What is a numeral base?

A base (or radix) is the number of distinct digits a positional number system uses. Base 10 (decimal) uses 0-9; base 2 (binary) uses 0 and 1; base 16 (hexadecimal) uses 0-9 then A-F.

Why does the converter stop at base 36?

Base 36 is the largest base expressible with the standard ten digits 0-9 plus the 26 letters A-Z, giving 36 distinct single-character digits. Higher bases would need additional symbols with no universal convention.

How are letters used as digits?

After the digits 0-9, letters take over: A is 10, B is 11, and so on up to Z which is 35. Input is case-insensitive, so 'ff' and 'FF' both mean 255 in hexadecimal.

Will very large numbers lose precision?

No. The converter uses exact big-integer arithmetic rather than floating-point, so a 200-digit input converts to any base with no rounding error.

Can it convert negative numbers or fractions?

It handles negative integers via a leading minus sign. It does not convert fractional values; for fixed-point or floating-point bit layouts, use the dedicated fixed-point or IEEE-754 tools.

What is the Arbitrary Base Converter?

Convert integers between any two numeral bases from 2 to 36 using digits 0-9 and A-Z. Exact big-integer arithmetic handles huge numbers with no rounding, plus a decimal cross-check. Runs entirely in your browser. It runs free in your browser on Gera Tools, with nothing uploaded.

Arbitrary Base Converter

Name: Arbitrary Base Converter
Creator: Gera Tools
License: https://creativecommons.org/licenses/by/4.0/

A numeral base, or radix, defines how many unique digits a positional number system uses before it rolls over into the next place value. This arbitrary base converter translates a whole number from any base between 2 and 36 into any other, using the digit set 0-9 followed by A-Z. It is handy for programming, digital electronics, cryptography puzzles and understanding how the same quantity looks in different notations.

How it works

In any base b, a number is the sum of each digit multiplied by b raised to the power of its position, counting from zero on the right. The tool reads your input left to right, repeatedly multiplying the running total by the source base and adding each digit’s value — this yields the exact integer. It then re-encodes that integer in the target base by repeated division: divide by the base, record the remainder as the next digit (from right to left), and continue until the quotient reaches zero.

Because the maths is done with exact big integers rather than floating-point numbers, conversions stay precise no matter how many digits you enter. For example, FF in base 16 parses as 15 × 16 + 15 = 255, which in base 2 becomes 11111111.

Example

Convert 7G3 from base 17 to base 10. The digit G equals 16, so the value is 7 × 17² + 16 × 17 + 3 = 7 × 289 + 272 + 3 = 2023 + 275 = 2298. The decimal cross-check line confirms this independently.

Common base systems at a glance

Base	Name	Digits	Where used
2	Binary	0, 1	CPUs, memory addressing, logic gates
8	Octal	0–7	Unix permissions (chmod 755), legacy computing
10	Decimal	0–9	Everyday arithmetic
16	Hexadecimal	0–9, A–F	Colours (#FF5733), memory addresses, SHA hashes
32	Base 32	0–9, A–V	Crockford encoding, TOTP secrets
36	Base 36	0–9, A–Z	URL shorteners, compact IDs (YouTube video IDs once used a variant)

Worked examples

Binary to hexadecimal: 11111111 (binary, base 2) → decimal 255 → FF (hex, base 16). The hex representation of a byte is always exactly 2 characters, which is why programmers prefer hex for raw byte values.

Octal Unix permissions: 755 in octal → binary 111 101 101 → read/write/execute for owner, read/execute for group and others. Enter 755 with source base 8 and target base 2 to see the full bit pattern.

Base 36 compact IDs: The number 1,000,000 in base 10 converts to LFLS in base 36 — four characters instead of seven. This is why base 36 is popular for shortening numeric IDs in URLs and ticket systems.

Tips and notes

You can include spaces or underscores as visual grouping separators in your input — they are ignored. If you enter a digit that is too large for the chosen source base (for example the digit 8 in base 8), the tool flags exactly which character is invalid. Everything is computed locally; nothing you type is sent to a server.

Input is case-insensitive: ff, FF, and fF all equal 255 in base 16. For negative numbers, include a leading minus sign. Fractions are not supported — use a dedicated fixed-point or IEEE 754 floating-point converter for those.