RepresentingDatawithBits

bits,bytes,numbers,andnotation

positionalnumberrepresentation

• Base determines:– Maximumdigit(base– 1).Minimumdigitis0.– Weightofeachposition.

• Eachpositionholdsadigit.• Representedvalue=sumofallpositionvalues– Positionvalue=digitvaluexbaseposition

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2 4 0100 10 1102 101 1002 1 0

=2x102 +4x101 +0x100

positionweight

binary=base2

Whenambiguous,subscriptwithbase:10110 Dalmatians (movie)

1012-SecondRule (folkwisdomforfoodsafety)

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1 0 1 18 4 2 123 22 21 203 2 1 0

=1x23 +0x22 +1x21 +1x20

positionweight

irony

Powersof2:memorizeupto≥210 (inbaseten)

ex

conversionandarithmetic1910 =?2 10012 =?10

24010 =?2 110100112 =?10

1012 +10112 =?2 10010112 x210 =?2

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Showpowers,strategies.

ex

numbersand wires

Onewirecarriesonebit.Howmanywirestorepresentagivennumber?

WhatifIwanttobuildacomputer(andnotchangethehardwarelater)?

1001

10001001

byte =8bitsSmallestunitofdatausedbyatypicalmoderncomputer

Binary 000000002 -- 111111112Decimal00010-- 25510Hexadecimal 0016 -- FF16

Programmer’shexnotation(C,etc.):0xB4 =B416

Octal(base8)alsouseful.Whydo240studentsoftenconfuseHalloweenandChristmas?

0 0 00001 1 00012 2 00103 3 00114 4 01005 5 01016 6 01107 7 01118 8 10009 9 1001A 10 1010B 11 1011C 12 1100D 13 1101E 14 1110F 15 1111 10

Whatdoyoucall4bits?

a.k.a.octet

Byte=2hexdigits!

Hexencodingpracticeex

char:representingcharacters

AC-stylestringisrepresentedbyaseriesofbytes(chars).— One-byteASCIIcodes foreachcharacter.— ASCII=AmericanStandardCodeforInformationInterchange

32 space 48 0 64 @ 80 P 96 ` 112 p33 ! 49 1 65 A 81 Q 97 a 113 q34 ” 50 2 66 B 82 R 98 b 114 r35 # 51 3 67 C 83 S 99 c 115 s36 $ 52 4 68 D 84 T 100 d 116 t37 % 53 5 69 E 85 U 101 e 117 u38 & 54 6 70 F 86 V 102 f 118 v39 ’ 55 7 71 G 87 W 103 g 119 w40 ( 56 8 72 H 88 X 104 h 120 x41 ) 57 9 73 I 89 Y 105 I 121 y42 * 58 : 74 J 90 Z 106 j 122 z43 + 59 ; 75 K 91 [ 107 k 123 {44 , 60 < 76 L 92 \ 108 l 124 |45 - 61 = 77 M 93 ] 109 m 125 }46 . 62 > 78 N 94 ^ 110 n 126 ~47 / 63 ? 79 O 95 _ 111 o 127 del

word|wərd|, n.Naturalunitofdatausedbyprocessor.– Fixedsize (e.g.32bits,64bits)• DefinedbyISA:InstructionSetArchitecture

– machineinstructionoperands– wordsize=registersize =addresssize

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Java/Cint =4bytes:11,501,584

31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

0 0 0 0 0 0 0 0 1 0 1 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

MSB:mostsignificantbit LSB:leastsignificantbit

(size inbytes)JavaDataTypeCDataType 32-bit64-bit

boolean 1 1byte char 1 1char 2 2short shortint 2 2int int 4 4float float 4 4

longint 4 8double double 8 8long longlong 8 8

longdouble 8 16

fixed-sizedatarepresentations

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Dependsonwordsize!

bitwise operatorsBitwiseoperatorsonfixed-width bitvectors.

AND&OR| XOR^ NOT~

LawsofBooleanalgebraapplybitwise.e.g.,DeMorgan’s Law:~(A|B)=~A&~B

01101001&0101010101000001

01101001|01010101

01101001^01010101 ~01010101

01010101^01010101

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Aside:setsas bitvectorsRepresentation:n-bitvectorgivessubsetof{0,…,n–1}.

ai =1 ≡i Î A

01101001 {0,3,5,6}76543210

01010101 {0,2,4,6}76543210

BitwiseOperations SetOperations?& 01000001 {0,6} Intersection| 01111101 {0,2,3,4,5,6} Union^ 00111100 {2,3,4,5} Symmetricdifference~ 10101010 {1,3,5,7} Complement

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ex

bitwise operatorsinC& | ^ ~ applytoanyintegral datatype

long,int,short,char, unsigned

Examples(char)~0x41 =

~0x00 =

0x69 & 0x55 =

0x69 | 0x55 =

Manybit-twiddlingpuzzlesinupcomingassignment

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ex

logical operationsinC&&||! applytoany"integral"datatype

long,int,short,char, unsigned

0is false nonzerois true resultalways 0or1

earlytermination a.k.a.short-circuitevaluation

Examples(char)!0x41 =!0x00 =!!0x41 =

0x69 && 0x55 =0x69 || 0x55 =

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ex

Encodeplayingcards.52cardsin4suits

Howdoweencodesuits,facecards?

Whatoperationsshouldbeeasytoimplement?GetandcomparerankGetandcomparesuit

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Twopossiblerepresentations

52cards– 52bitswithbitcorrespondingtocardsetto1

“One-hot”encodingHardtocomparevaluesandsuitsindependentlyNotspaceefficient

4bitsforsuit,13bitsforcardvalue– 17bitswithtwosetto1

Pairofone-hotencodedvaluesEasiertocomparesuitsandvaluesindependentlySmaller,butstillnotspaceefficient

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52bitsin2x32-bitwords

TwobetterrepresentationsBinaryencodingofall52cards– only6bitsneeded

Numbercardsuniquelyfrom0Smallerthanone-hotencodings.Hardtocomparevalueandsuit

Binaryencodingofsuit(2bits)andvalue(4bits)separately

NumbereachsuituniquelyNumbereachvalueuniquelyStillsmallEasysuit,valuecomparisons

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low-order6bitsofabyte

suit value

mask: abitvectorthat,whenbitwiseANDed withanotherbitvectorv,turnsallbut thebitsofinterestinv to0

CompareCardSuits

char hand[5]; // represents a 5-card handchar card1, card2; // two cards to compare...if ( sameSuit(hand[0], hand[1]) ) { ... }

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#define SUIT_MASK 0x30

int sameSuit(char card1, char card2) {return !((card1 & SUIT_MASK) ^ (card2 & SUIT_MASK));

//same as (card1 & SUIT_MASK) == (card2 & SUIT_MASK);}

0 0 1 1 0 0 0 0

suit value

mask: abitvectorthat,whenbitwiseANDed withanotherbitvectorv,turnsallbut thebitsofinterestinv to0

CompareCardValues

#define VALUE_MASK

int greaterValue(char card1, char card2) {

}

char hand[5]; // represents a 5-card handchar card1, card2; // two cards to compare...if ( greaterValue(hand[0], hand[1]) ) { ... }

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suit value

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Bitshifting

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10011001

x<<2 1001100100fillwithzeroesonright

x

logicalshiftleft2

00 10011001losebitsonright

10011001

x>>2

x

losebitsonleft

logical shiftright2

11 1 0011001arithmetic shiftright2

fillwithzeroesonleft

fillwithcopiesofMSBonleftx>>2

ShiftgotchasLogicalorarithmeticshiftright:howdowetell?C:compilerchooses

Usuallybasedontype:raincheck!

Java: >>isarithmetic,>>>islogical

Shiftann-bittypebyatleast0andnomorethann-1.C:othershiftdistancesareundefined.

anything couldhappen

Java:shiftdistanceisusedmodulonumberofbitsinshiftedtypeGivenint x:x<<34==x<<2

!!!

ShiftandMask: extractabitfieldWriteCcode:extract2nd mostsignificantbyte froma32-bitinteger.

givenshouldreturn:

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01100001011000100110001101100100x =

00000000000000000000000001100010

Desiredbitsinleastsignificantbyte.Allotherbitsarezero.

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