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| [[Image:Dyadic rational.svg|thumb|300px|Dyadic rationals in the interval from 0 to 1.]]
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| In [[mathematics]], a '''dyadic fraction''' or '''dyadic rational''' is a [[rational number]] whose [[denominator]] is a [[power of two]], i.e., a number of the form <math>\frac{a}{2^b}</math> where ''a'' is an [[integer]] and ''b'' is a [[natural number]]; for example, 1/2 or 3/8, but not 1/3. These are precisely the numbers whose [[binary numeral system|binary]] expansion is finite.
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| ==Use in measurement==
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| The [[inch]] is customarily subdivided in dyadic rather than decimal fractions; similarly, the customary divisions of the [[gallon]] into half-gallons, [[quart]]s, and [[pint]]s are dyadic. The ancient Egyptians also used dyadic fractions in measurement, with denominators up to 1/64.<ref>{{citation
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| | title = Concept of the exponential law prior to 1900
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| | last = Curtis | first = Lorenzo J.
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| | journal = [[American Journal of Physics]]
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| | year = 1978
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| | volume = 46
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| | issue = 9
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| | pages = 896–906
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| | doi = 10.1119/1.11512}}.</ref>
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| ==Arithmetic==
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| The [[Addition|sum]], [[Multiplication|product]], or [[Subtraction|difference]] of any two dyadic fractions is itself another dyadic fraction:
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| :<math>\frac{a}{2^b}+\frac{c}{2^d}=\frac{2^{d-b}a+c}{2^d} \quad (d\ge b)</math>
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| :<math>\frac{a}{2^b}-\frac{c}{2^d}=\frac{2^{d-b}a-c}{2^d} \quad (d\ge b)</math>
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| :<math>\frac{a}{2^b}-\frac{c}{2^d}=\frac{a-2^{b-d}c}{2^b} \quad (d< b)</math>
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| :<math>\frac{a}{2^b}\times \frac{c}{2^d} = \frac{ a \times c}{2^{b+d}}.</math>
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| However, the result of [[Division (mathematics)|dividing]] one dyadic fraction by another is not necessarily a dyadic fraction.
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| ==Additional properties==
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| Because they are closed under addition, subtraction, and multiplication, but not division, the dyadic fractions form a [[subring]] of the rational numbers '''[[rational number|Q]]'''. Algebraically, this subring is the [[localization of a ring|localization]] of the integers '''Z''' with respect to the set of powers of two.
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| The set of all dyadic fractions is [[dense set|dense]] in the [[real line]]: any real number ''x'' can be arbitrarily closely approximated by dyadic rationals of the form <math>\lfloor 2^i x \rfloor / 2^i</math>.
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| Compared to other dense subsets of the real line, such as the rational numbers, the dyadic rationals are in some sense a relatively "small" dense set, which is why they sometimes occur in proofs. (See for instance [[Urysohn's lemma]].)
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| ==Dual group==
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| Considering only the addition and subtraction operations of the dyadic rationals gives them the structure of an additive [[abelian group]]. The dual group of a group consists of its [[Character (mathematics)|characters]], [[group homomorphism]]s to the multiplicative group of the [[complex number]]s, and in the spirit of [[Pontryagin duality]] the dual group of the additive dyadic rationals can also be viewed as a [[topological group]]. It is called the '''dyadic solenoid''' and is an example of a [[solenoid group]] and of a [[protorus]].
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| The dyadic rationals are the [[direct limit]] of [[infinite cyclic]] subgroups of the rational numbers,
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| :<math>\varinjlim \left\{2^{-i}\mathbb{Z}\mid i = 0, 1, 2, \dots \right\}</math> | |
| and their dual group can be constructed as the [[inverse limit]] of the [[unit circle]] group under the repeated squaring map
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| :<math>\zeta\mapsto\zeta^2.</math> | |
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| An element of the dyadic solenoid can be represented as an infinite sequence of complex numbers ''q''<sub>0</sub>, ''q''<sub>1</sub>, ''q''<sub>2</sub>, ..., with the properties that each ''q''<sub>i</sub> lies on the unit circle and that, for all ''i'' > 0, ''q''<sub>i</sub><sup>2</sup> = ''q''<sub>i − 1</sub>. The group operation on these elements multiplies any two sequences componentwise. Each element of the dyadic solenoid corresponds to a character of the dyadic rationals that maps ''a''/2<sup>''b''</sup> to the complex number ''q''<sub>''b''</sub><sup>''a''</sup>. Conversely, every character ''χ'' of the dyadic rationals corresponds to the element of the dyadic solenoid given by ''q''<sub>''i''</sub> = ''χ''(1/2<sup>''i''</sup>).
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| As a topological space the dyadic solenoid is a [[Solenoid (mathematics)|solenoid]], and an [[indecomposable continuum]].<ref>{{citation
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| | title = The indecomposability of the dyadic solenoid
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| | jstor = 2319174
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| | last = Nadler | first = S. B., Jr.
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| | journal = [[American Mathematical Monthly]]
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| | year = 1973
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| | volume = 80
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| | issue = 6
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| | pages = 677–679
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| | doi = 10.2307/2319174}}.</ref>
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| ==Related constructions==
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| The [[surreal number]]s are generated by an iterated construction principle which starts by generating all finite dyadic fractions, and then goes on to create new and strange kinds of infinite, infinitesimal and other numbers.
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| The binary [[van der Corput sequence]] is an [[equidistributed]] [[permutation]] of the positive dyadic rational numbers.
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| ==In music==
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| [[Time signature]]s in Western [[musical notation]] traditionally consist of dyadic fractions, although non-dyadic time signatures have been introduced by composers in the twentieth century. (Non-dyadic time signatures are often called "[[Time_signature#Irrational_meters|irrational]]" by musicians, but this is a [[misnomer]] because they still consist of ratios of integers.)
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| ==See also==
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| * [[p-adic number|2-adic number]], a number system that extends the dyadic rationals
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| == References ==
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| {{reflist}}
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| [[Category:Fractions]]
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| [[Category:Rational numbers]]
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