BCD Addition

Binary-coded decimal (BCD) addition is similar to normal binary addition, except the result must be corrected to BCD form whenever the binary value of any decimal digit exceeds $9$. The correction: add $6$ to any invalid BCD result.

Why a correction is needed

In BCD, each decimal digit is encoded in 4 bits. But 4 bits can represent $0$ through $15$ — so $1010$ through $1111$ are invalid BCD codes (they don’t correspond to any decimal digit). Whenever an addition produces one of these invalid codes, BCD arithmetic gets the wrong answer unless we correct.

Adding $6$ pushes the result past the invalid range and back into valid BCD, with the appropriate carry into the next digit position.

Worked example

Add $5+7$ in BCD:

$0101+0111=1100$

The binary result $1100$ equals $12$ — correct in pure binary, but $1100$ is an invalid BCD code. Add $6=0110$ to correct:

$1100+0110=00010010$

Read in BCD: $0001$ is the tens digit, $0010$ is the ones digit, so $12$. ✓

The correction rule

For the result $Z$ of a 4-bit addition, the true BCD sum bit pattern $S$ and the carry-out $C_{\text{out}}$:

• If $Z\le 9$: $S=Z$ and $C_{\text{out}}=0$. No correction.
• If $Z>9$: $S=Z+6$ and $C_{\text{out}}=1$. Correction applied.

A useful detection trick: $Z>9$ iff there’s a carry-out from the 4-bit adder OR ($Z_3=1$ AND ($Z_2=1$ OR $Z_1=1$)).

Why this works: the carry-out from a 4-bit adder fires when the sum is at least $16$, i.e. $Z\ge 16$ — that catches all sums of two BCD digits that overflow past 4 bits. The remaining "$Z>9$" cases without overflow are $Z\in \{10,11,12,13,14,15\}$, with bit patterns $1010,1011,1100,1101,1110,1111$. Each of these has $Z_3=1$ along with at least one of $Z_2$ or $Z_1$ set. Bit patterns $1000$ and $1001$ ($Z=8,9$) have $Z_3=1$ but $Z_2=Z_1=0$, so the trick correctly excludes them. Adding $6=0110$ only affects the middle bits $Z_2,Z_1$ — doesn’t disturb the LSB or MSB of $Z$.

Implementation

A single-digit BCD adder consists of:

1. A 4-bit binary adder for the initial sum.
2. Detection logic that flags whether $Z>9$.
3. A 4-bit corrective adder that adds $6$ when the flag is set.

For multi-digit BCD numbers, chain single-digit BCD adders just like ripple-carry binary adders, with each stage’s $C_{\text{out}}$ feeding the next stage’s $C_{\text{in}}$.

When BCD is used

BCD is wasteful — 4 bits represent only 10 of 16 patterns. Pure binary is more efficient. But BCD is preferred where decimal precision matters and rounding errors from binary representation are unacceptable:

• Financial calculations: dollar amounts must be exact to the cent.
• Calculator chips: traditionally use BCD for exact decimal display.
• Time/date calculations: hours, minutes, days are decimal-natural.
• Some database systems: SQL DECIMAL types use BCD-like encoding.

For the underlying adder hardware, see that note. For character codes like ASCII, the lower 4 bits of digit characters happen to encode BCD — see that note.