Chapter 1

1. Chapter 1 Section 4-1 Historical Numeration Systems

2. Roman Numeration Roman numerals are written as combinations of the seven letters in the table below. The letters can be written as capital (XVI) or lower-case letters (xvi).

3. Roman Numeration Roman Numeral Calculator http://www.novaroma.org/via_romana/numbers.html Note: see “Roman Numerals” on website for additional information

4. Roman Numeration (Additional HW Problems) • Change to Roman Numerals • 215 • 379 • 1995 • Change to Decimal Numbers • DCIV • CDXXIX • MCMXCVII

5. Ancient Egyptian Numeration – Simple Grouping The ancient Egyptian system is an example of a simplegrouping system. It used ten as its base and the various symbols are shown on the next slide.

6. Ancient Egyptian Numeration

7. Example: Egyptian Numeral Write the number below in our system. Solution 2 (100,000) = 200,000 3 (1,000) = 3,000 1 (100) = 100 4 (10) = 40 5 (1) = 5 Answer: 203,145

8. Example: Egyptian Numeral Convert 427 to Egyptian. Symbol Inventory: 8

9. Example: Egyptian Numeral Convert 427 to Egyptian. Solution 9

10. Symbols See word doc for more complete list of symbols. 10

11. Traditional Chinese Numeration – Multiplicative Grouping A multiplicative grouping system involves pairs of symbols, each pair containing a multiplier and then a power of the base. The symbols for a Chinese version are shown on the next slide.

12. Chinese Numeration

13. Example: Chinese Numeral Interpret each Chinese numeral. a) b)

14. Example: Chinese Numeral Solution a) b) 7000 200 400 0 (tens) 1 80 Answer: 201 2 Answer: 7482

15. Example: Convert to Chinese Numeral Symbol Inventory 2018 15

16. Example: Convert to Chinese Numeral 2018 2000 0 hundreds 10 8 16

17. Example: Roman Numerals See word doc from website 17

18. Positional Numeration The power associated with each multiplier can be understood by the position that the multiplier occupies in the numeral. To work successfully, a positional system must have a symbol for zero to serve as a placeholder in case one or more powers of the base are not needed.

19. Hindu-Arabic Numeration – Positional One such system that uses positional form is our system, the Hindu-Arabic system. The place values in a Hindu-Arabic numeral, from right to left, are 1, 10, 100, 1000, and so on. The three 4s in the number 45,414 all have the same face value but different place values.

20. Hindu-Arabic Numeration Hundred thousands Millions Ten thousands Thousands Decimal point Hundreds Tens Units 7, 5 4 1, 7 2 5 .

21. Hindu-Arabic Numeration In Expanded Notation 45, 414 = 4 x 10000 + 5 x 1000 + 4 x 100 + 1 x 10 + 4 x 1 = 4 x 104 + 5 x 103 + 4 x 102 + 1 x 101 + 4 x 100 5, 014 = 5 x 103 + 0 x 102 + 1 x 101 + 4 x 100 = 5 x 103 + 1 x 101 + 4 x 100

22. Hindu-Arabic Numeration – Scientific Notation Scientific Notation: Number written in powers of ten such that one digit is left of the decimal place. 45,414 = 4.5414 x 104 .045414 = 4.5414 x 10-2

23. Chapter 1 Section 4-2 Arithmetic in the Hindu-Arabic System

24. Historical Calculation Devices One of the oldest devices used in calculations is the abacus. It has a series of rods with sliding beads and a dividing bar. The abacus is pictured on the next slide.

25. Abacus Reading from right to left, the rods have values of 1, 10, 100, 1000, and so on. The bead above the bar has five times the value of those below. Beads moved towards the bar are in “active” position.

26. Example: Abacus Which number is shown below? 100 104 101 103 102 Solution 1000 + (500 + 200) + 0 + (5 + 1) = 1706

27. Example: Abacus Which number is shown below? 100 104 101 103 102 Solution 10000 + (5000 + 1000) + (500 + 200) + 30 + (5 + 2) = 16737

28. Example: Abacus Draw an abacus to show each number? See abacus.ppt on website for use with hw problems 31 - 34

29. Lattice Method The Lattice Method was an early form of a paper-and-pencil method of calculation. This method arranged products of single digits into a diagonalized lattice.

30. Example: Lattice Method Find the product 32 x 741 by the lattice method. Solution 7 4 1 Set up the grid to the right. 3 2

31. Example: Lattice Method Fill in products 7 4 1 3 2

32. Example: Lattice Method Add diagonally right to left and carry as necessary to the next diagonal. 1 2 3 7 1 2

33. Example: Lattice Method 1 2 3 7 1 2 Answer: 23,712 See Lattice Template on Website

34. Napier’s Rods (Napier’s Bones) John Napier’s invention, based on the lattice method of multiplication, is often acknowledged as an early forerunner to modern computers. Refer figure 2 on page 165

35. Russian Peasant Method Similar to the Egyptian Method of multiplication but dividing one column by 2 instead of doubling.

36. Chapter 1 Section 4-4 Properties of Mathematical Systems

37. An Abstract System The focus will be on elements and operations that have no implied mathematical significance. We can investigate the properties of the system without notions of what they might be.

38. Potential Properties of a Single Operation Symbol Let a, b, and c be elements from the set of any system, and ◘ represent the operation of the system. Closurea ◘ b is in the set Commutative a ◘ b = a ◘ b. Associativea ◘ (b ◘ c) = (a ◘ b) ◘ c Identity The system has an element e such that a ◘ e = a and e ◘ a = a. Inverse there exists an element x in the set such that a ◘ x = e and x ◘ a = e.

39. Operation Table Consider the mathematical system with elements {a, b, c, d} and an operation denoted by ☺. The operation table on the next slide shows how operation ☺ combines any two elements. To use the table to find c ☺ d, locate c on the left and d on the top. The row and column intersect at b, so c ☺ d = b.

40. Operation Table for ☺

41. Closure Property For a system to be closed under an operation, the answer to any possible combination of elements from the system must in the set of elements. This system is closed.

42. Identity Property For the identity property to hold, there must be an element E in the set such that any element X in the set, X ☺ E = X and E ☺ X = X. a is the identity element of the set.

43. Inverse Property If there is an inverse in the system then for any element X in the system there is an element Y (the inverse of X) in the system such that X ☺ Y = E and Y ☺ X = E,where E is the identity element of the set. You can inspect the table to see that every element has an inverse.

44. Commutative Property For a system to have the commutative property, it must be true that for any elements X and Y from the set, X ☺ Y = Y ☺ X. This system has the commutative property. The symmetry with respect to the diagonal line shows this property

45. Associative Property For a system to have the associative property, it must be true that for any elements X, Y, and Z from the set, X ☺ (Y ☺ Z) = (X ☺ Y) ☺ Z. This system has the associative property. There is no quick check – just work through cases.

46. Example 1: Identifying Properties Consider the system shown with elements {0, 1, 2, 3} and operation #. Which properties are satisfied by this system?

47. Example 1: Identifying Properties Solution The system satisfies the closure, associative, commutative, and identity properties, and inverse property.

48. Example 2: Identifying Properties Consider the system shown with elements {0, 1, 2, 3, 4} and operation Which properties are satisfied by this system?

49. Example 2: Identifying Properties Solution The system satisfies the closure, associative, commutative, and identity properties, but not the inverse property.

50. Example 3: Identifying Properties Construct a base 5 addition system of remainders. Which properties are satisfied by this system?