binary numbers logic operations
TRANSCRIPT
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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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Number of rows in a truth
table?
2nn = number of input variables
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Assignment # 3
A. Convert the following into binary numbers:i. The last three digits of your roll number
ii. 256
B. x, y & z are Boolean variables. Determine the trut
htables for the following combinations:
i. (x y) +y
ii. (x y)+ w
Consult the CS101 syllabus for the submissioninstructions & deadline
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What have we learnt today?
1. About the binary number system, and how it differsfrom the decimal system
2. Positional notation for representing binary and
decimal numbers
3. A process (or algorithm) which can be used to
convert decimal numbers to binary numbers
4. Basic logic operations for Boolean variables, i.e.
NOT, OR, AND, XOR, NOR, NAND, XNOR
5. Construction oftruth tables (How many rows?)
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