Custom Base Converter

Convert between any two arbitrary number bases from 2 to 36. Explore mathematical number systems with custom digits and rules.

From Base
Base
Input Number
To Base
Base

Common Number Bases

Decimal → Binary
Base 10: 0-9
Base 2: 0-1
Decimal → Hex
Base 10: 0-9
Base 16: 0-9,A-F
Hex → Decimal
Base 16: 0-9,A-F
Base 10: 0-9
Octal → Binary
Base 8: 0-7
Base 2: 0-1
Binary → Octal
Base 2: 0-1
Base 8: 0-7
Duodecimal → Decimal
Base 12: 0-9,A-B
Base 10: 0-9
Vigesimal → Decimal
Base 20: 0-9,A-J
Base 10: 0-9
Base36 → Decimal
Base 36: 0-9,A-Z
Base 10: 0-9

Understanding Number Bases

Positional Notation

All number bases use positional notation where each digit's position represents a power of the base. The rightmost digit is the 0th power, the next is the 1st power, etc.

Example: 1A2₁₆ (hexadecimal)
1 × 16² = 1 × 256 = 256
A × 16¹ = 10 × 16 = 160
2 × 16⁰ = 2 × 1 = 2
Total: 418₁₀

Base Conversion Methods

  • Division method: Convert to decimal first
  • Substitution method: Direct base conversion
  • Grouping method: For powers of 2 (2,4,8,16,32)
  • Calculator method: Using built-in functions
  • Table method: Pre-computed conversion tables
  • Algorithm method: Custom conversion algorithms

Advanced Concepts

  • Base independence: All bases can represent the same numbers, just with different digit sequences
  • Efficiency: Higher bases use fewer digits but require more unique symbols
  • Computational uses: Different bases excel at different types of calculations
  • Historical bases: Base-60 (Babylonian), base-20 (Mayan), base-12 (dozenal)
  • Optimal bases: Base-e is theoretically optimal for human communication
  • Fractional numbers: All bases support fractional numbers with radix points