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428:, we get zero. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide
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With these definitions, the gcd or lcm of infinitely many natural numbers (or supernatural numbers) is a supernatural number. We can also extend the usual
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938:{\displaystyle \displaystyle \operatorname {gcd} (\{\omega _{i}\})\displaystyle =\prod _{p}p^{\inf(v_{p}(\omega _{i}))}}
828:{\displaystyle \displaystyle \operatorname {lcm} (\{\omega _{i}\})\displaystyle =\prod _{p}p^{\sup(v_{p}(\omega _{i}))}}
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1216:. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd ed.).
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581:{\displaystyle \prod _{p}p^{n_{p}}\cdot \prod _{p}p^{m_{p}}=\prod _{p}p^{n_{p}+m_{p}}}
471:
There is no natural way to add supernatural numbers, but they can be multiplied, with
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1908:
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448:"infinitely often," by taking that prime's corresponding exponent to be the symbol
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of supernatural numbers; primes other than 2 and 3 are omitted for simplicity.
1116:
1249:
588:. Similarly, the notion of divisibility extends to the supernaturals with
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1717:
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381:
then we recover the positive integers. Slightly less intuitively, if all
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1185:. Mathematical Surveys and Monographs. Vol. 124. Providence, RI:
1921:
1285:
1254:
1258:
1044:
Supernatural numbers are used to define orders and indices of
728:
can also be generalized for supernatural numbers, by defining
15:
1150:. Contemporary Mathematics. Vol. 95. Providence, RI:
1063:
Supernatural numbers also arise in the classification of
693:{\displaystyle v_{p}(\omega _{1})\leq v_{p}(\omega _{2})}
1048:
and subgroups, in which case many of the theorems from
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order functions to supernatural numbers by defining
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may be too technical for most readers to understand
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1052:carry over exactly. They are used to encode the
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1146:Brawley, Joel V.; Schnibben, George E. (1989).
1105:Journal für die reine und angewandte Mathematik
354:and there are only a finite number of non-zero
1148:Infinite algebraic extensions of finite fields
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193:{\displaystyle \omega =\prod _{p}p^{n_{p}},}
621:{\displaystyle \omega _{1}\mid \omega _{2}}
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1212:Fried, Michael D.; Jarden, Moshe (2008).
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59:Learn how and when to remove this message
43:, without removing the technical details.
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1137:Brawley & Schnibben (1989) pp.25-26
1088:
41:make it understandable to non-experts
7:
1014:{\displaystyle v_{p}(\omega )=n_{p}}
1441:Set-theoretically definable numbers
1179:Valuations, orderings, and Milnor
455:
415:
341:
14:
1101:"Algebraische Theorie der Körper"
110:in 1910 as a part of his work on
1926:
1250:Planet Math: Supernatural number
20:
1065:uniformly hyperfinite algebras
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989:
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287:{\displaystyle v_{p}(\omega )}
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102:, are a generalization of the
1:
1775:Plane-based geometric algebra
1187:American Mathematical Society
1152:American Mathematical Society
347:{\displaystyle n_{p}=\infty }
254:is zero, a natural number or
1735:{\displaystyle \mathbb {S} }
1658:{\displaystyle \mathbb {C} }
1619:{\displaystyle \mathbb {R} }
1581:{\displaystyle \mathbb {O} }
1553:{\displaystyle \mathbb {H} }
1525:{\displaystyle \mathbb {C} }
1497:{\displaystyle \mathbb {R} }
1410:{\displaystyle \mathbb {A} }
1377:{\displaystyle \mathbb {Q} }
1349:{\displaystyle \mathbb {Z} }
1321:{\displaystyle \mathbb {N} }
96:generalized natural numbers
1979:
1917:
1765:Algebra of physical space
1821:Extended complex numbers
1804:Extended natural numbers
726:greatest common divisor
461:{\displaystyle \infty }
441:{\displaystyle \omega }
421:{\displaystyle \infty }
130:{\displaystyle \omega }
1877:Transcendental numbers
1736:
1713:Hyperbolic quaternions
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117:A supernatural number
83:
1809:Extended real numbers
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1630:Split-complex numbers
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1388:Constructible numbers
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722:least common multiple
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401:{\displaystyle n_{p}}
376:
374:{\displaystyle n_{p}}
349:
316:
314:{\displaystyle n_{p}}
289:
249:
247:{\displaystyle n_{p}}
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195:
132:
74:
1841:Supernatural numbers
1751:Multicomplex numbers
1724:
1708:Dual-complex numbers
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1608:
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1542:
1514:
1486:
1468:Composition algebras
1436:Arithmetical numbers
1399:
1366:
1338:
1310:
1054:algebraic extensions
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954:
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735:
720:. The notion of the
704:
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207:
151:
121:
106:. They were used by
92:supernatural numbers
1746:Split-biquaternions
1458:Eisenstein integers
1421:Closed-form numbers
1177:Efrat, Ido (2006).
1050:finite group theory
294:is used instead of
94:, sometimes called
1904:Profinite integers
1867:Irrational numbers
1732:
1655:
1616:
1578:
1550:
1522:
1494:
1451:Gaussian rationals
1431:Computable numbers
1407:
1374:
1346:
1318:
1154:. pp. 23–26.
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1851:Superreal numbers
1831:Levi-Civita field
1826:Hyperreal numbers
1770:Spacetime algebra
1756:Geometric algebra
1669:Bicomplex numbers
1635:Split-quaternions
1476:Division algebras
1446:Gaussian integers
1393:Algebraic numbers
1296:definable numbers
1227:978-3-540-77269-9
1077:Profinite integer
1034:{\displaystyle p}
963:{\displaystyle p}
881:
771:
713:{\displaystyle p}
538:
508:
478:
216:{\displaystyle p}
160:
69:
68:
61:
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1799:Cardinal numbers
1760:Clifford algebra
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1703:Dual quaternions
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1360:Rational numbers
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1214:Field arithmetic
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1093:
1046:profinite groups
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100:Steinitz numbers
64:
57:
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50:
44:
24:
23:
16:
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1846:Surreal numbers
1836:Ordinal numbers
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1722:
1721:
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1642:
1640:Split-octonions
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1605:
1597:
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1568:
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1508:Complex numbers
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1307:
1304:Natural numbers
1289:
1283:
1246:
1228:
1220:. p. 520.
1218:Springer-Verlag
1211:
1197:
1189:. p. 125.
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1097:Steinitz, Ernst
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409:
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355:
328:
323:
322:
301:
296:
295:
265:
260:
259:
234:
229:
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205:
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175:
170:
149:
148:
119:
118:
104:natural numbers
65:
54:
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37:help improve it
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1922:Classification
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1909:Normal numbers
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1791:infinitesimals
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1244:External links
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223:runs over all
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108:Ernst Steinitz
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1963:Factorization
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1872:Fuzzy numbers
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1674:Biquaternions
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1198:
1196:0-8218-4041-X
1192:
1188:
1184:
1180:
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1171:
1167:
1163:
1161:0-8218-5101-2
1157:
1153:
1149:
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1143:
1134:
1131:
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1122:
1118:
1114:
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1107:(in German).
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971:
957:
920:
916:
907:
903:
892:
886:
882:
878:
866:
862:
852:
849:
841:
840:
839:
810:
806:
797:
793:
782:
776:
772:
768:
756:
752:
742:
739:
731:
730:
729:
727:
723:
707:
682:
678:
669:
665:
661:
653:
649:
640:
636:
613:
609:
605:
600:
596:
571:
567:
563:
558:
554:
549:
543:
539:
535:
528:
524:
519:
513:
509:
505:
498:
494:
489:
483:
479:
469:
435:
393:
389:
366:
362:
338:
333:
329:
306:
302:
278:
270:
266:
257:
239:
235:
226:
225:prime numbers
210:
187:
180:
176:
171:
165:
161:
157:
154:
147:
146:
145:
143:
140:
124:
115:
113:
109:
105:
101:
97:
93:
89:
81:
77:
76:Hasse diagram
73:
63:
60:
52:
49:December 2022
42:
38:
32:
29:This article
27:
18:
17:
1893:
1883:
1840:
1698:Dual numbers
1690:hypercomplex
1480:Real numbers
1213:
1182:
1178:
1147:
1133:
1108:
1104:
1091:
1062:
1058:finite field
1043:
948:
837:
470:
258:. Sometimes
202:
116:
112:field theory
99:
95:
91:
85:
55:
46:
30:
1860:Other types
1679:Bioctonions
1536:Quaternions
1111:: 167–309.
227:, and each
88:mathematics
1947:Categories
1814:Projective
1787:Infinities
1236:1145.12001
1205:1103.12002
1170:0674.12009
1125:41.0445.03
1083:References
1898:solenoids
1718:Sedenions
1564:Octonions
1117:0075-4102
1021:for each
993:ω
917:ω
883:∏
863:ω
853:
807:ω
773:∏
753:ω
743:
679:ω
662:≤
650:ω
610:ω
606:∣
597:ω
540:∏
510:∏
506:⋅
480:∏
456:∞
436:ω
416:∞
342:∞
279:ω
162:∏
155:ω
125:ω
1958:Infinity
1332:Integers
1294:Sets of
1099:(1910).
1071:See also
700:for all
321:. If no
256:infinity
1953:Numbers
1888:numbers
1720: (
1566: (
1538: (
1510: (
1482: (
1426:Periods
1395: (
1362: (
1334: (
1306: (
1288:systems
1183:-theory
142:product
80:lattice
78:of the
35:Please
1688:Other
1286:Number
1234:
1224:
1203:
1193:
1168:
1158:
1123:
1115:
203:where
139:formal
90:, the
1896:-adic
1886:-adic
1643:Over
1604:Over
1598:types
1596:Split
1056:of a
970:-adic
137:is a
1932:List
1789:and
1222:ISBN
1191:ISBN
1156:ISBN
1113:ISSN
838:and
724:and
408:are
1232:Zbl
1201:Zbl
1166:Zbl
1121:JFM
1109:137
897:inf
850:gcd
787:sup
740:lcm
628:if
98:or
86:In
39:to
1949::
1478::
1230:.
1199:.
1164:.
1119:.
1103:.
1067:.
1060:.
1041:.
468:.
144::
114:.
1900:)
1894:p
1890:(
1884:p
1758:/
1742:)
1729:S
1665::
1652:C
1626::
1613:R
1588:)
1575:O
1560:)
1547:H
1532:)
1519:C
1504:)
1491:R
1417:)
1404:A
1384:)
1371:Q
1356:)
1343:Z
1328:)
1315:N
1278:e
1271:t
1264:v
1238:.
1207:.
1181:K
1172:.
1127:.
1029:p
1007:p
1003:n
999:=
996:)
990:(
985:p
981:v
958:p
945:.
929:)
926:)
921:i
913:(
908:p
904:v
900:(
893:p
887:p
879:=
875:)
872:}
867:i
859:{
856:(
819:)
816:)
811:i
803:(
798:p
794:v
790:(
783:p
777:p
769:=
765:)
762:}
757:i
749:{
746:(
708:p
688:)
683:2
675:(
670:p
666:v
659:)
654:1
646:(
641:p
637:v
614:2
601:1
572:p
568:m
564:+
559:p
555:n
550:p
544:p
536:=
529:p
525:m
520:p
514:p
499:p
495:n
490:p
484:p
394:p
390:n
367:p
363:n
339:=
334:p
330:n
307:p
303:n
282:)
276:(
271:p
267:v
240:p
236:n
211:p
188:,
181:p
177:n
172:p
166:p
158:=
62:)
56:(
51:)
47:(
33:.
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