DIGITAL DESIGN

Section 2.6 Two’s-Complement Addition and Subtraction 35
either
DO
must check for both
NOT
representations of zero, or must
COPY
always convert
11 ⋅⋅⋅11 to 00 ⋅⋅⋅00.
*2.5.7 Excess Representations
Yes,
DO
the number of different systems
NOT
for representing negative
COPY
numbers is excessive,
but there’s just one more for us to cover. In excess-B representation, an
m-bit string whose unsigned integer value is M (0 ≤ M < 2
DO NOT COPY
DO NOT COPY
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m ) represents the
signed integer M − B, where B is called the bias of the number system.
For example, an excess−2m−1 system represents any number X in the range
−2m−1 through +2m−1 − 1 by the m-bit binary representation of X + 2m−1 (which
is always nonnegative and less than 2m ). The range of this representation is
exactly the same as that of m-bit two’s-complement numbers. In fact, the representations
of any number in the two systems are identical except for the sign bits,
which are always opposite. (Note that this is true only when the bias is 2m−1 excess-B representation
bias
excess-2
.)
The most common use of excess representations is in floating-point number
systems (see References).
2.6 Two’s-Complement Addition and Subtraction
2.6.1 Addition Rules
A table of decimal numbers and their equivalents in different number systems,
Table 2-6, reveals why the two’s complement is preferred for arithmetic operations.
If we start with 10002 (−810) and count up, we see that each successive
two’s-complement number all the way to 01112 (+710) can be obtained by adding
1 to the previous one, ignoring any carries beyond the fourth bit position.
The same cannot be said of signed-magnitude and ones’-complement numbers.
Because ordinary addition is just an extension of counting, two’s-complement
numbers can thus be added by ordinary binary addition, ignoring any carries
beyond the MSB. The result will always be the correct sum as long as the range
of the number system is not exceeded. Some examples of decimal addition and
the corresponding 4-bit two’s-complement additions confirm this:
+3 0011
−2 1110
+ +4 + 0100 + −6 + 1010
+7 0111 −8 11000
+6 0110
+4 0100
+ −3 + 1101 + −7 + 1001
+3 10011 −3 1101
m−1 system
two’s-complement
addition
Copyright © 1999 by John F. Wakerly Copying Prohibited