Digital Electronics - Number Systems

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A digital number system is a positional number system that has some symbols called digits. It provides a complete set of digits, operators, and rules to perform operations.

In a digital number system, the number of digits used determines the base of the number system. For example, the binary number system has two digits (0 and 1), hence, the base of the binary number system is 2.

Digital number systems form the foundation of the modern computing technologies and digital electronics. They are used to represent, process, and manipulate the information using a digital system.

In this chapter, we will discuss the fundamental concepts of different types of digital number systems.

Types of Digital Number Systems

In digital electronics, the following four types of digital number systems are mainly used −

• Binary Number System
• Decimal Number System
• Octal Number System
• Hexadecimal Number System

Lets discuss each of these number systems in detail.

Binary Number System

Binary number system is the fundamental building block behind the implementation and working of all digital systems.

Binary number system has two symbols or digits, i.e., 0 and 1. Hence, these two digits are used to represent information and perform all the digital operations. Each binary digit is called a bit.

Since there are two digits are used in the binary number system, hence its base is 2. Therefore, the value of a binary number is calculated as the sum of powers of 2.

Binary digits are used in digital system to represent their ON and OFF states. Where, 0 is used to represent the OFF state of the digital system and 1 is used to represent the ON state of the system.

Overall, the binary number system forms the foundation of computation, digital communication, and digital information storage.

Example

Consider the binary number 1101.011. The integer part of this number is 1101 and the fractional part of this number is 0.011. The digits 1, 0, 1 and 1 of the integer part have weights of 2⁰, 2¹, 2², 2³ respectively. Similarly, the digits 0, 1 and 1 of fractional part have weights of 2^-1, 2^-2, 2^-3 respectively.

Mathematically, we can write it as,

$$\mathrm{1101.011 \: = \: (1 \: \times \: 2^{3}) \: + \:(1 \: \times \: 2^{2}) \: + \: (0 \: \times \: 2^{1}) \: + \: (1 \: \times \: 2^{0}) \: + \: (0 \: \times \: 2^{−1}) \: + \: (1 \: \times \: 2^{−2}) \: + \: (1 \: \times \: 2^{−3})}$$

After simplifying the right-hand side terms, we will get a decimal number, which is an equivalent of binary number on left-hand side.

Decimal Number System

Decimal number system is not inherently a digital number system. But it is widely used to represent the digital information in a human readable format.

Decimal number system is a base 10 number system having 10 unique digits i.e., 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. It is the standard number system used by human beings to represent information in a natural way. However, a digital system cannot directly process the information represented in decimal form, so it is converted into binary form and then processed.

The base of the decimal number system is 10. So, the value of a decimal number is calculated by the sum of powers of 10.

Example

Consider the decimal number 1358.246. The integer part of this number is 1358 and the fractional part of this number is 0.246. The digits 8, 5, 3 and 1 have weights of (10)⁰, (10)¹, (10)² and (10)³ respectively. Similarly, the digits 2, 4 and 6 have weights of (10)^-1, (10)^-2 and (10)^-3 respectively.

Mathematically, we can write it as,

$$\mathrm{1358.246 \: = \: (1 \: \times \: 10^{3}) \: + \:(3 \: \times \: 10^{2}) \: + \: (5 \: \times \: 10^{1}) \: + \: (8 \: \times \: 10^{0}) \: + \: (2 \: \times \: 10^{−1}) \: + \: (4 \: \times \: 10^{−2}) \: + \: (6 \: \times \: 10^{−3})}$$

After simplifying the right-hand side terms, we will get the decimal number, which is on the left-hand side.

Octal Number System

The octal number system is another type of digital number system used in the field of digital electronics to represent information. It is a base 8 number system having eight unique digits i.e., 0, 1, 2, 3, 4, 5, 6, and 7.

It is important note that the octal number system is equivalent to 3-bit binary number system as 2³ = 8. Hence, this number system can be used in computing and digital electronic applications.

The value of an octal number is obtained by the sum of powers of 8, as 8 is the base of the octal number system.

Octal number system is used in the field of digital electronics to represent binary information in compact form, permissions in Linux or Unix systems, IPv6 address, binary machine code instructions, in error detection algorithms, etc.

Example

Consider the octal number 1457.236. Integer part of this number is 1457 and fractional part of this number is 0.236. The digits 7, 5, 4 and 1 have weights of (8)⁰, (8)¹, (8)² and (8)³ respectively. Similarly, the digits 2, 3 and 6 have weights of (8)^-1, (8)^-2, (8)^-3 respectively.

Mathematically, we can write it as,

$$\mathrm{1457.236 \: = \: (1 \: \times \: 8^{3}) \: + \:(4 \: \times \: 8^{2}) \: + \: (5 \: \times \: 8^{1}) \: + \: (7 \: \times \: 8^{0}) \: + \: (2 \: \times \: 8^{−1}) \: + \: (3 \: \times \: 8^{−2}) \: + \: (6 \: \times \: 8^{−3})}$$

After simplifying the right-hand side terms, we will get a decimal number, which is an equivalent of octal number on the left-hand side.

Hexadecimal Number System

The hexadecimal number system is a base 16 number system. It has 16 digits, 0 to 9 and A to F. Where, A represents 10, B represents 11, C represents 12, D represents 13, E represents 14, and F represents 15. The hexadecimal number system is equivalent to a 4-bit binary number system as 2⁴ = 16. Thus, the value of a hexadecimal number can be calculated by the sum of powers of 16.

In the field of digital electronics, the hexadecimal number system is used in memory address representation, digital colors representation, low level computer programming, encoding, assembly language programming, microcontrollers, keyboards, etc. Hexadecimal number system creates a balance between digital representation and human readability.

Example

Consider the hexadecimal number 1A05.2C4. The integer part of this number is 1A05 and the fractional part of this number is 0.2C4. The digits 5, 0, A and 1 have weights of (16)⁰, (16)¹, (16)² and (16)³ respectively. Similarly, the digits 2, C and 4 have weights of (16)^-1 , (16)^-2 and (16)^-3 respectively.

Mathematically, we can write it as,

$$\mathrm{1A05.2C4 \: = \: (1 \: \times \: 16^{3}) \: + \:(10 \: \times \: 16^{2}) \: + \: (0 \: \times \: 16^{1}) \: + \: (5 \: \times \: 16^{0}) \: + \: (2 \: \times \: 16^{−1}) \: + \: (12 \: \times \: 16^{−2}) \: + \: (4 \: \times \: 16^{−3})}$$

After simplifying the right-hand side terms, we will get a decimal number, which is an equivalent of the hexadecimal number on the left-hand side.

Advantages of Digital Number Systems

The following are some key advantages of digital number systems −

• Digital number systems provide a simple and consistent way of representing and understanding information.
• Digital number systems allow to develop efficient methods for storage and transmission of digital information.
• Digital number systems provide methods of representing different types of information like text, numbers, images, etc.
• Digital number systems allow to convert information from one form to full fill the needs of applications.
• Digital number systems create compatibility between hardware and software.

Applications of Digital Number Systems

Digital number systems are used in various digital electronic fields such as computing, internet, communication, signal processing, and more. Here are a few examples of applications of digital number systems −

• Information Representation
• Digital Communication
• Storage and Transmission of Digital Data and Information
• Algorithm Development
• System Programming, etc.

Conclusion

In this chapter, we discussed the basic concepts of digital number systems. The understanding of digital number systems is essential for designing, implementing, and troubleshooting the digital systems. Digital number systems provide different methods of representing and manipulating information in digital systems.