When the 4-bit Excess-3 codewords for two decimal digits are added and the…

Concept

Excess-3 represents a decimal digit d by the 4-bit binary code for d + 3. Therefore, a decimal sum digit s must finally be represented by the code for s + 3.

Adding the codewords for digits x and y gives (x + 3) + (y + 3) = x + y + 6. A carry-out occurs exactly when x + y is at least 10; without a carry-out, the raw 4-bit sum retains one excess 3 too many.

Application

1. Take x = 2 and y = 4. Their Excess-3 codewords are 0101 and 0111.

2. Add the codewords: 0101 + 0111 = 1100. There is no carry-out, and 1100 represents 12 = x + y + 6.

3. Subtract 0011: 1100 − 0011 = 1001. The code 1001 represents 9 = 6 + 3, which is the Excess-3 encoding of the decimal sum 6.

Cross-check

Decoding 1001 removes the excess 3: 9 − 3 = 6, which agrees with 2 + 4. Hence, in the no-carry case, subtract 0011 from the raw 4-bit sum.