Number Base Converter (Binary, Octal, Hex)

This number base converter switches a whole number between binary, octal, decimal and hexadecimal instantly. Number bases are just different ways to write the same quantity. Enter a value, tell the converter which base it is in, and it shows the equivalent in binary, octal, decimal and hexadecimal all at once.

Place value in any base

decimal 255 = 2·100 + 5·10 + 5·1
binary  11111111 = 128+64+32+16+8+4+2+1
hex     FF = 15·16 + 15·1

Every base works the same way — each digit is worth the digit times the base raised to its position.

Why hex and binary pair up

One hexadecimal digit covers exactly four binary bits, so FF is 11111111. That is why hex is the shorthand of choice for colours, memory addresses and byte values.

Worked example

Entering 255 in decimal:

binary  11111111    = 128+64+32+16+8+4+2+1
octal   377         = 3×64 + 7×8 + 7×1 = 192+56+7
hex     FF          = 15×16 + 15×1 = 240+15
bits    8

Every line is the same quantity, which is the whole point of the tool. Run it the other way and 11111111 typed as base 2 returns 255, FF typed as base 16 returns 255, and 377 typed as base 8 returns 255.

A power of two shows the pattern even more clearly. 1024 is 10000000000 in binary — a single 1 followed by ten zeros, needing 11 bits — 2000 in octal and 400 in hexadecimal. Each trailing zero digit is one more factor of the base.

What the input accepts

Every base is validated against its own digit set, so a mistake is caught rather than quietly reinterpreted:

A leading 0x or 0X is stripped to match how hex is written in code, so 0xFF, FF and ff all give 255. Underscores and spaces inside the value are removed too, which keeps grouped digits readable: 1111_1111 and FF FF both parse.

Only non-negative whole numbers are supported. Negatives are refused, and so are fractions — and that is not an arbitrary restriction. 0.1 has no exact binary representation at all, being the infinitely repeating 0.0001100110011…, which is the same reason decimal money arithmetic goes wrong in floating-point code.

Where bases turn up, and the one real limit

Two hex digits are exactly one byte, so hexadecimal is how colours (#FF8800 is three bytes), memory dumps and raw byte values get written. Octal survives in Unix file permissions, where 755 reads as three groups of three bits — rwxr-xr-x. Binary is the language of flags and bitmasks, where each bit is one independent on/off setting.

The limit is precision, and it is worth knowing before you trust a very large answer. Conversion runs on JavaScript numbers, which are 64-bit doubles, so results are exact only up to 2⁵³ − 1 = 9,007,199,254,740,991 — thirteen hexadecimal digits or 53 binary digits. Past that, integer arithmetic begins losing the low-order digits and the answer becomes approximate; you would need a big-integer library rather than a browser calculator. Within that range every conversion here is exact, with no rounding at any step. Roman Numeral Converter does the same job for a non-positional system.

Private and instant

Conversions run locally in your browser using integer arithmetic — nothing is uploaded and no account is needed.