Dyadic Rationals And Surreal Number Theory
IOSR Journal of Mathematics (IOSR-JM)e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 16, Issue 5 Ser. IV (Sep. – Oct. 2020), PP 35-43www.iosrjournals.orgDyadic Rationals and Surreal Number TheoryC. Avalos-RamosC.U.C.E.I.Universidad de Guadalajara, Guadalajara, Jalisco, MéxicoJ. A. Félix-AlgandarFacultad de Ciencias Físico-Matemáticas de la Universidad Autónoma de Sinaloa,80010, Culiacán, Sinaloa, México.J. A. NietoFacultad de Ciencias Físico-Matemáticas de la Universidad Autónoma de Sinaloa,80010, Culiacán, Sinaloa, México.AbstractWe establish a number of properties of the dyadic rational numbers associated with surreal number theory. Inparticular, we show that a two parameter function of dyadic rationals can give all the trees of n-days in surrealnumber --------------------------------------------Date of Submission: 07-10-2020Date of Acceptance: ---------------------------------------------I. IntroductionIn mathematics, from number theory history [1], one learns that historically, roughlyspeaking, the starting point was the natural numbers N and after a centuries of thoughevolution one ends up with the real numbersfrom which one constructs thedifferential and integral calculus. Surprisingly in 1973 Conway [2] (see also Ref. [3])developed the surreal numbers structurewhich contains no only the real numbers ,but also the hypereals and other numerical structures.Consider the set(1)and callandthe left and right sets ofdefined in terms of two axioms:, respectively. Surreal numbers areAxiom 1. Every surreal number corresponds to two setscreated numbers, such that no member of the left setany memberof the right set.andof previouslyis greater or equal toLet us denote by the symbolthe notion of no greater or equal to. So the axiomestablishes that ifis a surreal number then for eachandone has. This isdenoted by.DOI: 10.9790/5728-1605043543www.iosrjournals.org35 Page
Dyadic Rationals and Surreal Number TheoryAxiom 2. One numberis less than or equal to another numberif and only if the two conditionsandare satisfied.This can be simplified by saying thatif and only ifand.Observe that Conway definition relies in an inductive method; before a surreal numberis introduced one needs to know the two setsandof surreal numbers. UsingConway algorithm one finds that at the -day one obtainsnumbers, all ofwhich are of form(2)whereis an integer andis a natural number. Of course, the numbers (2) aredyadic rationals which are dense in the real . It is also possible to show that the realnumbersare contained in the surreals(see Ref. [2,3] for details). Of course, insome sense the prove relies on the fact that the dyadic numbers (2) are dense in thereal .In 1986, Gonshor [4] introduced a different but equivalent definition of surreal numbers.II. Dyadic numbers in the surreal number theoryAs we mentioned earlier, in [4] Gonshor provided a surreal number definition equivalentto the one given by Conway; in this note we will work with the Gonshor's definition,so we begin by recalling it.Definition 2.1 A surreal number is a functioninto the setfrom initial segment of the ordinals.For instance, ifis the function so that,,,thenis the surreal number. In the Gonshor approach oneobtains the sequence:-day(3)DOI: 10.9790/5728-1605043543www.iosrjournals.org36 Page
Dyadic Rationals and Surreal Number Theoryin the -day(4)and -day(5)respectively.Here, we would like to propose that the different dyadic numbers in the surrealnumber theory can be obtained from the two parameter function:(6)Here,(6), with,and thereforeand, becomes. The positive sector of(7)while the negative sector, withand therefore, is given by(8)DOI: 10.9790/5728-1605043543www.iosrjournals.org37 Page
Dyadic Rationals and Surreal Number TheoryObserve that(9)Moreover, it is worth mentioning that Gonshor [4] derived the formula(10)which corresponds to (III) in (6).Example 2.2 Let us consider the Gonshor surreal number. One gets(11)By defining the orderif, whereis the first place whereanddiffer and the convention, it is possible to show that the Conwayand Gonshor definitions of surreal numbers are equivalent (see Ref. [4] for details).Let us focus in (7) withand thereforeexplicitly asNotice that according to (6) one always has. Also, we write. The first thingthat one observe is that (I) in (7) and (8) gives the integer numbersrespectively. By completeness one setsand,. While (II) provides with thedyadic rationalswhereis an odd element in the rationalsand. So,this suggests that both integerand rationalnumbers are contained in the surrealnumbers .Assume. In this case (7) becomes(12)This implies that from (I) and (II) one gets(III) one obtainsDOI: 10.9790/5728-1605043543,,and fromand so on. Sincewww.iosrjournals.org38 Page
Dyadic Rationals and Surreal Number Theoryonealsohasand,and so on.There must be many interesting combinations between (12) and (7) (and (8), butperhaps one of the most attractive isProposition 2.3 The functionsandare related by(13)This means that the tree(and) plays the role of a mainbuilding block; any other treewithcan be obtained from (13).Surprisingly,has been studied in the context of Zeno algorithm [5],Tompson group [6], Minkowski's question mark function [7] among others. In somesense ifwere added toand (3) was used the surreal numbersterms of dyadic rationals could be discovered for another routes, different than gametheory [2].Another interesting aspect of the tree structureandis that one can derive, in an alternative way, how many numbers are createdin the -day. It is worth to mention that this notion of “day” is used by themathematicians, in spire of their development of surreal numbers theory is consideredonly in the mathematical context. First, let us use Gonshor formalism to answer thisquestion. In the -day one starts with the numberand in the -day the numbersandnamelyare created, namelyand,. While in the,-daynumbers are created,, and so on.First, we shall need the propositionProposition 2.4 The identity(14)holds.Proof. By induction one assumes that (14) holds for an integerholds for. Thus, one needs to prove thatDOI: 10.9790/5728-1605043543www.iosrjournals.organd proves that also39 Page
Dyadic Rationals and Surreal Number Theory(15)is true. But (15) implies that(16)For assumption (14) holds and therefore (16) becomes(17)which is an identity.Proposition 2.5The total number of surreal numbers created at the-day are(18)Proof.The seriesone sees that (18) holds.determines . So, from the identity (14)Remark 2.6 Since the functionis two a parameter functionand , ifone fixesand changeone moves vertically producing the corresponding tree, assense. While if one fixesand changeone is moving horizontally. In thisdetermines the day parameter used by the mathematician.Example 2.7 Let us set. From (7) one obtains, and the corresponding negatives. So, in thenumbers and so one discovers the seriesobtains with Gonshor approach.Thenumbers.,,-day we havewhich is what one-day is defined as the limit when the surreal numbers reproduce the realDOI: 10.9790/5728-1605043543www.iosrjournals.org40 Page
Dyadic Rationals and Surreal Number TheoryProposition 2.8 In the-day the treetake values in the interval(19)over the real.Proof. Let us first proof that. From III in (7), one has(20)withand. The maximum of (20) is obtained when one takes only thepositive number value in each term. In this case (20) becomes(21)which leads to(22)But, sincewhen, one hasNow the minimum value ofnegative values (20). So, one has.is obtained when one takes all the(23)This implies(24)DOI: 10.9790/5728-1605043543www.iosrjournals.org41 Page
Dyadic Rationals and Surreal Number TheoryThis means thatwhensees that the proposition is verified.Corollary 2.9 In the. Since-day the functiononetakes values in the interval(25)over the realProof.Accordingto(13)oneandsincehas.one sees that. On the other hand,one learns thatwhichSinceand thereforemeansthat.Thisprovethat. The other part of the proof follows from the fact that.A connection between oriented matroid theory [8] (see also Refs. [9]-[15] andreferences therein) and surreal number theory has been developed [16]. So, one mayexpect that, in the context of surreal number theory, the mathematical notions of thisarticle may be useful for further developing of oriented matroid theory .AcknowledgmentsJ.A. Nieto would like to thank the Mathematical, Computational & Modeling SciencesCenter of the Arizona State University where part of this work was developed. Thiswork was partially supported by PROFAPI-UAS/2013.References[1] O. Ore, Number Theory and its History (McGraw Hill Book Company, 1948).[2] J. H. Conway, On Number and Games, London Mathematical Society Monographs(Academic Press, 1976).[3] D. E. Knuth,Surreal Numbers: How Two Ex-Students Turned on To PureMathematics and Found Total Happiness}: A Mathematical Novelette (AddisonWesley Publishing Co, 1974).DOI: 10.9790/5728-1605043543www.iosrjournals.org42 Page
Dyadic Rationals and Surreal Number Theory[4] H. Gonshor, An Introduction to the Theory of Surreal Numbers , LondonMathematical Society Lectures Notes Series, Vol. 110, (Cambridge Univ. Press,1986).[5] B. Hayes, Wagering with Zeno, American Scientist 96, 194 (2008).[6] J. W. Cannon and W. J. Floyd, Notices of the AMS 58, 1112 (2011).[7] L. Vepstas, The Minkowski Question Mark, GL(2;Z) and the Modular Group ,(2004).[8] A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. M. Ziegler, OrientedMatroids, (Cambridge University Press, Cambridge, 1993).[9] J. A. Nieto, Adv. Theor. Math. Phys. 8, 177 (2004); arXiv: hep-th/0310071.[10] J. A. Nieto, Adv. Theor. Math. Phys. 10, 747 (2006), arXiv: hep-th/0506106.[11] J. A. Nieto, J. Math. Phys. 45, 285 (2004); arXiv: hep-th/0212100.[12] J. A. Nieto, Nucl. Phys. B. 883, 350 (2014); arXiv:1402.6998 [hep-th].[13] J. A. Nieto and M. C. Marín, J. Math. Phys. 41, 7997 (2000); hep-th/0005117.[14] J. A. Nieto, Phys. Lett. B. 718, 1543 (2013); e-Print: arXiv:1210.0928 [hep-th].[15] J. A. Nieto, Phys. Lett. B. 692, 43 (2010); e-Print: arXiv:1004.5372 [hep-th].[16] J. A. Nieto, Front. Phys. 6 (2018) (article 106).DOI: 10.9790/5728-1605043543www.iosrjournals.org43 Page
numbers are contained in the surreals (see Ref. [2,3] for details). Of course, in some sense the prove relies on the fact that the dyadic numbers (2) are dense in the real . In 1986, Gonshor [4] introduced a different but equivalent definition of surreal numbers. II. Dyadic numbe
1 What Are Surreal Numbers? 1.1 Conway’s Two Rules As the stone states, every surreal number is created on a certain day and corresponds to two sets of numbers. For a surreal number, x, we write x fX LjX Rgand call X L and X R the left and right set of x, respectively. In this sect
the class of surreal numbers, the applications of surreal numbers to combinatorial game theory, as well as on some objects inspired by surreal numbers such as pseudo numbers. We also provide a brief introduction to the Field On 2 and its properties. 1. A Brief History and Introduction Surr
2.1 Conway’s Surreal Numbers A richly structured special class of two-player games are John H. Conway’s surreal numbers1 [Con01, Knu74, Gon86, All87], a vast generalization of the real and ordinal number systems. Basically, a surreal number {L R} is the “simples
Surreal numbers were discovered (or created?) in the 1970s by J. H. Con-way [23] and popularized by M. Gardner, and by D. E. Knuth [55] who coined the term \surreal number". The surreal numbers form a proper class containing all reals
of surreal numbers: Two recursive definitions that lead to an incredible wealth of structure. But let us explore things a bit further, and let us start by taking a closer look at the definitions. Definition 1 says that a surreal number is a pair of sets of sur
surreal.stanford.edu Abstract: Reproducibility has been a significant challenge in deep reinforcement learning and robotics research. Open-source frameworks and standardized bench-marks can serve an integral role in rigorous evaluation and reproducible research. We introduce SURREAL
Mar 08, 2021 · Surreal numbers with algebraic structure. Surreal numbers with commutative algebraic operations recursively defined. Conway’s original recursive definition (1976): Dedekind Von Neumann Conway’s surreals M. Ma
appointment. If you cancel: 25.1. any time up to 24 hours before the appointment you will receive a full refund; and 25.2. within 24 hours of the appointment, you will not receive any refund. 26. Refunds will only be paid to the cardholder or person who made the original payment and will be made through the same means as payment was made. 27 .
