计算机应用基础英文版课件：2-Data representation in computers

计算机应用基础FundamentalsofComputerandComputingFundamentalsofComputerandComputingChapter#1

ProblemsolvinginCSChapter#2

Datarepresentationincomputers

Chapter#3

HardwareChapter#4

OperatingsystemChapter#5

NetworkandinternetChapter#6

DatabasemanagementsystemChapter#7

ComputingasadisciplineChapter#8

AlgorithmandprogrammingThefocusofthischapter：

Whyisbinaryrepresentionadoptedincomputers？Howisdatastoredincomputermemory？Howiscomputationcarriedout？Howtodealwithcharactersincomputer？Howtorepresentmultimedeainformation？Howtoencodeandrecognizebarcodes？Whyisstandardizationofinformationneeded？Chapter#2Datarepresentationincomputers

2.1Numbersystemsandtheirconversion

Howisdatastoredincomputer

Numbersystems

Conversionbetweennumbersystems

2.2Binaryrepresentationandarithmetic

2.3Encodingofcharacter

2.4Encodingofmultimediainformation

2.5Barcodeand

RFIDChapter#2DatarepresentationincomputersComputersaremultimediadevices,dealingwithmanycategoriesofinformation.Computersstore,present,andhelpmodify:NumbersTextAudioImagesandgraphicsVideoDataandComputersInformationcanberepresentedinoneoftwoways:analogordigital.AnalogdataAcontinuousrepresentation,analogoustotheactualinformationitrepresents.DigitaldataAdiscreterepresentation,breakingtheinformationupintoseparateelements.

AnalogandDigitalInformationComputerscannotworkwellwithanaloginformation,sowedigitizeinformationbybreakingitintopiecesandrepresentingthosepiecesseparately.Whydoweusebinary?Moderncomputersaredesignedtouseandmanagebinaryvaluesbecausethedevicesthatstoreandmanagethedataarefarlessexpensiveandfarmorereliableiftheyonlyneedtorepresentoneoftwopossiblevalues.AnanalogandadigitalsignalAnalogandDigitalInformationHighvoltageLowvoltageBinarycomputershavestorageunitscalledbinarydigitsorbitsbit:ThesmallestelementEitherbe0(LowVoltage)or1(HighVoltage)Byte:Thesmalleststorageunit1Byte=8bitsStorageunit

(B,KB,MB,GB,TB):1KB=210B=1024B；1MB=220B=1024KB=10242B；1GB=230B=1024MB=10242KB=10243B；1TB=240B=1024GB=10242MBComputerStorage1Onebitcanbeeither0or1.Therefore,onebitcanrepresentonlytwothings.

32Torepresentmorethantwothings,weneedmultiplebits.2bitscanrepresent4things:00,01,10,11.Ifwewanttorepresentmorethanfourthings,weneedmorethantwobits.3bitscanrepresent8things.

CodeofnumericaldataIngeneral,nbitscanrepresent2nthingsbecausethereare2ncombinationsof0and1thatcanbemadefromnbits.Notethateverytimeweincreasethenumberofbitsby1,wedoublethenumberofthingswecanrepresent.EightDiagrams

2.1Numbersystemsandtheirconversion

Howisdatastoredincomputer

Numbersystems

Conversionbetweennumbersystems

2.2BinaryrepresentationandarithmeticChapter#2Datarepresentationincomputers1.

Numbersystemandbase

Numbersystem

Base

ThebaseofanumberdeterminesthenumberofdigitsandthevalueofdigitpositionsHowtoexpresstheamount?NumbersystemDecimalnumbers,e.g.,642TimeThereare12monthsinoneyearThereare7daysinoneweekThereare60minutesinonehour2.Categories

Decimalconsistsof10digits0，1，2，3，4，5，6，7，8，9Base10

Binaryconsistsof2digits0，1Base2

Octal

consistsof8digits0，1，2，3，4，5，6，7Base8

Hexadecimalconsistsof16digits0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,FBase16Numbersystem

Numbersystem

Base

ThebaseofanumberdeterminesthenumberofdigitsandthevalueofdigitpositionsCarryCarrywillbeperformedwhenthesumoftwonumbersisgreaterthanthelargestdigitinthenumbersystembeingused.Arithmeticofnumbers

Decimalconsistsof10digits0，1，2，3，4，5，6，7，8，9Base10

Binaryconsistsof2digits0，1Base2Binary:CarryValues 111111 1010111

+100101110100010

111 1010111

-111011

0011100 BorrowValuesArithmeticofnumbers(1)Subscript(101)2(101)8(101)10(101)16(N)rDigits

Binary

Octal

Decimal

HexadecimalNumbersystemHowtodistinguish

numbersystem?(2)Postfix101B101O

101D101HNumbersystem(S)R=KiRi

Ki{0,1,……,R-1}

ni=-m(25.6)1021015100610-1(52.6)858128068-1Whichisbigger?(25.6)10or(52.6)8

3.Power(1)Conceptn:Digitsofinteger–1m:DigitsofdecimalfractionR：Base Ri：Power(thepositionofthenumber)Numbersystem(2)Spreadnumbers(1234)10=1103+2102+3101+4100(3)PowersunderdifferentnumbersystemsInteger:fromrighttolefti=0,1,2,3……Decimalfraction

:fromlefttoright

i=-1,-2,-3…...PowerDecimal(D)10iOctal(O)Binary(B)Hexadecimal(H)16i8i2ii

(1234)16=1163+2162+3161+4160RiNumbersystem4.ComparisonofdifferentnumbersystemsNumbersystem

2.1Numbersystemsandtheirconversion

Howisdatastoredincomputer

Numbersystems

Conversionbetweennumbersystems

2.2Binaryrepresentationandarithmetic

2.3Encodingofcharacter

2.4Encodingofmultimediainformation

2.5Barcodeand

RFIDChapter#2DatarepresentationincomputersThreecasesNon-DecimaltoDecimalDecimaltoNon-DecimalNon-DecimaltoNon-DecimalConvertingnumberInteger：10010112

1001011266425322416238224212201Powercode1×26+0×25+0×24+1×23+0×22+1×21+1×20=75Spreadnumberbypower1BinaryDecimalDecimalfraction：0.10102

10102-10.52-20.252-30.1252-40.0625Powercode1×2-1+0×2-2+1×2-3+0×2-4=0.62510Non-DecimaltoDecimal2.OctalDecimal

(375.42)8

=(？)10(375.42)8 =3×82+7×81+5×80+4×8-1+2×8-2

=192

+56

+5+0.5+0.03 =(253.53)10253.53SpreadnumberbypowerNon-DecimaltoDecimal(ABC.DE)16=(？)10(ABC.DE)16=10×162+11×161+12×160+13×16-1+14×16-2

=2560

+176

+12+0.81+0.05=(2748.86)102748.86Spreadnumberbypower3.

HexadecimalDecimal

Non-DecimaltoDecimalDecimalinteger:numberis

dividedby2,theremaindersarranged

inreverseorder.example：75÷2=37…………137÷2=18…………118÷2=9…………09÷2=4…………14÷2=2…………02÷2=1…………01÷2=0…………1result： 7510=1001011

21.DecimalBinaryAlldecimalintegercanbeconvertedtobinaryexactly.

remainder(75)10=(？)21001011DecimaltoNon-DecimalDecimalfraction:

multiplybase2,arrangetheintegerpartinorder.Example：0.6531×2=1.3062……10.3062×2=0.6124……00.6124×2=1.2248……10.2248×2=0.4496……00.4496×2=0.8992……0……(0.6531)10≈(0.10100)2

Note:Ingeneral,decimalfraction

can’tbeconvertedtobinaryexactly.(0.6531)10=(？)2DecimaltoNon-Decimal(49.58)10=(？)2110001

0.58

21.16

20.32

20.64100Example：convert49.58tobinaryform.

Integerpart:Decimalfractionpart:

249224----1

212----026----023----021----10----1.DecimaltoNon-DecimalDecimalinteger:numberis

dividedby8,theremaindersarranged

inreverseorder.Decimalfraction:multiplybase8，arrangetheintegerpartinorder.

2.DecimalOctal

8

2538

31----5

8

3----7

0----3

0.53

84.24

81.92

87.36（253.53)10

=()8375.42DecimaltoNon-DecimalDecimalinteger:numberis

dividedby16,theremaindersarranged

inreverseorder.Decimalfraction:multiplybase16，arrangetheintegerpartinorder.

2.DecimalHexadecimal?DecimaltoNon-Decimal(1)BinaryOctal(10

111

011.110

1)2

=()8(2)OctalBinary(6754.32)8

=()2110111101100011010..2736423=8Keypoint:Groupsofthree

ConverteachgroupIflessthanthree,fillwith0Non-DecimaltoNon-Decimal(1)Binary

Hexadecimal(1011

1110

0110.11011)2

=()16(2)

HexadecimalBinary(A7B8.C9)16

=()2101001111011100011001001..BE6D824=16Keypoint:Groupsoffour

ConverteachgroupIflessthanfour,fillwith0Non-DecimaltoNon-Decimal

Octal

Hexadecimal?Non-DecimaltoNon-Decimal

2.1Numbersystemsandtheirconversion

2.2Binaryrepresentationandarithmetic

Integernumber

RealnumerLogicalRepresentation

Chapter#2DatarepresentationincomputersThetiptopdigitofbinarynumbershowthesign

“0”means“+” “1”means“–”

a=+10，b=–10，expressthemusing8-bitsbinary?00100100

a=+1001100100

b=-10Alsocalledoriginalcode.PositiveandNegativeNumberTherearetwooriginalcodesfor0[+0]original=00000000[-0]original=10000000Therangeoforiginalcode–(2n–1–1)~(2n–1–1)Totalnumberis2n–1Ifn=8,therangeis:-127～

+1271111111101111111OriginalCodePro：Simple&DirectCons：SubtractionoperationmayfailExample:

(－4)+(－3)=－7Originalcodeis

1000000000000100

plus

10000000000000111000000000000100+1000000000000011

100000000000001117OverflowSolution:ComplementCodeOriginalCode126171011543298126171011543298①Adjust7hoursclockwise②Adjust5hoursanticlockwise.126171011543298Correctclockfrom11’clockto6?ComplementCorrectclockfrom11’clockto6+711+7=12+6=6–511-5=6+7isacomplementof–5

whenthemodulusis12RemovethemodulusDefinition:Complement=Modulus+OriginalModulus-RnThecapacityofacounterRisthebaseandnisthenumberofdigitsComplementHowtogetthecomplementIfthenumberispositive,itscomplementcodeequalsitsoriginalcodeIfthenumberisnegative,thenStep1:GettheOriginalcode.Step2:Alldigitsareinreversebutthesign.

Step3:Plus1.

Complement

Q：a=+8,b=-8,whatis[a]comp

and[b]comp，wheren=8.

Solution：[a]comp

=[a]ori

=00001000Reversealldigitsbutthesign;00010001

get[b]ori:[b]ori：[b]comp

=1111

110

111111000Plus1;[b]comp：1110

011

0ComplementComplementofcomplement=originalThereareonlyonecomplementcodefor0[-0]complement=11111111+1=100000000=00000000=[+0]complementTherangeofcomplementcode–2n–1~(2n–1–1)Totalnumberis2nIfn=8,therangeis:-128～

+1271000000001111111ComplementAllarithmeticoperationsareexecutedincomplementcode.Subtractionisconvertedintoadditiona–b=a+(-b)Thesignistreatedthesameasotherdigits.Formulas:[a]comp+[b]comp=[a+b]comp[a]comp-[b]comp=[a-b]comp[[a]comp]comp=[a]oriArithmeticinComplementWhenn=8,a=4,b=16,computetheresultofa-b?[a-b]comp=[a]comp+[-b]comp[4]comp

=

00000100[–16]ori

=

10010000[–16]comp=

11110000[a-b]ori=[[a-b]comp]comp=100011001111010000000100+11110000Takethesignasavalue-12NegativevalueArithmeticinComplementWhenn=8,a=4,b=16,computetheresultofb-a?[b-a]comp=[b]comp+[-a]comp[16]comp

=

00010000[–4]ori

=

10000100[–4]comp=

11111100[b-a]ori=[[b-a]comp]comp=

00001100

1

0

0

0

0

110000010000+11111

10012overflowPositivevalueArithmeticinComplementPositivenumber：complement=originalNegativenumber：

Original:signbitis1,therestisitsabsolutevalueComplement:reversealldigitsexceptthesign,theplus1Complementofcomplement=originalSummaryforNumberRepresentation

2.1Numbersystemsandtheirconversion

2.2Binaryrepresentationandarithmetic

Integernumber

Realnumer

LogicalRepresentation

Chapter#2DatarepresentationincomputersAdecimalrealvalueisdefinedbytheformulasign*mantissa*10expFloatingpointnotation3.14159X1000.314159X1010.0314159X102M•REMantissaBaseexponentFloatingPointRepresentationFloatingpointnotationAnumericformatthatcanbeusedtorepresentverylarge

realnumbersandverysmallrealnumbers.Floating-pointnumbersarestoredintwoparts,amantissa

andanexponent.Mantissa：numericalvalueExponent：therealposition

signdigitssigndigitsmantissaexponentFloatingPointRepresentation1bit7bits1bit23bits32-bitsfloat:1.6875=(1.1011)2=(0.11011×21)2=(0.011011×22)2=(0.0011011×23)2000000010110110000000000000000000000001000110110000000000000000000000011000110110000000000000000NormalizeFloatingPointRepresentationThefirstbitofthemantissamustbe1topreventmultiplerepresentationsofthesamevalue.Normalize0.0001230.123*10-3(0.000101)2(0.101)2*2-3FloatingPointRepresentationExample:Howtorepresent1.6875incomputerasa32-bitsfloating-pointnumber?Assuming8bitsofexponentpartand24bitsofmantissapart.1.6875=(1.1011)2=(0.11011×21)200000001011011000000000000000000FloatingPointRepresentationExample:Howtorepresent-1.6875incomputerasa32-bitsfloating-pointnumber?Assuming8bitsofexponentpartand24bitsofmantissapart.-1.6875=(-1.1011)2=(-0.11011×21)200000001111011000000000000000000FloatingPointRepresentationIna32bitscomputer,if8bitsforexponentpartand24bitsformantissapart,therangeofnumberis1038~2-1500-1038~-2-150FloatingPointRepresentation

2.1Numbersystemsand