intro to comp lec 3 final 1

8/2/2019 Intro to Comp Lec 3 Final 1

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Introduction To ComputingIntroduction To Computing

Chapter 3Chapter 3

Number SystemNumber System

Marium Jalal ChaudhryMarium Jalal Chaudhry


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Learning Goals for TodayLearning Goals for Today

1.1. To become familiar with number system used byTo become familiar with number system used bythe microprocessors -the microprocessors - binary numbersbinary numbers

2.2. To become able to performTo become able to perform decimal-to-binarydecimal-to-binaryconversionsconversions

3.3. To understand the NOT, AND, OR and XORTo understand the NOT, AND, OR and XOR logiclogic

operationsoperations the fundamental operations that arethe fundamental operations that are

available in all microprocessorsavailable in all microprocessors


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BINARYBINARY(BASE(BASE 22))

numbersnumbers


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DECIMALDECIMAL(BASE(BASE 1010))

numbersnumbers


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Decimal (base 10) number systemDecimal (base 10) number system

consists of 10 symbols or digitsconsists of 10 symbols or digits

0 1 2 3 40 1 2 3 4

5 6 7 8 95 6 7 8 9


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Binary (base 2) number systemBinary (base 2) number system

consists of just twoconsists of just two

0 10 1


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Other popular number systemsOther popular number systems

OctalOctalbase = 8base = 8

8 symbols (0,1,2,3,4,5,6,7)8 symbols (0,1,2,3,4,5,6,7)

HexadecimalHexadecimalbase = 16base = 16

16 symbols (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F)16 symbols (0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F)


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Decimal (base 10) numbers areDecimal (base 10) numbers are

expressed in the pexpressed in the positional notationositional notation

4202 = 2x104202 = 2x1000 + 0x10+ 0x1011 + 2x10+ 2x1022 + 4x10+ 4x1033

The right-most is the least significant digit

The left-most is the most significant digit


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expressed in theexpressed in thepositional notationpositional notation

42042022 == 2x102x1000 + 0x10+ 0x1011 + 2x10+ 2x1022 + 4x10+ 4x1033

1s multiplier

1


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42420022 == 2x102x1000 ++ 0x100x1011 + 2x10+ 2x1022 + 4x10+ 4x1033

10s multiplier

10


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442202 = 2x1002 = 2x1000 + 0x10+ 0x1011 ++ 2x102x1022 + 4x10+ 4x1033

100s multiplier

100


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44202202 == 2x102x1000 + 0x10+ 0x1011 + 2x10+ 2x1022 ++ 4x104x1033

1000s multiplier

1000


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Binary (base 2) numbers areBinary (base 2) numbers are alsoalso


1001110011==1x21x200++1x21x211++0x20x222++0x20x233++1x21x244

The right-most is the least significant digit

The left-most is the most significant digit


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1001100111==1x21x200++1x21x211++0x20x222++0x20x233++1x21x244

1s multiplier

1


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1001001111==1x21x200++1x21x211++0x20x222++0x20x233++1x21x244

2s multiplier

2


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1010001111==1x21x200++1x21x211++0x20x222++0x20x233++1x21x244

4s multiplier

4


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1100011011==1x21x200++1x21x211++0x20x222++0x20x233++1x21x244

8s multiplier

8


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1100110011==1x21x200++1x21x211++0x20x222++0x20x233++1x21x244

16s multiplier

16


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CountingCounting

in Decimalin Decimal00

11

2233

44

55

66

77

88

99

10

11

1213

14

15

16

17

18

19

20

21

2223

24

25

26

27

28

29

30

31

3233

34

35

36

.

.

.

0

1

1011

100

101

110

111

1000

1001

1010

1011

11001101

1110

1111

10000

10001

10010

10011

10100

10101

1011010111

11000

11001

11010

11011

11100

11101

11110

11111

100000100001

100010

100011

100100

.

.

.

Counting

in Binary

?


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Why binaryWhy binary?

Because this system is natural for digital computers

The fundamental building block of a digital computer

the switch possesses two natural states, ON & OFF.

It is natural to represent those states in a number

system that has only two symbols, 1 and 0, i.e. the

binary number system

In some ways, the decimal number system is natural

to us humans. Why?


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bitbitbinary diginary digit


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Byte = 8 bitsByte = 8 bits


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Decimal Binaryconversion


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Convert 75 to BinaryConvert 75 to Binary752

37 12

18 12

9 02

4 12

2 02

1 02

0 1

1001011

remainder


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CheckCheck

10010111001011 == 1x21x200++1x21x211++0x20x222++1x21x233++

0x20x244++0x20x255++1x21x266

== 11 ++ 22 ++ 00 ++ 88 ++ 00 ++ 00 ++ 6464

== 7575


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Convert 100 to BinaryConvert 100 to Binary1002

50 02

25 02

12 12

6 02

3 02

1 12

0 1

1100100

remainder


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That finishes our first topic -That finishes our first topic - introductionintroduction toto

binary numbers and theirbinary numbers and theirconversionconversion toto

and from decimal numbersand from decimal numbers

Our next topic is Our next topic is


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Chapter Number 6Chapter Number 6

Boolean Algebra and Logic CirciutsBoolean Algebra and Logic Circiuts


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BooleanBooleanLogicLogic

OperationsOperations


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LetLet

xx,,yy,,zzbe Booleanbe Boolean

variables. Boolean variables canvariables. Boolean variables can

only have binary valuesonly have binary values i.e., they cani.e., they can

have values which are either 0 or 1have values which are either 0 or 1

For example, if we represent the state of aFor example, if we represent the state of a

light switchlight switch with a Boolean variablewith a Boolean variablexx, we, we

will assign a value of 0 towill assign a value of 0 toxx when thewhen the

switch is OFF, and 1 when it is ONswitch is OFF, and 1 when it is ON


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A few other names for the statesA few other names for the states

of these Boolean variablesof these Boolean variables0 1

Off On

Low High

False True


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We define the followingWe define the following logic operations orlogic operations or

functionsfunctions among the Boolean variablesamong the Boolean variables

Name Example Symbolically

NOT y = NOT(x) xAND z =x ANDy x y

OR z =x ORy x + y

XOR z =x XORy x y


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WellWell define these operationsdefine these operations with the help ofwith the help of

truth tablestruth tables

what is the truth tablewhat is the truth tableof a logic functionof a logic function

A truth table defines theA truth table defines the outputoutput of aof a

logic function forlogic function forallall possiblepossible inputsinputs

?


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Truth Table for theTruth Table for the NOTNOT OperationOperation

(y true whenever x is false)(y true whenever x is false)

x y = x

01


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Truth Table for theTruth Table for the NOTNOT OperationOperation

x y = x

0 11 0


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Truth Table for theTruth Table for theANDAND OperationOperation

(z true when both x & y true)(z true when both x & y true)

x y z = x y

0 00 1

1 01 1


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Truth Table for theTruth Table for theANDAND OperationOperation

x y z = x y

0 0 00 1 0

1 0 01 1 1


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Truth Table for theTruth Table for the OROR OperationOperation

(z true when x or y or both true)(z true when x or y or both true)

x y z = x + y

0 00 1

1 01 1


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Truth Table for theTruth Table for the OROR OperationOperation

x y z = x + y

0 0 00 1 1

1 0 11 1 1


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Truth Table for theTruth Table for the XORXOR OperationOperation

(z true when x or y true, but not both)(z true when x or y true, but not both)

x y z = x y0 00 1

1 01 1


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Truth Table for theTruth Table for the XORXOR OperationOperation

x y z = x y0 0 00 1 1

1 0 11 1 0

Those 4 were theThose 4 were the fundamentalfundamental logic operationslogic operations


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Those 4 were theThose 4 were the fundamentalfundamental logic operations.logic operations.

Here are examples of a few moreHere are examples of a few more complex situationscomplex situations

z =z = ((xx ++ yy))

z = yz = y

((xx++

yy))

z =z = ((yy xx)) ww

STRATEGY: Divide & Conquer


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x y x +y z = (x + y)

0 0 0 10 1 1 0

1 0 1 01 1 1 0

z =z = ((xx ++ yy))


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x y x + y z = y (x + y)

0 0 0 00 1 1 1

1 0 1 01 1 1 1

z = yz = y ((xx ++ yy))


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x y w y x z = (y x) w

0 0 0 0 00 0 1 0 1

0 1 0 0 0

0 1 1 0 11 0 0 0 0

1 0 1 0 1

1 1 0 1 1

z =z = ((yy xx)) ww


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Number of rows in a truth table?Number of rows in a truth table?

22nn

n = number of input variablesn = number of input variables

Assignment # 2Assignment # 2


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Assignment # 2Assignment # 2

A.A. Convert the following into binary numbers:Convert the following into binary numbers:i.i. TheThe last three digitslast three digits of your roll numberof your roll number

ii.ii. 256256

B.B. x, y & z are Boolean variables. Determinex, y & z are Boolean variables. Determine

thethe truth tablestruth tables for the followingfor the following

combinations:combinations:

i.i. ((x yx y))++ yy

ii.ii. ((xx yy))++ ww

Wh t h l t t d ?Wh t h l t t d ?


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What have we learnt today?What have we learnt today?

1.1.

About theAbout the binary number systembinary number system, and how it, and how itdiffersdiffers from the decimal systemfrom the decimal system

2.2. PositionalPositional notationnotation for representing binaryfor representing binary

and decimal numbersand decimal numbers

3.3. AA processprocess (or algorithm) which can be used(or algorithm) which can be usedtoto convertconvert decimal numbers to binarydecimal numbers to binary

numbersnumbers

4.4. BasicBasic logic operationslogic operations for Boolean variables,for Boolean variables,i.e. NOT, OR, AND, XOR, NOR, NAND,i.e. NOT, OR, AND, XOR, NOR, NAND,

XNORXNOR

Focus of the Next LectureFocus of the Next Lecture


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Focus of the Next LectureFocus of the Next Lecture

Next lecture will be theNext lecture will be the 3rd on Web dev3rd on Web dev

The focus of the oneThe focus of the one after that, the 10after that, the 10thth lecturelecture,,

however, will be onhowever, will be on softwaresoftware. During that. During thatlecture we will try:lecture we will try:

To understand theTo understand the role of software in computingrole of software in computing

To become able toTo become able to differentiatedifferentiate betweenbetween systemsystem

andand applicationapplication softwaresoftware


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QuestionsQuestions ????????????

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