TheInfoList

OR: In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, division by zero is
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
where the divisor (denominator) is
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
. Such a division can be formally expressed as $\tfrac$, where is the dividend (numerator). In ordinary
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19t ...
, the expression has no meaning, as there is no number that, when multiplied by , gives (assuming $a \neq 0$); thus, division by zero is undefined. Since any number multiplied by zero is zero, the expression $\tfrac$ is also undefined; when it is the form of a limit, it is an indeterminate form. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to $\tfrac$ is contained in
Anglo-Irish Anglo-Irish people () denotes an ethnic, social and religious grouping who are mostly the descendants and successors of the English Protestant Ascendancy in Ireland. They mostly belong to the Anglican Church of Ireland, which was the establish ...
philosopher
George Berkeley George Berkeley (; 12 March 168514 January 1753) – known as Bishop Berkeley ( Bishop of Cloyne of the Anglican Church of Ireland) – was an Anglo-Irish philosopher whose primary achievement was the advancement of a theory he called "immate ...
's criticism of
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arith ...
in 1734 in '' The Analyst'' ("ghosts of departed quantities"). There are mathematical structures in which $\tfrac$ is defined for some such as in the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
(a
model A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure. Models c ...
of the extended
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
) and the Projectively extended real line; however, such structures do not satisfy every ordinary rule of arithmetic (the field axioms). In
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, e ...
, a program error may result from an attempt to divide by zero. Depending on the programming environment and the type of number (e.g.,
floating point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be r ...
,
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
) being divided by zero, it may generate positive or negative infinity by the IEEE 754 floating-point standard, generate an exception, generate an
error message An error message is information displayed when an unforeseen occurs, usually on a computer or other device. On modern operating systems with graphical user interfaces, error messages are often displayed using dialog boxes. Error messages are use ...
, cause the program to terminate, result in a special not-a-number value, or crash.

# Elementary arithmetic

When division is explained at the
elementary arithmetic The operators in elementary arithmetic are addition, subtraction, multiplication, and division. The operators can be applied on both real numbers and imaginary numbers. Each kind of number is represented on a number line designated to the type. ...
level, it is often considered as splitting a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of objects into equal parts. As an example, consider having ten cookies, and these cookies are to be distributed equally to five people at a table. Each person would receive $\tfrac = 2$ cookies. Similarly, if there are ten cookies, and only one person at the table, that person would receive $\tfrac = 10$ cookies. So, for dividing by zero, what is the number of cookies that each person receives when 10 cookies are evenly distributed among 0 people at a table? Certain words can be pinpointed in the question to highlight the problem. The problem with this question is the "when". There is no way to distribute 10 cookies to nobody. Therefore, $\tfrac$at least in elementary arithmeticis said to be either meaningless or undefined. If there are, say, 5 cookies and 2 people, the problem is in "evenly distribute". In any integer partition of 5 things into 2 parts, either one of the parts of the partition will have more elements than the other or there will be a
remainder In mathematics, the remainder is the amount "left over" after performing some computation. In arithmetic, the remainder is the integer "left over" after dividing one integer by another to produce an integer quotient (integer division). In algeb ...
(written as = 2 r1). Or, the problem with 5 cookies and 2 people can be solved by cutting one cookie in half, which introduces the idea of
fractions A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
( = ) . The problem with 5 cookies and 0 people, on the other hand, cannot be solved in any way that preserves the meaning of "divides". In
elementary algebra Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values). This use of variables entai ...
, another way of looking at division by zero is that division can always be checked using multiplication. Considering the example above, setting ''x'' = , if ''x'' equals ten divided by zero, then ''x'' times zero equals ten, but there is no ''x'' that, when multiplied by zero, gives ten (or any number other than zero). If, instead of ''x'' = , ''x'' = , then every ''x'' satisfies the question "what number ''x'', multiplied by zero, gives zero?"

# Early attempts

The ''
Brāhmasphuṭasiddhānta The ''Brāhmasphuṭasiddhānta'' ("Correctly Established Doctrine of Brahma", abbreviated BSS) is the main work of Brahmagupta, written c. 628. This text of mathematical astronomy contains significant mathematical content, including a good underst ...
'' of
Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tre ...
(c. 598–668) is the earliest text to treat
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
as a number in its own right and to define operations involving zero. The author could not explain division by zero in his texts: his definition can be easily proven to lead to algebraic absurdities. According to Brahmagupta,
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.
In 830,
Mahāvīra Mahavira (Sanskrit: महावीर) also known as Vardhaman, was the 24th ''tirthankara'' (supreme preacher) of Jainism. He was the spiritual successor of the 23rd ''tirthankara'' Parshvanatha. Mahavira was born in the early part of the 6t ...
unsuccessfully tried to correct the mistake Brahmagupta made in his book ''Ganita Sara Samgraha'': "A number remains unchanged when divided by zero."

# Algebra

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 An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
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 In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...
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 or even
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. 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. 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's axiom system and then this is expanded to the
ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often de ...
. 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 In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s of integers, with , define a
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
on this set by if and only if . This relation is shown to be an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
and its
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
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 transitivity). The above explanation may be too abstract and technical for many purposes, but if one assumes the existence and properties of the rational numbers, as is commonly done in elementary mathematics, the "reason" that division by zero is not allowed is hidden from view. Nevertheless, a (non-rigorous) justification can be given in this setting. It follows from the properties of the number system we are using (that is, integers, rationals, reals, etc.), if then the equation is equivalent to . Assuming that is a number , then it must be that . However, the single number would then have to be determined by the equation , but every number satisfies this equation, so we cannot assign a numerical value to .

## Division as the inverse of multiplication

The concept that explains
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting ...
in algebra is that it is the inverse of multiplication. For example, $\frac=2$ since is the value for which the unknown quantity in $?\times 3 = 6$ is true. But the expression $\frac = \, x$ requires a value to be found for the unknown quantity in $x\times 0 = 6.$ But any number multiplied by is and so there is no number that solves the equation. The expression $\frac = \, x$ requires a value to be found for the unknown quantity in $x \times 0 = 0.$ Again, any number multiplied by is and so this time every number solves the equation instead of there being a single number that can be taken as the value of . In general, a single value can’t be assigned to a fraction where the denominator is so the value remains undefined.

## Fallacies

A compelling reason for not allowing division by zero is that, if it were allowed, many absurd results (i.e.,
fallacies A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves," in the construction of an argument which may appear stronger than it really is if the fallacy is not spotted. The term in the Western intellectual tradition was in ...
) would arise. When working with numerical quantities it is easy to determine when an illegal attempt to divide by zero is being made. For example, consider the following computation. With the assumptions: $\begin 0\times 1 &= 0, \\ 0\times 2 &= 0, \end$ the following is true: $0\times 1 = 0\times 2.$ Dividing both sides by zero gives: Simplified, this yields: $1 = 2.$ The fallacy here is the assumption that dividing 0 by 0 is a legitimate operation with the same properties as dividing by any other number. However, it is possible to disguise a division by zero in an
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
ic argument, leading to invalid proofs that, for instance, such as the following: The disguised division by zero occurs since when .

# Analysis

## Extended real line

At first glance it seems possible to define ''a''/0 by considering the limit of ''a''/''b'' as ''b'' approaches 0. For any positive ''a'', the limit from the right is $\lim_ = +\infty$ however, the limit from the left is $\lim_ = -\infty$ and so the $\lim_$ is undefined (the limit is also undefined for negative ''a''). Furthermore, there is no obvious definition of 0/0 that can be derived from considering the limit of a ratio. The limit $\lim_$ does not exist. Limits of the form $\lim_$ in which both ''f''(''x'') and ''g''(''x'') approach 0 as ''x'' approaches 0, may equal any real or infinite value, or may not exist at all, depending on the particular functions ''f'' and ''g''. For example, consider: $\lim_$ This initially appears to be indeterminate. However: $\begin &= \lim_ \\ &= \lim_ \\ &= 2 \end$ and so the limit exists, and is equal to $2$. These and other similar facts show that the expression $\frac$ cannot be
well-defined In mathematics, a well-defined expression or unambiguous expression is an expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A func ...
as a limit.

### Formal operations

A
formal calculation Formal, formality, informal or informality imply the complying with, or not complying with, some set of requirements ( forms, in Ancient Greek). They may refer to: Dress code and events * Formal wear, attire for formal events * Semi-formal at ...
is one carried out using rules of arithmetic, without consideration of whether the result of the calculation is well-defined. Thus, it is sometimes useful to think of ''a''/0, where ''a'' ≠ 0, as being $\infty$. This infinity can be either positive, negative, or unsigned, depending on context. For example, formally: $\lim_ = \infty.$ As with any formal calculation, invalid results may be obtained. A logically rigorous (as opposed to formal) computation would assert only that $\lim_ \frac = +\infty ~\text~ \lim_ \frac = -\infty.$ Since the one-sided limits are different, the two-sided limit does not exist in the standard framework of the real numbers. Also, the fraction 1/0 is left undefined in the extended real line, therefore it and $\frac$ are meaningless expressions.

## Projectively extended real line

The set $\mathbb\cup\$ is the projectively extended real line, which is a one-point compactification of the real line. Here $\infty$ means an unsigned infinity or
point at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. A ...
, an infinite quantity that is neither positive nor negative. This quantity satisfies $-\infty = \infty$, which is necessary in this context. In this structure, $\frac = \infty$ can be defined for nonzero , and $\frac = 0$ when is not $\infty$. It is the natural way to view the range of the tangent function and cotangent functions of
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
: approaches the single point at infinity as approaches either or from either direction. This definition leads to many interesting results. However, the resulting algebraic structure is not a field, and should not be expected to behave like one. For example, $\infty+\infty$ is undefined in this extension of the real line.

## Riemann sphere

The set $\mathbb^*=\mathbb\cup\$ is the
Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers pl ...
, which is of major importance in
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
. Here $\tilde\infty$ represents complex infinity, which is also a point at infinity. This set is analogous to the projectively extended real line, except that it is based on the field of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. In the Riemann sphere, $\frac=\tilde\infty$ and $\frac = 0$, but $\frac$, $\frac$, and $0\times\tilde\infty$ are undefined.

# Higher mathematics

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 In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains number ...
s and the
surreal number In mathematics, the surreal number system is a totally ordered proper class containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals ...
s, division by zero is still impossible, but division by non-zero
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refer ...
s is possible.

## Distribution theory

In distribution theory one can extend the function $\frac$ to a distribution on the whole space of real numbers (in effect by using
Cauchy principal value In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. Formulation Depending on the type of singularity in the integrand ...
s). It does not, however, make sense to ask for a "value" of this distribution at ''x'' = 0; a sophisticated answer refers to the singular support of the distribution.

## Linear algebra

In
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
algebra (or
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...
in general), one can define a pseudo-division, by setting ''a''/''b'' = ''ab''+, in which ''b''+ represents the
pseudoinverse In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. The purpose of constructing a generalized inv ...
of ''b''. It can be proven that if ''b''−1 exists, then ''b''+ = ''b''−1. If ''b'' equals 0, then b+ = 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 In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, 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 Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is ...
. However, the localization of every commutative ring at zero is the
trivial ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for ...
, where $0 = 1$, 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 In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
can be extended to a seldom used structure called a
wheel A wheel is a circular component that is intended to rotate on an axle bearing. The wheel is one of the key components of the wheel and axle which is one of the six simple machines. Wheels, in conjunction with axles, allow heavy objects to be ...
in which division by zero is always possible. 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 $1$, and if the original system was an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural se ...
, the multiplication in the wheel no longer results in a
cancellative semigroup In mathematics, a cancellative semigroup (also called a cancellation semigroup) is a semigroup having the cancellation property. In intuitive terms, the cancellation property asserts that from an equality of the form ''a''·''b'' = ''a''·''c'', w ...
. The concepts applied to standard arithmetic are similar to those in more general algebraic structures, such as
rings Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
and
fields Fields may refer to: Music *Fields (band), an indie rock band formed in 2006 *Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song by ...
. 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 Skew may refer to: In mathematics * Skew lines, neither parallel nor intersecting. * Skew normal distribution, a probability distribution * Skew field or division ring * Skew-Hermitian matrix * Skew lattice * Skew polygon, whose vertices do not ...
(which for this reason is called a
division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
). However, in other rings, division by nonzero elements may also pose problems. For example, the ring Z/6Z of integers mod 6. The meaning of the expression $\frac$ should be the solution ''x'' of the equation $2x = 2$. But in the ring Z/6Z, 2 is a
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
. This equation has two distinct solutions, and , so the expression $\frac$ is undefined. In field theory, the expression $\frac$ is only shorthand for the formal expression ''ab''−1, where ''b''−1 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 for fields (or its equivalent) so that the
zero ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which for ...
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 The IEEE floating-point standard, supported by almost all modern
floating-point unit In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be ...
s, specifies that every
floating-point arithmetic In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be r ...
operation, including division by zero, has a well-defined result. The standard supports signed zero, as well as
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
and NaN (''not a number''). There are two zeroes: +0 (''positive zero'') and −0 (''negative zero'') and this removes any ambiguity when dividing. In IEEE 754 arithmetic, ''a'' ÷ +0 is positive infinity when ''a'' is positive, negative infinity when ''a'' is negative, and NaN when ''a'' = ±0. The infinity signs change when dividing by −0 instead. The justification for this definition is to preserve the sign of the result in case of arithmetic underflow. For example, in the single-precision computation 1/(''x''/2), where , the computation ''x''/2 underflows and produces ±0 with sign matching ''x'', and the result will be ±∞ with sign matching ''x''. The sign will match that of the exact result ±2150, but the magnitude of the exact result is too large to represent, so infinity is used to indicate overflow. Integer division by zero is usually handled differently from floating point since there is no integer representation for the result. Some processors generate an exception when an attempt is made to divide an integer by zero, although others will simply continue and generate an incorrect result for the division. The result depends on how division is implemented, and can either be zero, or sometimes the largest possible integer. Because of the improper algebraic results of assigning any value to division by zero, many computer
programming language A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming ...
s (including those used by
calculator An electronic calculator is typically a portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was created in the early 1960s. Pocket-sized ...
s) explicitly forbid the execution of the operation and may prematurely halt a program that attempts it, sometimes reporting a "Divide by zero" error. In these cases, if some special behavior is desired for division by zero, the condition must be explicitly tested (for example, using an
if statement In computer science, conditionals (that is, conditional statements, conditional expressions and conditional constructs,) are programming language commands for handling decisions. Specifically, conditionals perform different computations or acti ...
). Some programs (especially those that use
fixed-point arithmetic In computing, fixed-point is a method of representing fractional (non-integer) numbers by storing a fixed number of digits of their fractional part. Dollar amounts, for example, are often stored with exactly two fractional digits, representi ...
where no dedicated floating-point hardware is available) will use behavior similar to the IEEE standard, using large positive and negative numbers to approximate infinities. In some programming languages, an attempt to divide by zero results in undefined behavior. The graphical programming language Scratch 2.0 and 3.0 used in many schools returns Infinity or −Infinity depending on the sign of the dividend. In
two's complement Two's complement is a mathematical operation to reversibly convert a positive binary number into a negative binary number with equivalent (but negative) value, using the binary digit with the greatest place value (the leftmost bit in big- endian ...
arithmetic, attempts to divide the smallest signed integer by −1 are attended by similar problems, and are handled with the same range of solutions, from explicit error conditions to undefined behavior. Most calculators will either return an error or state that 1/0 is undefined; however, some TI and HP graphing calculators will evaluate (1/0)2 to ∞. Microsoft Math and
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, symbolic computation, data manipulation, network analysis, time series analysis, NLP, optimiza ...
return ComplexInfinity for 1/0.
Maple ''Acer'' () is a genus of trees and shrubs commonly known as maples. The genus is placed in the family Sapindaceae.Stevens, P. F. (2001 onwards). Angiosperm Phylogeny Website. Version 9, June 2008 nd more or less continuously updated since http ...
and
SageMath SageMath (previously Sage or SAGE, "System for Algebra and Geometry Experimentation") is a computer algebra system (CAS) with features covering many aspects of mathematics, including algebra, combinatorics, graph theory, numerical analysis, numbe ...
return an error message for 1/0, and infinity for 1/0.0 (0.0 tells these systems to use floating-point arithmetic instead of algebraic arithmetic). Some modern calculators allow division by zero in special cases, where it will be useful to students and, presumably, understood in context by mathematicians. Some calculators, the online Desmos calculator is one example, allow arctangent(1/0). Students are often taught that the inverse cotangent function, arccotangent, should be calculated by taking the arctangent of the reciprocal, and so a calculator may allow arctangent(1/0), giving the output which is the correct value of arccotangent 0. The mathematical justification is that the limit as x goes to zero of arctangent 1/x is

# Historical accidents

* On September 21, 1997, a division by zero error in the "Remote Data Base Manager" aboard USS ''Yorktown'' (CG-48) brought down all the machines on the network, causing the ship's propulsion system to fail.

*
Asymptote In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
* Defined and undefined * ''
Division by Zero In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as \tfrac, where is the dividend (numerator). In ordinary arithmetic, the expression has no meaning, as there is ...
'', a short story by
Ted Chiang Ted Chiang (born 1967) is an American science fiction writer. His work has won four Nebula awards, four Hugo awards, the John W. Campbell Award for Best New Writer, and six Locus awards. His short story " Story of Your Life" was the basis of ...
* Indeterminate form *
Zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
*
Zero to the power of zero Zero to the power of zero, denoted by , is a mathematical expression that is either defined as 1 or left undefined, depending on context. In algebra and combinatorics, one typically defines  . In mathematical analysis, the expression is so ...

# References

## Sources

* * * * *
Patrick Suppes Patrick Colonel Suppes (; March 17, 1922 – November 17, 2014) was an American philosopher who made significant contributions to philosophy of science, the theory of measurement, the foundations of quantum mechanics, decision theory, psychology ...
1957 (1999 Dover edition), ''Introduction to Logic'', Dover Publications, Inc., Mineola, New York. (pbk.). This book is in print and readily available. Suppes's §8.5 ''The Problem of Division by Zero'' begins this way: "That everything is not for the best in this best of all possible worlds, even in mathematics, is well illustrated by the vexing problem of defining the operation of division in the elementary theory of arithmetic" (p. 163). In his §8.7 Five Approaches to Division by Zero he remarks that "...there is no uniformly satisfactory solution" (p. 166) * * Charles Seife 2000, '' Zero: The Biography of a Dangerous Idea'', Penguin Books, NY, (pbk.). This award-winning book is very accessible. Along with the fascinating history of (for some) an abhorrent notion and others a cultural asset, describes how zero is misapplied with respect to multiplication and division. *
Alfred Tarski Alfred Tarski (, born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...
1941 (1995 Dover edition), ''Introduction to Logic and to the Methodology of Deductive Sciences'', Dover Publications, Inc., Mineola, New York. (pbk.). Tarski's §53 Definitions whose definiendum contains the identity sign discusses how mistakes are made (at least with respect to zero). He ends his chapter "(A discussion of this rather difficult problem xactly one number satisfying a definienswill be omitted here.*)" (p. 183). The * points to Exercise #24 (p. 189) wherein he asks for a proof of the following: "In section 53, the definition of the number '0' was stated by way of an example. To be certain this definition does not lead to a contradiction, it should be preceded by the following theorem: ''There exists exactly one number x such that, for any number y, one has: y + x = y''"