Editing Division by zero (section)

=== Inverse of multiplication ===
Division is the inverse of [[multiplication]], meaning that multiplying and then dividing by the same non-zero quantity, or vice versa, leaves an original quantity unchanged; for example <math>(5 \times 3) / 3 = {}</math><math>(5 / 3) \times 3 = 5</math>.<ref>{{citation |last1=Robinson |first1=K. M. |last2=LeFevre |first2=J. A. |year=2012 |title=The inverse relation between multiplication and division: Concepts, procedures, and a cognitive framework |journal=[[Educational Studies in Mathematics]] |volume=79 |issue=3 |pages=409–428 |doi=10.1007/s10649-011-9330-5 |jstor=41413121 }}</ref> Thus a division problem such as <math>\tfrac{6}{3} = {?}</math> can be solved by rewriting it as an equivalent equation involving multiplication, <math>{?}\times 3 = 6,</math> where <math>{?}</math> represents the same unknown quantity, and then finding the value for which the statement is true; in this case the unknown quantity is <math>2,</math> because <math>2\times 3 = 6,</math> so therefore <math>\tfrac63 = 2.</math><ref>{{harvnb|Cheng|2023|page=78}}; {{harvnb|Zazkis|Liljedahl|2009|page=55}}</ref>

An analogous problem involving division by zero, <math>\tfrac{6}{0} = {?},</math> requires determining an unknown quantity satisfying <math>{?}\times 0 = 6.</math> However, any number multiplied by zero is zero rather than six, so there exists no number which can substitute for <math>{?}</math> to make a true statement.{{sfn|Zazkis|Liljedahl|2009|page=55}}

When the problem is changed to <math>\tfrac{0}{0} = {?},</math> the equivalent multiplicative statement is {{nobr|<math>{?}\times 0 = 0</math>;}} in this case ''any'' value can be substituted for the unknown quantity to yield a true statement, so there is no single number which can be assigned as the quotient <math>\tfrac{0}{0}.</math>

Because of these difficulties, quotients where the divisor is zero are traditionally taken to be ''undefined'', and division by zero is not allowed.{{sfn|Cheng|2023|pp=82–83}}<ref>{{harvnb|Bunch|1982|page=14}}</ref>