291:
1593:, 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, 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
454:, fractional numbers arising from halving and repeated halving are among the earliest forms of fractions to develop. This stage of development of the concept of fractions has been called "algorithmic halving". 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.
191:
1738:
1723:
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960:
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22:
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937:{\displaystyle {\begin{aligned}{\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)}}}\\{\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)}}}\\{\frac {a}{2^{b}}}\cdot {\frac {c}{2^{d}}}&={\frac {ac}{2^{b+d}}}\end{aligned}}}
279:
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
64:
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
239:
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".
1900:
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. This number system is foundational to
1175:, is a dyadic rational. 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.
434:), 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, although
555:
2745:
Ambos-Spies, Klaus; Zheng, Xizhong (2019), "On the differences and sums of strongly computably enumerable real numbers", in Manea, Florin; Martin, Barnaby; Paulusma, Daniël; Primiero, Giuseppe (eds.),
2454:
Miller, Heather M.-L. (2013), "Weighty matters: evidence for unity and regional diversity from the Indus civilization weights", in
Abraham, Shinu Anna; Gullapalli, Praveena; Raczek, Teresa P.;
5088:
3402:
5380:
3819:
1651:
1252:
995:
1300:
1126:
264:, which also use powers of two implicitly in the majority of cases. Because of the simplicity of computing with dyadic rationals, they are also used for exact real computing using
3614:
1994:
1785:
of these wavelets is non-smooth. Similarly, the dyadic rationals parameterize the discontinuities in the boundary between stable and unstable points in the parameter space of the
4398:
3319:
5333:
5296:
2328:
2100:
4365:
4336:
1074:
1497:. 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
215:
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
5042:
3579:
3557:
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3347:
3249:
1564:
119:
2062:
2181:
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1453:
1418:
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260:, are called its representable numbers. For most floating-point representations, the representable numbers are a subset of the dyadic rationals. The same is true for
256:
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
1936:
1883:
1488:
2291:
2020:
500:
57:. Dyadic rationals also have applications in weights and measures, musical time signatures, and early mathematics education. They can accurately approximate any
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2120:
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1685:, 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
4064:
2934:
Hiebert, James; Tonnessen, Lowell H. (November 1978), "Development of the fraction concept in two physical contexts: an exploratory investigation",
1566:, meaning that it can be generated by evaluating polynomials with integer coefficients, at the argument 1/2. As a ring, the dyadic rationals are a
1818:. The same group can also be represented by an action on rooted binary trees, or by an action on the dyadic rationals within the unit interval.
235:
are also dyadic. The ancient
Egyptians used dyadic rationals in measurement, with denominators up to 64. Similarly, systems of weights from the
4603:
5485:
2975:
2936:
951:
one dyadic rational by another is not necessarily a dyadic rational. For instance, 1 and 3 are both dyadic rational numbers, but 1/3 is not.
1084:
Real numbers with no unusually-accurate dyadic rational approximations. The red circles surround numbers that are approximated within error
4226:
4100:
Cvitanović, Predrag; Gunaratne, Gemunu H.; Procaccia, Itamar (1988), "Topological and metric properties of Hénon-type strange attractors",
438:
have also been used. The numeric value of the signature, interpreted as a fraction, describes the length of a measure as a fraction of a
4822:
4158:
2747:
Computing with
Foresight and Industry: 15th Conference on Computability in Europe, CiE 2019, Durham, UK, July 15–19, 2019, Proceedings
4570:
2905:
Yanakiev, Ivan K. (2020), "Mathematical devices in aid of music theory, composition, and performance", in
Bozhikova, Milena (ed.),
199:
1505:, giving two different representations for 3/4. The dyadic rationals are the only numbers whose binary expansions are not unique.
1385:
subset of the real numbers, this error bound is within a constant factor of optimal: for these numbers, there is no approximation
3038:
Uiterwijk, Jos W. H. M.; Barton, Michael (2015), "New results for
Domineering from combinatorial game theory endgame databases",
3040:
2796:
2604:
3523:
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as a type of fractional number that many computers can manipulate directly. In particular, as a data type used by computers,
129:
4942:
3099:
1689:. It is called the dyadic solenoid, and is isomorphic to the topological product of the real numbers and 2-adic numbers,
1665:
Considering only the addition and subtraction operations of the dyadic rationals gives them the structure of an additive
294:
4596:
1518:
Because they are closed under addition, subtraction, and multiplication, but not division, the dyadic rationals are a
219:
is customarily subdivided in dyadic rationals rather than using a decimal subdivision. The customary divisions of the
5478:
5430:
4432:
4295:
4039:
3984:
3895:
3217:
2672:
2522:
1455:. The existence of accurate dyadic approximations can be expressed by saying that the set of all dyadic rationals is
5507:
5052:
3372:
442:. Its numerator describes the number of beats per measure, and the denominator describes the length of each beat.
1759:
4263:, Lecture Notes in Logic, vol. 21, La Jolla, California: Association for Symbolic Logic, pp. 175–188,
5347:
4815:
3774:
1885:
that approximates the given real number. Defining real numbers in this way allows many of the basic results of
1754:
Because they are a dense subset of the real numbers, the dyadic rationals, with their numeric ordering, form a
1606:
970:
1206:
4589:
4214:
3879:
3460:
3416:
2073:
3488:, Wavelet Analysis and Its Applications, vol. 2, Boston, Massachusetts: Academic Press, pp. 3–13,
2659:, Progress in Theoretical Computer Science, Boston, Massachusetts: Birkhäuser Boston, Inc., pp. 41–43,
1279:
5471:
5338:
3663:
Lucyshyn-Wright, Rory B. B. (2018), "Convex spaces, affine spaces, and commutants for algebraic theories",
3463:, vol. 12, New York: John Wiley & Sons for the Mathematical Association of America, pp. 2–3,
2973:; Sawada, Daiyo (November 1983), "Partitioning: the emergence of rational number ideas in young children",
2294:
1905:, and dyadic rationals arise naturally in this theory as the set of values of certain combinatorial games.
1803:
to itself that have power-of-2 slopes and dyadic-rational breakpoints forms a group under the operation of
1087:
3584:
2406:
Note that binary measures (2, 4, 8, 16) are very common indeed. This is particularly obvious with volumes.
1941:
5188:
2420:
290:
4370:
3289:
4948:
5309:
5272:
4963:
2300:
2078:
1463:. More strongly, this set is uniformly dense, in the sense that the dyadic rationals with denominator
49:. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in
5512:
4808:
4769:
4341:
4312:
1902:
1793:
253:
236:
3097:
Krebbers, Robbert; Spitters, Bas (2013), "Type classes for efficient exact real arithmetic in Coq",
1035:
5423:
5226:
5176:
3720:
1706:
544:
of any two dyadic rationals produces another dyadic rational, according to the following formulas:
5025:
3562:
3540:
3352:
3332:
3234:
1529:
164:
of 2-adic numbers. Functions from natural numbers to dyadic rationals have been used to formalize
84:
5235:
4969:
4928:
4757:
1890:
1812:
1674:
435:
297:
2025:
522:
is a power of two. Another equivalent way of defining the dyadic rationals is that they are the
5522:
5392:
5243:
5194:
4975:
4639:
2545:
2145:
1702:
1575:
948:
261:
125:
2878:
2553:
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2355:
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can be arbitrarily closely approximated by dyadic rationals. In particular, for a real number
4862:
4253:
3254:
3207:
3010:
2921:
2654:
2502:
2381:
1886:
1673:
is a method for understanding abelian groups by constructing dual groups, whose elements are
1423:
1388:
1329:
1131:
1000:
527:
165:
124:
In advanced mathematics, the dyadic rational numbers are central to the constructions of the
54:
4026:, Texts and Readings in Mathematics, vol. 12, Berlin: Springer-Verlag, pp. 77–86,
2703:
Zheng, Xizhong; Rettinger, Robert (2004), "Weak computability and representation of reals",
5517:
5116:
4990:
4752:
4495:
4461:
4305:
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4268:
4239:
4189:
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4111:
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4049:
3961:
3905:
3759:
3739:
3694:
3650:
3493:
3468:
3439:
3182:
3130:
3071:
2819:
2762:
2724:
2682:
2627:
2532:
2498:
2429:
1915:
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1835:
1804:
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4522:
Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in
Computer Science (LICS 2021)
8:
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3326:
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2494:
2270:
1999:
1827:
1523:
477:
265:
257:
169:
74:
4115:
3743:
2749:, Lecture Notes in Computer Science, vol. 11558, Cham: Springer, pp. 310–322,
2433:
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3134:
3108:
3075:
3049:
2992:
2953:
2857:
2766:
2728:
2686:
2337:
utilizes the dyadic fractions for constructing the separating function from the lemma.
2334:
2250:
2230:
2206:
2186:
2125:
2105:
1841:
1778:
1744:
1678:
1670:
1519:
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1309:
1259:
1185:
505:
450:
In theories of childhood development of the concept of a fraction based on the work of
313:
133:
66:
5452:
5249:
5014:
4955:
4730:
4547:
4503:
4291:
4135:
4102:
4035:
3891:
3767:
3641:
3519:
3213:
3190:
3138:
3016:
2910:
2886:
2861:
2810:
2794:(1986), "Random generation of combinatorial structures from a uniform distribution",
2770:
2668:
2561:
2518:
2471:
2395:
2361:
1763:
1694:
1686:
269:
153:
4197:
3969:
2838:; Pearson, Dunn (May 2013), "Music: highly engaged students connect music to math",
2732:
2690:
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5444:
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4019:
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3702:
3682:
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3118:
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3059:
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2849:
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2750:
2712:
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2613:
2585:
2510:
2463:
2437:
2387:
2224:
964:
386:
249:
50:
4517:
4062:
Girgensohn, Roland (1996), "Constructing singular functions via Farey fractions",
3151:
O'Connor, Russell (2007), "A monadic, functional implementation of real numbers",
3122:
1782:
190:
5220:
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5008:
4831:
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4045:
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3901:
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3755:
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3435:
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3126:
3067:
2835:
2815:
2758:
2754:
2720:
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2623:
2528:
1808:
1594:
463:
308:
276:
161:
137:
78:
4031:
2589:
1838:
to dyadic rationals, where the value of one of these functions for the argument
1654:
141:
5264:
3619:
3092:
2970:
2791:
2787:
1909:
1897:
1690:
1682:
1601:
1303:
541:
382:
149:
145:
4181:
3887:
3751:
3686:
3515:
3430:
3174:
3063:
2664:
2618:
2514:
2391:
5501:
5405:
5301:
4916:
4789:
4710:
4123:
2853:
2455:
1800:
1796:
1666:
1590:
1494:
959:
157:
3414:
Nilsson, Johan (2009), "On numbers badly approximable by dyadic rationals",
5437:
5212:
5108:
4857:
4847:
4740:
4735:
4705:
4078:
2716:
2460:
Connections and
Complexity: New Approaches to the Archaeology of South Asia
2418:
Curtis, Lorenzo J. (1978), "Concept of the exponential law prior to 1900",
1815:
1579:
1172:
1167:
outside the circles, all dyadic rational approximations have larger errors.
1080:
471:
280:
use a random generation process whose time is itself random and unbounded.
203:
46:
4139:
3982:
Nadler, S. B. Jr. (1973), "The indecomposability of the dyadic solenoid",
2602:
van der Hoeven, Joris (2006), "Computations with effective real numbers",
2467:
1786:
1722:
1712:
5417:
5128:
5002:
4884:
4560:
4430:
Mauldon, J. G. (1978), "Num, a variant of Nim with no first-player win",
2783:
2549:
1831:
1774:
between the set of all rational numbers and the set of dyadic rationals.
1755:
1586:
1179:
537:
523:
451:
58:
42:
4474:
Flanigan, J. A. (1982), "A complete analysis of black-white
Hackendot",
1737:
5182:
4784:
4779:
4581:
4487:
4453:
4005:
3510:, Undergraduate Texts in Mathematics, New York: Springer, p. 186,
3322:
2069:
2065:
1306:
that rounds its argument down to an integer. These numbers approximate
1164:
439:
389:
traditionally are written in a form resembling fractions (for example:
232:
2996:
2957:
5142:
5047:
4172:
3944:
3537:
In the notation of Estes and Ohm for rings that are both subrings of
2584:(2nd ed.), Springer International Publishing, pp. 183–214,
1771:
1460:
1456:
4445:
4156:
in groups of piecewise linear homeomorphisms of the unit interval",
4017:
3997:
2441:
1076:
The height of the pink region above each approximation is its error.
5136:
5122:
4800:
4647:
4612:
4530:
4018:
Bhattacharjee, Meenaxi; Macpherson, Dugald; Möller, Rögnvaldur G.;
3677:
3457:
Statistical
Independence in Probability, Analysis and Number Theory
3452:
3165:
3054:
2988:
2949:
1698:
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1493:
The dyadic rationals are precisely those numbers possessing finite
533:
38:
3734:
3113:
21:
5020:
4904:
4842:
4747:
4643:
4626:
1697:
of the dyadic rationals into this product. It is an example of a
1567:
1382:
467:
70:
997:), found by rounding to the nearest smaller integer multiple of
4254:"Basic applications of weak König's lemma in feasible analysis"
220:
3012:
Motivating Mathematics: Engaging Teachers And Engaged Students
2558:
Programming Massively Parallel Processors: A Hands-on Approach
1758:. As with any two unbounded countable dense linear orders, by
4616:
3718:
Manners, Freddie (2015), "A solution to the pyjama problem",
3212:, Geometry and Computing, vol. 6, Springer, p. 51,
3091:
Equivalent formulas to these, written in the language of the
2580:
Kneusel, Ronald T. (2017), "Chapter 6: Fixed-point numbers",
1912:
are a subset of the dyadic rationals, the closure of the set
224:
4099:
3882:, vol. 198, New York: Springer-Verlag, pp. 40–43,
4564:
4152:
Brin, Matthew G. (1999), "The ubiquity of Thompson's group
228:
216:
4516:
Erickson, Jeff; Nivasch, Gabriel; Xu, Junyan (June 2021),
3484:
Pollen, David (1992), "Daubechies' scaling function on ",
3921:
de Cornulier, Yves; Guyot, Luc; Pitsch, Wolfgang (2007),
1597:
of 2-adic numbers, but this decomposition is not unique.
296:
Audio playback is not supported in your browser. You can
1777:
The dyadic rationals play a key role in the analysis of
3920:
2507:
Wavelet Analysis: The Scalable Structure of Information
1713:
Functions with dyadic rationals as distinguished points
4559:
4379:
4346:
4317:
4290:(Second ed.), Natick, Massachusetts: A K Peters,
3383:
1747:, showing points of non-smoothness at dyadic rationals
1619:
1542:
1209:
1092:
156:
to the rational numbers; they form a subsystem of the
97:
5350:
5312:
5275:
5055:
5028:
4406:
4373:
4344:
4315:
3862:
3840:
3777:
3587:
3565:
3543:
3375:
3369:
approaches zero. The illustration shows this set for
3355:
3335:
3292:
3257:
3251:, the set of real numbers that have no approximation
3237:
3209:
Analysis and Design of Univariate Subdivision Schemes
2782:
2303:
2273:
2253:
2233:
2209:
2189:
2148:
2128:
2108:
2081:
2028:
2002:
1944:
1918:
1864:
1844:
1609:
1532:
1469:
1426:
1391:
1367:
1332:
1312:
1282:
1262:
1188:
1134:
1090:
1038:
1003:
973:
553:
508:
480:
87:
1589:, the dyadic rational numbers form a subring of the
2560:(2nd ed.), Morgan Kaufmann, pp. 155–159,
1653:of the dyadic rationals by the integers) forms the
5374:
5327:
5290:
5082:
5036:
4412:
4392:
4359:
4330:
4252:Fernandes, António M.; Ferreira, Fernando (2005),
3868:
3846:
3813:
3608:
3573:
3551:
3396:
3361:
3341:
3313:
3278:
3243:
2883:The NPR Listener's Encyclopedia of Classical Music
2322:
2285:
2259:
2239:
2215:
2195:
2175:
2134:
2114:
2094:
2056:
2014:
1988:
1930:
1877:
1850:
1645:
1558:
1482:
1447:
1412:
1373:
1361:, which can be made arbitrarily small by choosing
1353:
1318:
1294:
1268:
1246:
1194:
1155:
1120:
1068:
1024:
989:
936:
514:
494:
113:
4515:
4251:
4065:Journal of Mathematical Analysis and Applications
2909:, Cambridge Scholars Publishing, pp. 35–62,
2122:the smallest fusible number that is greater than
1574:of the integers. Algebraically, this ring is the
5499:
3096:
2933:
2744:
2357:How Mathematics Happened: The First 50,000 Years
832:
803:
766:
682:
653:
616:
268:, and are central to some theoretical models of
202:Kitchen weights measuring dyadic fractions of a
3923:"On the isolated points in the space of groups"
3662:
3037:
160:as well as of the reals, and can represent the
3916:
3914:
3834:(2000), "5.4 Fractional and integral parts of
2702:
2601:
2493:
1821:
1526:. The ring of dyadic rationals may be denoted
5479:
4816:
4597:
4309:; for the dyadic rationals, see "The numbers
3231:More precisely, for small positive values of
2976:Journal for Research in Mathematics Education
2969:
2937:Journal for Research in Mathematics Education
2509:, New York: Springer-Verlag, pp. 17–18,
1811:, the first known example of an infinite but
457:
5083:{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }
4227:Notices of the American Mathematical Society
3601:
3595:
3397:{\displaystyle \varepsilon ={\tfrac {1}{6}}}
3150:
2834:
1925:
1919:
1289:
1283:
1226:
1210:
1202:, consider the dyadic rationals of the form
502:in simplest terms is a dyadic rational when
25:Dyadic rationals in the interval from 0 to 1
4209:
4093:
3911:
3153:Mathematical Structures in Computer Science
1889:to be proven within a restricted theory of
1600:Addition of dyadic rationals modulo 1 (the
1578:of the integers with respect to the set of
445:
53:because they are the only ones with finite
5486:
5472:
4823:
4809:
4604:
4590:
4509:
4159:Journal of the London Mathematical Society
4061:
4055:
3713:
3711:
3033:
3031:
2830:
2828:
2503:"2.2.1: Digital computers and measurement"
1508:
5375:{\displaystyle \mathbb {Z} (p^{\infty })}
5352:
5315:
5278:
5076:
5063:
5030:
4571:On-Line Encyclopedia of Integer Sequences
4529:
4377:
4171:
4077:
3943:
3814:{\displaystyle {\widehat {\mathbb {Z} }}}
3783:
3733:
3676:
3640:
3590:
3567:
3545:
3479:
3477:
3429:
3286:with error smaller than a constant times
3164:
3112:
3095:interactive theorem prover, are given by
3085:
3053:
2872:
2870:
2809:
2617:
1731:maps rational numbers to dyadic rationals
1646:{\displaystyle \mathbb {Z} /\mathbb {Z} }
1639:
1611:
1534:
1420:with error smaller than a constant times
1247:{\textstyle \lfloor 2^{i}x\rfloor /2^{i}}
990:{\displaystyle {\sqrt {2}}\approx 1.4142}
89:
4611:
4473:
4467:
3617:
3201:
3199:
2904:
2898:
1766:to the rational numbers. In this case,
1079:
958:
954:
20:
18:Fraction with denominator a power of two
4553:
4429:
4423:
4245:
4203:
3717:
3708:
3531:
3413:
3407:
3144:
3028:
2963:
2927:
2825:
2579:
2544:
2538:
2297:for large numbers) already larger than
1834:is to represent them as functions from
1490:are uniformly spaced on the real line.
37:is a number that can be expressed as a
5500:
4518:"Fusible numbers and Peano arithmetic"
4280:
4274:
3981:
3975:
3830:
3824:
3505:
3499:
3483:
3474:
2876:
2867:
2573:
2462:, Left Coast Press, pp. 161–177,
2453:
2417:
2379:
2353:
1858:is a dyadic rational with denominator
1513:
1295:{\displaystyle \lfloor \dots \rfloor }
963:Dyadic rational approximations to the
4804:
4585:
3508:An Invitation to Abstract Mathematics
3205:
3196:
3008:
3002:
2487:
2447:
2411:
2386:, Springer International Publishing,
2373:
2347:
1121:{\displaystyle {\tfrac {1}{6}}/2^{i}}
4943:Free product of associative algebras
4830:
4476:International Journal of Game Theory
4151:
4145:
4024:Notes on Infinite Permutation Groups
4011:
3771:; see section 6.2.1, "A model case:
3656:
3609:{\displaystyle \mathbb {Z} _{\{2\}}}
3581:, the dyadic rationals are the ring
3015:, World Scientific, pp. 32–33,
2648:
2646:
2644:
2642:
2640:
2638:
2636:
1989:{\displaystyle x,y\mapsto (x+y+1)/2}
1585:As well as forming a subring of the
3620:"Stable range in commutative rings"
3451:
3445:
3100:Logical Methods in Computer Science
2907:Music between Ontology and Ideology
2885:, Workman Publishing, p. 873,
2776:
2738:
2656:Complexity Theory of Real Functions
1893:called "feasible analysis" (BTFA).
1681:to the multiplicative group of the
13:
5364:
4393:{\displaystyle 1\,{\tfrac {1}{2}}}
3314:{\displaystyle \varepsilon /2^{i}}
2696:
2652:
2595:
2247:grows so rapidly as a function of
1768:Minkowski's question-mark function
1729:Minkowski's question-mark function
1660:
130:Minkowski's question-mark function
14:
5534:
5431:Noncommutative algebraic geometry
4433:The American Mathematical Monthly
3985:The American Mathematical Monthly
3618:Estes, Dennis; Ohm, Jack (1967),
2633:
2360:, Prometheus Books, p. 148,
1781:, as the set of points where the
1326:from below to within an error of
180:
5328:{\displaystyle \mathbb {Q} _{p}}
5291:{\displaystyle \mathbb {Z} _{p}}
4656:
4625:
2323:{\displaystyle 2\uparrow ^{9}16}
2095:{\displaystyle \varepsilon _{0}}
1736:
1721:
1570:of the rational numbers, and an
248:Dyadic rationals are central to
189:
4360:{\displaystyle {\tfrac {3}{4}}}
4331:{\displaystyle {\tfrac {1}{4}}}
3225:
1381:to be arbitrarily large. For a
243:
175:
5369:
5356:
4540:10.1109/lics52264.2021.9470703
3954:10.1016/j.jalgebra.2006.02.012
3801:
3787:
3665:Applied Categorical Structures
2308:
2044:
2030:
1975:
1957:
1954:
1830:, one way of constructing the
1630:
1615:
1553:
1538:
1069:{\displaystyle i=0,1,2,\dots }
847:
835:
818:
806:
781:
769:
697:
685:
668:
656:
631:
619:
108:
93:
1:
4219:"What is … Thompson's group?"
3880:Graduate Texts in Mathematics
3461:Carus Mathematical Monographs
3417:Israel Journal of Mathematics
2340:
1770:provides an order-preserving
1163:. For numbers in the fractal
466:that result from dividing an
258:IEEE floating-point datatypes
5037:{\displaystyle \mathbb {Z} }
4022:(1997), "Rational numbers",
3642:10.1016/0021-8693(67)90075-0
3574:{\displaystyle \mathbb {Z} }
3552:{\displaystyle \mathbb {Q} }
3362:{\displaystyle \varepsilon }
3342:{\displaystyle \varepsilon }
3244:{\displaystyle \varepsilon }
3041:Theoretical Computer Science
3009:Wells, David Graham (2015),
2811:10.1016/0304-3975(86)90174-X
2797:Theoretical Computer Science
2755:10.1007/978-3-030-22996-2_27
2705:Mathematical Logic Quarterly
2605:Theoretical Computer Science
1760:Cantor's isomorphism theorem
1559:{\displaystyle \mathbb {Z} }
293:
114:{\displaystyle \mathbb {Z} }
69:, lying between the ring of
7:
5189:Unique factorization domain
4032:10.1007/978-93-80250-91-5_9
2590:10.1007/978-3-319-50508-4_6
2554:"7.2 Representable numbers"
2421:American Journal of Physics
1822:Other related constructions
1762:, the dyadic rationals are
462:The dyadic numbers are the
283:
81:. This ring may be denoted
10:
5539:
4949:Tensor product of algebras
4561:Sloane, N. J. A.
2057:{\displaystyle |x-y|<1}
458:Definitions and arithmetic
436:non-dyadic time signatures
4838:
4696:
4665:
4654:
4632:
4623:
4182:10.1112/S0024610799007905
3888:10.1007/978-1-4757-3254-2
3752:10.1007/s00222-014-0571-7
3687:10.1007/s10485-017-9496-9
3516:10.1007/978-1-4614-6636-9
3431:10.1007/s11856-009-0042-9
3175:10.1017/S0960129506005871
3064:10.1016/j.tcs.2015.05.017
2877:Libbey, Theodore (2006),
2665:10.1007/978-1-4684-6802-1
2619:10.1016/j.tcs.2005.09.060
2515:10.1007/978-1-4612-0593-7
2392:10.1007/978-3-319-46831-0
2354:Rudman, Peter S. (2009),
2295:Knuth's up-arrow notation
2176:{\displaystyle n+1/2^{k}}
1903:combinatorial game theory
1171:Every integer, and every
237:Indus Valley civilisation
210:down to 1/64 lb (1/4 oz)
5227:Formal power series ring
5177:Integrally closed domain
4261:Reverse Mathematics 2001
4124:10.1103/PhysRevA.38.1503
3721:Inventiones Mathematicae
2854:10.1177/1048371313486478
1707:indecomposable continuum
526:that have a terminating
446:In mathematics education
318:showing time signatures
5508:Fractions (mathematics)
5236:Algebraic number theory
4929:Total ring of fractions
4565:"Sequence A188545"
4524:, IEEE, pp. 1–13,
4420:, and so on", pp. 10–12
3279:{\displaystyle n/2^{i}}
3206:Sabin, Malcolm (2010),
3123:10.2168/LMCS-9(1:1)2013
1891:second-order arithmetic
1677:of the original group,
1509:In advanced mathematics
1448:{\displaystyle 1/2^{i}}
1413:{\displaystyle n/2^{i}}
1354:{\displaystyle 1/2^{i}}
1276:can be any integer and
1156:{\displaystyle n/2^{i}}
1025:{\displaystyle 1/2^{i}}
947:However, the result of
298:download the audio file
5393:Noncommutative algebra
5376:
5329:
5292:
5244:Algebraic number field
5195:Principal ideal domain
5084:
5038:
4976:Frobenius endomorphism
4414:
4394:
4361:
4332:
4079:10.1006/jmaa.1996.0370
3870:
3848:
3815:
3610:
3575:
3553:
3398:
3363:
3343:
3315:
3280:
3245:
2717:10.1002/malq.200310110
2324:
2287:
2261:
2241:
2217:
2197:
2177:
2136:
2116:
2096:
2058:
2016:
1996:, restricted to pairs
1990:
1932:
1879:
1852:
1647:
1560:
1484:
1449:
1414:
1375:
1355:
1320:
1296:
1270:
1248:
1196:
1168:
1157:
1122:
1077:
1070:
1026:
991:
938:
516:
496:
254:floating-point numbers
115:
55:binary representations
26:
5377:
5330:
5293:
5085:
5039:
4863:Superparticular ratio
4415:
4395:
4362:
4333:
3871:
3849:
3816:
3611:
3576:
3554:
3506:Bajnok, Béla (2013),
3399:
3364:
3344:
3316:
3281:
3246:
2582:Numbers and Computers
2499:Wells, Raymond O. Jr.
2468:10.4324/9781315431857
2380:Barnes, John (2016),
2325:
2288:
2262:
2242:
2218:
2198:
2178:
2137:
2117:
2097:
2059:
2017:
1991:
1933:
1931:{\displaystyle \{0\}}
1887:mathematical analysis
1880:
1878:{\displaystyle 2^{i}}
1853:
1648:
1561:
1485:
1483:{\displaystyle 2^{i}}
1450:
1415:
1376:
1356:
1321:
1297:
1271:
1249:
1197:
1158:
1123:
1083:
1071:
1027:
992:
962:
955:Additional properties
939:
528:binary representation
517:
497:
262:fixed-point datatypes
166:mathematical analysis
116:
24:
5399:Noncommutative rings
5348:
5310:
5273:
5117:Non-associative ring
5053:
5026:
4983:Algebraic structures
4404:
4371:
4342:
4313:
4287:On Numbers and Games
3860:
3838:
3775:
3585:
3563:
3541:
3373:
3353:
3333:
3290:
3255:
3235:
2920:; see in particular
2495:Resnikoff, Howard L.
2481:; see in particular
2301:
2271:
2251:
2231:
2223:cannot be proven in
2207:
2187:
2146:
2126:
2106:
2079:
2026:
2000:
1942:
1938:under the operation
1916:
1862:
1842:
1805:function composition
1607:
1530:
1467:
1424:
1389:
1365:
1330:
1310:
1280:
1260:
1207:
1186:
1132:
1088:
1036:
1001:
971:
551:
506:
478:
474:. A rational number
152:. These numbers are
85:
5158:Commutative algebra
4997:Associative algebra
4879:Algebraic structure
4116:1988PhRvA..38.1503C
3744:2015InMat.202..239M
3616:. See section 7 of
3329:, as a function of
3327:Hausdorff dimension
2841:General Music Today
2434:1978AmJPh..46..896C
2333:The usual proof of
2286:{\displaystyle n=3}
2183:. The existence of
2102:. For each integer
2015:{\displaystyle x,y}
1828:reverse mathematics
1779:Daubechies wavelets
1679:group homomorphisms
1514:Algebraic structure
495:{\displaystyle p/q}
266:interval arithmetic
223:into half-gallons,
170:reverse mathematics
134:Daubechies wavelets
5412:Semiprimitive ring
5372:
5325:
5288:
5096:Related structures
5080:
5034:
4970:Inner automorphism
4956:Ring homomorphisms
4633:Division and ratio
4488:10.1007/BF01771244
4410:
4390:
4388:
4357:
4355:
4328:
4326:
3931:Journal of Algebra
3866:
3844:
3811:
3628:Journal of Algebra
3606:
3571:
3549:
3394:
3392:
3359:
3339:
3311:
3276:
3241:
2792:Vazirani, Vijay V.
2788:Valiant, Leslie G.
2653:Ko, Ker-I (1991),
2320:
2283:
2257:
2237:
2213:
2193:
2173:
2132:
2112:
2092:
2054:
2012:
1986:
1928:
1875:
1848:
1813:finitely presented
1745:Daubechies wavelet
1695:diagonal embedding
1671:Pontryagin duality
1643:
1628:
1556:
1551:
1480:
1445:
1410:
1371:
1351:
1316:
1292:
1266:
1244:
1192:
1169:
1153:
1118:
1101:
1078:
1066:
1022:
987:
934:
932:
512:
492:
314:The Rite of Spring
270:computable numbers
111:
106:
29:In mathematics, a
27:
5496:
5495:
5453:Geometric algebra
5164:Commutative rings
5015:Category of rings
4871:
4870:
4798:
4797:
4574:, OEIS Foundation
4413:{\displaystyle 3}
4387:
4354:
4325:
4162:, Second Series,
4103:Physical Review A
4020:Neumann, Peter M.
3869:{\displaystyle p}
3847:{\displaystyle p}
3808:
3559:and overrings of
3525:978-1-4614-6635-2
3391:
3349:, goes to one as
3022:978-1-78326-755-2
2916:978-1-5275-4758-2
2892:978-0-7611-2072-8
2567:978-0-12-391418-7
2477:978-1-59874-686-0
2401:978-3-319-46830-3
2367:978-1-61592-176-8
2260:{\displaystyle n}
2240:{\displaystyle k}
2216:{\displaystyle n}
2196:{\displaystyle k}
2135:{\displaystyle n}
2115:{\displaystyle n}
1851:{\displaystyle i}
1687:topological group
1627:
1550:
1495:binary expansions
1374:{\displaystyle i}
1319:{\displaystyle x}
1269:{\displaystyle i}
1195:{\displaystyle x}
1100:
979:
928:
893:
873:
852:
743:
723:
702:
593:
573:
515:{\displaystyle q}
302:
105:
5530:
5513:Rational numbers
5488:
5481:
5474:
5459:Operator algebra
5445:Clifford algebra
5381:
5379:
5378:
5373:
5368:
5367:
5355:
5334:
5332:
5331:
5326:
5324:
5323:
5318:
5297:
5295:
5294:
5289:
5287:
5286:
5281:
5259:Ring of integers
5253:
5250:Integers modulo
5201:Euclidean domain
5089:
5087:
5086:
5081:
5079:
5071:
5066:
5043:
5041:
5040:
5035:
5033:
4937:Product of rings
4923:Fractional ideal
4882:
4874:
4873:
4832:Rational numbers
4825:
4818:
4811:
4802:
4801:
4775:Musical interval
4688:
4687:
4685:
4684:
4681:
4678:
4660:
4659:
4629:
4606:
4599:
4592:
4583:
4582:
4576:
4575:
4557:
4551:
4550:
4533:
4513:
4507:
4506:
4471:
4465:
4464:
4427:
4421:
4419:
4417:
4416:
4411:
4399:
4397:
4396:
4391:
4389:
4380:
4366:
4364:
4363:
4358:
4356:
4347:
4337:
4335:
4334:
4329:
4327:
4318:
4308:
4278:
4272:
4271:
4258:
4249:
4243:
4242:
4234:(8): 1112–1113,
4223:
4207:
4201:
4200:
4175:
4155:
4149:
4143:
4142:
4110:(3): 1503–1520,
4106:, Third Series,
4097:
4091:
4090:
4081:
4059:
4053:
4052:
4015:
4009:
4008:
3979:
3973:
3972:
3947:
3927:
3918:
3909:
3908:
3875:
3873:
3872:
3867:
3854:-adic numbers",
3853:
3851:
3850:
3845:
3832:Robert, Alain M.
3828:
3822:
3820:
3818:
3817:
3812:
3810:
3809:
3804:
3797:
3786:
3780:
3770:
3737:
3715:
3706:
3705:
3680:
3660:
3654:
3653:
3644:
3624:
3615:
3613:
3612:
3607:
3605:
3604:
3593:
3580:
3578:
3577:
3572:
3570:
3558:
3556:
3555:
3550:
3548:
3535:
3529:
3528:
3503:
3497:
3496:
3481:
3472:
3471:
3449:
3443:
3442:
3433:
3411:
3405:
3403:
3401:
3400:
3395:
3393:
3384:
3368:
3366:
3365:
3360:
3348:
3346:
3345:
3340:
3320:
3318:
3317:
3312:
3310:
3309:
3300:
3285:
3283:
3282:
3277:
3275:
3274:
3265:
3250:
3248:
3247:
3242:
3229:
3223:
3222:
3203:
3194:
3193:
3168:
3148:
3142:
3141:
3116:
3089:
3083:
3082:
3057:
3035:
3026:
3025:
3006:
3000:
2999:
2967:
2961:
2960:
2931:
2925:
2919:
2902:
2896:
2895:
2879:"Time signature"
2874:
2865:
2864:
2836:Jones, Shelly M.
2832:
2823:
2822:
2813:
2804:(2–3): 169–188,
2780:
2774:
2773:
2742:
2736:
2735:
2711:(4–5): 431–442,
2700:
2694:
2693:
2650:
2631:
2630:
2621:
2599:
2593:
2592:
2577:
2571:
2570:
2542:
2536:
2535:
2491:
2485:
2480:
2451:
2445:
2444:
2415:
2409:
2408:
2377:
2371:
2370:
2351:
2329:
2327:
2326:
2321:
2316:
2315:
2292:
2290:
2289:
2284:
2266:
2264:
2263:
2258:
2246:
2244:
2243:
2238:
2225:Peano arithmetic
2222:
2220:
2219:
2214:
2202:
2200:
2199:
2194:
2182:
2180:
2179:
2174:
2172:
2171:
2162:
2141:
2139:
2138:
2133:
2121:
2119:
2118:
2113:
2101:
2099:
2098:
2093:
2091:
2090:
2063:
2061:
2060:
2055:
2047:
2033:
2021:
2019:
2018:
2013:
1995:
1993:
1992:
1987:
1982:
1937:
1935:
1934:
1929:
1884:
1882:
1881:
1876:
1874:
1873:
1857:
1855:
1854:
1849:
1809:Thompson's group
1794:piecewise linear
1783:scaling function
1764:order-isomorphic
1740:
1725:
1652:
1650:
1649:
1644:
1642:
1637:
1629:
1620:
1614:
1595:fractional parts
1565:
1563:
1562:
1557:
1552:
1543:
1537:
1489:
1487:
1486:
1481:
1479:
1478:
1454:
1452:
1451:
1446:
1444:
1443:
1434:
1419:
1417:
1416:
1411:
1409:
1408:
1399:
1380:
1378:
1377:
1372:
1360:
1358:
1357:
1352:
1350:
1349:
1340:
1325:
1323:
1322:
1317:
1301:
1299:
1298:
1293:
1275:
1273:
1272:
1267:
1255:
1253:
1251:
1250:
1245:
1243:
1242:
1233:
1222:
1221:
1201:
1199:
1198:
1193:
1162:
1160:
1159:
1154:
1152:
1151:
1142:
1127:
1125:
1124:
1119:
1117:
1116:
1107:
1102:
1093:
1075:
1073:
1072:
1067:
1031:
1029:
1028:
1023:
1021:
1020:
1011:
996:
994:
993:
988:
980:
975:
965:square root of 2
943:
941:
940:
935:
933:
929:
927:
926:
911:
903:
894:
892:
891:
879:
874:
872:
871:
859:
853:
851:
850:
826:
822:
821:
785:
784:
753:
744:
742:
741:
729:
724:
722:
721:
709:
703:
701:
700:
676:
672:
671:
635:
634:
603:
594:
592:
591:
579:
574:
572:
571:
559:
521:
519:
518:
513:
501:
499:
498:
493:
488:
464:rational numbers
433:
432:
431:
430:
418:
417:
416:
415:
403:
402:
401:
400:
387:musical notation
377:
376:
375:
374:
362:
361:
360:
359:
347:
346:
345:
344:
332:
331:
330:
329:
250:computer science
209:
193:
162:fractional parts
154:order-isomorphic
138:Thompson's group
120:
118:
117:
112:
107:
98:
92:
79:rational numbers
51:computer science
5538:
5537:
5533:
5532:
5531:
5529:
5528:
5527:
5498:
5497:
5492:
5463:
5462:
5395:
5385:
5384:
5363:
5359:
5351:
5349:
5346:
5345:
5319:
5314:
5313:
5311:
5308:
5307:
5282:
5277:
5276:
5274:
5271:
5270:
5251:
5221:Polynomial ring
5171:Integral domain
5160:
5150:
5149:
5075:
5067:
5062:
5054:
5051:
5050:
5029:
5027:
5024:
5023:
5009:Involutive ring
4894:
4883:
4877:
4872:
4867:
4853:Dyadic rational
4834:
4829:
4799:
4794:
4765:Just intonation
4692:
4682:
4679:
4676:
4675:
4673:
4672:
4661:
4657:
4652:
4630:
4619:
4610:
4580:
4579:
4558:
4554:
4514:
4510:
4472:
4468:
4446:10.2307/2320870
4428:
4424:
4405:
4402:
4401:
4378:
4372:
4369:
4368:
4345:
4343:
4340:
4339:
4316:
4314:
4311:
4310:
4298:
4279:
4275:
4256:
4250:
4246:
4221:
4208:
4204:
4153:
4150:
4146:
4098:
4094:
4060:
4056:
4042:
4016:
4012:
3998:10.2307/2319174
3980:
3976:
3925:
3919:
3912:
3898:
3861:
3858:
3857:
3839:
3836:
3835:
3829:
3825:
3821:", pp. 255–257.
3793:
3782:
3781:
3779:
3778:
3776:
3773:
3772:
3716:
3709:
3661:
3657:
3622:
3594:
3589:
3588:
3586:
3583:
3582:
3566:
3564:
3561:
3560:
3544:
3542:
3539:
3538:
3536:
3532:
3526:
3504:
3500:
3482:
3475:
3450:
3446:
3412:
3408:
3382:
3374:
3371:
3370:
3354:
3351:
3350:
3334:
3331:
3330:
3305:
3301:
3296:
3291:
3288:
3287:
3270:
3266:
3261:
3256:
3253:
3252:
3236:
3233:
3232:
3230:
3226:
3220:
3204:
3197:
3149:
3145:
3107:(1): 1:01, 27,
3090:
3086:
3036:
3029:
3023:
3007:
3003:
2971:Pothier, Yvonne
2968:
2964:
2932:
2928:
2917:
2903:
2899:
2893:
2875:
2868:
2833:
2826:
2784:Jerrum, Mark R.
2781:
2777:
2743:
2739:
2701:
2697:
2675:
2651:
2634:
2600:
2596:
2578:
2574:
2568:
2550:Hwu, Wen-mei W.
2543:
2539:
2525:
2492:
2488:
2478:
2452:
2448:
2442:10.1119/1.11512
2416:
2412:
2402:
2378:
2374:
2368:
2352:
2348:
2343:
2335:Urysohn's lemma
2311:
2307:
2302:
2299:
2298:
2272:
2269:
2268:
2252:
2249:
2248:
2232:
2229:
2228:
2208:
2205:
2204:
2188:
2185:
2184:
2167:
2163:
2158:
2147:
2144:
2143:
2127:
2124:
2123:
2107:
2104:
2103:
2086:
2082:
2080:
2077:
2076:
2043:
2029:
2027:
2024:
2023:
2001:
1998:
1997:
1978:
1943:
1940:
1939:
1917:
1914:
1913:
1910:fusible numbers
1898:surreal numbers
1869:
1865:
1863:
1860:
1859:
1843:
1840:
1839:
1824:
1752:
1751:
1750:
1749:
1748:
1741:
1733:
1732:
1726:
1715:
1683:complex numbers
1663:
1661:Dyadic solenoid
1638:
1633:
1618:
1610:
1608:
1605:
1604:
1541:
1533:
1531:
1528:
1527:
1516:
1511:
1504:
1500:
1474:
1470:
1468:
1465:
1464:
1439:
1435:
1430:
1425:
1422:
1421:
1404:
1400:
1395:
1390:
1387:
1386:
1366:
1363:
1362:
1345:
1341:
1336:
1331:
1328:
1327:
1311:
1308:
1307:
1281:
1278:
1277:
1261:
1258:
1257:
1238:
1234:
1229:
1217:
1213:
1208:
1205:
1204:
1203:
1187:
1184:
1183:
1147:
1143:
1138:
1133:
1130:
1129:
1112:
1108:
1103:
1091:
1089:
1086:
1085:
1037:
1034:
1033:
1016:
1012:
1007:
1002:
999:
998:
974:
972:
969:
968:
957:
931:
930:
916:
912:
904:
902:
895:
887:
883:
878:
867:
863:
858:
855:
854:
831:
827:
796:
792:
759:
755:
754:
752:
745:
737:
733:
728:
717:
713:
708:
705:
704:
681:
677:
646:
642:
609:
605:
604:
602:
595:
587:
583:
578:
567:
563:
558:
554:
552:
549:
548:
507:
504:
503:
484:
479:
476:
475:
460:
448:
429:
424:
423:
422:
421:
420:
414:
409:
408:
407:
406:
405:
399:
394:
393:
392:
391:
390:
383:Time signatures
380:
379:
378:
373:
368:
367:
366:
365:
364:
358:
353:
352:
351:
350:
349:
343:
338:
337:
336:
335:
334:
328:
323:
322:
321:
320:
319:
317:
309:Igor Stravinski
307:Five bars from
305:
304:
303:
301:
286:
277:random variable
246:
213:
212:
211:
207:
201:
196:
195:
194:
183:
178:
150:fusible numbers
146:surreal numbers
126:dyadic solenoid
96:
88:
86:
83:
82:
35:binary rational
31:dyadic rational
19:
12:
11:
5:
5536:
5526:
5525:
5520:
5515:
5510:
5494:
5493:
5491:
5490:
5483:
5476:
5468:
5465:
5464:
5456:
5455:
5427:
5426:
5420:
5414:
5408:
5396:
5391:
5390:
5387:
5386:
5383:
5382:
5371:
5366:
5362:
5358:
5354:
5335:
5322:
5317:
5298:
5285:
5280:
5268:-adic integers
5261:
5255:
5246:
5232:
5231:
5230:
5229:
5223:
5217:
5216:
5215:
5203:
5197:
5191:
5185:
5179:
5161:
5156:
5155:
5152:
5151:
5148:
5147:
5146:
5145:
5133:
5132:
5131:
5125:
5113:
5112:
5111:
5093:
5092:
5091:
5090:
5078:
5074:
5070:
5065:
5061:
5058:
5044:
5032:
5011:
5005:
4999:
4993:
4979:
4978:
4972:
4966:
4952:
4951:
4945:
4939:
4933:
4932:
4931:
4925:
4913:
4907:
4895:
4893:Basic concepts
4892:
4891:
4888:
4887:
4869:
4868:
4866:
4865:
4860:
4855:
4850:
4845:
4839:
4836:
4835:
4828:
4827:
4820:
4813:
4805:
4796:
4795:
4793:
4792:
4787:
4782:
4777:
4772:
4767:
4762:
4761:
4760:
4750:
4745:
4744:
4743:
4733:
4728:
4723:
4718:
4713:
4708:
4703:
4697:
4694:
4693:
4691:
4690:
4669:
4667:
4663:
4662:
4655:
4653:
4651:
4650:
4636:
4634:
4631:
4624:
4621:
4620:
4609:
4608:
4601:
4594:
4586:
4578:
4577:
4552:
4508:
4466:
4440:(7): 575–578,
4422:
4409:
4386:
4383:
4376:
4353:
4350:
4324:
4321:
4296:
4273:
4244:
4202:
4166:(2): 449–460,
4144:
4092:
4072:(1): 127–141,
4054:
4040:
4010:
3992:(6): 677–679,
3974:
3938:(1): 254–277,
3910:
3896:
3876:-adic Analysis
3865:
3843:
3823:
3807:
3803:
3800:
3796:
3792:
3789:
3785:
3728:(1): 239–270,
3707:
3671:(2): 369–400,
3655:
3635:(3): 343–362,
3603:
3600:
3597:
3592:
3569:
3547:
3530:
3524:
3498:
3473:
3444:
3406:
3390:
3387:
3381:
3378:
3358:
3338:
3308:
3304:
3299:
3295:
3273:
3269:
3264:
3260:
3240:
3224:
3218:
3195:
3159:(1): 129–159,
3143:
3084:
3027:
3021:
3001:
2989:10.2307/748675
2983:(5): 307–317,
2962:
2950:10.2307/748774
2944:(5): 374–378,
2926:
2915:
2897:
2891:
2866:
2824:
2775:
2737:
2695:
2673:
2632:
2594:
2572:
2566:
2546:Kirk, David B.
2537:
2523:
2486:
2476:
2456:Rizvi, Uzma Z.
2446:
2428:(9): 896–906,
2410:
2400:
2372:
2366:
2345:
2344:
2342:
2339:
2319:
2314:
2310:
2306:
2282:
2279:
2276:
2256:
2236:
2212:
2192:
2170:
2166:
2161:
2157:
2154:
2151:
2131:
2111:
2089:
2085:
2074:epsilon number
2053:
2050:
2046:
2042:
2039:
2036:
2032:
2011:
2008:
2005:
1985:
1981:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1947:
1927:
1924:
1921:
1872:
1868:
1847:
1823:
1820:
1797:homeomorphisms
1742:
1735:
1734:
1727:
1720:
1719:
1718:
1717:
1716:
1714:
1711:
1662:
1659:
1655:Prüfer 2-group
1641:
1636:
1632:
1626:
1623:
1617:
1613:
1602:quotient group
1591:2-adic numbers
1555:
1549:
1546:
1540:
1536:
1515:
1512:
1510:
1507:
1502:
1498:
1477:
1473:
1442:
1438:
1433:
1429:
1407:
1403:
1398:
1394:
1370:
1348:
1344:
1339:
1335:
1315:
1304:floor function
1291:
1288:
1285:
1265:
1241:
1237:
1232:
1228:
1225:
1220:
1216:
1212:
1191:
1150:
1146:
1141:
1137:
1115:
1111:
1106:
1099:
1096:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1019:
1015:
1010:
1006:
986:
983:
978:
956:
953:
945:
944:
925:
922:
919:
915:
910:
907:
901:
898:
896:
890:
886:
882:
877:
870:
866:
862:
857:
856:
849:
846:
843:
840:
837:
834:
830:
825:
820:
817:
814:
811:
808:
805:
802:
799:
795:
791:
788:
783:
780:
777:
774:
771:
768:
765:
762:
758:
751:
748:
746:
740:
736:
732:
727:
720:
716:
712:
707:
706:
699:
696:
693:
690:
687:
684:
680:
675:
670:
667:
664:
661:
658:
655:
652:
649:
645:
641:
638:
633:
630:
627:
624:
621:
618:
615:
612:
608:
601:
598:
596:
590:
586:
582:
577:
570:
566:
562:
557:
556:
542:multiplication
511:
491:
487:
483:
459:
456:
447:
444:
425:
410:
395:
369:
354:
339:
324:
306:
295:
292:
289:
288:
287:
285:
282:
245:
242:
198:
197:
188:
187:
186:
185:
184:
182:
181:In measurement
179:
177:
174:
158:2-adic numbers
142:Prüfer 2-group
110:
104:
101:
95:
91:
17:
9:
6:
4:
3:
2:
5535:
5524:
5523:Number theory
5521:
5519:
5516:
5514:
5511:
5509:
5506:
5505:
5503:
5489:
5484:
5482:
5477:
5475:
5470:
5469:
5467:
5466:
5461:
5460:
5454:
5450:
5449:
5448:
5447:
5446:
5441:
5440:
5439:
5434:
5433:
5432:
5425:
5421:
5419:
5415:
5413:
5409:
5407:
5406:Division ring
5403:
5402:
5401:
5400:
5394:
5389:
5388:
5360:
5344:
5342:
5336:
5320:
5306:
5305:-adic numbers
5304:
5299:
5283:
5269:
5267:
5262:
5260:
5256:
5254:
5247:
5245:
5241:
5240:
5239:
5238:
5237:
5228:
5224:
5222:
5218:
5214:
5210:
5209:
5208:
5204:
5202:
5198:
5196:
5192:
5190:
5186:
5184:
5180:
5178:
5174:
5173:
5172:
5168:
5167:
5166:
5165:
5159:
5154:
5153:
5144:
5140:
5139:
5138:
5134:
5130:
5126:
5124:
5120:
5119:
5118:
5114:
5110:
5106:
5105:
5104:
5100:
5099:
5098:
5097:
5072:
5068:
5059:
5056:
5049:
5048:Terminal ring
5045:
5022:
5018:
5017:
5016:
5012:
5010:
5006:
5004:
5000:
4998:
4994:
4992:
4988:
4987:
4986:
4985:
4984:
4977:
4973:
4971:
4967:
4965:
4961:
4960:
4959:
4958:
4957:
4950:
4946:
4944:
4940:
4938:
4934:
4930:
4926:
4924:
4920:
4919:
4918:
4917:Quotient ring
4914:
4912:
4908:
4906:
4902:
4901:
4900:
4899:
4890:
4889:
4886:
4881:→ Ring theory
4880:
4876:
4875:
4864:
4861:
4859:
4856:
4854:
4851:
4849:
4846:
4844:
4841:
4840:
4837:
4833:
4826:
4821:
4819:
4814:
4812:
4807:
4806:
4803:
4791:
4788:
4786:
4783:
4781:
4778:
4776:
4773:
4771:
4768:
4766:
4763:
4759:
4756:
4755:
4754:
4751:
4749:
4746:
4742:
4739:
4738:
4737:
4734:
4732:
4729:
4727:
4724:
4722:
4719:
4717:
4714:
4712:
4709:
4707:
4704:
4702:
4699:
4698:
4695:
4671:
4670:
4668:
4664:
4649:
4645:
4641:
4638:
4637:
4635:
4628:
4622:
4618:
4614:
4607:
4602:
4600:
4595:
4593:
4588:
4587:
4584:
4573:
4572:
4566:
4562:
4556:
4549:
4545:
4541:
4537:
4532:
4527:
4523:
4519:
4512:
4505:
4501:
4497:
4493:
4489:
4485:
4481:
4477:
4470:
4463:
4459:
4455:
4451:
4447:
4443:
4439:
4435:
4434:
4426:
4407:
4384:
4381:
4374:
4351:
4348:
4322:
4319:
4307:
4303:
4299:
4297:1-56881-127-6
4293:
4289:
4288:
4283:
4282:Conway, J. H.
4277:
4270:
4266:
4262:
4255:
4248:
4241:
4237:
4233:
4229:
4228:
4220:
4216:
4212:
4211:Cannon, J. W.
4206:
4199:
4195:
4191:
4187:
4183:
4179:
4174:
4169:
4165:
4161:
4160:
4148:
4141:
4137:
4133:
4129:
4125:
4121:
4117:
4113:
4109:
4105:
4104:
4096:
4089:
4085:
4080:
4075:
4071:
4067:
4066:
4058:
4051:
4047:
4043:
4041:81-85931-13-5
4037:
4033:
4029:
4025:
4021:
4014:
4007:
4003:
3999:
3995:
3991:
3987:
3986:
3978:
3971:
3967:
3963:
3959:
3955:
3951:
3946:
3941:
3937:
3933:
3932:
3924:
3917:
3915:
3907:
3903:
3899:
3897:0-387-98669-3
3893:
3889:
3885:
3881:
3877:
3863:
3841:
3833:
3827:
3805:
3798:
3794:
3790:
3769:
3765:
3761:
3757:
3753:
3749:
3745:
3741:
3736:
3731:
3727:
3723:
3722:
3714:
3712:
3704:
3700:
3696:
3692:
3688:
3684:
3679:
3674:
3670:
3666:
3659:
3652:
3648:
3643:
3638:
3634:
3630:
3629:
3621:
3598:
3534:
3527:
3521:
3517:
3513:
3509:
3502:
3495:
3491:
3487:
3480:
3478:
3470:
3466:
3462:
3458:
3454:
3448:
3441:
3437:
3432:
3427:
3423:
3419:
3418:
3410:
3388:
3385:
3379:
3376:
3356:
3336:
3328:
3324:
3306:
3302:
3297:
3293:
3271:
3267:
3262:
3258:
3238:
3228:
3221:
3219:9783642136481
3215:
3211:
3210:
3202:
3200:
3192:
3188:
3184:
3180:
3176:
3172:
3167:
3162:
3158:
3154:
3147:
3140:
3136:
3132:
3128:
3124:
3120:
3115:
3110:
3106:
3102:
3101:
3094:
3088:
3081:
3077:
3073:
3069:
3065:
3061:
3056:
3051:
3047:
3043:
3042:
3034:
3032:
3024:
3018:
3014:
3013:
3005:
2998:
2994:
2990:
2986:
2982:
2978:
2977:
2972:
2966:
2959:
2955:
2951:
2947:
2943:
2939:
2938:
2930:
2923:
2918:
2912:
2908:
2901:
2894:
2888:
2884:
2880:
2873:
2871:
2863:
2859:
2855:
2851:
2847:
2843:
2842:
2837:
2831:
2829:
2821:
2817:
2812:
2807:
2803:
2799:
2798:
2793:
2789:
2785:
2779:
2772:
2768:
2764:
2760:
2756:
2752:
2748:
2741:
2734:
2730:
2726:
2722:
2718:
2714:
2710:
2706:
2699:
2692:
2688:
2684:
2680:
2676:
2674:0-8176-3586-6
2670:
2666:
2662:
2658:
2657:
2649:
2647:
2645:
2643:
2641:
2639:
2637:
2629:
2625:
2620:
2615:
2611:
2607:
2606:
2598:
2591:
2587:
2583:
2576:
2569:
2563:
2559:
2555:
2551:
2547:
2541:
2534:
2530:
2526:
2524:0-387-98383-X
2520:
2516:
2512:
2508:
2504:
2500:
2496:
2490:
2484:
2479:
2473:
2469:
2465:
2461:
2457:
2450:
2443:
2439:
2435:
2431:
2427:
2423:
2422:
2414:
2407:
2403:
2397:
2393:
2389:
2385:
2384:
2376:
2369:
2363:
2359:
2358:
2350:
2346:
2338:
2336:
2331:
2317:
2312:
2304:
2296:
2280:
2277:
2274:
2254:
2234:
2226:
2210:
2190:
2168:
2164:
2159:
2155:
2152:
2149:
2142:has the form
2129:
2109:
2087:
2083:
2075:
2072:equal to the
2071:
2067:
2051:
2048:
2040:
2037:
2034:
2009:
2006:
2003:
1983:
1979:
1972:
1969:
1966:
1963:
1960:
1951:
1948:
1945:
1922:
1911:
1906:
1904:
1899:
1894:
1892:
1888:
1870:
1866:
1845:
1837:
1836:unary numbers
1833:
1829:
1819:
1817:
1814:
1810:
1806:
1802:
1801:unit interval
1798:
1795:
1790:
1788:
1784:
1780:
1775:
1773:
1769:
1765:
1761:
1757:
1746:
1739:
1730:
1724:
1710:
1708:
1704:
1700:
1696:
1692:
1688:
1684:
1680:
1676:
1672:
1668:
1667:abelian group
1658:
1656:
1634:
1624:
1621:
1603:
1598:
1596:
1592:
1588:
1583:
1581:
1580:powers of two
1577:
1573:
1569:
1547:
1544:
1525:
1521:
1506:
1496:
1491:
1475:
1471:
1462:
1458:
1440:
1436:
1431:
1427:
1405:
1401:
1396:
1392:
1384:
1368:
1346:
1342:
1337:
1333:
1313:
1305:
1286:
1263:
1239:
1235:
1230:
1223:
1218:
1214:
1189:
1181:
1176:
1174:
1166:
1148:
1144:
1139:
1135:
1113:
1109:
1104:
1097:
1094:
1082:
1063:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1017:
1013:
1008:
1004:
984:
981:
976:
966:
961:
952:
950:
923:
920:
917:
913:
908:
905:
899:
897:
888:
884:
880:
875:
868:
864:
860:
844:
841:
838:
828:
823:
815:
812:
809:
800:
797:
793:
789:
786:
778:
775:
772:
763:
760:
756:
749:
747:
738:
734:
730:
725:
718:
714:
710:
694:
691:
688:
678:
673:
665:
662:
659:
650:
647:
643:
639:
636:
628:
625:
622:
613:
610:
606:
599:
597:
588:
584:
580:
575:
568:
564:
560:
547:
546:
545:
543:
539:
535:
531:
529:
525:
509:
489:
485:
481:
473:
469:
465:
455:
453:
443:
441:
437:
428:
413:
398:
388:
384:
372:
357:
342:
327:
316:
315:
310:
299:
281:
278:
275:Generating a
273:
271:
267:
263:
259:
255:
251:
241:
238:
234:
230:
226:
222:
218:
205:
200:
192:
173:
171:
167:
163:
159:
155:
151:
147:
143:
139:
135:
131:
127:
122:
102:
99:
80:
76:
72:
68:
62:
60:
56:
52:
48:
44:
40:
36:
32:
23:
16:
5457:
5443:
5442:
5438:Free algebra
5436:
5435:
5429:
5428:
5397:
5340:
5302:
5265:
5234:
5233:
5213:Finite field
5162:
5109:Finite field
5095:
5094:
5021:Initial ring
4981:
4980:
4954:
4953:
4896:
4858:Half-integer
4852:
4848:Dedekind cut
4725:
4568:
4555:
4521:
4511:
4482:(1): 21–25,
4479:
4475:
4469:
4437:
4431:
4425:
4286:
4276:
4260:
4247:
4231:
4225:
4215:Floyd, W. J.
4205:
4173:math/9705205
4163:
4157:
4147:
4107:
4101:
4095:
4069:
4063:
4057:
4023:
4013:
3989:
3983:
3977:
3945:math/0511714
3935:
3929:
3856:A Course in
3855:
3826:
3725:
3719:
3668:
3664:
3658:
3632:
3626:
3533:
3507:
3501:
3485:
3456:
3447:
3421:
3415:
3409:
3227:
3208:
3156:
3152:
3146:
3104:
3098:
3087:
3045:
3039:
3011:
3004:
2980:
2974:
2965:
2941:
2935:
2929:
2906:
2900:
2882:
2848:(1): 18–23,
2845:
2839:
2801:
2795:
2778:
2746:
2740:
2708:
2704:
2698:
2655:
2612:(1): 52–60,
2609:
2603:
2597:
2581:
2575:
2557:
2540:
2506:
2489:
2459:
2449:
2425:
2419:
2413:
2405:
2383:Nice Numbers
2382:
2375:
2356:
2349:
2332:
2066:well-ordered
1907:
1895:
1832:real numbers
1825:
1816:simple group
1791:
1776:
1753:
1664:
1599:
1587:real numbers
1584:
1576:localization
1517:
1501:= 0.10111...
1492:
1302:denotes the
1177:
1173:half-integer
1170:
946:
532:
524:real numbers
472:power of two
461:
449:
426:
411:
396:
381:
370:
355:
340:
325:
312:
274:
247:
244:In computing
214:
176:Applications
123:
63:
47:power of two
34:
30:
28:
15:
5518:Ring theory
5418:Simple ring
5129:Jordan ring
5003:Graded ring
4885:Ring theory
4753:Irreducible
4683:Denominator
2064:. They are
1792:The set of
1756:dense order
1180:real number
538:subtraction
452:Jean Piaget
385:in Western
59:real number
43:denominator
5502:Categories
5424:Commutator
5183:GCD domain
4785:Percentage
4780:Paper size
4689:= Quotient
4531:2003.14342
3678:1603.03351
3424:: 93–110,
3323:Cantor set
3166:cs/0605058
3055:1506.03949
2341:References
2293:it is (in
2070:order type
1807:. This is
1691:quotiented
1675:characters
1522:but not a
1165:Cantor set
440:whole note
5365:∞
5143:Semifield
4758:Reduction
4716:Continued
4701:Algebraic
4677:Numerator
4613:Fractions
4548:214727767
4504:119964871
3806:^
3768:119148680
3735:1305.1514
3453:Kac, Mark
3377:ε
3357:ε
3337:ε
3294:ε
3239:ε
3191:221168970
3139:218627153
3114:1106.3448
3048:: 72–86,
2862:220604326
2771:195795492
2309:↑
2267:that for
2203:for each
2084:ε
2038:−
1955:↦
1799:from the
1787:Hénon map
1772:bijection
1705:, and an
1461:real line
1290:⌋
1287:…
1284:⌊
1227:⌋
1211:⌊
1064:…
982:≈
876:⋅
801:−
790:−
764:−
726:−
651:−
614:−
5137:Semiring
5123:Lie ring
4905:Subrings
4731:Egyptian
4666:Fraction
4648:Quotient
4640:Dividend
4284:(2001),
4217:(2011),
4198:14490692
3970:11566447
3486:Wavelets
3455:(1959),
3321:forms a
2733:15815720
2691:11758381
2552:(2013),
2501:(1998),
2458:(eds.),
1703:solenoid
1699:protorus
1572:overring
949:dividing
534:Addition
284:In music
73:and the
71:integers
39:fraction
5339:Prüfer
4941:•
4843:Integer
4748:Integer
4721:Decimal
4686:
4674:
4644:Divisor
4563:(ed.),
4496:0665515
4462:0503877
4454:2320870
4306:1803095
4269:2185433
4240:2856142
4190:1724861
4140:9900529
4132:0970237
4112:Bibcode
4088:1412484
4050:1632579
4006:2319174
3962:2278053
3906:1760253
3760:3402799
3740:Bibcode
3703:3743682
3695:3770912
3651:0217052
3494:1161245
3469:0110114
3440:2520103
3183:2311089
3131:3029087
3080:5899577
3072:3367582
2820:0855970
2763:3981892
2725:2090389
2683:1137517
2628:2201092
2533:1712468
2430:Bibcode
2068:, with
1693:by the
1568:subring
1459:in the
1383:fractal
468:integer
65:form a
4991:Module
4964:Kernel
4741:Silver
4736:Golden
4726:Dyadic
4711:Binary
4706:Aspect
4617:ratios
4546:
4502:
4494:
4460:
4452:
4304:
4294:
4267:
4238:
4196:
4188:
4138:
4130:
4086:
4048:
4038:
4004:
3968:
3960:
3904:
3894:
3766:
3758:
3701:
3693:
3649:
3522:
3492:
3467:
3438:
3325:whose
3216:
3189:
3181:
3137:
3129:
3078:
3070:
3019:
2997:748675
2995:
2958:748774
2956:
2913:
2889:
2860:
2818:
2769:
2761:
2731:
2723:
2689:
2681:
2671:
2626:
2564:
2531:
2521:
2483:p. 166
2474:
2398:
2364:
2227:, and
1256:where
1178:Every
985:1.4142
540:, and
363:, and
231:, and
225:quarts
221:gallon
148:, and
41:whose
5343:-ring
5207:Field
5103:Field
4911:Ideal
4898:Rings
4544:S2CID
4526:arXiv
4500:S2CID
4450:JSTOR
4257:(PDF)
4222:(PDF)
4194:S2CID
4168:arXiv
4002:JSTOR
3966:S2CID
3940:arXiv
3926:(PDF)
3764:S2CID
3730:arXiv
3699:S2CID
3673:arXiv
3623:(PDF)
3187:S2CID
3161:arXiv
3135:S2CID
3109:arXiv
3076:S2CID
3050:arXiv
2993:JSTOR
2954:JSTOR
2922:p. 37
2858:S2CID
2767:S2CID
2729:S2CID
2687:S2CID
2022:with
1524:field
1457:dense
470:by a
419:, or
229:pints
206:from
204:pound
75:field
45:is a
4790:Unit
4615:and
4569:The
4292:ISBN
4136:PMID
4036:ISBN
3892:ISBN
3520:ISBN
3214:ISBN
3017:ISBN
2911:ISBN
2887:ISBN
2669:ISBN
2562:ISBN
2519:ISBN
2472:ISBN
2396:ISBN
2362:ISBN
2049:<
1908:The
1896:The
1701:, a
1520:ring
1032:for
233:cups
217:inch
208:2 lb
67:ring
4770:LCD
4536:doi
4484:doi
4442:doi
4178:doi
4120:doi
4074:doi
4070:203
4028:doi
3994:doi
3950:doi
3936:307
3884:doi
3748:doi
3726:202
3683:doi
3637:doi
3512:doi
3426:doi
3422:171
3171:doi
3119:doi
3093:Coq
3060:doi
3046:592
2985:doi
2946:doi
2850:doi
2806:doi
2751:doi
2713:doi
2661:doi
2614:doi
2610:351
2586:doi
2511:doi
2464:doi
2438:doi
2388:doi
1826:In
1128:by
833:max
804:min
767:min
683:max
654:min
617:min
311:'s
168:in
77:of
33:or
5504::
5451:•
5422:•
5416:•
5410:•
5404:•
5337:•
5300:•
5263:•
5257:•
5248:•
5242:•
5225:•
5219:•
5211:•
5205:•
5199:•
5193:•
5187:•
5181:•
5175:•
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5141:•
5135:•
5127:•
5121:•
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5101:•
5046:•
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5007:•
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4974:•
4968:•
4962:•
4947:•
4935:•
4927:•
4921:•
4915:•
4909:•
4903:•
4646:=
4642:÷
4567:,
4542:,
4534:,
4520:,
4498:,
4492:MR
4490:,
4480:11
4478:,
4458:MR
4456:,
4448:,
4438:85
4436:,
4400:,
4367:,
4338:,
4302:MR
4300:,
4265:MR
4259:,
4236:MR
4232:58
4230:,
4224:,
4213:;
4192:,
4186:MR
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4176:,
4164:60
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4108:38
4084:MR
4082:,
4068:,
4046:MR
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4034:,
4000:,
3990:80
3988:,
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3928:,
3913:^
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3890:,
3878:,
3762:,
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3724:,
3710:^
3697:,
3691:MR
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3681:,
3669:26
3667:,
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3645:,
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3625:,
3518:,
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3198:^
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3044:,
3030:^
2991:,
2981:14
2979:,
2952:,
2940:,
2881:,
2869:^
2856:,
2846:27
2844:,
2827:^
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2814:,
2802:43
2800:,
2790:;
2786:;
2765:,
2759:MR
2757:,
2727:,
2721:MR
2719:,
2709:50
2707:,
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2667:,
2635:^
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2608:,
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2529:MR
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2497:;
2470:,
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2426:46
2424:,
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2394:,
2330:.
2318:16
1789:.
1743:A
1709:.
1669:.
1657:.
1582:.
536:,
530:.
404:,
356:16
348:,
341:16
333:,
326:16
272:.
227:,
172:.
144:,
140:,
136:,
132:,
128:,
121:.
61:.
5487:e
5480:t
5473:v
5370:)
5361:p
5357:(
5353:Z
5341:p
5321:p
5316:Q
5303:p
5284:p
5279:Z
5266:p
5252:n
5077:Z
5073:1
5069:/
5064:Z
5060:=
5057:0
5031:Z
4824:e
4817:t
4810:v
4680:/
4605:e
4598:t
4591:v
4538::
4528::
4486::
4444::
4408:3
4385:2
4382:1
4375:1
4352:4
4349:3
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4320:1
4180::
4170::
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4122::
4114::
4076::
4030::
3996::
3952::
3942::
3886::
3864:p
3842:p
3802:]
3799:2
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3791:1
3788:[
3784:Z
3750::
3742::
3732::
3685::
3675::
3639::
3633:7
3602:}
3599:2
3596:{
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3428::
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3307:i
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3272:i
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3173::
3163::
3121::
3111::
3105:9
3062::
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2948::
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2852::
2808::
2753::
2715::
2663::
2616::
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2513::
2466::
2440::
2432::
2390::
2313:9
2305:2
2281:3
2278:=
2275:n
2255:n
2235:k
2211:n
2191:k
2169:k
2165:2
2160:/
2156:1
2153:+
2150:n
2130:n
2110:n
2088:0
2052:1
2045:|
2041:y
2035:x
2031:|
2010:y
2007:,
2004:x
1984:2
1980:/
1976:)
1973:1
1970:+
1967:y
1964:+
1961:x
1958:(
1952:y
1949:,
1946:x
1926:}
1923:0
1920:{
1871:i
1867:2
1846:i
1640:Z
1635:/
1631:]
1625:2
1622:1
1616:[
1612:Z
1554:]
1548:2
1545:1
1539:[
1535:Z
1503:2
1499:2
1476:i
1472:2
1441:i
1437:2
1432:/
1428:1
1406:i
1402:2
1397:/
1393:n
1369:i
1347:i
1343:2
1338:/
1334:1
1314:x
1264:i
1254:,
1240:i
1236:2
1231:/
1224:x
1219:i
1215:2
1190:x
1149:i
1145:2
1140:/
1136:n
1114:i
1110:2
1105:/
1098:6
1095:1
1061:,
1058:2
1055:,
1052:1
1049:,
1046:0
1043:=
1040:i
1018:i
1014:2
1009:/
1005:1
977:2
967:(
924:d
921:+
918:b
914:2
909:c
906:a
900:=
889:d
885:2
881:c
869:b
865:2
861:a
848:)
845:d
842:,
839:b
836:(
829:2
824:c
819:)
816:d
813:,
810:b
807:(
798:b
794:2
787:a
782:)
779:d
776:,
773:b
770:(
761:d
757:2
750:=
739:d
735:2
731:c
719:b
715:2
711:a
698:)
695:d
692:,
689:b
686:(
679:2
674:c
669:)
666:d
663:,
660:b
657:(
648:b
644:2
640:+
637:a
632:)
629:d
626:,
623:b
620:(
611:d
607:2
600:=
589:d
585:2
581:c
576:+
569:b
565:2
561:a
510:q
490:q
486:/
482:p
427:8
412:4
397:2
371:8
300:.
109:]
103:2
100:1
94:[
90:Z
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