356:
Any rational number has infinitely many different possible expansions as a sum of unit fractions, and since the discovery of the Rhind
Mathematical Papyrus mathematicians have struggled to understand how the ancient Egyptians might have calculated the specific expansions shown in this table.
791:
678:
589:
502:
429:
1172:
966:
901:. The answers were checked by multiplying the initial divisor by the proposed solution and checking that the resulting answer was 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 5
699:
1071:
Abdulrahman A. Abdulaziz, On the
Egyptian method of decomposing 2/n into unit fractions, Historia Mathematica, Volume 35, Issue 1, 2008, Pages 1-18, ISSN 0315-0860,
602:
518:
434:
924:
The Rhind
Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations (2 vols.)
47:), the form the Egyptians used to write fractional numbers. The text describes the representation of 50 rational numbers. It was written during the
877:
in terms of sums of hekat rational numbers, 1/3, 1/7, 1/10, 1/11 and 1/13. In this document a two-part set of fractions was written in terms of
1035:
55:, the first writer of mathematics whose name is known. Aspects of the document may have been copied from an unknown 1850 BCE text.
48:
1157:
855:
366:
980:
933:
1167:
360:
Suggestions by
Gillings included five different techniques. Problem 61 in the Rhind Mathematical Papyrus gives one formula:
1177:
1152:
820:
denominators. Others have suggested only one method was used by Ahmes which used multiplicative factors similar to
17:
840:
843:, written around 1850 BCE, is about the age of one unknown source for the Rhind papyrus. The Kahun 2/
1126:
Vymazalova, H. (2002), "The wooden tablets from Cairo: The use of the grain unit HK3T in ancient Egypt",
821:
870:
786:{\displaystyle \,{\frac {2}{n}}\;=\,{\frac {1}{n}}+{\frac {1}{2n}}+{\frac {1}{3n}}+{\frac {1}{6n}}}
1004:
28:
673:{\displaystyle \!{\frac {2}{mn}}\!={\frac {1}{m}}\!{\frac {1}{k}}+{\frac {1}{n}}{\frac {1}{k}}}
1031:
976:
929:
926:, Classics in Mathematics Education, vol. 8, Oberlin: Mathematical Association of America
584:{\displaystyle \,{\frac {2}{n}}\;=\;{\frac {1}{3}}{\frac {1}{n}}+{\frac {5}{3}}{\frac {1}{n}}}
40:
972:
1162:
1107:
1085:
962:
817:
32:
862:
into other unit fractions. The table consisted of 26 unit fraction series of the form 1/
497:{\displaystyle {\frac {2}{n}}={\frac {1}{2}}{\frac {1}{n}}+{\frac {3}{2}}{\frac {1}{n}}}
1089:
847:
fractions were identical to the fraction decompositions given in the Rhind
Papyrus' 2/
839:
An older ancient
Egyptian papyrus contained a similar table of Egyptian fractions; the
995:
Spalinger, Anthony (1990), "The Rhind
Mathematical Papyrus as a Historical Document",
1146:
1024:
44:
21:
348:
This part of the Rhind
Mathematical Papyrus was spread over nine sheets of papyrus.
1026:
Ancient
Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics
878:
810:
24:
1030:, Memoirs of the American Philosophical Society, American Philosophical Society,
858:(EMLR), circa 1900 BCE, lists decompositions of fractions of the form 1/
1112:
The
Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook
1072:
1008:
828:
table may have been decomposed was provided by Abdulrahman Abdulaziz.
64:
928:. Reprint, Reston: National Council of Teachers of Mathematics, 1979,
52:
424:{\displaystyle {\frac {2}{3n}}={\frac {1}{2n}}+{\frac {1}{6n}}}
1114:, Princeton, NJ: Princeton University Press, pp. 1–56
1110:(2007), "Egyptian mathematics", in Katz, Victor J. (ed.),
948:
The Rhind Mathematical Papyrus: an Ancient Egyptian Text
63:
The following table gives the expansions listed in the
702:
605:
521:
437:
369:
897:and remainders expressed in terms of a unit called
1023:
968:Mathematics in Ancient Egypt: A Contextual History
785:
672:
583:
496:
423:
824:. A detailed and simple explanation of how the 2/
813:) by two methods, and three methods to convert 2/
636:
622:
606:
1173:Ancient Egyptian objects in the British Museum
8:
793:. This formula yields the decomposition for
881:fractions which were fractions of the form
866:written as sums of other rational numbers.
714:
537:
533:
768:
750:
732:
719:
718:
704:
703:
701:
660:
650:
637:
626:
607:
604:
571:
561:
548:
538:
523:
522:
520:
484:
474:
461:
451:
438:
436:
406:
388:
370:
368:
76:table from the Rhind Mathematical Papyrus
1073:https://doi.org/10.1016/j.hm.2007.03.002
1051:
1049:
1047:
801:Ahmes was suggested to have converted 2/
69:
1058:History of Mathematics: An Introduction
914:
508:divisible by 3 in the latter equation).
431:, which can be stated equivalently as
51:(approximately 1650–1550 BCE) by
340:2/101 = 1/101 + 1/202 + 1/303 + 1/606
7:
946:Robins, Gay; Shute, Charles (1987),
287:2/79 = 1/60 + 1/237 + 1/316 + 1/790
272:2/73 = 1/60 + 1/219 + 1/292 + 1/365
242:2/61 = 1/40 + 1/244 + 1/488 + 1/610
197:2/43 = 1/42 + 1/86 + 1/129 + 1/301
49:Second Intermediate Period of Egypt
922:Chace, Arnold Buffum (1927–1929),
856:Egyptian Mathematical Leather Roll
14:
997:Studien zur Altägyptischen Kultur
1116:. See in particular pages 21–22.
835:Comparison to other table texts
1134:(1), Charles U., Prague: 27–42
971:, Princeton University Press,
950:, London: British Museum Press
873:wrote fractions in the form 1/
314:= 1/60 + 1/356 + 1/534 + 1/890
299:= 1/60 + 1/332 + 1/415 + 1/498
1:
512:Other possible formulas are:
164:= 1/24 + 1/58 + 1/174 + 1/232
332:2/97 = 1/56 + 1/679 + 1/776
257:2/67 = 1/40 + 1/335 + 1/536
182:2/37 = 1/24 + 1/111 + 1/296
167:2/31 = 1/20 + 1/124 + 1/155
1096:, University College London
137:2/19 = 1/12 + 1/76 + 1/114
1194:
1022:Clagett, Marshall (1999),
122:2/13 = 1/8 + 1/52 + 1/104
18:Rhind Mathematical Papyrus
1158:Papyri from ancient Egypt
1094:Lahun Papyri: table texts
1056:Burton, David M. (2003),
841:Lahun Mathematical Papyri
1168:Mathematics manuscripts
1060:, Boston: Wm. C. Brown
822:least common multiples
787:
674:
585:
498:
425:
329:= 1/60 + 1/380 + 1/570
269:= 1/40 + 1/568 + 1/710
239:= 1/36 + 1/236 + 1/531
224:= 1/30 + 1/318 + 1/795
209:= 1/30 + 1/141 + 1/470
194:= 1/24 + 1/246 + 1/328
788:
675:
586:
499:
426:
1178:Egyptian mathematics
871:Akhmim wooden tablet
700:
603:
519:
435:
367:
317:2/91 = 1/70 + 1/130
302:2/85 = 1/51 + 1/255
227:2/55 = 1/30 + 1/330
212:2/49 = 1/28 + 1/196
134:= 1/12 + 1/51 + 1/68
797:= 101 in the table.
337:2/99 = 1/66 + 1/198
322:2/93 = 1/62 + 1/186
307:2/87 = 1/58 + 1/174
292:2/81 = 1/54 + 1/162
277:2/75 = 1/50 + 1/150
262:2/69 = 1/46 + 1/138
247:2/63 = 1/42 + 1/126
232:2/57 = 1/38 + 1/114
217:2/51 = 1/34 + 1/102
152:2/25 = 1/15 + 1/75
78:
1153:Egyptian fractions
905:, which equals 1.
783:
684:is the average of
670:
581:
494:
421:
202:2/45 = 1/30 + 1/90
187:2/39 = 1/26 + 1/78
172:2/33 = 1/22 + 1/66
157:2/27 = 1/18 + 1/54
142:2/21 = 1/14 + 1/42
127:2/15 = 1/10 + 1/30
70:
43:(sums of distinct
41:Egyptian fractions
29:mathematical table
1128:Archiv Orientální
1108:Imhausen, Annette
1037:978-0-87169-232-0
963:Imhausen, Annette
781:
763:
745:
727:
712:
668:
658:
645:
634:
620:
579:
569:
556:
546:
531:
492:
482:
469:
459:
446:
419:
401:
383:
346:
345:
27:work, includes a
1185:
1137:
1135:
1123:
1117:
1115:
1104:
1098:
1097:
1082:
1076:
1069:
1063:
1061:
1053:
1042:
1040:
1029:
1019:
1013:
1011:
992:
986:
985:
959:
953:
951:
943:
937:
927:
919:
896:
894:
893:
890:
887:
792:
790:
789:
784:
782:
780:
769:
764:
762:
751:
746:
744:
733:
728:
720:
713:
705:
679:
677:
676:
671:
669:
661:
659:
651:
646:
638:
635:
627:
621:
619:
608:
590:
588:
587:
582:
580:
572:
570:
562:
557:
549:
547:
539:
532:
524:
503:
501:
500:
495:
493:
485:
483:
475:
470:
462:
460:
452:
447:
439:
430:
428:
427:
422:
420:
418:
407:
402:
400:
389:
384:
382:
371:
328:
313:
298:
283:
268:
253:
238:
223:
208:
193:
178:
163:
148:
133:
118:
111:
102:
95:
92:
85:
79:
33:rational numbers
22:ancient Egyptian
1193:
1192:
1188:
1187:
1186:
1184:
1183:
1182:
1143:
1142:
1141:
1140:
1125:
1124:
1120:
1106:
1105:
1101:
1084:
1083:
1079:
1070:
1066:
1055:
1054:
1045:
1038:
1021:
1020:
1016:
994:
993:
989:
983:
961:
960:
956:
945:
944:
940:
921:
920:
916:
911:
891:
888:
885:
884:
882:
837:
831:
773:
755:
737:
698:
697:
612:
601:
600:
595:divisible by 5)
517:
516:
433:
432:
411:
393:
375:
365:
364:
354:
326:
311:
296:
281:
266:
251:
236:
221:
206:
191:
176:
161:
146:
131:
116:
109:
100:
93:
90:
83:
61:
31:for converting
12:
11:
5:
1191:
1189:
1181:
1180:
1175:
1170:
1165:
1160:
1155:
1145:
1144:
1139:
1138:
1118:
1099:
1077:
1064:
1043:
1036:
1014:
987:
981:
954:
938:
913:
912:
910:
907:
836:
833:
799:
798:
779:
776:
772:
767:
761:
758:
754:
749:
743:
740:
736:
731:
726:
723:
717:
711:
708:
694:
693:
667:
664:
657:
654:
649:
644:
641:
633:
630:
625:
618:
615:
611:
597:
596:
578:
575:
568:
565:
560:
555:
552:
545:
542:
536:
530:
527:
510:
509:
491:
488:
481:
478:
473:
468:
465:
458:
455:
450:
445:
442:
417:
414:
410:
405:
399:
396:
392:
387:
381:
378:
374:
353:
350:
344:
343:
341:
338:
334:
333:
330:
323:
319:
318:
315:
308:
304:
303:
300:
293:
289:
288:
285:
284:= 1/44 + 1/308
278:
274:
273:
270:
263:
259:
258:
255:
254:= 1/39 + 1/195
248:
244:
243:
240:
233:
229:
228:
225:
218:
214:
213:
210:
203:
199:
198:
195:
188:
184:
183:
180:
173:
169:
168:
165:
158:
154:
153:
150:
149:= 1/12 + 1/276
143:
139:
138:
135:
128:
124:
123:
120:
113:
105:
104:
97:
87:
60:
57:
45:unit fractions
35:of the form 2/
13:
10:
9:
6:
4:
3:
2:
1190:
1179:
1176:
1174:
1171:
1169:
1166:
1164:
1161:
1159:
1156:
1154:
1151:
1150:
1148:
1133:
1129:
1122:
1119:
1113:
1109:
1103:
1100:
1095:
1091:
1087:
1081:
1078:
1074:
1068:
1065:
1059:
1052:
1050:
1048:
1044:
1039:
1033:
1028:
1027:
1018:
1015:
1010:
1006:
1002:
998:
991:
988:
984:
982:9780691209074
978:
974:
970:
969:
964:
958:
955:
949:
942:
939:
935:
934:0-87353-133-7
931:
925:
918:
915:
908:
906:
904:
900:
880:
876:
872:
867:
865:
861:
857:
852:
850:
846:
842:
834:
832:
829:
827:
823:
819:
816:
812:
808:
804:
796:
777:
774:
770:
765:
759:
756:
752:
747:
741:
738:
734:
729:
724:
721:
715:
709:
706:
696:
695:
691:
687:
683:
665:
662:
655:
652:
647:
642:
639:
631:
628:
623:
616:
613:
609:
599:
598:
594:
576:
573:
566:
563:
558:
553:
550:
543:
540:
534:
528:
525:
515:
514:
513:
507:
489:
486:
479:
476:
471:
466:
463:
456:
453:
448:
443:
440:
415:
412:
408:
403:
397:
394:
390:
385:
379:
376:
372:
363:
362:
361:
358:
351:
349:
342:
339:
336:
335:
331:
324:
321:
320:
316:
309:
306:
305:
301:
294:
291:
290:
286:
279:
276:
275:
271:
264:
261:
260:
256:
249:
246:
245:
241:
234:
231:
230:
226:
219:
216:
215:
211:
204:
201:
200:
196:
189:
186:
185:
181:
179:= 1/30 + 1/42
174:
171:
170:
166:
159:
156:
155:
151:
144:
141:
140:
136:
129:
126:
125:
121:
114:
107:
106:
103:= 1/4 + 1/28
98:
88:
81:
80:
77:
75:
68:
66:
58:
56:
54:
50:
46:
42:
38:
34:
30:
26:
23:
19:
1131:
1127:
1121:
1111:
1102:
1093:
1086:Imhausen, A.
1080:
1067:
1057:
1025:
1017:
1000:
996:
990:
967:
957:
947:
941:
923:
917:
902:
898:
879:Eye of Horus
874:
868:
863:
859:
853:
848:
844:
838:
830:
825:
814:
811:prime number
806:
802:
800:
794:
689:
685:
681:
592:
511:
505:
359:
355:
352:Explanations
347:
119:= 1/6 + 1/66
112:= 1/6 + 1/18
96:= 1/3 + 1/15
73:
71:
62:
36:
25:mathematical
15:
1003:: 295–337,
86:= 1/2 + 1/6
1147:Categories
1090:"UC 32159"
909:References
818:composite
1088:(2002),
1009:25150159
965:(2016),
1163:Papyrus
895:
883:
851:table.
805:(where
680:(where
327:
312:
297:
282:
267:
252:
237:
222:
207:
192:
177:
162:
147:
132:
117:
110:
101:
94:
91:
84:
65:papyrus
1034:
1007:
979:
932:
809:was a
1005:JSTOR
973:p. 65
59:Table
53:Ahmes
39:into
20:, an
1032:ISBN
977:ISBN
930:ISBN
869:The
854:The
688:and
325:2/95
310:2/89
295:2/83
280:2/77
265:2/71
250:2/65
235:2/59
220:2/53
205:2/47
190:2/41
175:2/35
160:2/29
145:2/23
130:2/17
115:2/11
16:The
108:2/9
99:2/7
89:2/5
82:2/3
1149::
1132:70
1130:,
1092:,
1046:^
1001:17
999:,
975:,
903:ro
899:ro
815:pq
72:2/
67:.
1136:.
1075:.
1062:.
1041:.
1012:.
952:.
936:.
892:2
889:/
886:1
875:n
864:n
860:n
849:n
845:n
826:p
807:p
803:p
795:n
778:n
775:6
771:1
766:+
760:n
757:3
753:1
748:+
742:n
739:2
735:1
730:+
725:n
722:1
716:=
710:n
707:2
692:)
690:n
686:m
682:k
666:k
663:1
656:n
653:1
648:+
643:k
640:1
632:m
629:1
624:=
617:n
614:m
610:2
593:n
591:(
577:n
574:1
567:3
564:5
559:+
554:n
551:1
544:3
541:1
535:=
529:n
526:2
506:n
504:(
490:n
487:1
480:2
477:3
472:+
467:n
464:1
457:2
454:1
449:=
444:n
441:2
416:n
413:6
409:1
404:+
398:n
395:2
391:1
386:=
380:n
377:3
373:2
74:n
37:n
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