Number System
The technique
to represent and work with numbers is called number system. Decimal
number system is the most common number system. Other popular number
systems include binary number system, octal number system, hexadecimal
number system, etc.
Decimal Number System
Decimal
number system is a base 10 number system having 10 digits from 0 to
9. This means that any numerical quantity can be represented using these 10
digits. Decimal number system is also a positional value system. This
means that the value of digits will depend on its position. Let us take an
example to understand this.
Say we
have three numbers – 734, 971 and 207. The value of 7 in all three numbers is
different−
- In 734, value
of 7 is 7 hundreds or 700 or 7 × 100 or 7 × 102
- In 971, value
of 7 is 7 tens or 70 or 7 × 10 or 7 × 101
- In 207, value
0f 7 is 7 units or 7 or 7 × 1 or 7 × 100
The
weightage of each position can be represented as follows −
In
digital systems, instructions are given through electric signals; variation is
done by varying the voltage of the signal. Having 10 different voltages to
implement decimal number system in digital equipment is difficult. So, many
number systems that are easier to implement digitally have been developed.
Let’s look at them in detail.
Binary Number System
The
easiest way to vary instructions through electric signals is two-state system –
on and off. On is represented as 1 and off as 0, though 0 is not actually no
signal but signal at a lower voltage. The number system having just these two
digits – 0 and 1 – is called binary number system.
Each
binary digit is also called a bit. Binary number system is also positional
value system, where each digit has a value expressed in powers of 2, as
displayed here.
In any
binary number, the rightmost digit is called least significant bit (LSB) and
leftmost digit is called most significant bit (MSB).
And
decimal equivalent of this number is sum of product of each digit with its
positional value.
110102 =
1×24 + 1×23 + 0×22 + 1×21 +
0×20
= 16 +
8 + 0 + 2 + 0
= 2610
Computer
memory is measured in terms of how many bits it can store. Here is a chart for
memory capacity conversion.
- 1 byte (B) = 8
bits
- 1 Kilobytes
(KB) = 1024 bytes
- 1 Megabyte (MB)
= 1024 KB
- 1 Gigabyte (GB)
= 1024 MB
- 1 Terabyte (TB)
= 1024 GB
- 1 Exabyte (EB)
= 1024 PB
- 1 Zettabyte =
1024 EB
- 1 Yottabyte
(YB) = 1024 ZB
Octal Number System
Octal number system has eight digits – 0, 1, 2, 3, 4, 5, 6 and 7.
Octal number system is also a positional value system with where each digit has
its value expressed in powers of 8, as shown here −
Decimal
equivalent of any octal number is sum of product of each digit with its
positional value.
7268 =
7×82 + 2×81 + 6×80
= 448
+ 16 + 6
= 47010
Hexadecimal Number System
Octal number system has 16 symbols – 0 to 9 and A to F where A is
equal to 10, B is equal to 11 and so on till F. Hexadecimal number system is
also a positional value system with where each digit has its value expressed in
powers of 16, as shown here −
Decimal
equivalent of any hexadecimal number is sum of product of each digit with its
positional value.
27FB16 =
2×163 + 7×162 + 15×161 + 10×160
= 8192
+ 1792 + 240 +10
=
1023410
Number System Relationship
The
following table depicts the relationship between decimal, binary, octal and
hexadecimal number systems.
|
HEXADECIMAL
|
DECIMAL
|
OCTAL
|
BINARY
|
|
0
|
0
|
0
|
0000
|
|
1
|
1
|
1
|
0001
|
|
2
|
2
|
2
|
0010
|
|
3
|
3
|
3
|
0011
|
|
4
|
4
|
4
|
0100
|
|
5
|
5
|
5
|
0101
|
|
6
|
6
|
6
|
0110
|
|
7
|
7
|
7
|
0111
|
|
8
|
8
|
10
|
1000
|
|
9
|
9
|
11
|
1001
|
|
A
|
10
|
12
|
1010
|
|
B
|
11
|
13
|
1011
|
|
C
|
12
|
14
|
1100
|
|
D
|
13
|
15
|
1101
|
|
E
|
14
|
16
|
1110
|
|
F
|
15
|
17
|
1111
|
ASCII
Besides
numerical data, computer must be able to handle alphabets, punctuation marks,
mathematical operators, special symbols, etc. that form the complete character
set of English language. The complete set of characters or symbols are called
alphanumeric codes. The complete alphanumeric code typically includes −
- 26 upper case
letters
- 26 lower case
letters
- 10 digits
- 7 punctuation
marks
- 20 to 40
special characters
Now a
computer understands only numeric values, whatever the number system used. So
all characters must have a numeric equivalent called the alphanumeric code. The
most widely used alphanumeric code is American Standard Code for Information
Interchange (ASCII). ASCII is a 7-bit code that has 128 (27) possible codes.
ISCII
ISCII
stands for Indian Script Code for Information Interchange. IISCII was
developed to support Indian languages on computer. Language supported by IISCI
include Devanagari, Tamil, Bangla, Gujarati, Gurmukhi, Tamil, Telugu, etc.
IISCI is mostly used by government departments and before it could catch on, a
new universal encoding standard called Unicode was introduced.
Unicode
Unicode
is an international coding system designed to be used with different language
scripts. Each character or symbol is assigned a unique numeric value, largely
within the framework of ASCII. Earlier, each script had its own encoding
system, which could conflict with each other.
In
contrast, this is what Unicode officially aims to do − Unicode provides
a unique number for every character, no matter what the platform, no matter
what the program, no matter what the language.
