ancient egyptian mathematics - sacredheart.edu · the rhind mathematical papyrus (rmp)/ a.k.a....

Ancient Egyptian

Mathematics

Douglas Furman – [email protected] Professor of Mathematics & Mathematics Program Coordinator

SUNY Ulster, Stone Ridge, NY

Fairfield County Math Teachers’ Circle Summer Immersion Workshop

August 7-8, 2017 - Sacred Heart University, Fairfield, CT

mailto:[email protected]

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

• Who?

• What?

• Where?

• When?

• Why?


What?


• 64 “Problems”

• Various Tables & Calculations


Where?


When?

Ancient Egyptian HistoryKingdoms (Dynasties) Approx. Dates

• Early Dynastic (1 – 2) 3100 – 2600

• Old Kingdom (3 – 6) 2600 – 2150

• 1st Intermediate Period (7 – 10) 2150 – 2000

• Middle Kingdom (11 – 13) 2000 – 1650

• 2nd Intermediate Period (14 – 17) 1650 – 1550

• New Kingdom (18 – 20) 1550 – 1075

• 3rd 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 (2nd 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

http://www.britishmuseum.org/research/collection_online/collection_object_details.aspx?objectId=110036&partId=1

https://www.brooklynmuseum.org/opencollection/objects/118304/Fragments_of_Rhind_Mathematical_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 [Ne-ma’et-Re]. The scribe Ahmose writes this copy.

Ahmose (Scribe)


Where?

(again)

Brooklyn Museum

Brooklyn Fragments - RMP


Why?

Why was Mathematics Needed?

• Collect Taxes

• Surveyors

• Construct Silos

• Maintain Armies

• Building Programs

• Trade

Tomb of Menna – Chief Scribe (1420-1411)


A Closer Look…

A closer look…


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



– c. 3000 BCE


• Hieratic script

– Gk. hieratikos “priestly”

– c. 3000 BCE

– Ink on papyrus, leather, wood, ostraca

Types of Writing - Hieratic Script



– c. 3000 BCE


• Hieratic script

– Gk. hieratikos “priestly”

– c. 3000 BCE

– In ink on papyrus, leather, wood, ostraca


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)

https://commons.wikimedia.org/wiki/User:AdMeskens



– 2/5 = 1/3 + 1/15




• Hieratic fractions

– A dot placed over the denominator

Hieratic Fractions



– 2/5 = 1/3 + 1/15






• 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



– 2/5 = 1/3 + 1/15






• Special symbols

– 2/3, 1/2, 1/3, 1/4


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

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

23 times 14

Division

\ 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

1539 divided by 81

Division – You Try

\ 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

1961 divided by 53


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)


2 ÷ n Table

2 ÷ n Table

23

3

1 1 12 , 8 , 25

2 8 25

We’ll use the following common convention:

Recall 2/3 is the only exception to the use of unit fractions, so we’ll use the common convention of:

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

25 225

27 135

30 90

35 63

36 60

45 45

30 90

2 ÷ n Table

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


Problems 1 – 6Dividing 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


• Modern

6/10 = 3/5

• Egyptian

6/10 =

All 10 get the same 2 pieces

2 10


• Egyptian

6/10 =

The product of the same:

2 10

1 2 10

2 1 5

4 2 3 15

8 4 3 10 30

Total loaves 6, which is correct.

/

/


‘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 1Total 5

\ 1 5\ 2 10Total 3

1 32 6

\ 4 12

12¼ 3

Total 15

115

4

Let 4

14 (4) ?

4

4 1 5 Off by a factor of 3.

So scale up value of

by a factor of 3.

4 3

12

x x

x

x

x

x

False Position – You Try

• A quantity and its 1/6 added together become 56. What is the quantity?

156

6

Let 6

16 (6) ?

6

6 1 7 Off by a factor of 8.

So scale up value of

by a factor of 8.

6 8

48

x x

x

x

x

x

Area of a CircleProblem 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

2

2

2

1

9

8

9

64

81

A d d

A d

A d

2

2

2

642

81

644

81

256

81

A r

A r

A r

Thus,

2563.16 an error of approx. 0.6%

81E

Epilouge

Moscow Papyrus

• Pushkin State Museum of Fine Arts, Moscow

• 16’ 5” x 3”

• c. 1850 BCE

• Problem 14

– Truncated Prism (Frustum)

– 2 21

3V h a ab b

Moscow Papyrus – Prob. 14 2 21

3V h a ab b

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., 3rd 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, 3rd ed., 2011.

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