1005:
31:
986:
808:, in which the value of a particular digit depends both on the digit itself and its position within the number. This was an extremely important development because non-place-value systems require unique symbols to represent each power of a base (ten, one hundred, one thousand, and so forth), which can make calculations more difficult.
848:
system of bases 10 and 6, since the ten sub-base was used merely to facilitate the representation of the large set of digits needed, while the place-values in a digit string were consistently 60-based and the
92:
rather than
Sumerian lexical numbers. However, the use of a special Sumerian sign for 60 (beside two Semitic signs for the same number) attests to a relation with the Sumerian system.
826:
These digits were used to represent larger numbers in the base 60 (sexagesimal) positional system. For example, 𒁹𒁹 𒌋𒌋𒁹𒁹𒁹 𒁹𒁹𒁹 would represent 2×60+23×60+3 = 8583.
1320:
837:, so the place of the units had to be inferred from context: 𒌋𒌋𒁹𒁹𒁹 could have represented 23, 23×60 (𒌋𒌋𒁹𒁹𒁹␣), 23×60×60 (𒌋𒌋𒁹𒁹𒁹␣␣), or 23/60, etc.
80:
or the
Akkadian civilizations. Neither of the predecessors was a positional system (having a convention for which 'end' of the numeral represented the units).
1220:
786:
506:
1114:
1084:
1262:
1300:
339:
1201:
1178:
1232:
896:
130:
1315:
989:) to represent zero, but only in the medial positions, and not on the right-hand side of the number, as we do in numbers like
1275:
1130:
Scientific
American – Why is a minute divided into 60 seconds, an hour into 60 minutes, yet there are only 24 hours in a day?
573:
779:
1166:
699:
709:
57:
stylus to print a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record.
1129:
526:
64:, who were famous for their astronomical observations, as well as their calculations (aided by their invention of the
586:
1142:
805:
682:
451:
833:. Babylonians later devised a sign to represent this empty place. They lacked a symbol to serve the function of
1325:
1217:
772:
99:
762:
546:
143:
1033:
964:
446:
362:
564:
1004:
983:, it was not seen as a number—merely the lack of a number. Later Babylonian texts used a placeholder (
869:
823:; for example, the combination 𒌋𒌋𒁹𒁹𒁹 represented the digit for 23 (see table of digits above).
659:
520:
513:
394:
1147:
908:
816:
741:
606:
557:
369:
301:
156:
117:
73:
88:
This system first appeared around 2000 BC; its structure reflects the decimal lexical numerals of
1255:
1010:
654:
407:
244:
239:
186:
50:
1330:
1283:
1197:
1174:
1110:
1104:
1080:
1074:
1048:
1018:
811:
Only two symbols (𒁹 to count units and 𒌋 to count tens) were used to notate the 59 non-zero
736:
726:
714:
694:
649:
644:
580:
412:
384:
291:
224:
214:
201:
166:
161:
89:
967:
were represented identically—a radix point was not written but rather made clear by context.
639:
533:
286:
274:
219:
209:
176:
151:
1266:
1236:
1224:
812:
751:
721:
664:
634:
619:
379:
347:
319:
296:
279:
138:
1229:
731:
1271:
1043:
857:
820:
746:
689:
669:
624:
497:
229:
196:
181:
107:
1309:
1246:
552:
441:
374:
314:
249:
191:
171:
975:
The
Babylonians did not technically have a digit for, nor a concept of, the number
904:
881:
704:
629:
30:
17:
985:
1286:
980:
956:
952:
948:
944:
940:
936:
900:
892:
845:
834:
674:
539:
491:
481:
69:
61:
1241:
1038:
1000:
976:
932:
928:
924:
920:
916:
912:
850:
830:
829:
A space was left to indicate a place without value, similar to the modern-day
476:
234:
54:
1291:
1028:
877:
873:
486:
1190:
853:
needed to work with these digit strings was correspondingly sexagesimal.
1023:
960:
841:
471:
456:
46:
42:
861:
856:
The legacy of sexagesimal still survives to this day, in the form of
815:. These symbols and their values were combined to form a digit in a
461:
65:
865:
466:
428:
389:
77:
29:
885:
1247:
High resolution photographs, descriptions, and analysis of the
1171:
Number Words and Number
Symbols: A Cultural History of Numbers
888:, although both of these systems are actually mixed radix.
804:
The
Babylonian system is credited as being the first known
1251:
tablet (YBC 7289) from the Yale
Babylonian Collection
1301:CESCNC – a handy and easy-to use numeral converter
1192:Number: From Ancient Civilisations to the Computer
1189:
1256:Photograph, illustration, and description of the
1098:
1096:
1068:
1066:
1064:
844:to represent digits, but it was not really a
780:
8:
1109:. Cambridge University Press. p. 248.
1079:. Cambridge University Press. p. 247.
899:(the previous and next in the series being
1260:tablet from the Yale Babylonian Collection
787:
773:
123:
94:
1106:Numerical Notation: A Comparative History
1076:Numerical Notation: A Comparative History
1321:Non-standard positional numeral systems
1060:
979:. Although they understood the idea of
911:: 2×2×3×5, which makes it divisible by
106:
7:
840:Their system clearly used internal
25:
1049:Sumerian language § Numerals
1019:Akkadian language § Numerals
1141:Lamb, Evelyn (August 31, 2014),
1003:
984:
897:superior highly composite number
1276:Wolfram Demonstrations Project
1:
39:Babylonian cuneiform numerals
34:Babylonian cuneiform numerals
1103:Stephen Chrisomalis (2010).
1073:Stephen Chrisomalis (2010).
1347:
507:Non-standard radices/bases
76:inherited from either the
907:), was chosen due to its
819:quite similar to that of
806:positional numeral system
74:positional numeral system
891:A common theory is that
884:and the measurement of
763:List of numeral systems
53:, using a wedge-tipped
1316:Babylonian mathematics
1274:by Michael Schreiber,
1242:Babylonian Mathematics
1188:McLeish, John (1991).
1034:Babylonian mathematics
35:
131:Hindu–Arabic numerals
33:
1143:"Look, Ma, No Zero!"
870:equilateral triangle
660:Prehistoric counting
436:Common radices/bases
118:Place-value notation
1272:Babylonian Numerals
1218:Babylonian numerals
1148:Scientific American
909:prime factorization
817:sign-value notation
607:Sign-value notation
18:Babylonian numerals
1284:Weisstein, Eric W.
1265:2012-08-13 at the
1235:2020-06-27 at the
1223:2017-05-20 at the
1167:Menninger, Karl W.
1011:Mathematics portal
263:East Asian systems
49:, were written in
36:
1230:Cuneiform numbers
1196:. HarperCollins.
1116:978-0-521-87818-0
1086:978-0-521-87818-0
797:
796:
596:
595:
90:Semitic languages
16:(Redirected from
1338:
1297:
1296:
1207:
1195:
1184:
1153:
1152:
1151:, Roots of Unity
1138:
1132:
1127:
1121:
1120:
1100:
1091:
1090:
1070:
1013:
1008:
1007:
992:
988:
789:
782:
775:
578:
562:
544:
534:balanced ternary
531:
518:
124:
95:
21:
1346:
1345:
1341:
1340:
1339:
1337:
1336:
1335:
1326:Numeral systems
1306:
1305:
1282:
1281:
1267:Wayback Machine
1237:Wayback Machine
1225:Wayback Machine
1214:
1204:
1187:
1181:
1165:
1162:
1157:
1156:
1140:
1139:
1135:
1128:
1124:
1117:
1102:
1101:
1094:
1087:
1072:
1071:
1062:
1057:
1039:History of zero
1009:
1002:
999:
990:
973:
802:
793:
757:
756:
679:
665:Proto-cuneiform
610:
609:
598:
597:
592:
591:
576:
560:
542:
529:
516:
503:
432:
431:
419:
418:
399:
359:
344:
335:
334:
325:
324:
306:
265:
264:
255:
254:
206:
148:
134:
133:
121:
120:
108:Numeral systems
86:
41:, also used in
28:
23:
22:
15:
12:
11:
5:
1344:
1342:
1334:
1333:
1328:
1323:
1318:
1308:
1307:
1304:
1303:
1298:
1279:
1269:
1253:
1244:
1239:
1227:
1213:
1212:External links
1210:
1209:
1208:
1202:
1185:
1179:
1161:
1158:
1155:
1154:
1133:
1122:
1115:
1092:
1085:
1059:
1058:
1056:
1053:
1052:
1051:
1046:
1044:Numeral system
1041:
1036:
1031:
1026:
1021:
1015:
1014:
998:
995:
972:
969:
821:Roman numerals
801:
798:
795:
794:
792:
791:
784:
777:
769:
766:
765:
759:
758:
755:
754:
749:
744:
739:
734:
729:
724:
719:
718:
717:
712:
707:
697:
692:
686:
685:
678:
677:
672:
667:
662:
657:
652:
647:
642:
637:
632:
627:
622:
616:
615:
614:Non-alphabetic
611:
605:
604:
603:
600:
599:
594:
593:
590:
589:
584:
571:
555:
550:
537:
524:
510:
509:
502:
501:
494:
489:
484:
479:
474:
469:
464:
459:
454:
449:
444:
438:
437:
433:
426:
425:
424:
421:
420:
417:
416:
410:
404:
403:
398:
397:
392:
387:
382:
377:
372:
366:
365:
363:Post-classical
358:
357:
351:
350:
343:
342:
336:
332:
331:
330:
327:
326:
323:
322:
317:
311:
310:
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271:
270:
266:
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260:
257:
256:
253:
252:
247:
242:
237:
232:
227:
222:
217:
212:
205:
204:
199:
194:
189:
184:
179:
174:
169:
164:
159:
154:
147:
146:
144:Eastern Arabic
141:
139:Western Arabic
135:
129:
128:
127:
122:
116:
115:
114:
111:
110:
104:
103:
85:
82:
27:Numeral system
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1343:
1332:
1329:
1327:
1324:
1322:
1319:
1317:
1314:
1313:
1311:
1302:
1299:
1294:
1293:
1288:
1287:"Sexagesimal"
1285:
1280:
1277:
1273:
1270:
1268:
1264:
1261:
1259:
1254:
1252:
1250:
1245:
1243:
1240:
1238:
1234:
1231:
1228:
1226:
1222:
1219:
1216:
1215:
1211:
1205:
1203:0-00-654484-3
1199:
1194:
1193:
1186:
1182:
1180:0-262-13040-8
1176:
1173:. MIT Press.
1172:
1168:
1164:
1163:
1159:
1150:
1149:
1144:
1137:
1134:
1131:
1126:
1123:
1118:
1112:
1108:
1107:
1099:
1097:
1093:
1088:
1082:
1078:
1077:
1069:
1067:
1065:
1061:
1054:
1050:
1047:
1045:
1042:
1040:
1037:
1035:
1032:
1030:
1027:
1025:
1022:
1020:
1017:
1016:
1012:
1006:
1001:
996:
994:
987:
982:
978:
970:
968:
966:
962:
958:
954:
950:
946:
942:
938:
934:
930:
926:
922:
918:
914:
910:
906:
902:
898:
894:
889:
887:
883:
879:
875:
871:
867:
864:or 60° in an
863:
859:
854:
852:
847:
843:
838:
836:
832:
827:
824:
822:
818:
814:
809:
807:
799:
790:
785:
783:
778:
776:
771:
770:
768:
767:
764:
761:
760:
753:
750:
748:
745:
743:
740:
738:
735:
733:
730:
728:
725:
723:
720:
716:
713:
711:
708:
706:
703:
702:
701:
700:Alphasyllabic
698:
696:
693:
691:
688:
687:
684:
681:
680:
676:
673:
671:
668:
666:
663:
661:
658:
656:
653:
651:
648:
646:
643:
641:
638:
636:
633:
631:
628:
626:
623:
621:
618:
617:
613:
612:
608:
602:
601:
588:
585:
582:
575:
572:
569:
568:
559:
556:
554:
551:
548:
541:
538:
535:
528:
525:
522:
515:
512:
511:
508:
505:
504:
499:
495:
493:
490:
488:
485:
483:
480:
478:
475:
473:
470:
468:
465:
463:
460:
458:
455:
453:
450:
448:
445:
443:
440:
439:
435:
434:
430:
423:
422:
414:
411:
409:
406:
405:
401:
400:
396:
393:
391:
388:
386:
383:
381:
378:
376:
373:
371:
368:
367:
364:
361:
360:
356:
353:
352:
349:
346:
345:
341:
338:
337:
333:Other systems
329:
328:
321:
318:
316:
315:Counting rods
313:
312:
308:
307:
303:
300:
298:
295:
293:
290:
288:
285:
281:
278:
277:
276:
273:
272:
268:
267:
259:
258:
251:
248:
246:
243:
241:
238:
236:
233:
231:
228:
226:
223:
221:
218:
216:
213:
211:
208:
207:
203:
200:
198:
195:
193:
190:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
150:
149:
145:
142:
140:
137:
136:
132:
126:
125:
119:
113:
112:
109:
105:
101:
97:
96:
93:
91:
83:
81:
79:
75:
71:
67:
63:
58:
56:
52:
48:
44:
40:
32:
19:
1290:
1257:
1248:
1191:
1170:
1160:Bibliography
1146:
1136:
1125:
1105:
1075:
974:
890:
882:trigonometry
855:
839:
828:
825:
810:
803:
566:
527:Signed-digit
402:Contemporary
354:
269:Contemporary
87:
59:
38:
37:
981:nothingness
860:(360° in a
846:mixed-radix
835:radix point
705:Akṣarapallī
675:Tally marks
574:Non-integer
70:sexagesimal
68:), used a
62:Babylonians
1310:Categories
1055:References
878:arcseconds
874:arcminutes
851:arithmetic
742:Glagolitic
715:Kaṭapayādi
683:Alphabetic
587:Asymmetric
429:radix/base
370:Cistercian
355:Babylonian
302:Vietnamese
157:Devanagari
72:(base-60)
1292:MathWorld
1029:Babylonia
965:fractions
710:Āryabhaṭa
655:Kharosthi
547:factorial
514:Bijective
415:(Iñupiaq)
245:Sundanese
240:Mongolian
187:Malayalam
51:cuneiform
1331:Numerals
1263:Archived
1233:Archived
1221:Archived
1169:(1969).
997:See also
961:Integers
737:Georgian
727:Cyrillic
695:Armenian
650:Etruscan
645:Egyptian
553:Negative
413:Kaktovik
408:Cherokee
385:Pentadic
309:Historic
292:Japanese
225:Javanese
215:Balinese
202:Dzongkha
167:Gurmukhi
162:Gujarati
100:a series
98:Part of
78:Sumerian
1258:root(2)
1249:root(2)
1024:Babylon
858:degrees
842:decimal
800:Symbols
640:Chuvash
558:Complex
348:Ancient
340:History
287:Hokkien
275:Chinese
220:Burmese
210:Tibetan
197:Kannada
177:Sinhala
152:Bengali
47:Chaldea
43:Assyria
1200:
1177:
1113:
1083:
955:, and
876:, and
868:of an
862:circle
813:digits
752:Hebrew
722:Coptic
635:Brahmi
620:Aegean
577:
561:
543:
530:
517:
380:Muisca
320:Tangut
297:Korean
280:Suzhou
192:Telugu
84:Origin
66:abacus
866:angle
747:Greek
732:Geʽez
690:Abjad
670:Roman
630:Aztec
625:Attic
540:Mixed
498:table
390:Quipu
375:Mayan
230:Khmer
182:Tamil
1198:ISBN
1175:ISBN
1111:ISBN
1081:ISBN
977:zero
971:Zero
963:and
903:and
895:, a
886:time
831:zero
395:Rumi
250:Thai
172:Odia
60:The
55:reed
45:and
991:100
905:120
880:in
872:),
427:By
235:Lao
1312::
1289:.
1145:,
1095:^
1063:^
993:.
959:.
957:60
953:30
951:,
949:20
947:,
945:15
943:,
941:12
939:,
937:10
935:,
931:,
927:,
923:,
919:,
915:,
901:12
893:60
492:60
487:20
482:16
477:12
472:10
102:on
1295:.
1278:.
1206:.
1183:.
1119:.
1089:.
933:6
929:5
925:4
921:3
917:2
913:1
788:e
781:t
774:v
583:)
581:φ
579:(
570:)
567:i
565:2
563:(
549:)
545:(
536:)
532:(
523:)
521:1
519:(
500:)
496:(
467:8
462:6
457:5
452:4
447:3
442:2
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
