Subtraction by Addition

(also called the Complements Method)

Here we see how to do subtraction using addition!

(I don't recommend this for normal subtraction work, but it is still a valid and interesting way to subtract. And in some cases it may save time.)

Steps

Follow these steps:

  • take the "complement" of the number we are subtracting (we will see how soon)
  • add it to to the number we are subtracting from
  • discard the extra "1" on the left

Complement

The "complement" is the number to add to make 10 (or 100, 1000, etc, depending on how many digits we have)

Example The complement of 3 is 7, because 3+7=10 (we add 7 to make 10)

Example: the complement of 85 is 15, because 85+15=100

Example: the complement of 111 is 889, because 111+889=1000

Calculating the Complement

The complement is easy to find!

The basic idea is to find the difference between each digit and 9. That will get us to "999...", so we only need to add 1 to make it "1000..."

In practice it is easy to follow this method:

  • Start at the the "units" position
  • Skip over any zeros
  • Then:
For the first digit that isn't zero:   find what would make it to 10
For all other digits:   find what would make it to 9

Here are two examples:

(You can check it works by adding the number and its complement,
for example 372+628=1000)

With a bit of experience the "what adds to 10" or "what adds to 9" becomes automatic, and taking the complement becomes quick and easy.

Here is another example where we have to skip over some zeros:

Example: What is the complement of 1700?

  • Skip over the two zeros
  • The "10" complement of the 7 is 3,
  • The "9" complement of 1 is 8,

So the answer is:

8300

(Check: 1700+8300 = 10000)

Now Add Them!

Now add the two numbers (using column addition), but don't forget to discard the extra "1" on the left.

Here are the 3 steps (complement, add, discard):

What if the number we are subtracting has less digits?

How would we, for example, do 4567 - 56 ?

The complement should have as many digits as the number we are subtracting from. In practice this just means filling the missing spaces on the right with 9s.

Example: 4567 - 56

Well, the complement of 56 is 44 , but we need to "pad it" out to 4 digits, so we end up with 9944. Now we add them:

 4567
+9944
14511

Then discard the extra "1" on the left, and the answer is:

4511

In that case it would have been easier to use Quick Subtraction, but it does show how this "complement, add, discard" method works.

Now, you can practice with these Subtraction Worksheets