3.3.4 - Binary addition

Adding bits and carrying

Binary addition works from right to left, just as decimal column addition does, but each column has only the digits 0 and 1.

The small sums are 0 + 0 = 0, 0 + 1 = 1 and 1 + 0 = 1. For 1 + 1, the total is decimal 2, written 10 in binary: write 0 in the current column and carry 1 to the next column. If a carried 1 joins two further 1s, the total is decimal 3, written 11: write 1 and carry 1.

Consider 00101101 + 00010011. These are decimal 45 and 19. Start with the rightmost column. It contains 1 + 1, so write 0 and carry 1. The next column contains 0 + 1 + the carried 1, so again write 0 and carry 1. The next contains 1 + 0 + 1, again giving 0 and a carry. Continue rather than forgetting the carry after a difficult column.

The sequence of carries reaches the 64 column, giving 01000000. The decimal check is $45+19=64$. Keep the columns aligned; shifting one operand by one place would change its value.