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2.6 SIGNED BINARY NUMBERS 45

EXAMPLE 2.14

-0 10 = 1 0000000 25M
Although the sign-magnitude system is used in computers, it has two drawbacks. There
is no unique zero, as indicated by the previous examples, and addition and subtraction
calculations require time-consuming decisions regarding operation and sign, for example,
(—7) minus (—4). Even so, the sign-magnitude representation is commonly used in floating-
point number systems as discussed in Section 2.8.

2.6.2 Radix Complement Representation
n
The radix complement N rc of an n-digit number N r is obtained by subtracting N r from r ,
that is,
n
N rC = r -N r
(2.12)
= N r + \ LSD
where

N r = Digit complementation in radix r

This operation is equivalent to that of replacing each digit a, in N r by (r — 1) — a, and
adding 1 to the LSD of the result as indicated by Algorithm 2.5. The digit complements
N r for three commonly used number systems are given in Table 2.6. Notice that the digit
complement of a binary is formed simply by replacing the 1's with O's and O's with 1's
required by 2" — N 2 — 1 = W 2 as discussed in Subsection 2.6.3. The range of representable
]
1
numbers is —(r"" ) through +(r"~ — 1).
Application of Eq. (2.12) or Algorithm 2.5 to the binary and decimal number systems
requires that for 2's complement representation NIC = N^ + ILSB and for 10's complement
N[Q C = N\Q + I LSD, where A^ and N\Q are the binary and decimal digit complements
given in Table 2.6.

Table 2.6 Digit complements for three
commonly used number systems
Complement (N r)

Digit Binary Decimal Hexadecimal
0 1 9 F
1 0 8 E
2 7 D
3 6 C
4 5 B
5 4 A
6 3 9
7 2 8
8 1 7