Radix Complement and Diminished Radix complement

Complements are used in digital computers for simplifying the subtraction and for logical manipulations. There are two types of complement

• Radix complements ( r complement)
• diminished radix complements (r-1 complement)

For base 2 or binary number system the r’s complement is 2’s complement and (r-1)’s complement is 1’s complement

For base 10 or decimal system the r’s complement is 10’s complement and (r-1)’s complement is 9’s complement.

10’s Complement

Let us assume the number 3567890, for finding the 10’s complement, there is a rule

3	5	6	7	8	9	0
Most significant Digit						Least Significant Digit

10’s complement can be obtained by

• leaving all least significant 0’s unchanged
• subtracting the first non zero least significant digit from 10
• and subtracting all higher significant digits from 9

For the above example, The 10’s complement is

Given number	3	5	6	7	8	9	0
Process	9-3	9-5	9-6	9-7	9-8	10-9	unchanged
10’s Complement	6	4	3	2	1	1	0

so 10’s complement of 3567890 is 6432110

9’s Complement

9’s complement is a diminished radix complement and can be easily found out by subtracting all the given digits by 9.

For example, the 9’s complement of 3567890 is

9999999 – 3567890 = 6432109

given number	3	5	6	7	8	9	0
Process	9-3	9-5	9-6	9-7	9-8	9-9	9-0
9’s complement	6	4	3	2	1	0	9

In short,

10’s Complement = 9’s Complement + 1

2’s Complement

For binary numbers, there is 2’s complement and 1’s complement

2’s complement can be obtained by

• leaving the least significant 0’s unchanged and the first 1 unchanged
• replacing 1’s with 0’s and 0’s with 1’s in all other higher significant digits

given number	1	1	0	1	1	0	0
process	0	0	1	0	unchanged	unchanged	unchanged
2’s complement	0	0	1	0	1	0	0

ie.

2’s complement of 1101100 is 0010100

1’s complement

Finding 1’s complement is just to replace all 1’s by 0’s and all 0’s by 1’s

1’s complement of 1101100 is 0010011

in short

2’s complement = 1’s complement + 1