Editing Elementary arithmetic (section)

==Subtraction==
{{Main|Subtraction}}
[[Subtraction]] evaluates the difference between two numbers, where the minuend is the number being subtracted from, and the subtrahend is the number being subtracted. It is represented using the minus sign (<math>-</math>). The minus sign is also used to notate negative numbers.<ref>{{Cite web |title=Math Operations With Basic Rules and Clear Examples. |url=https://blendedlearningmath.com/pages/basic-operation-in-mathematics/ |access-date=2025-01-15 |website=blendedlearningmath |language=en-US}}</ref>

Subtraction is not commutative, which means that the order of the numbers can change the final value; <math>3-5</math> is not the same as <math>5-3</math>. In elementary arithmetic, the minuend is always larger than the subtrahend to produce a positive result.

Subtraction is also used to separate, [[Combination|combine]] (e.g., find the size of a subset of a specific set), and find quantities in other contexts.

There are several methods to accomplish subtraction. The [[traditional mathematics]] method subtracts using methods suitable for hand calculation.<ref>{{Cite web |title=Everyday Mathematics4 at Home |url=https://everydaymath.uchicago.edu/parents/4th-grade/em4-at-home/vocab/4-1-9-us-traditional-subtraction.html |website=Everyday Mathematics Online |access-date=December 26, 2022}}</ref> [[Reform mathematics]] is distinguished generally by the lack of preference for any specific technique, replaced by guiding students to invent their own methods of computation.

American schools teach a method of subtraction using borrowing.<ref>{{Cite web |title=Subtraction Algorithms - Department of Mathematics at UTSA |url=https://mathresearch.utsa.edu/wiki/index.php?title=Subtraction_Algorithms |access-date=2024-04-01 |website=mathresearch.utsa.edu |language=en}}</ref> A subtraction problem such as <math>86-39</math> is solved by borrowing a 10 from the tens place to add to the ones place in order to facilitate the subtraction. Subtracting 9 from 6 involves borrowing a 10 from the tens place, making the problem into <math>70+16-39</math>. This is indicated by crossing out the 8, writing a 7 above it, and writing a 1 above the 6. These markings are called "crutches", which were invented by [[William A. Brownell]], who used them in a study, in November 1937.<ref>{{Cite web |last=Ross |first=Susan |title=Subtraction in the United States: An Historical Perspective |url=http://math.coe.uga.edu/tme/issues/v10n2/5ross.pdf |url-status=dead |archive-url=https://web.archive.org/web/20170811133911/http://math.coe.uga.edu/tme/issues/v10n2/5ross.pdf |archive-date=August 11, 2017 |access-date=June 25, 2019 |website=Microsoft Word - Issue 2 -9/23/}}</ref>

The Austrian method, also known as the additions method, is taught in certain European countries{{Which|date=July 2024}}. In contrast to the previous method, no borrowing is used, although there are crutches that vary according to certain countries.<ref>{{Cite web |last=Klapper |first=Paul |date=1916 |title=The Teaching of Arithmetic: A Manual for Teachers. pp. 177 |url=https://archive.org/details/teachingarithme00klapgoog/page/n190/mode/2up |access-date=2016-03-11}}</ref><ref>{{Cite web |last=Smith |first=David Eugene |date=1913 |title=The Teaching of Arithmetic. pp. 77 |url=https://archive.org/details/bub_gb_A7NJAAAAIAAJ/page/n85/mode/2up |access-date=2016-03-11}}</ref> The method of addition involves augmenting the subtrahend. This transforms the previous problem into <math>(80+16)-(39+10)</math>. A small 1 is marked below the subtrahend digit as a reminder.

=== Example ===
Subtracting the numbers 792 and 308, starting with the ones column, 2 is smaller than 8. Using the borrowing method, 10 is borrowed from 90, reducing 90 to 80. This changes the problem to <math>12-8</math>.

{|style="border-collapse: collapse; border-spacing: 2px; text-align:center"
|
|{{verth|Hundreds}}  
|{{verth|Tens}}  
|{{verth|'''''Ones'''''}}
|-
| || ||'''8'''||'''<sup>1</sup>2'''
|-
| ||7 ||<s>9</s> ||<s>2</s>
|-
|style="border-bottom: 1px solid black;"|− ||style="border-bottom: 1px solid black;"|3 ||style="border-bottom: 1px solid black;"|0 ||style="border-bottom: 1px solid black;"|8
|-
| || || ||4
|}

In the tens column, the difference between 80 and 0 is 80.

{|style="border-collapse: collapse; border-spacing: 2px; text-align:center"
|
|{{verth|Hundreds}}  
|{{verth|'''''Tens'''''}}
|{{verth|Ones}}
|-
| || ||'''8'''||<sup>1</sup>'''2'''
|-
| ||7 ||<s>9</s> ||<s>2</s>
|-
|style="border-bottom: 1px solid black;"|− ||style="border-bottom: 1px solid black;"|3 ||style="border-bottom: 1px solid black;"|0 ||style="border-bottom: 1px solid black;"|8
|-
| || ||8||4
|}

In the hundreds column, the difference between 700 and 300 is 400.

{|style="border-collapse: collapse; border-spacing: 2px; text-align:center"
|
|{{verth|'''''Hundreds'''''}}  
|{{verth|Tens}}
|{{verth|Ones}}
|-
| || ||'''8'''||<sup>1</sup>'''2'''
|-
| ||7 ||<s>9</s> ||<s>2</s>
|-
|style="border-bottom: 1px solid black;"|− ||style="border-bottom: 1px solid black;"|3 ||style="border-bottom: 1px solid black;"|0 ||style="border-bottom: 1px solid black;"|8
|-
| ||4||8||4
|}

The result:

<math display="block">792 - 308 = 484</math>