2's complement is simply

arithmetic modulo 2

^{n}, where

`n` is the

number of

bits you have. You can take any

consecutive 2

^{n} numbers and claim your bits represent them, so just

pick the numbers starting at -2

^{n-1}. This explains why

addition and and

multiplication continue to work.

sockpuppet's fun example with the diverging infinite series is just an example of how natural p-adic numbers really are; here `p`=2.