Abstract Representation: Your Ancient Heritage

1. Abstract Representation: Your Ancient Heritage 9 2 7 4 5 3 6 1 8

2. Moral:BE PUNCTUAL! • DON'T miss the magic trick that happens at 3:00 pm sharp! • Sit close-up: some of the tricks are hard to see from the back.

3. How to play the 9 stone game? 2 1 3 5 9 • 9 stones, numbered 1-9 • Two players alternate moves. • Each move a player gets to take a new stone • Any subset of 3 stones adding to 15, wins. 4 6 7 8

4. For enlightenment, let’s look to ancient China.

5. 2 9 7 4 5 3 6 1 8 Magic Square: Brought to humanity on the back of a tortoise from the river Lo in the days of Emperor Yu

6. Magic Square: Any 3 in a vertical, horizontal, or diagonal line add up to 15.

7. Conversely, any 3 that add to 15 must be on a line.

8. TIC-TAC-TOE on a Magic SquareRepresents The Nine Stone GameAlternate taking squares 1-9. Get 3 in a row to win.

9. Don’t let the representation choose you – CHOOSE THE REPRESENTATION!

10. Course Staff • Steven Rudich • Avrim Blum • Jevan Saks • Noah Falk • Mike Klipper • Charlie Garod • Taka Osogami • Charlotte Yano

11. ((( ))) Please feel free to ask questions!

12. Course Document • Let’s examine it together:

13. Course Content • The definition and manipulation of fundamental representations for numbers, sets, sequences, functions, sums, probabilities, structures, computations, and proofs. • Compelling applications of these abstract representations. • The history and philosophy of mathematics.

14. For Example • Representations of number: • Unary • Binary, base-b • Fractions • Egyptian fractions • Continued Fractions • Chinese Remainder Representation • Pascal’s Triangle Representation • Combinatorial Representation • Algebraic Representation • Computational Representation • ….

15. For Example • Applications to: • Fast algorithms for arithmetic • Analysis of resources in algorithms • Secure protocols • ….

16. We assume little • High school understanding of numbers and algebra. • Intuitive understanding of a set, a sequence, and a function.

17. Your Ancient Heritage The history of mathematics can be cast as the discovery and manipulation of abstract representations. Nothing has changed – a good computer scientist will invent representations to make computation easy.

18. Abstract Representations • The study of abstract representations starts with prehistoric unary and follows the development of mathematics to the present day.

19. Mathematical Prehistory:30,000 BC • Paleolithic peoples in Europe record unary numbers on bones 1 2 3

20. Wait a minute! Isn’t calling unary an abstract representation pushing it a bit?

21. No! In fact, it is important to respect the status of each representation, no matter how primitive. Unary is a perfect object lesson.

22. Consider the problem of finding a formula for the sum of the first n numbers. First, we will give the standard high school algebra proof….

23. 1 + 2 + 3 + . . . + n-1 + n = S n + n-1 + n-2 + . . . + 2 + 1 = S (n+1) + (n+1) + (n+1) + . . . + (n+1) + (n+1) = 2S n (n+1) = 2S

24. 1 + 2 + 3 + . . . + n-1 + n = S n + n-1 + n-2 + . . . + 2 + 1 = S (n+1) + (n+1) + (n+1) + . . . + (n+1) + (n+1) = 2S n (n+1) = 2S

25. 1 + 2 + 3 + . . . + n-1 + n = S n + n-1 + n-2 + . . . + 2 + 1 = S (n+1) + (n+1) + (n+1) + . . . + (n+1) + (n+1) = 2S n (n+1) = 2S

26. 1 + 2 + 3 + . . . + n-1 + n = S n + n-1 + n-2 + . . . + 2 + 1 = S (n+1) + (n+1) + (n+1) + . . . + (n+1) + (n+1) = 2S n (n+1) = 2S

27. Algebraic argument Let’s restate this argument using a UNARY representation

28. = number of white dots. 1 2 . . . . . . . . n

29. = number of white dots = number of yellow dots n . . . . . . . 2 1 1 2 . . . . . . . . n

30. = number of white dots = number of yellow dots n n There are n(n+1) dots in the grid n n n n n+1 n+1 n+1 n+1 n+1

31. = number of white dots = number of yellow dots n n n n n n n+1 n+1 n+1 n+1 n+1

32. nth Triangular Number • n = 1 + 2 + 3 + . . . + n-1 + n • = n(n+1)/2

33. nth Square Number • n = n + n-1 • = n2

34. Very convincing! The unary representation brings out the geometry of the problem and makes each step look very natural. By the way, my name is Bonzo. And you are?

35. Odette. Yes, Bonzo. Let’s take it even further…

36. Breaking a square up in a new way.

37. 1 Breaking a square up in a new way.

38. 1 + 3 Breaking a square up in a new way.

39. 1 + 3 + 5 Breaking a square up in a new way.

40. 1 + 3 + 5 + 7 Breaking a square up in a new way.

41. 1 + 3 + 5 + 7 + 9 Breaking a square up in a new way.

42. 1 + 3 + 5 + 7 + 9 = 52 The sum of the first 5 odd numbers is 5 squared

43. The sum of the first n odd numbers is n squared.

44. nth Triangular Number • n = 1 + 2 + 3 + . . . + n-1 + n • = n(n+1)/2

45. nth Square Number • n = 1 + 3 + … + 2n-1 • = Sum of first n odd numbers

46. nth Square Number • n = n + n-1 • = n2

47. Alternatively… • n + n-1 = • 1 + 2 + 3 + 4 + 5 ... • + 1 + 2 + 3 + 4 ... • 1 + 3 + 5 + 7 + 9 … • Sum of Odd numbers

48. (n-1)2= area of square ( n-1)2 n-1

49. nn+ nn-1 = n (n + n-1) = n n = n (n)2=area of square n ( n-1)2 = area of pieces n-1 n

50. (n)2 =(n-1)2 + n n ( n-1)2 n-1 n