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Number Systems in Digital Electronics

In this article, we will explore different types of number systems used in digital electronics and computer architecture. The number system is an elementary concept that provides a set of rules to represent and express numerical quantities. Some commonly used types of number systems are decimal number systems, binary number systems, octal number systems, and hexadecimal number systems.

number systems

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 102 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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