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Information Representation
How to count like a computer?
Ashok K Nagawat
Joint Director, Centre for Converging Technologies
Coordinator, Infonet Center
Deputy Director, CDPE
Associate Professor, Deptt. of Physics
Associate Member, Institute of Informatics and Instrumentation
• In order to understand what happens inside a
computer we must understand how a
computer represents information internally.
Analogue Computer
Digital Computer
Quantum Computer
University of Rajasthan, Jaipur
[email protected]
Digital Computer
– A digital system performing
computational tasks.
– Represents information using variables
that take a limited number of discrete
values.
– Processes these values internally
Radix Number Systems
Each number system has a number of
different digits which is called the radix
or the base of the number system.
•
•
•
•
Decimal
Octal
Hexadecimal (Hex)
Binary
Positional Notation
Decimal Number System
Base (Radix)
Digits
e.g.
10
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
747510
The magnitude represented by a digit is decided by
the position of the digit within the number.
1000
100
10
1
7
4
7
5
For example the digit 7 in the left-most position of
7475 counts for 7000 and the digit 7 in the second
position from the right counts for 70.
Base = 10
Base = 8
Base = 16
Base = 2
642 in base 10 positional notation is:
6 x 102 = 6 x 100 = 600
+ 4 x 101 = 4 x 10 = 40
+ 2 x 10º = 2 x 1 = 2
= 642 in base 10
This number is in
base 10
The power indicates
the position of
the number
1
Positional Notation
R is the base
of the number
As a formula:
Positional Notation
What if 642 has the base of 13?
+ 6 x 132 = 6 x 169 = 1014
+ 4 x 131 = 4 x 13 = 52
+ 2 x 13º = 2 x 1 = 2
= 1068 in base 10
dn * Rn-1 + dn-1 * Rn-2 + ... + d2 * R + d1
n is the number of
digits in the number
d is the digit in the
ith position
in the number
642 is 63 * 102 + 42 * 10 + 21
642 in base 13 is equivalent to 1068
in base 10
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Bases Higher than 10
How are digits in bases higher than 10
represented?
Hexadecimal Number System
Base (Radix)
Digits
e.g.
With distinct symbols for 10 and above.
Base 16 has 16 digits:
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E, and F
16
0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
A, B, C, D, E, F
2F4D 16
4096=163 256=162
2
16=161
F
4
1=160
D
The digit F in the third position from the right
represents the value 3840 and the digit D in the
first position from the right represents the value 1.
Converting Hexadecimal to Decimal
What is the decimal equivalent of the
hexadecimal number DEF?
162
Dx
= 13 x 256 = 3328
+ E x 161 = 14 x 16 = 224
+ F x 16º = 15 x 1 = 15
= 3567 in base 10
Remember, the digits in base 16 are
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
Octal Number System
Base (Radix)
Digits
e.g.
512=83
1
8
0, 1, 2, 3, 4, 5, 6, 7
16238
64=82
6
8=81
2
1=80
3
The digit 2 in the second position from the right
represents the value 16 and the digit 1 in the
fourth position from the right represents the value
512.
2
Converting Octal to Decimal
What is the decimal equivalent of the octal
number 642?
6 x 82 = 6 x 64 = 384
+ 4 x 81 = 4 x 8 = 32
+ 2 x 8º = 2 x 1 = 2
= 418 in base 10
Binary Number System
Base (Radix)
Digits
e.g.
2
0, 1
11102
1
Decimal is base 10 and has 10 digits:
0,1,2,3,4,5,6,7,8,9
Binary is base 2 and has 2 digits:
0,1
For a number to exist in a given number system,
the number system must include those digits. For
example, the number 284 only exists in base 9 and
higher.
Converting Binary to Decimal
What is the decimal equivalent of the binary
number 1101110?
8=23 4=22 2=21 1=20
1
Binary
1
0
The digit 1 in the third position from the right
represents the value 4 and the digit 1 in the
fourth position from the right represents the
value 8.
Why Binary?
• For example, the text you are reading is
stored on a computer disk. It was entered into
a computer system using Powerpoint.
• Inside the computer’s memory and on the
disk this text (and all other information) can
be thought of as being represented by a very
long sequence of ones and zeros, i.e. in
binary form.
1 x 26
+ 1 x 25
+ 0 x 24
+ 1 x 23
+ 1 x 22
+ 1 x 21
+ 0 x 2º
=
=
=
=
=
=
=
1 x 64
1 x 32
0 x 16
1x8
1x4
1x2
0x1
= 64
= 32
=0
=8
=4
=2
=0
= 110 in base 10
Why Binary?
• The basic unit of storage in a computer is the bit (binary
digit), which can have one of just two values: 0 or 1.
• This is easier to implement in hardware than a unit that can
take on 10 different values.
– For instance, it can be represented by a transistor being
off (0) or on (1).
– Alternatively, it can be a magnetic stripe that is
magnetized with North in one direction (0) or the
opposite (1).
• Binary also has a convenient and natural association with
logical values of False (0) and True (1).
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Information Representation in Binary
Digital Computer
Information Representation
• It appears as text only on a computer’s monitor or on
printed output.
• A computer monitor’s hardware receives binary
information and transforms it to the symbols that are
displayed.
• Similarly, a printer’s hardware converts binary
information to the text (or graphic) that is printed.
• Inside the CPU and the computer’s memory,
information is actually stored and transmitted in
electrical form, while on disk it is stored in magnetic
form.
• A particular electrical signal represents
a one and another signal represents a
zero. Similarly, a specific magnetic
orientation represents a one and
another orientation represents a zero. A
group of these binary digits (bits) can be
considered to form a binary code and
information is encoded as a sequence
of individual units each of which
correspond to a distinct binary code.
Fundamental Principle
Building on Binary
• Binary bits are grouped together to allow them
to represent more information:
• All information is
represented by binary
codes in a binary digital
computer system.
Building on Binary
• Clearly, the number of possible combinations
of a group of N bits is 2N = 2x2x2 … x2 (N
2s). Thus:
– A nybble can form 24=16 combinations
– A byte can form 28=256 combinations
– A 32-bit word can form 232=4,294,967,296
combinations
– A nybble is a group of 4 bits, e.g. 1011
– A byte is a group of 8 bits, e.g. 11010010
– A word is a larger grouping: usually 32 bits. A
halfword is half as many bits as a word, thus
usually 16 bits. A doubleword is twice as many
bits, usually 64 bits. However, computers have
been designed with various word sizes, such as 36,
48, or 60 bits.
Converting Binary to Octal
• Groups of Three (from right)
• Convert each group
10101011
10 101 011
2 5 3
10101011 is 253 in base 8
4
Converting Binary to Hexadecimal
• Groups of Four (from right)
• Convert each group
10101011
1010 1011
A
B
10101011 is AB in base 16
Converting Decimal to Other Bases
Algorithm for converting base 10 to other
bases
While the quotient is not zero
Divide the decimal number by the new base
Make the remainder the next digit to the left in
the answer
Replace the original dividend with the quotient
Converting Decimal to Hexadecimal
Converting Decimal to Hexadecimal
222
16 3567
32
36
32
47
32
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Try a Conversion
The base 10 number 3567 is
what number in base 16?
13
16 222
16
62
48
14
0
16 13
0
13
F E D
Binary Arithmetic
• Addition
Binary Addition
(a)
•Complements
•Subtraction
(c)
0
+0
0
1
+0
1
0
+1
1
(b)
(d)
1
+1
10
Carry Bit
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Binary Complement
Binary Addition Examples
1011
+ 1100
10111
(a)
(b)
1010
+ 100
1110
(c)
(1s Complement) Operation
1011
+ 101
10000
1
0
0
1
Example
(d)
101
+ 1001
1110
110010110
10011001
+
101100
11000101
(e)
Two’s Complement
The Two’s complement of a binary number
is obtained by first complementing the
number and then adding 1 to the result.
1001110
0110001
+
1
One’s Complement
0110010
Two’s Complement
001101001
Binary Subtraction
Binary subtraction is implemented by adding
the Two’s complement of the number to be
subtracted.
Two’s
Example
complement
of 1001
1101
-1001
1101
+0111
10100
If there is a carry then it is ignored. Thus,
the answer is 0100.
BCD – Binary Coded Decimal
Binary Codes
A binary code is a group of n bits that
assume up to 2n distinct combinations of 1’s
and 0’s with each combination representing
one element of the set that is being coded.
• BCD
– Binary Coded Decimal
• ASCII – American Standard Code for
Information Interchange
When the decimal numbers are
represented in BCD, each
decimal digit is represented by
the equivalent BCD code.
Example :BCD Representation
of Decimal 6349
6
3
4
9
0110 0011 0100 1001
Decimal
Number
BCD
Number
0
1
2
3
4
5
6
7
8
9
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
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ASCII
Number ASCII
0
1
2
3
4
5
6
7
8
9
0110000
0110001
0110010
0110011
0110100
0110101
0110110
0110111
0111000
0111001
Letter
A
B
C
D
E
F
G
H
I
ASCII
1000001
1000010
1000011
1000100
1000101
1000110
1000111
1001000
1001001
Logic Gates
ASCII
Continued.
Letter ASCII
J
1001010
K
1001011
L
1001100
M 1001101
N
1001110
O 1001111
P
1010000
Q 1010001
R
1010010
Letter ASCII
S
1010011
T
1010100
U
1010101
V
1010110
W 1010111
X
1011000
Y
1011001
Z
1011010
Logic Gates Contd…
•
Binary information is represented in
digital computers by physical quantities
called signals.
• A particular logic operation can be
described in an algebraic or tabular form.
•
Two different electrical voltage levels
such as 3 volts and 0.5 volts may be
used to represent binary 1 and 0.
•
Binary logic deals with binary variables
and with operations that assume a logical
meaning.
• The manipulation of binary information is
done by the circuits called logic gates
which are blocks of hardware that
produce signals of binary 1 or 0 when
input logic requirements are satisfied.
Logic Gates Contd…
• Each gate has a distinct graphics
symbol and it’s operation can be
described by means of an algebraic
expression or in a form of a table
called the truth table.
• Each gate has one or more binary
inputs and one binary output.
Logic Gates
– AND
– OR
– NOT (Inverter)
– NAND
(Not AND)
– NOR (Not OR)
– XOR (Exclusive-OR)
– Exclusive-NOR
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Logic Gates
•AND
Table
Logic Gates
Cont.
Logic Gate
A
B
Truth
x
x=A.B
A, B Binary Input Variables
x
Binary Output Variable
Logic Gates
•NOT
Table
AB
0 0
0 1
1 0
1 1
A
B
x=A+B
Truth
•NAND
A x
0 1
1 0
A
B
x
Logic Gates
Logic Gate
A
B
x=A+B
Truth
x
x
0
1
1
1
AB
0 0
0 1
1 0
1 1
x
1
0
0
0
•XOR
Table
Truth
Table
x
Logic Gates
Cont.
Logic Gate
AB
0 0
0 1
1 0
1 1
Cont.
x=A.B
x=A
•NOR
Table
x
Logic Gates
Logic Gate
Truth
This is read as x
equals A or B.
Cont.
A
Logic Gate
•OR
Table
x
0
0
0
1
Cont.
x=A+B
x
1
1
1
0
Cont.
Logic Gate
A
B
AB
0 0
0 1
1 0
1 1
x
Truth
AB
0 0
0 1
1 0
1 1
x
0
1
1
0
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Logic Gates
Basic logic gates
Cont.
•Exclusive-NOR Logic Gate
A
B
x
x=A+B
Truth
Table
AB
0 0
0 1
1 0
1 1
x
1
0
0
1
• Not
• And
x
y
z
xyz
• Or
• Nand
• Nor
• Xor
Completeness of NAND
Using A Single Gate Type
• It is desirable to use only one type of gate
generate the whole circuit.
• Can use NAND or NOR gate.
• In order to do so, enough to show that
– NOT, AND, OR NAND can be generated by
NOR gates
– NOT, AND, OR, NOR ca be generated by
NAND gates.
• We say that NAND, NOR are complete for
Boolean circuits
Completeness of NOR
Integrated Circuits
• Most gates are attached together to build
Integrated Circuits (IC’s)
• They are mounted on Plastic or Ceramic with
some combination of pins for input/output, power
and ground connections
• Types base on number of gates:
–
–
–
–
(SSI) Small Scale Integrated Circuit: 1 to 10
(MSI) Medium Scale IC: 10 to 100
(LSI) Large Scale IC: 100 to 100,000
(VLSI) Very Large Scale IC: Over 100,100
• The current Intel Pentium IV processors have 55
million transistors!
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