Sum bit

Also called: sum output, S output

The output of an adder stage that stays in its own column: the low bit of the column's total. In a full adder, S = A ⊕ B ⊕ Cin, the odd-parity function.

Each column of a binary addition produces two things: a sum bit that stays in the column, and a carry that moves left. The sum bit is the low bit of the column's total.

Half adder (two inputs): S = . The total is 0, 1 or 2, and only a total of 1 leaves a 1 in the column.
Full adder (three inputs): S = , with C the carry-in. The total is 0 to 3, and S = 1 when it's odd (1 or 3).

So the sum bit is the odd-parity function of the column's inputs. That's why adders build S from XOR gates: the K-map of a 3-input XOR is a checkerboard, with no two 1s adjacent, so no AND-OR grouping helps.

In a ripple carry adder, Sᵢ = Pᵢ ⊕ Cᵢ, where Pᵢ = Aᵢ ⊕ Bᵢ is the propagate signal. So a sum bit can't be final until the carry into its column is final, which is why higher sum bits settle later.

A\BC	00	01	11	10
0	0m0	1m1	0m3	1m2
1	1m4	0m5	1m7	0m6

Worked example

Example

Sum bits of 1001 + 0101

9 + 5. Each sum bit is the parity of its column, carry included.

1.
Bit 0: 1 + 1 + 0, two 1s → S0 = 0, carry 1.
2.
Bit 1: 0 + 0 + 1, one 1 → S1 = 1, carry 0.
3.
Bit 2: 0 + 1 + 0, one 1 → S2 = 1, carry 0.
4.
Bit 3: 1 + 0 + 0, one 1 → S3 = 1, carry 0.
5.
Result: 1110 = 14 = 9 + 5.

Common mistakes

Using OR for the sum. 1 + 1 must leave 0 in the column.
Expecting the sum's K-map to simplify. It's a checkerboard.
Forgetting the carry-in when working out S.

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