Math 300

Alternative Subtraction Algorithms

The Subtract from the Base Algorithm

Dr. Wilson

In this method, if you have a place where the bottom number is bigger than the top number, subtract the bottom number from ten and add the result to the top number. Of course when you do this you will also have to make an adjustment in the next place by either decreasing the top digit by one as in the standard algorithm, or increasing the bottom digit by one as in the Austrian method. For example

In the one's place, 7 is bigger than 3. Subtract 7 from 10 getting 3. Add this 3 to the 3 which we find on top and get a 6 in the one's place. Then if you either decrease the 8 by 1 to get 7 and subtract 5 or increase the 5 by 1 to get 6 which you then take from 8, you will get a 2 in the ten's place.

The justification for this method is the same as the justification for the standard algorithm and the Austrian method. If we look at the picture which we used to justify both of those methods

we see that after regrouping to get the 13 ones, you could take all 7 of the ones that you are subtracting from the 10 ones that you got from the borrowing or regouping process. After you take the 7 ones from those 10 ones, the 3 ones that are left wind up going with the 3 ones that you started with on top.

This method is called the "Subtract from the Base" method because it works in any base. If you are working in base ten, as we are here, then you would subtract the bottom nuber from ten and add it to the top number. If you were working in another base like base 12, you would subtract the bottom number from 12 and add it to the top.

The advantage of this method is that it cuts down on the number of subtraction facts that the students have to learn. In the standard approach, students have to have facts like 13 - 7 = 6 memorized. These borrowing facts, as they are called, where the top number is bigger than 10 and the bottom number is bigger than the number in the one's place on top, are probably the most difficult for students to assimilate. One reason would be that the numbers are bigger than the facts that you use when you are not borrowing. Moreover, if the top number is bigger than 10, students will run out of fingers if they try to do it on their fingers. With this method, students need only memorize the borrowing facts where the top number is ten.