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.)
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
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 "ones" position
- Skip over any zeros
|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 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:
(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 ?
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:
Then discard the extra "1" on the left, and the answer is:
In that case it is easier to use Quick Subtraction, but it does show how this "complement, add, discard" method works.
Now, you can practice with these Subtraction Worksheets