Two's complement, explained

Binary on its own has no minus sign. Two’s complement is the trick computers use to represent negative numbers without needing a separate symbol for negative.

The core idea

To make a number negative in two’s complement, you invert every bit and then add one. That is the entire algorithm:

00001010   (10 in binary)
11110101   (every bit inverted)
11110110   (add 1)

11110110 is now the 8-bit two’s complement representation of −10.

Why add one at all

Inverting the bits alone gives you something called one’s complement, which has an awkward side effect: it produces two different representations of zero (00000000 and 11111111). Adding one collapses that down to a single, unambiguous zero, which is why two’s complement became the standard almost every processor uses today.

Reading the sign

In two’s complement, the leftmost bit tells you the sign: 0 means positive, 1 means negative. That is also why an 8-bit two’s complement number can represent −128 to 127, rather than a symmetric −127 to 127, one extra negative value comes from that single sign bit.

Try the full working steps in the two’s complement calculator.