Editing Subtraction (section)

==Subtraction by hand==
===Austrian method===
Example:{{Citation needed|date=February 2023}}
<gallery>
File:Vertical Subtraction Method B Step 1.JPG|1 + ... = 3
File:Vertical Subtraction Method B Step 2.JPG|The difference is written under the line.
File:Vertical Subtraction Method B Step 3.JPG|9 + ... = 5<br>The required sum (5) is too small.
File:Vertical Subtraction Method B Step 4.JPG|So, we add 10 to it and put a 1 under the next higher place in the subtrahend.
File:Vertical Subtraction Method B Step 5.JPG|9 + ... = 15<br>Now we can find the difference as before.
File:Vertical Subtraction Method B Step 6.JPG|(4 + 1) + ... = 7
File:Vertical Subtraction Method B Step 7.JPG|The difference is written under the line.
File:Vertical Subtraction Method B Step 8.JPG|The total difference.
</gallery>

===Subtraction from left to right===
Example:{{Citation needed|date=February 2023}}
<gallery>
File:LeftToRight Subtraction Step 1.JPG|7 − 4 = 3<br>This result is only penciled in.
File:LeftToRight Subtraction Step 2.JPG|Because the next digit of the minuend is smaller than the subtrahend, we subtract one from our penciled-in-number and mentally add ten to the next.
File:LeftToRight Subtraction Step 3.JPG|15 − 9 = 6
File:LeftToRight Subtraction Step 4.JPG|Because the next digit in the minuend is not smaller than the subtrahend, we keep this number.
File:LeftToRight Subtraction Step 5.JPG|3 − 1 = 2
</gallery>

===American method===
In this method, each digit of the subtrahend is subtracted from the digit above it starting from right to left. If the top number is too small to subtract the bottom number from it, we add 10 to it; this 10 is "borrowed" from the top digit to the left, which we subtract 1 from. Then we move on to subtracting the next digit and borrowing as needed, until every digit has been subtracted. Example:{{Citation needed|date=February 2023}}
<gallery>
File:Vertical Subtraction Method A Step 1.JPG|3 − 1 = ...
File:Vertical Subtraction Method A Step 2.JPG|We write the difference under the line.
File:Vertical Subtraction Method A Step 3.JPG|5 − 9 = ...<br> The minuend (5) is too small!
File:Vertical Subtraction Method A Step 4.JPG|So, we add 10 to it. The 10 is "borrowed" from the digit on the left, which goes down by 1.
File:Vertical Subtraction Method A Step 5.JPG|15 − 9 = ...<br> Now the subtraction works, and we write the difference under the line.
File:Vertical Subtraction Method A Step 6.JPG|6 − 4 = ...
File:Vertical Subtraction Method A Step 7.JPG|We write the difference under the line.
File:Vertical Subtraction Method A Step 8.JPG|The total difference.
</gallery>

===Trade first===
A variant of the American method where all borrowing is done before all subtraction.<ref>[https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction The Many Ways of Arithmetic in UCSMP Everyday Mathematics] {{Webarchive|url=https://web.archive.org/web/20140225135251/https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction |date=2014-02-25 }} Subtraction: Trade First</ref>

Example:
<gallery>
File:Trade First Subtraction Step 1.JPG|1 − 3 = not possible.<br>We add a 10 to the 1. Because the 10 is "borrowed" from the nearby 5, the 5 is lowered by 1.
File:Trade First Subtraction Step 2.JPG|4 − 9 = not possible.<br>So we proceed as in step 1.
File:Trade First Subtraction Step 3.JPG|Working from right to left:<br>11 − 3 = 8
File:Trade First Subtraction Step 4.JPG|14 − 9 = 5
File:Trade First Subtraction Step 5.JPG|6 − 4 = 2
</gallery>

===Partial differences===
The partial differences method is different from other vertical subtraction methods because no borrowing or carrying takes place. In their place, one places plus or minus signs depending on whether the minuend is greater or smaller than the subtrahend. The sum of the partial differences is the total difference.<ref>[http://ouronlineschools.org/Schools/NC/Demoschool/4thGrade/Math/PartialDifferences.htm Partial-Differences Subtraction] {{Webarchive|url=https://web.archive.org/web/20140623021239/http://ouronlineschools.org/Schools/NC/Demoschool/4thGrade/Math/PartialDifferences.htm |date=2014-06-23 }}; [https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction The Many Ways of Arithmetic in UCSMP Everyday Mathematics] {{Webarchive|url=https://web.archive.org/web/20140225135251/https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction |date=2014-02-25 }} Subtraction: Partial Differences</ref>

Example:
<gallery>
File:Partial-Differences Subtraction Step 1.JPG|The smaller number is subtracted from the greater:<br>700 − 400 = 300<br>Because the minuend is greater than the subtrahend, this difference has a plus sign.
File:Partial-Differences Subtraction Step 2.JPG|The smaller number is subtracted from the greater:<br>90 − 50 = 40<br>Because the minuend is smaller than the subtrahend, this difference has a minus sign.
File:Partial-Differences Subtraction Step 3.JPG|The smaller number is subtracted from the greater:<br>3 − 1 = 2<br>Because the minuend is greater than the subtrahend, this difference has a plus sign.
File:Partial-Differences Subtraction Step 4.JPG|+300 − 40 + 2 = 262
</gallery>

===Nonvertical methods===
====Counting up====
Instead of finding the difference digit by digit, one can count up the numbers between the subtrahend and the minuend.<ref>[https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction The Many Ways of Arithmetic in UCSMP Everyday Mathematics] {{Webarchive|url=https://web.archive.org/web/20140225135251/https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction |date=2014-02-25 }} Subtraction: Counting Up</ref>

Example:
1234 − 567 = can be found by the following steps:
* {{nowrap|1=567 + '''3''' = 570}}
* {{nowrap|1=570 + '''30''' = 600}}
* {{nowrap|1=600 + '''400''' = 1000}}
* {{nowrap|1=1000 + '''234''' = 1234}}
Add up the value from each step to get the total difference: {{nowrap|1=3 + 30 + 400 + 234 = 667}}.

====Breaking up the subtraction====
Another method that is useful for [[Mental calculation|mental arithmetic]] is to split up the subtraction into small steps.<ref>[https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction The Many Ways of Arithmetic in UCSMP Everyday Mathematics] {{Webarchive|url=https://web.archive.org/web/20140225135251/https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction |date=2014-02-25 }} Subtraction: Left to Right Subtraction</ref>

Example:
1234 − 567 = can be solved in the following way:
* 1234 − '''500''' = 734
* 734 − '''60''' = 674
* 674 − '''7''' = 667

====Same change====
The same change method uses the fact that adding or subtracting the same number from the minuend and subtrahend does not change the answer. One simply adds the amount needed to get zeros in the subtrahend.<ref>[https://sites.google.com/a/oswego308.org/msimester/home/math/algorithms/subtraction The Many Ways of Arithmetic in UCSMP Everyday Mathematics] Subtraction: Same Change Rule</ref>

Example:

"1234 − 567 =" can be solved as follows:
* {{nowrap|1=1234 − 567 = 1237 − 570 =}} {{nowrap|1=1267 − 600 = 667}}