Division by zero

The concept that explains division in algebra is that it is the inverse of multiplication. For example,[10]

$${\displaystyle {\frac {6}{3}}=2}$$

since 2 is the value for which the unknown quantity in

$${\displaystyle$$ ?\times 3=6}

is true. But the expression

$${\displaystyle {\frac {6}{0}}=\,x}$$

requires a value to be found for the unknown quantity in

$${\displaystyle x\times 0=6.}$$

But any number multiplied by 0 is 0 and so there is no number that solves the equation.

The expression

$${\displaystyle {\frac {0}{0}}=\,x}$$

requires a value to be found for the unknown quantity in

$${\displaystyle x\times 0=0.}$$

Again, any number multiplied by 0 is 0 and so this time every number solves the equation instead of there being a single number that can be taken as the value of 0/0.

In general, a single value can't be assigned to a fraction where the denominator is 0 so the value remains undefined.

A compelling reason for not allowing division by zero is that, if it were allowed, many absurd results (i.e., fallacies) would arise. When working with numerical quantities it is easy to determine when an illegal attempt to divide by zero is being made. For example, consider the following computation.

With the assumptions:

$${\displaystyle {\begin{aligned}0\times 1&=0,\\0\times 2&=0,\end{aligned}}}$$

the following is true:

$${\displaystyle 0\times 1=0\times 2.}$$

Dividing both sides by zero gives:

$${\displaystyle {\begin{aligned}{\frac {0\times 1}{0}}&={\frac {0\times 2}{0}}\\[6px]{\frac {0}{0}}\times 1&={\frac {0}{0}}\times 2.\end{aligned}}}$$

Simplified, this yields:

$${\displaystyle 1=2.}$$

The fallacy here is the assumption that dividing 0 by 0 is a legitimate operation with the same properties as dividing by any other number.

However, it is possible to disguise a division by zero in an algebraic argument,[3] leading to invalid proofs that, for instance, 1 = 2 such as the following:[11]

Let 1 = x.

Multiply by x to get

$${\displaystyle x=x^{2}.}$$

Subtract 1 from each side to get

$${\displaystyle x-1=x^{2}-1.}$$

Divide both sides by x − 1

$${\displaystyle {\begin{aligned}{\frac {x-1}{x-1}}&={\frac {x^{2}-1}{x-1}}\\[6pt]&={\frac {(x+1)(x-1)}{x-1}},\end{aligned}}}$$

which simplifies to

$${\displaystyle 1=x+1.}$$

But, since x = 1,

$${\displaystyle 1=1+1=2}$$

and therefore

$${\displaystyle 1=2.}$$

The disguised division by zero occurs since x − 1 = 0 when x = 1.

At first glance it seems possible to define a/0 by considering the limit of a/b as b approaches 0.

For any positive a, the limit from the right is

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

however, the limit from the left is

$${\displaystyle \lim _{b\to 0^{-}}{a \over b}=-\infty }$$

and so the ${\displaystyle \lim _{b\to 0}{a \over b}}$ is undefined (the limit is also undefined for negative a).

Furthermore, there is no obvious definition of 0/0 that can be derived from considering the limit of a ratio. The limit

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

does not exist. Limits of the form

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

in which both f(x) and g(x) approach 0 as x approaches 0, may equal any real or infinite value, or may not exist at all, depending on the particular functions f and g.

For example, consider:

$${\displaystyle \lim _{x\to 1}{x^{2}-1 \over x-1}}$$

This initially appears to be indeterminate. However:

$${\displaystyle {\begin{aligned}&=\lim _{x\to 1}{(x-1)(x+1) \over x-1}\\&=\lim _{x\to 1}{(x+1)}\\&=2\end{aligned}}}$$

and so the limit exists, and is equal to ${\displaystyle 2}$.

These and other similar facts show that the expression ${\displaystyle {\frac {0}{0}}}$ cannot be well-defined as a limit.

A formal calculation is one carried out using rules of arithmetic, without consideration of whether the result of the calculation is well-defined. Thus, it is sometimes useful to think of a/0, where a ≠ 0, as being ${\displaystyle \infty }$. This infinity can be either positive, negative, or unsigned, depending on context. For example, formally:

$${\displaystyle \lim _{x\to 0}{{\frac {1}{x}}={\frac {\lim \limits _{x\to 0}{1}}{\lim \limits _{x\to 0}{x}}}}=\infty .}$$

As with any formal calculation, invalid results may be obtained. A logically rigorous (as opposed to formal) computation would assert only that

$${\displaystyle \lim _{x\to 0^{+}}{\frac {1}{x}}=+\infty ~{\text{ and }}~\lim _{x\to 0^{-}}{\frac {1}{x}}=-\infty .}$$

Since the one-sided limits are different, the two-sided limit does not exist in the standard framework of the real numbers. Also, the fraction 1/0 is left undefined in the extended real line, therefore it and

$${\displaystyle {\frac {\lim \limits _{x\to 0}1}{\lim \limits _{x\to 0}x}}}$$

are meaningless expressions.

The set ${\displaystyle \mathbb {R} \cup \{\infty \}}$ is the projectively extended real line, which is a one-point compactification of the real line. Here ${\displaystyle \infty }$ means an unsigned infinity or point at infinity, an infinite quantity that is neither positive nor negative. This quantity satisfies ${\displaystyle -\infty =\infty }$, which is necessary in this context. In this structure, ${\displaystyle {\frac {a}{0}}=\infty }$ can be defined for nonzero a, and ${\displaystyle {\frac {a}{\infty }}=0}$ when a is not ${\displaystyle \infty }$. It is the natural way to view the range of the tangent function 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.

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, ${\displaystyle \infty +\infty }$ is undefined in this extension of the real line.

The set ${\displaystyle \mathbb {C} ^{*}=\mathbb {C} \cup \{{\tilde {\infty }}\}}$ is the Riemann sphere, which is of major importance in complex analysis. Here ${\displaystyle {\tilde {\infty }}}$ represents complex infinity, which is also a point at infinity. This set is analogous to the projectively extended real line, except that it is based on the field of complex numbers. In the Riemann sphere, ${\displaystyle {\frac {1}{0}}={\tilde {\infty }}}$ and ${\displaystyle {\frac {1}{\tilde {\infty }}}=0}$, but ${\displaystyle {\frac {0}{0}}}$, ${\displaystyle {\frac {\tilde {\infty }}{\tilde {\infty }}}}$, and ${\displaystyle 0\times {\tilde {\infty }}}$ are undefined.

In the hyperreal numbers and the surreal numbers, division by zero is still impossible, but division by non-zero infinitesimals is possible.

In distribution theory one can extend the function ${\textstyle {\frac {1}{x}}}$ 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.

In matrix algebra (or linear algebra in general), one can define a pseudo-division, by setting a/b = ab⁺, in which b⁺ represents the pseudoinverse of b. It can be proven that if b⁻¹ exists, then b⁺ = b⁻¹. If b equals 0, then b⁺ = 0.

In abstract algebra, 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, 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, where ${\displaystyle 0=1}$, 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 can be extended to a seldom used structure called a wheel 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 ${\displaystyle 1}$, and if the original system was an integral domain, 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). 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 ${\textstyle {\frac {2}{2}}}$ should be the solution x of the equation ${\displaystyle 2x=2}$. But in the ring Z/6Z, 2 is a zero divisor. This equation has two distinct solutions, x = 1 and x = 4, so the expression ${\textstyle {\frac {2}{2}}}$ is undefined.

In field theory, the expression ${\textstyle {\frac {a}{b}}}$ 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 0 ≠ 1 for fields (or its equivalent) so that the zero ring 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.