## Standard American Algorithm

The following algorithm for dealing with the case where a digit in
the bottom is bigger than the digit in the same place in the top is
taught practically universally in America, and has been for quite
some time. For example, suppose you want to subtract

When you go to subtract the 7 ones from the 3 ones, you don't have
enough ones. However, the top number is bigger than the bottom
number, so you can take 7 away from the top number. To get more ones,
you need to go to the ten's column.

If we realize that 83 is 8 tens and 3 ones we can represent the 83
with the following picture,

where the three little circles represent ones and the eight
cylinders represent containers which each contain ten of the ones. If
we try to take 7 ones from the 3 ones that we have, we will need some
more ones. To get more ones, we open one of the cylinders to get ten
more ones.

In the Standard American Algorithm this was originally called
borrowing. More recently people have maintained that borrowing is not
the best way of thinking of this process. You aren't going to give
the ten ones you borrowed back to the tens. There has been a move on
for quite some time to call this process "regrouping". We are
regrouping the 8 tens and 3 ones into 7 tens and 13 ones. The
notation which is used in the standard algorithm is

Now we can take 7 ones from the 13 ones and 3 tens from the 7 tens
that are left after borrowing or regrouping.

We write