Transcript Binary Numbers

Binary Numbers
• Numbering Systems
• Counting
• Symbolic
• Bases
• Common Bases (10, 2, 8, 16)
• Representing Information
• Binary to Decimal Conversions
• Bit Groupings
• Encoding Information
Craig Schock, 2003
Numbering Systems
• There are many different ways to represent numbers
• Counting Based Systems
• The number is represented by the same
number of counters
• Symbolic
• Uses symbols to represent values
7=
Craig Schock, 2003
Symbolic Numbering Systems
• The symbols within the numbering systems are an
“abstraction” of the actual number.
• The symbol represents a value
• Abstraction: A symbolic representation of a thing or
idea.
• In order for an abstraction to be useful, people
must agree on its meaning.
• Symbols are combined to represent larger
numbers
Craig Schock, 2003
Numbering Bases
• Each symbolic numbering system has a “base”
• Base: The number of symbols in the system
• eg. Decimal has 10 symbols (0-9)
0
1
2
3
4
Craig Schock, 2003
5
6
7
8
9
Common Bases
• The most common base used in western society is
base 10 (decimal)
• It is based on 10 symbols
• Humans have 10 “digits”
• In the computer world, there are other commonly
used bases:
• Binary (base 2, 0-1)
• Octal (base 8, 0-7)
• Hexadecimal (base 16, 0-9, A-F)
Craig Schock, 2003
Decimal: Representing Numbers
• Because decimal has 10 digits, it can easily
represent numbers 0-9.
0
1
2
3
4
5
6
7
8
9
However, representing numbers which are larger
than the number of symbols poses a problem.
Craig Schock, 2003
Decimal: Positional Notation
• Symbolic systems deal with representing large
numbers through a positional notation.
• The actual value of the symbol is based on its
position
• Multiplication and addition is employed to
increase the value
• The factor used in the multiplication is the base
Note: base 10
First column starts at “1”
x10x10
x10
x1
7 5 6
10
2
10
1
10
=
0
Craig Schock, 2003
7 x 10 x 10 +
5 x 10 +
6x1
Information
• Information: What is it?
A message received and understood that
reduces the recipient’s uncertainty
• “Uncertainty” implies that for information to be
present, there must be at least 2 possible outcomes.
• If there was only 1 outcome, there would be no
uncertainty.
• If we were to represent information as a
numbering system, it would be a system which
contained at least 2 symbols.
Craig Schock, 2003
Representing Information
• We could choose any numbering system to
represent information.
• Whatever system we choose, it must contain at
least 2 symbols.
• Electrically, it is easy to represent a system
containing 2 symbols using switches.
• Off
• On
• We can also think of philosophical systems based
on “truthness” and “falseness”
Craig Schock, 2003
Binary – Base 2
• Binary is a numbering system which contains 2
symbols (i.e. base 2)
• For easy interoperability with other bases (such as
decimal), we choose the digits “0” and “1” to
represent the two possible values
• A “bit” is a single BInary digiT
• A bit is the smallest unit of information
• What if the information we need to represent has
more than 2 possibilities?
Craig Schock, 2003
Binary – Positional Notation
• As with decimal, we can represent values which
are larger than the number of symbols we have.
• This is accomplished through the use of a
positional notation. Because the base is 2, the
multiplier for each position is x2
Note: base 2
x2x2
x2
x1
1 1 1
2
2
2
1
2
=
0
Craig Schock, 2003
1x2x2+
1x2+
1x1
Binary to Decimal Conversions
• Because humans work well with decimal, it is
useful to know how to convert between binary and
decimal:
0000 = 0
0001 = 1
0010 = 2
0011 = 3
0100 = 4
0101 = 5
0110 = 6
0111 = 7
Craig Schock, 2003
1000 = 8
1001 = 9
1010 = 10
1011 = 11
1100 = 12
1101 = 13
1110 = 14
1111 = 15
Binary to Decimal Conversions (contd)
8
4
1 1
2
1
0 1
=
1x8 + 1x4 + 1x1 =
8+4+1=
13
To convert to decimal, represent the column values in decimal
128
64
1 0
32
16
8
4
0 1 1 1
2
1
0 1
1x128 + 1x16 + 1x8 + 1x4 + 1x1 =
128 + 16 + 8 + 4 + 1 =
157
Craig Schock, 2003
=
How many bits to use?
• In the last few slides, we have seen binary
numbers which contain differing numbers of digits –
3 bits, 4 bits, and 8 bits
• How many bits should one use?
• That depends on what information one wishes
to represent.
• eg. How many bits are necessary to represent
the days of the week?
Craig Schock, 2003
How Many bits?
• Days of the week. Let’s let each bit combination
represent one day:
000 = Sunday
001 = Monday
010 = Tuesday
011 = Wednesday
100 = Thursday
101 = Friday
110 = Saturday
• Because there are 7 days of the week, we need
enough bits which have at least 7 different
combinations. (3 bits) Craig Schock, 2003
Bits and Combinations
• How can we tell how many combinations a given
number of bits will provide?
• The answer is based on a simple formula:
n
n bits provides 2 combinations
1 bit = 2 combinations
2 bits = 4 combinations
3 bits = 8 combinations
4 bits = 16 combinations
5 bits = 32 combinations
6 bits = 64 combinations
7 bits = 128 combinations
8 bits = 256 combinations
9 bits = 512 combinations
10 bits = 1024 combinations
11 bits = 2048 combinations
12 bits = 4096 combinations
13 bits = 8192 combinations
14 bits = 16384 combinations
15 bits = 32768 combinations
16 bits = 65536 combinations
Craig Schock, 2003
More combinations
• How many bits are necessary to represent
• The days of the month?
• Your age
• The year of your birth?
• The number of cars you have owned?
• Your salary?
• The salary of a CEO of a multinational
corporation?
• The Canadian national debt?
• The US national debt?
• The number of atoms in the universe?
Craig Schock, 2003
Bit groupings
• Relatively speaking, there are few cases where the
information content only requires 1 bit.
• Bits are usually grouped into larger units.
• Common groupings include:
4 bits = 1 nybble (not commonly used)
8 bits = 1 byte
1024 bytes = 1 kilobyte (or 1K)
1024 * 1024 bits = 1048576 bits = 1 megabits = 131072 bytes
1024 * 1024 bytes = 1048576 bytes = 1 Megabyte
1024 * 1024 * 1024 bytes = 1073741824 bytes = 1 Gigabyte
Craig Schock, 2003
What does it mean?
• Exercise: What do the following numbers mean?
102, 102, 102, 126, 126, 102, 102, 102
0, 24, 24, 0, 24, 24, 24, 24
24, 24, 24, 24, 24, 0, 24, 24
Craig Schock, 2003
What does it mean?
102 = 01100110
102 = 01100110
102 = 01100110
126 = 01111110
126 = 01111110
102 = 01100110
102 = 01100110
102 = 01100110
0 = 00000000
24 = 00011000
24 = 00011000
0 = 00000000
24 = 00011000
24 = 00011000
24 = 00011000
24 = 00011000
Craig Schock, 2003
24 = 00011000
24 = 00011000
24 = 00011000
24 = 00011000
24 = 00011000
0 = 00000000
24 = 00011000
24 = 00011000
What does it mean?
11 11
11 11
11 11
111111
111111
11 11
11 11
11 11
11
11
11
11
11
11
Craig Schock, 2003
11
11
11
11
11
11
11
Information Encoding
• The exercise illustrates and important concept in
computer science.
• We have learned that binary numbers can be used
to represent information. Information, as a series of
bits, can be represented as decimal numbers. The
example provided a series of decimal numbers. But
what do they mean?
• What the numbers mean is a matter of definition
• Information is encoded using bits
• Encoding is an abstraction.
Craig Schock, 2003
ASCII Codes
• ASCII – American Standard Code for Information
Interchange
• ASCII is an 7 bit encoding which is used to
represent the Roman alphabet, numbers, standard
symbols and printer control characters.
• Some companies expanded ASCII to 8 bits. They
used the extra combinations to encode special
characters
Craig Schock, 2003
The ASCII Standard
Dec Char
0 NUL
1 SOH
2 STX
3 ETX
4 EOT
5 ENQ
6 ACK
7 BEL
8 BS
9 HT
10 LF
11 VT
12 FF
13 CR
14 SO
15 SI
16 DLE
17 DC1
18 DC2
19 DC3
20 DC4
21 NAK
22 SYN
Dec Char
23 ETB
24 CAN
25 EM
26 SUB
27 ESC
28 FS
29 GS
30 RS
31 US
32 SPACE
33 !
34 "
35 #
36 $
37 %
38 &
39 '
40 (
41 )
42 *
43 +
44 ,
45 -
Dec Char
46 .
47 /
48 0
49 1
50 2
51 3
52 4
53 5
54 6
55 7
56 8
57 9
58 :
59 ;
60 <
61 =
62 >
63 ?
64 @
65 A
66 B
67 C
68 D
Dec Char
69 E
70 F
71 G
72 H
73 I
74 J
75 K
76 L
77 M
78 N
79 O
80 P
81 Q
82 R
83 S
84 T
85 U
86 V
87 W
88 X
89 Y
90 Z
91 [
Craig Schock, 2003
Dec Char
92 \
93 ]
94 ^
95 _
96 `
97 a
98 b
99 c
100 d
101 e
102 f
103 g
104 h
105 i
106 j
107 k
108 l
109 m
110 n
111 o
112 p
113 q
114 r
Dec Char
115 s
116 t
117 u
118 v
119 w
120 x
121 y
122 z
123 {
124 |
125 }
126 ~
127 DEL
Using ASCII
• Using the ASCII standard, we can encode an
English sentence as a series of ASCII values
I would like to buy some cheese.
73, 32, 119, 111, 117, 108, 100, 32, 108, 105, 107,
101, 32, 116, 111, 32, 98, 117, 121, 32, 115, 111,
109, 101, 32, 99, 104, 101, 101, 115, 101, 46
• How many bytes are needed to encode the above
sentence using ASCII?
32 byes
Craig Schock, 2003