Division by zero
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, division by zero, division where the divisor (denominator) is
zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
, is a unique and problematic special case. Using
fraction A fraction (from , "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, thre ...
notation, the general example can be written as \tfrac a0, where a is the dividend (numerator). The usual definition of the
quotient In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
in
elementary arithmetic Elementary arithmetic is a branch of mathematics involving addition, subtraction, multiplication, and Division (mathematics), division. Due to its low level of abstraction, broad range of application, and position as the foundation of all mathema ...
is the number which yields the dividend when multiplied by the divisor. That is, c = \tfrac ab is equivalent to c \cdot b = a. By this definition, the quotient q = \tfrac is nonsensical, as the product q \cdot 0 is always 0 rather than some other number a. Following the ordinary rules of
elementary algebra Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variable (mathematics ...
while allowing division by zero can create a
mathematical fallacy In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple ''mistake'' and a ''mathematical fallacy'' in a proof ...
, a subtle mistake leading to absurd results. To prevent this, the arithmetic of
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s and more general numerical structures called fields leaves division by zero undefined, and situations where division by zero might occur must be treated with care. Since any number multiplied by zero is zero, the expression \tfrac is also undefined.
Calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
studies the behavior of functions in the limit as their input tends to some value. When a
real function In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...
can be expressed as a fraction whose denominator tends to zero, the output of the function becomes arbitrarily large, and is said to " tend to infinity", a type of
mathematical singularity In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For exa ...
. For example, the
reciprocal function In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/'' ...
, f(x) = \tfrac 1x, tends to infinity as x tends to 0. 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 is a mathematical expression that can obtain any value depending on circumstances. In calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corres ...
, 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 \tfrac can be defined to equal zero; it can be defined to equal a new explicit
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. Ad ...
, sometimes denoted by the
infinity symbol The infinity symbol () is a mathematical symbol representing the concept of infinity. This symbol is also called a ''lemniscate'', after the lemniscate curves of a similar shape studied in algebraic geometry, or "lazy eight", in the terminolo ...
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
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computer, computing machinery. It includes the study and experimentation of algorithmic processes, and the development of both computer hardware, hardware and softw ...
, 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 positive or negative infinity, return a special not-a-number value, or crash the program, among other possibilities.


Elementary arithmetic


The meaning of division

The division N/D = Q can be conceptually interpreted in several ways. In ''quotitive division'', the dividend N is imagined to be split up into parts of size D (the divisor), and the quotient Q 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 Now imagine instead that zero slices of bread are required per sandwich (perhaps a lettuce wrap). Arbitrarily many such sandwiches can be made from ten slices of bread, as the bread is irrelevant. The quotitive concept of division lends itself to calculation by repeated
subtraction Subtraction (which is signified by the minus sign, –) is one of the four Arithmetic#Arithmetic operations, arithmetic operations along with addition, multiplication and Division (mathematics), division. Subtraction is an operation that repre ...
: 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 never terminates. Such an interminable division-by-zero
algorithm In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
is physically exhibited by some
mechanical calculator A mechanical calculator, or calculating machine, is a mechanical device used to perform the basic operations of arithmetic automatically, or a simulation like an analog computer or a slide rule. Most mechanical calculators were comparable in si ...
s. In ''partitive division'', the dividend N is imagined to be split into D parts, and the quotient Q 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 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. In another interpretation, the quotient Q represents the
ratio In mathematics, a ratio () shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
N:D. For example, a cake recipe might call for ten
cup A cup is an open-top vessel (container) used to hold liquids for drinking, typically with a flattened hemispherical shape, and often with a capacity of about . Cups may be made of pottery (including porcelain), glass, metal, wood, stone, pol ...
s of flour and two cups of sugar, a ratio of 10:2 or, proportionally, 5:1. To scale this recipe to larger or smaller quantities of cake, a ratio of flour to sugar proportional to 5:1 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. Now imagine a sugar-free cake recipe calls for ten cups of flour and zero cups of sugar. The ratio 10:0, or proportionally 1:0, is perfectly sensible: 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 In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...
of a straight line in the Cartesian plane. 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 0:1 and a vertical line has slope 1:0. However, if the slope is taken to be a single
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
then a horizontal line has slope \tfrac01 = 0 while a vertical line has an undefined slope, since in real-number arithmetic the quotient \tfrac10 is undefined. The real-valued slope \tfrac of a line through the origin is the vertical coordinate of the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
between the line and a vertical line at horizontal coordinate 1, dashed black in the figure. The vertical red and dashed black lines are parallel, so they have no intersection in the plane. Sometimes they are said to intersect at a
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. Ad ...
, and the ratio 1:0 is represented by a new number see below. Vertical lines are sometimes said to have an "infinitely steep" slope.


Inverse of multiplication

Division is the inverse of
multiplication Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...
, meaning that multiplying and then dividing by the same non-zero quantity, or vice versa, leaves an original quantity unchanged; for example (5 \times 3) / 3 = (5 / 3) \times 3 = 5. Thus a division problem such as \tfrac = can be solved by rewriting it as an equivalent equation involving multiplication, \times 3 = 6, where represents the same unknown quantity, and then finding the value for which the statement is true; in this case the unknown quantity is 2, because 2\times 3 = 6, so therefore \tfrac63 = 2. An analogous problem involving division by zero, \tfrac = , requires determining an unknown quantity satisfying \times 0 = 6. However, any number multiplied by zero is zero rather than six, so there exists no number which can substitute for to make a true statement. When the problem is changed to \tfrac = , the equivalent multiplicative statement is 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 \tfrac. Because of these difficulties, quotients where the divisor is zero are traditionally taken to be ''undefined'', and division by zero is not allowed.


Fallacies

A compelling reason for not allowing division by zero is that allowing it leads to
fallacies A fallacy is the use of invalid or otherwise faulty reasoning in the construction of an argument that may appear to be well-reasoned if unnoticed. The term was introduced in the Western intellectual tradition by the Aristotelian '' De Sophis ...
. When working with numbers, it is easy to identify an illegal division by zero. For example: :From 0\times 1 = 0 and 0\times 2 = 0 one gets 0\times 1 = 0\times 2. Cancelling from both sides yields 1 = 2, a false statement. The fallacy here arises from the assumption that it is legitimate to cancel like any other number, whereas, in fact, doing so is a form of division by . Using
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, it is possible to disguise a division by zero to obtain an invalid proof. For example: This is essentially the same fallacious computation as the previous numerical version, but the division by zero was obfuscated because we wrote as .


Early attempts

The '' Brāhmasphuṭasiddhānta'' 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, do ...
(c. 598–668) is the earliest text to treat
zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...
as a number in its own right and to define operations involving zero. 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 (Devanagari: महावीर, ), also known as Vardhamana (Devanagari: वर्धमान, ), was the 24th ''Tirthankara'' (Supreme Preacher and Ford Maker) of Jainism. Although the dates and most historical details of his lif ...
unsuccessfully tried to correct the mistake Brahmagupta made in his book '' Ganita Sara Samgraha'': "A number remains unchanged when divided by zero."
Bhāskara II Bhāskara II ('; 1114–1185), also known as Bhāskarāchārya (), was an Indian people, Indian polymath, Indian mathematicians, mathematician, astronomer and engineer. From verses in his main work, Siddhānta Śiromaṇi, it can be inferre ...
's '' Līlāvatī'' (12th century) proposed that division by zero results in an infinite quantity,
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.
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 State rel ...
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, writer, and clergyman who is regarded as the founder of "immaterialism", a philos ...
's criticism of
infinitesimal calculus Calculus 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 arithmetic operations. Originally called infinitesimal calculus or "the calculus of ...
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 ...
'' ("ghosts of departed quantities").


Calculus

Calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
studies the behavior of functions using the concept of a limit, the value to which a function's output tends as its input tends to some specific value. The notation \lim_ f(x) = L means that the value of the function f can be made arbitrarily close to L by choosing x sufficiently close to c. In the case where the limit of the
real function In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...
f increases without bound as x tends to c, the function is not defined at x, a type of
mathematical singularity In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity. For exa ...
. Instead, the function is said to " tend to infinity", denoted \lim_ f(x) = \infty, and its
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
has the line x=c as a vertical
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, ...
. While such a function is not formally defined for x = c, and the
infinity symbol The infinity symbol () is a mathematical symbol representing the concept of infinity. This symbol is also called a ''lemniscate'', after the lemniscate curves of a similar shape studied in algebraic geometry, or "lazy eight", in the terminolo ...
\infty in this case does not represent any specific
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
, 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", -\infty. In some cases a function tends to two different values when x tends to c from above and below ; such a function has two distinct
one-sided limit In calculus, a one-sided limit refers to either one 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. The ...
s. A basic example of an infinite singularity is the
reciprocal function In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/'' ...
, f(x) = 1/x, which tends to positive or negative infinity as x tends to \lim_ \frac1x = +\infty,\qquad \lim_ \frac1x = -\infty. In most cases, the limit of a quotient of functions is equal to the quotient of the limits of each function separately, \lim_ \frac = \frac. However, when a function is constructed by dividing two functions whose separate limits are both equal to 0, then the limit of the result cannot be determined from the separate limits, so is said to take an
indeterminate form Indeterminate form is a mathematical expression that can obtain any value depending on circumstances. In calculus, it is usually possible to compute the limit of the sum, difference, product, quotient or power of two functions by taking the corres ...
, informally written \tfrac00. (Another indeterminate form, \tfrac \infty \infty, 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 \lim_ \dfrac, the separate limits of the numerator and denominator are 0, so we have the indeterminate form \tfrac00, but simplifying the quotient first shows that the limit exists: \lim_ \frac = \lim_ \frac = \lim_ (x + 1) = 2.


Alternative number systems


Extended real line

The affinely extended real numbers are obtained from the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s \R by adding two new numbers +\infty and -\infty, read as "positive infinity" and "negative infinity" respectively, and representing points at infinity. With the addition of \pm \infty, 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 1/0 is usually left undefined. However, in contexts where only non-negative values are considered, it is often convenient to define 1/0 = +\infty.


Projectively extended real line

The set \mathbb\cup\ is the
projectively extended real line In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the set of the real numbers, \mathbb, by a point denoted . It is thus the set \mathbb\cup\ with the standard ...
, which is a
one-point compactification In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Al ...
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. Ad ...
, 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 In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...
and cotangent functions of
trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...
: 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 subject of
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 ...
applies the concepts of calculus in the
complex numbers 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 a ...
. Of major importance in this subject is the extended complex numbers \C \cup\, the set of complex numbers with a single additional number appended, usually denoted by the
infinity symbol The infinity symbol () is a mathematical symbol representing the concept of infinity. This symbol is also called a ''lemniscate'', after the lemniscate curves of a similar shape studied in algebraic geometry, or "lazy eight", in the terminolo ...
\infty and representing a
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. Ad ...
, which is defined to be contained in every exterior domain, making those its
topological Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...
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 \infty, a
one-point compactification In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Al ...
, making the extended complex numbers topologically equivalent to a
sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
. This equivalence can be extended to a metrical equivalence by mapping each complex number to a point on the sphere via inverse
stereographic projection In mathematics, a stereographic projection is a perspective transform, perspective projection of the sphere, through a specific point (geometry), point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (th ...
, 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 In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...
. The set is usually denoted by the symbol for the complex numbers decorated by an asterisk, overline, tilde, or circumflex, for example \hat\C = \C \cup\. In the extended complex numbers, for any nonzero complex number z, ordinary complex arithmetic is extended by the additional rules \tfrac=\infty, \tfrac = 0, \infty + 0 = \infty, \infty + z = \infty, \infty \cdot z = \infty. However, \tfrac, \tfrac, and 0\cdot\infty 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 An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 (for example, The set of all ...
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 for ...
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 Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
. 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, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...
s of integers, with , define a
binary relation In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
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. A simpler example is equ ...
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 ...
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). 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, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, for some integer n
s, division by zero is still impossible, but division by non-zero
infinitesimal In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the " ...
s is possible. The same holds true in the
surreal number In mathematics, the surreal number system is a total order, totally ordered proper class containing not only the real numbers but also Infinity, infinite and infinitesimal, infinitesimal numbers, respectively larger or smaller in absolute value th ...
s.


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 values). 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 (: matrices or matrixes) 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 m ...
algebra, square or rectangular blocks of numbers are manipulated as though they were numbers themselves: matrices can be added and multiplied, and in some cases, a version of division also exists. Dividing by a matrix means, more precisely, multiplying by its inverse. Not all matrices have inverses. For example, a matrix containing only zeros is not invertible. 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 In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, 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 (mathematics), 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 prope ...
, 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. 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 fo ...
, 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 (mathematics), 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 prope ...
can be extended to a structure called a
wheel A wheel is a rotating component (typically circular in shape) that is intended to turn on an axle Bearing (mechanical), bearing. The wheel is one of the key components of the wheel and axle which is one of the Simple machine, six simple machin ...
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, 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 setting for studying divisibilit ...
, 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 rings and 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 In algebra, a division ring, also called a skew field (or, occasionally, a sfield), 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 multiplicativ ...
). 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 ze ...
. 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 fo ...
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 In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that ba ...
, which since the 1980s has been standardized by the
IEEE 754 The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic originally established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard #Design rationale, add ...
specification. In IEEE floating-point arithmetic, numbers are represented using a sign (positive or negative), a fixed-precision
significand The significand (also coefficient, sometimes argument, or more ambiguously mantissa, fraction, or characteristic) is the first (left) part of a number in scientific notation or related concepts in floating-point representation, consisting of its s ...
and an integer
exponent In mathematics, exponentiation, denoted , is an operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, i ...
. Numbers whose exponent is too large to represent instead "overflow" to positive or negative
infinity Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophic ...
(+∞ or −∞), while numbers whose exponent is too small to represent instead " underflow" to 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 Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, so that −0, +0 and 0 are equivalent. However, in computing, some number representations allow for the existence of two zeros, often denoted by ...
(−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 The term arithmetic underflow (also floating-point underflow, or just underflow) is a condition in a computer program where the result of a calculation is a number of more precise absolute value than the computer can actually represent in memory ...
. For example, using single-precision IEEE arithmetic, if , then ''x''/2 underflows to −0, and dividing 1 by this result produces 1/(''x''/2) = −∞. The exact result −2150 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

Integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
division by zero is usually handled differently from floating point since there is no integer representation for the result.
CPUs A central processing unit (CPU), also called a central processor, main processor, or just processor, is the primary Processor (computing), processor in a given computer. Its electronic circuitry executes Instruction (computing), instructions ...
differ in behavior: for instance
x86 x86 (also known as 80x86 or the 8086 family) is a family of complex instruction set computer (CISC) instruction set architectures initially developed by Intel, based on the 8086 microprocessor and its 8-bit-external-bus variant, the 8088. Th ...
processors trigger a hardware exception, while
PowerPC PowerPC (with the backronym Performance Optimization With Enhanced RISC – Performance Computing, sometimes abbreviated as PPC) is a reduced instruction set computer (RISC) instruction set architecture (ISA) created by the 1991 Apple Inc., App ...
processors silently generate an incorrect result for the division and continue, and ARM processors can either cause a hardware exception or return zero. Because of this inconsistency between platforms, the C and C++
programming language A programming language is a system of notation for writing computer programs. Programming languages are described in terms of their Syntax (programming languages), syntax (form) and semantics (computer science), semantics (meaning), usually def ...
s consider the result of dividing by zero
undefined behavior In computer programming, a program exhibits undefined behavior (UB) when it contains, or is executing code for which its programming language specification does not mandate any specific requirements. This is different from unspecified behavior, ...
. In typical higher-level programming languages, such as Python, an exception is raised for attempted division by zero, which can be handled in another part of the program.


In proof assistants

Many
proof assistant In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human–machine collaboration. This involves some sort of interactive proof edi ...
s, such as Coq and Lean, define 1/0 = 0. This is due to the requirement that all functions are total. Such a definition does not create contradictions, as further manipulations (such as cancelling out) still require that the divisor is non-zero.


Historical accidents

* On 21 September 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

*
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 ze ...
*
Zero to the power of zero Zero to the power of zero, denoted as , is a mathematical expression with different interpretations depending on the context. In certain areas of mathematics, such as combinatorics and algebra, is conventionally defined as 1 because this assignmen ...
*
L'Hôpital's rule L'Hôpital's rule (, ), also known as Bernoulli's rule, is a mathematical theorem that allows evaluating limits of indeterminate forms using derivatives. Application (or repeated application) of the rule often converts an indeterminate form ...


Notes


Sources

* (Dover reprint 1997) * * * * * *


Further reading

* * * (Dover reprint, 1999) * {{DEFAULTSORT:Division by zero 0 (number) Computer arithmetic Division (mathematics) Computer errors Fractions (mathematics) Infinity Mathematical analysis Mathematical fallacies