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Bits Bytes and Number Systems- Binary, Denary, Hexadecimal, Octal and ASCII Codes tutorial Back
Programming Tutorial
Updated 22 Feb 2004 Tutorials: HTML and Web Design | C programming | Smalltalk Programming
Contents Number Systems How do numbering systems work Denary Sometimes called Decimal Binary How to Convert binary to denary Hexadecimal Octal
Introduction Bit Byte Kilobyte Megabyte ASCII Codes This table is useful to compare values of the different numbering systems.
Binary System
Introduction This tutorial is going to assist you with the different numbering systems that are used in programming. Programmers will need to be familiar with most of them. You are already familiar with our every day denary numbering system based on the unit 10. Computers only understand the binary number system based on the unit 2 denary. Programming often uses the hexadecimal system base on a unit of 16 denary. For example HTML uses hexadecimal numbering in the color attribute, e.g. color="FF00FF". Another less common system is octal based on 8 denary. You should know how to convert between the different systems. The ASCII table at the foot of this page is useful for a quick conversions up to 127 denary. The windows calculator (in scientific mode) can be used for conversions. College and University students may find that calculators are not allowed in the exam, OU course T223 is an example of this, therefore manual conversion calculations must be made. This should be practiced to speed up the process, the calculator only used to check your manual conversion.
Bit A binary digit is called a bit. Usually expressed as 0 and 1 the two numbers of the binary numbering system. A bit is the smallest unit of information a computer can use. A 16 bit computer would process a series of 16 bits,such as 0100111101011000 in one go, repeating the process thousands or millions of times per second. Reading a series of bits is very difficult and to make this process easier they are often displayed in groups of 4 bits 0100 1111 0101 1000 This grouping is quite interesting in that a group of 4 bits can be replaced by a single hexadecimal digit Two groups of 4 bits, i.e. 8 bits ( a byte) can be replaced by 2 hexadecimal digits, and 4 hexadecimal digits are required to replace all 16 bits. Binary
0100
1111
0101
1000
4
F
5
8
Hexadecimal
Byte A group of 8 bits are in a byte. With 8 bits ( binary digits ), there exists 256 possible denary combinations. If you remember that 1 byte can store one alphabetical letter, single digit, or a single character/symbol, such as #. Large numbers of bytes can be expressed by kilobytes, megabytes etc 1 byte of memory can normally hold one of the following: a single alphabetical letter (upper or lower case), a single number 0-9 a symbol ( _ + £ # > etc a further 127 alternative characters. These could be the letters used in foreign languages, lines to produce boxes etc. See the ASCII codes at the foot of this page which shows how the first 127 characters of 256 characters are used
Kilobyte The value of a kilobyte is 1024. Worked out as 2^ 10. Normally Kilo refers to 1000 but in computing kilobyte is 1024.
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Megabyte (MB) Likewise, 1024Kb is referred to as a "Megabyte". Normally a Mega refers to a million. In computing 1 Mega byte is 1,048,576 bytes. Worked out as 220, or 1024*1024. A megabyte can store roughly 4 books of 200 pages
Gigabyte (GB) A Gigabyte is 1,073,741,824 (230) bytes. 1,024 Megabytes, or 1,048,576 Kilobytes. A gigabyte can store almost 4500 books of 200 pages You could store 1 but not 2 650MB CDs
Terabyte (TB) A Terabyte is 1,099,511,627,776 (240) bytes, 1,024 Gigabytes, or 1,048,576 Megabytes.
Petabyte (PB) 250 bytes
Exabyte (EB) 260 bytes bytes
Number Systems The every day number system we use is the denary system, sometimes called the decimal system. In programming three number systems are commonly used, binary, hexadecimal and to a lesser extent octal . In every day life we Usually use the denary number system which has a base of 10. But we also use other number systems, think of time (base 60, and base 10 within it), imperial distance yards and feet (base 3), there are many others, dates probably being the hardest number system to do calculations with. In the denary binary octal hexadecimal systems, the value of any digit in a number depends on its position within that number. I.e. which column it is in.
How do numbering systems work To understand this we will examine the Denary system in more detail. Because you are so used to the denary system and because it is very easy to multiply by 10, 100, or a 1000 etc you calculate the number in your head. Lets use the number 256 as an example. The calculation that is automatically done is the following The most important calculation to do is to work out the positional values for that system. The positional value is based on the powers of the number systems base value 104
103
102
10x10x10x10
10x10x10
10x10
10,000
1000
100
10
10,000
1000
100
10
1
10,000
1000
100
10
1
2
5
6
100
10
1
2
5
6
Power of the base Calculation = Position value
101
100
The value of the base number Any number to the power 0 is always 1 1
Write down the Positional values for the number system you are using so for Denary we would write Position value Underneath the correct positional value write in your number Position value Enter Number The calculation that is done Position value
10,000
1000
Enter Number Required Calculation
100x2
10x5
1x6
This equals
200
50
6
Add the 3 results 200 + 50 + 6 200 50 6 ____ 256 You can convert a number in any number system to a denary number using this calculation. Ensure you use the positional values for the number system you are using
Denary Numbering (base 10) sometimes called Decimal system The decimal system name should not be used because of confusion that this could be thought as introducing the decimal point and money systems Uses numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, that's 10 numbers. Hence base 10. Radix is another name for base 10. Adding 1 to 9 we must introduce an additional column to the left i.e. 10 104
103
102
101
100
10,000
1000
100
10
1
Power of the base Position value A byte of memory can store a number in the range 0 to 255 Denary.
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Binary Numbering (base 2) A series of eight bits strung together makes a byte, much as 12 makes a dozen. With 8 bits, or 8 binary digits, there exist 2^8=256 possible combinations Uses numbers 0, 1,, that's 2 numbers. Hence base 2. Binary numbering is the number system that is used by computers. Note: The positional value doubles as you go to the next positional on the left Power of the base
24
23
22
21
20
Positional value
16
8
4
2
1
A binary digit is called a bit. There are two possible states in a bit, Usually expressed as 0 and 1, the two numbers used in the binary number system. But the bit could represent on / off of an electrical circuit, yes / no, true / false, -1 / 0, -1 / +1, zero / non zero, or similar 2 state binary wording. A byte of memory can store a number in the range 00000000 to 11111111 binary. Numbers are often displayed in groups of 4, as follows, to make them easier to read. 0000 0000 to 1111 1111 binary.
How to Convert Binary to Denary Convert 1011 binary to denary Positional value
16
8
4
2
1
Enter Number
1
0
1
1
Required Calculation
8x1
4x0
2x1
1x1
This equals
8
0
2
1
Add the 3 results 8 + 0 + 2 + 1 8 0 2 1 ____ 11 Therefore 1011(One Zero One One) binary = 11 (Eleven) Denary
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Hexadecimal Numbering (base 16) Uses numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, that's 16 numbers. Hence base 16.. A=10 Dec, B=11 DEC, C=12 DEC, D=13 DEC, E=14 DEC, F=15 Dec. By using the letters A-F a single digit only requires a single position (column). Power of the base Positional value
164
163
162
161
160
65536
4,096
256
16
1
A byte of memory can store a number in the range 00 to FF Hex Hexadecimal numbers are often displayed in groups of 2, to make them easier to read. E.g. 10 AF 3C 9F A single hexadecimal number requires 4 units of binary numbers. This makes it reasonably easy to convert between these two numbering systems. E.g. 1 = 0001 9 = 0101 A = 1010
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Octal (base 8) Uses numbers 0, 1, 2, 3, 4, 5, 6, 7, , that's 8 numbers. Hence base 8.. A byte of memory can store a number in the range 0 to 377 Oct. 84
83
82
81
80
4096
512
64
8
1
Power of the base Positional value
Binary Systems The Binary system is used by digital devices such as computers. Consists of two possible states such as On - Off, Yes - No, True - False, Zero - Non Zero and possibly more confusing in programming -1 and 0. These states are usually represented by either the number 0 or 1 of the binary number system. Back Top
ASCII Codes ASCII DEC Hex Octal
Binary
ASCII Dec Hex Octal
Binary
ASCII Dec Hex Octal
Binary
NULL
000
00
000
0000 0000
+
043
2B
053
0010 1011
V
086
56
126
0101 0110
SOH
001
01
001
0000 0001
,
044
2C
054
0010 1100
W
087
57
127
0101 0111
STX
002
02
002
0000 0010
-
045
2D
055
0010 1101
X
088
58
130
0101 1000
ETX
003
03
003
0000 0011
.
046
2E
056
0010 1110
Y
089
59
131
0101 1001
EOT
004
04
004
0000 0100
/
047
2F
057
0010 1111
Z
090
5A
132
0101 1010
ENQ
005
05
005
0000 0101
0
048
30
060
0011 0000
[
091
5B
133
0101 1011
ACK
006
06
006
0000 0110
1
049
31
061
0011 0001
\
092
5C
134
0101 1100
BEL
007
07
007
0000 0111
2
050
32
062
0011 0010
]
093
5D
135
0101 1101
BS
008
08
010
0000 1000
3
051
33
063
0011 0011
^
094
5E
136
0101 1110
HT
009
09
011
0000 1001
4
052
34
064
0011 0100
_
095
5F
137
0101 1111
LF
010
0A
012
0000 1010
5
053
35
065
0011 0101
`
096
60
140
0110 0000
VT
011
0B
013
0000 1011
6
054
36
066
0011 0110
a
097
61
141
0110 0001
FF
012
0C
014
0000 1100
7
055
37
067
0011 0111
b
098
62
142
0110 0010
CR
013
0D
015
0000 1101
8
056
38
070
0011 1000
c
099
63
143
0110 0011
SO
014
0E
016
0000 1110
9
057
39
071
0011 1001
d
100
64
144
0110 0100
SI
015
0F
017
0000 1111
:
058
3A
072
0011 1010
e
101
65
145
0110 0101
DLE
016
10
020
0001 0000
;
059
3B
073
0011 1011
f
102
66
146
0110 0110
DC1
017
11
021
0001 0001
a
060
3C
074
0011 1100
g
103
67
147
0110 0111
DC2
018
12
022
0001 0010
=
061
3D
075
0011 1101
h
104
68
150
0110 1000
DC3
019
13
023
0001 0011
>
062
3E
076
0011 1110
i
105
69
151
0110 1001
DC4
020
14
024
0001 0100
?
063
3F
077
0011 1111
j
106
6A
152
0110 1010
NAK
021
15
025
0001 0101
@
064
40
100
0100 0000
k
107
6B
153
0110 1011
SYN
022
16
026
0001 0110
A
065
41
101
010 00001
l
108
6C
154
0110 1100
ETB
023
17
027
0001 0111
B
066
42
102
0100 0010
m
109
6D
155
0110 1101
CAN
024
18
030
0001 1000
C
067
43
103
0100 0011
n
110
6E
156
0110 1110
EM
025
19
031
0001 1001
D
068
44
104
0100 0100
o
111
6F
157
0110 1111
SUB
026
1A
032
0001 1010
E
069
45
105
0100 0101
p
112
70
160
0111 0000
ESC
027
1B
033
0001 1011
F
070
46
106
0100 0110
q
113
71
161
0111 0001
FS
028
1C
034
0001 1100
G
071
47
107
0100 0111
r
114
72
162
0111 0010
GS
029
1D
035
0001 1101
H
072
48
110
0100 1000
s
115
73
163
0111 0011
RS
030
1E
036
0001 1110
I
073
49
111
0100 1001
t
116
74
164
0111 0100
US
031
1F
037
0001 1111
J
074
4A
112
0100 1010
u
117
75
165
0111 0101
space
032
20
040
0010 0000
K
075
4B
113
0100 1011
v
118
76
166
0111 0110
!
033
21
041
0010 0001
L
076
4C
114
0100 1100
w
119
77
167
0111 0111
"
034
22
042
0010 0010
M
077
4D
115
0100 1101
x
120
78
170
0111 1000
#
035
23
043
0010 0011
N
078
4E
116
0100 1110
y
121
79
171
0111 1001
$
036
24
044
0010 0100
O
079
4F
117
0100 1111
z
122
7A
172
0111 1010
%
037
25
045
0010 0101
P
080
50
120
010 10000
{
123
7B
173
0111 1011
&
038
26
046
0010 0110
Q
081
51
121
0101 0001
|
124
7C
174
0111 1100
'
039
27
047
0010 0111
R
082
52
122
0101 0010
}
125
7D
175
0111 1101
(
040
28
050
0010 1000
S
083
53
123
0101 0011
~
126
7E
176
0111 1110
)
041
29
051
0010 1001
T
084
54
124
0101 0100
DEL
127
7F
177
0111 1111
*
042
2A
052
0010 1010
U
085
55
125
0101 0101
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