2adic

2-adic refers to the 2-adic number system, a completion of the rational numbers with respect to the 2-adic absolute value. For a nonzero rational x, write x = 2^v2(x)·m with m odd; the 2-adic absolute value is |x|2 = 2^−v2(x). The 2-adic valuation v2(x) counts the exponent of 2 in x. The resulting metric makes two rationals close when their difference is divisible by a large power of 2. The field of 2-adic numbers, denoted Q2, consists of all formal series that converge under this metric, and it is a locally compact, totally disconnected field.

The 2-adic integers, Z2, form the subring of Q2 with |x|2 ≤ 1. Every element of Z2 can be written uniquely as an infinite series ∑i≥0 ai2^i with digits ai in {0,1}. In particular, Z2 contains all limits of sequences of ordinary integers under the 2-adic metric. The units of Z2 are the 2-adic integers with a0 = 1, i.e., the odd 2-adic integers; these are precisely the elements whose 2-adic absolute value is 1.

Arithmetic in the 2-adics is defined by limits of finite sums, compatible with the ultrametric inequality |x+y|2 ≤ max(|x|2, |y|2). This non-Archimedean structure yields contrasts with real numbers: convergence is governed by divisibility by higher powers of 2 rather than by magnitude. The residue field of Z2 is F2, and Z2 is a compact, complete topological ring.

Q2 is obtained by allowing a finite number of negative powers of 2 in the series, giving a field that contains fractions and encodes information about congruences modulo 2n. 2-adic methods are central in number theory, enabling, for example, Hensel lifting to lift solutions of congruences modulo 2^n to solutions modulo higher powers, and they underpin various local-global techniques and p-adic analyses.