excess3

Excess-3 is a four-bit binary-coded decimal (BCD) encoding in which each decimal digit is represented by adding a fixed bias of three to the digit and encoding the result in binary. In other words, the code for a decimal digit d is the binary representation of (d + 3).

The mapping for digits 0 through 9 is as follows: 0 maps to 0011, 1 maps to 0100, 2 to 0101, 3 to 0110, 4 to 0111, 5 to 1000, 6 to 1001, 7 to 1010, 8 to 1011, and 9 to 1100.

Decoding Excess-3 is simply the reverse operation: subtract three from the 4-bit value to recover the original decimal digit.

Excess-3 is a BCD-compatible code and can be understood as a fixed offset version of the standard 8421 BCD. Because digits are stored as (d + 3), binary addition of two Excess-3 digits effectively performs decimal addition with a constant bias, requiring a correction step (subtracting 3) to obtain the Excess-3 result for the sum. The code has the property that the bitwise complement of an Excess-3 code for d corresponds to the Excess-3 code for 9 − d.

Applications of Excess-3 have included decimal arithmetic in digital logic circuits, calculators, and other hardware where straightforward binary addition was advantageous. It has largely been superseded by other encoding schemes in general-purpose computing but remains a useful example of a biased BCD code.