Question

So let’s say I want to add -7 and -7 together using 2s complement binary notation. I tried this:

-7 -> -00000111 -> 11111001
  1 1 1 1 1 0 0 1
+ 1 1 1 1 1 0 0 1
__________________
1 0 0 0 0 0 0 1 0

But after removing the extra digit, I get 00000010 = 2 instead of 11110001 = 14. What did I do wrong?

Answer 1

Somewhere along the line, you lost (and kept losing) the carry flag:

          +- Here, 1 + 1 = 0 and sets carry to 1
          |   (everything up to here isokay).
          v
  1 1 1 1 1 0 0 1
+ 1 1 1 1 1 0 0 1
__________________
1 1 1 1 1 0 0 1 0
| | | ^ ^
| | | | |
| | | | +- Here, 1 + 1 + carry(1) is 1, not 0.
| | | +--- Ditto.
| | +----- And again.
| +------- Now it's just getting tedious.
+--------- Lastly, here is the final carry, just discard.

Negating 11110010 (invert bits and add 1) gives you 00001110, which is definitely 14 (8 + 4 + 2).