Number System
A set of values used to represent different quantities is known asNumber
System". For example, a number system can be used to represent the number of students in
a class or number of viewers watching a certain TV program etc. The digital
computer represents all kinds of data and information in binary numbers. It includes audio, graphics,
video, text and numbers. The total number of digits used in
a number
systemis called its base or radix. The base is written after the
number as subscript such as 51210.
Decimal number system
Decimal number system
·
Binary number system
·
Octal number system
·
Hexadecimal number
system
The decimal number system is used in general. However, the computers use binarynumber system. The octal and hexadecimal number systems are used in the computer.
Decimal number System
The Decimal Number System consists of ten digits from 0 to 9. These
digits can beused to represent any numeric value. The base of
decimal number system is 10. It is the most
widely used number system. The
value represented by individual digit depends on weight and position of the digit.
Each number in this
system consists of digits which are located at different positions. The
position of first digit towards left side of the decimal point is 0. The
position of second digit towards left side of the decimal point is 1.
Similarly, the position of first digit towards right side of decimal point is
-1. The position of second digit towards right side of decimal point is -2 and
so on.
The value of the
number is determined by multiplying the digits with the weight of their
position and adding the results. This method is known as expansion method. The
rightmost digit of number has the lowest weight. This digit is called Least
Significant Digit (LSD). The leftmost digit of a number has the highest weight.
This digit is called Most Significant Digit (MSD). The digit 7 in the number
724 is most significant digit and 4 is the least significant digit.
Example:
The weights and
positions of each digit of the number 453 are as follows:
|
Position
|
2
|
1
|
0
|
|
Weights
|
102
|
101
|
100
|
|
Face value
|
4
|
5
|
3
|
The above table
indicates that:
The value of digit
4
= 4x102
= 400
The value of digit
4
= 5x10
=
50
The value of digit
3
= 3x10
= 3
The actual number can
be found by adding the values obtained by the digits as follows:
400 + 50 +
3 =45310
Binary Number System
Digital
computer represents all kinds of data and information in the binary system.Binary Number System consists of two digits 0 and 1. Its
base is 2. Each digit or bit inbinary number
system can be 0 or 1. A
combination of binary numbers may be used to
represent different quantities like 1001. The positional value of each digit in binarynumber is twice the
place value or face
value of the digit of its right side. The weight of each position is a power of
2.
The place value of the digits according to
position and weight is as follows:
|
Position
|
3
|
2
|
1
|
0
|
|
Weights
|
23
|
22
|
21
|
20
|
Example:
Convert 101112 decimal number
|
Position
|
2
|
1
|
0
|
-1
|
-2
|
|
Weights
|
102
|
101
|
100
|
10-1
|
10-2
|
|
Face
Value
|
1
|
3
|
9
|
7
|
8
|
101112
= 1 x 24 + 0 x 23 + 1 x 22 + 1
x 21 + 1 x 20
= 1 x
16 + 0 + 1 x 4 + 1
x 2 + 1 x 1
= 16 + 0 + 4 2 + 1
= 2310
Octal Number System
Octal Number System consists of eight digits from 0 to 7.
The base of octal system is 8. Each digit position in this system represents a
power of 8. Any digit in this system is always less than 8. Octal number system is used as a shorthand representation of
longbinary numbers. The number
6418 is not valid in this number
system as 8 is not a valid digit.
The place value of each digit according to
position and weight is as follows.
|
Position
|
4
|
3
|
2
|
1
|
0
|
|
Weight
|
84
|
83
|
82
|
81
|
80
|
Example:
convert 458 to decimal number
458
= 4 x 81 + 5 x 80
= 4 x 8 + 5 x 1
= 32 + 5
=
3710
Hexadecimal
number system
The
Hexadecimal Number System consists of 16 digits from 0 to 9 and
A to F. The alphabets A to F represent decimal numbers from 10 to 15. The base
of this number system is 16. Each digit position in
hexadecimal system represents a power of 16. The number 76416 is valid
hexadecimal number. It is different from 76410 which is sevenhundred and sixty four. This number system provides shortcut method to represent
long binary numbers.
The place value of each digit according to
position and weight is as follows:
|
Position
|
4
|
3
|
2
|
1
|
0
|
|
Weights
|
164
|
163
|
162
|
161
|
160
|
Example:
Convert 3A16 to decimal number
3A16
= 3 x 161 + A x
160
= 3 x 16 +
10 x 1
= 48 + 10
= 5810
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