Ancient Egyptian Mathematics

Transcription

1 Fairfield County Math Teachers Circle Summer Immersion Workshop August 7-8, Sacred Heart University, Fairfield, CT Ancient Egyptian Mathematics Douglas Furman furmand@sunyulster.edu Associate Professor of Mathematics & Mathematics Program Coordinator SUNY Ulster, Stone Ridge, NY

2 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Who? What? Where? When? Why?

3 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus What?

4 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus

5 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus 64 Problems Various Tables & Calculations

6 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus

7 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Where?

8 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus

9

10 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus When?

11 Ancient Egyptian History Kingdoms (Dynasties) Approx. Dates Early Dynastic (1 2) Old Kingdom (3 6) st Intermediate Period (7 10) Middle Kingdom (11 13) nd Intermediate Period (14 17) New Kingdom (18 20) rd Intermediate Period (21 25) Late Period (26 31) BCE

12 When was the RMP Written? 1999; 1553; Clagett, Ancient Egyptian Science: A Source Book, Volume Three: Ancient Egyptian Mathematics, [16] 2007; (Middle Kingdom); Imhausen, Egyptian Mathematics in Katz s The Mathematics of Egypt, Mesopotamia, China, India and Islam: A Sourcebook, [12] (Dates from Shaw) 2009; c. 1650; Katz, A History of Mathematics: An Introduction, [3] 2016; c. 1550; Imhausen, Mathematics in Ancient Egypt: A Contextual History, [66] 2017; c (2 nd Intermediate); British Musuem, ; (New Kingdom); Brooklyn Museum,

13 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Who?

14 Alexander Henry Rhind ( )

15 Title Page

16 Title Page Accurate reckoning for inquiring into things, and the knowledge of all things, mysteries... all secrets. This book was copied in regnal year 33, month 4 of Akhet [the inundation season], under the majesty of the King of Upper and Lower Egypt, Awserre [A-user-Re], given life, from an ancient copy made in the time of the King of Upper and Lower Egypt Nymatre [Nema et-re]. The scribe Ahmose writes this copy.

17 Ahmose (Scribe)

18 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Where? (again)

19 Brooklyn Museum

20 Brooklyn Fragments - RMP

21 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Why?

22 Why was Mathematics Needed? Collect Taxes Surveyors Construct Silos Maintain Armies Building Programs Trade

23 Tomb of Menna Chief Scribe ( )

24 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus A Closer Look

25 A closer look

26 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Types of Writing

27 Types of Writing Hieroglyphic (Gk. Sacred Carving) Gk. hieros sacred, glyphe carving c BCE Usually carved in stone

28 Types of Writing - Hieroglyphic

29 Types of Writing Hieroglyphic (Gk. Sacred Carving) Gk. hieros sacred, glyphe carving c BCE Usually carved in stone Hieratic script Gk. hieratikos priestly c BCE Ink on papyrus, leather, wood, ostraca

30 Types of Writing - Hieratic Script

31 Types of Writing Hieroglyphic (Gk. Sacred Carving) Gk. hieros sacred, glyphe carving c BCE Usually carved in stone Hieratic script Gk. hieratikos priestly c BCE In ink on papyrus, leather, wood, ostraca

32 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Types of Numerals

33 Egyptian Numerals - Hieratic Hieratic Numerals Personalized by scribe Chace, RMP Prob 41 facsimile [107/46] [line 3] 640 [line 4] 960 Prob 41 photo [142]

34 Hieroglyphic Numerals 1,333,331

35 Hieroglyphic Numerals 3,244 21,237

36 Egyptian Numerals - Fractions Unit fractions 1/n 2/5 = 1/3 + 1/15 One exception 2/3 Hieroglyphic fractions An oval part placed over the denominator

37 Hieroglyphic Fractions Author of photograph: Ad Meskens. (Wikipedia)

38 Egyptian Numerals - Fractions Unit fractions 1/n 2/5 = 1/3 + 1/15 One exception 2/3 Hieroglyphic fractions An oval part placed over the denominator Hieratic fractions A dot placed over the denominator

39 Hieratic Fractions

40 Egyptian Numerals - Fractions Unit fractions 1/n 2/5 = 1/3 + 1/15 One exception 2/3 Hieroglyphic fractions An oval part placed over the denominator Hieratic fractions A dot placed over the denominator Special symbols 2/3, 1/2, 1/3, 1/4

41 Special Fractions Imhausen, 2016, Mathematics in Ancient Egypt: A Contextual History, p. 53

42 Hieratic Fractions Special Symbols

43 Egyptian Numerals - Fractions Unit fractions 1/n 2/5 = 1/3 + 1/15 One exception 2/3 Hieroglyphic fractions An oval part placed over the denominator Hieratic fractions A dot placed over the denominator Special symbols 2/3, 1/2, 1/3, 1/4

44 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Multiplication & Division

45 Multiplication 31 times 11 \ 1 31 \ \ Pick bigger factor (31) to pair with 1 Then keep doubling both columns until a double would exceed the smaller factor (11) Note: 16 exceeds 11 Then starting with the biggest power of 2 mark the powers of 2 that add up to the smaller factor Add the doubles (i.e. the big factors) associated with the marked small factors, this is the product.

46 Multiplication You Try 23 times \ 2 46 \ 4 92 \ Pick bigger factor to pair with 1 Then keep doubling both columns until a double would exceed the smaller factor Then starting with the biggest power of 2 mark the powers of 2 that add up to the smaller factor Add the doubles of the marked big factors

47 Division 1539 divided by 81 \ 1 81 \ \ Pick the divisor (81) to pair with 1 Then keep doubling both columns until a double would exceed the dividend (1539) Note: doubling 1296 exceeds 1539 Then starting with the biggest doubled divisor, add the doubled divisors until they sum to the dividend (1539), marking the corresponding powers of 2 Add marked powers of 2

48 Division You Try 1961 divided by 53 \ \ \ Pick the divisor to pair with 1 Then keep doubling both columns until a double would exceed the dividend Then starting with the biggest doubled divisor, add the doubled divisors until they sum to the dividend (1961), marking the corresponding powers of 2 Add marked powers of 2

49 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Problems from the RMP

50 Problems from Original Sources Ahmose Papyrus 2/n Table Pr. 3 (6 loaves among 10) Pr. 26 ( aha - false position) Pr. 50 (area of a circle)

51 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus 2 n Table

52 2 n Table We ll use the following common convention: , 8, Recall 2/3 is the only exception to the use of unit fractions, so we ll use the common convention of: 2 3 3

53 2 n Table 2 divided by 3 2 divide by 5 2 divided by 7 2 divided by Do these numbers look familiar?

54 Hieratic Fractions

55 Title Page

56 2 n Table 2 divided by

57 2 n Table Note: See b/w handout Table 6.1 (Gillings 1972)

58 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Problems 1 6 Dividing n loaves of bread among 10 men

59 RMP Problem 3: Divide 6 loaves among 10 men Modern 6/10 = 3/5 6 men get 1 piece 4 men get 2 pieces

60 RMP Problem 3: Divide 6 loaves among 10 men Modern 6/10 = 3/5 Egyptian 6/10 = 2 10 All 10 get the same 2 pieces

61 RMP Problem 3: Divide 6 loaves among 10 men Egyptian 6/10 = 2 10 The product of the same: / / Total loaves 6, which is correct.

62 The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus aha Problems

63 The Rule of False Position aha (quantity) problem Problem 26 A quantity and its ¼ added together become 15. What is the quantity? Assume 4 \ 1 4 \ ¼ 1 Total 5 \ 1 5 \ 2 10 Total \ ¼ 3 Total 15 1 x x 15 4 Let x (4)? Off by a factor of 3. So scale up value of by a factor of 3. x 4 3 x 12 x

64 False Position You Try A quantity and its 1/6 added together become 56. What is the quantity? 1 x x 56 6 Let x (6)? Off by a factor of 8. So scale up value of by a factor of 8. x 68 x 48 x

65 Area of a Circle Problem 50: Example of a round field of diameter 9 khet. What is its area? Take away 1/9 of the diameter, namely 1; the remainder is 8. Multiply 8 times 8; it makes 64. Therefore it contains 64 setat of land. Do it thus: 1 9 1/9 1; This taken away leaves \ 8 64 Its area is 64 setat 1 A d d 9 8 A d A d 81 Thus, 2 2 A A A r r 81 r E 3.16 an error of approx. 0.6% 81 2

66 Epilouge

67 Pushkin State Museum of Fine Arts, Moscow 16 5 x 3 c BCE Problem 14 Truncated Prism (Frustum) 1 3 V h a 2 ab b 2 Moscow Papyrus

68 Moscow Papyrus Prob. 14 V h a 2 ab b 2 1 3

69 THANK YOU

70 Bilbiography Chace, Arnold B. et al. The Rhind Mathematical Papyrus, Reston, VA: NCTM, 1979 (originally published by MAA, ). Clagett, Marshall. Ancient Egyptian Science: A Source Book, Vol. 3: Ancient Egyptian Mathematics, Philadelphia: American Philosophical Society, 1999 Gillings, Richard J. Problems 1 to 6 of the Rhind Mathematical Papyrus. The Mathematics Teacher, Vol. 55, No. 1 (January 1962), pp Gillings, Richard J., Mathematics in the Time of the Pharaohs, New York: Dover, 1982 (originally published by MIT Press, 1972). Imhausen Annette. Mathematics in Ancient Egypt: A Contextual History, Princeton: Princeton University Press, Imhausen Annette. Egyptian Mathematics in Katz, Victor J., ed., The Mathematics of Egypt Mesopotamia, China, India and Islam: A Sourcebook, Princeton: Princeton University Press, 2007, pp Katz, Victor J. A History of Mathematics: An Introduction, Boston: Pearson Education, Inc., 3 rd ed., Robins, Gay & Shute, Charles. The Rhind Mathematical Papyrus: an ancient Egyptian text, New York: Dover reprint; London: British Museum Publications, 1987.

71 Further Reading Reimer, David. Count Like an Egyptian. Princeton: Princeton University Press, Joseph, George Gheverghese. The Crest of the Peacock: Non-European Roots of Mathematics, Princeton: Princeton University Press, 3 rd ed., 2011.