We need to learn four basic rules in order to perform binary addition. These rules are listed in the table below. Notice that the first three rules are quite simple since there is no difference between these binary rules and the corresponding decimal rules. The fourth rule, however, is different from decimal. Any time that we add two 1s together in binary, we generate a carry to the next column since the binary system only has two digits. In decimal we have 10 digits, so we do not generate a carry until the sum of two digits is greater than or equal to 10 (e.g. 5 + 7 = 12).

Rule 1 Rule 2 Rule 3 Rule 4

    0
 +  0
    0

    0
 +  1
    1

    1
 +  0
    1

    1
 +  1
   10

We can use these rules to derive another important rule for binary arithmetic. Consider what happens when we try to add three 1s together in binary. Let's split the problem into two addition problems and apply our rule for binary addition to find the answer.



  1. First, we apply Rule 4 to find the sum of the first two 1s.
      1
   +  1
     10
  1. Next, we take the previous result of 102 and add the final 1 to it. Notice that we use Rule 2 (0 + 1 = 1) to find the answer to the first column, and we use Rule 3 (1 + 0 = 1) to find the answer to the second column.
     10
   +  1
     11
  1. Now we have derived another rule for binary arithmetic. The sum of three 1s in binary is 112.
      1
      1
   +  1
     11

It is important to remember that in binary addition, two 1s always generate a carry to the next column. We can see this happening in the problem above. Adding the first two 1s gives us a carry to the next column and the remaining 1 becomes the value for the current column.