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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

1

The use of Mathematics in AncientEgypt

• Partitioning of fertile grounds

1

The use of Mathematics in AncientEgypt

• Partitioning of fertile grounds

• Construction of pyramids

1

The use of Mathematics in AncientEgypt

• Partitioning of fertile grounds

• Construction of pyramids

• Administration

1

The use of Mathematics in AncientEgypt

• Partitioning of fertile grounds

• Construction of pyramids

• Administration

• Calendar. . .

2

Sources

• Wrote mainly on papyrus

2

Sources

• Wrote mainly on papyrus

• Not suited for long conservation

2

Sources

• Wrote mainly on papyrus

• Not suited for long conservation

• Hieroglyphics and wall-paintings

2

Sources

• Wrote mainly on papyrus

• Not suited for long conservation

• Hieroglyphics and wall-paintings

• Babylonians had clay-tablets

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!

3

Rhind papyrus

4

• 1650 BC

• Henry Rhind (1833–1863)

• 87 problems with answer

• Main source of information

• Tables with fractions

• Ahmes

• Leather Roll

5

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

7

Theorem

Every fraction can be written as a sum of unitfractions.

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. . .

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

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

8

Remark

There are many ways to write a fraction as sum of

unit fractions.

1n

=1

n+ 1+

1n(n+ 1)

9

Multiplication 1

7× 22 =?

9

Multiplication 1

7× 22 =?

1 7

✔ 2 14

✔ 4 28

8 56

✔ 16 112

22 154

10

Division

154÷ 7 =?

10

Division

154÷ 7 =?

1 7

2 14 ✔

4 28 ✔

8 56

16 112 ✔

22 154

11

Multiplication 2

5 + 7/8× 12 + 2/3 =?

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

12

Writing of numbers

• Decimal

• But not positional

1 10 100 1000 10000 100000 1000000

� � � � µ � �

������� � �

� � �� � = 275

� µ µ µ µ µ ����� � � = 152023

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!!

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

14

Other papyri

• Reisner papyri (1880 BC)

• Kahun papyri (1800 BC)

• Rollin papyrus (1350 BC)

• Harris papyrus (1167 BC)

15

Mathematics used for building pyramids=

GEOMETRY

16

Weights and measures

• 1 cubit = 52.3 cm

• Long distances: ropes with knots

• Shorter distance: ruler

17

Egyptian ruler

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

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.

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

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.

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!!!

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?

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.

23

Rhind problem 57

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

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

25

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

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.

27

Construction of right angles 1

p

p

p

p

k0

k0k0

28

Property of a rectangle

• The diagonals of a rectangle meet eachotherhalfway.

28

Property of a rectangle

• The diagonals of a rectangle meet eachotherhalfway.

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

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

30

Area of a triangle?

Rhind problem 49: Multiply half a side with the

other side.

30

Area of a triangle?

Rhind problem 49: Multiply half a side with the

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

� � � �� � � �� � � �� � � �

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� � � �� � � �� � � �� � � �� � � �

1/2

b

1/2

c

a

31

Area of a quadrilateral?

b

c

d

a

a+ c

2· b+ d

2

32

Rhind problem 48

A = (d− d9)2

33

The area of a circle 1

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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.

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

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 . . .

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

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

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

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

36

Volume of a truncated pyramid

a

b

h

V =13h(a2 + ab+ b2)

37

How did they do it?

h’

a a

b

bh

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

39

40

Charles Piazzi Smyth

• 1819–1900

• Astronomer

• Professor at university of Edinburgh

• Fellow of the Royal Society

41

Verification

William Matthew Flinders Petrie (1853–1942)

42

Other pyramidologists

• Charles Lagrange

• David Davison

• Georges Barbarin

• Robert Bauval

• . . .

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