Transcript ece252.ece.wisc.edu

Introduction to Computer
Engineering
ECE/CS 252, Fall 2010
Prof. Mikko Lipasti
Department of Electrical and Computer Engineering
University of Wisconsin – Madison
Chapter 2
Bits, Data Types,
and Operations
- Part 1
Slides based on set prepared by
Gregory T. Byrd, North Carolina State University
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How do we represent data in a computer?
At the lowest level, a computer is an electronic machine.
• works by controlling the flow of electrons
Easy to recognize two conditions:
1. presence of a voltage – we’ll call this state “1”
2. absence of a voltage – we’ll call this state “0”
Could base state on value of voltage,
but control and detection circuits more complex.
• compare turning on a light switch to
measuring or regulating voltage
We’ll see examples of these circuits in the next chapter.
2-3
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Computer is a binary digital system.
Digital system:
• finite number of symbols
Binary (base two) system:
• has two states: 0 and 1
Basic unit of information is the binary digit, or bit.
Values with more than two states require multiple bits.
• A collection of two bits has four possible states:
00, 01, 10, 11
• A collection of three bits has eight possible states:
000, 001, 010, 011, 100, 101, 110, 111
• A collection of n bits has 2n possible states.
2-4
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What kinds of data do we need to represent?
• Numbers – signed, unsigned, integers, floating point,
complex, rational, irrational, …
• Text – characters, strings, …
• Images – pixels, colors, shapes, …
• Sound
• Logical – true, false
• Instructions
• …
Data type:
• representation and operations within the computer
We’ll start with numbers…
2-5
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Unsigned Integers
Non-positional notation
• could represent a number (“5”) with a string of ones (“11111”)
• problems?
Weighted positional notation
• like decimal numbers: “329”
• “3” is worth 300, because of its position, while “9” is only worth 9
329
102 101 100
3x100 + 2x10 + 9x1 = 329
most
significant
22
101
21
least
significant
20
1x4 + 0x2 + 1x1 = 5
2-6
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Unsigned Integers (cont.)
An n-bit unsigned integer represents 2n values:
from 0 to 2n-1.
22
21
20
0
0
0
0
0
0
1
1
0
1
0
2
0
1
1
3
1
0
0
4
1
0
1
5
1
1
0
6
1
1
1
7
2-7
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Adding Binary Numbers
Just like decimal arithmetic
Arithmetic tables:
1+1 = 2
1+2 = 3
…
9+9 = 18
If sum > 9 we have to carry
Binary table is much smaller
If sum > 1 we have to carry
A
B
C
A+B+C
(carry|sum)
0
0
0
00
0
1
0
01
1
0
0
01
1
1
0
10
0
0
1
01
0
1
1
10
1
0
1
10
1
1
1
11
2-8
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Unsigned Binary Arithmetic
Base-2 addition – just like base-10!
• add from right to left, propagating carry
carry
10010
+ 1001
11011
10010
+ 1011
11101
1111
+
1
10000
10111
+ 111
11110
Subtraction, multiplication, division,…
2-9
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Converting Binary to Decimal
Add powers of 2 that have “1” in the
corresponding bit positions.
n 2n
X = 01101000two
= 26+25+23 = 64+32+8
= 104ten
0
1
2
3
4
5
6
7
8
9
10
1
2
4
8
16
32
64
128
256
512
1024
2-10
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Converting Decimal to Binary
n 2n
First Method: Subtract Powers of Two
1. Subtract largest power of two
less than or equal to number.
2. Put a one in the corresponding bit position.
3. Keep subtracting until result is zero.
X = 104ten
104 - 64 = 40
40 - 32 = 8
8-8 = 0
0
1
2
3
4
5
6
7
8
9
10
1
2
4
8
16
32
64
128
256
512
1024
bit 6
bit 5
bit 3
X = 01101000two
2-11
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Converting Decimal to Binary
Second Method: Division
1. Divide by two – remainder is least significant bit.
2. Keep dividing by two until answer is zero,
writing remainders from right to left.
X = 104ten
X = 01101000two
104/2
52/2
26/2
13/2
6/2
3/2
=
=
=
=
=
=
52 r0
26 r0
13 r0
6 r1
3 r0
1 r1
1/2 = 0 r1
bit 0
bit 1
bit 2
bit 3
bit 4
bit 5
bit 6
2-12
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Operations: Arithmetic and Logical
Recall:
a data type includes representation and operations.
We have a good representation for unsigned integers,
and one operation: Addition
• Will look at other operations later
Logical operations are also useful:
• AND
• OR
• NOT
2-13
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Logical Operations
Operations on logical TRUE or FALSE
• two states -- takes one bit to represent: TRUE=1, FALSE=0
View n-bit number as a collection of n logical values
• operation applied to each bit independently
A
0
0
1
1
B
0
1
0
1
A AND B
0
0
0
1
A
0
0
1
1
B
0
1
0
1
A OR B
0
1
1
1
A
0
1
NOT A
1
0
2-14
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Examples of Logical Operations
AND
• useful for clearing bits
AND with zero = 0
AND with one = no change
11000101
AND 00001111
00000101
OR
• useful for setting bits
OR with zero = no change
OR with one = 1
NOT
• unary operation -- one argument
• flips every bit
OR
NOT
11000101
00001111
11001111
11000101
00111010
2-15
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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
2-16
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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.
2-17
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Text: ASCII Characters
ASCII: Maps 128 characters to 7-bit code.
• both printable and non-printable (BEL, 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
2-18
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Interesting Properties of ASCII Code
What is relationship between a decimal digit ('0', '1', …)
and its ASCII code?
What is the difference between an upper-case letter
('A', 'B', …) and its lower-case equivalent ('a', 'b', …)?
Given two ASCII characters, how do we tell which comes
first in alphabetical order?
Are 128 characters enough?
(http://www.unicode.org/)
No new operations -- integer arithmetic and logic.
2-19
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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
2-20
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Summary
Binary digital system
Data type: representation and operations
Unsigned integers
Weighted positional notation
Addition
Conversion from binary to decimal
Converstion from decimal to binary (2 methods)
Logical operations: AND/OR/NOT
Hexadecimal notation
ASCII representation for characters/text
Other data types
2-21