Editing Division by zero (section)

==Elementary arithmetic==
===The meaning of division===
{{also|Quotition and partition}}

The [[division (mathematics)|division]] <math>N/D = Q</math> can be conceptually interpreted in several ways.{{sfn|Cheng|2023|pp=75–83}}

In ''quotitive division'', the dividend <math>N</math> is imagined to be split up into parts of size <math>D</math> (the divisor), and the quotient <math>Q</math> is the number of resulting parts. For example, imagine ten slices of bread are to be made into sandwiches, each requiring two slices of bread. A total of five sandwiches can be made {{nobr|(<math>\tfrac{10}{2}=5</math>).}} Now imagine instead that zero slices of bread are required per sandwich (perhaps a [[lettuce sandwich|lettuce wrap]]). Arbitrarily many such sandwiches can be made from ten slices of bread, as the bread is irrelevant.{{sfn|Zazkis|Liljedahl|2009|page=52–53}}

The quotitive concept of division lends itself to calculation by repeated [[subtraction]]: dividing entails counting how many times the divisor can be subtracted before the dividend runs out. Because no finite number of subtractions of zero will ever exhaust a non-zero dividend, calculating division by zero in this way [[infinite loop|never terminates]].{{sfn|Zazkis|Liljedahl|2009|page=55–56}} Such an interminable division-by-zero [[algorithm]] is physically exhibited by some [[mechanical calculator]]s.<ref>{{citation|last1=Kochenburger |first1=Ralph J. |last2=Turcio |first2=Carolyn J. |year=1974 |title=Computers in Modern Society |place=Santa Barbara |publisher=Hamilton |url=https://archive.org/details/computersinmoder00koch/page/147/mode/1up?q=%22don%27t+try+to+divide+by+zero%22 |quote=Some other operations, including division, can also be performed by the desk calculator (but don't try to divide by zero; the calculator never will stop trying to divide until stopped manually).}} {{pb}} For a video demonstration, see: {{citation |title=What happens when you divide by zero on a mechanical calculator? | date=7 March 2021 |url=https://www.youtube.com/watch?v=s_hbvRTGcUI |access-date=2024-01-06 |language=en |via=YouTube }}</ref>

In ''partitive division'', the dividend <math>N</math> is imagined to be split into <math>D</math> parts, and the quotient <math>Q</math> is the resulting size of each part. For example, imagine ten cookies are to be divided among two friends. Each friend will receive five cookies {{nobr|(<math>\tfrac{10}{2}=5</math>).}} Now imagine instead that the ten cookies are to be divided among zero friends. How many cookies will each friend receive? Since there are no friends, this is an absurdity.<ref>{{harvnb|Zazkis|Liljedahl|2009|pages=53–54}}, give an example of a king's heirs equally dividing their inheritance of 12 diamonds, and ask what would happen in the case that all of the heirs died before the king's will could be executed.</ref>

[[File:Slopes as ratios.png|thumb|right|The slope of line in the plane is a ratio of vertical to horizontal coordinate differences. For a vertical line, this is {{math|1&thinsp;:&thinsp;0}}, a kind of division by zero.]]
In another interpretation, the quotient <math>Q</math> represents the [[ratio]] <math>N:D.</math><ref>In China, Taiwan, and Japan, school textbooks typically distinguish between the ''ratio'' <math>N:D</math> and the ''value of the ratio'' <math>\tfrac ND.</math> By contrast in the USA textbooks typically treat them as two notations for the same thing. {{pb}} {{citation |last1=Lo |first1=Jane-Jane |last2=Watanabe |first2=Tad |last3=Cai |first3=Jinfa |year=2004 |title=Developing Ratio Concepts: An Asian Perspective |journal=Mathematics Teaching in the Middle School |volume=9 |number=7 |pages=362–367 |doi=10.5951/MTMS.9.7.0362 |jstor=41181943 }}</ref> For example, a cake recipe might call for ten cups of flour and two cups of sugar, a ratio of <math>10:2</math> or, proportionally, <math>5:1.</math> To scale this recipe to larger or smaller quantities of cake, a ratio of flour to sugar proportional to <math>5:1</math> could be maintained, for instance one cup of flour and one-fifth cup of sugar, or fifty cups of flour and ten cups of sugar.<ref>{{citation |last1=Cengiz |first1=Nesrin |last2=Rathouz |first2=Margaret |year=2018 |title=Making Sense of Equivalent Ratios |journal=Mathematics Teaching in the Middle School |volume=24 |number=3 |pages=148–155 |doi=10.5951/mathteacmiddscho.24.3.0148 |jstor=10.5951/mathteacmiddscho.24.3.0148 |s2cid=188092067 }}</ref> Now imagine a sugar-free cake recipe calls for ten cups of flour and zero cups of sugar. The ratio <math>10:0,</math> or proportionally <math>1:0,</math> is perfectly sensible:<ref>{{citation |last1=Clark |first1=Matthew R. |last2=Berenson |first2=Sarah B. |last3=Cavey |first3=Laurie O. |year=2003 |title=A comparison of ratios and fractions and their roles as tools in proportional reasoning |journal=The Journal of Mathematical Behavior |volume=22 |number=3 |pages=297–317 |doi=10.1016/S0732-3123(03)00023-3 }}</ref> it just means that the cake has no sugar. However, the question "How many parts flour for each part sugar?" still has no meaningful numerical answer.

A geometrical appearance of the division-as-ratio interpretation is the [[slope]] of a [[straight line]] in the [[Cartesian plane]].<ref>{{citation |last=Cheng |first=Ivan |title=Fractions: A New Slant on Slope |journal=Mathematics Teaching in the Middle School |year=2010 |volume=16 |number=1 |pages=34–41 |doi=10.5951/MTMS.16.1.0034 |jstor=41183440}}</ref> The slope is defined to be the "rise" (change in vertical coordinate) divided by the "run" (change in horizontal coordinate) along the line. When this is written using the symmetrical ratio notation, a horizontal line has slope <math>0:1</math> and a vertical line has slope <math>1:0.</math> However, if the slope is taken to be a single [[real number]] then a horizontal line has slope <math>\tfrac01 = 0</math> while a vertical line has an undefined slope, since in real-number arithmetic the quotient <math>\tfrac10</math> is undefined.<ref>{{citation |last1=Cavey |first1=Laurie O. |last2=Mahavier |first2=W. Ted |year=2010 |title=Seeing the potential in students' questions |journal=The Mathematics Teacher |volume=104 |number=2 |pages=133–137 |jstor=20876802 |doi=10.5951/MT.104.2.0133 }}</ref> The real-valued slope <math>\tfrac{y}{x}</math> of a line through the origin is the vertical coordinate of the [[intersection (geometry)|intersection]] between the line and a vertical line at horizontal coordinate <math>1,</math> dashed black in the figure. The vertical red and dashed black lines are [[parallel (geometry)|parallel]], so they have no intersection in the plane. Sometimes they are said to intersect at a [[point at infinity]], and the ratio <math>1:0</math> is represented by a new number {{nobr|<math>\infty</math>;<ref>{{citation |last1=Wegman |first1=Edward J. |last2=Said |first2=Yasmin H. |year=2010 |title=Natural homogeneous coordinates |journal=Wiley Interdisciplinary Reviews: Computational Statistics |volume=2 |number=6 |pages=678–685 |doi=10.1002/wics.122 |s2cid=121947341 }}</ref>}} see {{slink|#Projectively extended real line}} below. Vertical lines are sometimes said to have an "infinitely steep" slope.

=== 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>

===Fallacies===
{{further|Mathematical fallacy}}
A compelling reason for not allowing division by zero is that allowing it leads to [[fallacies]]. 

When working with numbers, it is easy to identify an illegal division by zero. For example:

:From <math>0\times 1 = 0</math> and <math>0\times 2 = 0</math> one gets <math>0\times 1 = 0\times 2.</math>  Cancelling {{math|0}} from both sides yields <math>1 = 2</math>, a false statement.

The fallacy here arises from the assumption that it is legitimate to cancel {{math|0}} like any other number, whereas, in fact, doing so is a form of division by {{math|0}}.

Using [[elementary algebra|algebra]], it is possible to disguise a division by zero<ref name="Kaplan" /> to obtain an  [[invalid proof]]. For example:<ref>{{harvnb|Bunch|1982|page=15}}</ref>
{{block indent|em=1.6|text=Let <math>x = 1</math>. Multiply both sides by <math>x</math> to get <math>x = x^2</math>. Subtract {{math|1}} from each side to get <math display=block>x - 1 = x^2 - 1.</math> The right side can be factored, <math display=block>x - 1 = (x + 1)(x - 1).</math> Dividing both sides by {{math|''x'' − 1}} yields <math display=block>1 = x + 1.</math> Substituting {{math|1=''x'' = 1}} yields <math display=block>1 = 2.</math>
}}
This is essentially the same fallacious computation as the previous numerical version, but the division by zero was obfuscated because we wrote {{math|0}} as {{math|1=''x'' − 1}}.