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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, 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 o ...
where the divisor (denominator) is
zero 0 (zero) is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in languag ...

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 no number which, when multiplied by , gives (assuming a \neq 0), and so 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 Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
, it is an
indeterminate formIn calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...
. Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to \tfrac is contained in
Anglo-Irish Anglo-Irish () is a term which was more commonly used in the 19th and early 20th centuries to identify an ethnic group An ethnic group or ethnicity is a grouping of people A people is any plurality of person A person (plural people ...
philosopher
George Berkeley George Berkeley (; 12 March 168514 January 1753) – known as Bishop Berkeley (Bishop of Cloyne The Bishop of Cloyne is an episcopal title that takes its name after the small town of Cloyne in County Cork, Republic of Ireland Irela ...

George Berkeley
’s criticism of
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...
in 1734 in ''
The Analyst ''The Analyst'' (subtitled ''A Discourse Addressed to an Infidel Mathematician: Wherein It Is Examined Whether the Object, Principles, and Inferences of the Modern Analysis Are More Distinctly Conceived, or More Evidently Deduced, Than Religious M ...
'' ("ghosts of departed quantities"). There are mathematical structures in which \tfrac is defined for some such as in the
Riemann sphere In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

Riemann sphere
(a
model In general, 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. ...
of the extended
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
) and the
Projectively extended real line Image:Real projective line.svg, The projectively extended real line can be visualized as the real number line wrapped around a circle (by some form of stereographic projection) with an additional point at infinity. In real analysis, the projectiv ...
; however, such structures do not satisfy every ordinary rule of arithmetic (the
field axioms In mathematics, a field is a set (mathematics), set on which addition, subtraction, multiplication, and division (mathematics), division are defined and behave as the corresponding operations on rational number, rational and real numbers do. A ...
). 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 computer hardware , hardware and software. It has sci ...

computing
, 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 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 computer hardware , hardware and soft ...
,
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
) being divided by zero, it may generate positive or negative infinity by the
IEEE 754 The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard A technical standard is an established norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral ...
floating point standard, generate an exception, generate an
error message An error message is information displayed when an unforeseen problem occurs, usually on a computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically ...

error message
, cause the program to terminate, result in a special
not-a-number
not-a-number
value, or
crash Crash or CRASH may refer to: Common meanings * Collision, an impact between two or more objects * Crash (computing), a condition where a program ceases to respond * Cardiac arrest, a medical condition in which the heart stops beating * Couch su ...
.


Elementary arithmetic

When division is explained at the
elementary arithmetic Elementary arithmetic is the simplified portion of arithmetic that includes the operations of addition, subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is sig ...
level, it is often considered as splitting a set 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 amongst 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. So \tfrac, at least in elementary arithmetic, is 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(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 Latin ', "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-fifths, ...
( = ) . 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 some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with spec ...
, 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 Brahma ( sa, ब्रह्मा, Brahmā) is one of the Hindu deities, principal deities of Hinduism, though his importance has declined in recent centuries. He i ...
'' of
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, doc ...

Brahmagupta
(c. 598–668) is the earliest text to treat
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

zero
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 Sanskrit (, attributively , ''saṃskṛta-'', nominalization, nominally , ''saṃskṛtam'') is a classical language of South Asia belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. ...
unsuccessfully tried to correct Brahmagupta's mistake in his book in ''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 (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
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 (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
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 Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...
or even
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
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 integersIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

ordered pair
s of integers, with , define a
binary relation Binary may refer to: Science and technology Mathematics * Binary number In mathematics and digital electronics Digital electronics is a field of electronics The field of electronics is a branch of physics and electrical engineeri ...
on this set by if and only if . This relation is shown to be an
equivalence relation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
and its
equivalence class In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
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 o ...
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 reason Reason is the capacity of consciously applying logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of ...
) 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: \begin \frac &= \frac \\ \frac\times 1 &= \frac\times 2. \end 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 (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
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 Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
as a limit.


Formal operations

A
formal calculationIn mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algeb ...
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 limit In calculus, a one-sided limit is either of the two Limit of a function, limits of a function (mathematics), function ''f''(''x'') of a real number, real variable ''x'' as ''x'' approaches a specified point either from the left or from the right. ...
s 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 Expression may refer to: Linguistics * Expression (linguistics), a word, phrase, or sentence * Fixed expression, a form of words with a specific meaning * Idiom, a type of fixed expression * Metaphor#Common types, Metaphorical expression, a parti ...
.


Projectively extended real line

The set \mathbb\cup\ is the
projectively extended real line Image:Real projective line.svg, The projectively extended real line can be visualized as the real number line wrapped around a circle (by some form of stereographic projection) with an additional point at infinity. In real analysis, the projectiv ...
, which is a
one-point compactificationIn the mathematical field of topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematic ...
of the real line. Here \infty means an unsigned infinity, 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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...

tangent
function and cotangent functions of
trigonometry Trigonometry (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is ...

trigonometry
: 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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
, 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\cup\ is the
Riemann sphere In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

Riemann sphere
, 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 Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
. Here too \infty is an unsigned infinity – or, as it is often called in this context, the
point at infinity 150px, The real line with the point at infinity; it is called the real projective line. In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic ...
. This set is analogous to the projectively extended real line, except that it is based on the
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s. In the Riemann sphere, \frac=\infty and \frac = 0, but \frac and 0\times\infty are undefined.


Extended non-negative real number line

The negative real numbers can be discarded, and infinity introduced, leading to the set , where division by zero can be naturally defined as for positive . While this makes division defined in more cases than usual, subtraction is instead left undefined in many cases, because there are no negative numbers.


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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
s and the
surreal number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s, division by zero is still impossible, but division by non-zero
infinitesimal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 Mathematical singularity, singul ...
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 In mathematics, the support of a Real number, real-valued function (mathematics), function ''f'' is the subset of the Domain of a function, domain containing the elements which are not mapped to zero. If the domain of ''f'' is a topological space, ...
of the distribution.


Linear algebra

In
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
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 matrix (mat ...
in general), one can define a pseudo-division, by setting ''a''/''b'' = ''ab''+, in which ''b''+ represents the
pseudoinverse In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of ''b''. It can be proven that if ''b''−1 exists, then ''b''+ = ''b''−1. If ''b'' equals 0, then b+ = 0.


Abstract algebra

Any number system that forms a
commutative ring In ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...
—for instance, the integers, the real numbers, and the complex numbers—can be extended to a
wheel File:Roue primitive.png, An early wheel made of a solid piece of wood A wheel is a circular component that is intended to rotate on an axle An axle or axletree is a central shaft for a rotating wheel or gear. On wheeled vehicles, the ...
in which division by zero is always possible; however, in such a case, "division" has a slightly different meaning. The concepts applied to standard arithmetic are similar to those in more general algebraic structures, such as
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
and
fields File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe'' FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...
. 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 fieldIn algebra, a division ring, also called a skew field, is a ring (mathematics), ring in which division (mathematics), division is possible. Specifically, it is a zero ring, nonzero ring in which every nonzero element has a multiplicative inverse, th ...
(which for this reason is called a
division ringIn algebra, a division ring, also called a skew field, is a ring (mathematics), ring in which division (mathematics), division is possible. Specifically, it is a zero ring, nonzero ring in which every nonzero element has a multiplicative inverse, th ...
). 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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
. 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 In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies ...
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 The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard A technical standard is an established norm or requirement for a repeatable technical task. It is usually a formal document that establishes uniform engineering ...
, supported by almost all modern
floating-point unit A floating-point unit (FPU, colloquially a math coprocessor) is a part of a computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations automatically. Modern comp ...
s, specifies that every
floating point 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 computer hardware , hardware and soft ...
arithmetic operation, including division by zero, has a well-defined result. The standard supports
signed zero Signed zero is zero 0 (zero) is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be r ...
, as well as
infinity Infinity is that which is boundless, endless, or larger than any number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything ...

infinity
and
NaN 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 computer hardware , hardware and softwar ...

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 The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard A technical standard is an established norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral ...
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 The term arithmetic underflow (also floating point underflow, or just underflow) is a condition in a computer program A computer program is a collection of instructions that can be executed by a computer to perform a specific task. A compute ...
. 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 formal language In logic, mathematics, computer science, and linguistics, a formal language consists of string (computer science), words whose symbol (formal), letters are taken from an alphabet (computer science) ...

programming language
s (including those used by
calculator An electronic calculator is typically a portable device used to perform s, ranging from basic to complex . The first calculator was created in the early 1960s. Pocket-sized devices became available in the 1970s, especially after the , the f ...

calculator
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 Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algo ...

if statement
). Some programs (especially those that use
fixed-point arithmetic 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 computer hardware , hardware and sof ...
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 In computer programming Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, generati ...
. 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 In mathematics, an operation is a Function (mathematics), function which takes zero or more input values (called ''operands'') to a well-defined output value. The number of operands is the arity of the ...
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 In computer programming Computer programming is the process of designing and building an executable computer program to accomplish a specific computing result or to perform a particular task. Programming involves tasks such as analysis, generati ...
. 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 Microsoft Math Solver (formerly Microsoft Mathematics and Microsoft Math) is an entry-level educational app that solves math and science problems. Developed and maintained by Microsoft Microsoft Corporation is an American multinational cor ...
and
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, Computer algebra, symbolic computation, manipulating Matrix (mathematics), matrices, plotting Fun ...

Mathematica
return ComplexInfinity for 1/0.
Maple ''Acer'' is a genus Genus /ˈdʒiː.nəs/ (plural genera /ˈdʒen.ər.ə/) is a taxonomic rank In biological classification In biology, taxonomy () is the scientific study of naming, defining (Circumscription (taxonomy), circumscr ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such t ...
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.


See also

*
Asymptote In analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέ ...

Asymptote
*
Defined and undefined In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
* ''
Division by Zero In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
'', a short story by
Ted Chiang Ted Chiang (born 1967) is an American science fiction File:Imagination 195808.jpg, Space exploration, as predicted in August 1958 in the science fiction magazine ''Imagination (magazine), Imagination.'' Science fiction (sometimes shortened to ...
*
Indeterminate formIn calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...
*
Zero divisor In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...


References


Notes


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 a ...
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 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 [exactly one number satisfying a definiens] will 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''"


Further reading

* Jakub Czajko (July 2004) "", ''Chaos, Solitons and Fractals'', volume 21, number 2, pages 261–271. *
To Continue with Continuity
''Metaphysica'' 6, pp. 91–109, a philosophy paper from 2005, reintroduced the (ancient Indian) idea of an applicable whole number equal to 1/0, in a more modern (Cantorian) style. {{DEFAULTSORT:Division by zero 0 (number) Computer arithmetic Division (mathematics) Computer errors Fractions (mathematics) Infinity Mathematical analysis