1             0             1
  0             1             1
  1             0             0  
 10            01            10

N-bit Binary Addition Algorithm

Adding up a column of bits is as easy as counting. It is also easy to do electronically. When you take a course in digital logic you will probably build a circuit to do it.

Now let us look at the full, n-bit, binary addition algorithm. The algorithm takes two operands and produces one result. An operand is the data that an algorithm operates on.

To add two N-bit (representations of) integers: Proceed from right-to-left, column-by-column, until you reach the leftmost column. For each column, perform 1-bit addition. Write the carry-out of each column above the column to its left. The bit is the left column's carry-in.

The example adds two 4-bit operands. The initial conditions are shown on the left.

First, add up the bits in the right-most column. There is no carry-in to this column, so all you need to do is count the bits. The result is 1 with a carry of 0. Write the 1 as the result for the column, and put the carry above the next column.

Continue right-to-left column by column until all the columns have been added. The carry-out of the left-most column does not go into the sum, but should be written to the left of the other carry bits.

With 4-bit arithmetic, the inputs are 4 bits and the output is 4 bits. With paper and pencil, the inputs and output be any size, but with electronics operations are performed with a fixed number of bits (usually). The left-most carry-bit does not go into the result. (But it is examined by the electronics to determine if overflow happened.)

 0110     =      (base 10) 

 0111     =      (base 10)  
______
 
 1101     =      (base 10)