## Austrian Subtraction

This algorithm is used in Europe, and if you have students who
first learned how to subtract in Europe, they may be using this
method. The difference is that, if you have to borrow or regroup,
instead of decreasing the digit in the next place on top by one, you
increase the digit in the next place on the bottom by one. In our
example

instead of changing the 8 to a 7, you change the 5 to a 6. In both
methods you change the 3 to a 13. the way this is accomplished
notationally is to put the 1 that you would use to make the 13
between the 3 and the 5.

The little one serves two purposes. It tells you that you are
subtracting 7 from 13, and you can think of it as being added to the
5 to make a 6 which you subtract from the 8

The justification for the Austrian method is exactly the same as
the justification for the standard algorithm. If we look at the
picture we used to justify the standard
algorithm,

the one ten that we regrouped into 10 ones is an extra ten that
has been taken away. As a result, if you count the number of tens
that are x'ed out, you will see that there are a total of 6 big x's.

People who use the Austrian method are impressed by how much
neater it is than the standard algorithm, particularly if you have a
long problem. If you consider

the standard algorithm looks like

which is pretty messy with all of the cross outs and rewrites.
Even if you do all the cross outs and rewrites in your head so that
your paper doesn't get messed up, its a lot to keep in your head.
With the Austrian method the problem looks like

and you subtract 8 from 12 to get 4 in the one's place, 7 from 13
to get 6 in the ten's place, 6 from 11 to get 5 in the hundred's
place, and 5 from 7 to get 2 in the thousand's place.