In mathematics, a division is called a division by zero if the divisor is zero. Such a division can be formally expressed as where a is the dividend. Whether this expression can be assigned a meaningful (well-defined) value depends upon the mathematical setting. In ordinary (real number) arithmetic, the expression has no meaning. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... 0 (zero) is both a number — or, more precisely, a numeral representing a number — and a numerical digit. ... In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ...

In computer programming, integer division by zero often causes a program to terminate (floating point generally does not, see below). Computer programming (often simply programming or coding) is the craft of writing a set of commands or instructions that can later be compiled and/or interpreted and then inherently transformed to an executable that an electronic machine can execute or run. Programming requires mainly logic, but has elements of science... The integers are commonly denoted by the above symbol. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ...

Interpretation in elementary arithmetic

When division is explained at the elementary level, it is often considered as a description of dividing a set of objects into equal parts. As an example, if you have 10 blocks, and you make subsets of 5 blocks, then you have created 2 equal sets. This would be a demonstration that 10/5 = 2. The divisor is the number of blocks in each set. The result of division answers the question, "If I have equal sets of 5, how many of those sets will combine to make a set of 10?"

We can apply this to show the problems of dividing by zero. It is not meaningful for us to ask, "If I have equal sets of 0, how many of those sets will combine to give me a set of 10?", because adding many sets of zero will never amount to 10. Therefore, as far as elementary arithmetic is concerned, division by zero cannot be defined. Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. ...

Another method of describing division is a repeated subtraction, e.g. to divide 13 by 5, we can subtract 5 two times, which leaves a remainder of 3. The divisor is subtracted until the remainder is less than the divisor. The result is often reported as, 13/5 = 2 remainder 3. But in the case of zero, repeated subtraction of zero will never yield a remainder less than or equal to zero, so dividing by zero is not defined. In mathematics, the result of the division of two integers usually cannot be expressed with an integer quotient, unless a remainder —an amount left over— is also acknowledged. ...

Early attempts

The Brahmasphutasiddhanta of Brahmagupta is the earliest known text to treat zero as a number in its own right and to define operations involving zero. The author failed, however, in his attempt to explain division by zero – his definition can be easily proven to lead to algebraic absurdities. According to Brahmagupta, The main work of Brahmagupta, Brahmasphutasiddhanta (The Opening of the Universe), written in 628, contains some remarkably advanced ideas, including a good understanding of the mathematical role of zero, rules for manipulating both positive and negative numbers, a method for computing square roots, methods of solving linear and some quadratic... Brahmagupta (ब्रह्मगुप्त) (598-668) was an Indian mathematician and astronomer. ... 0 (zero) is both a number — or, more precisely, a numeral representing a number — and a numerical digit. ...

"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, Mahavira tried unsuccessfully to correct Brahmagupta's mistake in his book in Ganita Sara Samgraha: Mahavira, 9th century Indian mathematician, asserted that the square root of a negative number did not exist. ...

"A number remains unchanged when divided by zero."

Bhaskara II tried to solve the problem by defining . This definition makes a certain degree of sense, as discussed below, but can lead to paradoxes if not treated carefully. It is unlikely that he understood all the intricacies involved, so his solution cannot be considered successful. [1] Bhāskara (1114-1185), also called Bhāskara II and Bhāskarācārya (Bhaskara the teacher) was an Indian mathematician. ...

Algebraic interpretation

It is generally regarded among mathematicians that a natural way to interpret division by zero is to first define division in terms of other arithmetic operations. Under the standard rules for arithmetic on integers, rational numbers, real numbers and complex numbers, division by zero is undefined. Division by zero must be left undefined in any mathematical system that obeys the axioms of a field. The reason is that division is defined to be the inverse operation of multiplication. This means that the value of a / b is the solution x of the equation bx = a whenever such a value exists and is unique. Otherwise the value is left undefined. The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... Please refer to Real vs. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... In mathematics, multiplication is an elementary arithmetic operation. ...

For b = 0, the equation bx = a can be rewritten as 0x = a or simply 0 = a. Thus, in this case, the equation bx = a has no solution if a is not equal to 0, and has any x as a solution if a equals 0. In either case, there is no unique value, so a / b is undefined. Conversely, in a field, the expression a / b is always defined if b is not equal to zero. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Fallacies based on division by zero

It is possible to disguise a special case of division by zero in an algebraic argument, leading to spurious proofs that 2 = 1 such as the following: Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... In mathematics, there are a variety of spurious proofs of obvious contradictions. ...

• 1) For any real number x:

x² − x² = x² − x²

• 2) Factoring both sides in two different ways:

(x − x)(x + x) = x(x − x)

• 3) Divide both sides by x - x:

• 4) Simplified, yields:

(1)(x + x) = x(1)

(x + x) = x

• 5) Which is:

2x = x

• 6) Since this is valid for any value of x, we can plug in x = 1:

2 = 1

This argument is sometimes presented as a riddle; in such cases the 3rd step is usually omitted in an attempt to trick the listener.

The fallacy is the assumption in step 4 that , which is , simplifies to 1. This proof is for the special case of dividing by zero when the numerator is zero. The fallacy results from the assumption that ⁰/₀ = 1, an assumption that generates the absurdity that 2 = 1. This article or section does not cite its references or sources. ...

Any other non-zero value assigned to ⁰/₀ leads to similar contradictions. In practice, division by a term in any algebraic argument requires an explicit assumption that the term is not zero or a justification that the term can never be zero.

Abstract algebra

Similar statements are true in more general algebraic structures, such as rings and fields. In a field, every nonzero element is invertible under multiplication, so as above, division poses problems only when attempting to divide by zero. However, in other rings, division by nonzero elements may also pose problems. Consider, for example, the ring Z/6Z of integers mod 6. What meaning should we give to the expression 2 / 2? This should be the solution x of the equation 2x = 2. But in the ring Z/6Z, 2 is not invertible under multiplication. This equation has two distinct solutions, x = 1 and x = 4, so the expression 2 / 2 is undefined. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Limits and division by zero

The function y = 1/x. As x approaches 0, y approaches infinity (and vice versa).

At first glance it seems possible to define by considering the limit of as b approaches 0. Image File history File links Hyperbola_one_over_x. ... Image File history File links Hyperbola_one_over_x. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...

For any positive a, it is known that

and for any negative a, A negative number is a number that is less than zero, such as −3. ...

Therefore, we might consider defining as +∞ for positive a, and −∞ for negative a. However, this definition can be inconvenient for two reasons.

First, positive and negative infinity are not real numbers. So as long as we wish to remain in the context of real numbers, we have not defined anything meaningful. If we want to use such a definition, we will have to extend the real number line, as discussed below. The infinity symbol ∞ in several typefaces The word infinity comes from the Latin infinitas or unboundedness. ... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ...

Second, taking the limit from the right is arbitrary. We could just as well have taken limits from the left and defined to be −∞ for positive a, and +∞ for negative a. This can be further illustrated using the equation (assuming that several natural properties of reals extend to infinities)

which does not make much sense. This means that the only workable extension is introducing an unsigned infinity, discussed below.

Furthermore, there is no obvious definition of that can be derived from considering the limit of a ratio. The limit

does not exist. Limits of the form

in which both f(x) and g(x) approach 0 as x approaches 0, may converge to any value or may not converge at all (see l'Hôpital's rule for discussion and examples of limits of ratios). So, this particular approach cannot lead us to a useful definition of . In calculus, lHôpitals rule (alternatively, lHospitals rule) uses derivatives to help compute limits with indeterminate forms. ...

Formal interpretation

A formal calculation is one which is carried out using rules of arithmetic, without consideration of whether the result of the calculation is well-defined. Thus, as a "rule of thumb", it is sometimes useful to think of as being , provided a is not zero. This infinity can be either positive, negative or unsigned, depending on context. For example, formally: In mathematical logic, a formal calculation is sometimes defined as a calculation which is systematic, but without a rigorous justification. ...

As with any formal calculation, invalid results may be obtained.

Other number systems

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

Real projective line

The set is the real projective line. Here means an unsigned infinity, an infinite quantity which is neither positive nor negative. This quantity satisfies which, as we have seen, is necessary in this context. In this structure, we can define for nonzero a, and . These definitions lead to many interesting results. However, this structure is not a field, and should not be expected to behave like one. For example, has no meaning in the projective line. In mathematics, a projective line is a one-dimensional projective space. ...

This is a one-point compactification of the real line. In mathematics, compactification is applied to topological spaces to make them compact spaces. ...

It is the natural way to view the range of the tangent and cotangent functions of trigonometry: tan(x) approaches the single point at infinity as x approaches either π/2 or −π/2 from either direction. Wikibooks has a book on the topic of Trigonometry Trigonometry (from the Greek trigonon = three angles and metron = measure [1]) is a branch of mathematics which deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right triangles). ...

Riemann sphere

The set is the Riemann sphere, of major importance in complex analysis. In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ... Complex analysis is the branch of mathematics investigating functions of complex numbers, and is of enormous practical use in many branches of mathematics, including applied mathematics. ...

Here, too, is an unsigned infinity, or, as it is often called in this context, the point at infinity. The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ...

This set is analogous to the real projective line, except that it is based on the field of complex numbers; and this set is also not a field. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

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 a.

Non-standard analysis

In hyperreal numbers and surreal numbers, division by zero is still impossible, but division by non-zero infinitesimals is possible. In mathematics, particularly in non-standard analysis and mathematical logic, hyperreal numbers or nonstandard reals (usually denoted as *R) denote an ordered field which is a proper extension of the ordered field of real numbers R and which satisfies the transfer principle. ... In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ...

Abstract algebra

Any number system which forms a commutative ring, as do the integers, the real numbers, and the complex numbers, for instance, can be extended to a wheel in which division by zero is always possible, but division has then a slightly different meaning. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... Wheels are a kind of algebras where division is always defined. ...

In mathematical analysis

In distribution theory one can extend the function 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. This page deals with mathematical distributions. ... In mathematics, the Cauchy principal value of certain improper integrals is defined as either the finite number where b is a point at which the behavior of the function f is such that for any a < b and for any c > b (one sign is + and the other is −). or... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...

Division by zero in computer arithmetic

The IEEE floating-point standard, supported by almost all modern processors, specifies that every floating point arithmetic operation, including division by zero, has a well-defined result. In IEEE 754 arithmetic, is positive infinity when a is positive, negative infinity when a is negative, and NaN (not a number) when a = 0. The infinity signs change when dividing by −0 instead. This is possible because in IEEE 754 there are two zero values, plus zero and minus zero, and thus no ambiguity. The IEEE Standard for Binary Floating-Point Arithmetic (IEEE 754) is the most widely-used standard for floating-point computation, and is followed by many CPU and FPU implementations. ... A floating point unit (FPU) is a part of a CPU specially designed to carry out operations on floating point numbers. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... // Acronym NaN, an abbreviation of Not a Number used in computer arithmetic and defined in the IEEE floating-point standard. ... −0 is the representation of negative zero, a number that exists in computing in some signed number representations for integers and in most floating point number representations. ... −0 is the representation of negative zero, a number that exists in computing in some signed number representations for integers and in most floating point number representations. ...

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. (That result is often zero.) Exception handling is a programming language construct or computer hardware mechanism designed to handle the occurrence of some condition that changes the normal flow of execution. ...

Because of the improper algebraic results of assigning any value to division by zero, many computer programming languages (including those used by calculators) explicitly forbid the execution of the operation and may prematurely halt a program that attempts it, sometimes reporting a "Divide by zero" error. Some programs (especially those that use fixed-point arithmetic 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. A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ... A calculator is a device for performing calculations. ... In computing, a fixed-point number representation is a real data type for a number that has a fixed number of digits before and after the radix point (e. ... In computer science, undefined behavior is a feature of some programming languages — most famously C. In these languages, to simplify the specification and allow some flexibility in implementation, the specification leaves the results of certain operations specifically undefined. ...

In two's complement 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. Twos complement is the most popular method of representing signed integers in computer science. ... In computer science, undefined behavior is a feature of some programming languages — most famously C. In these languages, to simplify the specification and allow some flexibility in implementation, the specification leaves the results of certain operations specifically undefined. ...

Historical accidents

On September 21, 1997 a divide by zero error in the USS Yorktown (CG-48) Remote Data Base Manager brought down all the machines on the network, causing the ship's propulsion system to fail. [2] USS Yorktown (DDG-48/CG-48) was a Ticonderoga-class cruiser in the United States Navy from 1984 to 2004, named for the American Revolutionary War Battle of Yorktown. ...

In popular culture

• As a result of the errors often seen in computers and calculators when an operator attempts to divide by zero, an Internet meme has developed where dividing by zero is seen as synonymous with the "end of the world", universe, forum, etc. The meme inspired the short film "The Last Denominator" , where a division by zero is followed by the sudden realisation of what this means, as the planet explodes.

• E_DIV is an error code generated by some programming languages as a result of division by zero, and can be used in internet slang as an indication of confusion or impossibility.

• One of the satirical Chuck Norris Facts states that "Chuck Norris can divide by zero".

• A short story by Ted Chiang is titled Division by Zero.

This article or section does not cite its references or sources. ... The end of the world is a phrase used to most commonly refer to the death of all life on planet earth. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Chuck Norris Facts in Rolling Stone. ... Chuck Norris reciving the Veteran of the Year award by the U.S Air Force Carlos Ray Chuck Norris (born March 10, 1940) is an American martial artist, action star, and Hollywood actor. ... Ted Chiang Ted Chiang (born 1967) is an American science fiction writer. ...

See also

• Defined and undefined
• Indeterminate form