Base Converter

How the base converter works

A base, or radix, is how many different digits a number system uses before it carries to the next place. Decimal (base 10) uses 0 to 9, binary (base 2) uses 0 and 1, and hexadecimal (base 16) uses 0 to 9 and then A to F for 10 to 15.

Any whole number can be written in any base from 2 to 36, where the letters A to Z run out as extra digits. The converter always goes through decimal: it reads your number in the "from" base, works out its decimal value, then writes that value in the "to" base.

How to convert a number from one base to another

Convert 1011 from binary to hexadecimal.

1. Multiply each digit by its place value and add: 1×8 + 0×4 + 1×2 + 1×1 = 11 in decimal.
2. Divide by the new base: 11 ÷ 16 = 0 remainder 11.
3. Write each remainder as a digit of the new base and read them from the bottom up: 11 is B, so 1011₂ = B₁₆.

Step 1 on its own is what the binary to decimal converter does, and steps 2 and 3 on their own are the decimal to hexadecimal converter. Between binary and hex there is also a shortcut that skips decimal: the binary to hex converter groups the bits in fours.

Custom bases and digit checks

Choosing "Custom" on either side opens a box for any base from 2 to 36. Before converting, the tool checks every character against the digits allowed in the "from" base: 0 to 7 for base 8, 0 to 9 and A to J for base 20.

It names the character that does not belong instead of failing silently. If both bases are the same, it says no conversion is needed.

Where other number bases are used

Binary and hexadecimal are everywhere in programming: memory addresses, colour codes and bit masks. Octal survives in Unix file permissions, and the hex converter covers it next to hex and binary.

Bases outside those four are mostly for coursework on positional notation or for short codes such as base-36 IDs. All the conversion tools are listed on the number system converters page.