Page 76 - Engineering Digital Design

2.6 SIGNED BINARY NUMBERS 47

Table 2.7 Examples of eight-bit 2's and 1 's
complement representations (MSB = sign bit)
Decimal 2's 1's
Value Complement Complement
-128 10000000
-127 10000001 10000000
-31 11100001 11100000
-16 11110000 11101111
-15 11110001 11110000
_3 11111101 11111100
-0 00000000 11111111
+o 00000000 00000000
+3 00000011 00000011
+ 15 00001111 00001111
+ 16 00010000 00010000
+31 00011111 00011111
+ 127 01111111 01111111
+ 128

a decimal value of

4
7
6
2
(Af 2 c)io = -1 x 2 + 1 x 2 + 1 x 2 + 1 x 2' + 1 x 2~ + 1 x 2~ 3
= -128 + 64 + 16 + 2 + 0.25 + 0.125
= -45.625i 0.
But the same result could have easily been obtained by negation of A^c followed by the
use of positional weighting to obtain the decimal value. Negation is the reapplication of
Eq. (2.12) or Algorithms 2.5 or 2.6 to any 2's complement number A^c to obtain its true
value. Thus, from the forgoing example the negation of NIC is given by

Akc)2C= 00101101.101
= 32 + 8 + 5 + 0.5 + 0.125

= 45.625 10,

which is known to be a negative number, —45.625i 0.
Negative BCD numbers are commonly represented in 10's complement notation with
consideration of how BCD is formed from binary. As an example, — 59.24io = 40.76io is
represented in BCD 10's complement (BCD,IOC) by

-0101 1001.0010 0100) BCD = 0100 0000.0111 0110) BCD,ioc,

where application of Eq. (2.12), or Algorithm 2.5 or 2.6, has been applied in radix 10 fol-
lowed by the BCD representation as in Subsections 2.4.1. Alternatively, the sign-magnitude
(SM) representation of a negative BCD number simply requires the addition of a sign bit