mathematics in ancient egypt - vrije universiteit...
TRANSCRIPT
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
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33
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33
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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
• . . .