Editing Division (mathematics) (section)

== Introduction ==
The simplest way of viewing division is in terms of [[quotition and partition]]: from the quotition perspective, {{math|20 / 5}} means the number of 5s that must be added to get&nbsp;20. In terms of partition, {{math|20 / 5}} means the size of each of 5 parts into which a set of size 20 is divided. For example, 20 apples divide into five groups of four apples, meaning that "twenty divided by five is equal to four". This is denoted as {{math|1=20 / 5 = 4}}, or {{math|1={{sfrac|20|5}} = 4}}.<ref name="mwdiv">{{MathWorld|id=Division|title=Division}}</ref> In the example, 20 is the dividend, 5 is the divisor, and 4 is the quotient.

Unlike the other basic operations, when dividing natural numbers there is sometimes a [[remainder]] that will not go evenly into the dividend; for example, {{math|10 / 3}} leaves a remainder of 1, as 10 is not a multiple of 3. Sometimes this remainder is added to the quotient as a [[fractional part]], so {{math|10 / 3}} is equal to {{math|{{sfrac|3|1|3}}}} or {{math|3.33...}}, but in the context of [[integer]] division, where numbers have no fractional part, the remainder is kept separately (or exceptionally, discarded or [[rounding|rounded]]).<ref name="mwintdiv">{{MathWorld|id=IntegerDivision|title=Integer Division}}</ref> When the remainder is kept as a fraction, it leads to a [[rational number]]. The set of all rational numbers is created by extending the integers with all possible results of divisions of integers.

Unlike multiplication and addition, division is not [[commutative]], meaning that {{math|''a'' / ''b''}} is not always equal to {{math|''b'' / ''a''}}.<ref>http://www.mathwords.com/c/commutative.htm {{Webarchive|url=https://web.archive.org/web/20181028172101/http://www.mathwords.com/c/commutative.htm |date=2018-10-28 }} Retrieved October 23, 2018</ref> Division is also not, in general, [[associative]], meaning that when dividing multiple times, the order of division can change the result.<ref>http://www.mathwords.com/a/associative_operation.htm {{Webarchive|url=https://web.archive.org/web/20181028042107/http://mathwords.com/a/associative_operation.htm |date=2018-10-28 }} Retrieved October 23, 2018</ref> For example, {{math|(24 / 6) / 2 {{=}} 2}}, but {{math|24 / (6 / 2) {{=}} 8}} (where the use of parentheses indicates that the operations inside parentheses are performed before the operations outside parentheses).

Division is traditionally considered as [[Left associative operator|left-associative]]. That is, if there are multiple divisions in a row, the order of calculation goes from left to right:<ref name="Order of arithmetic operations">George Mark Bergman: [https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html Order of arithmetic operations] {{Webarchive|url=https://web.archive.org/web/20170305004813/https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html |date=2017-03-05 }}</ref><ref name="The Order of Operations">Education Place: [http://eduplace.com/math/mathsteps/4/a/index.html The Order of Operations] {{Webarchive|url=https://web.archive.org/web/20170608144614/http://eduplace.com/math/mathsteps/4/a/index.html |date=2017-06-08 }}</ref>
: <math>a / b / c = (a / b) / c = a / (b \times c) \;\ne\; a/(b/c)= (a\times c)/b.</math>

Division is [[right-distributive]] over addition and subtraction, in the sense that 
: <math>\frac{a \pm b}{c} = (a \pm b) / c = (a/c)\pm (b/c) =\frac{a}{c} \pm \frac{b}{c}.</math>

This is the same for [[multiplication]], as <math>(a + b) \times c = a \times c + b \times c</math>. However, division is ''not'' [[left-distributive]], as
: <math>\frac{a}{b + c} = a / (b + c) \;\ne\; (a/b) + (a/c) = \frac{ac+ab}{bc}.</math> &nbsp; For example  <math>\frac{12}{2+4} = \frac{12}{6} = 2 ,</math> but <math>\frac{12}{2} + \frac{12}{4} = 6+3 = 9 .</math>
This is unlike the case in multiplication, which is both left-distributive and right-distributive, and thus [[distributive law|distributive]].