4.2 Place-Value or Positional-Value Numeration Systems 183 To explain the procedure used to convert from a Hindu–Arabic numeral to a Babylonian numeral, we will consider a length of time. How can we change 9820 seconds into hours, minutes, and seconds? Since there are 3600 seconds in an hour (60 seconds to a minute and 60 minutes to an hour), we can determine the number of hours in 9820 seconds by dividing 9820 by 60 ,2 or 3600. ) 3600 9820 2 7200 2620 Hours Remaining seconds Now we can determine the number of minutes by dividing the remaining seconds by 60, the number of seconds in a minute. ) 60 2620 2400 220 180 43 40 Minutes Remaining seconds Since the remaining number of seconds, 40, is less than the number of seconds in a minute, our task is complete. 9820 sec 2 hr, 43 min, 40 sec = The same procedure is used to convert a decimal (base 10) numeral to a Babylonian numeral or any numeral in a different base. (2 60 ) (11 60) (18 1) (2 3600) (11 60) (18 1) 7200 660 18 7878 2 × + × + × = × + × + × = + + = 7 Now try Exercise 23 Did You Know? On the High Seas Vestiges of the Babylonian sexagesimal system are still with us today, especially in navigation. Navigators use degrees, minutes, and seconds of longitude and latitude and global positioning systems (GPS) to chart their course. If you look at a globe or world map, you will see that Earth is divided into 360° of latitude and 360° of longitude. Each degree can be divided into 60 minutes, and each minute can be divided into 60 seconds. Early explorers had to have an easy means of computing angles as they guided their ships by the stars. Base 60 easily divides into halves, thirds, fourths, fifths, sixths, tenths, twelfths, fifteenths, twentieths, and thirtieths, making such computations easier. Hence, the use of a base 60 system became popular. Wavebreak Media Ltd/123RF Example 4 From a Hindu–Arabic to a Babylonian Numeral Write 2519 as a Babylonian numeral. Solution The Babylonian numeration system has positional values of , (60) , (60) , 60, 1 3 2 … which can be expressed as , 216000, 3600, 60, 1 … The largest positional value less than or equal to 2519 is 60. To determine how many groups of 60 are in 2519, divide 2519 by 60. ) 60 2519 240 119 60 59 41 Groups of 60 Units remaining Thus, 2519 60 41 ÷ = with remainder 59. There are 41 groups of 60 and 59 units remaining. Because the remainder, 59, is less than the base, 60, no further division
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