Editing Dyadic rational (section)

==In advanced mathematics==
===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}}

===Dyadic solenoid===
Considering only the addition and subtraction operations of the dyadic rationals gives them the structure of an additive [[abelian group]]. [[Pontryagin duality]] is a method for understanding abelian groups by constructing dual groups, whose elements are [[Character (mathematics)|characters]] of the original group, [[group homomorphism]]s to the multiplicative group of the [[complex number]]s, with pointwise multiplication as the dual group operation. The dual group of the additive dyadic rationals, constructed in this way, can also be viewed as a [[topological group]]. It is called the dyadic solenoid, and is isomorphic to the topological product of the real numbers and 2-adic numbers, [[Quotient group|quotiented]] by the [[Diagonal morphism|diagonal embedding]] of the dyadic rationals into this product.{{r|manners}} It is an example of a [[protorus]], a [[Solenoid (mathematics)|solenoid]], and an [[indecomposable continuum]].{{r|nadler}}

===Functions with dyadic rationals as distinguished points===
{{multiple image|total_width=480
|image1=Minkowski question mark.svg|caption1=[[Minkowski's question-mark function]] maps rational numbers to dyadic rationals|alt1=Graph of the question mark function
|image2=Daubechies4-functions.svg|caption2=A [[Daubechies wavelet]], showing points of non-smoothness at dyadic rationals|alt2=Graph of the scaling and wavelet functions of Daubechies' wavelet
}}
Because they are a dense subset of the real numbers, the dyadic rationals, with their numeric ordering, form a [[dense order]]. As with any two unbounded countable dense linear orders, by [[Cantor's isomorphism theorem]],{{r|bmmn}} the dyadic rationals are [[Order isomorphism|order-isomorphic]] to the rational numbers.  In this case, [[Minkowski's question-mark function]] provides an order-preserving [[bijection]] between the set of all rational numbers and the set of dyadic rationals.{{r|girgensohn}}

The dyadic rationals play a key role in the analysis of [[Daubechies wavelet]]s, as the set of points where the [[Wavelet#Scaling function|scaling function]] of these wavelets is non-smooth.{{r|pollen}} Similarly, the dyadic rationals parameterize the discontinuities in the boundary between stable and unstable points in the parameter space of the [[Hénon map]].{{r|cvi-gun-pro}}

The set of [[piecewise linear function|piecewise linear]] [[homeomorphism]]s from the [[unit interval]] to itself that have power-of-2 slopes and dyadic-rational breakpoints forms a group under the operation of [[function composition]]. This is [[Thompson groups|Thompson's group]], the first known example of an infinite but [[Presentation of a group|finitely presented]] [[simple group]].{{r|brin}} The same group can also be represented by an action on rooted binary trees,{{r|can-flo}} or by an action on the dyadic rationals within the unit interval.{{r|cor-guy-pit}}

===Other related constructions===
In [[reverse mathematics]], one way of constructing the [[real number]]s is to represent them as functions from [[Unary numeral system|unary numbers]] to dyadic rationals, where the value of one of these functions for the argument <math>i</math> is a dyadic rational with denominator <math>2^i</math> that approximates the given real number. Defining real numbers in this way allows many of the basic results of [[mathematical analysis]] to be proven within a restricted theory of [[second-order arithmetic]] called "feasible analysis" (BTFA).{{r|fer-fer}}

The [[surreal number]]s are generated by an iterated construction principle which starts by generating all finite dyadic rationals, and then goes on to create new and strange kinds of infinite, infinitesimal and other numbers.{{r|conway}} This number system is foundational to [[combinatorial game theory]], and dyadic rationals arise naturally in this theory as the set of values of certain combinatorial games.{{r|mauldon|flanigan|uit-bar}}

The [[fusible number]]s are a subset of the dyadic rationals, the closure of the set <math>\{0\}</math> under the operation <math>x,y\mapsto(x+y+1)/2</math>, restricted to pairs <math>x,y</math> with <math>|x-y|<1</math>. They are [[well-order]]ed, with [[order type]] equal to the [[Epsilon numbers (mathematics)|epsilon number]] <math>\varepsilon_0</math>. For each integer <math>n</math> the smallest fusible number that is greater than <math>n</math> has the form <math>n+1/2^k</math>. The existence of <math>k</math> for each <math>n</math> cannot be proven in [[Peano arithmetic]],{{r|eri-niv-xu}} and <math>k</math> grows so rapidly as a function of <math>n</math> that for <math>n=3</math> it is (in [[Knuth's up-arrow notation]] for large numbers) already larger than <math>2\uparrow^9 16</math>.{{r|A188545}}

The usual proof of [[Urysohn's lemma]] utilizes the dyadic fractions for constructing the separating function from the lemma.