(rn-1)-N and rn − N=[(rn − 1) − N] + 1.

1. 1
Complements
BY UNSA SHAKIR

2. 2
Complements
 Complements are used for simplifying the
subtraction operation and for logical manipulation.
 There are two types of complements for each
base-r system: the radix and the diminished radix
complements.
 Binary numbers: 2’s complement
1’s complement
Decimal numbers:10’s complement
9’s complement

3. 3
Diminished radix complement
 Given a number N in base-r having n digits, the
(r-1)’s complement of N is defined as
(rn-1)-N.
Decimal numbers: 012398 have 6 digits and present below
(106 - 1) – 012398 = 999999 – 012398 = 987601
Binary numbers: 1011000=(88)10
(27 - 1) – 1011000 = 1111111 – 1011000 = 0100111(39)
shortcut(1<-->0) 0100111

4. Where
R = base of number system
N = given numbers
n = number of digits given
4

5. 5
Radix complement
 The r’s complement of an n-digit number in
base-r is defined as rn − N, for N=0 and 0 for N=0.
 Compare with (r − 1)’s complement, the r’s
complement is (r − 1)’s + 1 since
rn − N=[(rn − 1) − N] + 1.
Decimal number: 012398
106 − 012398 = 987602 = 999999 − 012398 + 1
Binary number: 1011000(88)
27 − 1011000=0101000(38)=1111111 − 1011000+1

6. 6
Summary:

7. 7
Find 1’s complement and 2’s complement of the binary
number 10110012?
1’s complement of 1011001
= {(27)10-1)} – 10110012
= 12710– 10110012
= 11111112– 10110012
= 01001102
2’s complement of 1011001
= (27)10-N
= 12810– 10110012
= 100000002– 10110012
= 01001112
Example:

8. 8
Find the 3’s complement and 4’s complement of 1304?
= {(43)10-1)} – 1304
= 6310– 1304
= 6310– 2810
= 3510
= 2034
4’s complement of 1304
= (43)10– 1304
= 6410– 1304
= 6410– 2810
= 3610
= 2044
Example: