In mathematics, division by zero is division where the divisor is zero. Such a division
can be formally expressed as a/0 where a is the dividend. In ordinary arithmetic, the
expression has no meaning, as there is no number which, multiplied by 0, gives a, and
so division by zero is undefined. Since any number multiplied by zero is zero, the expression
0/0 has no defined value and is called an indeterminate form. Historically, one of the
earliest recorded references to the mathematical impossibility of assigning a value to a/0
is contained in George Berkeley's criticism of infinitesimal calculus in The Analyst.
There are mathematical structures in which a/0 is defined for some a; however, such structures
cannot satisfy every ordinary rule of arithmetic. In computing, a program error may result from
an attempt to divide by zero. Depending on the programming environment and the type of
number being divided by zero, it may generate positive or negative infinity by the IEEE
754 floating point standard, generate an exception, generate an error message, cause the program
to terminate, or result in a special not-a-number value.
In elementary arithmetic When division is explained at the elementary
arithmetic level, it is often considered as splitting a set of objects into equal parts.
As an example, consider having ten cookies, and these cookies are to be distributed equally
to five people at a table. Each person would receive = 2 cookies. Similarly, if there are
ten cookies, and only one person at the table, that person would receive = 10 cookies.
So, for dividing by zero, what is the number of cookies that each person receives when
10 cookies are evenly distributed amongst 0 people at a table? Certain words can be
pinpointed in the question to highlight the problem. The problem with this question is
the "when". There is no way to evenly distribute 10 cookies to nobody. In mathematical jargon,
a set of 10 items cannot be partitioned into 0 subsets. So , at least in elementary arithmetic,
is said to be either meaningless, or undefined. Similar problems occur if one has 0 cookies
and 0 people, but this time the problem is in the phrase "the number". A partition is
possible, but since the partition has 0 parts, vacuously every set in our partition has a
given number of elements, be it 0, 2, 5, or 1000.
If there are, say, 5 cookies and 2 people, the problem is in "evenly distribute". In
any integer partition of a 5-set into 2 parts, one of the parts of the partition will have
more elements than the other. But the problem with 5 cookies and 2 people can be solved
by cutting one cookie in half. The problem with 5 cookies and 0 people cannot be solved
in any way that preserves the meaning of "divides". Another way of looking at division by zero
is that division can always be checked using multiplication. Considering the 10/0 example
above, setting x = 10/0, if x equals ten divided by zero, then x times zero equals ten, but
there is no x that, multiplied by zero, gives ten. If instead of x=10/0 we have x=0/0, then
every x satisfies the question 'what number x, multiplied by zero, gives zero?'
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,
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."
In algebra 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 is undefined. Conversely,
in a field, the expression is always defined if b is not equal to zero.
Division as the inverse of multiplication The concept that explains division in algebra
is that it is the inverse of multiplication. For example,
since 2 is the value for which the unknown quantity in
is true. But the expression
requires a value to be found for the unknown quantity in
But any number multiplied by 0 is 0 and so there is no number that solves the equation.
The expression
requires a value to be found for the unknown quantity in
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. 0/0 is known as indeterminate.
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 1 = 2 such as the following:
With the following assumptions:
The following must be true:
Dividing by zero gives:
Simplified, yields:
The fallacy is the implicit assumption that dividing by 0 is a legitimate operation with
the same properties as dividing by any other number.
In calculus Extended real line
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
however, the limit from the left is
and so the is undefined. Furthermore, there is no obvious definition
of 0/0 that can be derived from considering the limit of a ratio. The limit
does not exist. Limits of the form
in which both ƒ(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 ƒ and g. These
and other similar facts show that the expression 0/0 cannot be well-defined as a limit.
Formal operations 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
. This infinity can be either positive, negative, or unsigned, depending on context. For example,
formally:
As with any formal calculation, invalid results may be obtained. A logically rigorous computation
would assert only that
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
are meaningless expressions. Real projective line
The set is the real projective line, which is a one-point compactification of the real
line. Here means an unsigned infinity, an infinite quantity that is neither positive
nor negative. This quantity satisfies , which is necessary in this context. In this structure,
can be defined for nonzero a, and . 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 or 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, is undefined in the projective line.
Riemann sphere The set is the Riemann sphere, which is of
major importance in complex analysis. Here too 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. In the Riemann
sphere, , but is undefined, as is . Extended non-negative real number line
The negative real numbers can be discarded, and infinity introduced, leading to the set
[0, ∞], where division by zero can be naturally defined as a/0 = ∞ for positive a. While
this makes division defined in more cases than usual, subtraction is instead left undefined
in many cases, because there are no negative numbers.
In higher mathematics Although division by zero cannot be sensibly
defined with real numbers and integers, it is possible to consistently define it, or
similar operations, in other mathematical structures.
Non-standard analysis In the hyperreal numbers and the surreal numbers,
division by zero is still impossible, but division by non-zero infinitesimals is possible.
Distribution theory In distribution theory one can extend the
function to a distribution on the whole space of real numbers. 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. Linear algebra
In matrix algebra, 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; see Generalized inverse. Abstract algebra
Any number system that forms a commutative ring—for instance, the integers, the real
numbers, and the complex numbers—can be extended to a wheel in which division by zero
is always possible; however, in such a case, "division" has a slightly different meaning.
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. 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 should be the solution x of the equation . 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
is undefined. In field theory, the expression 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 include the axiom 0 ≠ 1 for
fields so that the zero ring is excluded from being a field.
In computer arithmetic
The IEEE floating-point standard, supported by almost all modern floating-point units,
specifies that every floating point arithmetic operation, including division by zero, has
a well-defined result. The standard supports signed zero, as well as infinity and NaN.
There are two zeroes, +0 and −0 and this removes any ambiguity when dividing. In IEEE
754 arithmetic, a ÷ +0 is positive infinity when a is positive, negative infinity when
a is negative, and NaN when a = ±0. The infinity signs change when dividing by −0
instead. The justification for this definition is to
preserve the sign of the result in case of arithmetic underflow. For example, in the
double-precision computation 12), where x = ±2−149, the computation x/2 underflows
and produces ±0 with sign matching x, and the result will be ±∞ with sign matching
x. The sign will match that of the exact result ±2150, but the magnitude of the exact result
is too large to represent, so infinity is used to indicate overflow.
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. The result depends on how division is implemented,
and can either be zero, or sometimes the largest possible integer.
Because of the improper algebraic results of assigning any value to division by zero,
many computer programming languages explicitly forbid the execution of the operation and
may prematurely halt a program that attempts it, sometimes reporting a "Divide by zero"
error. In these cases, if some special behavior is desired for division by zero, the condition
must be explicitly tested. Some programs 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. The graphical programming
language Scratch 2 used in many schools returns Infinity or -Infinity depending on the sign
of the dividend. In two's complement arithmetic, attempts to
divide the smallest signed integer by are attended by similar problems, and are handled
with the same range of solutions, from explicit error conditions to undefined behavior.
Most calculators will either return an error or state that 1/0 is undefined, however some
TI and HP graphing calculators will evaluate2 to ∞.
Microsoft Math and Mathematica return ComplexInfinity for 1/0. Maple and Sage return an error message
for 1/0, and infinity for 1/0.0. Historical accidents
On September 21, 1997, a division by zero error on board the USS Yorktown Remote Data
Base Manager brought down all the machines on the network, causing the ship's propulsion
system to fail. See also
Asymptote Defined and undefined
Indeterminate form Zero divisor
References Notes
Sources
Further reading