CS64 Lecture2 BinaryArithmeticLecture #2 Ziad Matni Dept. of Computer Science, UCSB Adding this...

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Binary Arithmetic CS 64: Computer Organization and Design Logic Lecture #2 Ziad Matni Dept. of Computer Science, UCSB

Transcript of CS64 Lecture2 BinaryArithmeticLecture #2 Ziad Matni Dept. of Computer Science, UCSB Adding this...

Page 1: CS64 Lecture2 BinaryArithmeticLecture #2 Ziad Matni Dept. of Computer Science, UCSB Adding this Class • The class is full – I will not be adding more ppl L ... • In C/C++, it’s

BinaryArithmetic

CS64:ComputerOrganizationandDesignLogicLecture#2

ZiadMatni

Dept.ofComputerScience,UCSB

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AddingthisClass

•  Theclassisfull–IwillnotbeaddingmorepplL– Evenifothersdrop

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LectureOutline

•  Reviewofpositionalnotation,binarylogic•  Bitwiseoperations•  Bitshiftoperations•  Two’scomplement•  Additionandsubtractioninbinary•  Multiplicationinbinary

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PositionalNotationinDecimal

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Continuing with our example… 642 in base 10 positional notation is:

= 6 x 100 = 600

= 4 x 10 = 40 = 2 x 1 = 2 = 642 in base 10

6x102+4x101+2x100

6 4 2100 10 1

642(base10)=600+40+2

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Whatif“642”isexpressedinthebaseof13?

6 x 132 = 6 x 169 = 1014 + 4 x 131 = 4 x 13 = 52 + 2 x 13º = 2 x 1 = 2

PositionalNotationThisishowyouconvertanybasenumberintodecimal!

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6 4 2132 131 130

642(base13)=1014+52+2 =1068(base10)

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PositionalNotationinBinary

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11101 in base 2 positional notation is:

1 x 24 = 1 x 16 = 16 + 1 x 23 = 1 x 8 = 8 + 1 x 22 = 1 x 4 = 4 + 0 x 21 = 1 x 2 = 0 + 1 x 20 = 1 x 1 = 1

So, 11101 in base 2 is 16 + 8 + 4+ 0 + 1 = 29 in base 10

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ConvenientTable…HEXADECIMAL BINARY

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

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HEXADECIMAL(Decimal)

BINARY

A(10) 1010B(11) 1011C(12) 1100D(13) 1101E(14) 1110F(15) 1111

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AlwaysHelpfultoKnow…N 2N

1 22 43 84 165 326 647 1288 2569 51210 1024=1kilobits

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N 2N

11 2048=2kb12 4kb13 8kb14 16kb15 32kb16 64kb17 128kb18 256kb19 512kb20 1024kb=1megabits

N 2N

21 2Mb22 4Mb23 8Mb24 16Mb25 32Mb26 64Mb27 128Mb28 256Mb29 512Mb30 1Gb

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ConvertingBinarytoOctalandHexadecimal

(oranybasethat’sapowerof2)NOTETHEFOLLOWING:•  Binaryis 1bit•  Octalis 3bits•  Hexadecimalis 4bits

•  Usethe“groupthebits”technique– Alwaysstartfromtheleastsignificantdigit– Groupevery3bitstogetherforbinàoct– Groupevery4bitstogetherforbinàhex

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ConvertingBinarytoOctalandHexadecimal

•  Taketheexample:10100110…tooctal:10100110…tohexadecimal:10100110

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2 4 6

10 6

246inoctal

A6inhexadecimal

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While (the quotient is not zero) 1.  Divide the decimal number by the new base 2.  Make the remainder the next digit to the left in the answer 3.  Replace the original decimal number with the quotient 4.  Repeat until your quotient is zero

Algorithmforconvertingnumberinbase10tootherbases

ConvertingDecimaltoOtherBases

Example:Whatis98(base10)inbase8?

98/8=12R2

12/8=1R4

1/8=0R1

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In-ClassExercise:ConvertingDecimalintoBinary&HexConvert54(base10)intobinaryandhex:•  54/2=27R0•  27/2=13R1•  13/2=6R1•  6/2=3R0•  3/2=1R1•  1/2=0R1

54(decimal)=110110(binary)=36(hex)

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Sanitycheck:110110=2+4+16+32=54

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BinaryLogicRefresherNOT,AND,OR

X NOTXX

0 11 0

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X Y XORYX||YX+Y

0 0 00 1 11 0 11 1 1

X Y XANDYX&&YX.Y

0 0 00 1 01 0 01 1 1

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BinaryLogicRefresherExclusive-OR(XOR)

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X Y XXORYXOY

0 0 00 1 11 0 11 1 0

+

Theoutputis“1”onlyiftheinputsareopposite

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BitwiseNOT

•  SimilartologicalNOT(!),exceptitworksonabit-by-bitmanner

•  InC/C++,it’sdenotedbyatilde:~ ~(1001)=0110

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Exercises

•  Sometimeshexadecimalnumbersarewritteninthe0xhhnotation,soforexample: Thehex3Bwouldbewrittenas0x3B

•  Whatis~(0x04)?– Ans:0xFB

•  Whatis~(0xE7)?– Ans:0x18

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BitwiseAND

•  SimilartologicalAND(&&),exceptitworksonabit-by-bitmanner

•  InC/C++,it’sdenotedbyasingleampersand:&(1001&0101)=1001 &0101

=0001

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Exercises

•  Whatis(0xFF)&(0x56)?–  Ans:0x56

•  Whatis(0x0F)&(0x56)?–  Ans:0x06

•  Whatis(0x11)&(0x56)?–  Ans:0x10

•  Notehow&canbeusedasa“masking”function

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Page 19: CS64 Lecture2 BinaryArithmeticLecture #2 Ziad Matni Dept. of Computer Science, UCSB Adding this Class • The class is full – I will not be adding more ppl L ... • In C/C++, it’s

BitwiseOR

•  SimilartologicalOR(||),exceptitworksonabit-by-bitmanner

•  InC/C++,it’sdenotedbyasinglepipe:|(1001|0101)=1001 |0101

=1101

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Exercises

•  Whatis(0xFF)|(0x92)?– Ans:0xFF

•  Whatis(0xAA)|(0x55)?– Ans:0xFF

•  Whatis(0xA5)|(0x92)?– Ans:B7

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Page 21: CS64 Lecture2 BinaryArithmeticLecture #2 Ziad Matni Dept. of Computer Science, UCSB Adding this Class • The class is full – I will not be adding more ppl L ... • In C/C++, it’s

BitwiseXOR

•  Worksonabit-by-bitmanner

•  InC/C++,it’sdenotedbyasinglecarat:^(1001^0101)=1001 ^0101

=1100

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Exercises

•  Whatis(0xA1)^(0x13)?– Ans:0xB2

•  Whatis(0xFF)^(0x13)?– Ans:0xEC

•  Notehow(1^b)isalways~bandhow(0^b)isalwaysb

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Page 23: CS64 Lecture2 BinaryArithmeticLecture #2 Ziad Matni Dept. of Computer Science, UCSB Adding this Class • The class is full – I will not be adding more ppl L ... • In C/C++, it’s

BitShiftLeft

•  MoveallthebitsNpositionstotheleft•  Whatdoyoudothepositionsnowempty?

– YouputinN0s

•  Example:Shift“1001”2positionstotheleft1001<<2=100100

•  Whyisthisusefulasaformofmultiplication?

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MultiplicationbyBitLeftShifting

•  VeeeeryusefulinCPU(ALU)design– Why?

•  Becauseyoudon’thavetodesignamultiplier•  Youjusthavetodesignawayforthebitstoshift

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BitShiftRight•  MoveallthebitsNpositionstotheright,subbing-in

eitherN0sorN1sontheleft•  Takesontwodifferentforms

•  Example:Shift“1001”2positionstotheright1001>>2=either0010or1110

•  Theinformationcarriedinthelast2bitsislost.•  IfShiftLeftdoesmultiplication,

whatdoesShiftRightdo?–  Itdivides,butittruncatestheresult

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Page 26: CS64 Lecture2 BinaryArithmeticLecture #2 Ziad Matni Dept. of Computer Science, UCSB Adding this Class • The class is full – I will not be adding more ppl L ... • In C/C++, it’s

TwoFormsofShiftRight

•  Subbing-in0smakessense•  Whataboutsubbing-intheleftmostbitwith1?

•  It’scalled“arithmetic”shiftright:1100(arithmetic)>>1=1110

•  It’susedfortwos-complementpurposes

– What?

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NegativeNumbersinBinary

•  Soweknowthat,forexample,6(10)=110(2)•  Butwhatabout–6(10)???

•  Whatifweaddedonemorebitonthefarlefttodenote“negative”?–  i.e.becomesthenewMSB

•  So:110(+6)becomes1110(–6)•  Butthisleavesalottobedesired

–  Baddesignchoice…4/5/18 Matni,CS64,Sp18 27

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TwosComplementMethod

•  ThisishowTwosComplementfixesthis.•  Let’swriteout-6(10)in2s-Complementbinaryin4bits:

So,–6(10)=1010(2)

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011010011010

Firsttaketheunsigned(abs)value(i.e.6)andconverttobinary:

Thennegateit(i.e.doa“NOT”functiononit):Nowadd1:

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Let’sdoitBackwards…BydoingitTHESAMEEXACTWAY!

•  2s-ComplementtoDecimalmethodisthesame!

•  Take1010fromourpreviousexample•  Negateitanditbecomes0101•  Nowadd1toit&itbecomes0110,whichis6(10)

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Page 30: CS64 Lecture2 BinaryArithmeticLecture #2 Ziad Matni Dept. of Computer Science, UCSB Adding this Class • The class is full – I will not be adding more ppl L ... • In C/C++, it’s

AnotherViewof2sComplement

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NOTE:InTwo’sComplement,ifthenumber’sMSBis“1”,thenthatmeansit’sanegativenumberandifit’s“0”thenthenumberispositive.

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AnotherViewof2sComplement

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NOTE:Oppositenumbersshowupassymmetricallyoppositeeachotherinthecircle.

NOTEAGAIN:Whenwetalkof2scomplement,wemustalsomentionthenumberofbitsinvolved

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Ranges

•  Therangerepresentedbynumberofbitsdiffersbetweenpositiveandnegativebinarynumbers

•  GivenNbits,therangerepresentedis:0to

and

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+2N–1forpositivenumbers–2N-1to+2N-1–1for2’sComplementnegativenumbers

Page 33: CS64 Lecture2 BinaryArithmeticLecture #2 Ziad Matni Dept. of Computer Science, UCSB Adding this Class • The class is full – I will not be adding more ppl L ... • In C/C++, it’s

Addition•  Wehaveanelementarynotionofaddingsingledigits,along

withanideaofcarryingdigits–  Example:whenadding3to9,weputforward2andcarrythe1

(i.e.tomean12)

•  Wecanbuildonthisnotiontoaddnumberstogetherthataremorethanonedigitlong

•  Example: 123 + 389

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11

215

Page 34: CS64 Lecture2 BinaryArithmeticLecture #2 Ziad Matni Dept. of Computer Science, UCSB Adding this Class • The class is full – I will not be adding more ppl L ... • In C/C++, it’s

AdditioninBinary

•  Samemathematicalprincipalapplies

0011+1101

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1

00

11

00

3+13

161

1

Page 35: CS64 Lecture2 BinaryArithmeticLecture #2 Ziad Matni Dept. of Computer Science, UCSB Adding this Class • The class is full – I will not be adding more ppl L ... • In C/C++, it’s

Exercises

Implementingan8-bitadder:•  Whatis(0x52)+(0x4B)?

– Ans:0x9D,outputcarrybit=0

•  Whatis(0xCA)+(0x67)?– Ans:0x31,outputcarrybit=1

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Page 36: CS64 Lecture2 BinaryArithmeticLecture #2 Ziad Matni Dept. of Computer Science, UCSB Adding this Class • The class is full – I will not be adding more ppl L ... • In C/C++, it’s

YOURTO-DOs

•  Assignment#1– Lookforitovertheweekendontheclasswebsite– LAB#1isonMONDAY!

•  Nextweek,wewilldiscussmoreArithmetictopicsandstartexploringAssemblyLanguage– Doyourreadings!

(again:foundontheclasswebsite)

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Multiplexers, ALU Design - GitHub Pages · 2021. 1. 14. · Simplified ALU •We can string 1-bit ALUs together to make bigger-bit ALUs (e.g. 32b ALU) 2/26/20 Matni, CS64, Wi20 23

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