Loving heavenly Father we come to you this hour asking for your blessing and help as we are gathered to-gether.

We pray for guidance in the mat-ters at hand and ask that you would clearly show us how to conduct our work with a spirit of joy and enthusiasm.

Give us the desire to find ways to excel in our work. Help us to work together and en-courage each other to excellence.

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.

Ice Breaker

Mathematics inEarly Civilization

Prepared by:Daniel KoEileen M. Pagaduan

EgyptianMathemat-

ics(Introduction)

Egyptian Numeric Symb ols

The Egyptian zero symbolize beauty, complete and ab-straction.

The Egyptian zero’s conso-nant sounds are “nfr” and the vowel sounds of it are un-known.

The “nfr” symbol is used to expressed zero remainders in an account sheet from the Middle Kingdom dynasty 13.

Numeric Symbols

1= simple stroke10= hobble for cattle100= coiled rope1000= lotus flower10000= finger100000= frog1000000= a god raising his adoration

Egyptian Fraction

The Eye of Horus

EgyptianArithmetic

Addition and Subtraction in Egyptian Numerals

365

+ 257

= 622

Egyptian Multiplication

• Doubling the number to be multiplied (multiplicand) and adding of the dou-blings 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

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

Multiply 23 x 13

1+4+8 = 13

Result: 23+92+ 184= 299

23

46

92

184

12

3

8

Divide 299/23=?

Dividend: 1+4+8 = 13

Result: 23+92+ 184= 299

23

46

92

184

12

3

8

19

Numbers that cannot divide evenly e.g.:35 divide by 8

8 1

16 2

√ 32 4

4 1/2

√ 2 1/4

√ 1 1/8

35 4 + 1/4 + 1/8

doubling

half

EgyptianGeometry

Reporter: Eileen M. Pagad-uan

Egyptian Geometry• Discusses a spans of time period rang-

ing from ca 3000 BC to ca. 300 BC.

• Geometric problems appear both the Moscow Mathematical Papyrus( MMP) and Rhind Mathe-matical Papyrus (RMP).

• Used many sacred geometric shapes like squares, triangles and obelisks.

Moscow Mathematical Papyrus

• Golenishchev Mathemati-cal

Papyrus• Written down 13th century

based on the older mate-rial dating Twelfth dy-nasty of Egypt.

• 18 feet long, 1 ½ and 3 inches

wide and divided into 25 problems with solution.• Older than the Rhind Mathematical Papryus.

Rhind Mathematical Papyrus

• Named after Alexander Henry

Rhind• Dates back during Second Intermediate Period of Egypt• 33 cm tall and 5 m long • Transliterated and mathe-

matically translated in the late 19th

century.• Larger than the Moscow

Mathematical Papyrus

AREAObject Source Formula (using

modern nota-tion)

Triangle Problem 51 in RMP and prob-lem 4,7 and 17 in MMP.

A= ½ bh

Rectangle Problem 49 in RMP and prob-lem 6 in MMP and Lahin LV.4.,problem1

A= bh

Circle Problem 51 in RMP and prob-lems 4,7 and 17 in MMP

A= ¼( 256/81)d^2

25

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 rectan-gle.

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. For the breadth. Calculate 1/2 1/4 until you get 1. Result 1

1/34. Reckon with these 12, 1 1/3 times. Result 165. Calculate thou its angle (square root). Result 4 for the

length.6. 1/2 1/4 is 3 for the breadth.

26

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 rectan-gles of land, 1 khet by 1 cubit.

27

Area of triangle

Problem: 51 of RMPThe 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 MMPThe 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.

28

Area of CircleComputing 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

29

Area of CircleComputing 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

VolumeObject Source Formula

Cylindrical granaries RMP 41

Cylindrical granaries RMP 42, Lahun IV.3

Rectangular gra-naries

RMP 44-46 and MMP 14

Truncated granaries MMP 14

ENDنهاية

Babylo-nianArith-metic

Reporter: Ko, Doungjun [Daniel]

Babylonian~!

• The Babylonian number system is old. (1900 BC to 1800 BC)

• But it was developed from a number system be-longing 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.

After 3000 B.C, Babylonians devel-oped a system of writing.

Pictograph-a kind of picture writ-ing

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 .

• Like the Egyptians, the Babylonians used to ones to represent two, and so on, up to nine.

• However, they tended to arrange the sym-bols 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.

• The symbol for sixty seems to be ex-actly the same as that for one.

• However, the Babylonians were work-ing their way towards a positional system

• The Babylonians had a very advanced number system even for today's stan-dards.

• It was a base 60 system (sexigesi-

mal) rather than a base 10 (deci-mal).

• 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.

• A positional number system is one where the numbers are arranged in columns. We use a posi-tional system, and our columns represent powers of ten. So the right hand column is units, the next is tens, the next is hundreds, and so on.

(7 ∙ 100) + (4 ∙ 10) + 5 = 745

Positional number system

10^2 = 100 10^1 = 10 10^0 = 1

7 4 5

123

• The Babylonians used powers of sixty rather than ten. So the left-hand column were units, the second, multiples of 60, the third, multiplies of 3,600, and so on.

(2*602) + (1*60) + (10 + 1) = 7271

Positional number system

(2*3600)+

(1*60) +

(10 + 1)

=7271

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

They had no symbol for zero. We use zero to dis-tinguish 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 sym-bol is the zero.

Lack of zero

123

• 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. 

FRACTION!!! =)

• Similarly the Babylonian Sexagesi-mal fraction 0;7,30 represented

7/60 + 30/3600

which again written in our notation is 1/8.

FRACTION!!! =)

• 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".

FRACTION!!! =)

References• From Book:

Burton, D. M. (2003). Burton’s The History of Mathematics: An Introduction, 5th 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_arit

h.html