The p-adic numbers are an alternative number system where "closeness" is measured by divisibility by p. The p-adic tree visualizes the ultrametric structure — two numbers are close if they agree modulo large powers of p.
The p-adic valuation v_p(n) = largest k such that p^k divides n.
p-adic norm: |n|_p = p^(−v_p(n))
Large p-adic norm = NOT divisible by p
In ℚ_5: the 5-adic expansion of -1 is ...44444 (all 4s)
In ℚ_2: -1 = ...11111 (all 1s)
This is why two's complement works for negative numbers in computers!