What? You mean my ancestors helped invent math? Gary Rubinstein Stuyvesant HS, New York City Math for America

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1 What? You mean my ancestors helped invent math? Gary Rubinstein Stuyvesant HS, New York City Math for America

2 Topics studied in ancient Dmes: MulDplicaDon and Division FracDons and Decimals Linear EquaDons QuadraDc EquaDons CalculaDon of square and cube roots Area of circles Pre- Trigonometry Number theory

3 Egypt

4 EgypDan MulDplicaDon 23 x 21 ü ü ü = =483

5 EgypDan Division 483 / 23 ü ü ü = =483

6 EgypDan Division 15 / 4 ü 1 4 ü 2 8 ü 1/2 2 ü 1/ /2+1/4 =3 + 1/2 + 1/ =15

7 EgypDan Unit FracDons Only of form 1/n except for 2/3 3/4=1/2+1/4 3/5=1/2+1/10 Why?

8 Divide 6 loaves among 10 men 6/10=3/5=1/2+1/10 3/5 per man or 1/2 + 1/10 per man.

9 MulDplying fracdons

10 EgypDan MulDplicaDon 2/3 + 1/30 x 10 ü 1 2/3 + 1/30 ü /3 + 1/ /3 + 1/10 + 1/30 ü /3 + 1/5 + 1/ /3 + 1/5 + 1/15 (5 + 1/5 + 2/5 = 5 + 3/5) /2 + 1/10

11 False PosiDon x x = 16 Try convenient value x = 7. 1 But 7+ g7 = 8 not Off by a factor of 16 8 = 2 x = 7 2= 14

12 A more involved example 1 8 x /2 4 x 1/4 2 x 1/ /4+1/8 19 x x = 19 Try x = 7. Becomes 8 not 19. 1) Need to do ) Then need to multiply 7 by that answer. x 1 2+1/4+1/8 x 2 4+1/2+1/4 x / /2+1/2+1/4+1/4+1/ /2+1/8

13 Base in base 60 is 119 in base in base 60 is 120 in base 10. Where have you seen = 2 00?

14 Base 60 decimals.2 5 in base 10 is 2/10 + 5/100 ;07 30 in base 60 is 7/ /3600 Fewer repeadng decimals in base 60!

15 Fewer repeadng decimals! 1/2=;30 1/6=;10 1/12=;05 1/3=;20 1/8=; /15=;04 1/4=;15 1/9=; /18=; /5=;12 1/10=;06 1/20=;03

16 ConverDng fracdons to base = x 60 9x = 60 x = = = =;

17 Babylonian MulDplicaDon ( ) 2 (26 19) = 4 = = 494

18 3 Babylonian IdenDDes (a + b) 2 (a b) 2 = 4ab (a + b) 2 4ab = (a b) 2 (a b) 2 + 4ab = (a + b) 2

19 Babylonian QuadraDcs In a rectangle, the length exceeds the width by 7. If the area of the rectangle is 60, what are the dimensions of the rectangle? x y = 7 xy = 60

20 Babylonian QuadraDcs 7 2 = = = x y = 7 xy = 60 (x y) 2 x y 2 x y xy = x + y = = = 5 x + y 2 2 = (x + y) 2 (x + y) (x y) + = x 2 2 (x + y) (x y) = y 2 2

21 Babylonian QuadraDcs The area of a square is reduced by 7 times the length and the result is 60. What is the length? x 2 7x = 60 x(x 7) = 60 y = x 7 xy = 60 x y = 7 x = 12 y = 5

22 Babylonian QuadraDcs 3z 2 7z = 20 9z 2 21z = 60 (3z) 2 7(3z) = 60 3z = x x 2 7x = 60 x(x 7) = 60 y = x 7 xy = 60 x y = 7 x = 12 3z = 12 z = 4

23 Plimpton 322

24 Pythagorean Triples x + 1 x 2 x + 1 x 4 x + 1 x x 1 x 2 x 1 x 4 x 1 x = 4 = 1 = 1 c 2 a 2 = b 2 c = x + 1 x 2 x 1 a = x 2 b = 1 Pick x so that x and 1/x are terminating decimals x = x = c = = a = = 9 40 b = 1 = Pythagorean Triple: 9, 40, 41

25 China

26 Broken Bamboo Problem Now given a bamboo 10 chi high, which is broken so that its Dp touches the ground 3 chi away from the base. Tell: what is the height of the break?

27 Broken Bamboo Problem x 2 = (10 x) x 2 = x + x 2 20x = 91 x =

28 Now given a tree 20 chi high, and 3 chi in circumference. A kudzu vine winds around it 7 Dmes from its root to its top. Tell: what is the length of the vine? Vine Problem

29 Vine Problem = x = x = x 2 29 = x

30 The Sea Island Problem Set up two poles of the same height, 5 bu, the distance between the two poles being 1000 bu. Move away 123 bu from the front pole and observe the peak of the island from ground level; it is seen that the Dp of the front pole coincides with the peak. Move backward 127 bu from the rear pole and observe the peak of the island from ground level; the Dp of the back pole also coincides with the peak. What is the height of the island?

31 h = bd + b = 5g1000 a 1 a = = 1255

32 Square Root Algorithm = = = 6 6?? = 621 > = 544 > = 469

33 Why it works 30x + 30x + x 2 = x + x 2 = 469 (60 + x)x = 469

34 Square Root Algorithm = = = 4 4?? > = 176 > = 129 < = ?? > = 3269

35 Why it works

36 EsDmaDon of Pi Hexagon=6*AOB=2.598 COB=1/2*CO*DB=1/2*1*1/2=1/4 12-gon=12*COB= <3<pi

37 EsDmaDon of Pi 12-gon-hexagon=6 triangles 12-gon-hexagon=3 rectangles 2*12-gon-2*hexagon=6 rectangles 12-gon<pi<hexagon+6 rectangles 12-gon<pi<hexagon+2*12-gon-2*hexagon 12-gon<pi<2*12-gon-1*hexagon

38 The Chinese Remainder Theorem The Emperor of Qin secretly counts his soldiers in an enigmatic manner. When he counts his soldiers in groups of three, two soldiers are left. When he counts in groups of five, three soldiers are left, and when the soldiers are counted in groups of seven, two soldiers are left. What is the smallest number of soldiers possible? x 2(mod 3) x 3(mod 5) x 2(mod 7)

39 x 2(mod 3) x 3(mod 5) x 2(mod 7) The Chinese Remainder Theorem Find a multiple of 5*7 which is one more than a multiple of 3: 70 Find a multiple of 3*7 which is one more than a multiple of 5: 21 Find a multiple of 3*5 which is one more than a multiple of 7: 15 Multiply the three desired remainders by these three numbers and add: 233 Divide by 3*5*7=105 and take the remainder: ggg ggg ggg

40 Pascal s Triangle?

41 India

42 Pythagorean Theorem proof in India

43 From??? Indian square root of 2

44 Indian square root of 2

45 Indian square root of 2

46 Indian square root of x = g x = 12 2

2 > 3 2 1 12 = 17 12 17 1 1 x = g 12 2 12 1 x = 408

47 2 > = x = g x = 408 Indian square root of 2 2 > =

48 The Gregory- Liebniz formula? Arctan x = x- x/3+x/5- x/7+

49 Ramanujan Considered to be one of top mathemadcians of all Dme. Died at 33 years old.

50 Taxicab number Hardy said that his taxicab number, 1729, was not a very interesdng number. Ramanujan pointed out that 1729 is the sum of two perfect cubes: =1729 in two different ways: =1729

51 Ramanujan nearly squares the circle

52 Ramanujan nearly squares the circle Works because π

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