Hex to Decimal to Binary Converter, Plus Octal

How the Base Converter works

Hex to decimal to binary converter: positional notation

A base is nothing more than how many distinct digits you have before you carry. Decimal has ten, binary two, octal eight, hexadecimal sixteen. The number itself never changes — 255, 0xFF and 11111111 are the same quantity written three ways — and being fluent in the translation is most of what makes low-level code readable.

Hexadecimal exists because it maps cleanly onto bits. One hex digit is exactly four bits, so a byte is always two hex digits and a 32-bit word is always eight. That fixed relationship is why colours, memory addresses, hashes and byte dumps are all written in hex: you can read the bit pattern straight off the page without arithmetic. Decimal has no such alignment, which is why 4278190080 tells you nothing and 0xFF000000 immediately reads as a full alpha channel.

Octal survives mostly in one place. Three bits per digit made it natural on early machines with word sizes divisible by three, and it lost to hexadecimal when eight-bit bytes won. It remains in Unix file permissions, where read, write and execute form a perfect three-bit group and 755 is faster to say than rwxr-xr-x. It also lurks as a hazard: in C and several languages a leading zero means octal, so 010 is eight, which has caused real bugs in date handling.

Prefixes tell the reader and the parser which base you meant. 0b for binary, 0o for octal and 0x for hexadecimal are near-universal in modern languages, and this page reads all three. Without a prefix, a string of digits is genuinely ambiguous, which is why auto-detect here falls back to decimal unless it sees a letter from a to f. When you write a literal in code, use the prefix even where the language would guess correctly.

Bit width is the practical question behind most conversions. A value needing nine bits will not fit in a byte, and a 33-bit value overflows a 32-bit signed integer and wraps to a negative number. The significant-bit count above is the number of bits after leading zeroes, which is what determines whether a value fits a field, a colour channel, a subnet mask or a database column. It is also how you check a bit-shift did what you expected.

Grouping makes long binary readable. Four-bit groups line up with hex digits and eight-bit groups line up with bytes; this page groups by four so you can read each nibble against the hexadecimal row directly underneath. Underscores serve the same purpose in source code and most modern languages accept them in numeric literals.

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