➕ BCD Addition

• BCD (Binary-Coded Decimal) represents decimal digits 0–9 using 4-bit binary codes.
• Although 4 bits can encode 16 values (0–15), only 0000 to 1001 are valid in BCD.
• Values from 1010 to 1111 (10–15) are unused and considered invalid BCD.

🔢 BCD Encoding Table

🧠 Addition Logic

BCD addition mimics place-value decimal addition:

• Add digits column-wise (units, tens, etc.)
• If the result exceeds 9, it spills into the invalid BCD zone (10–15) or overflows into 16–18
• These results cannot be represented directly in BCD

⚠️ Correction Mechanism

To restore valid BCD:

• Add 6 (0110) to any sum > 9
• This bumps the result into the next valid BCD digit and triggers a carry to the next place-value

Example: 7 + 8

  0111 (7)
+ 1000 (8)
= 1111 (15) → invalid
+ 0110 (6)
= 1 0101 → result = 0101 (5), carry = 1

🧩 Symbolic Interpretation

There are two ways to interpret the correction:

1. Recount from Zero

• Find the difference between 9 (1001) and the overflow value (e.g. 11)
• Recount from 0 to find the valid BCD digit
• Carry the excess into the next place-value

2. Bump by the Gap

• Add the difference between the last valid BCD (9) and the last 4-bit value (15)
• This difference is always 6
• Adding 6 pushes the result into the next BCD digit and triggers a carry

🔢 Carry Boundaries

• Max sum of two single-digit decimals: 9 + 9 = 18
• Binary: 10010 (5 bits)
• Only 1 carry bit is needed
• This rule holds for any base: the sum of two max digits always fits within one carry

🧠 Philosophical Note: Overflow as Symbolic Transition

• BCD overflow isn’t just arithmetic—it’s a semantic shift across digit boundaries
• The unused zone (10–15) acts as a symbolic buffer between valid digits and overflow
• Correction by adding 6 is a structural bridge between binary encoding and decimal semantics

✅ Summary Table