Chapter 3.3, Problem 12P

The standard subtraction algorithm described in the text is not the only correct subtraction algorithm. Some people use the following algorithm instead: Line up the numbers as in the standard algorithm, and subtract column by column, proceeding from right to left. The only difference between this new algorithm and our standard one is in regrouping. To regroup with the new algorithm, move one column to the left and cross out the digit at the bottom of the column, replacing it with that digit plus 1. (So replace an 8 with a 9, with a 10, etc.) Then , as in the standard algorithm, replace the digit at the top of the original column with the original number plus 10. The following example shows the steps of this new algorithm:

 

1. Use the new algorithm to solve $524-198$ and $1003-95$ . Verify that the new algorithm gives correct answers.
2. Explain why it makes sense that this new algorithm gives correct answers to subtraction problems. What is the reasoning behind this new algorithm?
3. What are some advantages and disadvantages of this new algorithm compared with our common one?