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Binary Values Chapter 2

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Why Binary? Electrical devices are most reliable when they are built with 2 states that are hard to confuse: gate open / gate closed

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Why Binary? Electrical devices are most reliable when they are built with 2 states that are hard to confuse: gate open / gate closed full on / full off fully charged / fully discharged charged positively / charged negatively magnetized / nonmagnetized magnetized clockwise / magnetized ccw These states are separated by a huge energy barrier.

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Punch Cards holeNo hole

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Jacquard Loom Invented in 1801

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Jacquard Loom Invented in 1801

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Why Weaving is Binary

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Holes Were Binary But Encodings Were Not

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11111111111101111111111111111110

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Everyday Binary Things Examples:

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Everyday Binary Things Examples: Light bulb on/off Door locked/unlocked Garage door up/down Refrigerator door open/closed A/C on/off Dishes dirty/clean Alarm set/unset

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Binary (Boolean) Logic If:customers account is at least five years old, and customer has made no late payments this year or customers late payments have been forgiven, and customers current credit score is at least 700 Then:Approve request for limit increase.

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Exponential Notation 4 2 = 4 * 4 = 4 3 = 4 * 4 * 4 = 10 3 = 10 11 = 100,000,000,000

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Powers of Two

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01 12 24 38 416 532 664 7128 8256 9512 101024 112048 124096 138192 1416384

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Positional Notation 2473 = 2 * 1000(10 3 ) = 2000 +4 * 100(10 2 )= 400 +7 * 10(10 1 )= 70 +3 * 1(10 0 )= 3 2473 = 2 * 10 3 + 4 * 10 2 + 7 * 10 1 + 3 * 10 0 Base 10

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Base 8 (Octal) 93 = 1 * 64(8 2 )= 6429 +3 * 8(8 1 )= 24 5 +5 * 1(8 0 )= 5 0 93 93 = 135 8 remainder 512

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Base 3 (Ternary) 95 = 1 * 81(3 4 ) = 8114 +0 * 27(3 3 )= 014 +1 * 9(3 2 )= 9 5 +1 * 3(3 1 )= 3 2 +2 * 1(10 0 )= 0 0 93 93 = 10112 3 remainder

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Base 2 (Binary) 93 = 1 * 64(2 6 ) = 6429 +0 * 32(2 5 )= 029 +1 * 16(2 4 )= 1613 +1 * 8(2 3 )= 8 5 +1 * 4(2 2 )= 4 1 +0 * 2(3 1 )= 0 1 +1 * 1(10 0 )= 1 0 93 93 = 1011101 2 remainder 128

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Counting in Binary http://www.youtube.com/watch?v=zELAfmp3fXY

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A Conversion Algorithm def dec_to_bin(n): answer = "" while n != 0: remainder = n % 2 n = n //2 answer = str(remainder) + answer return(answer)

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Running the Tracing Algorithm Try: 13 64 1234 345731

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An Easier Way to Do it by Hand 1 2 4 8 16 32 64 128 256 512 1,024 2,048 4,096 8,192 16,384

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The Powers of 2 1 2 4 8 16 32 64 128 256 512 1,024 2,048 4,096 8,192 16,384 Now you try the examples on the handout.

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My Android Phone

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Naming the Quantities See Dale and Lewis, page 124. 10 3 = 1000 2 10 = 1024

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How Many Bits Does It Take? To encode 12 values: To encode 52 values: To encode 3 values:

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A Famous 3-Value Example

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One, if by land, and two, if by sea; And I on the opposite shore will be,

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Braille

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With six bits, how many symbols can be encoded?

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Braille Escape Sequences Indicates that the next symbol is capitalized.

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Binary Strings Can Get Really Long 111111110011110110010110

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Binary Strings Can Get Really Long 111111110011110110010110

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Base 16 (Hexadecimal) 52 = 110100already hard for us to read

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Base 16 (Hexadecimal) 52 = 110100already hard for us to read = 11 0100 3 4

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Base 16 (Hexadecimal) 52 = 110100

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Base 16 (Hexadecimal) 52 = 110100 =3 * 16(16 1 )= 48 4 +4 * 1(16 0 )= 4 0 52 52 = 34 16 256

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Base 16 (Hexadecimal) 2337 = 9 * 256(16 2 )= 230433 +2 * 16(16 1 )= 32 1 +1 * 1(16 0 )= 1 0 2337 2337 = 921 16 2337 = 1001 0010 0001 2 4096

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Base 16 (Hexadecimal) We need more digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 31 = 1 * 16(16 1 )= 16 15 +? * 1(16 0 )= 15 0 31 31 = 3 16 ?

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Base 16 (Hexadecimal) We need more digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F 31 = 1 * 16(16 1 )= 16 15 +? * 1(16 0 )= 15 0 31 31 = 3 16 ? 31 = 1F 16

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Base 16 (Hexadecimal) F F 3 D 9 6 1111 1111 0011 1101 1001 0110

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A Very Visible Use of Hex http://easycalculation.com/color-coder.php http://lectureonline.cl.msu.edu/~mmp/applist/RGBColor/c.htm

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Binary, Octal, Hex 8 = 2 3 So one octal digit corresponds to three binary ones. Binary to octal:1 011 11195 1 3 7

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Binary, Octal, Hex 8 = 2 3 So one octal digit corresponds to three binary ones. Binary to octal:1 011 11195 1 3 7 Octal to binary:2 1 5 10 001 101

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Binary, Octal, Hex 16 = 2 4 So one hex digit corresponds to four binary ones. Binary to hex:101 111195 5 F

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Binary, Octal, Hex 16 = 2 4 So one hex digit corresponds to four binary ones. Binary to hex:101 1111 95 5 F Binary to hex: 101 1110 1111 5 E F

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Binary, Octal, Hex 16 = 2 4 So one hex digit corresponds to four binary ones. Binary to hex:1011111 95 5 F Binary to hex: 0101 1110 11111519 5 E F byte

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Binary, Octal, Hex 16 = 2 4 So one hex digit corresponds to four binary ones. Hex to decimal: 5 F 0101 1111 then to decimal: 95

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Binary Arithmetic Addition: 11010 + 1001

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Binary Arithmetic Multiplication: 11010 * 11

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Binary Arithmetic Multiplication by 2: 11010 * 10

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Binary Arithmetic Multiplication by 2: 11010 * 10 Division by 2: 11010 // 10

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Computer Humor http://www.youtube.com/watch?v=WGWmh1fK87A

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