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#### Two's-Complement, Integer Fixed-Point Numbers

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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