## The Subtract from the Base Algorithm

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.