How to subtract in binary

Remember how subtraction works. We work from the least-significant bits to the most-significant, and pairwise subtract them. 1-1=0, 1-0=1, 0-0=0.

But 0-1 presents a problem: it’s negative 1. We represent this by outputting 1 and carrying a 1 to be subtracted from the next bits. We end up with the following logic table. It shows inputs CI (carry in), A (the bit being subtracted from), and B (the bit to subtract). These are mapped to outputs CO (carry out) and O (output).

(Inputs)  | (Outputs)
CI  A  B  | CO  O
== == ==  | == ==
 0  0  0  |  0  0
 0  0  1  |  1  1
 0  1  0  |  0  1
 0  1  1  |  0  0
 1  0  0  |  1  1
 1  0  1  |  1  0
 1  1  0  |  0  0
 1  1  1  |  1  1

The meaning here is that CO and O together represent the result of A - (B+CI). This is a number between -2 and 1. Here’s the meaning:

CO  O | Meaning
== == | =======
 0  0 |       0
 0  1 |       1
 1  0 |      -2
 1  1 |      -1

This is not arbitrary. The relation is that Meaning = ( CO * -2) + O. This is precisely “two’s complement representation”: CO and O placed side-by-side are the two’s complement representation of the number.