Transcript of "History of Mathematics (Egyptian and Babylonian) "
1.
Loving heavenly Father
we come to you this hour
asking for your blessing
and help as we are
gathered together.
2.
We
pray
for
guidance in the matters
at hand and ask that
you would clearly show
us how to conduct our
work with a spirit of joy
and enthusiasm.
3.
Give us the desire to
find ways to excel in our
work. Help us to work
together and encourage
each other to excellence.
4.
We ask that we would
challenge each other to
reach higher and farther to
be the best we can be.
We ask this in the name of the
Lord Jesus Christ
Amen.
8.
Egyptian Numeric Symb ols
The Egyptian zero symbolize
beauty, complete and abstraction.
The Egyptian zero’s consonant
sounds are “nfr” and the vowel
sounds of it are unknown.
The “nfr” symbol is used to
expressed zero remainders in an
account sheet from the Middle
Kingdom dynasty 13.
9.
Numeric Symbols
1= simple stroke
10= hobble for cattle
100= coiled rope
1000= lotus flower
10000= finger
100000= frog
1000000= a god raising his adoration
12.
Addition and Subtraction in Egyptian Numerals
365
+ 257
= 622
13.
Egyptian Multiplication
• Doubling the number to be multiplied
(multiplicand) and adding of the doublings
to add together.
• Starting with a doubling of numbers from
1,2,4,8,16,32,64 and so on.
• Doubling of numbers appears only once.
Examples:
a. 11= 1+2+8
b. 23= 1+2+4+16
c. 44= 4+8+32
14.
Applying the distribution law:
a x (b+c)=(a x b) + (a x c)
Example:
23 x 13= 23x (1+4+8)
= 23 +92 +184
= 299
18.
Egyptian
Geometry
Reporter: Eileen M. Pagaduan
19.
Egyptian Geometry
• Discusses a spans of time period ranging
from ca 3000 BC to ca. 300 BC.
• Geometric problems appear both the
Moscow Mathematical Papyrus( MMP)
and Rhind Mathematical Papyrus (RMP).
• Used many sacred geometric shapes like
squares, triangles and obelisks.
20.
• Golenishchev Mathematical
Papyrus
• Written down 13th century based
on the older material dating Twe
lfth dynasty of Egypt.
• 18 feet long, 1 ½ and 3 inches
wide and divided into 25
problems with solution.
• Older than the Rhind
Mathematical Papryus.
Moscow
Mathematical Papyrus
21.
• Named after Alexander Henry
Rhind
• Dates back during Second
Intermediate Period of Egypt
• 33 cm tall and 5 m long
• Transliterated and mathematically
translated in the late 19th
century.
• Larger than the Moscow Mathemati
cal Papyrus
Rhind
Mathematical
Papyrus
22.
AREA
Object
Source
Formula (using m
odern notation)
Triangle
Problem 51 in RM
P and problem 4,7
and 17 in MMP.
A= ½ bh
Rectangle
Problem 49 in RM
P and problem 6 in
MMP and Lahin LV
.4.,problem1
A= bh
Circle
Problem 51 in RM
P and problems 4,
7 and 17 in MMP
A= ¼( 256/81)d^2
23.
Area of Rectangle
Problem: 6 of MMP
Calculation of the area of a rectangle is used in a problem
of simultaneous equations.
The following text accompanied the drawn rectangle.
1. Method of calculating area of rectangle.
2. If it is said to thee a rectangle in 12 in the area is 1/2 1/4 of the length.
3.
4.
5.
6.
For the breadth. Calculate 1/2 1/4 until you get 1. Result 1 1/3
Reckon with these 12, 1 1/3 times. Result 16
Calculate thou its angle (square root). Result 4 for the length.
1/2 1/4 is 3 for the breadth.
25
24.
Area of Rectangle
Problem: 49 of RMP
• The area of a rectangle of length 10 khet
(1000 cubits) and breadth 1 khet (100
cubits) is to be found 1000x100= 100,000
square cubits.
• The area was given by the scribe as 1000
cubits strips, which are rectangles of land,
1 khet by 1 cubit.
26
25.
Area of triangle
Problem: 51 of RMP
The scribe shows how to find the area of a triangle of land
of side 10 khet and of base 4 khet.
The scribe took the half of 4, then multiplied 10 by 2
obtaining the area as 20 setats of land.
Problem: 4 of MMP
The same problem was stated as finding the area of a
triangle of height (meret) 10 and base (teper) 4.
No units such as khets or setats were mentioned.
27
26.
Area of Circle
Computing π
Archimedes of Syracuse (250BC) was known as the
first person to calculate π to some accuracy; however, the
Egyptians already knew Archimedes value of
π = 256/81 = 3 + 1/9 + 1/27 + 1/81
28
27.
Area of Circle
Computing π
Problem: 50 of RMP
A circular field has diameter 9 khet. What is its area?
The written solution says, subtract 1/9 of the diameter
which leaves 8 khet. The area is 8 multiplied by 8 or 64
khet.
This will lead us to the value of
π = 256/81 = 3 + 1/9 + 1/27 + 1/81 = 3.1605
But the suggestion that the Egyptian used is
π = 3 = 1/13 + 1/17 + 1/160 = 3.1415
29
31.
Babylonian~!
• The Babylonian number system is old. (1900 BC to 1800 BC)
• But it was developed from a number system
belonging to a much older civilization called the
Sumerians.
• It is quite a complicated system, but it was used by
other cultures, such as the Greeks, as it had
advantages over their own systems.
• Eventually it was replaced by Arabic Number.
32.
After 3000 B.C, Babylonians developed
a system of writing.
Pictograph-a kind of picture writing
Cuneiform - Latin word “cuneus” which means “wedge”
Sharp edge of a stylus made a vertical stroke (ǀ)
and the base made a more or less deep impression (∆).
The combined effect was a head-and-tail figure resembling a wedge .
33.
• Like the Egyptians, the Babylonians used to
ones to represent two, and so on, up to nine.
• However, they tended to arrange the symbols
in to neat piles. Once they got to ten, there
were too many symbols, so they turned the
stylus on its side to make a different symbol.
• This is a unary system.
34.
• The symbol for sixty seems to be exactly
the same as that for one.
• However, the Babylonians were working
their way towards a positional system
35.
• The Babylonians had a very advanced
number system even for today's standards.
• It was a base 60 system (sexigesimal)
rather than a base 10 (decimal).
36.
• When they wrote "60", they would put a
single wedge mark in the second place of
the numeral.
• When they wrote "120", they would put two
wedge marks in the second place.
37.
• A positional number system is one where the numbers ar
e arranged in columns. We use a positional system, and
our columns represent powers of ten. So the right hand c
olumn is units, the next is tens, the next is hundreds, and
so on.
3
2
1
10^2 = 100
10^1 = 10
10^0 = 1
7
4
5
(7 ∙ 100) + (4 ∙ 10) + 5 = 745
38.
• The Babylonians used powers of sixty rather than ten. S
o the left-hand column were units, the second, multiples
of 60, the third, multiplies of 3,600, and so on.
(2*3600)+
(1*60) +
(10 + 1)
=7271
(2*602)
+
(1*60)
+
(10 + 1)
= 7271
39.
x 3600
x 60
Units
Value
1
1+1=2
10
10 + 1 = 11
10 + 10 = 20
60
60 + 1 = 61
60 + 1 + 1 = 62
60 + 10 = 70
60 + 10 + 1 = 71
2 x 60 = 120
2 x 60 + 1 = 121
10 x 60 = 600
10 x 60 + 1 = 601
10 x 60 + 10 = 660
3600 (60 x 60)
2 x 3600 = 7200
40.
They had no symbol for zero.
We use zero to distinguish
between 10 (one ten and no
units) and 1 (one unit).
The number 3601 is not too
different from 3660, and they
are both written as two ones.
The strange slanting symbol is
the zero.
3
2
1
41.
• The Babylonians used a system of
Sexagesimal fractions similar to our
decimal fractions.
For example:
if we write 0.125 then this is
1/10 + 2/100 + 5/1000 = 1/8.
42.
• Similarly the Babylonian Sexagesimal
fraction 0;7,30 represented
7/60 + 30/3600
which again written in our notation is 1/8.
43.
• We have introduced the notation of
the semicolon to show where the
integer part ends and the fractional
part begins.
• It is the “Sexagesimal point".
44.
References
•
From Book:
,
5th
Burton, D. M. (2003). Burton’s The History of Mathematics: An Introduction
Edition. New York: McGraw Hill.
•
From online
•
http://www.math.wichita.edu/history/topics/num-sys.html#babylonian
•
http://www.math.wichita.edu/history/topics/num-sys.html#babylonian
•
•
•
http://www.slideshare.net/Mabdulhady/egyptian-mathematics?from_search=1
http://en.wikipedia.org/wiki/Egyptian_mathematics
http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egypt_arith.html
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