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Data Representation
CS2052 Computer Architecture
Computer Science & Engineering
University of Moratuwa
Dilum Bandara
Dilum.Bandara@uom.lk
Outline
 Representing numbers
 Unsigned
 Signed
 Floating point
 Representing characters & symbols
 ASCII
 Unicode
2
Data Representation in Computers
 Data are stored in Registers
 Registers are limited in number & size
 With a n-bit register
 Min value 0
 Max value 2n-1
3
n-1 n-2 ...… ... 2 01
n-bits
4
Data Representation
Data
Representation
• Represents quality or
characteristics
• Not proportional to a
value
• Name, NIC no, index no,
Address
Qualitative
Quantitative
• Quantifiable
• Proportional to value α
• No of students, marks for
CS2052, GPA
Data Representation (Cont.)
5
Data
Quantitative
Integers
Signed
Unsigned
Non-
integers
Signed
Unsigned
Qualitative
Number Systems
 Decimal number system
 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
 Binary number system
 0, 1
 Octal number system
 0, 1, 2, 3, 4, 5, 6, 7
 Hexadecimal number system
 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
6
Quantitative Numbers
 Integers
 Unsigned 20
 Signed +20, -20
 Non-integers
 Floating point numbers - 10.25, 3.33333…, 1/8 = 0.125
7
Signed Integers
 We need a way to represent negative values
 3 representations
 Sign & Magnitude representation (S&M)
 Complement method
 Bias notation or Excess notation
8
1. Sign & Magnitude Representation
 n-bit unsigned magnitude & sign bit (S)
 If S
 0 – Integer is positive or zero
 1 – Integer is negative or zero
 Range –(2n-1) to +(2n-1)
9
sign n-1 ...n-2 ... 2 01
Magnitude (n-bits)
Example – Sign & Magnitude
 If 8-bit register is used what are min & max
numbers?
 What are 0000 0000 and 1000 0000 in decimal?
 Representation of zero is not unique
10
Sign & Magnitude (Cont.)
 Advantages
 Sign reversal
 Finding absolute value |a|
 Flip sign bit
 Disadvantage
 Adding a negative of a number is not the same as
subtraction
 e.g., add 2 and -3
 Need different operations
 Zero is not unique
11
2. Complement Method
 Base = Radix
 Radix r system means r number of symbols
 e.g., binary numbers have symbols 0, 1
 2 types
 r’s complement
 (r – 1)’s complement
 Where r is radix (base) of number system
 Examples
 Decimal 9’s & 10’s complement
 Binary 1’s & 2’s complement
12
Complement Method – Definition
 Given a number m in base/radix r & having n
digits
 (r – 1)’s complement of m is
(rn – 1) – m
 r ’s complement of n is
(rn – 1) – m + 1 = rn – m
13
Example – Complement Method
 If m = 5982 & n = 4 digits
 9’s complement is
9 9 9 9 - maximum representable no
5 9 8 2 –
4 0 1 7
 10’s complement
9 9 9 9 or 1 0 0 0 0
5 9 8 2 – 5 9 8 2 –
4 0 1 7 4 0 1 8
1 +
4 0 1 8
14
Example – Complement Method
 If m = 382 & n = 3
 -n = -382 =
9 9 9 9 9 9
3 8 2 – 3 8 2 -
6 1 7 6 1 7
1 +
6 1 8
 -382 = 617 or 618
 Depending on which complement we use
 These are called complementary pair
15
1’s complement
 Calculated by
 (2n – 1) – m
 If m = 0101
 1’s complement of m on a 4-bit system
1 1 1 1
0 1 0 1 –
1 0 1 0
 This represents -5 in 1’s complement
16
Finding 1’s Complement – Short Cut
 Invert each bit of m
 Example
 m = 0 0 1 0 1 0 1 1
 1’s complement of m = 1 1 0 1 0 1 0 0
 m = 0 0 0 0 0 0 0 0 = 0
 1’s complement of m = 1 1 1 1 1 1 1 1 = -0
 Values range from -127 to +127
17
Addition with 1’s Complement
 If results has a carry add it to LSB (Least
Significant Bit)
 Example
 Add 6 and -3 on a 3-bit system
 6 = 1 1 0
 -3 = 1 0 0 +
= 1 0 1 0
1 +
0 1 1
18
2’s Complement
 Doesn’t require end-around carry operation as in
1’s complement
 2’s complement is formed by
 Finding 1’s complement
 Add 1 to LSB
 New range is from -128 to +127
 -128 because of +1 to negative value
19
Example – 2’s Complement
 Find 2’s complement of 0101011
 m = 0 1 0 1 0 1 1
 = 1 0 1 0 1 0 0
________ 1 +
 2’s = 1 0 1 0 1 0 1
 Short-cut
1. Search for the 1st bit with value 1 starting from LSB
2. Inver all bits after 1st one
20
Example – 2’s Complement (Cont.)
 Add 6 and -5 on a 4-bit system
5 = 0101
-5 = 1011
6 = 0 1 1 0
-5 = 1 0 1 1+
1 0 0 0 1  0 0 0 1
21
Discard
1’s vs. 2’s Complement
 1’s complement has 2 zeros (+0, -0)
 Value range is less than 2’s complement
 2’s complement only has a single zero
 Value range is unequal
 No need of a separate subtract circuit
 Doing a NOT operation is much more cost effective
interms of circuit design
 However, multiplication & division is slow
22
S&M, 1’s, & 2’s
23
Source: www3.ntu.edu.sg/home/ehchua/programming/java/DataRepresentation.html
4-bit, 2’s Complement Adder
Subtractor
24
Source: www.instructables.com/id/How-to-Build-an-8-Bit-Computer/step7/The-ALU/
Detecting Negative Numbers &
Overflow
 Check for MSB
 To find magnitude
 1’s complement
 Flip all bits
 2’s complement
 Flip all bits + 1
 Rules to detect overflows in 2’s complement
 If sum of 2 positive numbers yields a negative result,
sum has overflowed
 If sum of 2 negative numbers yields a positive result,
sum has overflowed
 Else, no overflow 25
3. Bias Notation
 Number range to be represented is given a
positive bias so that smallest unsigned value is
equal to -B
 If bias is B
 Value B in number range becomes zero
 If bias is 127 for 8-bit number range is
 -127 (0 - 127) to 128 (255 - 127) 26


n
i
i
i Bb
0
2
Bias Notation (Cont.)
27
Bit pattern Unsigned Biased-127
0000 0000 +0 -127
0000 0001 +1 -126
0000 0010 +2 -125
… … …
0111 1110 +126 -1
0111 1111 +127 +0
1000 0000 +128 +1
… … …
1111 1111 +255 +128
Floating Point Numbers
 We needed to represent fractional values &
values beyond 2n – 1
 +3207.23 = 3.20723x103
 -0.000321 = -3.21x10-4
28
Sign
Mantissa
Radix
(base)
Exponent
Floating Point Numbers (Cont.)
29
es
mN 2.)1(
Sign
Mantissa
Radix
Exponent
sign en-1 ...… e0 … m0m1mn-1 …mn-2
MantissaExponent
IEEE Floating Point Standard (FPS)
 2 standards
1. Single precision
 32-bits
 23-bit mantissa
 8-bit exponent
 1-bit sign
2. Double precision
 64-bits
 52-bit mantissa
 11-bit exponent
 1-bit sign
30
31 30 23 22 0
S Exponent Mantissa (bits 0-22)
63 62 52 51 0
S Exponent Mantissa (bits 0-51)
IEEE Floating Point Standard (Cont.)
 E – 127 = e
 E = e + 127
 What is stored is E
31
s31 e30 ...… e23 … m0m1m22 …mn-2
MantissaExponent
127
2].1[)1( 
 Es
mN
Single Precision
 23-bit mantissa
 m ranges from 1 to (2 – 2-23 )
 8-bit exponent
 E ranges from 1 to 254
 0 & 255 reserved to represent -∞, +∞, NaN
 e ranges from -126 to +127
 Maximum representable number
 2 x 2127 = 2128
32
Review – Binary Fractions
 What is 101.11012 in decimal?
 22 x 1 + 21 x 0 + 20 x 1 + 2-1 x 1 + 2-2 x 1 + 2-3 x 0 + 2-4 x 1
 = 4 x 1 + 2 x 0 + 1 x 1 + 0.5 x 1 + 0.25 x 1 + 0.125 x 0 + 0.0625 x 1
 = 4 + 1 + 0.5 + 0.25 + 0.0625
 = 5.8125
 What is 5.812510 in binary?
 Integer – 5 = 101
 Fraction – Keep multiplying fraction until answer is 1
 0.8125 x 2 = 1.625
 0.625 x 2 = 1.25
 0.25 x 2 = 0.5
 0.5 x 2 = 1.0
 0.812510 = .11012
33
Example – Single Precision
 IEEE Single Precision Representation of 17.1510
 Find 17.1510 in binary
17.1510 = 10001.00100112
= 1.00010010011 x 24
E = e + 127
E = 4 + 127 = 131
34
0 1000 0011 0001 0010 0110 0000 ……..
MantissaExponent
Character Representation
 Belong to category of qualitative data
 Represent quality or characteristics
 Includes
 Letters a-z, A-Z
 Digits 0-9
 Symbols !, @ , *, /, &, #, $
 Control characters <CR>, <BEL>, <ESC>, <LF>
35
Character Representation (Cont.)
 With a single byte (8-bits) 256 characters can be
represented
 Standards
 ASCII – American Standard Code for Information
Interchange
 EBCDIC – Extended Binary-Coded Decimal
Interchange Code
 Unicode
36
ASCII
 De facto world-wide standard
 Used to represent
 Upper & lower-case Latin letters
 Numbers
 Punctuations
 Control characters
 There are 128 standard ASCII codes
 Can be represented by a 7 digit binary number
 000 0000 through 111 1111
 Plus parity bit
37
ASCII Hex Symbol
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
20
21
22
23
24
25
26
27
28
29
2A
2B
2C
2D
2E
2F
(space)
!
"
#
$
%
&
'
(
)
*
+
,
-
.
/
38
ASCII Table
ASCII
Hex Symbol
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
NUL
SOH
STX
ETX
EOT
ENQ
ACK
BEL
BS
TAB
LF
VT
FF
CR
SO
SI
ASCII Hex Symbol
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
30
31
32
33
34
35
36
37
38
39
3A
3B
3C
3D
3E
3F
0
1
2
3
4
5
6
7
8
9
:
;
<
=
>
?
ASCII Hex Symbol
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
50
51
52
53
54
55
56
57
58
59
5A
5B
5C
5D
5E
5F
P
Q
R
S
T
U
V
W
X
Y
Z
[

]
^
_
39
ASCII Table (Cont.)
ASCII Hex Symbol
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
40
41
42
43
44
45
46
47
48
49
4A
4B
4C
4D
4E
4F
@
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
ASCII Hex Symbol
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
60
61
62
63
64
65
66
67
68
69
6A
6B
6C
6D
6E
6F
`
a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
ASCII – Things to Note
 ASCII codes for digits aren’t equal to numeric
value
 Uppercase & lowercase alphabetic codes differ
by 0x20
 Shift key clears bit 5
 0x20 = 32 = 0010 0000
 Example:
 A = 65 = 0100 0001
 a = 97 = 0110 0001
 Most languages need more than 128 characters
40
Unicode (www.unicode.org)
 Designed to overcome limitation of number of
characters
 Assigns unique character codes to characters in
a wide range of languages
 A 16-bit character set
 UCS-2
 UCS-4 is 32-bit
 65,536 (216) distinct Unicode characters
Unicode provides a unique number for every character,
no matter what the platform,
no matter what the program,
no matter what the language
41
Unicode (Cont.)
42
Source: www3.ntu.edu.sg/home/ehchua/programming/java/DataRepresentation.html
Unicode – Sinhala & Tamil
43

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Data Representation

  • 1. Data Representation CS2052 Computer Architecture Computer Science & Engineering University of Moratuwa Dilum Bandara Dilum.Bandara@uom.lk
  • 2. Outline  Representing numbers  Unsigned  Signed  Floating point  Representing characters & symbols  ASCII  Unicode 2
  • 3. Data Representation in Computers  Data are stored in Registers  Registers are limited in number & size  With a n-bit register  Min value 0  Max value 2n-1 3 n-1 n-2 ...… ... 2 01 n-bits
  • 4. 4 Data Representation Data Representation • Represents quality or characteristics • Not proportional to a value • Name, NIC no, index no, Address Qualitative Quantitative • Quantifiable • Proportional to value α • No of students, marks for CS2052, GPA
  • 6. Number Systems  Decimal number system  0, 1, 2, 3, 4, 5, 6, 7, 8, 9  Binary number system  0, 1  Octal number system  0, 1, 2, 3, 4, 5, 6, 7  Hexadecimal number system  0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F 6
  • 7. Quantitative Numbers  Integers  Unsigned 20  Signed +20, -20  Non-integers  Floating point numbers - 10.25, 3.33333…, 1/8 = 0.125 7
  • 8. Signed Integers  We need a way to represent negative values  3 representations  Sign & Magnitude representation (S&M)  Complement method  Bias notation or Excess notation 8
  • 9. 1. Sign & Magnitude Representation  n-bit unsigned magnitude & sign bit (S)  If S  0 – Integer is positive or zero  1 – Integer is negative or zero  Range –(2n-1) to +(2n-1) 9 sign n-1 ...n-2 ... 2 01 Magnitude (n-bits)
  • 10. Example – Sign & Magnitude  If 8-bit register is used what are min & max numbers?  What are 0000 0000 and 1000 0000 in decimal?  Representation of zero is not unique 10
  • 11. Sign & Magnitude (Cont.)  Advantages  Sign reversal  Finding absolute value |a|  Flip sign bit  Disadvantage  Adding a negative of a number is not the same as subtraction  e.g., add 2 and -3  Need different operations  Zero is not unique 11
  • 12. 2. Complement Method  Base = Radix  Radix r system means r number of symbols  e.g., binary numbers have symbols 0, 1  2 types  r’s complement  (r – 1)’s complement  Where r is radix (base) of number system  Examples  Decimal 9’s & 10’s complement  Binary 1’s & 2’s complement 12
  • 13. Complement Method – Definition  Given a number m in base/radix r & having n digits  (r – 1)’s complement of m is (rn – 1) – m  r ’s complement of n is (rn – 1) – m + 1 = rn – m 13
  • 14. Example – Complement Method  If m = 5982 & n = 4 digits  9’s complement is 9 9 9 9 - maximum representable no 5 9 8 2 – 4 0 1 7  10’s complement 9 9 9 9 or 1 0 0 0 0 5 9 8 2 – 5 9 8 2 – 4 0 1 7 4 0 1 8 1 + 4 0 1 8 14
  • 15. Example – Complement Method  If m = 382 & n = 3  -n = -382 = 9 9 9 9 9 9 3 8 2 – 3 8 2 - 6 1 7 6 1 7 1 + 6 1 8  -382 = 617 or 618  Depending on which complement we use  These are called complementary pair 15
  • 16. 1’s complement  Calculated by  (2n – 1) – m  If m = 0101  1’s complement of m on a 4-bit system 1 1 1 1 0 1 0 1 – 1 0 1 0  This represents -5 in 1’s complement 16
  • 17. Finding 1’s Complement – Short Cut  Invert each bit of m  Example  m = 0 0 1 0 1 0 1 1  1’s complement of m = 1 1 0 1 0 1 0 0  m = 0 0 0 0 0 0 0 0 = 0  1’s complement of m = 1 1 1 1 1 1 1 1 = -0  Values range from -127 to +127 17
  • 18. Addition with 1’s Complement  If results has a carry add it to LSB (Least Significant Bit)  Example  Add 6 and -3 on a 3-bit system  6 = 1 1 0  -3 = 1 0 0 + = 1 0 1 0 1 + 0 1 1 18
  • 19. 2’s Complement  Doesn’t require end-around carry operation as in 1’s complement  2’s complement is formed by  Finding 1’s complement  Add 1 to LSB  New range is from -128 to +127  -128 because of +1 to negative value 19
  • 20. Example – 2’s Complement  Find 2’s complement of 0101011  m = 0 1 0 1 0 1 1  = 1 0 1 0 1 0 0 ________ 1 +  2’s = 1 0 1 0 1 0 1  Short-cut 1. Search for the 1st bit with value 1 starting from LSB 2. Inver all bits after 1st one 20
  • 21. Example – 2’s Complement (Cont.)  Add 6 and -5 on a 4-bit system 5 = 0101 -5 = 1011 6 = 0 1 1 0 -5 = 1 0 1 1+ 1 0 0 0 1  0 0 0 1 21 Discard
  • 22. 1’s vs. 2’s Complement  1’s complement has 2 zeros (+0, -0)  Value range is less than 2’s complement  2’s complement only has a single zero  Value range is unequal  No need of a separate subtract circuit  Doing a NOT operation is much more cost effective interms of circuit design  However, multiplication & division is slow 22
  • 23. S&M, 1’s, & 2’s 23 Source: www3.ntu.edu.sg/home/ehchua/programming/java/DataRepresentation.html
  • 24. 4-bit, 2’s Complement Adder Subtractor 24 Source: www.instructables.com/id/How-to-Build-an-8-Bit-Computer/step7/The-ALU/
  • 25. Detecting Negative Numbers & Overflow  Check for MSB  To find magnitude  1’s complement  Flip all bits  2’s complement  Flip all bits + 1  Rules to detect overflows in 2’s complement  If sum of 2 positive numbers yields a negative result, sum has overflowed  If sum of 2 negative numbers yields a positive result, sum has overflowed  Else, no overflow 25
  • 26. 3. Bias Notation  Number range to be represented is given a positive bias so that smallest unsigned value is equal to -B  If bias is B  Value B in number range becomes zero  If bias is 127 for 8-bit number range is  -127 (0 - 127) to 128 (255 - 127) 26   n i i i Bb 0 2
  • 27. Bias Notation (Cont.) 27 Bit pattern Unsigned Biased-127 0000 0000 +0 -127 0000 0001 +1 -126 0000 0010 +2 -125 … … … 0111 1110 +126 -1 0111 1111 +127 +0 1000 0000 +128 +1 … … … 1111 1111 +255 +128
  • 28. Floating Point Numbers  We needed to represent fractional values & values beyond 2n – 1  +3207.23 = 3.20723x103  -0.000321 = -3.21x10-4 28 Sign Mantissa Radix (base) Exponent
  • 29. Floating Point Numbers (Cont.) 29 es mN 2.)1( Sign Mantissa Radix Exponent sign en-1 ...… e0 … m0m1mn-1 …mn-2 MantissaExponent
  • 30. IEEE Floating Point Standard (FPS)  2 standards 1. Single precision  32-bits  23-bit mantissa  8-bit exponent  1-bit sign 2. Double precision  64-bits  52-bit mantissa  11-bit exponent  1-bit sign 30 31 30 23 22 0 S Exponent Mantissa (bits 0-22) 63 62 52 51 0 S Exponent Mantissa (bits 0-51)
  • 31. IEEE Floating Point Standard (Cont.)  E – 127 = e  E = e + 127  What is stored is E 31 s31 e30 ...… e23 … m0m1m22 …mn-2 MantissaExponent 127 2].1[)1(   Es mN
  • 32. Single Precision  23-bit mantissa  m ranges from 1 to (2 – 2-23 )  8-bit exponent  E ranges from 1 to 254  0 & 255 reserved to represent -∞, +∞, NaN  e ranges from -126 to +127  Maximum representable number  2 x 2127 = 2128 32
  • 33. Review – Binary Fractions  What is 101.11012 in decimal?  22 x 1 + 21 x 0 + 20 x 1 + 2-1 x 1 + 2-2 x 1 + 2-3 x 0 + 2-4 x 1  = 4 x 1 + 2 x 0 + 1 x 1 + 0.5 x 1 + 0.25 x 1 + 0.125 x 0 + 0.0625 x 1  = 4 + 1 + 0.5 + 0.25 + 0.0625  = 5.8125  What is 5.812510 in binary?  Integer – 5 = 101  Fraction – Keep multiplying fraction until answer is 1  0.8125 x 2 = 1.625  0.625 x 2 = 1.25  0.25 x 2 = 0.5  0.5 x 2 = 1.0  0.812510 = .11012 33
  • 34. Example – Single Precision  IEEE Single Precision Representation of 17.1510  Find 17.1510 in binary 17.1510 = 10001.00100112 = 1.00010010011 x 24 E = e + 127 E = 4 + 127 = 131 34 0 1000 0011 0001 0010 0110 0000 …….. MantissaExponent
  • 35. Character Representation  Belong to category of qualitative data  Represent quality or characteristics  Includes  Letters a-z, A-Z  Digits 0-9  Symbols !, @ , *, /, &, #, $  Control characters <CR>, <BEL>, <ESC>, <LF> 35
  • 36. Character Representation (Cont.)  With a single byte (8-bits) 256 characters can be represented  Standards  ASCII – American Standard Code for Information Interchange  EBCDIC – Extended Binary-Coded Decimal Interchange Code  Unicode 36
  • 37. ASCII  De facto world-wide standard  Used to represent  Upper & lower-case Latin letters  Numbers  Punctuations  Control characters  There are 128 standard ASCII codes  Can be represented by a 7 digit binary number  000 0000 through 111 1111  Plus parity bit 37
  • 38. ASCII Hex Symbol 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 20 21 22 23 24 25 26 27 28 29 2A 2B 2C 2D 2E 2F (space) ! " # $ % & ' ( ) * + , - . / 38 ASCII Table ASCII Hex Symbol 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 A B C D E F NUL SOH STX ETX EOT ENQ ACK BEL BS TAB LF VT FF CR SO SI ASCII Hex Symbol 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 30 31 32 33 34 35 36 37 38 39 3A 3B 3C 3D 3E 3F 0 1 2 3 4 5 6 7 8 9 : ; < = > ?
  • 39. ASCII Hex Symbol 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 50 51 52 53 54 55 56 57 58 59 5A 5B 5C 5D 5E 5F P Q R S T U V W X Y Z [ ] ^ _ 39 ASCII Table (Cont.) ASCII Hex Symbol 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 40 41 42 43 44 45 46 47 48 49 4A 4B 4C 4D 4E 4F @ A B C D E F G H I J K L M N O ASCII Hex Symbol 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 60 61 62 63 64 65 66 67 68 69 6A 6B 6C 6D 6E 6F ` a b c d e f g h i j k l m n o
  • 40. ASCII – Things to Note  ASCII codes for digits aren’t equal to numeric value  Uppercase & lowercase alphabetic codes differ by 0x20  Shift key clears bit 5  0x20 = 32 = 0010 0000  Example:  A = 65 = 0100 0001  a = 97 = 0110 0001  Most languages need more than 128 characters 40
  • 41. Unicode (www.unicode.org)  Designed to overcome limitation of number of characters  Assigns unique character codes to characters in a wide range of languages  A 16-bit character set  UCS-2  UCS-4 is 32-bit  65,536 (216) distinct Unicode characters Unicode provides a unique number for every character, no matter what the platform, no matter what the program, no matter what the language 41
  • 43. Unicode – Sinhala & Tamil 43