Excess-3 code with examples

Excess-3 (XS-3) code is a non-weighted BCD code in which each decimal digit is coded as the 4-bit binary of (digit + 3). It was used in early computers and calculators because it makes decimal arithmetic easier.

Excess-3 table

Codes 0000, 0001, 0010, 1101, 1110, 1111 are invalid (unused).

How to convert

Add 3 to each decimal digit separately, then write each result in 4-bit binary.

Example 1: $(39)_{10}$

• 3 + 3 = 6 → 0110
• 9 + 3 = 12 → 1100
• $(39)_{10} = (0110\ 1100)_{XS-3}$

Example 2: $(258)_{10}$

• 2 → 0101, 5 → 1000, 8 → 1011
• $(258)_{10} = (0101\ 1000\ 1011)_{XS-3}$

Properties

1. Unweighted code: bit positions have no fixed weights.
2. Self-complementing: the 9's complement of a digit is obtained by simply inverting all bits. For example, 4 = 0111; inverting gives 1000 = 5 in XS-3, and $9 - 4 = 5$. This made subtraction simple in old machines.
3. No all-zero code: every valid code has at least one 1, which helps detect a dead line.
4. Easy addition: when two XS-3 digits are added, a carry occurs exactly when the decimal sum exceeds 9. Correction: if there is a carry, add 0011 to the digit; if no carry, subtract 0011.

Addition example: $4 + 5$

• $0111 + 1000 = 1111$, no carry
• subtract 0011: $1111 - 0011 = 1100$ = 9 in XS-3

Excess-3 is therefore a convenient BCD code for decimal arithmetic, though it needs more bits than pure binary.