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.