Binary addition generalises the idea of base-10 addition to the binary number system. As usual, we have a carry bit if there’s an overflow in our operation, and a sum bit. Numbers can only be 0 or 1, so in any case where we get a sum of 2, then we actually have a carry of 1 and sum of 0.
A full adder is a digital circuit that carries out binary addition for a single bit. We can represent the sum of each binary add operation and the carry bit as:
and with standard gates:
Full adders that are chained together to do addition of multiple-bit numbers are called ripple carry adders, from the way the carry ripples from right to left.
A variant of this is the carry lookahead adder, which doesn’t propagate the carry in the same way.