Editing Dyadic rational (section)

== References ==
{{reflist|refs=

<ref name=A188545>{{cite OEIS|A188545|mode=cs2}}</ref>

<ref name=asz>{{citation
 | last1 = Ambos-Spies | first1 = Klaus
 | last2 = Zheng | first2 = Xizhong
 | editor1-last = Manea | editor1-first = Florin
 | editor2-last = Martin | editor2-first = Barnaby
 | editor3-last = Paulusma | editor3-first = Daniël
 | editor4-last = Primiero | editor4-first = Giuseppe
 | contribution = On the differences and sums of strongly computably enumerable real numbers
 | doi = 10.1007/978-3-030-22996-2_27
 | mr = 3981892
 | pages = 310–322
 | publisher = Springer | location = Cham
 | series = Lecture Notes in Computer Science
 | title = Computing with Foresight and Industry: 15th Conference on Computability in Europe, CiE 2019, Durham, UK, July 15–19, 2019, Proceedings
 | volume = 11558
 | year = 2019| isbn = 978-3-030-22995-5
 | s2cid = 195795492
 }}</ref>

<ref name=bajnok>{{citation
 | last = Bajnok | first = Béla
 | doi = 10.1007/978-1-4614-6636-9
 | location = New York
 | page = 186
 | publisher = Springer
 | title = An Invitation to Abstract Mathematics
 | series = Undergraduate Texts in Mathematics
 | year = 2013| isbn = 978-1-4614-6635-2
 }}</ref>

<ref name=barnes>{{citation
 | last = Barnes | first = John
 | doi = 10.1007/978-3-319-46831-0
 | publisher = Springer International Publishing
 | quote = Note that binary measures (2, 4, 8, 16) are very common indeed. This is particularly obvious with volumes. 
 | title = Nice Numbers
 | url = https://books.google.com/books?id=6l2XDQAAQBAJ&pg=PA10
 | year = 2016| isbn = 978-3-319-46830-3
 }}</ref>

<ref name=bmmn>{{citation
 | last1 = Bhattacharjee | first1 = Meenaxi
 | last2 = Macpherson | first2 = Dugald
 | last3 = Möller | first3 = Rögnvaldur G.
 | last4 = Neumann | first4 = Peter M. | author4-link = Peter M. Neumann
 | contribution = Rational numbers
 | doi = 10.1007/978-93-80250-91-5_9
 | isbn = 81-85931-13-5
 | mr = 1632579
 | pages = 77–86
 | publisher = Springer-Verlag | location = Berlin
 | series = Texts and Readings in Mathematics
 | title = Notes on Infinite Permutation Groups
 | volume = 12
 | year = 1997}}</ref>

<ref name=brin>{{citation
 | last = Brin | first = Matthew G.
 | arxiv = math/9705205
 | doi = 10.1112/S0024610799007905
 | issue = 2
 | journal = [[Journal of the London Mathematical Society]]
 | mr = 1724861
 | pages = 449–460
 | series = Second Series
 | title = The ubiquity of Thompson's group {{mvar|F}} in groups of piecewise linear homeomorphisms of the unit interval
 | volume = 60
 | year = 1999| s2cid = 14490692
 }}</ref>

<ref name=can-flo>{{citation
 | last1 = Cannon | first1 = J. W. | author1-link = James W. Cannon
 | last2 = Floyd | first2 = W. J. | author2-link = William Floyd (mathematician)
 | issue = 8
 | journal = [[Notices of the American Mathematical Society]]
 | mr = 2856142
 | pages = 1112–1113
 | title = What is … Thompson's group?
 | url = https://www.ams.org/notices/201108/rtx110801112p.pdf
 | volume = 58
 | year = 2011}}</ref>

<ref name=cor-guy-pit>{{citation
 | last1 = de Cornulier | first1 = Yves
 | last2 = Guyot | first2 = Luc
 | last3 = Pitsch | first3 = Wolfgang
 | arxiv = math/0511714
 | doi = 10.1016/j.jalgebra.2006.02.012
 | issue = 1
 | journal = [[Journal of Algebra]]
 | mr = 2278053
 | pages = 254–277
 | title = On the isolated points in the space of groups
 | url = https://www.normalesup.org/~cornulier/CoGuPi.pdf
 | volume = 307
 | year = 2007 | s2cid = 11566447
 }}</ref>

<ref name=cvi-gun-pro>{{citation
 | last1 = Cvitanović | first1 = Predrag
 | last2 = Gunaratne | first2 = Gemunu H.
 | last3 = Procaccia | first3 = Itamar
 | doi = 10.1103/PhysRevA.38.1503
 | issue = 3
 | journal = [[Physical Review A]]
 | mr = 970237
 | pages = 1503–1520
 | series = Third Series
 | title = Topological and metric properties of Hénon-type strange attractors
 | volume = 38
 | year = 1988| pmid = 9900529
 | bibcode = 1988PhRvA..38.1503C
 }}</ref>

<ref name=conway>{{citation
 | last = Conway | first = J. H. | author-link = John Horton Conway
 | edition = Second
 | isbn = 1-56881-127-6
 | mr = 1803095
 | publisher = A K Peters | location = Natick, Massachusetts
 | title = On Numbers and Games | title-link = On Numbers and Games
 | year = 2001}}; for the dyadic rationals, see "The numbers <math>\tfrac{1}{4}</math>, <math>\tfrac{3}{4}</math>, <math>1\,\tfrac{1}{2}</math>, <math>3</math>, and so on", pp. 10–12</ref>

<ref name=curtis>{{citation
  | title = Concept of the exponential law prior to 1900
  | last = Curtis | first = Lorenzo J.
  | journal = [[American Journal of Physics]]
  | year = 1978
  | volume = 46
  | issue = 9
  | pages = 896–906
  | doi = 10.1119/1.11512| bibcode = 1978AmJPh..46..896C }}</ref>

<ref name=est-ohm>In the notation of Estes and Ohm for rings that are both subrings of <math>\Q</math> and overrings of <math>\Z</math>, the dyadic rationals are the ring <math>\Z_{\{2\}}</math>. See section 7 of {{citation
 | last1 = Estes | first1 = Dennis
 | last2 = Ohm | first2 = Jack
 | doi = 10.1016/0021-8693(67)90075-0 | doi-access = free
 | journal = [[Journal of Algebra]]
 | mr = 217052
 | pages = 343–362
 | title = Stable range in commutative rings
 | url = https://core.ac.uk/download/pdf/82241374.pdf
 | volume = 7
 | year = 1967| issue = 3
 }}</ref>

<ref name=eri-niv-xu>{{citation
 | last1 = Erickson | first1 = Jeff
 | last2 = Nivasch | first2 = Gabriel
 | last3 = Xu | first3 = Junyan
 | arxiv = 2003.14342
 | contribution = Fusible numbers and Peano arithmetic
 | contribution-url = https://jeffe.cs.illinois.edu/pubs/fusible.html
 | date = June 2021
 | doi = 10.1109/lics52264.2021.9470703
 | pages = 1–13
 | publisher = IEEE
 | title = Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2021)| isbn = 978-1-6654-4895-6
 | s2cid = 214727767
 }}</ref>

<ref name=fer-fer>{{citation
 | last1 = Fernandes | first1 = António M.
 | last2 = Ferreira | first2 = Fernando
 | contribution = Basic applications of weak König's lemma in feasible analysis
 | contribution-url = https://webpages.ciencias.ulisboa.pt/~fjferreira/basic.pdf
 | mr = 2185433
 | pages = 175–188
 | publisher = Association for Symbolic Logic | location = La Jolla, California
 | series = Lecture Notes in Logic
 | title = Reverse Mathematics 2001
 | volume = 21
 | year = 2005}}</ref>

<ref name=flanigan>{{citation
 | last = Flanigan | first = J. A.
 | doi = 10.1007/BF01771244
 | issue = 1
 | journal = International Journal of Game Theory
 | mr = 665515
 | pages = 21–25
 | title = A complete analysis of black-white Hackendot
 | volume = 11
 | year = 1982| s2cid = 119964871
 }}</ref>

<ref name=girgensohn>{{citation
 | last = Girgensohn | first = Roland
 | doi = 10.1006/jmaa.1996.0370 | doi-access = free
 | issue = 1
 | journal = [[Journal of Mathematical Analysis and Applications]]
 | mr = 1412484
 | pages = 127–141
 | title = Constructing singular functions via Farey fractions
 | volume = 203
 | year = 1996}}</ref>

<ref name=hie-ton>{{citation
 | last1 = Hiebert | first1 = James
 | last2 = Tonnessen | first2 = Lowell H.
 | date = November 1978
 | doi = 10.2307/748774
 | issue = 5
 | journal = [[Journal for Research in Mathematics Education]]
 | jstor = 748774
 | pages = 374–378
 | title = Development of the fraction concept in two physical contexts: an exploratory investigation
 | volume = 9}}</ref>

<ref name=jon-pea>{{citation
 | last1 = Jones | first1 = Shelly M. | author1-link = Shelly M. Jones
 | last2 = Pearson | first2 = Dunn
 | date = May 2013
 | doi = 10.1177/1048371313486478
 | issue = 1
 | journal = [[General Music Today]]
 | pages = 18–23
 | title = Music: highly engaged students connect music to math
 | volume = 27| s2cid = 220604326 }}</ref>

<ref name=jvv>{{citation
 | last1 = Jerrum | first1 = Mark R. | author1-link = Mark Jerrum
 | last2 = Valiant | first2 = Leslie G. | author2-link = Leslie Valiant
 | last3 = Vazirani | first3 = Vijay V. | author3-link = Vijay Vazirani
 | doi = 10.1016/0304-3975(86)90174-X | doi-access = free
 | issue = 2–3
 | journal = [[Theoretical Computer Science (journal)|Theoretical Computer Science]]
 | mr = 855970
 | pages = 169–188
 | title = Random generation of combinatorial structures from a uniform distribution
 | volume = 43
 | year = 1986}}</ref>

<ref name=kac>{{citation
 | last = Kac | first = Mark | author-link = Mark Kac
 | mr = 0110114
 | pages = 2–3
 | publisher = John Wiley & Sons for the Mathematical Association of America | location = New York
 | series = [[Carus Mathematical Monographs]]
 | title = Statistical Independence in Probability, Analysis and Number Theory
 | volume = 12
 | year = 1959}}</ref>

<ref name=kirk-hwu>{{citation
 | last1 = Kirk | first1 = David B. | author1-link = David Kirk (scientist)
 | last2 = Hwu | first2 = Wen-mei W. | author2-link = Wen-mei Hwu
 | contribution = 7.2 Representable numbers
 | contribution-url = https://books.google.com/books?id=E0Uaag8qicUC&pg=PA155
 | edition = 2nd
 | isbn = 978-0-12-391418-7
 | pages = 155–159
 | publisher = Morgan Kaufmann
 | title = Programming Massively Parallel Processors: A Hands-on Approach
 | year = 2013}}</ref>

<ref name=kneusel>{{citation
 | last = Kneusel | first = Ronald T.
 | contribution = Chapter 6: Fixed-point numbers
 | doi = 10.1007/978-3-319-50508-4_6
 | edition = 2nd
 | pages = 183–214
 | publisher = Springer International Publishing
 | title = Numbers and Computers
 | year = 2017}}</ref>

<ref name=ko>{{citation
 | 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]]