Edu_Vault

  Introduction In digital electronics, the concept of complements plays an important role in performing arithmetic operations, representing negative numbers, and simplifying circuit design. Complements are alternative representations of numbers that make it possible to carry out subtraction by addition, which is much easier to implement in hardware. They are also used in error detection techniques. There are two types of complements generally considered for any number system with base r : (r – 1)’s complement, known as the diminished radix complement r’s complement, known as the radix complement Depending on the base of the number system, these complements take different forms such as 9’s and 10’s complements for decimal numbers, 1’s and 2’s complements for binary numbers, and 15’s and 16’s complements for hexadecimal numbers. Complements in the Binary System (Base 2) 1’s Complement The 1’s complement of a binary number is obtained by replacing every 0 with 1 and every 1 ...

  Hexadecimal Number System Conversion – Complete Guide Introduction to the Hexadecimal Number System The hexadecimal number system is a base-16 system that uses 16 symbols : 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , A , B , C , D , E , F Here, A = 10, B = 11, C = 12, D = 13, E = 14, F = 15 (in decimal). It is widely used in digital electronics, computer programming, memory addressing, and color codes in web design because it provides a compact way to represent binary numbers. Types of Hexadecimal Number Conversions Hexadecimal numbers are often converted to and from other number systems: Hexadecimal to Decimal Decimal to Hexadecimal Hexadecimal to Binary Binary to Hexadecimal Hexadecimal to Octal (via binary as an intermediate step) 1. Hexadecimal to Decimal Conversion To convert from hexadecimal to decimal: Multiply each digit by 16 raised to the power of its position (starting from 0 at the rightmost digit). Add the results. ...

  Decimal Number System Conversion – Complete Guide Introduction to the Decimal Number System The decimal number system is the most commonly used number system in our daily life. It is a base-10 system that uses ten digits : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Each digit’s value depends on its position (place value) and the base (10). For example: 345 = (3 × 10²) + (4 × 10¹) + (5 × 10⁰) Types of Decimal Number Conversions We often convert decimal numbers into other number systems in digital electronics, computer science, and mathematics . The main conversions are: Decimal to Binary (Base-2) Decimal to Octal (Base-8) Decimal to Hexadecimal (Base-16) Reverse Conversions (Binary/Octal/Hexadecimal to Decimal) 1. Decimal to Binary Conversion The binary number system uses only two digits: 0 and 1 . Steps for Integer Part: Divide the decimal number by 2. Write down the remainder (0 or 1). Repeat division with the quotient until it becomes 0. Read the ...

Scroll down to ger pdf What Is the Octal Number System? The Octal (base-8) number system uses digits from 0 to 7 . It's used in computing as a shorthand for binary because 1 octal digit = 3 binary digits (bits) . Example: Octal: 237₈ This is different from Decimal ( 237₁₀ ) or Binary ( 101011₂ ).  1. Octal to Decimal – Explained We convert each octal digit into a power of 8. Decimal (base-10) is what we normally use. So, we want to rewrite the base-8 number into base-10 , by evaluating powers of 8.  Integral Part Let’s take: 237₈ Break it down: 2 × 8² = 2 × 64 = 128 3 × 8¹ = 3 × 8 = 24 7 × 8⁰ = 7 × 1 = 7 Now, add them: 128 + 24 + 7 = 159 So, 237₈ = 159₁₀ Fractional Part Example: 0.52₈ We apply powers of 8 in negative , starting from −1. 5 × 8⁻¹ = 5 × 0.125 = 0.625 2 × 8⁻² = 2 × 0.015625 = 0.03125 Add: 0.625 + 0.03125 = 0.65625 So, 0.52₈ = 0.65625₁₀ 2. Decimal to Octal – Explained We reverse the above process. Integral Part Let’s convert: 1...

 What is a Number System? A number system is a way to represent numbers using a consistent set of symbols or digits. The base (or radix ) of a number system determines how many unique digits it uses. Types of Number Systems 1. Binary Number System (Base-2) Digits Used : 0, 1 Base : 2 Application : Widely used in digital electronics, computers, and communication systems. 2. Octal Number System (Base-8) Digits Used : 0 to 7 Base : 8 Application : Sometimes used in computing as a compact form of binary. 3. Decimal Number System (Base-10) Digits Used : 0 to 9 Base : 10 Application : Most commonly used in daily human activities and calculations. 4. Hexadecimal Number System (Base-16) Digits Used : 0–9 and A–F (where A=10, B=11, ..., F=15) Base : 16 Application : Used in programming and debugging as a human-friendly representation of binary-coded values. Binary Conversion 1. Decimal to Binary (Integral Part) Steps : Divide the decimal num...