How To Convert Number Bases

A number base, also called a radix, is the number of distinct digit symbols a counting system uses. Decimal uses ten, binary uses two, octal uses eight and hexadecimal uses sixteen.

In any base, the value of a number is the sum of each digit multiplied by the base raised to the power of its position, counting from zero at the rightmost column. That single rule is the whole of place-value notation.

Decimal 255 means two hundreds, five tens and five ones. The same quantity in binary is 11111111, which is eight columns of one, because each column is a power of two rather than a power of ten.

Binary is the natural base for digital hardware because a wire either carries a voltage or it does not. Eight binary digits, a byte, can represent 256 distinct values, which is why 255 is the largest value a single byte holds.

Octal groups binary digits in threes, so one octal digit stands for exactly three bits. It was common on older systems with word lengths that were multiples of three, and it survives in file permission codes on Unix systems.

Hexadecimal groups binary digits in fours, so one hex digit stands for exactly four bits. Two hex digits make a byte, which is why hex is the standard shorthand for memory addresses, colour codes and byte dumps.

Hexadecimal needs sixteen digit symbols but only ten exist as numerals, so the letters A to F stand for the values ten through fifteen. Case is irrelevant, so FF and ff mean the same thing.

Converting from another base to decimal is the easiest direction: multiply each digit by its place value and add. Converting from decimal to another base is done by repeated division, taking the remainders in reverse order.

Converting between binary, octal and hex never needs decimal at all. Because eight and sixteen are powers of two, you can regroup the bits directly: three bits per octal digit, four bits per hex digit.

The base is usually written as a subscript to remove ambiguity, because 11 means three in binary, nine in decimal and seventeen in hex. Where subscripts are unavailable, a prefix such as 0x for hex or 0b for binary is used.

Negative numbers need a convention of their own. A leading minus sign is the simple approach, while two's complement is what hardware uses to represent signed integers without a separate sign bit.

Fractional values extend the same rule to the right of the point, with negative powers of the base. Binary fractions cannot represent one tenth exactly, which is why decimal arithmetic on a computer sometimes shows small rounding artefacts.

The calculator takes a value and the base it is written in, and shows the same quantity in decimal, binary, octal and hexadecimal along with its bit length. It accepts a leading minus sign and ignores spaces and underscores used as digit separators.