View source for Dyadic rational

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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 signatures, 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]].

==Applications==
===In measurement===
{{CSS image crop|Image=Kitchen weights metric imperial.jpg|bSize=427|cWidth=256|oLeft=165|oTop=27|cHeight=256|Description=Kitchen weights measuring dyadic fractions of a [[Pound (mass)|pound]] from {{nowrap|2 lb}} down to 1/64 lb (1/4 oz)|Alt=Photo of metal disks used as kitchen weights}}
Many traditional systems of weights and measures are based on the idea of repeated halving, which produces dyadic rationals when measuring fractional amounts of units. The [[inch]] is customarily subdivided in dyadic rationals rather than using a decimal subdivision.{{r|rudman}} The customary divisions of the [[gallon]] into half-gallons, [[quart]]s, [[pint]]s, and [[Cup (unit)|cups]] are also dyadic.{{r|barnes}} The ancient Egyptians used dyadic rationals in measurement, with denominators up to 64.{{r|curtis}} Similarly, systems of weights from the [[Indus Valley civilisation]] are for the most part based on repeated halving; anthropologist Heather M.-L. Miller writes that "halving is a relatively simple operation with beam balances, which is likely why so many weight systems of this time period used binary systems".{{r|miller}}

===In computing===
Dyadic rationals are central to [[computer science]] as a type of fractional number that many computers can manipulate directly.{{r|reswel}} In particular, as a data type used by computers, [[Floating-point arithmetic|floating-point numbers]] are often defined as integers multiplied by positive or negative powers of two. The numbers that can be represented precisely in a floating-point format, such as the [[IEEE floating point|IEEE floating-point datatypes]], are called its representable numbers. For most floating-point representations, the representable numbers are a subset of the dyadic rationals.{{r|kirk-hwu}} The same is true for [[fixed-point arithmetic|fixed-point datatypes]], which also use powers of two implicitly in the majority of cases.{{r|kneusel}} Because of the simplicity of computing with dyadic rationals, they are also used for exact real computing using [[interval arithmetic]],{{r|vdh}} and are central to some theoretical models of [[computable number]]s.{{r|ko|zr|asz}}

Generating a [[random variable]] from random bits, in a fixed amount of time, is possible only when the variable has finitely many outcomes whose probabilities are all dyadic rational numbers. For random variables whose probabilities are not dyadic, it is necessary either to approximate their probabilities by dyadic rationals, or to use a random generation process whose time is itself random and unbounded.{{r|jvv}}

===In music===
{{Image frame|content=<score sound="1"> { \new PianoStaff << \new Staff \relative c'' { \set Staff.midiInstrument = #"violin" \clef treble \tempo 8 = 126 \time 3/16 r16 <d c a fis d>\f-! r16\fermata | \time 2/16 r <d c a fis d>-! \time 3/16 r <d c a fis d>8-! | r16 <d c a fis d>8-! | \time 2/8 <d c a fis>16-! <e c bes g>->-![ <cis b aes f>-! <c a fis ees>-!] } \new Staff \relative c { \set Staff.midiInstrument = #"violin" \clef bass \time 3/16 d,16-! <bes'' ees,>-! r\fermata | \time 2/16 <d,, d,>-! <bes'' ees,>-! | \time 3/16 d16-! <ees cis>8-! | r16 <ees cis>8-! | \time 2/8 d16\sf-! <ees cis>-!->[ <d c>-! <d c>-!] } >> } </score>|caption=Five bars from [[Igor Stravinski]]'s ''[[The Rite of Spring]]''<br/>showing time signatures {{music|time|3|16}}, {{music|time|2|16}}, {{music|time|3|16}}, and {{music|time|2|8}}}}
[[Time signature]]s in Western [[musical notation]] traditionally are written in a form resembling fractions (for example: {{music|time|2|2}}, {{music|time|4|4}}, or {{music|time|6|8}}),{{r|jon-pea}} although the horizontal line of the musical staff that separates the top and bottom number is usually omitted when writing the signature separately from its staff. As fractions they are generally dyadic,{{r|libbey}} although [[Time signature#Irrational meters|non-dyadic time signatures]] have also been used.{{r|yanakiev}} The numeric value of the signature, interpreted as a fraction, describes the length of a measure as a fraction of a [[whole note]]. Its numerator describes the number of beats per measure, and the denominator describes the length of each beat.{{r|jon-pea|libbey}}

===In mathematics education===
In theories of childhood development of the concept of a fraction based on the work of [[Jean Piaget]], fractional numbers arising from halving and repeated halving are among the earliest forms of fractions to develop.{{r|hie-ton}} This stage of development of the concept of fractions has been called "algorithmic halving".{{r|pot-saw}} Addition and subtraction of these numbers can be performed in steps that only involve doubling, halving, adding, and subtracting integers. In contrast, addition and subtraction of more general fractions involves integer multiplication and factorization to reach a common denominator. Therefore, dyadic fractions can be easier for students to calculate with than more general fractions.{{r|wells}}

==Definitions and arithmetic==
The dyadic numbers are the [[rational number]]s that result from dividing an [[integer]] by a [[power of two]].{{r|ko}} A rational number <math>p/q</math> in simplest terms is a dyadic rational when <math>q</math> is a power of two.{{r|uit-bar}} Another equivalent way of defining the dyadic rationals is that they are the [[real number]]s that have a terminating [[binary representation]].{{r|ko}}

[[Addition]], [[subtraction]], and [[multiplication]] of any two dyadic rationals produces another dyadic rational, according to the following formulas:{{r|kreb-spit}}
: <math>\begin{align}
\frac{a}{2^b}+\frac{c}{2^d}&=\frac{2^{d - \min(b,d)}a + 2^{b - \min(b,d)}c} {2^{\max(b,d)} } \\[6px]
\frac{a}{2^b}-\frac{c}{2^d}&=\frac{2^{d - \min(b,d)}a - 2^{b - \min(b,d)}c} {2^{\max(b,d)} } \\[6px]
\frac{a}{2^b}\cdot \frac{c}{2^d} &= \frac{ a c}{2^{b+d}}
\end{align}</math>
However, the result of [[Division (mathematics)|dividing]] one dyadic rational by another is not necessarily a dyadic rational.{{r|oconnor}} For instance, 1 and 3 are both dyadic rational numbers, but 1/3 is not.

==Additional properties==
[[File:Dyadic sqrt2 approximation.svg|thumb|Dyadic rational approximations to the [[square root of 2]] (<math>\sqrt{2}\approx 1.4142</math>), found by rounding to the nearest smaller integer multiple of <math>1/2^i</math> for <math>i=0,1,2,\dots</math> The height of the pink region above each approximation is its error.]]
[[File:Bad dyadic approximation.svg|thumb|upright=1.2|Real numbers with no unusually-accurate dyadic rational approximations. The red circles surround numbers that are approximated within error <math>\tfrac16/2^i</math> by <math>n/2^i</math>. For numbers in the fractal [[Cantor set]] outside the circles, all dyadic rational approximations have larger errors.]]
Every integer, and every [[half-integer]], is a dyadic rational.{{r|sabin}} They both meet the definition of being an integer divided by a power of two: every integer is an integer divided by one (the zeroth power of two), and every half-integer is an integer divided by two.

Every [[real number]] can be arbitrarily closely approximated by dyadic rationals. In particular, for a real number <math>x</math>, consider the dyadic rationals of the form {{nowrap|<math display=inline>\lfloor 2^i x \rfloor / 2^i</math>,}} where <math>i</math> can be any integer and <math>\lfloor\dots\rfloor</math> denotes the [[floor function]] that rounds its argument down to an integer. These numbers approximate <math>x</math> from below to within an error of <math>1/2^i</math>, which can be made arbitrarily small by choosing <math>i</math> to be arbitrarily large. For a [[fractal]] subset of the real numbers, this error bound is within a constant factor of optimal: for these numbers, there is no approximation <math>n/2^i</math> with error smaller than a constant times <math>1/2^i</math>.<ref>More precisely, for small positive values of <math>\varepsilon</math>, the set of real numbers that have no approximation <math>n/2^i</math> with error smaller than a constant times <math>\varepsilon/2^i</math> forms a [[Cantor set]] whose [[Hausdorff dimension]], as a function of <math>\varepsilon</math>, goes to one as <math>\varepsilon</math> approaches zero. The illustration shows this set for <math>\varepsilon=\tfrac16</math>.</ref>{{r|nilsson}} The existence of accurate dyadic approximations can be expressed by saying that the set of all dyadic rationals is [[dense set|dense]] in the [[real line]].{{r|sabin}} More strongly, this set is uniformly dense, in the sense that the dyadic rationals with denominator <math>2^i</math> are uniformly spaced on the real line.{{r|ko}}

The dyadic rationals are precisely those numbers possessing finite [[binary number|binary expansions]].{{r|ko}} Their binary expansions are not unique; there is one finite and one infinite representation of each dyadic rational other than 0 (ignoring terminal 0s). For example, 0.11<sub>2</sub> = 0.10111...<sub>2</sub>, giving two different representations for 3/4.{{r|ko|kac}} The dyadic rationals are the only numbers whose binary expansions are not unique.{{r|ko}}

==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.

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 | last = Ko | first = Ker-I
 | doi = 10.1007/978-1-4684-6802-1
 | isbn = 0-8176-3586-6
 | location = Boston, Massachusetts
 | mr = 1137517
 | pages = 41–43
 | publisher = Birkhäuser Boston, Inc.
 | series = Progress in Theoretical Computer Science
 | title = Complexity Theory of Real Functions
 | url = https://books.google.com/books?id=QJnVBwAAQBAJ&pg=PA41
 | year = 1991| s2cid = 11758381
 }}</ref>

<ref name=kreb-spit>Equivalent formulas to these, written in the language of the [[Coq (software)|Coq]] interactive theorem prover, are given by {{citation
 | last1 = Krebbers | first1 = Robbert
 | last2 = Spitters | first2 = Bas
 | arxiv = 1106.3448
 | doi = 10.2168/LMCS-9(1:1)2013
 | issue = 1
 | journal = [[Logical Methods in Computer Science]]
 | mr = 3029087
 | page = 1:01, 27
 | title = Type classes for efficient exact real arithmetic in Coq
 | volume = 9
 | year = 2013| s2cid = 218627153
 }}</ref>

<ref name=libbey>{{citation
 | last = Libbey | first = Theodore
 | contribution = Time signature
 | contribution-url = https://books.google.com/books?id=lE8R56pksxEC&pg=PA873
 | isbn = 978-0-7611-2072-8
 | page = 873
 | publisher = Workman Publishing
 | title = The NPR Listener's Encyclopedia of Classical Music
 | year = 2006}}</ref>

<ref name=lucy>{{citation
 | last = Lucyshyn-Wright | first = Rory B. B.
 | arxiv = 1603.03351
 | doi = 10.1007/s10485-017-9496-9
 | issue = 2
 | journal = Applied Categorical Structures
 | mr = 3770912
 | pages = 369–400
 | title = Convex spaces, affine spaces, and commutants for algebraic theories
 | volume = 26
 | year = 2018| s2cid = 3743682
 }}</ref>

<ref name=manners>{{citation
 | last = Manners | first = Freddie
 | arxiv = 1305.1514
 | doi = 10.1007/s00222-014-0571-7
 | issue = 1
 | journal = [[Inventiones Mathematicae]]
 | mr = 3402799
 | pages = 239–270
 | title = A solution to the pyjama problem
 | volume = 202
 | year = 2015| bibcode = 2015InMat.202..239M
 | s2cid = 119148680
 }}; see section 6.2.1, "A model case: <math>\widehat{\Z[1/2]}</math>", pp. 255–257.</ref>

<ref name=mauldon>{{citation
 | last = Mauldon | first = J. G.
 | doi = 10.2307/2320870
 | issue = 7
 | journal = [[The American Mathematical Monthly]]
 | jstor = 2320870
 | mr = 503877
 | pages = 575–578
 | title = Num, a variant of Nim with no first-player win
 | volume = 85
 | year = 1978}}</ref>

<ref name=miller>{{citation
 | last = Miller | first = Heather M.-L.
 | editor1-last = Abraham | editor1-first = Shinu Anna
 | editor2-last = Gullapalli | editor2-first = Praveena
 | editor3-last = Raczek | editor3-first = Teresa P.
 | editor4-last = Rizvi | editor4-first = Uzma Z. | editor4-link = Uzma Z. Rizvi
 | contribution = Weighty matters: evidence for unity and regional diversity from the Indus civilization weights
 | date = 2013
 | doi = 10.4324/9781315431857
 | isbn = 978-1-59874-686-0
 | pages = 161–177
 | publisher = Left Coast Press
 | title = Connections and Complexity: New Approaches to the Archaeology of South Asia}}; see in particular [https://books.google.com/books?id=ddRmDAAAQBAJ&pg=PA166 p. 166]</ref>

<ref name=nadler>{{citation
  | title = The indecomposability of the dyadic solenoid
  | jstor = 2319174
  | last = Nadler | first = S. B. Jr.
  | journal = [[The American Mathematical Monthly]]
  | year = 1973
  | volume = 80
  | issue = 6
  | pages = 677–679
  | doi = 10.2307/2319174}}</ref>

<ref name=nilsson>{{citation
 | last = Nilsson | first = Johan
 | doi = 10.1007/s11856-009-0042-9 | doi-access=free
 | journal = [[Israel Journal of Mathematics]]
 | mr = 2520103
 | pages = 93–110
 | title = On numbers badly approximable by dyadic rationals
 | volume = 171
 | year = 2009}}</ref>

<ref name=oconnor>{{citation
 | last = O'Connor | first = Russell
 | doi = 10.1017/S0960129506005871
 | issue = 1
 | journal = Mathematical Structures in Computer Science
 | mr = 2311089
 | pages = 129–159
 | title = A monadic, functional implementation of real numbers
 | volume = 17
 | year = 2007| s2cid = 221168970
 | arxiv = cs/0605058
 }}</ref>

<ref name=pollen>{{citation
 | last = Pollen | first = David
 | contribution = Daubechies' scaling function on [0,3]
 | mr = 1161245
 | pages = 3–13
 | publisher = Academic Press | location = Boston, Massachusetts
 | series = Wavelet Analysis and Its Applications
 | title = Wavelets
 | volume = 2
 | year = 1992}}</ref>

<ref name=pot-saw>{{citation
 | last1 = Pothier | first1 = Yvonne | author1-link = Yvonne Pothier
 | last2 = Sawada | first2 = Daiyo
 | date = November 1983
 | doi = 10.2307/748675
 | issue = 5
 | journal = [[Journal for Research in Mathematics Education]]
 | jstor = 748675
 | pages = 307–317
 | title = Partitioning: the emergence of rational number ideas in young children
 | volume = 14}}</ref>

<ref name=reswel>{{citation
 | last1 = Resnikoff | first1 = Howard L. | author1-link = Howard L. Resnikoff
 | last2 = Wells | first2 = Raymond O. Jr. | author2-link = Raymond O. Wells Jr.
 | contribution = 2.2.1: Digital computers and measurement
 | contribution-url = https://books.google.com/books?id=jrvkBwAAQBAJ&pg=PA17
 | doi = 10.1007/978-1-4612-0593-7
 | isbn = 0-387-98383-X
 | location = New York
 | mr = 1712468
 | pages = 17–18
 | publisher = Springer-Verlag
 | title = Wavelet Analysis: The Scalable Structure of Information
 | year = 1998}}</ref>

<ref name=robert>{{citation
 | last = Robert | first = Alain M. | author-link = Alain M. Robert
 | contribution = 5.4 Fractional and integral parts of <math>p</math>-adic numbers
 | doi = 10.1007/978-1-4757-3254-2
 | isbn = 0-387-98669-3
 | mr = 1760253
 | pages = 40–43
 | publisher = Springer-Verlag | location = New York
 | series = [[Graduate Texts in Mathematics]]
 | title = A Course in <math>p</math>-adic Analysis
 | volume = 198
 | year = 2000}}</ref>

<ref name=rudman>{{citation
 | last = Rudman | first = Peter S.
 | isbn = 978-1-61592-176-8
 | page = 148
 | publisher = Prometheus Books
 | title = How Mathematics Happened: The First 50,000 Years
 | url = https://books.google.com/books?id=BtcQq4RUfkUC&pg=PA148
 | year = 2009}}</ref>

<ref name=sabin>{{citation
 | last = Sabin | first = Malcolm
 | isbn = 9783642136481
 | page = 51
 | publisher = Springer
 | series = Geometry and Computing
 | title = Analysis and Design of Univariate Subdivision Schemes
 | url = https://books.google.com/books?id=18UC7d7h0LQC&pg=PA51
 | volume = 6
 | year = 2010}}</ref>

<ref name=uit-bar>{{citation
 | last1 = Uiterwijk | first1 = Jos W. H. M.
 | last2 = Barton | first2 = Michael
 | arxiv = 1506.03949
 | doi = 10.1016/j.tcs.2015.05.017
 | journal = [[Theoretical Computer Science (journal)|Theoretical Computer Science]]
 | mr = 3367582
 | pages = 72–86
 | title = New results for Domineering from combinatorial game theory endgame databases
 | volume = 592
 | year = 2015| s2cid = 5899577
 }}</ref>

<ref name=vdh>{{citation
 | last = van der Hoeven | first = Joris
 | doi = 10.1016/j.tcs.2005.09.060 | doi-access = free
 | issue = 1
 | journal = [[Theoretical Computer Science (journal)|Theoretical Computer Science]]
 | mr = 2201092
 | pages = 52–60
 | title = Computations with effective real numbers
 | volume = 351
 | year = 2006}}</ref>

<ref name=wells>{{citation
 | last = Wells | first = David Graham
 | isbn = 978-1-78326-755-2
 | pages = 32–33
 | publisher = World Scientific
 | title = Motivating Mathematics: Engaging Teachers And Engaged Students
 | url = https://books.google.com/books?id=suY7DQAAQBAJ&pg=PA32
 | year = 2015}}</ref>

<ref name=yanakiev>{{citation
 | last = Yanakiev | first = Ivan K.
 | editor-last = Bozhikova | editor-first = Milena
 | contribution = Mathematical devices in aid of music theory, composition, and performance
 | isbn = 978-1-5275-4758-2
 | pages = 35–62
 | publisher = Cambridge Scholars Publishing
 | title = Music between Ontology and Ideology
 | year = 2020}}; see in particular [https://books.google.com/books?id=dQbUDwAAQBAJ&pg=PA37 p. 37].</ref>

<ref name=zr>{{citation
 | last1 = Zheng | first1 = Xizhong
 | last2 = Rettinger | first2 = Robert
 | doi = 10.1002/malq.200310110
 | issue = 4–5
 | journal = Mathematical Logic Quarterly
 | mr = 2090389
 | pages = 431–442
 | title = Weak computability and representation of reals
 | volume = 50
 | year = 2004| s2cid = 15815720
 }}</ref>

}}

{{Fractions and ratios}}
{{Rational numbers}}
{{Ring theory sidebar}}

[[Category:Fractions (mathematics)]]
[[Category:Rational numbers]]
[[Category:Ring theory]]
[[Category:Number theory]]