1. Feljone G. Ragma, Ed.D.

2. Egyptian Mathematical Papyri Mathematics arose from practical needs. The Egyptians required ordinary arithmetic in the daily transactions of commerce and to construct a workable calendar. Simple geometric rules were applied to determine boundaries if fields and the contents of granaries. According to Herodotus, Egypt if the gift of the Nile and Geometry is the second gift. Due to annual inundation of the Nile valley, it became necessary for purposes of taxation to determine how much land had been gained or lost.

3. Egyptian Mathematical Papyri The initial emphasis was on utilitarian mathematics. Algebra evolved ultimately from the techniques of calculation, and theoretical geometry began with land measurement. Most of our knowledge of early mathematics in Egypt comes from two sizable- each named after its former owner- the RHIND PAPYRUS and the GOLENISCHEV or the MOSCOW PAPYRUS.

4. The RHIND Papyrus It was found in Thebes, in the ruins of a small building near the Ramessuem. It was purchased in Luxor, Egypt in 1858 by Henry Rhind. Later on, it was willed to the British Museum. It was written in hieratic script by Ahmes It was a scroll 18 feet long and 13 inches high.

5. The Rosetta Stone It is a slab of polished black basalt. It was found during the Napoleons expedition. It was uncovered by Napoleons army near the Rosetta branch of the Nile in 1799. It is made up of 3 panels, each inscribed in a differeny type of writing: Greek down the bottom third, demotic script of Egyptian in the middle, and ancient hieroglyphic in the broken upper third. It is now laid in the British Museum, where 4 plaster cats were made for Oxford, Cambridge, Edinburgh and Dublin Universities.

6. Jean Francois Champollion The greatest of all names associated to the study of Egypt and the Rosetta stone. An Egyptologist. At 13, he was reading 3 languages, 17 he was appointed to the faculty of the University of Grenoble. When he was older, he had compiled a hieroglyphic vocabulary and given a complete reading of the upper panel of the Rosetta stone. He established correlations bet. Ind hieroglyphics and greek letters (Ptolemy and Cleopatra)- cartouches cartridge.

7. He formulated a system of grammar and general decipherment that is the foundation on which all later Egyptologists have worked. In general, the Rosetta stone had provided the key to understanding one of the great civilization of the past.

8. The Egyptian Arithmetic E.A. was essentially additive, meaning that its tendency was to reduce multiplication and division to repeated additions. Multiplication of two numbers was accomplished by successively doubling one of the numbers (multiplier) and then adding the appropriate duplication to form the product.

9. The Egyptian Arithmetic: Multiplication Multiply 19 and 71. 1 71 2 142 4 284 8 568 16 1136 Total = 19

10. The Egyptian Arithmetic Multiply 19 and 71. 1 71 2 142 4 284 8 568 16 1136 Total = 19

11. The Egyptian Arithmetic Multiply 19 and 71. Multiply 23 and 40. Multiply 13 and 15. Multiply 23 and 88. Multiply 15 and 21. Multiply 8 and 49. Multiply 73 and 88. Multiply 113 and 140.

12. The unit fractions Unit fractions were the only ones recognized. But 2/3 was recognized with a special symbol . Doubling is not the only procedure; other numbers are allowed in Egyptian Arithmetic: Division Decomposing fractions fractions expressed as sum of unit fractions ( fractions whose numerator is 1)

13. Decomposing fractions Rule: NO repetition of fractions Rewrite 6/7 as a sum of unit fractions. 1 7 3.5 0.875 1/8 0.4375 1/7 1 1/14 0.5 1/28 0.25 1/56 0.125

14. Decomposing fractions Rule: NO repetition of fractions Rewrite 6/7 as a sum of unit fractions. 1 7 3.5 0.875 1/8 0.4375 1/7 1 1/14 0.5 1/28 0.25 1/56 0.125 6

15. Decomposing fractions Rule: NO repetition of fractions Rewrite 6/7 as a sum of unit fractions. Check using your calculator if the result is correct! 1 7 3.5 1.75 1/7 1 1/14 0.5 1/28 0.25 1/2+ + 1/14 + 1/28 6

16. Decompose the following 3/5 7/9 5/8 2/9 3/8 2/3 2/7 3/7 3/4

17. Discovered Unit Fraction table The Rhind Papyrus- it contained a fraction table with 2 as the numerator and an odd number between 5 and 101 in the denominator. (e.g. 2/5, 2/7, 2/11, 2/13) The general rule 2/3k = 1/2k + 1/6k; USE ONLY FOR MULTIPLES OF 5 Decompose 2/15 1/10 + 1/30

18. Decompose the following using the general rule 2/10 2/20 2/25 2/30

19. Additional Rules: Small denominators were preferred, with none greater than 1000. The fewer the unit fractions, the better; and there were never more than four. Denominators that were even were more desirable than odd ones, esp. for the initial term A small first denominator might be increased if the size of the others was thereby reduced.

20. Decompose the following 2/5 2/7 2/11 2/13 2/17 2/19 2/23 2/25

21. FRACTIONAL TABLE

22. Multiply the following with the aid of the fraction table (2 + ) x (1 + + 1/7) 1 1 + + 1/7 2 2 + 1 + 2/7 + 4 + 1/14 + 1/8 + 1/28 Not a unit fraction (use table)

23. Multiply the following with the aid of the fraction table (2 + ) x (1 + + 1/7) 1 1 + + 1/7 2 2 + 1 +1/4 + 1/28 + 4 + 1/14 + 1/8 + 1/28 2 + 2 +1 +1/4 + 1/28+ + 1/8 + 1/28 Or 3 + + 1/8 +1/14

24. Multiply the following (11 + + 1/8 ) (37) (1 + + ) ( 9 + + ) (2 + ) ( 1 + + ) Show that the product of (1 + + ) is equal to 1/8 Show that the product of (1/32 + 1/224) and (1 + + ) is equal to 1/16

25. Divide 37 and ( 1 + 2/3 + + 1/7) 1 1 + 2/3 + + 1/7 2 2 + 1 + 1/3 + 1 + + 1/28 or 4 + 1/3 + + 1/28 4 8 + 2/3 +1/2 + 1/14 8 16 + 4/3 + 1 + 1/7 or 16 +1 + 1/3 + 1 + 1/7 or 18 + 1/3 + 1/7 16 36 + 2/3 + + 1/28

26. 2/3 + + 1/28 + ____ = 1 2/3 + + 1/28 + x/84= 1 x= 4; thus the fraction is 4/84 or 1/21 So, y (1 + 2/3 + + 1/7) = 1/21 Y = 2/97 ( from the table = 1/56 + 1/67 + 1/776)

27. Divide 37 and ( 1 + 2/3 + + 1/7) 1 1 + 2/3 + + 1/7 2 2 + 1 + 1/3 + 1 + + 1/28 or 4 + 1/3 + + 1/28 4 8 + 2/3 +1/2 + 1/14 8 16 + 4/3 + 1 + 1/7 or 16 +1 + 1/3 + 1 + 1/7 or 18 + 1/3 + 1/7 16 36 + 2/3 + + 1/28 1/56 + 1/679 + 1/776 1/21

28. Representing rational numbers Two methods: Splitting method and Fibonaccis method Splitting Method: 1/n = 1/(n+1) + 1/ (n(n+1)) 1/9 Where n = 9 1/9 = 1/(10) + 1/(90)

29. Represent the following using splitting method 1/6 2/5 5/7 4/7

30. Leonardo of Pisa (Fibonacci) First step: find n1 satisfying 1/n1 a/b < 1/(n1-1) Then, a/b- 1/n1 = (n1a-b)/(bn1) = a1/b1 Formula: a/b = 1/n1 + 1/n2 + 1/nk + 1/bk Thus, a/b = 1/n1 + a1/b1

31. Fibonacci Method 2/19 Find n1 9< 19/2