Binary & Hexadecimal Calculator

Binary Arithmetic and Bitwise Operations Guide

Arithmetic and bitwise operations form the foundational mathematical backbone of microprocessor execution units (ALUs), system compilers, cryptographic engines, and network packet processors. Unlike standard decimal calculators, performing operations directly across Binary (base 2), Octal (base 8), Decimal (base 10), and Hexadecimal (base 16) allows software engineers to inspect register states, manipulate bitflags, and design hardware circuits.

1. Mixed-Base Arithmetic

In low-level software engineering, operands frequently originate from different bases (e.g. adding a decimal offset 16 to a hexadecimal memory address 0x00FF). This calculator converts each operand to native arbitrary-precision BigInt representations, executes the mathematical operation, and simultaneously renders the output across all 4 numbering bases.

• Addition (+): Sums both values. Example: 1010 (Bin = 10) + 5 (Dec) = 15 (Dec) / F (Hex).
• Subtraction (-): Subtracts operand B from operand A with full signed-integer support.
• Multiplication (×): Computes integer products. Example: 100 (Bin = 4) × A (Hex = 10) = 40 (Dec) / 28 (Hex).
• Integer Division (÷) & Modulo (%): Returns the integer quotient and the exact division remainder.

2. Bitwise Logic Operators

• AND (&): Sets each bit to 1 if and only if both corresponding bits are 1. Commonly used for bitmask filtering (e.g. extracting status flags).
• OR (|): Sets each bit to 1 if at least one corresponding bit is 1. Commonly used to enable configuration bits.
• XOR (^): Sets each bit to 1 if exactly one bit is 1 and the other is 0. Widely used in cryptography (stream ciphers), checksum parity calculations, and toggle mechanisms.
• NOT (~): Unary bit inversion. Inverts all bits within the selected word size (8, 16, 32, or 64 bits). For instance, in an 8-bit word, ~0x0F produces 0xF0 (240).
• Bit Shifts (<< / >>): Left shift multiplies by powers of 2 ($x \times 2^n$) by pushing zeros into lower bit positions. Right shift divides integers by powers of 2 ($x \div 2^n$).

3. Word Sizes & Bit Width Clamping

In CPU architectures, registers are bounded by fixed word lengths: 8-bit (Byte, max 255), 16-bit (Word, max 65,535), 32-bit (DWord, max 4,294,967,295), and 64-bit (QWord). Choosing the correct bit width is crucial when evaluating unary NOT or overflow-sensitive bit masks.