Inmathematics,division by zero,division where the divisor (denominator) iszero, is a problematic special case. Usingfraction notation, the general example can be written as, where is the dividend (numerator).
The usual definition of thequotient inelementary arithmetic is the number which yields the dividend whenmultiplied by the divisor. That is, is equivalent to. By this definition, the quotient is nonsensical, as the product is always rather than some other number. Following the ordinary rules ofelementary algebra while allowing division by zero can create amathematical fallacy, a subtle mistake leading to absurd results. To prevent this, the arithmetic ofreal numbers and more general numerical structures calledfields leaves division by zeroundefined, and situations where division by zero might occur must be treated with care. Since any number multiplied by zero is zero, the expression is also undefined.
Calculus studies the behavior offunctions in thelimit as their input tends to some value. When areal function 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 ofmathematical singularity. For example, thereciprocal function,, tends to infinity as tends to. When both the numerator and the denominator tend to zero at the same input, the expression is said to take anindeterminate form, as the resulting limit depends on the specific functions forming the fraction and cannot be determined from their separate limits.
As an alternative to the common convention of working with fields such as the real numbers and leaving division by zero undefined, it is possible to define the result of division by zero in other ways, resulting in different number systems. For example, the quotient can be defined to equal zero; it can be defined to equal a new explicitpoint at infinity, sometimes denoted by theinfinity symbol; 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.
Incomputing, an error may result from an attempt to divide by zero. Depending on the context and thetype of number involved, dividing by zero may evaluate topositive or negative infinity, return a specialnot-a-number value, orcrash the program, among other possibilities.
Thedivision can be conceptually interpreted in several ways.[1]
Inquotitive division, the dividend is imagined to be split up into parts of size (the divisor), and the quotient 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 alettuce wrap). Arbitrarily many such sandwiches can be made from ten slices of bread, as the bread is irrelevant.[2]
The quotitive concept of division lends itself to calculation by repeatedsubtraction: 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 waynever terminates.[3] Such an interminable division-by-zeroalgorithm is physically exhibited by somemechanical calculators.[4]
Inpartitive division, the dividend is imagined to be split into parts, and the quotient 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.[5]
In another interpretation, the quotient represents theratio.[6] For example, a cake recipe might call for tencups of flour and two cups of sugar, a ratio of or, proportionally,. To scale this recipe to larger or smaller quantities of cake, a ratio of flour to sugar proportional to 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.[7] Now imagine a sugar-free cake recipe calls for ten cups of flour and zero cups of sugar. The ratio, or proportionally, is perfectly sensible:[8] 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 theslope of astraight line in theCartesian plane.[9] 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 and a vertical line has slope. However, if the slope is taken to be a singlereal number then a horizontal line has slope while a vertical line has an undefined slope, since in real-number arithmetic the quotient is undefined.[10] The real-valued slope of a line through the origin is the vertical coordinate of theintersection between the line and a vertical line at horizontal coordinate, dashed black in the figure. The vertical red and dashed black lines areparallel, so they have no intersection in the plane. Sometimes they are said to intersect at apoint at infinity, and the ratio is represented by a new number;[11] see§ Projectively extended real line below. Vertical lines are sometimes said to have an "infinitely steep" slope.
Division is the inverse ofmultiplication, meaning that multiplying and then dividing by the same non-zero quantity, or vice versa, leaves an original quantity unchanged; for example.[12] Thus a division problem such as can be solved by rewriting it as an equivalent equation involving multiplication,, where represents the same unknown quantity, and then finding the value for which the statement is true; in this case the unknown quantity is, because, so therefore.[13]
An analogous problem involving division by zero,, requires determining an unknown quantity satisfying. 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.[14]
When the problem is changed to, the equivalent multiplicative statement is; in this caseany 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.
Because of these difficulties, quotients where the divisor is zero are traditionally taken to beundefined, and division by zero is not allowed.[15][16]
A compelling reason for not allowing division by zero is that allowing it leads tofallacies.
When working with numbers, it is easy to identify an illegal division by zero. For example:
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.
Usingalgebra, it is possible to disguise a division by zero[17] to obtain aninvalid proof. For example:[18]
This is essentially the same fallacious computation as the previous numerical version, but the division by zero was obfuscated because we wrote as.
TheBrāhmasphuṭasiddhānta ofBrahmagupta (c. 598–668) is the earliest text to treatzero as a number in its own right and to define operations involving zero.[17] 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 unsuccessfully tried to correct the mistake Brahmagupta made in his bookGanita Sara Samgraha: "A number remains unchanged when divided by zero."[17]
Bhāskara II'sLīlāvatī (12th century) proposed that division by zero results in an infinite quantity,[19]
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 is contained inAnglo-Irish philosopherGeorge Berkeley's criticism ofinfinitesimal calculus in 1734 inThe Analyst ("ghosts of departed quantities").[20]
Calculus studies the behavior offunctions using the concept of alimit, the value to which a function's output tends as its input tends to some specific value. The notation means that the value of the function can be made arbitrarily close to by choosing sufficiently close to.
In the case where the limit of thereal function increases without bound as tends to, the function is not defined at, a type ofmathematical singularity. Instead, the function is said to "tend to infinity", denoted, and itsgraph has the line as a verticalasymptote. While such a function is not formally defined for, and theinfinity symbol in this case does not represent any specificreal number, such limits are informally said to "equal infinity". If the value of the function decreases without bound, the function is said to "tend to negative infinity",. In some cases a function tends to two different values when tends to from above () and below (); such a function has two distinctone-sided limits.[21]
A basic example of an infinite singularity is thereciprocal function,, which tends to positive or negative infinity as tends to:
In most cases, the limit of a quotient of functions is equal to the quotient of the limits of each function separately,
However, when a function is constructed by dividing two functions whose separate limits are both equal to, then the limit of the result cannot be determined from the separate limits, so is said to take anindeterminate form, informally written. (Another indeterminate form,, 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 the separate limits of the numerator and denominator are, so we have the indeterminate form, but simplifying the quotient first shows that the limit exists:
Theaffinely extended real numbers are obtained from thereal numbers by adding two new numbers and, read as "positive infinity" and "negative infinity" respectively, and representingpoints at infinity. With the addition of, 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 is usually left undefined. However, in contexts where only non-negative values are considered, it is often convenient to define.
The set is theprojectively extended real line, which is aone-point compactification of the real line. Here means an unsigned infinity orpoint at infinity, an infinite quantity that is neither positive nor negative. This quantity satisfies, which is necessary in this context. In this structure, can be defined for nonzero, and when is not. It is the natural way to view the range of thetangent function and cotangent functions oftrigonometry: 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 afield, and should not be expected to behave like one. For example, is undefined in this extension of the real line.
The subject ofcomplex analysis applies the concepts of calculus in thecomplex numbers. Of major importance in this subject is theextended complex numbers, the set of complex numbers with a single additional number appended, usually denoted by theinfinity symbol and representing apoint at infinity, which is defined to be contained in everyexterior domain, making those itstopologicalneighborhoods.
This can intuitively be thought of as wrapping up the infinite edges of the complex plane and pinning them together at the single point, aone-point compactification, making the extended complex numbers topologically equivalent to asphere. This equivalence can be extended to a metrical equivalence by mapping each complex number to a point on the sphere via inversestereographic projection, with the resultingspherical 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 theRiemann sphere. The set is usually denoted by the symbol for the complex numbers decorated by an asterisk, overline, tilde, or circumflex, for example.
In the extended complex numbers, for any nonzero complex number, ordinary complex arithmetic is extended by the additional rules,,,,. However,,, and are left undefined.
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 ofintegers in order to incorporate the negative integers. Similarly, to support division of any integer by any other, the realm of numbers must expand to therational numbers. 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 (isundefined) in the whole number setting, this remains true as the setting expands to thereal or evencomplex numbers.[22]
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.[23] 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 onset theory. First, the natural numbers (including zero) are established on an axiomatic basis such asPeano's axiom system and then this is expanded to thering of integers. The next step is to define the rational numbers keeping in mind that this must be done using only the sets and operations that have already been established, namely, addition, multiplication and the integers. Starting with the set ofordered pairs of integers, with, define abinary relation on this set by if and only if. This relation is shown to be anequivalence relation and itsequivalence classes 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 verifyingtransitivity).[24][25][26]
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.
In thehyperreal numbers, division by zero is still impossible, but division by non-zeroinfinitesimals is possible.[27] The same holds true in thesurreal numbers.[28]
Indistribution theory one can extend the function to a distribution on the whole space of real numbers (in effect by usingCauchy principal values). It does not, however, make sense to ask for a "value" of this distribution at; a sophisticated answer refers to thesingular support of the distribution.
Inmatrix algebra, square or rectangular blocks of numbers are manipulated as though they were numbers themselves: matrices can beadded andmultiplied, and in some cases, a version of division also exists. Dividing by a matrix means, more precisely, multiplying by itsinverse. Not all matrices have inverses.[29] For example, amatrix containing only zeros is not invertible.
One can define a pseudo-division, by setting, in which represents thepseudoinverse of. It can be proven that if exists, then. If, then.
Inabstract algebra, the integers, the rational numbers, the real numbers, and the complex numbers can be abstracted to more general algebraic structures, such as acommutative ring, which is a mathematical structure where addition, subtraction, and multiplication behave as they do in the more familiar number systems, but division may not be defined. Adjoining a multiplicative inverses to a commutative ring is calledlocalization. However, the localization of every commutative ring at zero is thetrivial ring, where, 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 acommutative ring can be extended to a structure called awheel in which division by zero is always possible.[30] 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, and if the original system was anintegral domain, the multiplication in the wheel no longer results in acancellative semigroup.
The concepts applied to standard arithmetic are similar to those in more general algebraic structures, such asrings andfields. 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 askew field (which for this reason is called adivision ring). However, in other rings, division by nonzero elements may also pose problems. For example, the ring of integersmodulo 6. The meaning of the expression should be the solution of the equation. But in the ring, is azero divisor. This equation has two distinct solutions, and, so the expression isundefined.
In field theory, the expression is only shorthand for the formal expression, where is the multiplicative inverse of. Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning when is zero. Modern texts, which define fields as a special type of ring, include the axiom for fields (or its equivalent) so that thezero ring is excluded from being a field. In the zero ring, division by zero is possible, which shows that the other field axioms alone are not sufficient to exclude division by zero in defining a field.
In computing, most numerical calculations are done withfloating-point arithmetic, which since the 1980s has been standardized by theIEEE 754 specification. In IEEE floating-point arithmetic, numbers are represented using a sign (positive or negative), a fixed-precisionsignificand and an integerexponent. Numbers whose exponent is too large to represent instead "overflow" to positive or negativeinfinity (+∞ or −∞), while numbers whose exponent is too small to represent instead "underflow" topositive or negative zero (+0 or −0). ANaN (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 bynegative zero (−0) results in an infinity of the opposite sign as the dividend. This definition preserves the sign of the result in case ofarithmetic underflow.[31]
For example, using single-precision IEEE arithmetic, if, then underflows to, and dividing by this result produces. The exact result is too large to represent as a single-precision number, so an infinity of the same sign is used instead to indicate overflow.
Integer division by zero is usually handled differently from floating point since there is no integer representation for the result.CPUs differ in behavior: for instancex86 processors trigger ahardware exception, whilePowerPC processors silently generate an incorrect result for the division and continue, andARM processors can either cause a hardware exception or return zero.[32] Because of this inconsistency between platforms, theC andC++programming languages consider the result of dividing by zeroundefined behavior.[33] In typicalhigher-level programming languages, such asPython,[34] anexception is raised for attempted division by zero, which can be handled in another part of the program.
Manyproof assistants, such asRocq andLean, define 1/0 = 0. This is due to the requirement that all functions aretotal. Such a definition does not create contradictions, as further manipulations (such ascancelling out) still require that the divisor is non-zero.[35][36]
Some other operations, including division, can also be performed by the desk calculator (but don't try to divide by zero; the calculator never will stop trying to divide until stopped manually).For a video demonstration, see:What happens when you divide by zero on a mechanical calculator?, 7 Mar 2021, retrieved2024-01-06 – via YouTube
With appropriate care to be certain that the algebraic signs are not determined by rounding error, the affine mode preserves order relations while fixing up overflow. Thus, for example, the reciprocal of a negative number which underflows is still negative.
ZeroDivisionError", retrieved2024-01-22The standard division function on natural numbers in Coq, div, is total and pure, but incorrect: when the divisor is 0, the result is 0.