Math 300

Alternative Subtraction Algorithms

Standard American Algorithm

Dr. Wilson

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.

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