Two's Complement

In two’s complement arithmetic, a negative number is stored by taking the bitwise inverse of the positive magnitude and then adding 1. This works because, for a fixed bit-width, the result is the additive inverse modulo 2^n: when you add a number to this transformed version, the carry wraps around and produces 0 in the low n bits. The formula only makes sense relative to a chosen register size, since overflow beyond that width is intentionally discarded.

Use this when converting signed integers to binary for CPUs, digital circuits, low-level debugging, or manually checking machine code behavior. It is especially important when reasoning about overflow, subtraction, or how signed values are represented in a fixed number of bits.

• Forgetting that the bit-width is fixed, so extra carry bits are discarded
• Inverting the bits without adding 1, which gives one’s complement instead
• Using the formula on a decimal number directly instead of first writing its binary form