Editing Dyadic rational (section)

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{{Short description|Fraction with denominator a power of two}}
[[File:Dyadic rational.svg|thumb|upright=1.2|Dyadic rationals in the interval from 0 to 1|alt=Unit interval subdivided into 1/128ths]]
In mathematics, a '''dyadic rational''' or '''binary rational''' is a number that can be expressed as a [[fraction]] whose [[denominator]] is a [[power of two]]. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in [[computer science]] because they are the only ones with finite [[binary representation]]s. Dyadic rationals also have applications in weights and measures, musical [[time signature]]s, and early mathematics education. They can accurately approximate any [[real number]].

The sum, difference, or product of any two dyadic rational numbers is another dyadic rational number, given by a simple formula. However, division of one dyadic rational number by another does not always produce a dyadic rational result. Mathematically, this means that the dyadic rational numbers form a [[Ring (mathematics)|ring]], lying between the ring of [[integer]]s and the [[Field (mathematics)|field]] of [[rational number]]s. This ring may be denoted <math>\Z[\tfrac12]</math>.

In advanced mathematics, the dyadic rational numbers are central to the constructions of the [[Solenoid (mathematics)|dyadic solenoid]], [[Minkowski's question-mark function]], [[Daubechies wavelet]]s, [[Thompson groups|Thompson's group]], [[Prüfer group|Prüfer 2-group]], [[surreal number]]s, and [[fusible number]]s. These numbers are [[order isomorphism|order-isomorphic]] to the rational numbers; they form a subsystem of the [[p-adic number|2-adic numbers]] as well as of the reals, and can represent the [[fractional part]]s of 2-adic numbers. Functions from natural numbers to dyadic rationals have been used to formalize [[mathematical analysis]] in [[reverse mathematics]].