Division by zero

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In mathematics, the result of division by zero, for some , is undefined and not allowed in real numbers and integers. The reason is that division ought to be the inverse operation of multiplication, which means that should be the solution of , but for this has no solution if , and any as solution if also . In both cases, cannot be defined meaningfully.

It is sometimes incorrectly stated that, within the real number system, is equal to infinity (). The origin of the statement is due to the property that grows larger as approaches zero, for positve. More precisely, as a correct statement in the language of mathematical limits, we have

However, it is not true that equals infinity, because infinity is not a real number and so cannot be the result of a mathematical operation within the real number system. Furthermore, the limit of as approaches zero from the left (i.e. when is negative) is negative infinity, not positive infinity. Here the point is that ∞ is a symbol only defined within the notation for a limit; it doesn't have a free-standing sense.

Note that dividing 0 by 0 produces even greater problems. Consider the limits of

as , which show that any value for 0/0 is possible.

Table of contents

1 Examples of fallacies
2 In mathematical analysis
3 Other number systems
4 Extension to complex numbers
5 Computers

Examples of fallacies

Another way to see why division by zero does not work is to work backwards from multiplication, remembering that anything multiplied by zero is zero. So

which, if we are allowed to divide by zero, means that

But

suggesting that , which is nonsense.

It is possible to disguise a division by zero in a long algebraic argument, leading to such things as a spurious proof that 2 equals 1. More practically, division by a variable in any algebraic argument will typically require an assumption that the variable cannot be equal to zero.

In mathematical analysis

It is both possible and meaningful to find the limit as x approaches 0 of some divisions by x; see l'Hopital's rule for some examples; see also indeterminate form. 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 .

Other number systems

Although division by zero is undefined with real numbers and integers, it is possible to consistently define division by zero in other number systems, for instance in the Riemann sphere. See also hyperreal numbers and surreal numbers. If a number system forms a commutative ring, as does the integers, the real numbers, and the complex numbers, for instance, it can be extended to a wheel in which division by zero is always possible, but division has then a slightly different meaning.

Extension to complex numbers

For the complex plane, see also pole (complex analysis).

Computers

Many computer architectures produce a runtime exception when an attempt is made to divide an integer by zero. However, most programss that use user input for calculations perform checks to make sure a divide by zero operation is not attempted. The IEEE standard for computer floating-point numbers states that is Infinity when is positive, negative Infinity when is negative, and NaN (Not a Number) when is zero.