Digital Design Fundamentals·Combinational Circuits
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Adding in binary is the same column method you use in decimal. Work right to left, and carry when a column overflows. The only difference: a binary column overflows the moment it passes 1.
The four single-bit sums are the whole of it:
| A | B | Sum bit | Carry |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 1 | 0 |
| 1 | 0 | 1 | 0 |
| 1 | 1 | 0 | 1 |
The last row is the key one: , which is in binary, so the column writes 0 and carries 1.
Figure: A hand-worked binary column addition of 0011 plus 0110 equals 1001
Add (3) and (6):
| Column (value) | 8 | 4 | 2 | 1 |
|---|---|---|---|---|
| A | 0 | 0 | 1 | 1 |
| B | 0 | 1 | 1 | 0 |
| Carry in | 0 | 1 | 1 | 0 |
| Sum | 1 | 0 | 0 | 1 |
Reading the sum row: , and . The carries ripple left exactly as they do in decimal.
A fixed-width adder has nowhere to put a carry off the top end. Add (9) and (8) in 4 bits and the true answer is 17, which needs 5 bits. The result wraps to and a carry-out of 1.
That carry-out is not junk. It is the signal that the result did not fit, and real hardware uses it to flag overflow or to extend the sum into more bits. A byte adder that ignores its carry-out silently computes .
Hardware has no minus sign, only bits. To represent negatives, digital systems use two's complement: to make , flip every bit of and add 1.
Figure: A worked two's complement of 0011: first the bits are inverted to 1100 labelled "flip every bit"
For in 4 bits: start with , flip to , add 1 to get . The top bit acts as a sign: 0 for non-negative, 1 for negative.
Two's complement earns its keep here: subtraction becomes addition. To compute , add to the two's complement of . No separate subtractor circuit is needed, just the adder you are about to build plus some inverters.
The carry off the top is dropped, and is 2, the right answer. One adder does both jobs.
