[sweetjesus]

this calculator nicely illustrates an insight that I feel gets lost the way most people learn and reteach some simple CS concepts:

"two's complement" is not a different system for arithmetic that includes a "sign bit", it's just a different encoding or labelling of states which happens to have a bit that reflects the sign. So, inputs to this calculator can be said to go from 0-15, but more interestingly it can also add numbers in the range -8 to +7 (and therefore, it can also subtract, though it can't negate so you'd have to manually do that to your input by performing a different encoding table lookup).

(edit: now I'm realizing you could negate by performing a two's complement multiplication by -1, performed using this calculator via a sequence of 3 (shift+adds) of your input number to itself... that's correct at least up to some fencepost)

And then by extension, you could test "what about treating the range as -10 to +5", would that encoding succeed or break down? for starters, you would no longer have a sign bit...