1054:
1026:
1151:"1,000". A late-thirteenth-century Cistercian doodle had differentiated horizontal digits for lower powers of ten from vertical digits for higher powers of ten, but that potentially productive convention is not known to have been exploited at the time; it could have covered numbers into the tens of millions (horizontal 10 to 10, vertical 10 to 10). A sixteenth-century mathematician used vertical digits for the traditional values, horizontal digits for millions, and rotated them a further 45° counter-clockwise for billions and another 90° for trillions, but it is not clear how the intermediate powers of ten were to be indicated and this convention was not adopted by others.
1066:
1190:
829:
734:
1042:
22:
1861:
1053:
824:
The two dozen or so surviving
Cistercian manuscripts that use the system date from the thirteenth to the fifteenth century, and cover an area from England to Italy, Normandy to Sweden. The numbers were not used for arithmetic, fractions or accounting, but indicated years, foliation (numbering pages),
836:
Although mostly confined to the
Cistercian order, there was some usage outside it. A late-fifteenth-century Norman treatise on arithmetic used both Cistercian and Indo-Arabic numerals. In one known case, Cistercian numerals were inscribed on a physical object, indicating the calendrical, angular and
883:
A horizontal stave was most common while the numerals were in use among the
Cistercians. A vertical stave was attested only in Northern France in the fourteenth and fifteenth centuries. However, eighteenth- and twentieth-century revivals of the system in France and Germany used a vertical stave.
1171:
The book received mixed reviews. Historian Ann Moyer lauded King for re-introducing the numerical system to a larger audience, since many had forgotten about it. Mathematician Detlef Spalt claimed that King exaggerated the system's importance and made mistakes in applying the system in the book
791:(units, tens, hundreds or thousands). These digits are compounded on a single stave to indicate more complex numbers. The Cistercians eventually abandoned the system in favor of the Arabic numerals, but marginal use outside the order continued until the early twentieth century.
1126:
When the system spread outside the order in the fifteenth and sixteenth centuries, numbers into the millions were enabled by compounding with the digit for "thousand". For example, a late-fifteenth century Norman treatise on arithmetic indicated 10,000 as a ligature of
1059:
A fourteenth-century Norman manuscript that used only
Cistercian numerals. These were horizontal to fit the flow of the text. Note the round form of the digit 9. Numbers were later retranscribed with Hindu-Arabic digits in the margin notes: here we see 4,484, 715 and
1025:
1013:(a digit 1 at each corner) is the number 1,111. (The exact forms varied by date and by monastery. For example, the digits shown here for 3 and 4 were in some manuscripts swapped with those for 7 and 8, and the 5's may be written with a lower dot (
741:
from
Brussels. Each character is a page/column number. These early Cistercian forms, with 3 and 4 swapped for 7 and 8, plus single and double dots for 5 and 6 and a triangular 9, are found in only one other surviving manuscript. The numbers
884:
There is also some historical variation as to which corner of the number represented which place value. The place-values shown here were the most common among the
Cistercians and the only ones used later.
25:
Numbers written with
Cistercian numerals. From left to right: 1 in units place, 2 in tens place (20), 3 in hundreds place (300), 4 in thousands place (4,000), then compound numbers 5,555, 6,789, 9,394.
1370:
King, David A. (1992). "The
Ciphers of the Monks and the Astrolabe of Berselius Reconsidered". In Demidov, Sergei S.; Rowe, David; Folkerts, Menso & Scriba, Christoph J. (eds.).
1345:
King, David (1993). "Rewriting history through instruments: The secrets of a medieval astrolabe from
Picardy". In Anderson, R. G. W.; Bennett, J. A. & Ryan, W. F. (eds.).
1081:
and an adaptation of
Cistercian numerals; the year 1834 at bottom left is written with the four characters for 1000, 800, 30, and 4, rather than the single character for 1834.
1874:
at GitHub. Uses the Private Use Area, since Unicode has declined to assign character codes. Font characters are segments, to be combined into the complete numerals.
1172:
devoted to it. Moritz Wedell, however, called the book a "lucid description" and a "comprehensive review of the history of research" concerning the monks' ciphers.
1915:
825:
divisions of texts, the numbering of notes and other lists, indexes and concordances, arguments in Easter tables, and the lines of a staff in musical notation.
1193:
A copy of the ciphers in a treatise on penmanship (c. 1300 CE) commonly attributed to John of Tilbury, with the corresponding Basingstoke numerical values.
1031:
The vertical forms of the digits (1–9, 10–90, 100–900 and 1,000–9,000), with an innovative form of 5 as engraved on an early-sixteenth-century Norman
1209:
1118:
Omitting a digit from a corner meant a value of zero for that power of ten, but there was no digit zero. (That is, an empty stave was not defined.)
1019:
etc.), with a short vertical stroke in place of the dot, or even with a triangle joining to the stave, which in other manuscripts indicated a 9.)
832:
Samples of mixed alphabetic-Cistercian notation used for foliation in a late thirteenth-century manuscript. Shown are a1 to a6 and g1 to g7.
821:, the Cistercian system was not used for numbers greater than 99, but it was soon expanded to four places, enabling numbers up to 9,999.
1041:
720:
1196:
Basingstoke's biographer claimed that he learned his system from his teacher in Athens. However, there is no known parallel among
440:
1387:
1278:
1865:
1404:
1354:
273:
1733:
1604:
1497:
1316:
1230:
851:
64:
1229:. Depending on the fonts you have installed, it may be that only the ones and twos will display properly. (The
787:
Digits are based on a horizontal or vertical stave, with the position of the digit on the stave indicating its
507:
1889:
713:
288:
1905:
1537:
1347:
Making Instruments Count: Essays on Historical Scientific Instruments Presented to Gerard L'Estrange Turner
633:
1147:
1141:
1135:
1129:
1111:
1105:
1099:
1093:
1087:
1015:
1009:
1003:
997:
991:
985:
979:
973:
967:
961:
955:
949:
943:
937:
931:
925:
919:
913:
907:
901:
895:
889:
460:
1484:
Sesiano, Jacques (1985). "Un système artificiel de numérotation au Moyen Age". In Folkerts, Menso &
643:
1605:"Character Encodings - Private Use Agreements - Under-ConScript Unicode Registry - Cistercian Numerals"
1554:
846:
520:
845:. After the Cistercians had abandoned the system, marginal use continued outside the order. In 1533,
1201:
1165:
1065:
872:
871:
The modern definitive expert on Cistercian numerals is the mathematician and historian of astronomy,
616:
385:
1910:
1307:
803:
were apparently based on a two-place (1–99) numeral system introduced into the Cistercian Order by
738:
706:
33:
696:
480:
77:
1139:"10" (and similarly for higher numbers), and Noviomagus in 1539 wrote "million" by subscripting
341:
1883:
380:
296:
1423:
1422:
Meskens, Ad; Bonte, Germain; De Groot, Jacques; De Jonghe, Mieke & King, David A. (1999).
800:
498:
1454:
864:
of Paris briefly adopted the numerals for mystical use, and in the early twentieth century
593:
454:
447:
328:
8:
804:
675:
665:
540:
491:
235:
90:
51:
130:
1813:
1776:
773:
588:
178:
173:
120:
1189:
1085:
Horizontal numbers were the same, but rotated 90 degrees counter-clockwise. (That is,
125:
1729:
1533:
1503:
1493:
1383:
1350:
1322:
1312:
1284:
1274:
818:
670:
660:
648:
628:
583:
578:
514:
346:
318:
225:
158:
148:
135:
100:
95:
1880:
at dCode. Uses digit shapes similar to the astrolabe (vertical stave, triangular 5).
1630:
563:
1768:
1466:
1435:
1375:
1266:
1226:
860:
area at least until the early eighteenth century. In the late eighteenth century,
769:
573:
467:
220:
208:
153:
143:
110:
85:
1723:
1379:
1078:
828:
777:
733:
685:
655:
598:
568:
553:
313:
281:
253:
230:
213:
72:
168:
1197:
861:
781:
680:
623:
603:
558:
431:
163:
115:
41:
1871:
1772:
1308:
The Ciphers of the Monks : a forgotten number-notation of the Middle Ages
1899:
1288:
1270:
780:
were introduced to northwestern Europe. They are more compact than Arabic or
486:
375:
308:
248:
183:
105:
1507:
1424:"Wine-Gauging at Damme [The evidence of a late medieval manuscript]"
1326:
1570:
King, David (1995). "A forgotten Cistercian system of numerical notation".
1485:
1439:
887:
Using graphic substitutes with a vertical stave, the first five digits are
638:
1877:
1470:
813:
788:
608:
473:
425:
415:
1892:
for use on web pages. Includes a live updating Cistercian numeral clock.
1817:
1801:
1780:
1725:
The Ciphers of the Monks: A Forgotten Number-notation of the Middle Ages
1162:
The Ciphers of the Monks: A Forgotten Number-notation of the Middle Ages
1756:
1070:
410:
1559:. New York Public Library. Philadelphia : Lippincott. p. 92.
21:
1032:
838:
808:
768:, or "ciphers" in nineteenth-century parlance, were developed by the
420:
1539:
The Great Book of Magical Art, Hindu Magic and East Indian Occultism
1233:
has tentatively assigned the units to PUA values U+EBA1 to U+EBAF.)
784:, with a single glyph able to indicate any integer from 1 to 9,999.
811:, who it seems based them on a twelfth-century English shorthand (
1455:"Les soi-disant chiffres grecs ou chaldéens (XIIe – XVIe siècle)"
1222:
842:
405:
390:
1168:
and published in 2001, describes the Cistercian numeral system.
1860:
857:
395:
1200:. It seems more likely that Basingstoke picked up the idea of
1421:
1074:
865:
737:
The entry for the word 'aqua' in an early-thirteenth-century
400:
362:
323:
1886:
Background for Unicode consideration of Cistercian numerals
817:). In its earliest attestations, in the monasteries of the
1047:
All Cistercian numerals from 1 to 9999 (open to enlarge).
1402:
1265:. Cambridge: Cambridge University Press. p. 350.
1631:"Cistercian numerals in Fōrmulæ programming language"
1556:
The American boys' book of signs, signals and symbols
1403:Agrippa von Nettesheim, Heinrich Cornelius (1533).
1208:, such as the one at right, commonly attributed to
1311:. Stuttgart: F. Steiner. pp. 16, 29, 34, 41.
868:considered using the numerals as Aryan symbolism.
1897:
1490:Mathemata : Festschrift für Helmuth Gericke
1263:Numerical notation : a comparative history
849:included a description of these ciphers in his
1916:Writing systems introduced in the 13th century
856:The numerals were used by wine-gaugers in the
1256:
1254:
1252:
1250:
714:
18:Numeral system developed by Cistercian monks
1532:
1492:(in French). Stuttgart: F. Steiner Verlag.
1260:
1154:
1837:Wedell, Moritz (2003). "Buchbesprechung".
1721:
1340:
1338:
1336:
1300:
1298:
1247:
1182:
721:
707:
1452:
1221:Cistercian numerals are not supported by
977:500, and doing both forms the thousands,
847:Heinrich Cornelius Agrippa von Nettesheim
799:The digits and idea of forming them into
1622:
1188:
1064:
837:other numbers on the fourteenth-century
827:
732:
20:
1542:. Chicago: De Laurence Co. p. 174.
1483:
1374:. Basel: Birkhäuser. pp. 375–388.
1333:
1295:
1215:
1204:in Greece and applied it to an English
947:50. Inverting them forms the hundreds,
841:of Berselius, which was made in French
1898:
1836:
1715:
1702:
1587:
1585:
1799:
1754:
1597:
1552:
1703:Høyrup, Jens (2008). "Book review".
1690:
1678:
1666:
1654:
1628:
1591:
1569:
1520:
1405:"De notis Hebraeorum et Chaldaeorum"
1369:
1344:
1304:
1582:
13:
917:5. Reversing them forms the tens,
14:
1927:
1853:
1572:Citeaux Commentarii Cistercienses
1133:"1,000" wrapped under and around
1121:
1859:
1231:Under-ConScript Unicode Registry
1225:, and are here substituted with
1052:
1040:
1024:
852:Three Books of Occult Philosophy
1830:
1793:
1748:
1696:
1684:
1672:
1660:
1648:
1563:
1546:
1526:
1514:
1477:
1446:
1415:
1396:
1363:
1:
1553:Beard, Daniel Carter (1918).
1459:Revue d'histoire des sciences
1261:Chrisomalis, Stephen (2010).
1240:
1202:alphabetic numerical notation
1069:A 19th-century gravestone in
760:779, 783, 803, 818, 834, 858.
758:659, 678, 686, 697, 724, 759,
756:566, 591, 601, 604, 628, 635,
754:436, 446, 476, 506, 508, 552,
752:403, 404, 405, 420, 434, 435,
750:296, 317, 343, 368, 378, 387,
748:267, 268, 272, 281, 284, 295,
746:146, 148, 150, 169, 194, 198,
1380:10.1007/978-3-0348-8599-7_18
7:
1878:Cistercian number generator
1839:Zeitschrift für Germanistik
1534:De Laurence, Lauron William
1115:for 1,000, as seen above.)
862:Chevaliers de la Rose-Croix
10:
1932:
1349:. University of Michigan.
794:
441:Non-standard radices/bases
1872:FRB Cistercian font (OTF)
1773:10.1017/S0038713400132002
770:Cistercian monastic order
744:21, 41, 81, 85, 106, 115,
1890:Cistercian Web Component
1411:(in Latin). p. 141.
1271:10.1017/CBO9780511676062
1175:
1156:The Ciphers of the Monks
1453:Beaujouan, Guy (1950).
1305:King, David A. (2001).
1198:Greek numbering systems
878:
776:at about the time that
697:List of numeral systems
1800:Spalt, Detlef (2004).
1440:10.3406/hism.1999.1501
1409:De Occulta Philosophia
1194:
1145:"1,000" under another
1082:
833:
761:
26:
1471:10.3406/rhs.1950.2795
1428:Histoire & Mesure
1192:
1068:
831:
736:
65:Hindu–Arabic numerals
24:
1868:at Wikimedia Commons
1629:R.Ugalde, Laurence.
1609:www.kreativekorp.com
594:Prehistoric counting
370:Common radices/bases
52:Place-value notation
1906:Cistercian numerals
1866:Cistercian numerals
1755:Moyer, Ann (2003).
1722:King, D.A. (2001).
1077:, inscribed in the
805:John of Basingstoke
766:Cistercian numerals
541:Sign-value notation
1195:
1083:
834:
774:thirteenth century
762:
197:East Asian systems
27:
1864:Media related to
1705:Annals of Science
1389:978-3-0348-8599-7
1280:978-0-511-67683-3
1227:Chao tone letters
819:County of Hainaut
731:
730:
530:
529:
1923:
1863:
1847:
1846:
1834:
1828:
1827:
1825:
1824:
1797:
1791:
1790:
1788:
1787:
1752:
1746:
1745:
1743:
1742:
1719:
1713:
1712:
1700:
1694:
1688:
1682:
1676:
1670:
1664:
1658:
1652:
1646:
1645:
1643:
1641:
1626:
1620:
1619:
1617:
1615:
1601:
1595:
1589:
1580:
1579:
1567:
1561:
1560:
1550:
1544:
1543:
1530:
1524:
1518:
1512:
1511:
1481:
1475:
1474:
1450:
1444:
1443:
1419:
1413:
1412:
1400:
1394:
1393:
1367:
1361:
1360:
1342:
1331:
1330:
1302:
1293:
1292:
1258:
1234:
1219:
1213:
1186:
1149:
1143:
1137:
1131:
1113:
1107:
1101:
1095:
1089:
1056:
1044:
1028:
1017:
1011:
1005:
999:
993:
987:
981:
975:
969:
963:
957:
951:
945:
939:
933:
927:
921:
915:
909:
903:
897:
891:
807:, archdeacon of
723:
716:
709:
512:
496:
478:
468:balanced ternary
465:
452:
58:
57:
29:
28:
1931:
1930:
1926:
1925:
1924:
1922:
1921:
1920:
1911:Numeral systems
1896:
1895:
1856:
1851:
1850:
1835:
1831:
1822:
1820:
1806:Sudhoffs Archiv
1798:
1794:
1785:
1783:
1753:
1749:
1740:
1738:
1736:
1720:
1716:
1701:
1697:
1689:
1685:
1677:
1673:
1665:
1661:
1653:
1649:
1639:
1637:
1627:
1623:
1613:
1611:
1603:
1602:
1598:
1590:
1583:
1578:(3–4): 183–217.
1568:
1564:
1551:
1547:
1531:
1527:
1519:
1515:
1500:
1482:
1478:
1451:
1447:
1420:
1416:
1401:
1397:
1390:
1368:
1364:
1357:
1343:
1334:
1319:
1303:
1296:
1281:
1259:
1248:
1243:
1238:
1237:
1220:
1216:
1210:John of Tilbury
1187:
1183:
1178:
1159:
1124:
1061:
1057:
1048:
1045:
1036:
1029:
881:
797:
778:Arabic numerals
759:
757:
755:
753:
751:
749:
747:
745:
743:
727:
691:
690:
613:
599:Proto-cuneiform
544:
543:
532:
531:
526:
525:
510:
494:
476:
463:
450:
437:
366:
365:
353:
352:
333:
293:
278:
269:
268:
259:
258:
240:
199:
198:
189:
188:
140:
82:
68:
67:
55:
54:
42:Numeral systems
19:
12:
11:
5:
1929:
1919:
1918:
1913:
1908:
1894:
1893:
1887:
1881:
1875:
1869:
1855:
1854:External links
1852:
1849:
1848:
1829:
1812:(1): 108–109.
1792:
1767:(3): 919–921.
1747:
1734:
1728:. F. Steiner.
1714:
1695:
1683:
1671:
1659:
1647:
1621:
1596:
1581:
1562:
1545:
1525:
1513:
1498:
1476:
1465:(2): 170–174.
1445:
1414:
1395:
1388:
1362:
1356:978-0860783947
1355:
1332:
1317:
1294:
1279:
1245:
1244:
1242:
1239:
1236:
1235:
1214:
1180:
1179:
1177:
1174:
1158:
1153:
1123:
1122:Higher numbers
1120:
1063:
1062:
1058:
1051:
1049:
1046:
1039:
1037:
1030:
1023:
880:
877:
796:
793:
782:Roman numerals
729:
728:
726:
725:
718:
711:
703:
700:
699:
693:
692:
689:
688:
683:
678:
673:
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619:
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611:
606:
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591:
586:
581:
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566:
561:
556:
550:
549:
548:Non-alphabetic
545:
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538:
537:
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533:
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471:
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351:
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337:
332:
331:
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321:
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311:
306:
300:
299:
297:Post-classical
292:
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285:
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277:
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205:
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191:
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181:
176:
171:
166:
161:
156:
151:
146:
139:
138:
133:
128:
123:
118:
113:
108:
103:
98:
93:
88:
81:
80:
78:Eastern Arabic
75:
73:Western Arabic
69:
63:
62:
61:
56:
50:
49:
48:
45:
44:
38:
37:
17:
9:
6:
4:
3:
2:
1928:
1917:
1914:
1912:
1909:
1907:
1904:
1903:
1901:
1891:
1888:
1885:
1882:
1879:
1876:
1873:
1870:
1867:
1862:
1858:
1857:
1845:(3): 671–673.
1844:
1841:(in German).
1840:
1833:
1819:
1815:
1811:
1808:(in German).
1807:
1803:
1802:"Book review"
1796:
1782:
1778:
1774:
1770:
1766:
1762:
1758:
1757:"Book review"
1751:
1737:
1735:9783515076401
1731:
1727:
1726:
1718:
1711:(2): 306–308.
1710:
1706:
1699:
1692:
1687:
1680:
1675:
1668:
1663:
1656:
1651:
1636:
1632:
1625:
1610:
1606:
1600:
1593:
1588:
1586:
1577:
1573:
1566:
1558:
1557:
1549:
1541:
1540:
1535:
1529:
1522:
1517:
1509:
1505:
1501:
1499:3-515-04324-1
1495:
1491:
1487:
1486:Lindgren, Uta
1480:
1472:
1468:
1464:
1461:(in French).
1460:
1456:
1449:
1441:
1437:
1433:
1429:
1425:
1418:
1410:
1406:
1399:
1391:
1385:
1381:
1377:
1373:
1366:
1358:
1352:
1348:
1341:
1339:
1337:
1328:
1324:
1320:
1318:3-515-07640-9
1314:
1310:
1309:
1301:
1299:
1290:
1286:
1282:
1276:
1272:
1268:
1264:
1257:
1255:
1253:
1251:
1246:
1232:
1228:
1224:
1218:
1211:
1207:
1203:
1199:
1191:
1185:
1181:
1173:
1169:
1167:
1166:David A. King
1163:
1157:
1152:
1150:
1144:
1138:
1132:
1119:
1116:
1114:
1108:
1103:for 100—thus
1102:
1096:
1090:
1080:
1079:Theban script
1076:
1072:
1067:
1055:
1050:
1043:
1038:
1034:
1027:
1022:
1021:
1020:
1018:
1012:
1006:
1000:
994:
988:
982:
976:
970:
964:
958:
952:
946:
940:
934:
928:
922:
916:
910:
904:
898:
892:
885:
876:
874:
873:David A. King
869:
867:
863:
859:
855:
853:
848:
844:
840:
830:
826:
822:
820:
816:
815:
810:
806:
802:
792:
790:
785:
783:
779:
775:
772:in the early
771:
767:
764:The medieval
740:
735:
724:
719:
717:
712:
710:
705:
704:
702:
701:
698:
695:
694:
687:
684:
682:
679:
677:
674:
672:
669:
667:
664:
662:
659:
657:
654:
650:
647:
645:
642:
640:
637:
636:
635:
634:Alphasyllabic
632:
630:
627:
625:
622:
621:
618:
615:
614:
610:
607:
605:
602:
600:
597:
595:
592:
590:
587:
585:
582:
580:
577:
575:
572:
570:
567:
565:
562:
560:
557:
555:
552:
551:
547:
546:
542:
536:
535:
522:
519:
516:
509:
506:
503:
502:
493:
490:
488:
485:
482:
475:
472:
469:
462:
459:
456:
449:
446:
445:
442:
439:
438:
433:
429:
427:
424:
422:
419:
417:
414:
412:
409:
407:
404:
402:
399:
397:
394:
392:
389:
387:
384:
382:
379:
377:
374:
373:
369:
368:
364:
357:
356:
348:
345:
343:
340:
339:
335:
334:
330:
327:
325:
322:
320:
317:
315:
312:
310:
307:
305:
302:
301:
298:
295:
294:
290:
287:
286:
283:
280:
279:
275:
272:
271:
267:Other systems
263:
262:
255:
252:
250:
249:Counting rods
247:
246:
242:
241:
237:
234:
232:
229:
227:
224:
222:
219:
215:
212:
211:
210:
207:
206:
202:
201:
193:
192:
185:
182:
180:
177:
175:
172:
170:
167:
165:
162:
160:
157:
155:
152:
150:
147:
145:
142:
141:
137:
134:
132:
129:
127:
124:
122:
119:
117:
114:
112:
109:
107:
104:
102:
99:
97:
94:
92:
89:
87:
84:
83:
79:
76:
74:
71:
70:
66:
60:
59:
53:
47:
46:
43:
40:
39:
35:
31:
30:
23:
16:
1842:
1838:
1832:
1821:. Retrieved
1809:
1805:
1795:
1784:. Retrieved
1764:
1760:
1750:
1739:. Retrieved
1724:
1717:
1708:
1704:
1698:
1686:
1674:
1662:
1650:
1638:. Retrieved
1634:
1624:
1612:. Retrieved
1608:
1599:
1575:
1571:
1565:
1555:
1548:
1538:
1528:
1516:
1489:
1479:
1462:
1458:
1448:
1434:(1): 51–77.
1431:
1427:
1417:
1408:
1398:
1371:
1365:
1346:
1306:
1262:
1217:
1205:
1184:
1170:
1161:
1160:
1155:
1146:
1140:
1134:
1128:
1125:
1117:
1110:
1109:for 101—and
1104:
1098:
1092:
1086:
1084:
1014:
1008:
1007:5,000. Thus
1002:
996:
990:
984:
978:
972:
966:
960:
954:
948:
942:
936:
930:
924:
918:
912:
906:
900:
894:
888:
886:
882:
870:
850:
835:
823:
812:
798:
786:
765:
763:
500:
461:Signed-digit
336:Contemporary
303:
203:Contemporary
15:
1206:ars notaria
814:ars notaria
789:place value
739:concordance
639:Akṣarapallī
609:Tally marks
508:Non-integer
1900:Categories
1823:2021-01-08
1786:2021-01-08
1741:2015-08-13
1691:King (2001
1679:King (2001
1669::156, 214)
1667:King (2001
1655:King (2001
1592:King (2001
1523::243, 251)
1521:King (2001
1241:References
1071:Llanfyllin
676:Glagolitic
649:Kaṭapayādi
617:Alphabetic
521:Asymmetric
363:radix/base
304:Cistercian
289:Babylonian
236:Vietnamese
91:Devanagari
1884:L2/20-290
1681::182–185)
1289:630115876
1033:astrolabe
839:astrolabe
809:Leicester
801:ligatures
644:Āryabhaṭa
589:Kharosthi
481:factorial
448:Bijective
349:(Iñupiaq)
179:Sundanese
174:Mongolian
121:Malayalam
1818:20777934
1781:20060835
1761:Speculum
1640:July 29,
1536:(1915).
1508:12644728
1488:(eds.).
1327:48254993
1097:for 10,
671:Georgian
661:Cyrillic
629:Armenian
584:Etruscan
579:Egyptian
487:Negative
347:Kaktovik
342:Cherokee
319:Pentadic
243:Historic
226:Japanese
159:Javanese
149:Balinese
136:Dzongkha
101:Gurmukhi
96:Gujarati
34:a series
32:Part of
1635:Fōrmulæ
1614:6 April
1372:Amphora
1223:Unicode
1091:for 1,
1001:4,000,
995:3,000,
989:2,000,
983:1,000,
843:Picardy
795:History
574:Chuvash
492:Complex
282:Ancient
274:History
221:Hokkien
209:Chinese
154:Burmese
144:Tibetan
131:Kannada
111:Sinhala
86:Bengali
1816:
1779:
1732:
1506:
1496:
1386:
1353:
1325:
1315:
1287:
1277:
1060:5,199.
858:Bruges
686:Hebrew
656:Coptic
569:Brahmi
554:Aegean
511:
495:
477:
464:
451:
314:Muisca
254:Tangut
231:Korean
214:Suzhou
126:Telugu
1814:JSTOR
1777:JSTOR
1693::210)
1657::427)
1176:Notes
1164:, by
1075:Wales
971:400,
965:300,
959:200,
953:100,
866:Nazis
681:Greek
666:Geʽez
624:Abjad
604:Roman
564:Aztec
559:Attic
474:Mixed
432:table
324:Quipu
309:Mayan
164:Khmer
116:Tamil
1730:ISBN
1642:2021
1616:2021
1594::39)
1504:OCLC
1494:ISBN
1384:ISBN
1351:ISBN
1323:OCLC
1313:ISBN
1285:OCLC
1275:ISBN
941:40,
935:30,
929:20,
923:10,
879:Form
742:are,
329:Rumi
184:Thai
106:Odia
1769:doi
1467:doi
1436:doi
1376:doi
1267:doi
911:4,
905:3,
899:2,
893:1,
361:By
169:Lao
1902::
1843:13
1810:88
1804:.
1775:.
1765:78
1763:.
1759:.
1709:65
1707:.
1633:.
1607:.
1584:^
1576:46
1574:.
1502:.
1457:.
1432:14
1430:.
1426:.
1407:.
1382:.
1335:^
1321:.
1297:^
1283:.
1273:.
1249:^
1073:,
998:˩˨
992:˨˩
968:꜕꜖
962:꜖꜕
938:˥˦
932:˦˥
908:꜓꜒
902:꜒꜓
875:.
426:60
421:20
416:16
411:12
406:10
36:on
1826:.
1789:.
1771::
1744:.
1644:.
1618:.
1510:.
1473:.
1469::
1463:3
1442:.
1438::
1392:.
1378::
1359:.
1329:.
1291:.
1269::
1212:.
1148:¬
1142:¬
1136:⌉
1130:⌋
1112:¬
1106:⏘
1100:⏗
1094:⌐
1088:⌙
1035:.
1016:꜎
1010:⌶
1004:꜌
986:˨
980:˩
974:꜑
956:꜕
950:꜖
944:꜈
926:˦
920:˥
914:꜍
896:꜓
890:꜒
854:.
722:e
715:t
708:v
517:)
515:φ
513:(
504:)
501:i
499:2
497:(
483:)
479:(
470:)
466:(
457:)
455:1
453:(
434:)
430:(
401:8
396:6
391:5
386:4
381:3
376:2
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