Editing Arithmetic (section)

== Operations ==
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|image1            = Apple addition.svg
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|footer            = Arithmetic operations underlie many everyday occurrences, like when putting four apples from one bag together with three apples from another bag (top image) or when distributing nine apples equally among three children (bottom image).
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Arithmetic operations are ways of combining, transforming, or manipulating numbers. They are [[Function (mathematics)|functions]] that have numbers both as input and output.<ref>{{multiref | {{harvnb|Nagel|2002|p=179}} | {{harvnb|Husserl|Willard|2012|pp=[https://books.google.com/books?id=lxftCAAAQBAJ&pg=PR44 XLIV–XLV]}} | {{harvnb|O'Leary|2015|p=[https://books.google.com/books?id=Ci6kBgAAQBAJ&pg=PA190 190]}} }}</ref> The most important operations in arithmetic are [[addition]], [[subtraction]], [[multiplication]], and [[Division (mathematics)|division]].<ref>{{multiref | {{harvnb|Rising|Matthews|Schoaff|Matthew|2021|p=[https://books.google.com/books?id=hjVZEAAAQBAJ&pg=PA110 110]}} | {{harvnb|Bukhshtab|Pechaev|2020}} | {{harvnb|Nagel|2002|pp=177, 179–180}} }}</ref> Further operations include [[exponentiation]], extraction of [[nth root|roots]], and [[logarithm]].<ref>{{multiref | {{harvnb|Bukhshtab|Pechaev|2020}} | {{harvnb|Burgin|2022|pp=57, 77}} | {{harvnb|Adamowicz|1994|p=299}} | {{harvnb|Nagel|2002|pp=177, 179–180}} }}</ref> If these operations are performed on variables rather than numbers, they are sometimes referred to as [[algebraic operations]].<ref>{{multiref | {{harvnb|Khan|Graham|2018|pp=[https://books.google.com/books?id=vy73DwAAQBAJ&pg=PA9 9–10]}} | {{harvnb|Smyth|1864|p=[https://books.google.com/books?id=BqQZAAAAYAAJ&pg=PA55 55]}} }}</ref>

Two important concepts in relation to arithmetic operations are [[identity element]]s and [[inverse element]]s. The identity element or neutral element of an operation does not cause any change if it is applied to another element. For example, the identity element of addition is 0 since any sum of a number and 0 results in the same number. The inverse element is the element that results in the identity element when combined with another element. For instance, the [[additive inverse]] of the number 6 is -6 since their sum is 0.<ref>{{multiref | {{harvnb|Tarasov|2008|pp=[https://books.google.com/books?id=pHK11tfdE3QC&pg=PA57 57–58]}} | {{harvnb|Mazzola|Milmeister|Weissmann|2004|p=[https://books.google.com/books?id=CkFCCA-2sRgC&pg=PA66 66]}} | {{harvnb|Krenn|Lorünser|2023|p=[https://books.google.com/books?id=RRi2EAAAQBAJ&pg=PA8 8]}} }}</ref>

There are not only inverse elements but also [[inverse function|inverse operations]]. In an informal sense, one operation is the inverse of another operation if it undoes the first operation. For example, subtraction is the inverse of addition since a number returns to its original value if a second number is first added and subsequently subtracted, as in <math>13 + 4 - 4 = 13</math>. Defined more formally, the operation "<math>\star</math>" is an inverse of the operation "<math>\circ</math>" if it fulfills the following condition: <math>t \star s = r</math> if and only if <math>r \circ s = t</math>.<ref>{{multiref | {{harvnb|Kay|2021|pp=[https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA44 44–45]}} | {{harvnb|Wright|Ellemor-Collins|Tabor|2011|p=[https://books.google.com/books?id=3yqdEAAAQBAJ&pg=PA136 136]}} }}</ref>

[[Commutativity]] and [[associativity]] are laws governing the order in which some arithmetic operations can be carried out. An operation is commutative if the order of the arguments can be changed without affecting the results. This is the case for addition, for instance, <math>7 + 9</math> is the same as <math>9 + 7</math>. Associativity is a rule that affects the order in which a series of operations can be carried out. An operation is associative if, in a series of two operations, it does not matter which operation is carried out first. This is the case for multiplication, for example, since <math>(5 \times 4) \times 2</math> is the same as <math>5 \times (4 \times 2)</math>.<ref>{{multiref | {{harvnb|Krenn|Lorünser|2023|p=[https://books.google.com/books?id=RRi2EAAAQBAJ&pg=PA8 8]}} | {{harvnb|Mazzola|Milmeister|Weissmann|2004|p=[https://books.google.com/books?id=CkFCCA-2sRgC&pg=PA66 66]}} }}</ref>

=== Addition and subtraction ===
{{main|Addition|Subtraction}}

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|alt2              = Diagram showing subtraction
|footer            = Addition and subtraction
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Addition is an arithmetic operation in which two numbers, called the addends, are combined into a single number, called the sum. The symbol of addition is <math>+</math>. Examples are <math>2 + 2 = 4</math> and <math>6.3 + 1.26 = 7.56</math>.<ref>{{multiref | {{harvnb|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA87 87]}} | {{harvnb|Romanowski|2008|p=303}} }}</ref> The term [[summation]] is used if several additions are performed in a row.<ref>{{harvnb|Burgin|2022|p=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA25 25]}}</ref> Counting is a type of repeated addition in which the number 1 is continuously added.<ref>{{harvnb|Confrey|1994|p=[https://books.google.com/books?id=4MwZlzgvaGYC&pg=PA308 308]}}</ref>

Subtraction is the inverse of addition. In it, one number, known as the subtrahend, is taken away from another, known as the minuend. The result of this operation is called the difference. The symbol of subtraction is <math>-</math>.<ref>{{multiref | {{harvnb|Romanowski|2008|p=303}} | {{harvnb|Musser|Peterson|Burger|2013|pp=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA93 93–94]}} | {{harvnb|Kay|2021|pp=[https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA44 44–45]}} | {{harvnb|Wright|Ellemor-Collins|Tabor|2011|p=[https://books.google.com/books?id=3yqdEAAAQBAJ&pg=PA136 136]}} }}</ref> Examples are <math>14 - 8 = 6</math> and <math>45 - 1.7 = 43.3</math>. Subtraction is often treated as a special case of addition: instead of subtracting a positive number, it is also possible to add a negative number. For instance <math>14 - 8 = 14 + (-8)</math>. This helps to simplify mathematical computations by reducing the number of basic arithmetic operations needed to perform calculations.<ref>{{multiref | {{harvnb|Wheater|2015|p=[https://books.google.com/books?id=Q7R3EAAAQBAJ&pg=PP19 19]}} | {{harvnb|Wright|Ellemor-Collins|Tabor|2011|pp=[https://books.google.com/books?id=3yqdEAAAQBAJ&pg=PA136 136–137]}} | {{harvnb|Achatz|Anderson|2005|p=[https://books.google.com/books?id=YOdtemSmzQQC&pg=PA18 18]}} }}</ref>

The additive identity element is 0 and the additive inverse of a number is the negative of that number. For instance, <math>13 + 0 = 13</math> and <math>13 + (-13) = 0</math>. Addition is both commutative and associative.<ref>{{multiref | {{harvnb|Mazzola|Milmeister|Weissmann|2004|p=[https://books.google.com/books?id=CkFCCA-2sRgC&pg=PA66 66]}} | {{harvnb|Romanowski|2008|p=303}} | {{harvnb|Nagel|2002|pp=179–180}} }}</ref>

=== Multiplication and division ===
{{main|Multiplication|Division (mathematics)}}

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Multiplication  is an arithmetic operation in which two numbers, called the multiplier and the multiplicand, are combined into a single number called the [[Product (mathematics)|product]].<ref>{{multiref | {{harvnb|Romanowski|2008|p=303}} | {{harvnb|Musser|Peterson|Burger|2013|pp=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA101 101–102]}} }}</ref>{{efn|Some authors use a different terminology and refer to the first number as multiplicand and the second number as the multiplier.<ref>{{harvnb|Cavanagh|2017|p=275}}</ref> }} The symbols of multiplication are <math>\times</math>, <math>\cdot</math>, and *. Examples are <math>2 \times 3 = 6</math> and <math>0.3 \cdot 5 = 1.5</math>. If the multiplicand is a natural number then multiplication is the same as repeated addition, as in <math>2 \times 3 = 2 + 2 + 2</math>.<ref>{{multiref | {{harvnb|Romanowski|2008|p=304}} | {{harvnb|Wright|Ellemor-Collins|Tabor|2011|p=[https://books.google.com/books?id=3yqdEAAAQBAJ&pg=PA136 136]}} | {{harvnb|Musser|Peterson|Burger|2013|pp=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA101 101–102]}} }}</ref>

Division is the inverse of multiplication. In it, one number, known as the dividend, is split into several equal parts by another number, known as the divisor. The result of this operation is called the [[quotient]]. The symbols of division are <math>\div</math> and <math>/</math>. Examples are <math>48 \div 8 = 6</math> and <math>29.4 / 1.4 = 21</math>.<ref>{{multiref | {{harvnb|Romanowski|2008|p=303}} | {{harvnb|Wheater|2015|p=[https://books.google.com/books?id=Q7R3EAAAQBAJ&pg=PP19 19]}} | {{harvnb|Wright|Ellemor-Collins|Tabor|2011|p=[https://books.google.com/books?id=3yqdEAAAQBAJ&pg=PA136 136]}} }}</ref> Division is often treated as a special case of multiplication: instead of dividing by a number, it is also possible to multiply by its [[Multiplicative inverse|reciprocal]]. The reciprocal of a number is 1 divided by that number. For instance, <math>48 \div 8 = 48 \times \tfrac{1}{8}</math>.<ref>{{multiref | {{harvnb|Kay|2021|p=[https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA117 117]}} | {{harvnb|Wheater|2015|p=[https://books.google.com/books?id=Q7R3EAAAQBAJ&pg=PP19 19]}} | {{harvnb|Wright|Ellemor-Collins|Tabor|2011|pp=[https://books.google.com/books?id=3yqdEAAAQBAJ&pg=PA136 136–137]}} }}</ref>

The [[multiplicative identity]] element is 1 and the multiplicative inverse of a number is the reciprocal of that number. For example, <math>13 \times 1 = 13</math> and <math>13 \times \tfrac{1}{13} = 1</math>. Multiplication is both commutative and associative.<ref>{{multiref | {{harvnb|Mazzola|Milmeister|Weissmann|2004|p=[https://books.google.com/books?id=CkFCCA-2sRgC&pg=PA66 66]}} | {{harvnb|Romanowski|2008|pp=303–304}} | {{harvnb|Nagel|2002|pp=179–180}} }}</ref>

=== Exponentiation and logarithm ===
{{main|Exponentiation|Logarithm}}

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|footer            = Exponentiation and logarithm
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Exponentiation is an arithmetic operation in which a number, known as the base, is raised to the power of another number, known as the exponent. The result of this operation is called the power. Exponentiation is sometimes expressed using the symbol ^ but the more common way is to write the exponent in [[superscript]] right after the base. Examples are <math>2^4 = 16</math> and <math>3</math>^<math>3 = 27</math>. If the exponent is a natural number then exponentiation is the same as repeated multiplication, as in <math>2^4 = 2 \times 2 \times 2 \times 2</math>.<ref>{{multiref | {{harvnb|Musser|Peterson|Burger|2013|pp=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA117 117–118]}} | {{harvnb|Kay|2021|pp=[https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA27 27–28]}} }}</ref>{{efn|If the exponent is 0 then the result is 1, as in <math>7^0 = 1</math>. The only exception is <math>0^0</math>, which is not defined.<ref>{{harvnb|Musser|Peterson|Burger|2013|p=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA120 120]}}</ref>}}

Roots are a special type of exponentiation using a fractional exponent. For example, the [[square root]] of a number is the same as raising the number to the power of <math>\tfrac{1}{2}</math> and the [[cube root]] of a number is the same as raising the number to the power of <math>\tfrac{1}{3}</math>. Examples are <math>\sqrt{4} = 4^{\frac{1}{2}} = 2</math> and <math>\sqrt[3]{27} = 27^{\frac{1}{3}} = 3</math>.<ref>{{multiref | {{harvnb|Kay|2021|p=[https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA118 118]}} | {{harvnb|Klose|2014|p=[https://books.google.com/books?id=rG7iBQAAQBAJ&pg=PA105 105]}} }}</ref>

Logarithm is the inverse of exponentiation. The logarithm of a number <math>x</math> to the base <math>b</math> is the [[exponent]] to which <math>b</math> must be raised to produce <math>x</math>. For instance, since <math>1000 = 10^3</math>, the logarithm base 10 of 1000 is 3. The logarithm of <math>x</math> to base <math>b</math> is denoted as <math>\log_b (x)</math>, or without parentheses, <math>\log_b x</math>, or even without the explicit base, <math>\log x</math>, when the base can be understood from context. So, the previous example can be written <math>\log_{10} 1000 = 3</math>.<ref>{{multiref | {{harvnb|Kay|2021|pp=[https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA121 121–122]}} | {{harvnb|Rodda|Little|2015|p=[https://books.google.com/books?id=cb_dCgAAQBAJ&pg=PA7 7]}} }}</ref>

Exponentiation and logarithm do not have general identity elements and inverse elements like addition and multiplication. The neutral element of exponentiation in relation to the exponent is 1, as in <math>14^1 = 14</math>. However, exponentiation does not have a general identity element since 1 is not the neutral element for the base.<ref>{{multiref | {{harvnb|Kay|2021|p=[https://books.google.com/books?id=aw81EAAAQBAJ&pg=PA117 117]}} | {{harvnb|Mazzola|Milmeister|Weissmann|2004|p=[https://books.google.com/books?id=CkFCCA-2sRgC&pg=PA66 66]}} }}</ref> Exponentiation and logarithm are neither commutative nor associative.<ref>{{multiref | {{harvnb|Sally|Sally (Jr.)|2012|p=[https://books.google.com/books?id=Ntjq07-FA_IC&pg=PA3 3]}} | {{harvnb|Klose|2014|pp=[https://books.google.com/books?id=rG7iBQAAQBAJ&pg=PA107 107–108]}} }}</ref>