Hex Binary Converter

Every positional numeral system evaluates a string of digits as a sum of digit times base raised to position, so the only thing that changes between base 2, base 8, base 10 and base 16 is the size of the digit alphabet. Hexadecimal earns its place in computing through a purely arithmetic coincidence: 16 equals 2 to the fourth power, so one hex digit maps onto exactly four bits with no carry across the boundary. That group of four bits is a nibble, two nibbles make a byte, and a byte is therefore always exactly two hex characters. The mapping is a lookup rather than a computation, which is why 0xC0A80101 and 11000000 10101000 00000001 00000001 are the same 32-bit quantity written two ways. Octal, at three bits per digit, does not align with the 8-bit byte and survives now mainly in Unix file permission masks. Width and signedness are where conversions go wrong. An 8-bit field holds 0 to 255 unsigned, or -128 to 127 in two's complement, where the top bit carries a weight of negative 2^(n-1); negating a value means inverting all bits and adding one. The same bit pattern 0xFF is 255 or -1 depending entirely on the declared type, which is why this tool reports the signed reading separately. Practical limits also apply: JavaScript bitwise operations truncate to 32 bits, and integers above 2^53 lose exactness in the double-precision format, so 64-bit values should be handled with BigInt rather than parseInt. Conventional prefixes are 0x for hexadecimal, 0b for binary and 0o for octal.

Binary arithmetic was formalized by Gottfried Wilhelm Leibniz, whose 1703 paper Explication de l'Arithmetique Binaire set out addition and multiplication using only 0 and 1. It remained a mathematical curiosity until George Boole published his algebra of logic in 1854 and Claude Shannon showed in his 1937 MIT master's thesis that Boolean algebra describes the behavior of relay switching circuits, which connected binary notation to physical computing hardware. Two's complement, the representation this converter reports for negative values, was described in John von Neumann's 1945 First Draft of a Report on the EDVAC and became universal because it lets one adder circuit handle both signed and unsigned addition. Hexadecimal notation for programmers is a product of byte-oriented machine design. Early word-oriented computers used octal, which suits the 6-, 12- and 36-bit word sizes of machines like the PDP-8 and the IBM 7090. When IBM shipped System/360 in 1964 with an 8-bit byte and 32-bit words, octal no longer divided the word evenly and IBM adopted base 16 with the letters A through F, popularizing both the notation and the term hexadecimal. The 0x prefix comes from the C language developed by Dennis Ritchie at Bell Labs in the early 1970s and was inherited by C++, Java, JavaScript, Python and most of their descendants, which is why the same convention appears in memory dumps, MAC addresses and CSS color codes today.