Carry

Also called: carry bit

The 1 that moves into the next column when a column's sum is too big for one digit. In binary, a column carries when its total reaches 2.

A carry is what happens when a column of an addition fills up. The column keeps the part that fits and passes 1 to the next column on the left.

In decimal a column carries when it reaches 10. In binary, a column can only hold 0 or 1, so it carries when the total reaches 2. A binary column adds at most three bits (two operand bits plus the carry from the right), so there are just four cases:

total 0: write 0, carry 0
total 1: write 1, carry 0
total 2 = 10₂: write 0, carry 1
total 3 = 11₂: write 1, carry 1

Read the pair as a 2-bit number, carry first, and you have the column's total. That's exactly what a full adder outputs. The carry is 1 when at least two of the three inputs are 1.

Names to know:

the carry entering a column is its carry in
the carry leaving the top column is the carry out, the bit that doesn't fit in the result
in an adder circuit, carries pass from bit to bit, and how fast they travel sets the adder's speed (ripple carry adder, carry lookahead adder)

The same idea works in any base: hex addition carries at 16.


0	0	0	0	0
0	0	1	1	0
0	1	0	1	0
0	1	1	0	1
1	0	0	1	0
1	0	1	0	1
1	1	0	0	1
1	1	1	1	1

Worked example

Example

Tracking carries in 1011 + 0111

11 + 7, column by column from the right.

1.
Bit 0: 1 + 1 = 10₂. Write 0, carry 1.
2.
Bit 1: 1 + 1 + 1 = 11₂. Write 1, carry 1.
3.
Bit 2: 0 + 1 + 1 = 10₂. Write 0, carry 1.
4.
Bit 3: 1 + 0 + 1 = 10₂. Write 0, carry 1.
5.
The last carry becomes a fifth bit: 10010 = 18 = 11 + 7.

Common mistakes

Writing 2 in a binary column. 1 + 1 is 10₂: write 0, carry 1.
Forgetting the incoming carry when a column holds 1 + 1: with the carry it's 1 + 1 + 1, so write 1 and carry 1.
Confusing a carry with overflow. Carries between columns are normal; only the carry-out of the top column can signal unsigned overflow.