Addition of Binary Numbers

Addition of binary numbers is basically the same as addition of decimal numbers. Eachsystem has an augend, an addend, a sum, and carries. The following example will refreshyour memory:

Since only two symbols, 0 and 1, are used with the binary system, only four combinations of addition are possible.

The sum of each of the first three combinations is obvious:

The fourth combination presents a different situation. The sum of 1 and 1 in any other number system is 2, but the numeral 2 does not exist in the binary system. Therefore, the sum of 1₂ and 1₂ is 10₂ (spoken as one zero base two), which is equal to 2₁₀.

Study the following examples using the four combinations mentioned above:

When a carry is produced, it is noted in the column of the next higher value or in the column immediately to the left of the one that produced the carry.

Example: Add 1011₂ and 1101₂.

Solution: Write out the problem as shown:

As we noted previously, the sum of 1 and 1 is 2, which cannot be expressed as a single digit in the binary system. Therefore, the sum of 1 and 1 produces a carry:

The following steps, with the carry indicated, show the completion of the addition:

When the carry is added, it is marked through to prevent adding it twice.

In the final step the remaining carry is brought down to the sum.

In the following example you will see that more than one carry may be produced by a single column. This is something that does not occur in the decimal system.

Example: Add 1₂, 1₂, 1₂, and 1₂

The sum of the augend and the first addend is 0 with a carry. The sum of the second and third addends is also 0 with a carry. At this point the solution resembles the following example:

The sum of the carries is 0 with a carry, so the sum of the problem is as follows:

The same situation occurs in the following example:

Add 100₂, 101₂, and 111₂

As in the previous example, the sum of the four 1s is 0 with two carries, and the sum of the two carries is 0 with one carry. The final solution will look like this:

In the addition of binary numbers, you should remember the following binary addition rules:

Now practice what you've learned by solving the following problems:

Q.9

Q.10

Q.11

Q.12

Q.13

Q.14