Number Base Converter

Number Base Converter (Base 2–36)

Number Base Converter — Any Base from 2 to 36

This free number base converter handles any combination of bases from base 2 to base 36. Unlike simple binary-to-decimal converters, it supports any arbitrary radix — base 3 (ternary), base 5, base 12 (duodecimal), base 36 (alphanumeric), and everything between. Step-by-step working is shown for each conversion.

Positional Notation Explained

All positional number systems work the same way: each digit's value is the digit multiplied by the base raised to its positional power. In decimal (base 10), the number 345 means 3×10² + 4×10¹ + 5×10⁰ = 300+40+5. In binary (base 2), 1011 means 1×2³ + 0×2² + 1×2¹ + 1×2⁰ = 8+0+2+1 = 11 decimal.

Digits Used by Each Base

Bases 2–10 use standard digit characters 0–9. Bases 11 and above use letters to represent digit values above 9: A=10, B=11, C=12, D=13, E=14, F=15, and so on up to Z=35 (for base 36). Hexadecimal (base 16) uses 0–9 and A–F. Base 36 uses all ten digits plus all 26 letters.

Common Bases in Computing and Science

  • Base 2 (binary): Digital electronics, computer memory, boolean logic.
  • Base 8 (octal): Unix file permissions, some embedded systems.
  • Base 16 (hexadecimal): Memory addressing, colour codes, networking.
  • Base 12 (duodecimal): Proposed as a better alternative to decimal — 12 has more factors (1,2,3,4,6,12) than 10 (1,2,5,10).
  • Base 36: URL shorteners, compact identifiers (e.g., Git short hashes in case-insensitive contexts).
  • Base 60 (sexagesimal): Used in time (60 seconds, 60 minutes) and angles (360°), inherited from ancient Babylonia.

The Division Algorithm for Base Conversion

To convert a decimal number to any other base: repeatedly divide by the target base and collect the remainders. Read the remainders from last to first. For example, converting 255 to base 16: 255÷16=15 r15 (F), 15÷16=0 r15 (F). Reading bottom to top: FF. This is exactly what the step-by-step output shows.

Frequently Asked Questions

What is base 36 used for?
Base 36 uses all ten digits and all 26 letters, making it the most compact case-insensitive alphanumeric encoding. It is used in URL shorteners, ticket systems, serial number generation, and any application that needs compact, human-readable, case-insensitive identifiers.
Why does base 60 have 60 seconds in a minute?
The Babylonians used base 60 (sexagesimal) because 60 has many integer divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), making fractions simpler to express. This system was adopted for time and angular measurement and remains in use over 4,000 years later.
Is there a limit to how large a number I can convert?
This converter uses JavaScript's floating-point arithmetic, which is precise for integers up to 2^53 (about 9 quadrillion). For cryptographic applications or very large numbers, dedicated arbitrary-precision libraries are required.
How do I convert between non-decimal bases without going through decimal?
For related bases (e.g., binary to hex where 16 = 2^4), you can group digits directly: group binary digits in fours for hex. For unrelated bases, the standard method is to convert to decimal first as an intermediate step, then convert decimal to the target base.