Editing Dyadic rational (section)

===Algebraic structure===
Because they are closed under addition, subtraction, and multiplication, but not division, the dyadic rationals are a [[ring (mathematics)|ring]] but not a [[field (mathematics)|field]].{{r|pollen}} The ring of dyadic rationals may be denoted <math>\Z[\tfrac12]</math>, meaning that it can be generated by evaluating polynomials with integer coefficients, at the argument 1/2.{{r|bajnok}} As a ring, the dyadic rationals are a [[subring]] of the rational numbers, and an [[overring]] of the integers.{{r|est-ohm}} Algebraically, this ring is the [[localization of a ring|localization]] of the integers with respect to the set of [[power of two|powers of two]].{{r|lucy}}

As well as forming a subring of the [[real number]]s, the dyadic rational numbers form a subring of the [[p-adic number|2-adic number]]s, a system of numbers that can be defined from binary representations that are finite to the right of the binary point but may extend infinitely far to the left. The 2-adic numbers include all rational numbers, not just the dyadic rationals. Embedding the dyadic rationals into the 2-adic numbers does not change the arithmetic of the dyadic rationals, but it gives them a different topological structure than they have as a subring of the real numbers. As they do in the reals, the dyadic rationals form a dense subset of the 2-adic numbers,{{r|manners}} and are the set of 2-adic numbers with finite binary expansions. Every 2-adic number can be decomposed into the sum of a 2-adic integer and a dyadic rational; in this sense, the dyadic rationals can represent the [[fractional part]]s of 2-adic numbers, but this decomposition is not unique.{{r|robert}}

Addition of dyadic rationals modulo 1 (the [[quotient group]] <math>\Z[\tfrac12]/\Z</math> of the dyadic rationals by the integers) forms the [[Prüfer group|Prüfer 2-group]].{{r|cor-guy-pit}}