Binary Calculator

About Binary Calculator

The Binary Calculator is a powerful online tool for performing mathematical operations with binary numbers as well as converting between binary, decimal, and hexadecimal number systems. It can execute all basic arithmetic operations (addition, subtraction, multiplication, division) and bitwise operations (AND, OR, XOR, NOT) on binary numbers.

How Binary Numbers Work

Binary is a base-2 number system that uses only two digits: 0 and 1. Each digit in a binary number represents a power of 2. For example:

1011 (binary) = 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8 + 0 + 2 + 1 = 11 (decimal)

Binary Arithmetic Operations

The calculator performs these basic operations on binary numbers:

• Addition: Follows the same rules as decimal addition but with only two digits (0 and 1)
• Subtraction: Uses the borrow method similar to decimal subtraction
• Multiplication: Based on the same principles as decimal multiplication
• Division: Similar to long division in the decimal system

Bitwise Operations

Bitwise operations work on the binary representation of numbers at the bit level:

• AND: Each bit is 1 if both corresponding bits are 1
• OR: Each bit is 1 if at least one corresponding bit is 1
• XOR: Each bit is 1 if exactly one corresponding bit is 1
• NOT: Inverts all bits (1 becomes 0 and 0 becomes 1)

Number System Conversions

The calculator can convert between these number systems:

• Binary to Decimal: Sum of powers of 2 for each '1' bit
• Decimal to Binary: Repeated division by 2 and recording remainders
• Hexadecimal to Binary: Each hex digit converts to 4 binary digits
• Binary to Hexadecimal: Groups of 4 binary digits convert to one hex digit

Practical Applications

Binary calculations are fundamental in computer science and digital electronics:

• Computer programming and algorithm design
• Digital circuit design and analysis
• Data encryption and compression
• Network protocols and data transmission
• Memory addressing and storage