Subtraction by Addition

Here we see how to do subtraction using addition!

(also called the Complements Method)

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 10s 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 10s complement is the number to add to make 10 (or 100, 1000, and so on, depending on how many digits we have)

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

Example: the 10s complement of 85 is 15, because 85+15=100

Example: the 10s complement of 111 is 889, because 111+889=1000

Calculating the Complement

The 10s 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 "ones" position
Skip over any zeros
Then:

For the first digit that isn't zero:

find what makes it to 10

For all other digits:

find what makes 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 10s 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):

And we have found that 653 − 372 = 281 (check it if you want!)

What if the number we are subtracting has fewer digits?

How can we, for example, do 4567 − 56 ?

After taking the complement we just fill any missing spaces with 9s.

Example: 4567 − 56

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

4567
+9944
14511

Discard the 1 on the left and we get: 4511

That case would be easier using Quick Subtraction, but it does show how this "complement, add, discard" method works.

Extra 1?

Why discard the 1? When we find the complement, we are actually adding a large number (like 1,000). By discarding the "1" at the end, we are simply subtracting that 1,000 back out!

This method assumes the result is not negative (the top number is bigger).

If the number we subtract is bigger, there will be no extra 1 to discard, and the answer will be negative (we need a different approach).

Now, you can practice with Subtraction Worksheets, or the questions below.

1596, 1597, 3420, 3421, 1598, 1599, 3422, 3423, 5037, 5038