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Chapter 2
Bits, Data Types,
and Operations
Hexadecimal Notation
It is often convenient to write binary (base-2) numbers
as hexadecimal (base-16) numbers instead.
• fewer digits -- four bits per hex digit
• less error prone -- easy to corrupt long string of 1’s and 0’s
Binary
Hex
Decimal
Binary
Hex
Decimal
0000
0001
0010
0011
0100
0101
0110
0111
0
1
2
3
4
5
6
7
0
1
2
3
4
5
6
7
1000
1001
1010
1011
1100
1101
1110
1111
8
9
A
B
C
D
E
F
8
9
10
11
12
13
14
15
2-2
Converting from Binary to Hexadecimal
Every four bits is a hex digit.
• start grouping from right-hand side
011101010001111010011010111
3
A
8
F
4
D
7
This is not a new machine representation,
just a convenient way to write the number.
2-3
Converting from Hexadecimal to Binary
Hexadecimal to binary conversion:
Remember that hex is a 4-bit representation.
FA91hex or xFA91
F A 9 1
1111 1010 1001 0001
2DEhex or x2DE
2 D E
0010 1011 1100
2-4
Convert Hexadecimal to Decimal
Hexadecimal to decimal is performed the same as binary
to decimal, positional notation.
• Binary to decimal uses base 2
• Decimal is base 10
• Hexadecimal is base 16
3AF4hex = 3x163 + Ax162 + Fx161 + 4x160
= 3x163 + 10x162 + 15x161 + 4x160
= 3x4096 + 10x256 + 15x16 + 4x1
= 12,288 + 2,560 + 240 + 4
= 19,092ten
2-5
Fractions: Fixed-Point
How can we represent fractions?
• Use a “binary point” to separate positive
from negative powers of two -- just like “decimal point.”
• 2’s comp addition and subtraction still work.
if binary points are aligned
2-1 = 0.5
2-2 = 0.25
2-3 = 0.125
00101000.101 (40.625)
+ 11111110.110 (-1.25)
00100111.011 (39.375)
No new operations -- same as integer arithmetic.
2-6
Fractions: Fixed-Point
How is -6 5/8 represented in the floating point data type?
• Break problem into two parts
Whole: 6 = 1x22 + 1x21 + 0x20 => 110
Fraction: 5/8 = ½ (4/8) + 1/8 => 1x2-1 + 0x2-2 + 1x2-3 = .101
-6 5/8
ten
= - 110.101two
2-7
Very Large and Very Small: Floating-Point
Large values: 6.023 x 1023 -- requires 79 bits
Small values: 6.626 x 10-34 -- requires >110 bits
Use equivalent of “scientific notation”: F x 2E
Need to represent F (fraction), E (exponent), and sign.
IEEE 754 Floating-Point Standard (32-bits):
1b
8b
S Exponent
23b
Fraction
N  ( 1)S  1.fraction  2exponent 127 , 1  exponent  254
N  ( 1)S  0.fraction  2126 , exponent  0
2-8
Floating Point Example
Single-precision IEEE floating point number:
10111111010000000000000000000000
sign exponent
fraction
• Sign is 1 – number is negative.
• Exponent field is 01111110 = 126 (decimal).
• Fraction is 0.100000000000… = 0.5 (decimal).
Value = -1.5 x 2(126-127) = -1.5 x 2-1 = -0.75.
2-9
Floating Point Example
Single-precision IEEE floating point number:
00111111110010000000000000000000
sign exponent
fraction
• Sign is 0 – number is positive.
• Exponent field is 01111111 = 127 (decimal).
• Fraction is 0.100100000000… = 0.5625 (decimal).
Value = 1.5625 x 2(127-127) = 1.5625 x 20 = 1.5625.
2-10
Floating Point Example
Single-precision IEEE floating point number:
00000000011110000000000000000000
sign exponent
fraction
• Sign is 0 – number is positive.
• Exponent field is 00000000 = 0 (decimal) special case.
• Fraction is 0.111100000000… = 0.9375 (decimal).
Value = 0.9375 x 2(-126) = = 0.9375 x 2-126.
2-11
Text: ASCII Characters
ASCII: Maps 128 characters to 7-bit code.
• both printable and non-printable (ESC, DEL, …) characters
00
01
02
03
04
05
06
07
08
09
0a
0b
0c
0d
0e
0f
nul
soh
stx
etx
eot
enq
ack
bel
bs
ht
nl
vt
np
cr
so
si
10
11
12
13
14
15
16
17
18
19
1a
1b
1c
1d
1e
1f
dle
dc1
dc2
dc3
dc4
nak
syn
etb
can
em
sub
esc
fs
gs
rs
us
20
21
22
23
24
25
26
27
28
29
2a
2b
2c
2d
2e
2f
sp
!
"
#
$
%
&
'
(
)
*
+
,
.
/
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
:
;
<
=
>
?
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
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
[
\
]
^
_
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
70
71
72
73
74
75
76
77
78
79
7a
7b
7c
7d
7e
7f
p
q
r
s
t
u
v
w
x
y
z
{
|
}
~
del
2-12
Interesting Properties of ASCII Code
What is relationship between a decimal digit ('0', '1', …)
and its ASCII code? x30 -> ‘0’, x31 -> ’1’, … x39 -> ’9’
What is the difference between an upper-case letter
('A', 'B', …) and its lower-case equivalent ('a', 'b', …)?
Difference of x20
Given two ASCII characters, how do we tell which comes
first in alphabetical order? Compare ASCII values, the
lowest value is the first in alphabetical order
Are 128 characters enough?
(http://www.unicode.org/)
2-13
Other Data Types
Text strings
• sequence of characters, terminated with NULL (0)
• typically, no hardware support
Image
• array of pixels
monochrome: one bit (1/0 = black/white)
color: red, green, blue (RGB) components (e.g., 8 bits each)
other properties: transparency
• hardware support:
typically none, in general-purpose processors
MMX -- multiple 8-bit operations on 32-bit word
Sound
• sequence of fixed-point numbers
2-14
Another use for bits: Logic
Beyond numbers
• logical variables can be true or false, on or off, etc., and so are
readily represented by the binary system.
• A logical variable A can take the values false = 0 or true = 1 only.
• The manipulation of logical variables is known as Boolean
Algebra, and has its own set of operations - which are not to be
confused with the arithmetical operations of the previous
section.
• Some basic operations: NOT, AND, OR, XOR
2 - 15
LC-3 Data Types
Some data types are supported directly by the
instruction set architecture.
For LC-3, there is only one hardware-supported data type:
• 16-bit 2’s complement signed integer
• Operations: ADD, AND, NOT
Other data types are supported by interpreting
16-bit values as logical, text, fixed-point, etc.,
in the software that we write.
2-16