Binary / Decimal / Hex Converter

Gottfried Wilhelm Leibniz formally described binary arithmetic in 1703, but its connection to digital electronics came much later. Claude Shannon\'s 1938 master\'s thesis at MIT — "A Symbolic Analysis of Relay and Switching Circuits" — showed that Boolean algebra and binary numbers could describe any digital circuit. This work, performed when Shannon was 21, essentially founded digital design as a mathematical discipline.

Octal (base 8) dominated early computing because many mainframes had word widths that were multiples of 3 bits: the PDP-1 (18-bit), PDP-10 (36-bit), and various IBM machines used octal for easy word-level representation. Unix inherited the convention (its "chmod 755" file-permission syntax is octal), but octal largely died with 16/32/64-bit architectures whose byte boundaries don\'t align with 3-bit groupings.

Hexadecimal (base 16) replaced octal as the standard engineering notation when 8-bit bytes became universal in the 1970s. Each hex digit maps to exactly 4 bits (half a byte, i.e. a nibble), so a byte is always 2 hex characters — visually clean and free of byte-boundary confusion. The C programming language\'s 0x prefix (1972) became the de-facto standard across most subsequent languages. Today, memory addresses, register values, and binary blobs are almost universally displayed in hex.

Type a value in any base — decimal, hex, binary, or octal — and all the other fields update live. Prefixes are respected (0x, 0b, 0o) and stripped for conversion. Pick a bit width (8, 16, 32, or 64) to constrain the signed/unsigned range; pick signed or unsigned to change how negative values are displayed.

Also shown: two\'s complement interpretation, bit-count (popcnt — useful for debugging bitmap operations), and ASCII character if the value falls in printable range. Useful for register decoding, packed-data debugging, color-code conversion, and general bit-level analysis. All math runs client-side.