Base Number Systems
Each number system employs a
number of different digits which is called the base of the number
system.
We learn to count at school but most people take it for granted
why we have only the following numbers:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Using the numbers above, we can create any number that we care to. If
we count, there are 10 digits in the number sequence above. We call
this the base
or radix
of the number system.
Some Commonly used
number systems
Decimal
Base = 10
Binary
Base = 2
Octal
Base = 8
Hexadecimal
(Hex) Base
= 16
Decimal Number
System
Base 10
Digits 0, 1, 2, 3, 4, 5, 6,
7, 8, 9
e.g. 747510
The magnitude represented by a digit is decided by the position of the
digit within the number.
For example the digit 7 in the left-most position of 7475 counts for
7000 and the digit 7 in the second position from the right counts for
70.
Binary Number System
Base 2
Digits 0, 1
e.g. 11102
The digit 1 in the third position from the right represents the value 4
and the digit 1 in the fourth position from the right represents the
value 8.
Octal Number System
Base 8
Digits 0, 1, 2, 3, 4, 5, 6, 7
e.g. 16238
The digit 2 in the second position from the right represents the value
16 and the digit 1 in the fourth position from the right represents the
value 512.
Hexadecimal Number
System
Base 16
Digits 0, 1, 2, 3, 4, 5, 6, 7, 8,
9, A, B, C, D, E, F
e.g. 2F4D
16
The digit F in the third position from the right represents the value
3840 and the digit D in the first position from the right represents
the value 1.
Binary Codes
A binary code is a group of n bits that
assume up to 2n
distinct
combinations of 1’s and 0’s with each combination representing one
element of the set that is being coded.
BCD – Binary Coded Decimal
ASCII – American Standard Code for Information
Interchange
BCD – Binary Coded
Decimal
When the decimal numbers are represented in BCD, each decimal digit is
represented by the equivalent BCD code.
Example. BCD Representation of Decimal 6349
6
3
4 9
0110 0011
0100 1001
Decimal BCD
Number Number
0
0000
1
0001
2
0010
3
0011
4
0100
5
0101
6
0110
7
0111
8
1000
9
1001
ASCII
Number ASCII
0 0110000
1 0110001
2 0110010
3 0110011
4 0110100
5 0110101
6 0110110
7 0110111
8 0111000
9 0111001
Letter ASCII
A 1000001
B 1000010
C 1000011
D 1000100
E 1000101
F 1000110
G 1000111
H 1001000
I 1001001
J 1001010
K 1001011
L 1001100
M 1001101
N 1001110
O 1001111
P 1010000
Q 1010001
R 1010010
S 1010011
T 1010100
U 1010101
V 1010110
W 1010111
X 1011000
Y 1011001
Z 1011010
Hexadecimal to Decimal
Hexadecimal number B4F
Convert B4F to Decimal
BF4= 11 x 162 + 4 x 161 + 15 x 160
=11 x 256
+ 4 x 16 + 15
= 2, 816 + 64 +
15=
289510
Decimal to Hexadecimal
Convert decimal number 20385 to hexadecimal.
16 ) 20385 remainder =
1
1’s place
16 ) 1274 remainder =
10 16 ’s
place
16 ) 79 remainder =
15 162 ’s
place
16 ) 4 remainder
= 4 163
’s place
2038510
=4FA116
DECIMAL HEXADECIMAL
0
0
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
A
11
B
12
C
13
D
14
E
15
F