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

Concept

A decimal code is self-complementing when the 1’s complement of the code word for a digit d is the code word for its 9’s-complement digit, 9 − d.

Excess-3 represents d as the four-bit binary value of d + 3. Complementing four bits changes a value x to 15 − x.

Application

Apply the definition to Excess-3:

1. For d = 2, add the bias: 2 + 3 = 5, so XS-3(2) = 0101.

2. Invert every bit: 0101 becomes 1010.

3. The binary value 1010 is 10; removing the bias gives 10 − 3 = 7.

4. Since 9 − 2 = 7, the inverted word is XS-3(7), exactly the code for the 9’s complement of 2.

Cross-check and contrast

Starting with 7 gives XS-3(7) = 1010; inversion gives 0101, whose Excess-3 digit is 5 − 3 = 2. Thus the operation works in both directions.

• ASCII maps characters to code points; that purpose does not define decimal complementation.

• Financial calculation is an application area, not a bitwise transformation rule.

• Gray code is defined by one-bit changes between adjacent values, a different property from decimal self-complementation.

Result

Therefore, Excess-3 is considered self-complementing because the 9’s complement of each decimal digit is obtained by inverting all bits of its Excess-3 code word.