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Section 4.2 Place-Value or Positional-Value Numeration Systems

Section 4.2 Place-Value or Positional-Value Numeration Systems. What You Will Learn. Place-Value or Position-Value Numeration Systems. Place-Value System (or Positional-Value System). The value of the symbol depends on its position in the representation of the number.

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Section 4.2 Place-Value or Positional-Value Numeration Systems

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  1. Section 4.2Place-Value or Positional-Value Numeration Systems

  2. What You Will Learn • Place-Value or Position-Value Numeration Systems

  3. Place-Value System(or Positional-Value System) • The value of the symbol depends on its position in the representation of the number. • It is the most common type of numeration system in the world today. • The most common place-value system is the Hindu-Arabic numeration system. • This is used in the United States.

  4. Place-Value System • A true positional-value system requires a base and a set of symbols, including a symbol for zero and one for each counting number less than the base. • The most common place-value system is the base 10 system. It is called the decimal number system.

  5. Hindu-Arabic System • Digits: In the Hindu-Arabic system, the digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 • Positions: In the Hindu-Arabic system, the positional values or place values are … 105, 104, 103, 102, 10, 1

  6. Expanded Form • To evaluate a numeral in this system, multiply the first digit on the right by 1. • Multiply the second digit from the right by base 10. • Multiply the third digit from the right by base 102 or 100, and so on. • In general, we multiply the digit n places from the right by 10n–1 to show a number in expanded form.

  7. Expanded Form • In expanded form, 1234 is written • 1234 = (1 × 103) + (2 × 102) + (3 × 10) + (4 × 1) • or • = (1 × 1000) + (2 × 100) + (3 × 10) + 4

  8. Babylonian Numerals • Oldest known numeration system that resembled a place-value system • Developed in about 2500 B.C. • Resembled a place-value system with a base of 60, a sexagesimal system • Not a true place-value system because it lacked a symbol for zero • The lack of a symbol for zero led to a great deal of ambiguity and confusion

  9. Babylonian Numerals • The positional values in the Babylonian system are • …, (60)3, (60)2, 60, 1

  10. Babylonian Numerals • A gap is left between characters to distinguish place values. • From right to left, the sum of the first group of numerals is multiplied by 1. • The sum of the second group is multiplied by 60. • The sum of the third group is multiplied by 602, and so on.

  11. Example 1: The Babylonian System: A Positional-Value System • Write as a Hindu-Arabic numeral. Solution 10 + 10 + 1 10 + 10 + 10 + 1 (21 × 60) + (31 × 1) 1260 + 31 = 1291

  12. Example 5: A Babylonian Numeral with a Blank Space • Write 7223 as a Babylonian numeral. Solution Divide 7223 by the largest positional value less than or equal to 7223: 3600 7223 ÷ 3600 = 2 remainder 23 There are 2 groups of 3600 The next positional value is 60, but 23 is less than 60, so there are zero groups of 60 with 23 units remaining

  13. Example 5: A Babylonian Numeral with a Blank Space Solution Now write 7223 as follows: = (2 × 602) + (0 × 60) + (23 + 1) The Babylonian numeration system does not contain a symbol for 0, so leave a larger blank space to indicate there are no 60s present in 7223

  14. Mayan Numerals • Numerals are written vertically. • Units position is on the bottom. • Numeral in bottom row is multiplied by1. • Numeral in second row is multiplied by 20. • Numeral in third row is multiplied by18 × 20, or 360. • Numeral in fourth row is multiplied by18 × 202, or 7200, and so on.

  15. Mayan Numerals • The positional values in the Mayan system are • …, 18 × (20)3, 18 × (20)2, 20, 1 • or …, 144,000, 7200, 20, 1

  16. Example 7: From Mayan to Hindu-Arabic Numerals • Write as a Hindu-Arabic numeral.

  17. Example 7: From Mayan to Hindu-Arabic Numerals Solution = (2 × [18 × (20)2] = 14,400 = (8 × (18 × 20) = 2 880 = 11 × 20 = 220 = 4 × 1 = 4 = 17,504

  18. Example 8: From Hindu-Arabic to Mayan Numerals • Write 4025 as a Mayan numeral. Solution Divide 4025 by the largest positional value less than or equal to 4025: 360. 4025 ÷ 360 = 11 with remainder 65 There are 11 groups of 360. The next positional value is 20. 65 ÷ 20 = 3 with remainder 5 3 groups of 20 with 5 units remaining

  19. Example 8: From Hindu-Arabic to Mayan Numerals Solution Now write 4025 as follows: = (11 × 360) + (3 × 20) + (5 × 1) 11 × 360 3 × 20 = 5 × 1

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