Computer representation of numbers & characters

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

5. Data Representation (Cont.)
5
Data
Quantitative
Integers
Signed
Unsigned
Non-
integers
Signed
Unsigned
Qualitative

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

42. Unicode (Cont.)
42
Source: www3.ntu.edu.sg/home/ehchua/programming/java/DataRepresentation.html

43. Unicode – Sinhala & Tamil
43