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Mathematics in Ancient Egypt Philippe Cara Department of Mathematics http://www.vub.ac.be/DWIS BEST Summerschool “Pyramids in the Cosmos” VUB, 7th September 2001

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Page 1: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

Mathematics in Ancient Egypt

Philippe Cara

Department of Mathematics

http://www.vub.ac.be/DWIS

BEST Summerschool “Pyramids in the Cosmos”

VUB, 7th September 2001

Page 2: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

1

The use of Mathematics in AncientEgypt

• Partitioning of fertile grounds

Page 3: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

1

The use of Mathematics in AncientEgypt

• Partitioning of fertile grounds

• Construction of pyramids

Page 4: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

1

The use of Mathematics in AncientEgypt

• Partitioning of fertile grounds

• Construction of pyramids

• Administration

Page 5: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

1

The use of Mathematics in AncientEgypt

• Partitioning of fertile grounds

• Construction of pyramids

• Administration

• Calendar. . .

Page 6: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

2

Sources

• Wrote mainly on papyrus

Page 7: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

2

Sources

• Wrote mainly on papyrus

• Not suited for long conservation

Page 8: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

2

Sources

• Wrote mainly on papyrus

• Not suited for long conservation

• Hieroglyphics and wall-paintings

Page 9: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

2

Sources

• Wrote mainly on papyrus

• Not suited for long conservation

• Hieroglyphics and wall-paintings

• Babylonians had clay-tablets

Page 10: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

2

Sources

• Wrote mainly on papyrus

• Not suited for long conservation

• Hieroglyphics and wall-paintings

• Babylonians had clay-tablets

• Mainly “real life” problems, no general methods!

Page 11: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

3

Rhind papyrus

Page 12: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

4

• 1650 BC

• Henry Rhind (1833–1863)

• 87 problems with answer

• Main source of information

• Tables with fractions

• Ahmes

• Leather Roll

Page 13: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

5

Page 14: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

6

Back of Rhind papyrus

2/5 = 1/3 + 1/15

2/7 = 1/4 + 1/28

2/9 = 1/6 + 1/18

2/11 = 1/6 + 1/66

2/13 = 1/8 + 1/52 + 1/104

2/15 = 1/10 + 1/30...

2/101 = 1/101 + 1/202 + 1/303 + 1/606

Page 15: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

7

Theorem

Every fraction can be written as a sum of unitfractions.

Page 16: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

7

Theorem

Every fraction can be written as a sum of unitfractions.

Given p/q with p < q, we subtract the largest unit

fraction 1/n. Repeat till a unit fraction is left. . .

Page 17: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

7

Theorem

Every fraction can be written as a sum of unitfractions.

Given p/q with p < q, we subtract the largest unit

fraction 1/n. Repeat till a unit fraction is left. . .

p

q− 1n

=np− qnq

Page 18: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

7

Theorem

Every fraction can be written as a sum of unitfractions.

Given p/q with p < q, we subtract the largest unit

fraction 1/n. Repeat till a unit fraction is left. . .

p

q− 1n

=np− qnq

Suppose np− q > p then np− p > q or (n− 1)p > q

1n<

1n− 1

6p

q

Page 19: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

8

Remark

There are many ways to write a fraction as sum of

unit fractions.

1n

=1

n+ 1+

1n(n+ 1)

Page 20: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

9

Multiplication 1

7× 22 =?

Page 21: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

9

Multiplication 1

7× 22 =?

1 7

✔ 2 14

✔ 4 28

8 56

✔ 16 112

22 154

Page 22: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

10

Division

154÷ 7 =?

Page 23: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

10

Division

154÷ 7 =?

1 7

2 14 ✔

4 28 ✔

8 56

16 112 ✔

22 154

Page 24: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

11

Multiplication 2

5 + 7/8× 12 + 2/3 =?

Page 25: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

11

Multiplication 2

5 + 7/8× 12 + 2/3 =?

✔ 1 12+2/3

2 25+1/3

✔ 4 50+2/3

✔ 1/2 6+1/3

✔ 1/4 3+1/6

✔ 1/8 1+1/2+1/12

5+7/8 99+1/2+1/4

Page 26: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

12

Writing of numbers

• Decimal

• But not positional

1 10 100 1000 10000 100000 1000000

� � � � µ � �

������� � �

� � �� � = 275

� µ µ µ µ µ ����� � � = 152023

Page 27: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

12

Writing of numbers

• Decimal

• But not positional

1 10 100 1000 10000 100000 1000000

� � � � µ � �

������� � �

� � �� � = 275

� µ µ µ µ µ ����� � � = 152023No zero symbol needed!!

Page 28: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

13

Moscow papyrus

• 1850 BC

• Golenischev

• 5 metres long, 8 cm high

• 25 problems

• Bad handwriting

• Volume of truncated pyramid

• Surface of half a sphere

Page 29: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

14

Other papyri

• Reisner papyri (1880 BC)

• Kahun papyri (1800 BC)

• Rollin papyrus (1350 BC)

• Harris papyrus (1167 BC)

Page 30: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

15

Mathematics used for building pyramids=

GEOMETRY

Page 31: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

16

Weights and measures

• 1 cubit = 52.3 cm

• Long distances: ropes with knots

• Shorter distance: ruler

Page 32: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

17

Egyptian ruler

The cubit of King Amenhotep I (1559 – 1539 BC)

Page 33: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

18

Other units for distances

• 1 palm = 1/7 of a cubit

• 1 finger = 1/4 of a palm

• 1 hayt = 1 khet = 100 cubits

• 1 remen = half the length of the diagonal of a

square with side one cubit. That is√

22 cubit.

Useful when measuring land areas.

• 1 double remen =√

2 cubit.

• 1 aura = 1 setat = area of square with side 100

cubits, hence 10000 square cubits.

Page 34: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

19

Weights or volumes

• 1 hekat = 1/30 of a cubic cubit of grain

• 1 hinu = 1/10 of a hekat

• 1 ro = 1/320 of a hekat

• Horus eye to write fractions of a hekat

Page 35: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

20

Beer and Pesu

• 1 des = approx. half a liter

• The pesu is a unit for measuring the strength of

beer, bread or cakes.

• If one hekat of grain was used to make 5 des of

beer, it was said to have a pesu of 5.

Page 36: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

20

Beer and Pesu

• 1 des = approx. half a liter

• The pesu is a unit for measuring the strength of

beer, bread or cakes.

• If one hekat of grain was used to make 5 des of

beer, it was said to have a pesu of 5.

• The less the pesu, the stronger the beer!!!

Page 37: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

21

Rhind problem 76

If you want to trade 1000 des of beer of pesu10 for beer of pesu 20, how many des do youget?

Page 38: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

22

Sekhed of a pyramid

is the inclination of any one of the four triangular

faces to the horizontal plane of its base.

a/2

a

α

h h

sekhed =a

2hIn fact the sekhed is the cotangent of the slope α.

Page 39: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

23

Rhind problem 57

The sekhed of a pyramid is 5 palms and 1finger and the side is 140 cubit. What is theheight?

Page 40: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

24

Sekheds of some well-known pyramids

Name Sekhed

Chephren, Ouserkaf 3/4 cubit

Neferirka-Re, Teti, Pepi 21 fingers

Cheops, Snofru 11/14 cubit

Neouser-Re 22 fingers

Sesostris 6/7 cubit

Amenemhat III 9/14 cubit

Page 41: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

25

Page 42: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

26

Pythagoras’ theorem

• The sum of the squares of the right anglesides of a rectangular triangle is the square ofthe remaining side.

a2 + b2 = c2a

bc

Page 43: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

26

Pythagoras’ theorem

• The sum of the squares of the right anglesides of a rectangular triangle is the square ofthe remaining side.

a2 + b2 = c2a

bc

• If a triangle fulfills the above then it is rect-angular.

Page 44: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

27

Construction of right angles 1

p

p

p

p

k0

k0k0

Page 45: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

28

Property of a rectangle

• The diagonals of a rectangle meet eachotherhalfway.

Page 46: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

28

Property of a rectangle

• The diagonals of a rectangle meet eachotherhalfway.

• If a quadrilateral fulfills the above then it is arectangle.

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29

Construction of right angles 2

m

m

mm

a

bc

d

a

b

p pa

d

b pa

a d a

b pa

d

R R

R R

Page 48: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

30

Area of a triangle?

Rhind problem 49: Multiply half a side with the

other side.

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30

Area of a triangle?

Rhind problem 49: Multiply half a side with the

other side.� � � � �� � � � �� � � � �� � � � �

� � � �� � � �� � � �� � � �

� � � �� � � �� � � �� � � �� � � �� � � �� � � �

� � � �� � � �� � � �� � � �� � � �� � � �� � � �

� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �

� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �

� � � �� � � �� � � �� � � �� � � �

� � � �� � � �� � � �� � � �� � � �

1/2

b

1/2

c

a

Page 50: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

31

Area of a quadrilateral?

b

c

d

a

a+ c

2· b+ d

2

Page 51: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

32

Rhind problem 48

A = (d− d9)2

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33

The area of a circle 1

��

����

����

����

��

����

Page 53: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

33

The area of a circle 1

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Page 54: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds

33

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33

The area of a circle 1

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34

The number π

Definition: The ratio of the area of a circle and the

square of its radius is denoted by π.

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34

The number π

Definition: The ratio of the area of a circle and the

square of its radius is denoted by π.

πd2

4≈ 64

81d2

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34

The number π

Definition: The ratio of the area of a circle and the

square of its radius is denoted by π.

πd2

4≈ 64

81d2

π ≈ 46481

=25681

= 3.1604938 . . .

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34

The number π

Definition: The ratio of the area of a circle and the

square of its radius is denoted by π.

πd2

4≈ 64

81d2

π ≈ 46481

=25681

= 3.1604938 . . .

• Babylonians: π ≈ 3.125

• Bible: π ≈ 3

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35

Area of a basket

Moscow problem 10: Find the area of a basket

with given “mouth” d.

d

They use the formula

A = 26481d2

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35

Area of a basket

Moscow problem 10: Find the area of a basket

with given “mouth” d.

d

They use the formula

A = 26481d2

This is

A = 225681r2

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35

Area of a basket

Moscow problem 10: Find the area of a basket

with given “mouth” d.

d

They use the formula

A = 26481d2

This is

A = 225681r2

and since π ≈ 25681 we find

A = 2πr2

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36

Volume of a truncated pyramid

a

b

h

V =13h(a2 + ab+ b2)

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37

How did they do it?

h’

a a

b

bh

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38

Mysticism. . .

• The ratio of twice the side to the height of the

Great pyramid is π.

• In the dimensions of the Great pyramid are en-

coded: the radius of th earth, the density of earth,

the distance between earth and sun, . . .

• All important dates of human history

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40

Charles Piazzi Smyth

• 1819–1900

• Astronomer

• Professor at university of Edinburgh

• Fellow of the Royal Society

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41

Verification

William Matthew Flinders Petrie (1853–1942)

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Other pyramidologists

• Charles Lagrange

• David Davison

• Georges Barbarin

• Robert Bauval

• . . .