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Chapter 4

Chapter 4. Numeration Systems. Chapter 4: Numeration Systems. 4.1 Historical Numeration Systems 4.2 More Historical Numeration Systems 4.3 Arithmetic in the Hindu-Arabic System 4.4 Conversion Between Number Bases. Section 4-1. Historical Numeration Systems.

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Chapter 4

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  1. Chapter 4 Numeration Systems 2012 Pearson Education, Inc.

  2. Chapter 4: Numeration Systems • 4.1 Historical Numeration Systems • 4.2 More Historical Numeration Systems • 4.3 Arithmetic in the Hindu-Arabic System • 4.4 Conversion Between Number Bases 2012 Pearson Education, Inc.

  3. Section 4-1 • Historical Numeration Systems 2012 Pearson Education, Inc.

  4. Historical Numeration Systems • Basics of Numeration • Ancient Egyptian Numeration • Ancient Roman Numeration • Classical Chinese Numeration 2012 Pearson Education, Inc.

  5. 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. 2012 Pearson Education, Inc.

  6. 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 the following: 2012 Pearson Education, Inc.

  7. 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. 2012 Pearson Education, Inc.

  8. Ancient Egyptian Numeration – Simple Grouping The ancient Egyptian system is an example of a simplegrouping system. It uses ten as its base and the various symbols are shown on the next slide. 2012 Pearson Education, Inc.

  9. Ancient Egyptian Numeration 2012 Pearson Education, Inc.

  10. 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 2012 Pearson Education, Inc.

  11. Ancient Roman Numeration The ancient Roman method of counting is a modified grouping system. It uses ten as its base, but also has symbols for 5, 50, and 500. The Roman system also has a subtractive feature which allows a number to be written using subtraction. A smaller-valued symbol placed immediately to the left of the larger value indicated subtraction. 2012 Pearson Education, Inc.

  12. Ancient Roman Numeration The ancient Roman numeration system also has a multiplicative feature to allow for bigger numbers to be written. A bar over a number means multiply the number by 1000. A double bar over the number means multiply by 10002 or 1,000,000. 2012 Pearson Education, Inc.

  13. Ancient Roman Numeration 2012 Pearson Education, Inc.

  14. Example: Roman Numeral Write the number below in our system. MCMXLVII Solution M= 1000 CM= -100 + 1000 XL = -10 + 50 V= 5 I= 1 I= 1 Answer: 1000 + 900 + 40 + 5 + 1 + 1= 1947 2012 Pearson Education, Inc.

  15. 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. 2012 Pearson Education, Inc.

  16. Chinese Numeration 2012 Pearson Education, Inc.

  17. Example: Chinese Numeral Interpret each Chinese numeral. a) b) 2012 Pearson Education, Inc.

  18. Example: Chinese Numeral Solution 7000 200 400 0 (tens) 1 80 Answer: 201 2 Answer: 7482 2012 Pearson Education, Inc.

  19. Section 4-2 • More Historical Numeration Systems 2012 Pearson Education, Inc.

  20. More Historical Numeration Systems • Basics of Positional Numeration • Hindu-Arabic Numeration • Babylonian Numeration • Mayan Numeration • Greek Numeration 2012 Pearson Education, Inc.

  21. 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. 2012 Pearson Education, Inc.

  22. 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. 2012 Pearson Education, Inc.

  23. 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 is not needed. 2012 Pearson Education, Inc.

  24. 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. 2012 Pearson Education, Inc.

  25. Hindu-Arabic Numeration Hundred thousands Millions Ten thousands Thousands Decimal point Hundreds Tens Units 7, 5 4 1, 7 2 5 . 2012 Pearson Education, Inc.

  26. Babylonian Numeration The ancient Babylonians used a modified base 60 numeration system. The digits in a base 60 system represent the number of 1s, the number of 60s, the number of 3600s, and so on. The Babylonians used only two symbols to create all the numbers between 1 and 59. ▼ = 1 and ‹ =10 2012 Pearson Education, Inc.

  27. Example: Babylonian Numeral Interpret each Babylonian numeral. a) ‹‹‹▼▼▼▼ b) ▼▼‹‹‹▼▼▼▼ ▼ 2012 Pearson Education, Inc.

  28. Example: Babylonian Numeral Solution ‹‹‹▼▼▼▼ Answer: 34 ▼▼‹‹‹▼▼▼▼ ▼ Answer: 155 2012 Pearson Education, Inc.

  29. Mayan Numeration The ancient Mayans used a base 20 numeration system, but with a twist. Normally the place values in a base 20 system would be 1s, 20s, 400s, 8000s, etc. Instead, the Mayans used 360s as their third place value. Mayan numerals are written from top to bottom. Table 1 2012 Pearson Education, Inc.

  30. Example: Mayan Numeral Write the number below in our system. Solution Answer: 3619 2012 Pearson Education, Inc.

  31. Greek Numeration The classical Greeks used a ciphered counting system. They had 27 individual symbols for numbers, based on the 24 letters of the Greek alphabet, with 3 Phoenician letters added. The Greek number symbols are shown on the next slide. 2012 Pearson Education, Inc.

  32. Greek Numeration Table 2 Table 2 (cont.) 2012 Pearson Education, Inc.

  33. Example: Greek Numerals Interpret each Greek numeral. a) ma b) cpq 2012 Pearson Education, Inc.

  34. Example: Greek Numerals Solution a) ma b) cpq Answer: 41 Answer: 689 2012 Pearson Education, Inc.

  35. Section 4-3 • Arithmetic in the Hindu-Arabic System 2012 Pearson Education, Inc.

  36. Arithmetic in the Hindu-Arabic System • Expanded Form • Historical Calculation Devices 2012 Pearson Education, Inc.

  37. 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. 2012 Pearson Education, Inc.

  38. Example: Expanded Form Write the number 23,671 in expanded form. Solution 2012 Pearson Education, Inc.

  39. Distributive Property For all real numbers a, b, and c, For example, 2012 Pearson Education, Inc.

  40. Example: Expanded Form Use expanded notation to add 34 and 45. Solution 2012 Pearson Education, Inc.

  41. 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. 2012 Pearson Education, Inc.

  42. 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. 2012 Pearson Education, Inc.

  43. 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. 2012 Pearson Education, Inc.

  44. Example: Abacus Which number is shown below? Solution 1000 + (500 + 200) + 0 + (5 + 1) = 1706 2012 Pearson Education, Inc.

  45. 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. 2012 Pearson Education, Inc.

  46. Example: Lattice Method Find the product by the lattice method. Solution Set up the grid to the right. 7 9 4 3 8 2012 Pearson Education, Inc.

  47. Example: Lattice Method Fill in products 7 9 4 3 8 2012 Pearson Education, Inc.

  48. Example: Lattice Method Add diagonally right to left and carry as necessary to the next diagonal. 1 2 3 0 1 7 2 2012 Pearson Education, Inc.

  49. Example: Lattice Method 1 2 3 0 1 7 2 Answer: 30,172 2012 Pearson Education, Inc.

  50. Nines Complement • Used for subtracting • We first agree that the nines complement of a digit n is 9 – n. For example, the nines complement of 0 is 9, of 1 is 8, of 2 is 7, etc. • Step 1 – Align the digits as in a standard subtraction problem. • 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 – Finally, delete the leading digit (1), and add 1 to the remaining part of the sum. 2012 Pearson Education, Inc.

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