Section 4.2 Place-Value or Positional-Value Numeration Systems

Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 4.2

Place-Value or

Positional-Value

Numeration Systems


What You Will Learn

Place-Value or Position-Value

Numeration Systems

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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.

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Place-Value SystemA 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.

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Hindu-Arabic SystemDigits: 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, 14.2-5


Expanded FormTo 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.

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Expanded FormIn expanded form, 1234 is written

1234 = (1 × 103) + (2 × 102)+ (3 × 10) + (4 × 1)

or = (1 × 1000) + (2 × 100)

+ (3 × 10) + 4

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Babylonian Numerals

Oldest known numeration system that resembled a place-value systemDeveloped in about 2500 B.C.Resembled a place-value system with a base of 60, a sexagesimal systemNot a true place-value system because it lacked a symbol for zeroThe lack of a symbol for zero led to a great deal of ambiguity and confusion

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Babylonian Numerals

The positional values in the Babylonian system are

…, (60)3, (60)2, 60, 1

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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.

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Example 1: The Babylonian System: A Positional-Value SystemWrite as a Hindu-Arabic numeral.

Solution

10 + 10 + 1 10 + 10 + 10 + 1(21 × 60) + (31 × 1)1260 + 31 = 1291

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Example 5: A Babylonian Numeral with a Blank SpaceWrite 7223 as a Babylonian numeral.

SolutionDivide 7223 by the largest positional value less than or equal to 7223: 36007223 ÷ 3600 = 2 remainder 23There are 2 groups of 3600The next positional value is 60, but 23 is less than 60, so there are zero groups of 60 with 23 units remaining

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Example 5: A Babylonian Numeral with a Blank SpaceSolutionNow 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

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Mayan NumeralsNumerals 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.

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Mayan NumeralsThe positional values in the Mayan system are

…, 18 × (20)3, 18 × (20)2, 20, 1or …, 144,000, 7200, 20, 1

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Example 7: From Mayan to Hindu-Arabic Numerals

Write as a Hindu-Arabic numeral.

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Example 7: From Mayan to Hindu-Arabic NumeralsSolution

= 4 × 1 = 4

= 11 × 20 = 220

= (8 × (18 × 20) = 2 880

= (2 × [18 × (20)2]= 14,400

= 17,504

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Example 8: From Hindu-Arabic to Mayan NumeralsWrite 4025 as a Mayan numeral.SolutionDivide 4025 by the largest positional value less than or equal to 4025: 360.4025 ÷ 360 = 11 with remainder 65There are 11 groups of 360.The next positional value is 20.65 ÷ 20 = 3 with remainder 53 groups of 20 with 5 units remaining

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Example 8: From Hindu-Arabic to Mayan Numerals

SolutionNow write 4025 as follows:= (11 × 360) + (3 × 20) + (5 × 1)

11 × 360

5 × 1

3 × 20 =

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Section 4.2 Place-Value or Positional-Value Numeration Systems