Fairfield County Math Teachers Circle Summer Immersion Workshop August 7-8, 2017 - 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
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Who? What? Where? When? Why?
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus What?
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus 64 Problems Various Tables & Calculations
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Where?
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus When?
Ancient Egyptian History Kingdoms (Dynasties) Approx. Dates Early Dynastic (1 2) 3100 2600 Old Kingdom (3 6) 2600 2150 1 st Intermediate Period (7 10) 2150 2000 Middle Kingdom (11 13) 2000 1650 2 nd Intermediate Period (14 17) 1650 1550 New Kingdom (18 20) 1550 1075 3 rd Intermediate Period (21 25) 1075 675 Late Period (26 31) 675 332 BCE
When was the RMP Written? 1999; 1553; Clagett, Ancient Egyptian Science: A Source Book, Volume Three: Ancient Egyptian Mathematics, [16] 2007; 2025-1773 (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. 1550 (2 nd Intermediate); British Musuem, http://www.britishmuseum.org/research/collection_online/collection_object_details.aspx?objectid=110036&partid=1 2017; 1493-1481 (New Kingdom); Brooklyn Museum, https://www.brooklynmuseum.org/opencollection/objects/118304/fragments_of_rhind_mathematical_papyrus
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Who?
Alexander Henry Rhind (1833-1863)
Title Page
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.
Ahmose (Scribe)
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Where? (again)
Brooklyn Museum
Brooklyn Fragments - RMP
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Why?
Why was Mathematics Needed? Collect Taxes Surveyors Construct Silos Maintain Armies Building Programs Trade
Tomb of Menna Chief Scribe (1420-1411)
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus A Closer Look
A closer look
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Types of Writing
Types of Writing Hieroglyphic (Gk. Sacred Carving) Gk. hieros sacred, glyphe carving c. 3000 BCE Usually carved in stone
Types of Writing - Hieroglyphic
Types of Writing Hieroglyphic (Gk. Sacred Carving) Gk. hieros sacred, glyphe carving c. 3000 BCE Usually carved in stone Hieratic script Gk. hieratikos priestly c. 3000 BCE Ink on papyrus, leather, wood, ostraca
Types of Writing - Hieratic Script
Types of Writing Hieroglyphic (Gk. Sacred Carving) Gk. hieros sacred, glyphe carving c. 3000 BCE Usually carved in stone Hieratic script Gk. hieratikos priestly c. 3000 BCE In ink on papyrus, leather, wood, ostraca
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Types of Numerals
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]
Hieroglyphic Numerals 1,333,331
Hieroglyphic Numerals 3,244 21,237
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
Hieroglyphic Fractions Author of photograph: Ad Meskens. (Wikipedia)
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
Hieratic Fractions
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
Special Fractions Imhausen, 2016, Mathematics in Ancient Egypt: A Contextual History, p. 53
Hieratic Fractions Special Symbols
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
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Multiplication & Division
Multiplication 31 times 11 \ 1 31 \ 2 62 4 124 \ 8 248 341 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.
Multiplication You Try 23 times 14 1 23 \ 2 46 \ 4 92 \ 8 184 322 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
Division 1539 divided by 81 \ 1 81 \ 2 162 4 324 8 648 \ 16 1296 19 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
Division You Try 1961 divided by 53 \ 1 53 2 106 \ 4 212 8 424 16 848 \ 32 1696 37 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
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Problems from the RMP
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)
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus 2 n Table
2 n Table We ll use the following common convention: 1 1 1 2, 8, 25 2 8 25 Recall 2/3 is the only exception to the use of unit fractions, so we ll use the common convention of: 2 3 3
2 n Table 2 divided by 3 2 divide by 5 2 divided by 7 2 divided by 9 3 3 15 4 28 6 18 Do these numbers look familiar?
Hieratic Fractions
Title Page
2 n Table 2 divided by 45 24 360 30 90 25 225 27 135 30 90 35 63 36 60 45 45
2 n Table Note: See b/w handout Table 6.1 (Gillings 1972)
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus Problems 1 6 Dividing n loaves of bread among 10 men
RMP Problem 3: Divide 6 loaves among 10 men Modern 6/10 = 3/5 6 men get 1 piece 4 men get 2 pieces
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
RMP Problem 3: Divide 6 loaves among 10 men Egyptian 6/10 = 2 10 The product of the same: 1 2 10 / 2 1 5 4 2 3 15 / 8 4 3 10 30 Total loaves 6, which is correct.
The Rhind Mathematical Papyrus (RMP)/ a.k.a. Ahmose (Ahmes) Papyrus aha Problems
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 3 1 3 2 6 \ 4 12 12 ¼ 3 Total 15 1 x x 15 4 Let x 4 1 4 (4)? 4 4 1 5 Off by a factor of 3. So scale up value of by a factor of 3. x 4 3 x 12 x
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 1 6 (6)? 6 6 1 7 Off by a factor of 8. So scale up value of by a factor of 8. x 68 x 48 x
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 1 8 2 16 4 32 \ 8 64 Its area is 64 setat 1 A d d 9 8 A d 9 64 2 A d 81 Thus, 2 2 A A A 64 2 81 64 4 r 81 256 2 r 81 r 2 256 E 3.16 an error of approx. 0.6% 81 2
Epilouge
Pushkin State Museum of Fine Arts, Moscow 16 5 x 3 c. 1850 BCE Problem 14 Truncated Prism (Frustum) 1 3 V h a 2 ab b 2 Moscow Papyrus
Moscow Papyrus Prob. 14 V h a 2 ab b 2 1 3
THANK YOU
Bilbiography Chace, Arnold B. et al. The Rhind Mathematical Papyrus, Reston, VA: NCTM, 1979 (originally published by MAA, 1927-9). 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. 61-69 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, 2016. 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. 7-56. Katz, Victor J. A History of Mathematics: An Introduction, Boston: Pearson Education, Inc., 3 rd ed., 2009. Robins, Gay & Shute, Charles. The Rhind Mathematical Papyrus: an ancient Egyptian text, New York: Dover reprint; London: British Museum Publications, 1987.
Further Reading Reimer, David. Count Like an Egyptian. Princeton: Princeton University Press, 2014. Joseph, George Gheverghese. The Crest of the Peacock: Non-European Roots of Mathematics, Princeton: Princeton University Press, 3 rd ed., 2011.