Section 4.2 Place-Value or Positional- Value Numeration Systems
What You Will Learn Place-Value or Position-Value Numeration Systems 4.2-2
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
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
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
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
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
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
Babylonian Numerals The positional values in the Babylonian system are, (60) 3, (60) 2, 60, 1 4.2-9
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
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 4.2-11
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 4.2-12
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 4.2-13
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. 4.2-14
Mayan Numerals The positional values in the Mayan system are, 18 (20) 3, 18 (20) 2, 20, 1 or, 144,000, 7200, 20, 1 4.2-15
Example 7: From Mayan to Hindu- Arabic Numerals Write as a Hindu-Arabic numeral. 4.2-16
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 4.2-17
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 4.2-18
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 4.2-19