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- 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.
- 5. Ice Breaker
- 6. Mathematics in Early Civilization Prepared by: Daniel Ko Eileen M. Pagaduan
- 7. Egyptian Mathematics (Introduction)
- 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
- 10. Egyptian Fraction The Eye of Horus
- 11. Egyptian Arithmetic
- 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
- 15. Multiply 23 x 13 23 46 92 184 1 2 3 8 1+4+8 = 13 Result: 23+92+ 184= 299
- 16. Divide 299/23=? 23 46 92 184 Result: 23+92+ 184= 299 1 2 3 8 Dividend: 1+4+8 = 13
- 17. 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 19
- 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
- 28. Volume Object Source Cylindrical granaries RMP 41 Cylindrical granaries RMP 42, Lahun IV.3 Rectangular granaries RMP 44-46 and MMP 14 Truncated granaries MMP 14 Formula
- 29. END
- 30. Babylonian Arithmetic Reporter: Ko, Doungjun [Daniel]
- 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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