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In
mathematics, a
division is called a '''division by zero''' if the divisor is zero. Such a division can be formally expressed as <math>\frac{a}{0}</math>, where ''a'' is the dividend. Whether or not this expression can be assigned a meaningful (
well-defined) value depends upon how the expression is interpreted.
==Algebraic interpretation==
One of the most common and fruitful ways to approach the problem of division by zero is to formulate in terms of arithmetic, i.e. in terms of algebraic operations. Under the standard rules for arithmetic on
integers,
rational numbers,
real numbers and
complex numbers, the value of a division by zero is undefined. The reason is that
division is defined to be the inverse operation of
multiplication. This means that the value of
: <math>{a \over b}</math>
is the solution ''x'' of the equation
: <math>b x = a \quad </math>
whenever such a value exists and is unique. Otherwise the expression <math>{a \over b}</math> is undefined.
For ''b'' = 0, the equation ''bx'' = ''a'' can be rewritten as 0''x'' = ''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, <math>{a \over b}</math> is undefined. Conversely, for the number systems mentioned above, the expression <math>{a \over b}</math> is ''always'' defined if ''b'' is not equal to zero.
===Fallacies based on division by zero===
It is possible to disguise a division by zero in an
algebraic argument, leading to
spurious proofs that 2 = 1 such as the following:
* For any real number ''x'':
::<math> x^2 - x^2 = x^2 - x^2 \quad </math>
* Factoring both sides in two different ways:
::<math> (x - x) \times (x + x) = x \times (x - x)\quad </math>
* Dividing both sides by ''x'' − ''x'':
::<math> x + x = x \quad </math>
* Since this is valid for any value of ''x'', we can plug in ''x'' = 1.
::<math> 2 = 1 \quad </math>
The fallacy is in the assumption that division by ''x'' − ''x'' = 0 is defined.
In practice, division by a term in any algebraic argument will require either an explicit assumption that the term is not zero, or a separate justification showing 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 only poses problems when attempting to divide by zero. However, in other rings, division by nonzero elements may also pose problems. Consider, for example, the ring '''Z'''/6'''Z''' of integers mod 6. What meaning should we give to the expression
: <math>{2 \over 2}</math>
This should be the solution ''x'' of the equation
: <math>2x = 2 \quad </math>
But this equation has two distinct solutions, ''x'' = 1 and ''x'' = 4, so the expression is undefined. The problems occurs because 2 is not invertible under multiplication.
== Limits and division by zero ==
At first glance it seems possible to define <math>{a \over 0}</math> by considering the
limit of <math>{a \over b}</math> as ''b'' approaches 0. For any nonzero ''a'', it is known that
:<math>\lim_{b \to 0{+}} {a \over b} = \infty</math>
From this we may consider defining <math>{a \over 0}</math> as infinity for nonzero ''a''. However, with this definition, division is no longer the inverse of multiplication: setting ''x'' equal to infinity does not solve
:<math>0 \, x = a</math>
Without the inverse relation between division and multiplication, the definition <math>{a \over 0} = \infty</math> isn't generally useful.
As for the case ''a'' = 0, there is not any obvious definition of <math>{0 \over 0}</math> that can be derived from considering the limit of a ratio. The limit
:<math> \lim_{(a,b) \to (0,0)} {a \over b} </math>
does not exist. In addition, limits of the form
:<math> \lim_{x \to 0} {f(x) \over g(x)} </math>
in which both ''f(x)'' and ''g(x)'' approach 0 as ''x'' approaches zero, may converge to any value or may not converge at all. See
l'Hopital's rule for discussion and examples of limits of ratios.
==In
mathematical analysis==
In
distribution theory one can extend the function
:<math>{1 \over x}</math>
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 <math>x = 0</math>; a sophisticated answer refers to the
singular support of the distribution.
==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 mathematical structures, for instance on the
Riemann sphere (see also
poles in complex analysis). In
hyperreal numbers and
surreal numbers, division by non-zero
infinitesimals is possible. If a number system forms a
commutative ring, as do 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.
== Division by zero in computer arithmetic ==
IEEE 754 specifies that every
floating point arithmetic operation, including division by zero, has a well-defined result. In IEEE 754 arithmetic, ''a''/0 is positive infinity when ''a'' is positive, negative infinity when ''a'' is negative, and
NaN ("not a number") when ''a'' = 0. These definitions are derived from the properties of limits of ratios, as discussed above. At present, IEEE 754 is the most common floating point specification, as it is implemented by
Intel processors and others.
Integer division by zero may be handled differently than floating point. Intel processors generate an
interrupt when an attempt is made to divide an integer by zero.
To ensure that every operation yields a finite, numerical result (floating point), and avoids an interrupt (integer), a computer program may refuse to execute a division if the divisor is zero.
