In Excess-3 Code, why is the code considered self-complementing?

Excess-3 code is called self-complementing because the 1’s complement of the Excess-3 code of a decimal digit gives the Excess-3 code of the 9’s complement of that digit.

Why Excess-3 Code is Called Self-Complementing

Excess-3 (XS-3) code is called a: Self-Complementing Code

because:

the 1’s complement of the Excess-3 code of a decimal digit gives the Excess-3 code of the 9’s complement of that digit.

Main Concept

If a decimal digit is: N

then:
1’s complement of XS-3(N) = XS-3(9-N)

Example

Take decimal digit: 2

Step 1: Find XS-3 of 2

Add 3:
2+3=5

Binary of 5: 0101

So:
2 -----> 0101

Step 2: Take 1’s Complement
0101 ----> 1010

Step 3: Convert Back
10102 = 10

Subtract 3:
10-3=7

And:
9-2=7

So:
1010 is XS-3 code of: 7

Hence Proven
1’s complement of XS-3(2) = {XS-3}(7)

where:
7=9-2

Therefore Excess-3 is: Self-complementing

Why This Happens

Because Excess-3 uses: +3 bias.

The code structure becomes symmetric around: 9

which creates the self-complementing property.

Important Advantage

Because of self-complementing nature:

• decimal subtraction becomes easier,

• complement arithmetic becomes simpler,

• digital circuit design becomes easier.