Adding Binary Numbers

Adding two binary numbers works in the same way as column addition with denary: write the numbers one above the other so the place values line up, then add one column at a time, starting at the rightmost bit (the 1s column) and moving left.

The rules for a single column

Each column holds only 1s and 0s, so there are just four single-bit cases when nothing is carried in. The answer to 1 + 1 is the surprising one: you cannot write 2 in binary, so the column drops back to 0 and you carry 1 into the column on its left, where it is worth 2.

The last row matters because a column may already have a carry sitting above it, giving three bits to add. Work through it: 1 + 1 makes 0 carry 1, and adding the third 1 turns that 0 back into 1, still carrying 1.

Keeping track of carries

Write each carry in a row above the numbers, positioned over the next column to the left, so you cannot forget it. If a carry travels through several columns in a row, the row above fills with 1s, and each of those columns needs its carry included when you add it.

A worked 8-bit example

Add 01011010 (90) and 00110110 (54), using the place values 128 down to 1 as the column headings.

Reading the result row gives 10010000, which is 128 + 16 = 144. The denary check confirms it, because 90 + 54 = 144. Converting your answer back to denary like this is the quickest way to spot a carry you missed.

A carry leaving the 128 column has no column of its own inside eight bits; that situation is called overflow and is covered in Overflow in Binary Addition.