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ECE201Notes(2) - ECE 201 Digital Logic Chapter 2 Number...

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ECE 201 Digital Logic Chapter 2 – Number Systems and Arithmetic 1 of 37 Additive Number Systems Each Numeral (Number Symbol) is added together. Tally (“Caveman”): 1 5 = 5 + 5 + 3 = 13 Egyptian: 1 10 100 1,000 10,000 100,000 1,000,000 Staff Heel Bone Scroll Lotus Flower Pointing Finger Whale Astonished Person = 30,000 + 4,000 + 300 + 10 + 6 = 34,316 Roman: I V X L C D M 1 5 10 50 100 500 1000 CLXII = 100 + 50 + 10 + 1 + 1 = 162 DCXLVI = 500 + 100 + (50 – 10) + 5 + 1 = 646
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ECE 201 Digital Logic Chapter 2 – Number Systems and Arithmetic 2 of 37 Ciphered Number Systems Have many more characters (symbols) that are added together. Hebrew: } P H I U G T R E W 1 2 3 4 5 6 7 8 9 10 " - : L K Y F D A - > < - M N : C Z 20 30 40 50 60 70 80 90 100 200 300 400 Gen 4:25 Adam K I} = 40+4+1 = 45 Gen 2:18 the Man K I}U = 40+4+1+5 = 50 Gen 2:19 to the Man K I}UW L } = 40+4+1+5+10+30+1 = 91 Gen 2:21 on the Man K I}UW L D = 40+4+1+5+10+30+70 = 160 Ionic Greek: α β γ δ ε ς ' ζ η θ ι κ λ μ 1 2 3 4 5 6 7 8 9 10 20 30 40 ν ξ ο π ο ρ σ-ς τ υ ϕ χ ψ ϖ 50 60 70 80 90 100 200 300 400 500 600 700 800 Luke 3:38 Adam Αδαμ = 1 + 4 + 1 + 40 = 46 Rev 13:18 the number of [a] man (the beast) uniF020 χξς ' = 666
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ECE 201 Digital Logic Chapter 2 – Number Systems and Arithmetic 3 of 37 Russian: A B C D E F G H I Ä J K L 1 2 3 4 5 6 7 8 9 10 20 30 40 M N O P Q R S T U V W X Y 50 60 70 80 90 100 200 300 400 500 600 700 800 Z [ ¡ ¢ ^ _ 900 1000 2000 3000 4000 5000 M I K H A I L S. G O R B A C H E V LI U A I K Q . D N P B A W F C = 40 + 9 + 400 + 1 + 9 + 30 + 90 + 4 + 60 + 80 + 2 + 1 + 600 + 6 + 3 = 489 + 90 + 756 =1335 ASCII: ! # $ % & ( ) * + , 32 33 34 35 36 37 38 39 40 41 42 43 44 - . / 0 1 2 3 4 5 6 7 8 9 45 46 47 48 49 50 51 52 53 54 55 56 57 : ; < = > ? @ 58 59 60 61 62 63 64 A B C D E F G H I J K L M 65 66 67 68 69 70 71 72 73 74 75 76 77 N O P Q R S T U V W X Y Z 78 79 80 81 82 83 84 85 86 87 88 89 90 [ \ ] ^ _ ` 91 92 93 94 95 96 a b c d e f g h i j k l m 97 98 99 100 101 102 103 104 105 106 107 108 109 n o p q r s t u v w x y z 110 111 112 113 114 115 116 117 118 119 120 121 122 B I L L G A T E S 3 66 73 76 76 71 65 84 69 83 3 = 666
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ECE 201 Digital Logic Chapter 2 – Number Systems and Arithmetic 4 of 37 Multiplicative Number Systems Symbols are multiplied by powers of ten and then added together. Chinese: 1 2 3 4 5 6 7 8 9 10 100 1000 2 x 10 = 20 4 x 100 = 400 2 8 10 = 428 4 100
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ECE 201 Digital Logic Chapter 2 – Number Systems and Arithmetic 5 of 37 2.1 Positional (Place-Value) Number Systems Value of symbols depends on place (position) of the symbol Babylonian: Each grouping represents a multiplier times a power of 60. 1 10 Subtraction = (20 + 1) x 60 + (30 + 3) x 60 = 1293 1 0 = 1 x 3600 + 11 x 60 + (20 - 2) x 1 = 4278 Mayan: In the Mayan system, the first numeral is multiplied by 18 x 20 (360), the second by 20 and the last by 1. 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 = 8 x (18 x 20) + 11 x 20 + 4 x 1 = 3104
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ECE 201 Digital Logic Chapter 2 – Number Systems and Arithmetic 6 of 37 Base Conversions Decimal Numbers (Base 10) : Ten Symbols/Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 1492 = 1 . 10 3 + 4 . 10 2 + 9 . 10 1 + 2 . 10 0 = 1 . 1000 + 4 . 100 + 9 . 10 + 2 . 1 = 1000 + 400 + 90 + 2 0.185 = 0 . 1 + 1 . 10 -1 + 8 . 10 -2 + 5 . 10 -3 = 0 + 0.1 + 0.08 + 0.005 Binary Numbers (Base 2): Two Digits/Bits (0, 1) ( 1011011.11 ) 2 = 1 . 2 6 + 0 . 2 5 + 1 . 2 4 + 1 . 2 3 + 0 . 2 2 + 1 . 2 1 + 1 . 2 0 + 1 . 2 -1 + 1 . 2 -2 = 64 + 0 + 16 + 8 + 0 + 2 + 1 + 0.5 + 0.25 = (91.75) 10 Base 6: Six Digits (0, 1, 2, 3, 4, 5) (5034.2) 6 = 5 .
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