1. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 4.2 Place-Value or Positional- Value Numeration Systems

2. Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Place-Value or Position-Value Numeration Systems 4.2-2

3. Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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. 4.2-3

4. Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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. 4.2-4

5. Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 … 10 5, 10 4, 10 3, 10 2, 10, 1 4.2-5

6. Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 10 2 or 100, and so on. In general, we multiply the digit n places from the right by 10 n–1 to show a number in expanded form. 4.2-6

7. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Expanded Form In expanded form, 1234 is written 1234 = (1 × 10 3 ) + (2 × 10 2 ) + (3 × 10) + (4 × 1) or = (1 × 1000) + (2 × 100) + (3 × 10) + 4 4.2-7

8. Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 4.2-8

9. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Babylonian Numerals The positional values in the Babylonian system are …, (60) 3, (60) 2, 60, 1 4.2-9

10. Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 60 2, and so on. 4.2-10

11. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: The Babylonian System: A Positional-Value System Write as a Hindu-Arabic numeral. Solution 10 + 10 + 110 + 10 + 10 + 1 (21 × 60) + (31 × 1) 1260 + 31 = 1291 4.2-11

12. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Try This Write the following as a Hindu-Arabic numeral 4.2-12

13. Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 4.2-13

14. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 5: A Babylonian Numeral with a Blank Space Solution Now write 7223 as follows: = (2 × 60 2 ) + (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 4.2-14

15. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Try This Write 2,562 as a Babalonian Numeral 4.2-15

16. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Mayan Numerals Numerals are written vertically. Units position is on the bottom. Numeral in bottom row is multiplied by 1. Numeral in second row is multiplied by 20. Numeral in third row is multiplied by 18 × 20, or 360. Numeral in fourth row is multiplied by 18 × 20 2, or 7200, and so on. 4.2-16

17. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Mayan Numerals The positional values in the Mayan system are …, 18 × (20) 3, 18 × (20) 2, 20, 1 or …, 144,000, 7200, 20, 1 4.2-17

18. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 7: From Mayan to Hindu- Arabic Numerals Write as a Hindu-Arabic numeral. 4.2-18

19. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 7: From Mayan to Hindu- Arabic Numerals Solution = 4 × 1= 4 = 11 × 20= 220 = (8 × (18 × 20)= 2 880 = (2 × [18 × (20) 2 ]= 14,400 = 17,504 4.2-19

20. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Try This Change from Mayan to Hindu-Arabic 4.2-20

21. Copyright 2013, 2010, 2007, Pearson, Education, Inc. 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 4.2-21

22. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 8: From Hindu-Arabic to Mayan Numerals Solution Now write 4025 as follows: = (11 × 360) + (3 × 20) + (5 × 1) 11 × 360 5 × 1 3 × 20= 4.2-22

23. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Try This Write 3,202 in Mayan 4.2-23

24. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Homework P 184# 15 – 48 (x3) 4.2-24