Math 300

Alternative Subtraction Algorithms

Three Digit Trick

Dr. Wilson

This method works for subtracting three digit numbers. Subtract the bottom number from the top and add it to 999. Of course, if you add the difference between 999 and the bottom number to the top number, you will have a bigger number than the top number, and subtraction is supposed to give you smaller answers. But you can get from here to the correct answer by subtracting 1000 and adding 1. For example

Subtract the bottom number from 999.

Now add 761 to the top number

At this point subtracting 1000 and adding 1 is quite easy

which you should be able to verify as being the correct answer.

The reason this works is if you look at what we did

999 - 238 + 516 - 1000 + 1

999 + 1 = 1000, which is cancelled out by the step where you subtract 1000 leaving us with our original problem.

The advantage with doing this is that you will never have to borrow or regroup if you subtract a 3 digit number from 999. This method can be modified for subtracting numbers of any length. However many digits are in the top number, subtract the bottom number by the number you get by taking that many 9's, and add the difference to the top number. Since the bottom number, being smaller than the top number is further away from the number with all the 9's than the top number, the sum will have one more digit than the top number and that one more digit will be a 1. Split off that 1 and add it to the resulting number and you have the correct answer to the original problem.

This method is used in base two to develop a subtraction algorithm for computers.