Chapter 4

1. Chapter 4 Numeration and Mathematical Systems © 2008 Pearson Addison-Wesley. All rights reserved

2. Chapter 4: Numeration and Mathematical Systems 4.1 Historical Numeration Systems 4.2 Arithmetic in the Hindu-Arabic System 4.3 Conversion Between Number Bases 4.4 Clock Arithmetic and Modular Systems 4.5 Properties of Mathematical Systems 4.6 Groups © 2008 Pearson Addison-Wesley. All rights reserved

3. Chapter 1 Section 4-1 Historical Numeration Systems © 2008 Pearson Addison-Wesley. All rights reserved

4. Historical Numeration Systems • Mathematical and Numeration Systems • Ancient Egyptian Numeration – Simple Grouping • Traditional Chinese Numeration – Multiplicative Grouping • Hindu-Arabic Numeration - Positional © 2008 Pearson Addison-Wesley. All rights reserved

5. Mathematical and Numeration Systems • A mathematical system is made up of • three components: • 1. a set of elements; • one or more operations for combining • the elements; • 3. one or more relations for comparing the • elements. © 2008 Pearson Addison-Wesley. All rights reserved

6. Mathematical and Numeration Systems The various ways of symbolizing and working with the counting numbers are called numeration systems. The symbols of a numeration system are called numerals. © 2008 Pearson Addison-Wesley. All rights reserved

7. Example: Counting by Tallying Tally sticks and tally marks have been used for a long time. Each mark represents one item. For example, eight items are tallied by writing © 2008 Pearson Addison-Wesley. All rights reserved

8. Counting by Grouping Counting by grouping allows for less repetition of symbols and makes numerals easier to interpret. The size of the group is called the base (usually ten) of the number system. © 2008 Pearson Addison-Wesley. All rights reserved

9. 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. © 2008 Pearson Addison-Wesley. All rights reserved

10. Ancient Egyptian Numeration © 2008 Pearson Addison-Wesley. All rights reserved

11. Example: Egyptian Numeral Write the number below in our system. © 2008 Pearson Addison-Wesley. All rights reserved

12. 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. © 2008 Pearson Addison-Wesley. All rights reserved

13. Chinese Numeration © 2008 Pearson Addison-Wesley. All rights reserved

14. Example: Chinese Numeral Interpret each Chinese numeral. a) b) © 2008 Pearson Addison-Wesley. All rights reserved

15. Positional Numeration A positional system is one where the various powers of the base require no separate symbols. The power associated with each multiplier can be understood by the position that the multiplier occupies in the numeral. © 2008 Pearson Addison-Wesley. All rights reserved

16. Positional Numeration In a positional numeral, each symbol (called a digit) conveys two things: 1. Face value – the inherent value of the symbol. 2. Place value – the power of the base which is associated with the position that the digit occupies in the numeral. © 2008 Pearson Addison-Wesley. All rights reserved

17. Positional Numeration 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. © 2008 Pearson Addison-Wesley. All rights reserved

18. 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. © 2008 Pearson Addison-Wesley. All rights reserved

19. Hindu-Arabic Numeration Hundred thousands Millions Ten thousands Thousands Decimal point Units (Ones) Hundreds Tens 7, 5 4 1, 7 2 5 . © 2008 Pearson Addison-Wesley. All rights reserved

20. Section 4.1: Historical Numerical Systems • A mathematical system has • a) Elements • b) Operations • c) Relations • d) All of the above

21. Section 4.1: Historical Numerical Systems • 2. Our numeration system is an example of a • simple grouping system • multiplicative grouping system • positional system • d) complex grouping

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

23. Arithmetic in the Hindu-Arabic System • Expanded Form • Historical Calculation Devices

24. Expanded Form By using exponents, numbers can be written in expanded form in which the value of the digit in each position is made clear.

25. Example: Expanded Form Write the number 23,671 in expanded form.

26. Distributive Property For all real numbers a, b, and c, For example,

27. Example: Expanded Form Use expanded notation to add 34 and 45.

28. Decimal System Because our numeration system is based on powers of ten, it is called the decimal system, from the Latin word decem, meaning ten.

29. 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.

30. 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.

31. Example: Abacus Which number is shown below?

32. 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. The method is shown in the next example.

33. Example: Lattice Method Find the product by the lattice method. 7 9 4 3 8

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. The rods are pictured on the next slide.

35. Napier’s Rods See figure 2 on page 174

36. Russian Peasant Method Method of multiplication which works by expanding one of the numbers to be multiplied in base two.

37. Nines Complement Method Step 1 Align the digits as in the standard subtraction algorithm. Step 2 Add leading zeros, if necessary, in the subtrahend so that both numbers have the same number of digits. Step 3 Replace each digit in the subtrahend with its nines complement, and then add. Step 4 Delete the leading (1) and add 1 to the remaining part of the sum.

38. Example: Nines Complement Method Use the nines complement method to subtract 2803 – 647. Solution Step 1 Step 2 Step 3 Step 4

39. Section 4.2: Arithmetic in the Hindu-Arabic System • 1. Which of the following is an example of expanded form? • 205 • b) • c) 5(40 + 1)

40. Section 4.2: Arithmetic in the Hindu-Arabic System 2. Which of the following is an example of the distributive property? a) b) c) 5(3 + 4) = 5(7)

41. Chapter 1 Section 4-3 Conversion Between Number Bases

42. Conversion Between Number Bases • General Base Conversions • Computer Mathematics

43. General Base Conversions We consider bases other than ten. Bases other than ten will have a spelled-out subscript as in the numeral 54eight. When a number appears without a subscript assume it is base ten. Note that 54eight is read “five four base eight.” Do not read it as “fifty-four.”

44. Powers of Alternative Bases

45. Example: Converting Bases Convert 2134five to decimal form.

46. Calculator Shortcut for Base Conversion To convert from another base to decimal form: Start with the first digit on the left and multiply by the base. Then add the next digit, multiply again by the base, and so on. The last step is to add the last digit on the right. Do not multiply it by the base.

47. Example: Use the calculator shortcut to convert 432134five to decimal form.

48. Example: Converting Bases Convert 7508 to base seven.

49. Converting Between Two Bases Other Than Ten Many people feel the most comfortable handling conversions between arbitrary bases (where neither is ten) by going from the given base to base ten and then to the desired base.

50. Computer Mathematics There are three alternative base systems that are most useful in computer applications. These are binary (base two), octal (base eight), and hexadecimal (base sixteen) systems. Computers and handheld calculators use the binary system.