What is a Number System?
A number
system can be defined as a systematic way of representing numerical values and
quantities. It consists of a set of various symbols and rules that provide a
way to represent numerical values in a meaningful sense. Also, the number
systems allow us to perform various operations on these numerical values.
A number
system has a fixed base or radix assigned to it. The base of a number system
depends on the number of symbols used in it to represent different numerals
uniquely. For example, a decimal number system has a base 10, therefore, it is
also known as a base 10 number system. This is because the decimal number system
has 10 unique symbols to represent each digit of the system.
Components of a Number System
A number
system has the following four major components:
(1). Base
or Radix – The base or radix of a number system is the factor that determines
the number of unique symbols or digits used in the number system. It also
represents the value at which the positional weight of a digit within a number
changes. For example, the base of the decimal number system is 10, the base of
the binary number system is 2, etc.
(2). Digit
– In a number system, the digit is simply a symbol that represents a numeral. A
number system has a number of unique digits depending on the base. For example,
the decimal number system has ten symbols or digits, i.e. 0, 1, 2, 3, 4, 5, 6,
7, 8, and 9. Here, each symbol represents a numeral of the decimal number
system.
(3). Positional
Weight – The positional weight, also called place weight, is the value
associated with a particular position in a number. It is specified by the power
of base or radix. For example, in the decimal number system, the positional weight
is specified as the power of 10 like 10^{2} represents the hundreds
place or positional weight in a number.
(4). Rules
– A number system has a set of rules to perform various operations like
addition, subtraction, multiplication, conversion, and more. There are
different rules associated with different types of number systems.
Types of Number Systems
The
following four number systems are most commonly used in digital electronics and
computer engineering:
- Decimal Number System
- Binary Number System
- Octal Number System
- Hexadecimal Number System
Let us now
discuss each type of number system with examples.
(1). Decimal Number System:
The decimal
number system is a type of number system that contains 10 unique symbols or
digits. Thus, the decimal number system has a base or radix equal to 10. The
symbols used in the decimal number system to represent its numerals are 0, 1,
2, 3, 4, 5, 6, 7, 8, and 9.
In the
decimal number system, the position of a digit in a number is specified by the
power of 10.
For
example, in a decimal number ‘759’, the position of different digits is as
follows:
9 = `10^0` = 1, i.e. 9 is at the 1’s place.
5 = `10^1` = 10, i.e. 5 is at the 10’s place.
7 = `10^2` = 100, i.e. 7 is at the 100’s place.
In the
decimal number system, the rightmost digit of a number that has the least
positional weight is called the LSD (Least Significant Digit). The leftmost digit of a decimal number that has the greatest positional weight among
all the digits is called the MSD (Most Significant Digit).
(2). Binary Number System:
A binary number system has only two unique symbols or digits to represent numerals.
Therefore, the base of the binary number system is 2. The symbols used in the binary
number system are 0 and 1. Each of these binary digits, i.e. a 0 or a 1 is
called a bit. Thus, a binary number
is nothing but a sequence of multiple bits.
Another important
fact to note about the binary number system is that it is a weighted number
system which means the value of a bit in a number depends on its position within
the number. Where, the weight of each position is given by `2^n`, where
n is an integer.
In a binary
number, the rightmost bit is termed LSB
(Least Significant Bit), and the leftmost bit is called MSB (Most Significant Bit).
The binary
number system is widely used in digital electronic systems like digital
computers, data transmission, digital communication systems, internet
technology, and more.
(3). Octal Number System:
The octal
number system is a base-8 number system that has 8 unique symbols to represent
its numerals. These symbols are 0, 1, 2, 3, 4, 5, 6, and 7. Similar to decimal
and binary number systems, the octal number system is also a positional
weighted number system. The octal number system was very popular in early
minicomputers.
(4). Hexadecimal Number System:
The
hexadecimal number system is another commonly used type of number system. It is
a number system having 16 unique symbols to represent different numerals of the
system. Thus, it has a base or radix equal to 16. The symbols of the
hexadecimal number system are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A = 10, B = 11, C =
12, D = 13, E = 14, and F = 15.
The hexadecimal
number system was primarily developed with the intent to represent long
binary numbers in a short form. The hexadecimal number system is also known as the hex number system. Each hexadecimal digit
can be represented as a group of 4 bits (`2^4` = 16), called a nibble.
The hexadecimal number system is widely used in various types of digital computers,
keyboards, terminal machines, etc.
Table of Number Systems
The following
table shows the relationship among different number systems:
Decimal Number System |
Binary Number System |
Octal Number System |
Hexadecimal Number System |
0 |
0 |
0 |
0 |
1 |
1 |
1 |
1 |
2 |
10 |
2 |
2 |
3 |
11 |
3 |
3 |
4 |
100 |
4 |
4 |
5 |
101 |
5 |
5 |
6 |
110 |
6 |
6 |
7 |
111 |
7 |
7 |
8 |
1000 |
10 |
8 |
9 |
1001 |
11 |
9 |
10 |
1010 |
12 |
A |
11 |
1011 |
13 |
B |
12 |
1100 |
14 |
C |
13 |
1101 |
15 |
D |
14 |
1110 |
16 |
E |
15 |
1111 |
17 |
F |
In conclusion, this article describes the concept of number systems in digital electronics and different types of number systems along with their relationship table.
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