Number System PDF

Number System What is a Number system? A number system is a method for representing numbers mathematically, using a specific base or radix. This system allows for a unique representation of every number and provides arithmetic operations. A number system in digital electronics is a system...

Number System What is a Number system? A number system is a method for representing numbers mathematically, using a specific base or radix. This system allows for a unique representation of every number and provides arithmetic operations. A number system in digital electronics is a systematic way of representing and expressing numbers using specific symbols or digits. In digital electronics, number systems are used to represent binary data, which is the foundation of all digital systems, including computers, microprocessors, and other digital devices.  The most commonly used number systems in digital electronics are binary, decimal, octal, and hexadecimal. Each of these systems has its own base, which determines the range of symbols used and the value of each digit.  Base (Radix): The base of a number system determines the number of unique digits, including zero, used in the system. For example, the binary system has a base of 2, meaning it uses only two digits (0 and 1). Positional Value: In any number system, the value of a digit depends on its position within the number, with each position representing a power of the base. Applications: Number systems are used in various digital electronics applications, including data storage, processing, communication, and hardware design. Types of Number System Decimal Number System (Base 10) Binary Number System (Base 2) Octal Number System (Base 8) Hexadecimal Number System (Base 16) Decimal Number System The Decimal Number System is the most familiar number system, as it is the one we use in our everyday life. It has a base of 10, meaning it uses ten digits (0-9) to represent numbers. Each digit's positional value is a power of 10. Characteristics: Base: 10 Digits Used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Positional Value: Each digit in a decimal number is multiplied by 10 raised to the power of its position. Example: Binary Number System The Binary Number System is the foundation of all digital electronics. It has a base of 2, meaning it uses only two digits: 0 and 1. Every digital device, from simple calculators to complex computers, relies on the binary system to perform operations and store data. Characteristics: Base: 2 Digits Used: 0, 1 Positional Value: Each digit in a binary number is multiplied by 2 raised to the power of its position. Octal Number System A number system with a base of 8 is known as the octal number system. This system utilizes 8 symbols—0, 1, 2, 3, 4, 5, 6, and 7—to represent numbers.  Base: 8 Digits Used: 0, 1, 2, 3, 4, 5, 6, 7 Positional Value: Each digit in an octal number is multiplied by 8 raised to the power of its position. Hexadecimal Number System The Hexadecimal Number System is widely used in digital electronics and computing. It has a base of 16, meaning it uses sixteen digits, including 0-9 and A-F, where A represents 10, B represents 11, up to F represents 15. Characteristics: Base: 16 Digits Used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Positional Value: Each digit in a hexadecimal number is multiplied by 16 raised to the power of its position. Conversion Table The ability to convert between different number systems is essential in digital electronics as data represented internally in one system may need to be viewed or transmitted externally in another form. Here are the general step-by-step approaches: Conversion Methods: 1.Binary to Decimal: Multiply each binary digit by 2 raised to its position power and sum the results. 2.Decimal to Binary: Divide the decimal number by 2 repeatedly, recording the remainder, until you reach 0. The binary number is the sequence of remainders read in reverse. 3.Binary to Octal: Group the binary digits into sets of three, starting from the right, and convert each group to its octal equivalent. 4.Octal to Binary: Convert each octal digit to its three-digit binary equivalent. 5.Binary to Hexadecimal: Group the binary digits into sets of four, starting from the right, and convert each group to its hexadecimal equivalent. 6.Hexadecimal to Binary: Convert each hexadecimal digit to its four-digit binary equivalent. Application of Number System Data Representation: Binary numbers are used to represent and store data in digital systems. Programming: Hexadecimal numbers are commonly used in programming and debugging. Digital Circuit Design: Binary, octal, and hexadecimal numbers are used in the design and analysis of digital circuits. Communication Protocols: Number systems are used in protocols for data transmission and encoding.