Arbitrary Base Converter
Convert a number from any base to any other, from 2 up to 1000, including fractions. Use the standard digits, a colon-separated digit list for large bases such as 60, or your own alphabet.
How to Use
- Enter the base you are converting from and the base you want.
- Choose the digits: standard (0–9, A–Z, a–z, + /, up to base 64), a digit list such as 33:46 for any base, or your own alphabet.
- Type the number, with a point for fractions if needed.
- Read the result, the decimal value and the number of digits; repeating fraction digits are shown in brackets.
- Show Work reads the input’s place values and divides by the new base.
Worked Example
2026 in base 60. 2026 ÷ 60 = 33 remainder 46, and 33 ÷ 60 = 0 remainder 33. Reading upwards, the base-60 digits are 33 and 46, written 33:46. Check: 33 × 60 + 46 = 1,980 + 46 = 2,026.
2026 in base 64. 2026 = 31 × 64 + 42. With the standard digits (A = 10 … Z = 35, a = 36 …), 31 is V and 42 is g: Vg. With the Base64 alphabet A–Z a–z 0–9 + /, 31 is f and 42 is q: fq. Same number, different alphabet.
The common mistake: converting digit by digit. You cannot turn base-7 digits straight into base-5 digits one at a time, as you can between binary and hex (where 16 = 2⁴). In general the whole value has to be computed, then rewritten in the new base.
Show Work
Formulas
One Algorithm, Every Base
Place-value notation works in any base, a point that Simon Stevin, Leibniz and others made explicit in the 16th and 17th centuries as they compared decimal with binary and sexagesimal arithmetic. Every conversion is the same two steps, reading the value and rewriting it, which is why the classic algorithms appear in the first volume of Donald Knuth’s The Art of Computer Programming.
Very large bases are not just curiosities. Big-number libraries store integers as arrays of digits in base 2³² or 2⁶⁴, and the Babylonians’ base 60 survives in every clock. Writing such digits as colon-separated numbers keeps them readable whatever the base.
About This Calculator
This converter changes a whole number or fraction from any base between 2 and 1000 to any other, exactly, marking repeating fraction digits in brackets. Digits can be the standard 0–9, A–Z, a–z, + and / up to base 64, colon-separated numbers for any base, or a custom alphabet of your own, such as the Base64 order.
Everything runs in your browser; nothing is sent anywhere. For the everyday bases side by side, use the number base converter.
Related calculators: Number Base Converter, Base58 and Base62 Encoder, and Babylonian Numerals.
Frequently Asked Questions
How do I convert between two bases?
Go through the value. First read the number in its own base by multiplying each digit by its place value and adding. Then write that value in the new base by dividing repeatedly by the new base and reading the remainders from last to first. Fractions use repeated multiplication instead.
What digits are used above base 10?
Letters: A = 10 up to Z = 35, then lower-case a = 36 up to z = 61, then + and / for 62 and 63, which covers base 64. Bases up to 36 ignore letter case. Above 64 there are no conventional symbols, so the digits are written as decimal numbers separated by colons, the way base 60 is written for time and angles.
Why does base 64 here differ from Base64 encoding?
Base64 encoding (RFC 4648) uses a different order, A–Z, a–z, 0–9, + and /, and encodes bytes in groups of three, not a single number. Choose the custom alphabet and paste ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz0123456789+/ to use its digits: 2026 is then fq instead of Vg.
Can fractions be converted exactly?
Yes, as long as you allow repeating digits. A fraction ends in a base only if its denominator’s prime factors all divide the base; otherwise its digits repeat, and the repeating block is shown in brackets. 0.1 in base 3 is 0.(0022), because 10 has the prime 5, which does not divide 3.
Why not just use one base?
Different bases suit different jobs: 2 and 16 for computers, 60 for time and angles, 12 for divisibility, 36 to 62 for short identifiers. Converting between them is the same algorithm every time, which is why one converter can handle all of them.
How do I use the Arbitrary Base Converter?
Just type or paste your value. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Do I need to install or sign up for anything?
Not at all — it runs in the browser with nothing to install and no account. After it loads once, it even works without an internet connection.
Is my information private?
Yes. Everything happens in your browser. Nothing you type is sent to a server or saved anywhere.
Common Use Cases
Sexagesimal
2026 in base 60 is 33:46, the way Babylonians would write it.
Base 64 digits
2026 is Vg with the standard digits and fq with the RFC 4648 alphabet.
Unusual bases
1,000 in base 7 is 2626.
Fractions
0.1 in base 3 is 0.(0022), repeating for ever.
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