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Representation of Numbers 253
rule to compute addition, another rule to compute subtraction, and a
way tocompare two magnitudes to determine their relative value before
subtraction.
MATLAB Implementation MATLAB is a 64-bit floating-point computation
engine that provides results in decimal numbers. Therefore,
fixed-point binary operations must be simulated in MATLAB. It provides
a function, dec2bin, toconvert a positive decimal integer into a B-bit
representation, which is a symbol (or a code) and not a number. Hence
it cannot be used in computation. Similarly, the function bin2dec converts
a B-bit binary character code into a decimal integer. For example,
dec2bin(3,3) gives 011 and bin2dec(’111’) results in 7. To obtain a
sign-magnitude format, a sign bit must be prefixed. Similarly, to convert a
sign-magnitude format, the leading bit must be used to impart a positive
or negative value. These functions are explored in Problem P9.1.
One’s-complement format In this format, the negation (or complementation)
of an integer x is obtained by complementing every bit (i.e., a
0isreplaced by 1 and a 1 by 0) in the binary representation of x. Suppose
the B-bit binary representation of x is b B−1 b B−2 ··· b 0 ; then the B-bit
one’s-complement, ¯x, ofx is given by
¯x △ = ¯b B−1 ¯bB−2 ··· ¯b 0
where each bit ¯b i is a complement of bit b i . Clearly then
x +¯x ≡ 11 ... 1=2 B − 1 (6.29)
The MSB of the representation once again represents the sign bit,
because the positive integer has the MSB of 0 so that its negation (or a
negative integer) has the MSB of 1. The remaining bits represent either
the number x (if positive) or its one’s-complement (if negative). Thus,
using (6.29) the one’s-complement format representation 2 is given by
x (1)
∆
=
{
x, x ≥ 0
|x|, x