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Chapter 2 Numeration Systems and Sets Copyright © 2013, 2010, and 2007, Pearson Education, Inc. 2-1 Numeration Systems Hindu-Arabic Numeration System Tally Numeration System Egyptian Numeration System Babylonian Numeration System Mayan Numeration System Roman Numeration System Other Number Base Systems Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Definition Numerals: written symbols to represent cardinal numbers. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Definition Numeration system: a collection of properties and symbols agreed upon to represent numbers systematically. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Hindu-Arabic Numeration System 1. All numerals are constructed from the 10 digits 2. Place value is based on powers of 10 Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Place value assigns a value to a digit depending on its placement in a numeral. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Expanded form 3 2 1 6789 = 6 ´ 10 + 7 ´ 10 + 8 ´ 10 + 9 ´ 1 Factor If a is any number and n is any natural number, then n factors an = a ×a ×a × ×a Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Base-ten blocks 1 long →101 = 1 row of 10 units 1 flat →102 = 1 row of 10 longs, or 100 units 1 block→103 = 1 row of 10 flats, or 100 longs, or 1000 units Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-1 What is the fewest number of pieces you can receive in a fair exchange for 11 flats, 17 longs, and 16 units? 11 flats 11 flats 17 longs 1 long 18 longs 16 units (16 units = 1 long 6 units and 6 units) 6 units (after the first trade) Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-1 11 flats 1 flat 12 flats 18 longs 8 longs 8 longs (continued) 6 units (18 longs = 1 flat) and 8 longs) 6 units (after the second trade) Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-1 (continued) (12 flats = 1 block and 2 flats) 1 block 1 block 12 flats 2 flats 2 flats 8 longs 6 units 8 longs 6 units The fewest number of pieces = 1 + 2 + 8 + 6 = 17. This is analogous to rewriting as Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Tally Numeration System Uses single strokes (tally marks) to represent each object that is counted. = 13 Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Egyptian Numeration System Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example Use the Egyptian numeration system to represent 2,345,123. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Babylonian Numeration System Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example Use the Babylonian numeration system to represent 305,470. 1 603 216,000 + 24 602 + 86,400 + + 51 60 3060 + 10 1 = 10 + = 305,470 Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Mayan Numeration System Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example Use the Mayan numeration system to represent 305,470. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Roman Numeration System Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Roman Numeration System Copyright © 2013, 2010, and 2007, Pearson Education, Inc. In the Middle Ages, a bar was placed over a Roman number to multiply it by 1000. V represents 5 1000 5000 CDX represents 410 1000 410,000 Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example Use the Roman numeration system to represent 15,478. XVCDLXXVIII Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Other Number Base Systems Quinary (basefive) system Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-2 Convert 11244five to base 10. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Base Two Binary system – only two digits Base two is especially important because of its use in computers. One of the two digits is represented by the presence of an electrical signal and the other by the absence of an electrical signal. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-3a Convert 10111two to base ten. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-3b Convert 27 to base two. 16 8 4 2 1 27 –16 11 –8 3 –0 3 –2 1 –1 0 1 1 or 0 1 1 Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Base Twelve Duodecimal system – twelve digits Use T to represent a group of 10. Use E to represent a group of 11. The base-twelve digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, T, and E. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-4a Convert E2Ttwelve to base ten. Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-4b Convert 1277 to base twelve. 144 1277 –1152 12 125 –120 1 5 –5 0 8 T 5 1277ten = 8T5twelve Copyright © 2013, 2010, and 2007, Pearson Education, Inc. Example 2-5 What is the value of g in g36twelve = 1050ten? Copyright © 2013, 2010, and 2007, Pearson Education, Inc.