b_0 \cdot 2^{N-1} + \sum_{i=1}^{N - 2} b_i \cdot 2^{N - 1 - i},\quad b_i\in\{0,1\} \protect$$ (G.1)

We visualize the binary word containing these bits as

$$x = [b_0 b_1 \cdots b_{N-1}]$$

Each bit $b_i$ is of course either 0 or 1. Check that the $N=3$ table above is computed correctly using this formula. As an example, the number 3 is expressed as

$$3 =[ 0 1 1 ] = - 0\cdot 4 + 1\cdot 2 + 1 \cdot 1$$

while the number -3 is expressed as

$$-3 =[ 1 0 1 ] = - 1\cdot 4 + 0\cdot 2 + 1 \cdot 1$$

and so on.

The most-significant bit in the word, $b_0$, can be interpreted as the ``sign bit''. If $b_0$ is ``on'', the number is negative. If it is ``off'', the number is either zero or positive.

The least-significant bit is $b_{N-1}$. ``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 $b_0$, in which case $x=2^{N-1}-1$. The largest (in magnitude) negative number is $10\cdots0$, i.e., $b_0=1$ and $b_i=0$ for all $i>0$. Table G.4 shows some of the most common cases.

Table: Numerical range limits in $N$-bit two's-complement.

$N$	$x_{\mbox{\small min}}$	$x_{\mbox{\small max}}$
8	-128	127
16	-32768	32767
24	-8,388,608	8,388,607
32	-2,147,483,648	2,147,483,647