Number Base Converter

How positional number systems work

In a positional system with base b, each digit is worth its symbol multiplied by a power of b that depends on its position. In decimal, 352 is 3·10² + 5·10¹ + 2·10⁰. In binary, 1101 is 1·2³ + 1·2² + 0·2¹ + 1·2⁰ = 13. Bases above 10 use letters: A = 10, B = 11… up to Z = 35 in base 36.

From any base to decimal

Apply the definition: multiply each digit by the base raised to its position and add everything up. To the right of the radix point the positions are negative (b⁻¹, b⁻², …).

From decimal to any base

The integer part is converted by repeated division: divide by the base, write down the remainder and repeat with the quotient until it is 0. The number is the list of remainders read from bottom to top. The fractional part is converted by repeated multiplication: multiply by the base, the integer part is the next digit and you continue with the remaining fraction; here the digits are read from top to bottom.

Shortcut between binary, octal and hexadecimal

Since 8 = 2³ and 16 = 2⁴, each octal digit is exactly 3 bits and each hex digit is exactly 4. There is no need to go through decimal: group the bits from the radix point (padding with zeros on the left of the integer part and on the right of the fraction) and translate each group.

Decimal, binary, octal and hex conversion table

The first 16 values, worth memorizing to convert quickly between binary and hexadecimal:

Useful powers of 2: 2⁸ = 256 (one byte), 2¹⁰ = 1,024, 2¹⁶ = 65,536 and 2³² = 4,294,967,296.

Frequently asked questions

Why can't 0.1 be represented exactly in binary?

A fraction has a finite expansion in base b only if its reduced denominator contains no prime factors other than those of b. 0.1 = 1/10 and 10 = 2·5: 5 is not a factor of 2, so in binary it repeats forever (0.000110011…). That is why 0.1 + 0.2 is not exactly 0.3 in floating point; see it in the IEEE 754 converter.

How many bits do I need for a decimal number N?

⌊log₂ N⌋ + 1 bits. With n bits you can represent unsigned values from 0 to 2ⁿ − 1. For negative numbers, see the two's complement calculator.

Is there a size limit?

Not in practice: the tool works with arbitrary-precision integers, so you can convert numbers with hundreds of digits without losing accuracy. The fractional part is cut when it does not terminate.

Can I convert hex to binary directly?

Yes. Choose base 16 as the source and base 2 as the target: each hex digit becomes four bits, and the tool shows the grouping shortcut.