Complementary Subtraction

If you do any work with computers, you will soon find out that most digital systems cannot subtract - they can only add. You are going to need a method of adding that gives the results of subtraction. Does that sound confusing? Really, it is quite simple. A COMPLEMENT is used for our subtractions. A complement is something used to complete something else.

In most number systems you will find two types of complements. The first is the amount necessary to complete a number up to the highest number in the number system. In the decimal system, this would be the difference between a given number and all 9s. This is called the nines complement or the radix-1 or R's-1 complement. As an example, the nines complement of 254 is 999 minus 254, or 745.

The second type of complement is the difference between a number and the next higher power of the number base. As an example, the next higher power of 10 above 999 is 1,000. The difference between 1,000 and 254 is 746. This is called the tens complement in the decimal number system. It is also called the radix or R's complement. We will use complements to subtract. Let's look at the magic of this process. There are three important points we should mention before we start: (1) Never complement the minuend in a problem, (2) always disregard any carry beyond the number of positions of the largest of the original numbers, and (3) add the R's complement of the original subtrahend to the original minuend. This will have the same effect as subtracting the original number. Let's look at a base ten example in which we subtract 38 from 59:

Now let's look at the number system that most computers use, the binary system. Just as the decimal system, had the nines (R's-1) and tens (R's) complement, the binary system has two types of complement methods. These two types are the ones (R's-1) complement and the twos (R's) complement. The binary system R's-1 complement is the difference between the binary number and all 1s. The R's complement is the difference between the binary number and the next higher power of 2.

Let's look at a quick and easy way to form the R's-1 complement. To do this, change each 1 in the original number to 0 and each 0 in the original number to 1 as has been done in the example below.

1011011₂

0100100₂ R's-1 complement

There are two methods of achieving the R's complement. In the first method we perform the R's-1 complement and then add 1. This is much easier than subtracting the original number from the next higher power of 2. If you had subtracted, you would have had to borrow.

Saying it another way, to reach the R's complement of any binary number, change all 1s to 0s and all 0s to 1s, and then add 1.

As an example let's determine the R's complement of 10101101₂:

The second method of obtaining the R's complement will be demonstrated on the binary number 00101101100₂.

Step 1 - Start with the LSD, working to the MSD, writing the digits as they are up to and including the first one.

Step 2-Now R's-1 complement the remaining digits:

Now let's R's complement the same number using both methods:

Now let's do some subtracting by using the R's complement method. We will go through the subtraction of 3₁₀ from 9₁₀ (0011₂from 1001₂):

Step 1 - Leave the minuend alone:

1001₂ remains 1001₂

Step 2 - Using either method, R's complement the subtrahend:

1101₂ R's complement of subtrahend

Step 3 - Add the R's complement found in step 2 to the minuend of the original problem:

Step 4 - Remember to discard any carry beyond the size of the original number. Our original problem had four digits, so we discard the carry that expanded the difference to five digits. This carry we disregard is significant to the computer. It indicates that the difference is positive. Because we have a carry, we can read the difference directly without any further computations. Let's check our answer:

If we do not have a carry, it indicates the difference is a negative number. In that case, the difference must be R's complemented to produce the correct answer.

Let's look at an example that will explain this for you.

Subtract 9₁₀ from 5₁₀ (1001₂ from 0101₂):

Step 1 - Leave the minuend alone:

0101₂ remains 0101₂

Step 2 - R's complement the subtrahend:

0111₂ R's complement of subtrahend

Step 3 - Add the R's complement found in step 2 to the minuend of the original problem:

Step 4 - We do not have a carry; and this tells us, and any computer, that our difference (answer) is negative. With no carry, we must R's complement the difference in step 3. We will then have arrived at the answer (difference) to our original problem. Let's do this R's complement step and then check our answer:

0100₂ R's complement of difference in step 3

Remember, we had no carry in step 3. That showed us our answer was going to be negative. Make sure you indicate the difference is negative. Let's check the answer to our problem:

Try solving a few subtraction problems by using the complement method:

Q.21 Subtract:

Q.22 Subtract:

Q.23 Subtract: