I'm curious why some old architectures didn't use two's complement for signed numbers. What advantage did one's complement or signed magnitude have over two's complement?
https://en.wikipedia.org/wiki/Signed_overpunch
On the other hand (no pun intended), early mechanical (decimal) manual adding machines made use of complement arithmetic too:
Except the "natural" way also recognizes a single zero with no sign, so it's still not accurately modeling that.
If you wanted to model natural arithmetic accurately you'd need 2 bits for the sign (positive, negative, unsigned). At that point, all of single bit signed magnitude, and complements are compromises.
Burroughs had a unique numeric representation. Numbers were 48 bits. Sign, sign of exponent, exponent, mantissa, with the binary point at the low end. Integers were thus valid floating point numbers. The math operations would maintain a value as an integer, with a zero exponent, if possible.
IEEE floating point also maintains integer values as integers until they don't fit, but the representation is not integer-like.
Most software doesn't handle this properly, they don't realise abs doesn't always return a positive number (as abs(INT_MIN)=INT_MIN), and many other similar problems.
In an ideal world, I would only use unsigned when you care about things like being able to use all bit representations, then have made the all-1s number something like NaN, for ints.
Interestingly, posits as originally proposed do have this property (except for infinity).
Negabinary operations are extremely simple and elegant. Like 2s complement and 1s complement, it suffers from asymmetry in its range, though even more so.