View source for Division by zero

{{Short description|Class of mathematical expression}}
{{other uses}}
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[[File: Reciprocal function.png |thumb|alt= Graph showing the diagrammatic representation of limits tending to infinity|The reciprocal function {{math|1=''y'' = {{sfrac|1|''x''}}}}. As {{mvar|x}} approaches zero from the right, {{mvar|y}} tends to positive infinity. As {{mvar|x}} approaches zero from the left, {{mvar|y}} tends to negative infinity.]]

In [[mathematics]], '''division by zero''', [[division (mathematics)|division]] where the divisor (denominator) is [[0|zero]], is a unique and problematic special case. Using [[fraction]] notation, the general example can be written as <math>\tfrac a0</math>, where <math>a</math> is the dividend (numerator).

The usual definition of the [[quotient]] in [[elementary arithmetic]] is the number which yields the dividend when [[multiplication|multiplied]] by the divisor. That is, <math>c = \tfrac ab</math> is equivalent to <math>c \cdot b = a.</math> By this definition, the quotient <math>q = \tfrac{a}{0}</math> is nonsensical, as the product <math>q \cdot 0</math> is always <math>0</math> rather than some other number <math>a.</math> Following the ordinary rules of [[elementary algebra]] while allowing division by zero can create a [[mathematical fallacy]], a subtle mistake leading to absurd results. To prevent this, the arithmetic of [[real number]]s and more general numerical structures called [[field (mathematics)|field]]s leaves division by zero [[undefined (mathematics)|undefined]], and situations where division by zero might occur must be treated with care.  Since any number multiplied by zero is zero, the expression [[0/0|<math>\tfrac{0}{0}</math>]] is also undefined.

[[Calculus]] studies the behavior of [[function (mathematics)|functions]] in the [[limit (mathematics)|limit]] as their input tends to some value. When a [[real function]] can be expressed as a fraction whose denominator tends to zero, the output of the function becomes arbitrarily large, and is said to "[[Limit (mathematics)#Infinity in limits of functions|tend to infinity]]", a type of [[mathematical singularity]]. For example, the [[reciprocal function]], <math>f(x) = \tfrac 1x,</math> tends to infinity as <math>x</math> tends to <math>0.</math> When both the numerator and the denominator tend to zero at the same input, the expression is said to take an [[Indeterminate form#Indeterminate form 0/0|indeterminate form]], as the resulting limit depends on the specific functions forming the fraction and cannot be determined from their separate limits.

As an alternative to the common convention of working with fields such as the real numbers and leaving division by zero undefined, it is possible to define the result of division by zero in other ways, resulting in different number systems. For example, the quotient <math>\tfrac{a}{0}</math> can be defined to equal zero; it can be defined to equal a new explicit [[point at infinity]], sometimes denoted by the [[infinity symbol]] {{nobr|<math>\infty</math>;}} or it can be defined to result in signed infinity, with positive or negative sign depending on the sign of the dividend. In these number systems division by zero is no longer a special exception per se, but the point or points at infinity involve their own new types of exceptional behavior.

In [[computer|computing]], an error may result from an attempt to divide by zero. Depending on the context and the type of number involved, dividing by zero may evaluate to [[Extended real number line|positive or negative infinity]], return a special [[NaN|not-a-number]] value, or [[Crash (computing)|crash]] the program, among other possibilities.

==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|1= ''x'' = 1}}. Multiply both sides by {{mvar|x}} 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}}.

==Early attempts==
The ''[[Brāhmasphuṭasiddhānta]]'' of [[Brahmagupta]] (c. 598–668) is the earliest text to treat [[0|zero]] as a number in its own right and to define operations involving zero.<ref name="Kaplan">{{citation |last=Kaplan |first=Robert |title=The Nothing That Is: A Natural History of Zero |url=https://archive.org/details/nothingthatisnat00kapl |url-access=registration |publisher=Oxford University Press |year=1999 |location=New York |pages=[https://archive.org/details/nothingthatisnat00kapl/page/68 68–75] |isbn=978-0-19-514237-2}}</ref> According to Brahmagupta,

<blockquote>A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.</blockquote>

In 830, [[Mahāvīra (mathematician)|Mahāvīra]] unsuccessfully tried to correct the mistake Brahmagupta made in his book ''Ganita Sara Samgraha'': "A number remains unchanged when divided by zero."<ref name="Kaplan"/>

[[Bhāskara II]]'s ''Līlāvatī'' (12th century) proposed that division by zero results in an infinite quantity,<ref>{{citation |last=Roy |first=Rahul |journal=Resonance |volume=8 |number=1 |date=January 2003 |title=Babylonian Pythagoras' Theorem, the Early History of Zero and a Polemic on the Study of the History of Science |pages=30–40 |url=https://www.ias.ac.in/describe/article/reso/008/01/0030-0040 |doi=10.1007/BF02834448 }}</ref>

<blockquote>A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.</blockquote>

Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to <math display="inline">\tfrac{a}{0}</math> is contained in [[Anglo-Irish people|Anglo-Irish]] philosopher [[George Berkeley]]'s criticism of [[Calculus#Limits and infinitesimals|infinitesimal calculus]] in 1734 in ''[[The Analyst]]'' ("ghosts of departed quantities").<ref>{{citation
 | last = Cajori | first = Florian | author-link = Florian Cajori
 | journal = The Mathematics Teacher
 | volume = 22 | issue = 6 | jstor = 27951153
 | pages = 366–368
 | title = Absurdities due to division by zero: An historical note| year = 1929 | doi = 10.5951/MT.22.6.0366 }}.</ref>

== Calculus ==

[[Calculus]] studies the behavior of [[function (mathematics)|functions]] using the concept of a [[limit (mathematics)|limit]], the value to which a function's output tends as its input tends to some specific value. The notation <math display=inline> \lim_{x \to c} f(x) = L</math> means that the value of the function <math>f</math> can be made arbitrarily close to <math>L</math> by choosing <math>x</math> sufficiently close to <math>c.</math>

In the case where the limit of the [[real function]] <math>f</math> increases without bound as <math>x</math> tends to <math>c,</math> the function is not defined at <math>x,</math> a type of [[mathematical singularity]]. Instead, the function is said to "[[limit (mathematics)#Infinity in limits of functions|tend to infinity]]", denoted <math display=inline> \lim_{x \to c} f(x) = \infty,</math> and its [[Graph of a function|graph]] has the line <math>x=c</math> as a vertical [[asymptote]]. While such a function is not formally defined for <math>x = c,</math> and the [[infinity symbol]] <math>\infty</math> in this case does not represent any specific [[real number]], such limits are informally said to "equal infinity". If the value of the function decreases without bound, the function is said to "tend to negative infinity", <math>-\infty.</math> In some cases a function tends to two different values when <math>x</math> tends to <math>c</math> from above {{nobr|(<math>x \to c^+</math>)}} and below {{nobr|(<math>x \to c^-</math>)}}; such a function has two distinct [[one-sided limit]]s.<ref>{{citation |last1=Herman |first1=Edwin |chapter-url=https://openstax.org/books/calculus-volume-1/pages/2-2-the-limit-of-a-function |title=Calculus |chapter=2.2 The Limit of a Function |volume=1 |last2=Strang |first2=Gilbert |year=2023 |publisher=OpenStax |isbn=978-1-947172-13-5 |location=Houston |oclc=1022848630 |display-authors=etal |author2-link=Gilbert Strang |page=454}}</ref>

A basic example of an infinite singularity is the [[reciprocal function]], <math>f(x) = 1/x,</math> which tends to positive or negative infinity as <math>x</math> tends to {{nobr|<math>0</math>:}}

<math display=block>
\lim_{x \to 0^+} \frac1x = +\infty,\qquad 
\lim_{x \to 0^-} \frac1x = -\infty.
</math>

In most cases, the limit of a quotient of functions is equal to the quotient of the limits of each function separately,

<math display=block>
\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\displaystyle \lim_{x \to c} f(x)}{\displaystyle \lim_{x \to c} g(x)}.
</math>

However, when a function is constructed by dividing two functions whose separate limits are both equal to <math>0,</math> then the limit of the result cannot be determined from the separate limits, so is said to take an [[indeterminate form]], informally written <math>\tfrac00.</math> (Another indeterminate form, <math>\tfrac \infty \infty,</math> results from dividing two functions whose limits both tend to infinity.) Such a limit may equal any real value, may tend to infinity, or may not converge at all, depending on the particular functions.  For example, in 

<math display=block> \lim_{x \to 1} \dfrac{x^2 - 1}{x - 1},</math> 

the separate limits of the numerator and denominator are <math>0</math>, so we have the indeterminate form <math>\tfrac00</math>, but simplifying the quotient first shows that the limit exists: 

<math display=block>
\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1} \frac{(x - 1)(x + 1)}{x - 1}
= \lim_{x \to 1} (x + 1)
= 2.
</math>

== Alternative number systems ==

===Extended real line===
The [[affinely extended real numbers]] are obtained from the [[real number]]s <math>\R</math> by adding two new numbers <math>+\infty</math> and <math>-\infty,</math> read as "positive infinity" and "negative infinity" respectively, and representing [[point at infinity|points at infinity]]. With the addition of <math>\pm \infty,</math> the concept of a "limit at infinity" can be made to work like a finite limit. When dealing with both positive and negative extended real numbers, the expression <math>1/0</math> is usually left undefined. However, in contexts where only non-negative values are considered, it is often convenient to define <math>1/0 = +\infty</math>.

===Projectively extended real line===
The set <math>\mathbb{R}\cup\{\infty\}</math> is the [[projectively extended real line]], which is a [[one-point compactification]] of the real line. Here <math>\infty</math> means an unsigned infinity or [[point at infinity]], an infinite quantity that is neither positive nor negative. This quantity satisfies <math>-\infty = \infty</math>, which is necessary in this context. In this structure, <math>\frac{a}{0} = \infty</math> can be defined for nonzero {{math|''a''}}, and <math>\frac{a}{\infty} = 0</math> when {{math|''a''}} is not <math>\infty</math>. It is the natural way to view the range of the [[tangent function]] and cotangent functions of [[trigonometry]]: {{math|tan(''x'')}} approaches the single point at infinity as {{math|''x''}} approaches either {{math|+{{sfrac|π|2}}}} or {{math|−{{sfrac|π|2}}}} from either direction.

This definition leads to many interesting results. However, the resulting algebraic structure is not a [[Field (mathematics)|field]], and should not be expected to behave like one. For example, <math>\infty+\infty</math> is undefined in this extension of the real line.

===Riemann sphere===
The subject of [[complex analysis]] applies the concepts of calculus in the [[complex numbers]]. Of major importance in this subject is the [[extended complex numbers]] <math>\C \cup\{\infty\},</math> the set of complex numbers with a single additional number appended, usually denoted by the [[infinity symbol]] <math>\infty</math> and representing a [[point at infinity]], which is defined to be contained in every [[Domain (mathematical analysis)|exterior domain]], making those its [[topology|topological]] [[neighborhood (topology)|neighborhoods]].

This can intuitively be thought of as wrapping up the infinite edges of the complex plane and pinning them together at the single point <math>\infty,</math> a [[one-point compactification]], making the extended complex numbers topologically equivalent to a [[sphere]]. This equivalence can be extended to a metrical equivalence by mapping each complex number to a point on the sphere via inverse [[stereographic projection]], with the resulting [[spherical distance]] applied as a new definition of distance between complex numbers; and in general the geometry of the sphere can be studied using complex arithmetic, and conversely complex arithmetic can be interpreted in terms of spherical geometry. As a consequence, the set of extended complex numbers is often called the [[Riemann sphere]]. The set is usually denoted by the symbol for the complex numbers decorated by an asterisk, overline, tilde, or circumflex, for example <math>\hat\C = \C \cup\{\infty\}.</math>

In the extended complex numbers, for any nonzero complex number <math>z,</math> ordinary complex arithmetic is extended by the additional rules <math>\tfrac{z}{0}=\infty,</math> <math>\tfrac{z}{\infty} = 0,</math> <math>\infty + 0 = \infty,</math> <math>\infty + z = \infty,</math> <math>\infty \cdot z = \infty.</math> However, <math>\tfrac{0}{0}</math>, <math>\tfrac{\infty}{\infty}</math>, and <math>0\cdot\infty</math> are left undefined.

==Higher mathematics==
The four basic operations – addition, subtraction, multiplication and division – as applied to whole numbers (positive integers), with some restrictions, in elementary arithmetic are used as a framework to support the extension of the realm of numbers to which they apply. For instance, to make it possible to subtract any whole number from another, the realm of numbers must be expanded to the entire set of [[integer]]s in order to incorporate the negative integers. Similarly, to support division of any integer by any other, the realm of numbers must expand to the [[rational number]]s. During this gradual expansion of the number system, care is taken to ensure that the "extended operations", when applied to the older numbers, do not produce different results. Loosely speaking, since division by zero has no meaning (is ''undefined'') in the whole number setting, this remains true as the setting expands to the [[real number|real]] or even [[complex number]]s.<ref>{{harvnb|Klein|1925|page=63}}</ref>

As the realm of numbers to which these operations can be applied expands there are also changes in how the operations are viewed. For instance, in the realm of integers, subtraction is no longer considered a basic operation since it can be replaced by addition of signed numbers.<ref>{{harvnb|Klein|1925|page=26}}</ref> Similarly, when the realm of numbers expands to include the rational numbers, division is replaced by multiplication by certain rational numbers. In keeping with this change of viewpoint, the question, "Why can't we divide by zero?", becomes "Why can't a rational number have a zero denominator?". Answering this revised question precisely requires close examination of the definition of rational numbers.

In the modern approach to constructing the field of real numbers, the rational numbers appear as an intermediate step in the development that is founded on [[set theory]]. First, the natural numbers (including zero) are established on an axiomatic basis such as [[Peano axioms|Peano's axiom system]] and then this is expanded to the [[ring of integers]]. The next step is to define the rational numbers keeping in mind that this must be done using only the sets and operations that have already been established, namely, addition, multiplication and the integers. Starting with the set of [[ordered pair]]s of integers, {{math|{(''a'', ''b'')}<nowiki/>}} with {{math|''b'' ≠ 0}}, define a [[binary relation]] on this set by {{math|(''a'', ''b'') ≃ (''c'', ''d'')}} if and only if {{math|1=''ad'' = ''bc''}}. This relation is shown to be an [[equivalence relation]] and its [[equivalence class]]es are then defined to be the rational numbers. It is in the formal proof that this relation is an equivalence relation that the requirement that the second coordinate is not zero is needed (for verifying [[Transitive relation|transitivity]]).<ref>{{harvnb|Schumacher|1996|page=149}}</ref><ref>{{harvnb|Hamilton|1982|page=19}}</ref><ref>{{harvnb|Henkin|Smith|Varineau|Walsh|2012|page=292}}</ref>

Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures.

===Non-standard analysis===
In the [[hyperreal number]]s, division by zero is still impossible, but division by non-zero [[infinitesimal]]s is possible.<ref>{{citation|last=Keisler |first=H. Jerome |title=Elementary Calculus: An Infinitesimal Approach |url=https://people.math.wisc.edu/~hkeisler/calc.html |orig-year=1986 |year=2023 |publisher=Prindle, Weber & Schmidt |pages=29–30}}</ref> The same holds true in the [[surreal number]]s.<ref>{{citation |url=https://books.google.com/books?id=tXiVo8qA5PQC |title=On Numbers and Games |edition=2nd |last=Conway |first=John H. |date=2000 |orig-year=1976 |publisher=CRC Press |isbn=9781568811277 |page=20}}</ref>

===Distribution theory===
In [[Distribution (mathematics)|distribution theory]] one can extend the function <math display="inline">\frac{1}{x}</math> to a distribution on the whole space of real numbers (in effect by using [[Cauchy principal value]]s). It does not, however, make sense to ask for a "value" of this distribution at ''x''&nbsp;=&nbsp;0; a sophisticated answer refers to the [[singular support]] of the distribution.

===Linear algebra===
In [[matrix (mathematics)|matrix]] algebra, square or rectangular blocks of numbers are manipulated as though they were numbers themselves: matrices can be [[matrix addition|added]] and [[matrix multiplication|multiplied]], and in some cases, a version of division also exists. Dividing by a matrix means, more precisely, multiplying by its [[Invertible matrix|inverse]]. Not all matrices have inverses.<ref>{{citation|last=Gbur |first=Greg |author-link=Greg Gbur |title=Mathematical Methods for Optical Physics and Engineering |pages=88–93 |year=2011 |isbn=978-0-521-51610-5 |publisher=Cambridge University Press|bibcode=2011mmop.book.....G }}</ref> For example, a [[zero matrix|matrix containing only zeros]] is not invertible.

One can define a pseudo-division, by setting ''a''/''b''&nbsp;=&nbsp;''ab''<sup>+</sup>, in which ''b''<sup>+</sup> represents the [[Moore–Penrose inverse|pseudoinverse]] of ''b''. It can be proven that if ''b''<sup>−1</sup> exists, then ''b''<sup>+</sup> = ''b''<sup>−1</sup>. If ''b'' equals 0, then b<sup>+</sup> = 0.

===Abstract algebra===
In [[abstract algebra]], the integers, the rational numbers, the real numbers, and the complex numbers can be abstracted to more general algebraic structures, such as a [[commutative ring]], which is a mathematical structure where addition, subtraction, and multiplication behave as they do in the more familiar number systems, but division may not be defined. Adjoining a multiplicative inverses to a commutative ring is called [[Localization (commutative algebra)|localization]]. However, the localization of every commutative ring at zero is the [[trivial ring]], where <math>0 = 1</math>, so nontrivial commutative rings do not have inverses at zero, and thus division by zero is undefined for nontrivial commutative rings.

Nevertheless, any number system that forms a [[commutative ring]] can be extended to a structure called a [[Wheel theory|wheel]] in which division by zero is always possible.<ref>{{citation |year=2004 |last=Carlström |first=Jesper |title=Wheels: On Division by Zero |journal=Mathematical Structures in Computer Science |doi=10.1017/S0960129503004110 |volume=14 |issue=1 |pages=143–184 |doi-broken-date=1 November 2024 |url=http://www2.math.su.se/reports/2001/11/ }}</ref> However, the resulting mathematical structure is no longer a commutative ring, as multiplication no longer distributes over addition. Furthermore, in a wheel, division of an element by itself no longer results in the multiplicative identity element <math>1</math>, and if the original system was an [[integral domain]], the multiplication in the wheel no longer results in a [[cancellative semigroup]].

The concepts applied to standard arithmetic are similar to those in more general algebraic structures, such as [[Ring (mathematics)|rings]] and [[Field (mathematics)|fields]]. In a field, every nonzero element is invertible under multiplication; as above, division poses problems only when attempting to divide by zero. This is likewise true in a [[skew field]] (which for this reason is called a [[division ring]]). However, in other rings, division by nonzero elements may also pose problems. For example, the ring '''Z'''/6'''Z''' of integers mod 6. The meaning of the expression <math display="inline">\frac{2}{2}</math> should be the solution ''x'' of the equation <math>2x = 2</math>. But in the ring '''Z'''/6'''Z''', 2 is a [[zero divisor]]. This equation has two distinct solutions, {{math|1=''x'' = 1}} and {{math|1=''x'' = 4}}, so the expression <math display="inline">\frac{2}{2}</math> is [[Defined and undefined|undefined]].

In field theory, the expression <math display="inline">\frac{a}{b}</math> is only shorthand for the formal expression ''ab''<sup>−1</sup>, where ''b''<sup>−1</sup> is the multiplicative inverse of ''b''. Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when ''b'' is zero. Modern texts, that define fields as a special type of ring, include the axiom {{math|0 ≠ 1}} for fields (or its equivalent) so that the [[zero ring]] is excluded from being a field. In the zero ring, division by zero is possible, which shows that the other field axioms are not sufficient to exclude division by zero in a field.

==Computer arithmetic==

=== Floating-point arithmetic ===

In computing, most numerical calculations are done with [[floating-point arithmetic]], which since the 1980s has been standardized by the [[IEEE 754]] specification. In IEEE floating-point arithmetic, numbers are represented using a sign (positive or negative), a fixed-precision [[significand]] and an integer [[exponent]]. Numbers whose exponent is too large to represent instead "overflow" to positive or negative [[infinity]] (+∞ or −∞), while numbers whose exponent is too small to represent instead "[[Arithmetic underflow|underflow]]" to [[signed zero|positive or negative zero]] (+0 or −0). A [[NaN]] (not a number) value represents undefined results.

In IEEE arithmetic, division of 0/0 or ∞/∞ results in NaN, but otherwise division always produces a well-defined result. Dividing any non-zero number by positive zero (+0) results in an infinity of the same sign as the dividend. Dividing any non-zero number by [[negative zero]] (−0) results in an infinity of the opposite sign as the dividend. This definition preserves the sign of the result in case of [[arithmetic underflow]].<ref>{{citation|last=Cody|first=W. J.|title=Analysis of Proposals for the Floating-Point Standard|journal=Computer|date=March 1981 |volume=14|issue=3|pages=65|doi=10.1109/C-M.1981.220379|s2cid=9923085|quote=With appropriate care to be certain that the algebraic signs are not determined by rounding error, the affine mode preserves order relations while fixing up overflow. Thus, for example, the reciprocal of a negative number which underflows is still negative.}}</ref>

For example, using single-precision IEEE arithmetic, if {{nowrap|1=''x'' = −2<sup>−149</sup>}}, then ''x''/2 underflows to −0, and dividing 1 by this result produces 1/(''x''/2) = −∞. The exact result −2<sup>150</sup> is too large to represent as a single-precision number, so an infinity of the same sign is used instead to indicate overflow.

=== Integer arithmetic ===
[[File:TI86 Calculator DivByZero.jpg|thumb|Handheld calculators, such as this [[TI-86]], typically halt and display an error message after an attempt to divide by zero.]]

[[Integer (computer science)|Integer]] division by zero is usually handled differently from floating point since there is no integer representation for the result. [[Central processing unit|CPUs]] differ in behavior: for instance [[x86]] processors trigger a [[Interrupt|hardware exception]], while [[PowerPC]] processors silently generate an incorrect result for the division and continue, and [[ARM architecture family|ARM]] processors can either cause a hardware exception or return zero.<ref>{{citation |title=Divide instructions |work=ARMv7-M Architecture Reference Manual |publisher=Arm Limited |url=https://developer.arm.com/documentation/ddi0403/d/Application-Level-Architecture/The-ARMv7-M-Instruction-Set/Data-processing-instructions/Divide-instructions |access-date=2024-06-12 |edition=Version D |year=2010 }}</ref> Because of this inconsistency between platforms, the [[C (programming language)|C]] and [[C++]] [[programming language]]s consider the result of dividing by zero [[undefined behavior]].<ref> {{cite conference |mode=cs2 |last1=Wang |first1=Xi |last2=Chen |first2=Haogang |last3=Cheung |first3=Alvin |last4=Jia |first4=Zhihao |last5=Zeldovich |first5=Nickolai |last6=Kaashoek |first6=M. Frans |contribution=Undefined behavior: what happened to my code? |title=APSYS '12: Proceedings of the Asia-Pacific Workshop on Systems |conference=APSYS '12, Seoul, 23–24 July 2012 |location=New York |publisher=Association for Computing Machinery |isbn=978-1-4503-1669-9 |doi=10.1145/2349896.2349905 |doi-access=free |hdl=1721.1/86949|hdl-access=free }}</ref> In typical [[high-level programming language|higher-level programming languages]], such as [[Python (programming language)|Python]],<ref>{{citation |title=Python 3 Library Reference |chapter=Built-in Exceptions |chapter-url=https://docs.python.org/3/library/exceptions.html#ZeroDivisionError |publisher=Python Software Foundation |at=§&nbsp;"Concrete exceptions – exception <code>ZeroDivisionError</code>" |access-date=2024-01-22 }}</ref> an [[Exception handling (programming)|exception]] is raised for attempted division by zero, which can be handled in another part of the program.

=== In proof assistants ===

Many [[proof assistant]]s, such as [[Coq (software)|Coq]] and [[Lean (proof assistant)|Lean]], define 1/0 = 0. This is due to the requirement that all functions are [[Total functional programming|total]]. Such a definition does not create contradictions, as further manipulations (such as [[cancelling out]]) still require that the divisor is non-zero.<ref>{{cite conference |mode=cs2 |last1=Tanter |first1=Éric |last2=Tabareau |first2=Nicolas |title=Gradual certified programming in coq |year=2015 |publisher=Association for Computing Machinery |book-title=DLS 2015: Proceedings of the 11th Symposium on Dynamic Languages |doi=10.1145/2816707.2816710 |quote=The standard division function on natural numbers in Coq, div, is total and pure, but incorrect: when the divisor is 0, the result is 0.|arxiv=1506.04205 }}</ref><ref>{{citation |last1=Buzzard |first1=Kevin |title=Division by zero in type theory: a FAQ |type=Blog |url=https://xenaproject.wordpress.com/2020/07/05/division-by-zero-in-type-theory-a-faq/ |website=Xena Project |date=5 July 2020 |access-date=2024-01-21}}</ref>

== Historical accidents ==
* On 21 September 1997, a division by zero error in the "Remote Data Base Manager" aboard [[USS Yorktown (CG-48)|USS ''Yorktown'' (CG-48)]] brought down all the machines on the network, causing the ship's propulsion system to fail.<ref>{{citation |last=Stutz |first=Michael |url=https://www.wired.com/1998/07/sunk-by-windows-nt/ |title=Sunk by Windows NT |date=1998-07-24 |work=[[Wired News]] |url-status=live |url-access=subscription |archive-url=https://web.archive.org/web/19990429091432/http://www.wired.com/news/technology/story/13987.html |archive-date=1999-04-29}}</ref><ref>{{citation|url=http://www.cs.berkeley.edu/~wkahan/Boulder.pdf|title=Desperately Needed Remedies for the Undebuggability of Large Floating-Point Computations in Science and Engineering|author=William Kahan|date=14 October 2011}}</ref>
{{clear}}

== See also ==
* [[Zero divisor]]
* [[Zero to the power of zero]]
* [[L'Hôpital's rule]]

==Notes==
{{reflist}}

==Sources==
* {{citation |first=Bryan |last=Bunch |title=Mathematical Fallacies and Paradoxes |year=1982 |url=https://archive.org/details/mathematicalfall0000bunc/ |url-access=limited |location=New York |publisher=Van Nostrand Reinhold |isbn=0-442-24905-5 }} (Dover reprint 1997)
* {{citation|last=Cheng |first=Eugenia |author-link=Eugenia Cheng |title=Is Math(s) Real? How Simple Questions Lead Us to Mathematics' Deepest Truths |publisher=Basic Books |year=2023 |isbn=978-1-541-60182-6}}
* {{citation|first=Felix|last=Klein|title=Elementary Mathematics from an Advanced Standpoint / Arithmetic, Algebra, Analysis|year=1925|edition=3rd|publisher=Dover|translator-first1=E. R.|translator-last1=Hedrick|translator-first2=C. A.|translator-last2=Noble}}
* {{citation|first=A. G.|last=Hamilton|title=Numbers, Sets, and Axioms|year=1982|publisher=Cambridge University Press|isbn=978-0521287616}}
* {{citation|first1=Leon|last1=Henkin|first2=Norman|last2=Smith|first3=Verne J.|last3=Varineau|first4=Michael J.|last4=Walsh|year=2012|title=Retracing Elementary Mathematics|publisher=Literary Licensing LLC|isbn=978-1258291488}}
* {{citation|first=Carol|last=Schumacher|author-link= Carol Schumacher |title=Chapter Zero : Fundamental Notions of Abstract Mathematics|year=1996|publisher=Addison-Wesley|isbn=978-0-201-82653-1}}
* {{citation|last1=Zazkis |first1=Rina |last2=Liljedahl |first2=Peter |year=2009 |title=Teaching Mathematics as Storytelling |chapter=Stories that explain |doi=10.1163/9789087907358_008 |pages=51–65 |publisher=Sense Publishers |isbn=978-90-8790-734-1 }}

== Further reading ==
* {{citation |last=Northrop |first=Eugene P. |year=1944 |title=Riddles in Mathematics: A Book of Paradoxes |location=New York |publisher=D. Van Nostrand |at=Ch. 5 "Thou Shalt Not Divide By Zero", {{pgs|77–96}} |url=https://archive.org/details/riddlesinmathema0000euge/ |url-access=limited}}
* {{citation |last=Seife |first=Charles |author-link=Charles Seife |year=2000 |title=[[Zero: The Biography of a Dangerous Idea]] |publisher=Penguin |location=New York |isbn=0-14-029647-6 }}
* {{citation|last=Suppes |first=Patrick |author-link=Patrick Suppes |year=1957 |title=Introduction to Logic |location=Princeton |publisher=D. Van Nostrand |url=https://archive.org/details/introductiontolo0000supp/ |url-access=limited |at=§8.5 "The Problem of Division by Zero" and §8.7 "Five Approaches to Division by Zero"}} (Dover reprint, 1999)
* {{citation |last=Tarski |first=Alfred |author-link=Alfred Tarski |year=1941 |title=Introduction to Logic and to the Methodology of Deductive Sciences |publisher=Oxford University Press |url=https://archive.org/details/introductiontolo0000alfr |url-access=limited |at=§53 "Definitions whose definiendum contains the identity sign"}}


{{DEFAULTSORT:Division by zero}}
[[Category:0 (number)]]
[[Category:Computer arithmetic]]
[[Category:Division (mathematics)]]
[[Category:Computer errors]]
[[Category:Fractions (mathematics)]]
[[Category:Infinity]]
[[Category:Mathematical analysis]]
[[Category:Mathematical fallacies]]