Number Base Converter

Formula

How this is calculated

Enter the number, the base it is written in (the source radix, 2-36) and the base you want it converted to (the target radix, 2-36). Each character is interpreted as a digit: 0-9 then A-Z represent values 0 through 35, so base 16 uses 0-9 and A-F, while base 36 uses every letter. The calculator first checks that every digit is valid for the source base — for example the digit 2 is illegal in binary and the letter G is illegal in hexadecimal — and reports an error if any digit is out of range.

Conversion happens in two steps. First the value is turned into an ordinary decimal (base-10) integer using positional notation: each digit is multiplied by the source base raised to the power of its position counted from the right, and these products are summed. The place-value chart shows exactly how much each digit contributes. Second, that decimal value is re-expressed in the target base by repeated division, and letters are shown in upper case.

Inputs are treated as non-negative integers; leading and trailing spaces are ignored and letters are case-insensitive. Because the intermediate decimal is a standard double-precision number, results are exact for values up to about 2^53 (roughly 9 quadrillion); beyond that, rounding can occur. The binary, octal, decimal and hexadecimal reference row always describes the same underlying value regardless of the chosen target base.