Arithmetic at Machine Level

Doing Sums in Binary

Computers add binary numbers with the same carry rules as decimal, but with only two digits. The core facts are $0+0=0$, $0+1=1$, $1+0=1$, and $1+1=10$ (that is, $0$ carry $1$). Adding $011 + 001$ gives $100$: the rightmost column $1+1$ is $0$ carry $1$, the next column $1+0+\text{carry }1$ is $0$ carry $1$, and the last column receives that carry to give $1$.

Because a register holds a fixed number of bits, a sum that needs more bits than are available overflows — the extra high bit is lost. In an 8-bit register, $255 + 1$ wraps around to $0$, because the true answer $256$ needs a 9th bit that does not exist. Overflow is not an error the hardware raises by default; it is silent, and reasoning about it is a real part of low-level programming.

Negative numbers use two's complement, the near-universal scheme for signed integers. To negate a number, invert all its bits and add 1. In 8-bit two's complement, the leftmost bit is the sign bit: $0$ means non-negative, $1$ means negative. This representation has one great virtue: the ordinary binary addition circuit works correctly for signed numbers too, with no special cases — subtraction becomes "add the two's complement."

Common pitfall: assuming an integer can grow without bound. Machine integers have a fixed width, so they wrap on overflow — an 8-bit unsigned value silently goes $255 \to 0$, and a signed one goes $+127 \to -128$. Forgetting this fixed range is the source of countless real-world bugs.