Two's Complement Calculator

Formula

How this is calculated

Two's complement is the standard way computers store signed integers in a fixed number of bits (w). The Mode selector chooses the direction: Decimal → Binary takes a signed decimal value, while Binary → Decimal takes a raw bit pattern. The Value field accepts either a signed decimal integer or a string of 0s and 1s, and Bit width sets w to 8, 16, 32, or 64.

To encode a decimal v, non-negative values are written directly in binary and zero-padded to w bits. Negative values are encoded as v + 2^w, which is equivalent to inverting the bits of |v| − 1 (the invert-and-add-one rule); the result is truncated to w bits. To decode a w-bit unsigned reading u, the most significant bit is the sign: if it is 1 the true value is u − 2^w, otherwise the value is simply u. The hexadecimal output is the same unsigned bit pattern written in base 16.

A w-bit field represents integers from −2^(w−1) to 2^(w−1) − 1; the calculator rejects decimals outside this range and rejects bit patterns longer than w. Because 64-bit values exceed JavaScript's safe integer limit, the math is done with BigInt for exact binary and hexadecimal output, though the headline decimal card may round very large 64-bit magnitudes when displayed.