Editing Subtraction (section)

===Comparing the two main methods===
Both these methods break up the subtraction as a process of one digit subtractions by place value. Starting with a least significant digit, a subtraction of the subtrahend:
:''s''<sub>''j''</sub> ''s''<sub>''j''−1</sub> ... ''s''<sub>1</sub>
from the minuend
:''m''<sub>''k''</sub> ''m''<sub>''k''−1</sub> ... ''m''<sub>1</sub>,
where each ''s''<sub>''i''</sub> and ''m''<sub>''i''</sub> is a digit, proceeds by writing down {{nowrap|''m''<sub>1</sub> − ''s''<sub>1</sub>}}, {{nowrap|''m''<sub>2</sub> − ''s''<sub>2</sub>}}, and so forth, as long as ''s''<sub>''i''</sub> does not exceed ''m''<sub>''i''</sub>. Otherwise, ''m''<sub>''i''</sub> is increased by 10 and some other digit is modified to correct for this increase. The American method corrects by attempting to decrease the minuend digit ''m''<sub>''i''+1</sub> by one (or continuing the borrow leftwards until there is a non-zero digit from which to borrow). The European method corrects by increasing the subtrahend digit ''s''<sub>''i''+1</sub> by one.

'''Example:''' 704 − 512.

{{equation|
   \begin{array}{rrrr}
        & \color{Red}-1 \\
        & C & D & U  \\
        & 7 & 0 & 4 \\   
        & 5 & 1 & 2 \\
      \hline
        & 1 & 9 & 2 \\
   \end{array}
   \begin{array}{l}
       { \color{Red}\longleftarrow \rm carry }\\
       \\
       \longleftarrow \; \rm Minuend\\
       \longleftarrow \; \rm Subtrahend\\
       \longleftarrow \rm{Rest \; or \; Difference}\\
   \end{array}
}}

The minuend is 704, the subtrahend is 512. The minuend digits are {{nowrap|1=''m''<sub>3</sub> = 7}}, {{nowrap|1=''m''<sub>2</sub> = 0}} and {{nowrap|1=''m''<sub>1</sub> = 4}}. The subtrahend digits are {{nowrap|1=''s''<sub>3</sub> = 5}}, {{nowrap|1=''s''<sub>2</sub> = 1}} and {{nowrap|1=''s''<sub>1</sub> = 2}}. Beginning at the one's place, 4 is not less than 2 so the difference 2 is written down in the result's one's place. In the ten's place, 0 is less than 1, so the 0 is increased by 10, and the difference with 1, which is 9, is written down in the ten's place. The American method corrects for the increase of ten by reducing the digit in the minuend's hundreds place by one. That is, the 7 is struck through and replaced by a 6. The subtraction then proceeds in the hundreds place, where 6 is not less than 5, so the difference is written down in the result's hundred's place. We are now done, the result is 192.

The Austrian method does not reduce the 7 to 6. Rather it increases the subtrahend hundreds digit by one. A small mark is made near or below this digit (depending on the school). Then the subtraction proceeds by asking what number when increased by 1, and 5 is added to it, makes 7. The answer is 1, and is written down in the result's hundreds place.

There is an additional subtlety in that the student always employs a mental subtraction table in the American method. The Austrian method often encourages the student to mentally use the addition table in reverse. In the example above, rather than adding 1 to 5, getting 6, and subtracting that from 7, the student is asked to consider what number, when increased by 1, and 5 is added to it, makes 7.