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    CS101 Introduction to Computing

    Lecture 8Binary Numbers & Logic Operations

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    The focus of the last lecture was

    on th

    e microprocessor During that lecture we learnt about the function ofthe central component of a computer, the

    microprocessor

    And its various sub-systems

    Bus interface unit

    Data & instruction cache memory

    Instruction decoder

    ALU

    Floating-point unit

    Control unit

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    Learning Goals forToday

    1. To become familiar with number system used by

    the microprocessors - binary numbers

    2. To become able to perform decimal-to-binaryconversions

    3. To understand the NOT, AND, OR and XOR logic

    operations the fundamental operations that are

    available in all microprocessors

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    BINARY(BASE 2)

    numbers

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    DECIMAL(BASE 10)

    numbers

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    Decimal (base 10) number system

    consists of 10 symbols or digits

    0 1 2 3 4

    5 6 7 8 9

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    Binary (base 2) number system

    consists of just two

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    Other popular number systems

    Octal

    base = 8

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

    Hexadecimal

    base = 16

    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 are

    expressed in the positional notation

    4202 = 2x100 +0x101 +2x102 + 4x103

    The right-most is the least significant digit

    The left-most is the most significant digit

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    Decimal (base 10) numbers are

    expressed in thepositional notation

    4202= 2x100 +0x101 +2x102 + 4x103

    1s multiplier

    1

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    Decimal (base 10) numbers are

    expressed in thepositional notation

    4202 = 2x100 +0x101 +2x102 + 4x103

    10s multiplier

    10

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    Decimal (base 10) numbers are

    expressed in thepositional notation

    4202 = 2x100 +0x101 +2x102 + 4x103

    100s multiplier

    100

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    Decimal (base 10) numbers are

    expressed in thepositional notation

    4202 = 2x100 +0x101 +2x102 +4x103

    1000s multiplier

    1000

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    Binary (base 2) numbers are also

    expressed in thepositional notation

    10011 = 1x20 +1x21 +0x22 +0x23 +1x24

    The right-most is the least significant digit

    The left-most is the most significant digit

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    Binary (base 2) numbers are also

    expressed in thepositional notation

    10011 = 1x20 +1x21 +0x22 +0x23 +1x24

    1s multiplier

    1

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    Binary (base 2) numbers are also

    expressed in thepositional notation

    10011 = 1x20 + 1x21 +0x22 +0x23 +1x24

    2s multiplier

    2

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    Binary (base 2) numbers are also

    expressed in thepositional notation

    10011 = 1x20 +1x21 + 0x22 +0x23 +1x24

    4s multiplier

    4

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    Binary (base 2) numbers are also

    expressed in thepositional notation

    10011 = 1x20 +1x21 +0x22 + 0x23 +1x24

    8s multiplier

    8

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    Binary (base 2) numbers are also

    expressed in thepositional notation

    10011 = 1x20 +1x21 +0x22 +0x23 + 1x24

    16s multiplier

    16

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    Counting

    in Decimal0

    1

    23

    4

    5

    6

    7

    8

    9

    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 binaryBecause this system is natural for digital computersThe 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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    Byte = 8 bits

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    Decimal Binaryconversion

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    Convert 75 to Binary

    75237 12

    18 12

    9 02 4 12

    2 02

    1 02

    0 1

    1001011

    remainder

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    Check

    1001011 = 1x20 + 1x21 + 0x22 + 1x23 +

    0x24 + 0x25 + 1x26

    = 1+ 2+ 0 + 8 +0+ 0+ 64

    = 75

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    Convert 100 to Binary

    100250 02

    25 02

    12 12 6 02

    3 02

    1 12

    0 1

    1100100

    remainder

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    That finishes our first topic - introduction

    to binary numbers and theirconversion

    to and from decimal numbers

    Our next topic is

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    Boolean

    Logic

    Operations

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    Let

    x,y,zbe Boolean

    variables. Boolean variables can

    only have binary values i.e., they can

    have values which are either 0 or 1

    For example, if we represent the state of

    a light switch with a Boolean variablex,

    we will assign a value of 0 tox when the

    switch is OFF, and 1 when it is ON

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    A few other names for the states

    of these Boolean variables

    0 1

    Off On

    Low High

    False True

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    Well define these operations with the help of

    truth tables

    what is the truth tableofalogic function

    A truth table defines the output of a

    logic function forall possible inputs

    ?

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    TruthTable for the NOTOperation

    (y true whenever x is false)

    x y = x

    0

    1

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    TruthTable for theAND Operation

    (z true wh

    en both

    x & y true)

    x y z = x y

    0 0

    0 1

    1 01 1

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    TruthTable for the OR Operation

    (z true wh

    en x or y or both

    true)

    x y z = x +y

    0 0

    0 1

    1 01 1

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    TruthTable for the OR Operation

    x y z = x +y

    0 0 0

    0 1 1

    1 0 11 1 1

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    TruthTable for the XOR Operation

    (z true when x or y true, but not bot

    h)

    x y z = x y

    0 0

    0 1

    1 01 1

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    TruthTable for the XOR Operation

    x y z = x y

    0 0 0

    0 1 1

    1 0 11 1 0

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    Those 4 were the fundamental logic operations.

    Here are examples ofa few more complex situations

    z = (x +y)

    z = y(x

    +y)

    z = (y x) w

    STRATEGY: Divide & Conquer

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

    0 0 0 1

    0 1 1 0

    1 0 1 01 1 1 0

    z = (x +y)

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

    0 0 0 0

    0 1 1 1

    1 0 1 01 1 1 1

    z = y (x +y)

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    x y w y x z = (y x) w0 0 0 0 0

    0 0 1 0 10 1 0 0 0

    0 1 1 0 1

    1 0 0 0 01 0 1 0 1

    1 1 0 1 1

    1 1 1 1 0

    z = (y x) w

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