In mathematics, a division is called a division by zero if the divisor is zero. Such a division can be formally expressed as $\backslash begin\{matrix\}\backslash frac\{a\}\{0\}\backslash end\{matrix\}$ 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.

In computer programming, division by zero often causes a program to terminate (see below).

Interpretation in elementary arithmetic

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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.

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 zero, so dividing by zero is not defined.

Early attempts

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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,

"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:

"A number remains unchanged when divided by zero."

Bhaskara II tried to solve the problem by defining $\backslash textstyle\backslash frac\{n\}\{0\}=\backslash infty$. 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. ^*

Algebraic interpretation

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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.

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.

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:

• 1) For any real number $x$:

$x^2\; -\; x^2\; =\; x^2\; -\; x^2$

• 2) Factoring both sides in two different ways:

$(x\; -\; x)(x\; +\; x)\; =\; x(x\; -\; x)$

• 3) Dividing both sides by $x\; -\; x$, giving $(0/0)$:

$(0/0)(x\; +\; x)\; =\; x(0/0)$

• 4) Simplified, yields:

$(1)(x\; +\; x)\; =\; x(1)$

• 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 $(x\; -\; x)/(x\; -\; x)$ -- which is $(0/0)$ -- 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 $2\; =\; 1$.

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.

Limits and division by zero

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At first glance it seems possible to define $\backslash begin\{matrix\}\backslash frac\{a\}\{0\}\backslash end\{matrix\}$ by considering the limit of $\backslash begin\{matrix\}\backslash frac\{a\}\{b\}\backslash end\{matrix\}$ as b approaches 0.

For any positive a, it is known that

$\backslash lim\_\{b\; \backslash to\; 0^\{+\}\}\; \{a\; \backslash over\; b\}\; =\; \{+\}\backslash infty$

and for any negative a,

$\backslash lim\_\{b\; \backslash to\; 0^\{+\}\}\; \{a\; \backslash over\; b\}\; =\; \{-\}\backslash infty$

Therefore, we might consider defining $\backslash begin\{matrix\}\backslash frac\{a\}\{0\}\backslash end\{matrix\}$ 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.

Second, taking the limit from the right is arbitrary. We could just as well have taken limits from the left and defined $\backslash begin\{matrix\}\backslash frac\{a\}\{0\}\backslash end\{matrix\}$ 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)

$+\backslash infty\; =\; \backslash frac\{1\}\{0\}\; =\; \backslash frac\{1\}\{-0\}\; =\; -\backslash frac\{1\}\{0\}\; =\; -\backslash infty$

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

Furthermore, there is no obvious definition of $\backslash begin\{matrix\}\backslash frac\{0\}\{0\}\backslash end\{matrix\}$ that can be derived from considering the limit of a ratio. The limit

$\backslash lim\_\{(a,b)\; \backslash to\; (0,0)\}\; \{a\; \backslash over\; b\}$

does not exist. Limits of the form

$\backslash lim\_\{x\; \backslash to\; 0\}\; \{f(x)\; \backslash over\; g(x)\}$

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 $\backslash begin\{matrix\}\backslash frac\{0\}\{0\}\backslash end\{matrix\}$.

Formal interpretation

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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 $\backslash begin\{matrix\}\backslash frac\{a\}\{0\}\backslash end\{matrix\}$ as being $\backslash infty$, provided a is not zero. This infinity can be either positive, negative or unsigned, depending on context. For example, formally:

$\backslash lim\_\{x\; \backslash to\; 0\}\; \{\backslash frac\{1\}\{x^2\}\; =\backslash frac\{\backslash lim\_\{x\; \backslash to\; 0\}\; \{1\}\}\{\backslash lim\_\{x\; \backslash to\; 0\}\; \{x^2\}\}\}\; =\; \backslash frac\{1\}\{+0\}\; =\; +\backslash infty$

Of course, as with any formal calculation, invalid results may be obtained.

Other number systems

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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 $\backslash mathbb\{R\}\backslash cup\backslash \{\backslash infty\backslash \}$ is the real projective line. Here $\backslash infty$ means an unsigned infinity, an infinite quantity which is neither positive nor negative. This quantity satisfies $-\backslash infty\; =\; \backslash infty$ which, as we have seen, is necessary in this context. In this structure, we can define $\backslash begin\{matrix\}\backslash frac\{a\}\{0\}\; =\; \backslash infty\backslash end\{matrix\}$ for nonzero a, and $\backslash begin\{matrix\}\backslash frac\{a\}\{\backslash infty\}\; =\; 0\backslash end\{matrix\}$. 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, $\backslash infty\; +\; \backslash infty$ has no meaning in the projective line.

Riemann sphere

The set $\backslash mathbb\{C\}\backslash cup\backslash \{\backslash infty\backslash \}$ is the Riemann sphere, of major importance in complex analysis.

Here, too, $\backslash infty$ is an unsigned infinity, or, as it is often called in this context, the point at infinity.

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.

Extended non-negative real number line

The negative real numbers can be discarded, and infinity introduced, leading to the set $\backslash infty$ - Where division by zero can be naturally defined as $\backslash begin\{matrix\}\backslash frac\{a\}\{0\}\; =\; \backslash infty\backslash end\{matrix\}$ 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.

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 mathematical analysis

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In distribution theory one can extend the function $\backslash begin\{matrix\}\backslash frac\{1\}\{x\}\backslash end\{matrix\}$ 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.

Division by zero in computer arithmetic

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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, $\backslash textstyle\backslash frac\{a\}\{0\}$ is positive infinity when a is positive, negative infinity when a is negative, and NaN (not a number) when a = 0.

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 (often 0).

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 popular culture

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Because of the errors often seen in computers and calculators when an operator attempts to divide by zero, an Internet meme has surfaced where dividing by zero is synonymous with the end of the world, universe, forum, etc. This meme has inspired the short film "The Last Denominator" ^*.

See also

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• Defined and undefined

Elementary arithmetic | Computer arithmetic | Fractions

Null#Division durch Null | División por cero | Divisione per zero | Delen door nul | ゼロ除算 | Dzielenie przez zero | Division by zero | Division med noll