Division by zero

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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 \textstyle\frac{a}{0} 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, integer division by zero may cause a program to terminate or, as in the case of floating point numbers, may result in a special not-a-number value (see below).

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[edit] Interpretation in elementary arithmetic

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 consisting of 5 blocks each, then you have divided the original set of 10 blocks into two parts. This would be a demonstration that \textstyle\frac{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, \textstyle\frac{13}{5} = 2 remainder 3. But in the case of zero, repeated subtraction of zero will never yield a remainder less than or equal to zero, so dividing by zero is not defined. Dividing by zero by repeated subtraction results in a series of subtractions that never ends.

[edit] Early attempts

The Brahmasphutasiddhanta of Brahmagupta (598–668) 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 \textstyle\frac{n}{0}=\infty. This definition makes some sense, as discussed below, but can lead to paradoxes if not treated carefully. These paradoxes were not treated until modern times.[1]

[edit] Algebraic interpretation

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 \textstyle\frac{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 \textstyle\frac{a}{b} is undefined. Conversely, in a field, the expression \textstyle\frac{a}{b} is always defined if b is not equal to zero.

[edit] 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:

With the following assumptions:

0\cdot 1 = 0
0\cdot 2 = 0

The following must be true:

0\cdot 1 = 0\cdot 2

Dividing by zero gives:

\textstyle \frac{0}{0}\cdot 1 = \frac{0}{0}\cdot 2

Simplified, yields:

1 = 2

The fallacy is the implicit assumption that dividing by 0 is a legitimate operation with 0 / 0 = 1.

Although most people would probably recognize the above "proof" as fallacious, the same argument can be presented in a way that makes it harder to spot the error. For example, if 1 is denoted by x, 0 can be hidden behind x − x and 2 behind x + x. The abovementioned proof can then be displayed as follows:

(x − x)x = x2 − x2
(x − x)(x + x) = x2 − x2

hence:

(x − x)x = (x − x)(x + x)

Dividing by x − x gives:

x = x + x

and dividing by x gives:

1 = 2

[edit] 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. 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. Consider, for example, the ring Z/6Z of integers mod 6. What meaning should we give to the expression \textstyle\frac{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 \textstyle\frac{2}{2} is undefined.

[edit] Limits and division by zero

The function y = . As x approaches 0, y approaches infinity (and vice versa).
The function y = \textstyle\frac{1}{x}. As x approaches 0, y approaches infinity (and vice versa).

At first glance it seems possible to define \textstyle\frac{a}{0} by considering the limit of \textstyle\frac{a}{b} as b approaches 0.

For any positive a, it is known that

\lim_{b \to 0^{+}} {a \over b} = {+}\infty

and for any negative a,

\lim_{b \to 0^{+}} {a \over b} = {-}\infty

Therefore, we might consider defining \textstyle\frac{a}{0} 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 \textstyle\frac{a}{0} 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)

+\infty = \frac{1}{0} = \frac{1}{-0} = -\frac{1}{0} = -\infty

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

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

\lim_{(a,b) \to (0,0)} {a \over b}

does not exist. Limits of the form

\lim_{x \to 0} {f(x) \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 \textstyle\frac{0}{0}.

[edit] Formal interpretation

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

\lim\limits_{x \to 0} {\frac{1}{x^2} =\frac{\lim\limits_{x \to 0} {1}}{\lim\limits_{x \to 0} {x^2}}} = \frac{1}{+0} = +\infty.

As with any formal calculation, invalid results may be obtained. A logically rigorous as opposed to formal computation might say only

\lim_{x \to 0} \frac{1}{x^2} = +\infty

(+∞ is not a number but an object that may be approached from within the real line; those familiar with point-set topology may call it a member of a two-point compactification of the line).

[edit] Pseudo-division by zero

One can define a pseudo-division, by setting \textstyle\frac{a}{b}=a b^+, in which b+ represents b's pseudoinverse. It can be proven that if b−1 exists, then b+ = b−1. If b equals 0, then 0+ = 0, see pseudoinverse.

[edit] Other number systems

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.

[edit] Real projective line

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

This is a one-point compactification of the real line.

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 \textstyle+\frac{\pi}{2} or \textstyle-\frac{\pi}{2} from either direction.

[edit] Riemann sphere

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

Here, too, \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.

[edit] Extended non-negative real number line

The negative real numbers can be discarded, and infinity introduced, leading to the set [0, \infty], where division by zero can be naturally defined as \textstyle\frac{a}{0} = \infty for positive a.

[edit] Non-standard analysis

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

[edit] 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.

[edit] In mathematical analysis

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.

[edit] Division by zero in computer arithmetic

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, a ÷ 0 is positive infinity when a is positive, negative infinity when a is negative, and NaN (not a number) when a = 0. The infinity signs change when dividing by −0 instead. This is possible because in IEEE 754 there are two zero values, plus zero and minus zero, and thus no ambiguity.

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. (That result is often zero.)

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 some programming languages, an attempt to divide by zero results in undefined behavior.

In two's complement arithmetic, attempts to divide the smallest signed integer by − 1 are attended by similar problems, and are handled with the same range of solutions, from explicit error conditions to undefined behavior.

[edit] Historical accidents

On September 21, 1997, a divide by zero error in the USS Yorktown (CG-48) Remote Data Base Manager brought down all the machines on the network, causing the ship's propulsion system to fail.[2]

[edit] In popular culture

  • E_DIV is an error code generated by some programming languages as a result of division by zero, and can be used in internet slang as an indication of confusion or impossibility.
  • As a result of the errors often seen in computers and calculators when an operator attempts to divide by zero, an Internet meme has developed where dividing by zero is seen as synonymous with the "end of the world", universe, forum, etc, usually preceded by a declaration of "OH SHI-". The meme inspired the short film The Last Denominator, where a division by zero is followed by the sudden realization of what this means as the planet explodes.
  • The webcomic 1/0 takes its title from equating division by zero (conceptually) with the metafictional idea of breaking the fourth wall.
  • A short story by Ted Chiang is titled Division by Zero.
  • One of the satirical Chuck Norris Facts states that "Chuck Norris can divide by zero".

[edit] References

  1. ^ Zero
  2. ^ "Sunk by Windows NT", Wired News, 1998-07-24.

[edit] Further reading

[edit] See also

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