2adics

2-adics, or the 2-adic numbers, are the completion of the rational numbers with respect to the 2-adic absolute value. They form the field Q2, with its ring of integers Z2. Z2 can be described as the inverse limit of the system Z/2^n Z, or equivalently as the set of all infinite series sum_{i=0}^∞ a_i 2^i with digits a_i ∈ {0,1}.

Every nonzero 2-adic number x can be written uniquely as x = 2^k u with k ∈ Z and u ∈ Z2^×, where u is a 2-adic unit. The 2-adic valuation v2(x) is the exponent k of the highest power of 2 dividing x, and the corresponding absolute value is |x|2 = 2^{-v2(x)}. This yields a non-Archimedean, ultrametric topology, in which convergent series and Cauchy sequences behave differently from real or complex numbers.

Topological and algebraic properties: Z2 is a compact, totally disconnected ring; Q2 is a locally compact field. The maximal ideal of Z2 is 2Z2, and its units are precisely the 2-adic integers with a0 = 1 in their digit expansion, i.e., the odd 2-adic integers. In Q2, every element can be expressed as 2^k times a 2-adic unit.

Arithmetic and analysis: computation in 2-adics follows non-Archimedean rules, with convergent sequences determined by their 2-adic proximity. Hensel’s lemma allows lifting solutions modulo 2^n to solutions in Z2, facilitating local methods for solving equations. The 2-adics provide a natural setting for p-adic analysis and the study of local fields in number theory.