DIGITAL DESIGN

Section 2.5 Representation of Negative Numbers 31
DO
The signed-magnitude
NOT
system has an equal number of positive
COPY
and negative
integers. An n-bit signed-magnitude integer lies within the range −(2
DO NOT COPY
DO NOT COPY
DO NOT COPY
DO NOT COPY
DO NOT COPY
DO NOT COPY
DO NOT COPY
DO NOT COPY
n−1−1) through +(2n−1−1), and there are two possible representations of zero.
Now suppose that we wanted to build a digital logic circuit that adds
signed-magnitude numbers. The circuit must examine the signs of the addends
to determine what to do with the magnitudes. If the signs are the same, it must
add the magnitudes and give the result the same sign. If the signs are different, it
must compare the magnitudes, subtract the smaller from the larger, and give the
result the sign of the larger. All of these “ifs,” “adds,” “subtracts,” and “compares”
translate into a lot of logic-circuit complexity. Adders for complement
number systems are much simpler, as we’ll show next. Perhaps the one redeeming
feature of a signed-magnitude system is that, once we know how to build a
signed-magnitude adder, a signed-magnitude subtractor is almost trivial to
build—it need only change the sign of the subtrahend and pass it along with the
minuend to an adder.
2.5.2 Complement Number Systems
While the signed-magnitude system negates a number by changing its sign, a
complement number system negates a number by taking its complement as
defined by the system. Taking the complement is more difficult than changing
the sign, but two numbers in a complement number system can be added or subtracted
directly without the sign and magnitude checks required by the signedmagnitude
system. We shall describe two complement number systems, called
the “radix complement” and the “diminished radix-complement.”
In any complement number system, we normally deal with a fixed number
of digits, say n. (However, we can increase the number of digits by “sign extension”
as shown in Exercise 2.23, and decrease the number by truncating highorder
digits as shown in Exercise 2.24.) We further assume that the radix is r, and
that numbers have the form
D = dn–1dn–2···d1d0 .
The radix point is on the right and so the number is an integer. If an operation
produces a result that requires more than n digits, we throw away the extra highorder
digit(s). If a number D is complemented twice, the result is D.
2.5.3 Radix-Complement Representation
In a radix-complement system, the complement of an n-digit number is obtained
by subtracting it from r n . In the decimal number system, the radix complement
is called the 10’s complement. Some examples using 4-digit decimal numbers
(and subtraction from 10,000) are shown in Table 2-4.
By definition, the radix complement of an n-digit number D is obtained by
subtracting it from r n . If D is between 1 and rn signed-magnitude
adder
signed-magnitude
subtractor
complement number
system
radix-complement
system
10’s complement
− 1, this subtraction produces
Copyright © 1999 by John F. Wakerly Copying Prohibited