The basic steps of binary addition are the same as decimal addition. To add two numbers we simply add together the corresponding digits in each number from right to left (l.s.d. to m.s.d.).
If the addition of any pair of digits produces a two digit number, we enter the l.s.d. in the column we are currently working on and carry the m.s.d. over into the addition of the next pair of digits.
The same process is applied in binary addition. We will limit our consideration to the addition of two binary numbers. So most of the time, we only need to add pairs of bits, at each step of the calculation. However, sometimes we will have to add three bits together, when the previous step resulted in a carry bit.
(Similar to decimal, if the addition of a pair of bits results in a two bit answer we carry the m.s.b. into the addition of the next pair of digits in the number.)
So to add two binary numbers we only need to be able to perform the binary additions shown below at any stage in the calculation.
a) All possible combinations of addition for a pair of bits
b) When the pair of bits are both 1 and there is also a carry bit from the previous stage.