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Number Base Converter

Math

Convert numbers between different bases (binary, octal, decimal, hexadecimal, and more). Learn about number systems and their practical applications in computing and mathematics.

Base Converter

Valid chars: 0-9
(Base 2-36)

Quick Examples

🔢 Conversion Results

Binary (Base 2)
Digital systems
101010
Octal (Base 8)
Unix permissions
52
Decimal (Base 10)
Human counting
42
Hexadecimal (Base 16)
Memory, colors
2A
Base 2
Custom conversion
2A

📊 Common Number Bases

Binary (Base 2)
Computer systems, digital logic
BIN
Octal (Base 8)
Unix permissions, legacy systems
OCT
Decimal (Base 10)
Human counting, everyday math
DEC
Hexadecimal (Base 16)
Memory addresses, colors
HEX

💡 Conversion Tips

  • • Binary: Use powers of 2 (1, 2, 4, 8, 16, 32...)
  • • Hexadecimal: Each digit represents 4 binary digits
  • • Octal: Each digit represents 3 binary digits
  • • Large bases: Use letters A-Z for digits 10-35

🤔 Why Do We Have Different Number Bases?

Historical Reasons

Different cultures developed various counting systems based on available tools and natural patterns. Base 10 likely came from counting on fingers, while base 60 was used by Babylonians for astronomy.

  • • Base 10: Ten fingers for counting
  • • Base 12: Finger segments (3 per finger × 4 fingers)
  • • Base 20: Fingers and toes combined
  • • Base 60: Babylonian astronomy and time

Technical Advantages

Modern applications use different bases for efficiency and convenience in specific domains. Each base has unique properties that make certain calculations easier.

  • • Binary: Matches digital on/off states
  • • Hexadecimal: Compact representation of binary
  • • Octal: Clean grouping of binary triplets
  • • Base 12: More divisors than base 10

📚 Understanding Number Systems

Binary (Base 2)

The foundation of all digital technology. Uses only 0 and 1, representing off and on states in electronic circuits.

Digits: 0, 1
Example: 1010₂ = 10₁₀
Uses:
  • • Computer memory and processing
  • • Digital logic circuits
  • • Data storage and transmission
  • • Boolean algebra and programming

Octal (Base 8)

Popular in early computing and still used in Unix/Linux systems for file permissions.

Digits: 0-7
Example: 755₈ = 493₁₀
Uses:
  • • Unix/Linux file permissions
  • • Legacy computer systems
  • • Compact binary representation
  • • Assembly language programming

Decimal (Base 10)

The standard human counting system. Most natural for everyday mathematics and general calculations.

Digits: 0-9
Example: 42₁₀ = 42₁₀
Uses:
  • • Everyday counting and arithmetic
  • • Financial calculations
  • • Scientific measurements
  • • Human-readable displays

Hexadecimal (Base 16)

Extremely popular in computing for its compact representation of binary data.

Digits: 0-9, A-F
Example: FF₁₆ = 255₁₀
Uses:
  • • Memory addresses and pointers
  • • Color codes in web design (#FF0000)
  • • Assembly language and debugging
  • • Cryptography and hashing

🌟 Special Number Bases

Base 12 (Duodecimal)

Has more divisors than base 10, making fractions easier.

Used in: Clocks (12 hours), measurements (dozen, gross), angles (360° = 12×30°)

Base 60 (Sexagesimal)

Ancient Babylonian system still used today.

Used in: Time (60 minutes/seconds), angles (360°), geographic coordinates

Base 36

Uses all digits and letters, compact for encoding.

Used in: URL shorteners, license plates, compact identifiers

🛠️ Real-World Applications

Computer Science

  • • Binary: All digital data storage and processing
  • • Hexadecimal: Memory dumps, machine code, color values
  • • Octal: File permissions in Unix/Linux systems
  • • Base64: Email encoding and data transmission

Other Fields

  • • Time: Base 60 for minutes and seconds
  • • Angles: Base 60 for degrees, minutes, seconds
  • • Commerce: Base 12 for dozens and gross
  • • Historical: Base 20 in Mayan mathematics