Numbers Converter

Binary (Base 2)

The fundamental number system in computing. Uses only digits 0 and 1. 1010₂ = 10₁₀.

Common uses: Computer programming, digital electronics, machine code, low-level operations.

Octal (Base 8)

Uses digits 0-7. Historically used in computing because it's a power of 2. 12₈ = 10₁₀.

Common uses: File permissions in Unix/Linux, older computer systems, compact representation of binary.

Decimal (Base 10)

The standard number system used in everyday life. Uses digits 0-9. 255₁₀.

Common uses: Universal standard, human readable, everyday calculations.

Hexadecimal (Base 16)

Uses digits 0-9 and letters A-F (representing 10-15). FF₁₆ = 255₁₀.

Common uses: Color codes (#FF5733), memory addresses, web design, digital systems.

Base Conversion Method

To convert from any base to decimal, multiply each digit by the base raised to its position:

• Example: 1010₂ = 1×2³ + 0×2² + 1×2¹ + 0×2⁰ = 8 + 0 + 2 + 0 = 10₁₀
• Reverse: To convert decimal to another base, repeatedly divide by the base

Common Number Systems in Computing

• Binary (Base 2): Bits, digital logic, boolean operations
• Octal (Base 8): File permissions (rwx = 7), legacy systems
• Decimal (Base 10): Human interface, general purpose
• Hexadecimal (Base 16): Colors, memory, microprocessor programming
• Base-12: Duodecimal, some historical and specialized applications
• Base-32 & Base-36: Data encoding, efficient representation

Digit Symbols in Different Bases

For bases higher than 10, letters represent larger digit values:

• 0-9: Represent values 0-9 (same in all bases)
• A: Represents value 10
• B: Represents value 11
• C-Z: Continue through value 35
• Example: In base-16, 'FF' = 15×16¹ + 15×16⁰ = 240 + 15 = 255₁₀

Efficient Base Conversions

Certain bases convert efficiently between each other because they are powers:

• Binary to Hexadecimal: Group binary digits in 4s (2⁴ = 16)
• Binary to Octal: Group binary digits in 3s (2³ = 8)
• Example: 1010₂ = 12₈ = A₁₆ (1010 binary)

Common Applications

Number base conversions are essential in:

• Programming: Bit manipulation, debugging, low-level operations
• Web Design: Color codes (#RGB hex format)
• Networking: IP addresses, port numbers
• Computer Hardware: Memory addresses, registers, instruction codes
• Data Encoding: Base-64 for email, Base-32 for URLs
• Cryptography: Hexadecimal representation of keys and hashes
• Digital Electronics: Binary logic gates and circuits

Quick Reference Table