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Number Bases in Positional Systems

Number Bases in Positional Systems. Section 4.2. Number Bases. base – the number of individual digit symbols used, along with the number whose powers define place values, denoted with a subscript. Our Hindu-Arabic system is base ten. There are ten digits:

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Number Bases in Positional Systems

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  1. Number Bases in Positional Systems Section 4.2

  2. Number Bases base – the number of individual digit symbols used, along with the number whose powers define place values, denoted with a subscript Our Hindu-Arabic system is base ten. There are ten digits: 0 1 2 3 4 5 6 7 8 9 In expanded form, ten is raised to various powers: 327 = 3102 + 2101 + 7100 Notation of base ten is 32710; no subscript implies 10.

  3. Number Bases Note that base 5 has no digit of 5! First, let’s count in various bases. Example Base 5 Base 5 has five digits: 0, 1, 2, 3, 4 Using only these digits, count in base 5. 2 3 4 10 12 13 14 1 11 20 22 23 24 30 32 33 34 21 31 40 42 43 44 100 102 103 104 41 101 etc.

  4. Number Bases First, let’s count in various bases. Example Base 8 Base 8 has eight digits: 0, 1, 2, 3, 4, 5, 6, 7 Using only these digits, count in base 8. 2 3 4 5 7 1 6 10 12 13 14 15 17 11 16 20 22 23 24 25 27 21 26 30 32 33 34 35 37 etc. 31 36

  5. Number Bases First, let’s count in various bases. Example Base 12 Base 12 has twelve digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B Base 12 has twelve digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ?, ? Using only these digits, count in base twelve. 2 3 4 5 7 8 9 B 1 6 A 10 12 13 14 15 17 18 19 1B 11 16 1A 20 22 23 24 25 27 28 29 2B 21 26 2A 30 32 33 34 35 37 38 39 3B 31 36 3A etc.

  6. Names of Bases for Number Systems hexadecimal for computers Hindu-Arabic Babylonian Mayan

  7. Conversions between bases Example Convert 1225 to base 10. To convert TO BASE 10, use place values. 37 1225 = ______10 = 152 + 251 + 250 1225 = 125 + 25 + 21 = 25 + 10 + 2 = 37

  8. Conversions between bases Example Convert 47268 to base 10. To convert TO BASE 10, use place values. 2518 47268 = ______10 = 483 + 782 + 281 + 680 47268 = 4512 + 764 + 28 + 61 = 2048 + 448 + 16 + 6 = 2518

  9. Conversions between bases Example Convert 1001012 to base 10. To convert TO BASE 10, use place values. 37 1001012 = ______10 = 125 + 024 + 023 + 122 + 021 + 120 = 125 + 024 + 023 + 122 + 021 + 120 1001012 = 125 + 122 + 120 = 132 + 14 + 11 = 32 + 4 + 1 = 37

  10. Conversions between bases Example Convert 12056 to base 10. To convert TO BASE 10, use place values. 293 12056 = ______10 = 163 + 262 + 061 + 560 = 163 + 262+ 061 + 560 12056 = 163 + 262 + 560 = 1216 + 236 + 51 = 216 + 72 + 5 = 293

  11. Conversions between bases Example Convert EC716 to base 10. 3,783 EC716 = ______10 = E162 + C161 + 7160 EC716 = 14162 + 12161 + 7160 = 14256+ 1216 + 71 = 3,584 + 192 + 7 = 3,783 10 11 12 13 14 15 Base 16 digits: 0 1 2 3 4 5 6 7 8 9 A B C D E F

  12. Conversions between bases There’s got to be a better way… Example Convert 4210 to base 5. 132 4210 = ______5 = ?52+ ?51 + ?50 4210 = ?25+ ?5 + ?1 = 125+ ?5 + ?1 = 125+ 35 + ?1 = 125 + 35 + 21 42 – 25 = 17 17 – 15 = 2 = 1325

  13. Conversions between bases Example Convert 4210 to base 5. To convert FROM BASE 10, divide repeatedly. To convert FROM BASE 10, divide. 3 1 2 4210 = ______5 Stop when cannot divide 1 R 3 5 8 R 2 5 42 Original Number New Base

  14. Conversions between bases Example Convert 80710 to base 4. To convert FROM BASE 10, divide repeatedly. 0 1 3 3 2 80710 = ______ 4 Stop when cannot divide 3 R 0 4 12 R 2 4 50 R 1 4 201 R 3 4 807 Original Number New Base

  15. Conversions between bases Example Convert 35710 to base 8. To convert FROM BASE 10, divide repeatedly. 4 5 5 35710 = ______ 8 Stop when cannot divide 5 R 4 8 44 R 5 8 357 Original Number New Base

  16. Conversions between bases Example Convert 35710 to base 2. 1 R 0 0 1 0 1 0 1 0 1 1 35710 = __________ 2 2 2 R 1 2 5 R 1 2 11 R 0 2 22 R 0 2 44 R 1 2 89 R 0 2 178 R 1 2 357

  17. Conversions between bases Example Convert 6393310 to base 16. 9 D F B 6393310 = ______ 16 F Stop when cannot divide 15 R 9 16 249 R 11 B 16 D 3995 R 13 16 63933 10 11 12 13 14 15 Base 16 digits: 0 1 2 3 4 5 6 7 8 9 A B C D E F

  18. Homework From the Cow book 4.1 pg 168 # 1 – 49 EOO, 61, 62 4.2 pg175 # 1 – 37 EOO, 49 NOTE: EOO means “every other odd”

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