651:
3282:
3683:
3226:
3178:
3294:
3215:
3246:
3256:
3236:
1635:
1592:
1546:
1500:
1443:
1365:
1243:
688:
2079:
Problem 56 of the RMP indicates an understanding of the idea of geometric similarity. This problem discusses the ratio run/rise, also known as the seqed. Such a formula would be needed for building pyramids. In the next problem (Problem 57), the height of a pyramid is calculated from the base length
127:
which depicts offerings of 400,000 oxen, 1,422,000 goats and 120,000 prisoners. Archaeological evidence has suggested that the
Ancient Egyptian counting system had origins in Sub-Saharan Africa. Also, fractal geometry designs which are widespread among Sub-Saharan African cultures are also found in
1450:
The table above can also be used to divide 1120 by 80. We would solve this problem by finding the quotient (80) as the sum of those multipliers of 80 that add up to 1120. In this example that would yield a quotient of 10 + 4 = 14. A more complicated example of the division
283:
or sums of such unit fractions. Scribes used tables to help them work with these fractions. The
Egyptian Mathematical Leather Roll for instance is a table of unit fractions which are expressed as sums of other unit fractions. The Rhind Mathematical Papyrus and some of the other texts contain
1913:. The technique is also called the method of false assumption. The scribe would substitute an initial guess of the answer into the problem. The solution using the false assumption would be proportional to the actual answer, and the scribe would find the answer by using this ratio.
1085:
arithmetic), a method that links to the Old
Kingdom. The multiplicand was written next to figure 1; the multiplicand was then added to itself, and the result written next to the number 2. The process was continued until the doublings gave a number greater than half of the
481:
for cattle, number 100 is represented by a coiled rope, the number 1000 is represented by a lotus flower, the number 10,000 is represented by a finger, the number 100,000 is represented by a frog, and a million was represented by a god with his hands raised in adoration.
189:
The Moscow
Mathematical Papyrus and Rhind Mathematical Papyrus are so called mathematical problem texts. They consist of a collection of problems with solutions. These texts may have been written by a teacher or a student engaged in solving typical mathematics problems.
2036:: Several problems compute the volume of cylindrical granaries (RMP 41–43), while problem 60 RMP seems to concern a pillar or a cone instead of a pyramid. It is rather small and steep, with a seked (reciprocal of slope) of four palms (per cubit). In section IV.3 of the
1643:
First the scribe would double 365 repeatedly until the largest possible multiple of 365 is reached, which is smaller than 3200. In this case 8 times 365 is 2920 and further addition of multiples of 365 would clearly give a value greater than 3200. Next it is noted that
2084:(Egyptian for the reciprocal of the slope), while problem 58 gives the length of the base and the height and uses these measurements to compute the seqed. In Problem 59 part 1 computes the seqed, while the second part may be a computation to check the answer:
477:. In either representation the number system was always given in base 10. The number 1 was depicted by a simple stroke, the number 2 was represented by two strokes, etc. The numbers 10, 100, 1000, 10,000 and 100,000 had their own hieroglyphs. Number 10 is a
355:
Current understanding of ancient
Egyptian mathematics is impeded by the paucity of available sources. The sources that do exist include the following texts (which are generally dated to the Middle Kingdom and Second Intermediate Period):
699:). In the middle register we see 835 horned cattle on the left, right behind them are some 220 animals (cows?) and on the right 2235 goats. In the bottom register we see 760 donkeys on the left and 974 goats on the right.
2012:
Problem 48 of the RMP compares the area of a circle (approximated by an octagon) and its circumscribing square. This problem's result is used in problem 50, where the scribe finds the area of a round field of diameter 9
1916:
The mathematical writings show that the scribes used (least) common multiples to turn problems with fractions into problems using integers. In this connection red auxiliary numbers are written next to the fractions.
1090:. Then the doubled numbers (1, 2, etc.) would be repeatedly subtracted from the multiplier to select which of the results of the existing calculations should be added together to create the answer.
2758:, edited by Jack M. Sasson, John R. Baines, Gary Beckman, and Karen S. Rubinson. Vol. 3 of 4 vols. New York: Charles Schribner's Sons. (Reprinted Peabody: Hendrickson Publishers, 2000). 1799–1813
1904:
430:, a literary text written as a (fictional) letter written by a scribe named Hori and addressed to a scribe named Amenemope. A segment of the letter describes several mathematical problems.
2416:
Clagett, Marshall
Ancient Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics (Memoirs of the American Philosophical Society) American Philosophical Society. 1999
1920:
The use of the Horus eye fractions shows some (rudimentary) knowledge of geometrical progression. Knowledge of arithmetic progressions is also evident from the mathematical sources.
1081:
Egyptian multiplication was done by a repeated doubling of the number to be multiplied (the multiplicand), and choosing which of the doublings to add together (essentially a form of
1955:
684:
The
Egyptian number system was additive. Large numbers were represented by collections of the glyphs and the value was obtained by simply adding the individual numbers together.
123:. These labels appear to have been used as tags for grave goods and some are inscribed with numbers. Further evidence of the use of the base 10 number system can be found on the
1832:
also contains four of these type of problems. Problems 1, 19, and 25 of the Moscow
Papyrus are Aha problems. For instance problem 19 asks one to calculate a quantity taken
785:
was represented by the glyph for a mouth with 2 (different sized) strokes. The rest of the fractions were always represented by a mouth super-imposed over a number.
2626:. 2 vols. Classics in Mathematics Education 8. Oberlin: Mathematical Association of America. (Reprinted Reston: National Council of Teachers of Mathematics, 1979).
439:
Ostraca from Deir el-Medina contain computations. Ostracon IFAO 1206 for instance shows the calculation of volumes, presumably related to the quarrying of a tomb.
677:
record the use of this number system. It is also common to see the numerals in offering scenes to indicate the number of items offered. The king's daughter
3408:
31:
3326:
2624:
The Rhind
Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations
2681:
Johnson, G., Sriraman, B., Saltztstein. 2012. "Where are the plans? A socio-critical and architectural survey of early
Egyptian mathematics"| In
2845:
3558:
3393:
2872:
2898:
3599:
2970:
889:
3625:
3199:
3027:
1828:
Aha problems involve finding unknown quantities (referred to as Aha) if the sum of the quantity and part(s) of it are given. The
1781:
1417:
1394:
1388:
1386:
1324:
1322:
1270:
1269:
1268:
1266:
1265:
1264:
1230:
1155:
1154:
1153:
1152:
1150:
1149:
1148:
1147:
1100:(RMP) provides the following illustration, as if Hieroglyphic symbols were used (rather than the RMP's actual hieratic script).
1057:
1047:
1044:
1016:
1010:
929:
636:
598:
579:
541:
3503:
1779:
1055:
972:
948:
908:
2826:
2745:
2715:
2421:
2329:
2283:
2211:
2122:
1776:
1416:
1415:
1414:
1413:
1391:
1352:
1351:
1350:
1349:
1330:
1328:
1327:
1293:
1292:
1273:
1211:
1210:
1209:
1208:
1206:
1205:
1204:
1203:
1174:
1053:
978:
977:
976:
975:
974:
953:
952:
951:
950:
910:
617:
560:
522:
148:
1966:
There are only a limited number of problems from ancient Egypt that concern geometry. Geometric problems appear in both the
3319:
3145:
2444:
17:
2000:
Problems regarding the area of a rectangular plot of land appear in the RMP and the MMP. A similar problem appears in the
3620:
3381:
3259:
3017:
3012:
2893:
2117:
997:
Steps of calculations were written in sentences in Egyptian languages. (e.g. "Multiply 10 times 100; it becomes 1000.")
3553:
3518:
2865:
2690:
367:
163:
3707:
2982:
2791:
2770:
2649:
2631:
2539:
2460:
2356:
119:
Written evidence of the use of mathematics dates back to at least 3200 BC with the ivory labels found in Tomb U-j at
3661:
1861:
1093:
As a shortcut for larger numbers, the multiplicand can also be immediately multiplied by 10, 100, 1000, 10000, etc.
3649:
3530:
3483:
3135:
2903:
3686:
3615:
3425:
3312:
3167:
3113:
2965:
1076:
650:
76:
68:
3476:
3464:
3239:
3140:
2564:
2236:
1031:
1027:
2840:
143:
which gives guidelines for the slope of the mastaba. The lines in the diagram are spaced at a distance of one
3712:
3630:
3459:
3398:
2926:
2921:
2858:
1692:
times 365 gives us the value of 280 we need. Hence we find that 3200 divided by 365 must equal 8 +
96:
2948:
2916:
2131:
1851:
times and added to 4 to make 10. In other words, in modern mathematical notation we are asked to solve the
1451:
algorithm is provided by Problem 66. A total of 3200 ro of fat are to be distributed evenly over 365 days.
3157:
3120:
3068:
2987:
2126:
2086:
If you construct a pyramid with base side 12 and with a seqed of 5 palms 1 finger; what is its altitude?
2049:
1967:
1959:
1759:
422:
From the New Kingdom there are a handful of mathematical texts and inscriptions related to computations:
361:
197:
mathematics is the use of unit fractions. The Egyptians used some special notation for fractions such as
159:
1034:), symbols for feet, were used to mean "to add" and "to subtract." These were presumably shorthands for
3272:
3007:
2737:
2053:
1971:
1937:
1829:
1755:
407:
403:
183:
179:
737:, which is frequently found in the mathematical texts. Very rarely a special glyph was used to denote
3491:
3130:
3098:
3083:
3078:
2977:
2931:
2037:
2001:
373:
167:
3420:
3249:
3125:
3046:
2997:
2644:. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society.
1910:
396:
155:
3454:
3442:
3413:
3339:
2656:
Mathématiques égyptiennes: Recherches sur les connaissances mathématiques de l'Égypte pharaonique
2585:
Mathematics before the Greeks Science, New Series, Vol.73, No. 1831, (Jan. 31, 1930), pp. 109–121
2582:
1978:
knew how to compute areas of several geometric shapes and the volumes of cylinders and pyramids.
415:
2801:. Quellen und Studien zur Geschichte der Mathematik; Abteilung A: Quellen 1. Berlin: J. Springer
3574:
3508:
3369:
3335:
3229:
3162:
3002:
2438:
2113:
2101:
769:
was represented by a glyph that may have depicted a piece of linen folded in two. The fraction
3656:
3668:
3523:
3498:
3430:
3376:
3194:
2346:
1082:
104:
3548:
3447:
3298:
3219:
3177:
3034:
2943:
2936:
2096:
1808:
1797:
696:
470:
386:
324:
56:
8:
3589:
3437:
3388:
3359:
3039:
2960:
2106:
2040:
the volume of a granary with a circular base is found using the same procedure as RMP 43.
1933:
379:
175:
3594:
3354:
3286:
3093:
3056:
2955:
2707:
2697:
2693:
Monographs in Mathematics Education 12, Information Age Publishing, Inc., Charlotte, NC
2600:
2527:
2520:
2482:
2466:
2179:
1929:
427:
328:
108:
72:
2728:. London: The University Press of Liverpool limited and Hodder & Stoughton limited
1936:
fragment. Additionally, the Egyptians solve first-degree algebraic equations found in
1928:
The ancient Egyptians were the first civilization to develop and solve second-degree (
1447:
denotes the intermediate results that are added together to produce the final answer.
186:(c. 1650 BC) is said to be based on an older mathematical text from the 12th dynasty.
3584:
3281:
3051:
2992:
2822:
2787:
2766:
2754:
Robins, R. Gay. 1995. "Mathematics, Astronomy, and Calendars in Pharaonic Egypt". In
2741:
2711:
2645:
2627:
2560:
2535:
2486:
2474:
2456:
2417:
2352:
2325:
2279:
2232:
2207:
2183:
1975:
1949:
670:
464:
460:
262:
84:
64:
60:
30:"Mathematics in Ancient Egypt" redirects here. For the book by Annette Imhausen, see
3304:
2434:
1962:. The problem includes a diagram indicating the dimensions of the truncated pyramid.
444:
3471:
3108:
3022:
2682:
2675:
2594:
2448:
2313:
2171:
2159:
1804:
1749:
194:
135:(c. 2690–2180 BC) is scarce, but can be deduced from inscriptions on a wall near a
100:
2701:
1852:
478:
392:
332:
124:
3088:
2515:
2317:
1087:
687:
347:
have been found that record volumes of dirt removed while quarrying the tombs.
340:
2478:
662:(dated 2590–2565 BC) from her tomb at Giza, painting on limestone, now in the
3701:
2881:
2271:
1097:
674:
302:
tables. These tables allowed the scribes to rewrite any fraction of the form
120:
83:. From these texts it is known that ancient Egyptians understood concepts of
45:
2322:
The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook
3150:
2799:
Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau
2663:
Cairo Museo des Antiquities Egyptiennes Catalogue General Ostraca hieraques
2470:
1992:
The scribes record problems computing the area of a triangle (RMP and MMP).
692:
678:
659:
336:
171:
88:
2162:(2006). "Ancient Egyptian Mathematics: New Perspectives on Old Sources".
655:
132:
41:
2452:
67:
for counting and solving written mathematical problems, often involving
3189:
2850:
2175:
52:
2734:
Count Like an Egyptian: A Hands-on Introduction to Ancient Mathematics
2348:
Count Like an Egyptian: A Hands-on Introduction to Ancient Mathematics
327:(c. 1550–1070 BC) mathematical problems are mentioned in the literary
75:. Evidence for Egyptian mathematics is limited to a scarce amount of
3061:
2559:(2nd ed.). Detroit, Mich.: Professional Educational Services.
2252:
Eglash, R. (1995). "Fractal Geometry in African Material Culture".
2231:. New Brunswick, N.J.: Rutgers University Press. pp. 89, 141.
474:
344:
2797:
Struve, Vasilij Vasil'eviÄŤ, and Boris AleksandroviÄŤ Turaev. 1930.
2687:
Crossroads in the History of Mathematics and Mathematics Education
2500:
Chace, Arnold Buffum; Bull, Ludlow; Manning, Henry Parker (1929).
3103:
2508:
2062:
2056:(numbers 44, 45, 46) compute the volume of a rectangular granary.
1954:
448:
136:
80:
663:
140:
92:
2726:
The Rhind Mathematical Papyrus, British Museum 10057 and 10058
2437:(February 15, 2019). "From Scotus Eriugena to Saint Bernard".
2229:
African fractals : modern computing and indigenous design
411:
144:
703:
The Egyptians almost exclusively used fractions of the form
681:
is shown with an offering of 1000 oxen, bread, beer, etc.
2678:. 2003. "Ă„gyptische Algorithmen". Wiesbaden: Harrassowitz
2763:
The Rhind Mathematical Papyrus: An Ancient Egyptian Text
2068:
The volume of a truncated pyramid is computed in MMP 14.
2311:
1909:
Solving these Aha problems involves a technique called
3334:
3270:
2021:
Problem 10 in the MMP finds the area of a hemisphere.
1864:
339:
records land measurements. In the workers village of
154:
The earliest true mathematical documents date to the
399:
and found in Nag el-Deir, the ancient town of Thinis
2819:
Die Pyramiden von Giza – Mathematik in Stein gebaut
2782:Strudwick, Nigel G., and Ronald J. Leprohon. 2005.
2504:. Vol. 2. Mathematical Association of America.
2519:
2440:History of Christian Philosophy in the Middle Ages
1898:
469:Ancient Egyptian texts could be written in either
418:papyrus. The RMP is the largest mathematical text.
170:which are a part of the much larger collection of
32:Mathematics in Ancient Egypt: A Contextual History
2578:
2576:
451:"taught the Egyptians arythmetic and astronomy".
3699:
2619:. John Wiley. Reprint Princeton U. Press (1985).
2499:
436:Ostracon Turin 57170, a text written in hieratic
2765:. London: British Museum Publications Limited.
2761:Robins, R. Gay, and Charles C. D. Shute. 1987.
433:Ostracon Senmut 153, a text written in hieratic
27:Mathematics developed and used in Ancient Egypt
2573:
1932:) equations. This information is found in the
1899:{\displaystyle {\frac {3}{2}}\times x+4=10.\ }
1070:
131:The evidence of the use of mathematics in the
128:Egyptian architecture and cosmological signs.
3320:
2866:
2276:Architecture and Mathematics in Ancient Egypt
1754:Egyptian algebra problems appear in both the
1000:In Rhind Papyrus Problem 28, the hieroglyphs
2806:Science Awakening". Oxford University Press.
2320:, Dauben JW, Plofker K, Berggren JL (2007).
2204:The History of Mathematics: An Introduction
1453:
1102:
3327:
3313:
2873:
2859:
2696:
2427:
2412:
2410:
2408:
2406:
2404:
2402:
2400:
2398:
2396:
2394:
2392:
2390:
2388:
2251:
261:, but other fractions were all written as
2386:
2384:
2382:
2380:
2378:
2376:
2374:
2372:
2370:
2368:
1974:(RMP). The examples demonstrate that the
2880:
2814:, Archiv OrientálnĂ, Vol 1, pages 27–42.
2672:. MIT Press. (Dover reprints available).
2197:
2195:
2193:
2158:
1953:
721:. One notable exception is the fraction
686:
649:
414:, identifies it as a copy of a now lost
2670:Mathematics in the Time of the Pharaohs
2638:Ancient Egyptian Science: A Source Book
2548:
95:of three-dimensional shapes useful for
14:
3700:
2777:Introduction to the History of Science
2756:Civilizations of the Ancient Near East
2731:
2514:
2493:
2433:
2365:
2344:
2307:
2305:
2303:
2301:
2299:
2297:
2295:
2226:
2201:
1923:
3308:
2854:
2554:
2270:
2190:
2123:Ancient Egyptian units of measurement
3235:
2445:Catholic University of America Press
2154:
2152:
2150:
2148:
2146:
1067:meaning "to go in" and "to go out."
486:Hieroglyphics for Egyptian numerals
3255:
2598:Digitalegypt website: Lahun Papyrus
2522:A History of Mathematical Notations
2292:
2118:Transliteration of Ancient Egyptian
669:Egyptian numerals date back to the
24:
2691:The Montana Mathematics Enthusiast
2608:
1762:as well as several other sources.
368:Egyptian Mathematical Leather Roll
164:Egyptian Mathematical Leather Roll
25:
3724:
2983:Ancient Egyptian race controversy
2846:Introduction to Early Mathematics
2834:
2779:, Vol 1. Willians & Williams.
2658:. Paris: Éditions Le Léopard d'Or
2622:Chace, Arnold Buffum. 1927–1929.
2143:
789:Hieroglyphics for some fractions
63:. The ancient Egyptians utilized
3682:
3681:
3292:
3280:
3254:
3244:
3234:
3225:
3224:
3213:
3176:
2557:The African roots of mathematics
1633:
1590:
1544:
1498:
1441:
1363:
1241:
3245:
2703:The Exact Sciences in Antiquity
2665:, vol 1901, number 25001-25385.
2588:
1096:For example, Problem 69 on the
1077:Ancient Egyptian multiplication
787:
59:until roughly the beginning of
44:that was developed and used in
2502:The Rhind Mathematical Papyrus
2351:. Princeton University Press.
2338:
2324:. Princeton University Press.
2278:. Cambridge University Press.
2264:
2245:
2220:
2164:The Mathematical Intelligencer
410:(c. 1650 BC), but its author,
13:
1:
2812:Wooden Tablets from Cairo....
2786:. Brill Academic Publishers.
2254:Symmetry: Culture and Science
2137:
1958:Image of Problem 14 from the
178:all date to this period. The
48:
2804:Van der Waerden, B.L. 1961.
2642:Ancient Egyptian Mathematics
2555:Moore, Deborah Lela (1994).
2345:Reimer, David (2014-05-11).
2132:Mathematics and architecture
2074:
2061:Truncated pyramid (frustum)
695:(copied by the Egyptologist
320:as a sum of unit fractions.
38:Ancient Egyptian mathematics
7:
3168:Egypt–Mesopotamia relations
2988:Population history of Egypt
2668:Gillings, Richard J. 1972.
2090:
2050:Moscow Mathematical Papyrus
1968:Moscow Mathematical Papyrus
1960:Moscow Mathematical Papyrus
1943:
1760:Moscow mathematical papyrus
1071:Multiplication and division
992:
454:
362:Moscow Mathematical Papyrus
160:Moscow Mathematical Papyrus
114:
10:
3729:
2821:. (2 ed) Books on Demand.
2784:Texts from the Pyramid Age
2738:Princeton University Press
2054:Rhind Mathematical Papyrus
1972:Rhind Mathematical Papyrus
1947:
1938:Rhind Mathematical Papyrus
1830:Rhind Mathematical Papyrus
1768:
1756:Rhind mathematical papyrus
1747:
1743:
1405:
1378:
1341:
1314:
1284:
1256:
1222:
1195:
1166:
1139:
1074:
1036:
1002:
964:
940:
921:
900:
881:
628:
609:
590:
571:
552:
533:
514:
458:
408:Second Intermediate Period
404:Rhind Mathematical Papyrus
350:
193:An interesting feature of
184:Second Intermediate Period
180:Rhind Mathematical Papyrus
87:, such as determining the
29:
3677:
3639:
3608:
3567:
3541:
3347:
3208:
3185:
3174:
2912:
2889:
2724:Peet, Thomas Eric. 1923.
2636:Clagett, Marshall. 1999.
2038:Lahun Mathematical Papyri
2002:Lahun Mathematical Papyri
1117:
1112:
1105:
382:, written around 1800 BC
374:Lahun Mathematical Papyri
168:Lahun Mathematical Papyri
147:and show the use of that
97:architectural engineering
3708:Ancient Egyptian society
3220:Ancient Egypt portal
2817:Wirsching, Armin. 2009.
2810:Vymazalova, Hana. 2002.
2654:Couchoud, Sylvia. 1993.
2052:(problem 14) and in the
2048:Several problems in the
1911:method of false position
1107:To multiply 80 × 14
397:Twelfth dynasty of Egypt
2661:Daressy, G. "Ostraca,"
158:(c. 1990–1800 BC). The
3600:Medieval Islamic world
3336:History of mathematics
2775:Sarton, George. 1927.
2732:Reimer, David (2014).
2617:History of Mathematics
2202:Burton, David (2005).
2114:Egyptian hieroglyphics
2102:History of mathematics
2033:Cylindrical (cylinder)
1963:
1900:
700:
666:
406:(RMP), dated from the
245:and in some texts for
3669:Future of mathematics
3646:Women in mathematics
2894:Glossary of artifacts
2615:Boyer, Carl B. 1968.
2045:Rectangular (Cuboid):
1957:
1901:
1113:Egyptian calculation
691:This scene depicts a
690:
653:
395:, dated to the early
105:false position method
3713:Egyptian mathematics
3621:Over Cantor's theory
2227:Eglash, Ron (1999).
2097:Red auxiliary number
1862:
1456:Dividing 3200 by 365
673:. Ivory labels from
387:Akhmim Wooden Tablet
57:Old Kingdom of Egypt
51:3000 to c. 300
18:Egyptian mathematics
3657:Approximations of π
3568:By ancient cultures
3040:Cursive hieroglyphs
2841:Egyptian Arithmetic
2453:10.2307/j.ctvdf0jnn
2107:History of geometry
1924:Quadratic equations
1813:(1550–1069 BC)
1458:
1118:Modern calculation
790:
487:
380:Berlin Papyrus 6619
182:which dates to the
176:Berlin Papyrus 6619
149:unit of measurement
109:quadratic equations
3460:Information theory
3013:Funerary practices
2708:Dover Publications
2528:Dover Publications
2176:10.1007/bf02986998
1964:
1896:
1454:
788:
701:
671:Predynastic period
667:
485:
428:Papyrus Anastasi I
329:Papyrus Anastasi I
3695:
3694:
3531:Separation axioms
3268:
3267:
3023:Great Royal Wives
2993:Prehistoric Egypt
2827:978-3-8370-2355-8
2747:978-0-691-16012-2
2736:. Princeton, NJ:
2717:978-0-486-22332-2
2676:Imhausen, Annette
2443:. Washington DC:
2422:978-0-87169-232-0
2331:978-0-691-11485-9
2285:978-0-521-69053-9
2213:978-0-07-305189-5
2160:Imhausen, Annette
1976:Ancient Egyptians
1970:(MMP) and in the
1950:Egyptian geometry
1895:
1873:
1826:
1825:
1814:
1789:
1788:
1785:
1784:
1641:
1640:
1437:
1436:
1425:
1424:
1421:
1420:
1402:
1401:
1398:
1397:
1360:
1359:
1356:
1355:
1338:
1337:
1334:
1333:
1301:
1300:
1297:
1296:
1281:
1280:
1277:
1276:
1238:
1237:
1234:
1233:
1219:
1218:
1215:
1214:
1182:
1181:
1178:
1177:
1163:
1162:
1159:
1158:
1065:
1064:
1061:
1060:
1024:
1023:
1020:
1019:
990:
989:
986:
985:
982:
981:
961:
960:
957:
956:
937:
936:
933:
932:
918:
917:
914:
913:
897:
896:
893:
892:
648:
647:
644:
643:
640:
639:
625:
624:
621:
620:
606:
605:
602:
601:
587:
586:
583:
582:
568:
567:
564:
563:
549:
548:
545:
544:
530:
529:
526:
525:
465:Egyptian fraction
461:Egyptian numerals
335:from the time of
77:surviving sources
61:Hellenistic Egypt
16:(Redirected from
3720:
3685:
3684:
3405:Category theory
3329:
3322:
3315:
3306:
3305:
3297:
3296:
3295:
3285:
3284:
3276:
3258:
3257:
3248:
3247:
3238:
3237:
3228:
3227:
3218:
3217:
3216:
3180:
2875:
2868:
2861:
2852:
2851:
2751:
2721:
2698:Neugebauer, Otto
2683:Bharath Sriraman
2603:
2599:
2595:Annette Imhausen
2592:
2586:
2580:
2571:
2570:
2552:
2546:
2545:
2525:
2512:
2506:
2505:
2497:
2491:
2490:
2431:
2425:
2414:
2363:
2362:
2342:
2336:
2335:
2309:
2290:
2289:
2268:
2262:
2261:
2249:
2243:
2242:
2224:
2218:
2217:
2199:
2188:
2187:
2156:
1905:
1903:
1902:
1897:
1893:
1874:
1866:
1850:
1848:
1847:
1844:
1841:
1837:
1812:
1773:
1772:
1769:
1765:
1764:
1750:Egyptian algebra
1739:
1737:
1736:
1733:
1730:
1723:
1721:
1720:
1717:
1714:
1707:
1705:
1704:
1701:
1698:
1691:
1689:
1688:
1685:
1682:
1675:
1673:
1672:
1669:
1666:
1659:
1657:
1656:
1653:
1650:
1637:
1636:
1630:
1628:
1627:
1624:
1621:
1613:
1611:
1610:
1607:
1604:
1594:
1593:
1587:
1585:
1584:
1581:
1578:
1574:
1567:
1565:
1564:
1561:
1558:
1548:
1547:
1541:
1539:
1538:
1535:
1532:
1528:
1521:
1519:
1518:
1515:
1512:
1502:
1501:
1459:
1445:
1444:
1410:
1409:
1406:
1383:
1382:
1379:
1367:
1366:
1346:
1345:
1342:
1319:
1318:
1315:
1289:
1288:
1285:
1261:
1260:
1257:
1245:
1244:
1227:
1226:
1223:
1200:
1199:
1196:
1171:
1170:
1167:
1144:
1143:
1140:
1103:
1041:
1040:
1037:
1007:
1006:
1003:
969:
968:
965:
945:
944:
941:
926:
925:
922:
905:
904:
901:
886:
885:
882:
876:
874:
873:
870:
867:
859:
857:
856:
853:
850:
842:
840:
839:
836:
833:
825:
823:
822:
819:
816:
808:
806:
805:
802:
799:
791:
784:
782:
781:
778:
775:
768:
766:
765:
762:
759:
752:
750:
749:
746:
743:
736:
734:
733:
730:
727:
720:
718:
717:
712:
709:
633:
632:
629:
614:
613:
610:
595:
594:
591:
576:
575:
572:
557:
556:
553:
538:
537:
534:
519:
518:
515:
488:
484:
319:
317:
316:
311:
308:
301:
299:
298:
293:
290:
282:
280:
279:
274:
271:
260:
258:
257:
254:
251:
244:
242:
241:
238:
235:
228:
226:
225:
222:
219:
212:
210:
209:
206:
203:
195:ancient Egyptian
65:a numeral system
50:
21:
3728:
3727:
3723:
3722:
3721:
3719:
3718:
3717:
3698:
3697:
3696:
3691:
3673:
3635:
3616:Brouwer–Hilbert
3604:
3563:
3542:Numeral systems
3537:
3399:Grandi's series
3343:
3333:
3303:
3293:
3291:
3279:
3271:
3269:
3264:
3214:
3212:
3204:
3181:
3172:
2908:
2885:
2879:
2837:
2832:
2748:
2718:
2611:
2609:Further reading
2606:
2597:
2593:
2589:
2581:
2574:
2567:
2553:
2549:
2542:
2516:Cajori, Florian
2513:
2509:
2498:
2494:
2463:
2447:. p. 265.
2435:Gilson, Étienne
2432:
2428:
2415:
2366:
2359:
2343:
2339:
2332:
2310:
2293:
2286:
2269:
2265:
2250:
2246:
2239:
2225:
2221:
2214:
2206:. McGraw–Hill.
2200:
2191:
2157:
2144:
2140:
2093:
2077:
1952:
1946:
1926:
1865:
1863:
1860:
1859:
1853:linear equation
1845:
1842:
1839:
1838:
1835:
1833:
1811:
1795:
1780:
1752:
1746:
1734:
1731:
1728:
1727:
1725:
1718:
1715:
1712:
1711:
1709:
1702:
1699:
1696:
1695:
1693:
1686:
1683:
1680:
1679:
1677:
1670:
1667:
1664:
1663:
1661:
1654:
1651:
1648:
1647:
1645:
1634:
1625:
1622:
1619:
1618:
1616:
1608:
1605:
1602:
1601:
1599:
1591:
1582:
1579:
1576:
1575:
1572:
1570:
1562:
1559:
1556:
1555:
1553:
1545:
1536:
1533:
1530:
1529:
1526:
1524:
1516:
1513:
1510:
1509:
1507:
1499:
1442:
1387:
1364:
1329:
1323:
1267:
1242:
1207:
1151:
1079:
1073:
1056:
1054:
995:
973:
949:
909:
871:
868:
865:
864:
862:
854:
851:
848:
847:
845:
837:
834:
831:
830:
828:
820:
817:
814:
813:
811:
803:
800:
797:
796:
794:
779:
776:
773:
772:
770:
763:
760:
757:
756:
754:
753:. The fraction
747:
744:
741:
740:
738:
731:
728:
725:
724:
722:
713:
710:
707:
706:
704:
467:
459:Main articles:
457:
393:Reisner Papyrus
353:
333:Papyrus Wilbour
312:
309:
306:
305:
303:
294:
291:
288:
287:
285:
275:
272:
269:
268:
266:
255:
252:
249:
248:
246:
239:
236:
233:
232:
230:
223:
220:
217:
216:
214:
207:
204:
201:
200:
198:
125:Narmer Macehead
117:
35:
28:
23:
22:
15:
12:
11:
5:
3726:
3716:
3715:
3710:
3693:
3692:
3690:
3689:
3678:
3675:
3674:
3672:
3671:
3666:
3665:
3664:
3654:
3653:
3652:
3643:
3641:
3637:
3636:
3634:
3633:
3628:
3626:Leibniz–Newton
3623:
3618:
3612:
3610:
3606:
3605:
3603:
3602:
3597:
3592:
3587:
3585:Ancient Greece
3582:
3577:
3571:
3569:
3565:
3564:
3562:
3561:
3556:
3551:
3545:
3543:
3539:
3538:
3536:
3535:
3534:
3533:
3528:
3527:
3526:
3513:
3512:
3511:
3506:
3496:
3495:
3494:
3488:Number theory
3486:
3481:
3480:
3479:
3469:
3468:
3467:
3457:
3452:
3451:
3450:
3445:
3435:
3434:
3433:
3423:
3418:
3417:
3416:
3411:
3403:
3402:
3401:
3396:
3386:
3385:
3384:
3374:
3373:
3372:
3364:
3363:
3362:
3351:
3349:
3345:
3344:
3332:
3331:
3324:
3317:
3309:
3302:
3301:
3289:
3266:
3265:
3263:
3262:
3252:
3242:
3232:
3222:
3209:
3206:
3205:
3203:
3202:
3197:
3192:
3186:
3183:
3182:
3175:
3173:
3171:
3170:
3165:
3160:
3155:
3154:
3153:
3148:
3138:
3133:
3128:
3123:
3118:
3117:
3116:
3111:
3101:
3096:
3091:
3086:
3081:
3076:
3071:
3066:
3065:
3064:
3059:
3049:
3044:
3043:
3042:
3032:
3031:
3030:
3020:
3015:
3010:
3005:
3000:
2995:
2990:
2985:
2980:
2975:
2974:
2973:
2963:
2958:
2953:
2952:
2951:
2941:
2940:
2939:
2934:
2929:
2919:
2913:
2910:
2909:
2907:
2906:
2901:
2896:
2890:
2887:
2886:
2878:
2877:
2870:
2863:
2855:
2849:
2848:
2843:
2836:
2835:External links
2833:
2831:
2830:
2815:
2808:
2802:
2795:
2780:
2773:
2759:
2752:
2746:
2729:
2722:
2716:
2706:(2 ed.).
2694:
2679:
2673:
2666:
2659:
2652:
2634:
2620:
2612:
2610:
2607:
2605:
2604:
2587:
2583:R.C. Archibald
2572:
2565:
2547:
2540:
2507:
2492:
2461:
2426:
2364:
2357:
2337:
2330:
2291:
2284:
2272:Rossi, Corinna
2263:
2244:
2237:
2219:
2212:
2189:
2141:
2139:
2136:
2135:
2134:
2129:
2120:
2111:
2110:
2109:
2099:
2092:
2089:
2076:
2073:
2072:
2071:
2070:
2069:
2057:
2041:
2024:
2023:
2022:
2014:
2005:
1993:
1948:Main article:
1945:
1942:
1934:Berlin Papyrus
1925:
1922:
1907:
1906:
1892:
1889:
1886:
1883:
1880:
1877:
1872:
1869:
1824:
1823:
1820:
1819:
1816:
1815:
1801:
1800:
1791:
1790:
1787:
1786:
1783:
1782:
1777:
1748:Main article:
1745:
1742:
1639:
1638:
1631:
1614:
1596:
1595:
1588:
1568:
1550:
1549:
1542:
1522:
1504:
1503:
1496:
1493:
1489:
1488:
1486:
1483:
1479:
1478:
1476:
1473:
1469:
1468:
1466:
1463:
1435:
1434:
1431:
1428:
1426:
1423:
1422:
1419:
1418:
1403:
1400:
1399:
1396:
1395:
1392:
1389:
1375:
1374:
1371:
1368:
1361:
1358:
1357:
1354:
1353:
1339:
1336:
1335:
1332:
1331:
1325:
1311:
1310:
1307:
1304:
1302:
1299:
1298:
1295:
1294:
1282:
1279:
1278:
1275:
1274:
1271:
1253:
1252:
1249:
1246:
1239:
1236:
1235:
1232:
1231:
1220:
1217:
1216:
1213:
1212:
1192:
1191:
1188:
1185:
1183:
1180:
1179:
1176:
1175:
1164:
1161:
1160:
1157:
1156:
1136:
1135:
1132:
1129:
1127:
1124:
1120:
1119:
1116:
1114:
1110:
1109:
1075:Main article:
1072:
1069:
1063:
1062:
1059:
1058:
1051:
1048:
1045:
1022:
1021:
1018:
1017:
1014:
1011:
994:
991:
988:
987:
984:
983:
980:
979:
962:
959:
958:
955:
954:
938:
935:
934:
931:
930:
919:
916:
915:
912:
911:
898:
895:
894:
891:
890:
878:
877:
860:
843:
826:
809:
654:Slab stela of
646:
645:
642:
641:
638:
637:
626:
623:
622:
619:
618:
607:
604:
603:
600:
599:
588:
585:
584:
581:
580:
569:
566:
565:
562:
561:
550:
547:
546:
543:
542:
531:
528:
527:
524:
523:
511:
510:
507:
504:
501:
498:
495:
492:
456:
453:
445:Étienne Gilson
441:
440:
437:
434:
431:
420:
419:
416:Middle Kingdom
400:
389:
383:
376:
370:
364:
352:
349:
341:Deir el-Medina
263:unit fractions
116:
113:
103:, such as the
69:multiplication
26:
9:
6:
4:
3:
2:
3725:
3714:
3711:
3709:
3706:
3705:
3703:
3688:
3680:
3679:
3676:
3670:
3667:
3663:
3660:
3659:
3658:
3655:
3651:
3648:
3647:
3645:
3644:
3642:
3638:
3632:
3631:Hobbes–Wallis
3629:
3627:
3624:
3622:
3619:
3617:
3614:
3613:
3611:
3609:Controversies
3607:
3601:
3598:
3596:
3593:
3591:
3588:
3586:
3583:
3581:
3580:Ancient Egypt
3578:
3576:
3573:
3572:
3570:
3566:
3560:
3557:
3555:
3552:
3550:
3547:
3546:
3544:
3540:
3532:
3529:
3525:
3522:
3521:
3520:
3517:
3516:
3514:
3510:
3507:
3505:
3502:
3501:
3500:
3497:
3493:
3490:
3489:
3487:
3485:
3484:Math notation
3482:
3478:
3475:
3474:
3473:
3470:
3466:
3463:
3462:
3461:
3458:
3456:
3453:
3449:
3446:
3444:
3441:
3440:
3439:
3436:
3432:
3429:
3428:
3427:
3424:
3422:
3421:Combinatorics
3419:
3415:
3412:
3410:
3407:
3406:
3404:
3400:
3397:
3395:
3392:
3391:
3390:
3387:
3383:
3380:
3379:
3378:
3375:
3371:
3368:
3367:
3365:
3361:
3358:
3357:
3356:
3353:
3352:
3350:
3346:
3341:
3337:
3330:
3325:
3323:
3318:
3316:
3311:
3310:
3307:
3300:
3299:Ancient Egypt
3290:
3288:
3283:
3278:
3277:
3274:
3261:
3253:
3251:
3243:
3241:
3233:
3231:
3223:
3221:
3211:
3210:
3207:
3201:
3198:
3196:
3195:Egyptologists
3193:
3191:
3188:
3187:
3184:
3179:
3169:
3166:
3164:
3161:
3159:
3156:
3152:
3149:
3147:
3144:
3143:
3142:
3139:
3137:
3134:
3132:
3129:
3127:
3124:
3122:
3119:
3115:
3112:
3110:
3107:
3106:
3105:
3102:
3100:
3097:
3095:
3092:
3090:
3087:
3085:
3082:
3080:
3077:
3075:
3072:
3070:
3067:
3063:
3060:
3058:
3055:
3054:
3053:
3050:
3048:
3045:
3041:
3038:
3037:
3036:
3033:
3029:
3026:
3025:
3024:
3021:
3019:
3016:
3014:
3011:
3009:
3006:
3004:
3001:
2999:
2996:
2994:
2991:
2989:
2986:
2984:
2981:
2979:
2976:
2972:
2969:
2968:
2967:
2964:
2962:
2959:
2957:
2954:
2950:
2947:
2946:
2945:
2942:
2938:
2935:
2933:
2930:
2928:
2925:
2924:
2923:
2920:
2918:
2915:
2914:
2911:
2905:
2902:
2900:
2897:
2895:
2892:
2891:
2888:
2883:
2882:Ancient Egypt
2876:
2871:
2869:
2864:
2862:
2857:
2856:
2853:
2847:
2844:
2842:
2839:
2838:
2828:
2824:
2820:
2816:
2813:
2809:
2807:
2803:
2800:
2796:
2793:
2792:90-04-13048-9
2789:
2785:
2781:
2778:
2774:
2772:
2771:0-7141-0944-4
2768:
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2749:
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2730:
2727:
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2657:
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2650:0-87169-232-5
2647:
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2639:
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2633:
2632:0-87353-133-7
2629:
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2621:
2618:
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2602:
2596:
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2579:
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2558:
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2541:0-486-67766-4
2537:
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2517:
2511:
2503:
2496:
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2484:
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2462:9780813231952
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2360:
2358:9781400851416
2354:
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2306:
2304:
2302:
2300:
2298:
2296:
2287:
2281:
2277:
2273:
2267:
2259:
2255:
2248:
2240:
2234:
2230:
2223:
2215:
2209:
2205:
2198:
2196:
2194:
2185:
2181:
2177:
2173:
2169:
2165:
2161:
2155:
2153:
2151:
2149:
2147:
2142:
2133:
2130:
2128:
2124:
2121:
2119:
2115:
2112:
2108:
2105:
2104:
2103:
2100:
2098:
2095:
2094:
2088:
2087:
2083:
2067:
2066:
2064:
2058:
2055:
2051:
2047:
2046:
2042:
2039:
2035:
2034:
2030:
2029:
2028:
2025:
2020:
2019:
2015:
2011:
2010:
2006:
2003:
1999:
1998:
1994:
1991:
1990:
1986:
1985:
1984:
1981:
1980:
1979:
1977:
1973:
1969:
1961:
1956:
1951:
1941:
1939:
1935:
1931:
1921:
1918:
1914:
1912:
1890:
1887:
1884:
1881:
1878:
1875:
1870:
1867:
1858:
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1854:
1831:
1822:
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1818:
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1810:
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1803:
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1799:
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1792:
1778:
1775:
1774:
1771:
1770:
1767:
1766:
1763:
1761:
1757:
1751:
1741:
1724: +
1708: +
1676: +
1660: +
1632:
1615:
1598:
1597:
1589:
1569:
1552:
1551:
1543:
1523:
1506:
1505:
1497:
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1490:
1487:
1484:
1481:
1480:
1477:
1474:
1471:
1470:
1467:
1464:
1461:
1460:
1457:
1452:
1448:
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1432:
1429:
1427:
1412:
1411:
1408:
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1404:
1393:
1390:
1385:
1384:
1381:
1380:
1377:
1376:
1372:
1369:
1362:
1348:
1347:
1344:
1343:
1340:
1326:
1321:
1320:
1317:
1316:
1313:
1312:
1308:
1305:
1303:
1291:
1290:
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1272:
1263:
1262:
1259:
1258:
1255:
1254:
1250:
1247:
1240:
1229:
1228:
1225:
1224:
1221:
1202:
1201:
1198:
1197:
1194:
1193:
1189:
1186:
1184:
1173:
1172:
1169:
1168:
1165:
1146:
1145:
1142:
1141:
1138:
1137:
1133:
1130:
1128:
1125:
1122:
1121:
1115:
1111:
1108:
1104:
1101:
1099:
1098:Rhind Papyrus
1094:
1091:
1089:
1084:
1078:
1068:
1052:
1049:
1046:
1043:
1042:
1039:
1038:
1035:
1033:
1029:
1015:
1012:
1009:
1008:
1005:
1004:
1001:
998:
971:
970:
967:
966:
963:
947:
946:
943:
942:
939:
928:
927:
924:
923:
920:
907:
906:
903:
902:
899:
888:
887:
884:
883:
880:
879:
861:
844:
827:
810:
793:
792:
786:
716:
698:
694:
689:
685:
682:
680:
676:
672:
665:
661:
657:
652:
635:
634:
631:
630:
627:
616:
615:
612:
611:
608:
597:
596:
593:
592:
589:
578:
577:
574:
573:
570:
559:
558:
555:
554:
551:
540:
539:
536:
535:
532:
521:
520:
517:
516:
513:
512:
508:
505:
502:
499:
496:
493:
490:
489:
483:
480:
476:
472:
466:
462:
452:
450:
446:
443:According to
438:
435:
432:
429:
425:
424:
423:
417:
413:
409:
405:
401:
398:
394:
390:
388:
384:
381:
377:
375:
371:
369:
365:
363:
359:
358:
357:
348:
346:
342:
338:
334:
330:
326:
321:
315:
297:
278:
264:
196:
191:
187:
185:
181:
177:
173:
169:
165:
161:
157:
152:
150:
146:
142:
138:
134:
129:
126:
122:
112:
110:
106:
102:
98:
94:
90:
86:
82:
78:
74:
70:
66:
62:
58:
54:
47:
46:Ancient Egypt
43:
39:
33:
19:
3579:
3559:Hindu-Arabic
3455:Group theory
3443:Trigonometry
3414:Topos theory
3073:
2922:Architecture
2818:
2811:
2805:
2798:
2783:
2776:
2762:
2755:
2733:
2725:
2702:
2686:
2669:
2662:
2655:
2641:
2640:. Volume 3:
2637:
2623:
2616:
2590:
2556:
2550:
2531:
2521:
2510:
2501:
2495:
2439:
2429:
2347:
2340:
2321:
2275:
2266:
2257:
2253:
2247:
2228:
2222:
2203:
2170:(1): 19–27.
2167:
2163:
2085:
2081:
2078:
2060:
2059:
2044:
2043:
2032:
2031:
2026:
2017:
2016:
2008:
2007:
1996:
1995:
1988:
1987:
1982:
1965:
1927:
1919:
1915:
1908:
1827:
1753:
1642:
1455:
1449:
1440:
1438:
1106:
1095:
1092:
1080:
1066:
1025:
999:
996:
714:
702:
693:cattle count
683:
679:Neferetiabet
668:
660:Neferetiabet
468:
442:
421:
354:
337:Ramesses III
322:
313:
295:
276:
265:of the form
192:
188:
172:Kahun Papyri
156:12th Dynasty
153:
130:
118:
89:surface area
37:
36:
3575:Mesopotamia
3549:Prehistoric
3509:Probability
3366:Algorithms
3287:Mathematics
3260:WikiProject
3074:Mathematics
3035:Hieroglyphs
2949:Portraiture
2917:Agriculture
2904:Main topics
2530:. pp.
2471:j.ctvdf0jnn
2018:Hemisphere:
1997:Rectangles:
1809:New Kingdom
1798:hieroglyphs
1134:Multiplier
1126:Multiplier
656:Old Kingdom
471:hieroglyphs
325:New Kingdom
323:During the
133:Old Kingdom
79:written on
55:, from the
42:mathematics
3702:Categories
3499:Statistics
3431:Logarithms
3377:Arithmetic
3190:Egyptology
3158:Technology
3121:Philosophy
3069:Literature
2961:Chronology
2685:, Editor.
2566:1884123007
2479:1080547285
2260:: 174–177.
2238:0813526140
2138:References
2127:technology
2004:in London.
1989:Triangles:
1525:243
1088:multiplier
509:1,000,000
331:, and the
3519:Manifolds
3515:Topology
3426:Functions
3094:Mythology
3018:Geography
3008:Dynasties
2956:Astronomy
2534:229–230.
2518:(1993) .
2487:170577624
2314:Imhasen A
2184:122060653
2075:The Seqed
1930:quadratic
1876:×
1571:36
658:princess
73:fractions
3687:Category
3662:timeline
3650:timeline
3524:timeline
3504:timeline
3492:timeline
3477:timeline
3465:timeline
3448:timeline
3438:Geometry
3409:timeline
3394:timeline
3389:Calculus
3382:timeline
3370:timeline
3360:timeline
3348:By topic
3340:timeline
3230:Category
3151:District
3146:Capitals
3131:Religion
3114:Titulary
3104:Pharaohs
3084:Military
3079:Medicine
3062:Hieratic
3052:Language
2978:Clothing
2932:Obelisks
2700:(1969).
2318:Robson E
2312:Katz V,
2274:(2007).
2091:See also
2080:and the
2027:Volumes:
2009:Circles:
1944:Geometry
1834:1
1758:and the
993:Notation
475:hieratic
455:Numerals
343:several
174:and the
115:Overview
85:geometry
3554:Ancient
3355:Algebra
3273:Portals
3250:Outline
3240:Commons
3200:Museums
3136:Scribes
3126:Pottery
3057:Demotic
3047:History
2998:Cuisine
2927:Revival
2063:Frustum
1849:
1744:Algebra
1738:
1726:
1722:
1710:
1706:
1694:
1690:
1678:
1674:
1662:
1658:
1646:
1629:
1617:
1612:
1600:
1586:
1566:
1554:
1540:
1520:
1508:
1131:Result
1123:Result
875:
863:
858:
846:
841:
829:
824:
812:
807:
795:
783:
771:
767:
755:
751:
739:
735:
723:
719:
705:
697:Lepsius
506:100,000
449:Abraham
351:Sources
345:ostraca
318:
304:
300:
286:
281:
267:
259:
247:
243:
231:
227:
215:
211:
199:
137:mastaba
101:algebra
81:papyrus
40:is the
3099:People
2966:Cities
2884:topics
2825:
2790:
2769:
2744:
2714:
2648:
2630:
2563:
2538:
2485:
2477:
2469:
2459:
2420:
2355:
2328:
2282:
2235:
2210:
2182:
1894:
1083:binary
675:Abydos
664:Louvre
503:10,000
479:hobble
473:or in
166:, the
162:, the
141:Meidum
121:Abydos
99:, and
93:volume
3640:Other
3595:India
3590:China
3472:Logic
3163:Trade
3141:Sites
3089:Music
3003:Dance
2937:Pylon
2899:Index
2483:S2CID
2467:JSTOR
2180:S2CID
2082:seked
2013:khet.
1983:Area:
1430:1120
412:Ahmes
145:cubit
3109:List
3028:List
2971:List
2823:ISBN
2788:ISBN
2767:ISBN
2742:ISBN
2712:ISBN
2646:ISBN
2628:ISBN
2601:IV.3
2561:ISBN
2536:ISBN
2475:OCLC
2457:ISBN
2418:ISBN
2353:ISBN
2326:ISBN
2280:ISBN
2233:ISBN
2208:ISBN
2125:and
2116:and
1735:2190
1687:2190
1609:2190
1495:2920
1485:1460
1439:The
1370:320
1306:160
1248:800
500:1000
463:and
426:The
402:The
391:The
385:The
378:The
372:The
366:The
360:The
229:and
107:and
91:and
71:and
2944:Art
2532:pp.
2449:doi
2258:6–1
2172:doi
1891:10.
1805:Era
1796:in
1794:Aha
1475:730
1465:365
1433:14
1251:10
1187:80
1050:and
1032:D55
1028:D54
1013:and
497:100
139:in
53:BCE
3704::
2740:.
2710:.
2689:.
2575:^
2526:.
2481:.
2473:.
2465:.
2455:.
2367:^
2316:,
2294:^
2256:.
2192:^
2178:.
2168:28
2166:.
2145:^
1940:.
1855::
1807::
1740:.
1719:10
1671:10
1563:10
1373:4
1309:2
1190:1
1030:,
494:10
447:,
213:,
151:.
111:.
49:c.
3342:)
3338:(
3328:e
3321:t
3314:v
3275::
2874:e
2867:t
2860:v
2829:.
2794:.
2750:.
2720:.
2569:.
2544:.
2489:.
2451::
2424:.
2361:.
2334:.
2288:.
2241:.
2216:.
2186:.
2174::
2065::
1888:=
1885:4
1882:+
1879:x
1871:2
1868:3
1846:2
1843:/
1840:1
1836:+
1732:/
1729:1
1716:/
1713:1
1703:3
1700:/
1697:2
1684:/
1681:1
1668:/
1665:1
1655:3
1652:/
1649:2
1626:6
1623:/
1620:1
1606:/
1603:1
1583:2
1580:/
1577:1
1573:+
1560:/
1557:1
1537:3
1534:/
1531:1
1527:+
1517:3
1514:/
1511:2
1492:8
1482:4
1472:2
1462:1
1026:(
872:5
869:/
866:1
855:4
852:/
849:1
838:3
835:/
832:2
821:3
818:/
815:1
804:2
801:/
798:1
780:3
777:/
774:2
764:2
761:/
758:1
748:4
745:/
742:3
732:3
729:/
726:2
715:n
711:/
708:1
491:1
314:n
310:/
307:1
296:n
292:/
289:2
277:n
273:/
270:1
256:4
253:/
250:3
240:3
237:/
234:2
224:3
221:/
218:1
208:2
205:/
202:1
34:.
20:)
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