Convert between binary, decimal, and hexadecimal number systems. Perform bitwise operations, binary arithmetic, and get step-by-step conversion solutions for computer science and digital electronics.
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Binary is a base-2 number system using only two digits: 0 and 1. Each binary digit (bit) represents a power of 2, making it fundamental for digital computers and electronics.
Method: Multiply each bit by 2 raised to its position power, then sum results.
Example: 1011₂ = (1×2³) + (0×2²) + (1×2¹) + (1×2⁰) = 8 + 0 + 2 + 1 = 11₁₀
Hexadecimal (base-16) uses digits 0-9 and letters A-F. It's compact for binary representation: each hex digit represents 4 binary bits.
Example: 1011 1101₂ = BD₁₆
Method for representing negative numbers in binary: invert bits and add 1. Essential for computer arithmetic with signed integers.
This binary calculator provides accurate number system conversions and bitwise operations for educational and programming purposes. While calculations are mathematically precise, real-world computer systems may have specific implementation details regarding integer sizes, signed number representations, and overflow handling that should be considered in practical applications.
This advanced binary calculator implements comprehensive number system conversions and bitwise operations essential for computer science, digital electronics, and programming. Each operation follows standard computer science principles with detailed step-by-step explanations.
Digits: 0, 1
Fundamental for digital systems. Each bit represents a power of 2. Used in computer memory and processing.
Digits: 0-9
Standard human number system. Each digit represents a power of 10. Used for everyday calculations.
Digits: 0-9, A-F
Compact binary representation. Each hex digit equals 4 binary bits. Used in programming and memory addressing.
Operations: AND, OR, XOR, NOT
Manipulate individual bits. Essential for low-level programming, cryptography, and digital circuit design.
Binary to decimal conversion uses positional notation: each binary digit represents a power of 2. For example, binary 1011 = (1×2³) + (0×2²) + (1×2¹) + (1×2⁰) = 8 + 0 + 2 + 1 = 11 in decimal. Our calculator shows each step of this calculation including the power of 2 for each bit position and the final summation process.
Binary uses base-2 (digits 0-1), decimal uses base-10 (digits 0-9), hexadecimal uses base-16 (digits 0-9, A-F). Binary is fundamental for computers, decimal is natural for humans, and hexadecimal provides compact binary representation. Each system has specific applications: binary for digital circuits, decimal for everyday math, hexadecimal for programming and memory addressing.
Bitwise operations manipulate individual bits: AND (result 1 only if both bits 1), OR (result 1 if either bit 1), XOR (result 1 if bits different), NOT (invert each bit). Our calculator shows step-by-step bit manipulation for each operation, demonstrating how each bit position is processed independently to produce the final result.
Yes, our calculator handles negative numbers using two's complement representation and fractions using binary point notation. For negative numbers, we use the standard two's complement method (invert bits and add 1). For fractions, we represent them using binary points similar to decimal points, with each position after the point representing negative powers of 2.
Binary calculations are essential for computer programming, digital electronics, networking protocols, cryptography, data compression, and any field involving digital data processing. Specific applications include microprocessor design, file format specifications, network addressing, encryption algorithms, and computer graphics rendering where precise bit manipulation is required.
Conversions use precise mathematical algorithms with exact results. Binary-decimal-hexadecimal conversions are lossless when working with integers within system limits. The calculator handles numbers up to 64-bit precision, ensuring accurate conversions for most practical applications. Step-by-step solutions provide complete transparency in the conversion process.