Editing Elementary arithmetic

{{Short description|Numbers and the basic operations on them}}
{{More citations needed|date=May 2023}}
[[File:Arithmetic symbols.svg|thumb|The symbols for elementary-level math operations. From top-left going clockwise: addition, division, multiplication, and subtraction.]]
'''Elementary arithmetic''' is a branch of [[mathematics]] involving [[addition]], [[subtraction]], [[multiplication]], and [[Division (mathematics)|division]]. Due to its low level of [[abstraction]], broad range of application, and position as the foundation of all mathematics, elementary arithmetic is generally the first branch of mathematics taught in schools.{{r|mw|bmk}}

==Numeral systems==
{{Main|Numeral system}}
In [[numeral system|numeral systems]], [[Numerical digit|digits]] are characters used to represent the value of numbers. An example of a numeral system is the predominantly used [[Hindu–Arabic numeral system|Indo-Arabic numeral system]] (0 to 9), which uses a [[Base 10|decimal]] [[positional notation]].<ref>{{Cite web |title=numeral system {{!}} mathematics {{!}} Britannica |url=https://www.britannica.com/science/numeral-system |access-date=2022-11-24 |website=www.britannica.com |at=Paragraph 2, sentence 4 |language=en |archive-date=2023-08-10 |archive-url=https://web.archive.org/web/20230810073412/https://www.britannica.com/science/numeral-system |url-status=live }}</ref> Other numeral systems include the [[Kaktovik numerals|Kaktovik system]] (often used in the [[Eskimo-Aleut]] languages of [[Alaska]], [[Canada]], and [[Greenland]]), and is a [[vigesimal]] [[positional notation]] system.<ref>{{cite web |last1=Tillinghast-Raby |first1=Amory |title=A Number System Invented by Inuit Schoolchildren Will Make Its Silicon Valley Debut |url=https://www.scientificamerican.com/author/amory-tillinghast-raby/ |website=Scientific American |access-date=24 July 2023 |archive-url=https://web.archive.org/web/20230719170919/https://www.scientificamerican.com/article/a-number-system-invented-by-inuit-schoolchildren-will-make-its-silicon-valley-debut1/ |archive-date=19 July 2023 |url-status=live}}</ref> Regardless of the numeral system used, the results of arithmetic operations are unaffected. 

==Successor function and ordering==
In elementary arithmetic, the [[Successor function|successor]] of a [[natural number]] (including zero) is the next natural number and is the result of adding one to that number. The predecessor of a natural number (excluding zero) is the previous natural number and is the result of subtracting one from that number. For example, the successor of zero is one, and the predecessor of eleven is ten ('''<math>0+1=1</math>''' and '''<math>11-1=10</math>'''). Every natural number has a successor, and every natural number except 0 has a predecessor.{{r|ma}}

The natural numbers have a [[Total order|total ordering]]. If one number is greater than (<math>></math>) another number, then the latter is less than (<math><</math>) the former. For example, three is less than eight (<math>3<8</math>), thus eight is greater than three (<math>8>3</math>). The natural numbers are also [[Well-order|well-ordered]], meaning that any subset of the natural numbers has a [[least element]].

==Counting==
{{Main|Counting#Counting in mathematics}}

Counting assigns a natural number to each object in a [[Set theory|set]], starting with 1 for the first object and increasing by 1 for each subsequent object. The number of objects in the set is the count. This is also known as the [[cardinality]] of the set.

Counting can also be the process of [[Tally marks|tallying]], the process of drawing a mark for each object in a set.

== Addition ==
[[File:Addition with carry.png|thumb|alt=Diagram of addition with carry|Example of [[Carry (arithmetic)|addition with carry]]. The black numbers are the addends, the green number is the carry, and the blue number is the sum. In the rightmost digit, the addition of 9 and 7 is 16, carrying 1 into the next pair of the digit to the left, making its addition 1 + 5 + 2 = 8. Therefore, 59 + 27 = 86.]]
{{Main|Addition}}
[[Addition]] is a mathematical operation that combines two or more numbers (called addends or summands) to produce a combined number (called the sum). The addition of two numbers is expressed with the plus sign (<math>+</math>).{{r|mpb}} It is performed according to these rules:
* The order in which the addends are added does not affect the sum. This is known as the [[commutative property]] of addition. (a + b) and (b + a) produce the same output.{{r|rosen|hall}}
* The sum of two numbers is unique; there is only one correct answer for a sums.{{r|hall}}

When the sum of a pair of digits results in a two-digit number, the "tens" digit is referred to as the "carry digit".{{r|rf}} In elementary arithmetic, students typically learn to add [[Integer|whole numbers]] and may also learn about topics such as [[Negative number|negative numbers]] and [[Fraction|fractions]].

==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>

==Multiplication==
{{Main|Multiplication}}[[Multiplication]] is a mathematical operation of repeated addition. When two numbers are multiplied, the resulting value is a product. The numbers being multiplied are multiplicands, multipliers, or factors. Multiplication can be expressed as "five times three equals fifteen," "five times three is fifteen," or "fifteen is the product of five and three."

Multiplication is represented using the multiplication sign (×), the asterisk (*), parentheses (), or a dot (⋅). The statement "five times three equals fifteen" can be written as "<math>5 \times 3 = 15</math>", "<math>5 \ast 3 = 15</math>", "<math>(5)(3) = 15</math>", or "<math>5 \cdot 3 = 15</math>".

In elementary arithmetic, multiplication satisfies the following properties{{efn|While elementary arithmetic mainly operates under the [[Set (mathematics)|set]] of [[Natural number|natural numbers]] (sometimes including 0), multiplication under other number sets can satisfy more or less properties than those listed here, such as having an [[inverse element]] in the [[Rational number|rational numbers]] and beyond, or lacking [[commutativity]] in the [[quaternions]] and higher order number sets.}}:
* [[Commutativity]]. Switching the order in a product does not change the result: <math>a \times b = b \times a</math>.
* [[Associativity]]. Rearranging the order of parentheses in a product does not change the result: <math>a \times (b \times c) = (a \times b) \times c</math>.
* [[Distributivity]]. Multiplication distributes over addition: <math>a \times (b + c) = a \times b + a \times c</math>.
* [[Identity element|Identity]]. Any number multiplied by 1 is itself: <math>a \times 1 = a</math>.
* [[Absorbing element|Zero]]. Any number multiplied by 0 is 0: <math>a \times 0 = 0</math>.

In the multiplication algorithm, the "tens" digit of the product of a pair of digits is referred to as the "carry digit".

=== Example of multiplication for a single-digit factor ===
Multiplying 729 and 3, starting on the ones column, the product of 9 and 3 is 27. 7 is written under the ones column and 2 is written above the tens column as a carry digit.

{| style="border-collapse: collapse; border-spacing: 2px; text-align:center"
|
|{{verth|Hundreds}}  
|{{verth|Tens}}  
|{{verth|'''''Ones'''''}}
|-
| || ||'''2'''||
|-
| ||7 ||2 ||9
|-
|style="border-bottom: 1px solid black;"|× ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"|3
|-
| || || ||7
|}

The product of 2 and 3 is 6, and the carry digit adds 2 to 6, so 8 is written under the tens column.

{| style="border-collapse: collapse; border-spacing: 2px; text-align:center"
|
|{{verth|Hundreds}}  
|{{verth|'''''Tens'''''}}
|{{verth|Ones}}
|-
|
|
|2
|
|-
| ||7 ||2 ||9
|-
|style="border-bottom: 1px solid black;"|× ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"|3
|-
| || ||8 ||7
|}

The product of 7 and 3 is 21, and since this is the last digit, 2 will not be written as a carry digit, but instead beside 1.

{|style="border-collapse: collapse; border-spacing: 2px; text-align:center"
|
|{{verth|'''''Hundreds'''''}}
|{{verth|Tens}}
|{{verth|Ones}}
|-
|
|
|2
|
|-
| || 7 || 2 || 9
|-
|style="border-bottom: 1px solid black;"|× ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"|3
|-
|2 ||1 ||8 ||7
|}

The result:
:<math>3 \times 729 = 2187</math>

=== Example of multiplication for multiple-digit factors ===
Multiplying 789 and 345, starting with the ones column, the product of 789 and 5 is 3945.
{|style="border-collapse: collapse; border-spacing: 2px; text-align:center"
| ||7 ||8 ||9
|-
|style="border-bottom: 1px solid black;"|× ||style="border-bottom: 1px solid black;"|3 ||style="border-bottom: 1px solid black;"|4 ||style="border-bottom: 1px solid black;"|5
|-
|3 ||9 ||4 ||5
|}

4 is in the tens digit. The multiplier is 40, not 4. The product of 789 and 40 is 31560.
{|style="border-collapse: collapse; border-spacing: 2px; text-align:center"
|-
|
| ||7 ||8 ||9
|-
|
| style="border-bottom: 1px solid black;" |× || style="border-bottom: 1px solid black;" |3 || style="border-bottom: 1px solid black;" |4 || style="border-bottom: 1px solid black;" |5
|-
|
|3 ||9 ||4 ||5
|-
|3
|1 ||5 ||6 ||0 
|}

3 is in the hundreds digits. The multiplier is 300. The product of 789 and 300 is 236700.
{| style="border-collapse: collapse; border-spacing: 2px; text-align:center"
|
|
| ||7 ||8 ||9
|-
|
|
|style="border-bottom: 1px solid black;"|× ||style="border-bottom: 1px solid black;"|3 ||style="border-bottom: 1px solid black;"|4 ||style="border-bottom: 1px solid black;"|5
|-
|
|
|3 ||9 ||4 ||5
|-
|
|3
|1 ||5 ||6 ||0 
|-
|2
|3
|6 ||7 ||0 ||0 
|}

Adding all the products,
{|style="border-collapse: collapse; border-spacing: 2px; text-align:center"
| || || || ||7 ||8 ||9
|-
|style="border-bottom: 1px solid black;"|× ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"|3 ||style="border-bottom: 1px solid black;"|4 ||style="border-bottom: 1px solid black;"|5
|-
| || || ||3 ||9 ||4 ||5
|-
| || ||3 ||1 ||5 ||6 ||0
|-
|style="border-bottom: 1px solid black;"|+ ||style="border-bottom: 1px solid black;"|2 ||style="border-bottom: 1px solid black;"|3 ||style="border-bottom: 1px solid black;"|6 ||style="border-bottom: 1px solid black;"|7 ||style="border-bottom: 1px solid black;"|0 ||style="border-bottom: 1px solid black;"|0
|-
| ||2 ||7 ||2 ||2 ||0 ||5
|}

The result:

:<math>789 \times 345 = 272205</math>

==Division==
{{Main|Division (mathematics)|Long division}}
[[Division (mathematics)|Division]] is an arithmetic operation, and the inverse of [[multiplication]], given that <math>c \times b = a</math>. 

Division can be written as <math>a \div b</math>, <math>\frac ab</math>, or {{frac|''a''|''b''}}. This can be read verbally as "''a'' divided by ''b''" or "''a'' over ''b''".

In some non-[[English language|English]]-speaking cultures{{Which|date=February 2024}}, "''a'' divided by ''b''" is written {{nowrap|''a'' : ''b''}}. In English usage, the [[colon (punctuation)|colon]] is restricted to the concept of [[ratio]]s ("''a'' is to ''b''").

In an equation ''<math>a \div b = c</math>, a'' is the dividend, ''b'' the divisor, and ''c'' the quotient. [[Division by zero]] is considered impossible at an elementary arithmetic level. 

Two numbers can be divided on paper using [[long division]]. An abbreviated version of long division, [[short division]], can be used for smaller divisors.

A less systematic method involves the concept of [[Chunking (division)|chunking]], involving subtracting more multiples from the partial remainder at each stage.

=== Example ===
Dividing 272 and 8, starting with the hundreds digit, 2 is not divisible by 8. Add 20 and 7 to get 27. The largest number that the divisor of 8 can be multiplied by without exceeding 27 is 3, so it is written under the tens column. Subtracting 24 (the product of 3 and 8) from 27 gives 3 as the [[remainder]].

{|style="border-collapse: collapse; border-spacing: 2px; text-align:center"
| ||2 ||7 ||2
|-
|style="border-bottom: 1px solid black;"|÷ ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"|8
|-
| || ||3||
|}

Going to the ones digit, the number is 2. Adding 30 (the remainder, 3, times 10) and 2 gets 32. The quotient of 32 and 8 is 4, which is written under the ones column.

{|style="border-collapse: collapse; border-spacing: 2px; text-align:center"
| ||2 ||7 ||2
|-
|style="border-bottom: 1px solid black;"|÷ ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"| ||style="border-bottom: 1px solid black;"|8
|-
| || ||3||4
|}

The result:
<math display="block">272 \div 8 = 34</math>

==== Bus stop method ====
Another method of dividing taught in some schools is the bus stop method, sometimes notated as
           <u>&nbsp;&nbsp;result&nbsp;&nbsp;</u>
   (divisor) dividend

The steps here are shown below, using the same example as above:
     <u> <span style="color: red;">0</span><span style="color: green;">3</span><span style="color: blue;">4</span> </u>     
    8|272
      <u>0</u>        ( 8 &times;  <span style="color: red;">0</span> =  0)
      <span style="color: darkorange;">2</span>7       ( 2 -  0 =  <span style="color: darkorange;">2</span>)
      <u>24</u>       ( 8 &times;  <span style="color: green;">3</span> = 24)
       <span style="color: darkcyan;">3</span>2      (27 - 24 =  <span style="color: darkcyan;">3</span>)
       <u>32</u>      ( 8 &times;  <span style="color: blue;">4</span> = 32)
        0      (32 - 32 =  0)
The result:

<math display="block">272 \div 8 = 34</math>

==Educational standards==
Elementary arithmetic is typically taught at the primary or secondary school levels and is governed by local educational standards. There has been debate about the content and methods used to teach elementary arithmetic in the United States and Canada.<ref>{{Cite web |title=Debate about Teaching style of Maths |url=https://edmontonjournal.com/news/local-news/the-great-canadian-math-debate-pt-6-math-prof-anna-stokke-responds-to-alberta-education |website=edmontonjournal.com}}</ref><ref>{{Cite web |last=Gollom |first=Mark |date=April 10, 2016 |title=Educators debate whether some math basics are 'a dead issue in the year 2016' |url=https://www.cbc.ca/news/canada/math-scores-students-canada-basics-discovery-1.3526188 }}</ref>

==See also==
*[[Early numeracy]]
*[[Elementary mathematics]]
*[[Chunking (division)]]
*[[Plus and minus signs]]
*[[Peano axioms]]
*[[Division by zero]]
*[[Real number]]
*[[Imaginary number]]
*[[Number sentence]]

==Notes==
{{notelist}}

==References==
{{Reflist|refs=

<ref name="bmk">{{cite journal
 | last1 = Björklund | first1 = Camilla
 | last2 = Marton | first2 = Ference
 | last3 = Kullberg | first3 = Angelika
 | year = 2021
 | title = What is to be learnt? Critical aspects of elementary arithmetic skills
 | journal = Educational Studies in Mathematics
 | language = en
 | volume = 107 | issue = 2 | pages = 261–284
 | doi = 10.1007/s10649-021-10045-0
 | issn = 0013-1954
 | doi-access = free
}}</ref>

<ref name="hall">{{cite book
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 | url = https://books.google.com/books?id=qqs8AAAAIAAJ&pg=PA171
 | page = 171
 | publisher = [[Cambridge University Press]]
| isbn = 978-0-521-08484-0
 }}</ref>

<ref name="ma">{{cite book
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 | last2 = Aubrey | first2 = Jason A.
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 | title = An Introduction to Proof through Real Analysis
 | url = https://books.google.com/books?id=6EkzDwAAQBAJ&pg=PA3
 | page = 3
 | publisher = John Wiley & Sons
 | isbn = 9781119314721
}}</ref>

<ref name="mpb">{{cite book
 | last1 = Musser | first1 = Gary L.
 | last2 = Peterson | first2 = Blake E.
 | last3 = Burger | first3 = William F.
 | title = Mathematics for Elementary Teachers: A Contemporary Approach
 | year = 2013
 | publisher = John Wiley & Sons
 | isbn = 978-1-118-48700-6
 | url = https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA87
 | page = 87
 |language = en
}}</ref>

<ref name="mw">{{cite book
 | last1 = Mitchelmore | first1 = Michael C.
 | last2 = White | first2 = Paul
 | title = Abstraction in Mathematics Learning
 | year = 2012
 | url = https://doi.org/10.1007/978-1-4419-1428-6_516
 | pages = 31–33
 | editor-last = Seel | editor-first = Norbert M.
 | place = Boston, MA
 | publisher = Springer US
 | language = en
 | doi = 10.1007/978-1-4419-1428-6_516
 | isbn = 978-1-4419-1428-6
 }}</ref>

<ref name="rf">{{cite book
 | last1 = Resnick | first1 = L. B.
 | last2 = Ford | first2= W. W.
 | title = Psychology of Mathematics for Instruction
 | year = 2012
 | publisher = Routledge
 | isbn = 978-1-136-55759-0
 | url = https://books.google.com/books?id=xj-j8pw2HN8C&pg=PA110
 | page = 110
 | language = en
}}</ref>


<ref name="rosen">{{cite book
 | last = Rosen | first = Kenneth
 | year = 2013
 | title = Discrete Maths and Its Applications Global Edition
 | publisher = McGraw Hill
 | isbn = 978-0-07-131501-2
}} See the [https://books.google.com/books?id=-oVvEAAAQBAJ&pg=SL1-PA1 Appendix I]. </ref>

}}
*

{{Elementary arithmetic}}
{{Authority control}}

[[Category:Elementary arithmetic| ]]
[[Category:Mathematics education]]
[[Category:Addition]]
[[Category:Subtraction]]
[[Category:Multiplication]]