Multiplication by Infinity: Division By Zero

In mathematics, division by zero is a term used if the divisor (denominator) is zero. Such a division can be formally expressed as a / 0 where a is the dividend (numerator). Whether this expression can be assigned a well-defined value depends upon the mathematical setting. In ordinary (real number) arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a (a≠0).
In computer programming, an attempt to divide by zero may, depending on the programming language and the type of number being divided by zero, generate an exception, generate an error message, crash the program being executed, generate either positive or negative infinity, or could result in a special not-a-number value (see below).
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; see Ghosts of departed quantities.

In elementary arithmetic

When division is explained at the elementary arithmetic level, it is often considered as a description of dividing a set of objects into equal parts. As an example, consider having ten apples, and these apples are to be distributed equally to five people at a table. Each person would receive $\textstyle\frac{10}{5}$ = 2 apples. Similarly, if there are 10 apples, and only one person at the table, that person would receive $\textstyle\frac{10}{1}$ = 10 apples.
So for dividing by zero – what is the number of apples that each person receives when 10 apples are evenly distributed amongst 0 people? 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 distribute 10 apples amongst 0 people. In mathematical jargon, a set of 10 items cannot be partitioned into 0 subsets. So $\textstyle\frac{10}{0}$ , at least in elementary arithmetic, is said to be meaningless, or undefined.
Similar problems occur if one has 0 apples and 0 people, but this time the problem is in the phrase "the number". A partition is possible (of a set with 0 elements into 0 parts), 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 apples 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.
In all of the above three cases, $\textstyle\frac{10}{0}$ , $\textstyle\frac{0}{0}$ and $\textstyle\frac{5}{2}$ , one is asked to consider an impossible situation before deciding what the answer will be, and that is why the operations are undefined in these cases.
To understand division by zero, one must check it with multiplication: multiply the quotient by the divisor to get the original number. However, no number multiplied by zero will produce a product other than zero. To satisfy division by zero, the quotient must be bigger than all other numbers, i.e., infinity. This connection of division by zero to infinity takes us beyond elementary arithmetic (see below).
A recurring theme even at this elementary stage is that for every undefined arithmetic operation, there is a corresponding question that is not well-defined. "How many apples will each person receive under a fair distribution of ten apples amongst three people?" is a question that is not well-defined because there can be no fair distribution of ten apples amongst three people.
There is another way, however, to explain the division: if one wants to find out how many people, who are satisfied with half an apple, can one satisfy by dividing up one apple, one divides 1 by 0.5. The answer is 2. Similarly, if one wants to know how many people, who are satisfied with nothing, can one satisfy with 1 apple, one divides 1 by 0. The answer is infinite; one can satisfy infinite people, that are satisfied with nothing, with 1 apple.
Clearly, one cannot extend the operation of division based on the elementary combinatorial considerations that first define division, but must construct new number systems.

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.^[1] 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."^[1]
Bhaskara II tried to solve the problem by defining (in modern notation) $\textstyle\frac{n}{0}=\infty$ .^[1] 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.

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

Division as the inverse of multiplication

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

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

since 2 is the value for which the unknown quantity in

$?\times 3=6$

is true. But the expression

$\frac{6}{0}=\,?$

requires a value to be found for the unknown quantity in

$?\times 0=6.$

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

$\frac{0}{0}=\,?$

requires a value to be found for the unknown quantity in

$?\times 0=0.$

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 (see below for other applications).

Fallacies based on division by zero

It is possible to disguise a special case of division by zero in an algebraic argument,^[1] leading to spurious proofs that 1 = 2 such as the following:
With the following assumptions:

$\begin{align} 0\times 1 &= 0 \\ 0\times 2 &= 0. \end{align}$

The following must be true:

$0\times 1 = 0\times 2.\,$

Dividing by zero gives:

$\textstyle \frac{0}{0}\times 1 = \frac{0}{0}\times 2.$

Simplified, yields:

$1 = 2.\,$

The fallacy is the implicit assumption that dividing by 0 is a legitimate operation.

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

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

however, the limit from the left is

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

and so the $\lim_{b \to 0} {a \over b}$ 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

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

does not exist. Limits of the form

$\lim_{x \to 0} {f(x) \over g(x)}$

in which both 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 (see l'Hôpital's rule for discussion and examples of limits of ratios). 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 $\infty$ . This infinity can be either positive, negative, or unsigned, depending on context. For example, formally:

$\lim\limits_{x \to 0} {\frac{1}{x} =\frac{\lim\limits_{x \to 0} {1}}{\lim\limits_{x \to 0} {x}}} = \frac{1}{0} = \infty.$

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

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

(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

$\frac{\lim\limits_{x \to 0} 1 }{\lim\limits_{x \to 0} x}$

are meaningless expressions.)

Real projective line

The set $\mathbb{R}\cup\{\infty\}$ is the real projective line, which is a one-point compactification of the real line. Here $\infty$ means an unsigned infinity, an infinite quantity that is neither positive nor negative. This quantity satisfies $-\infty = \infty$ , which is necessary in this context. In this structure, $\scriptstyle a/0 = \infty$ can be defined for nonzero a, and $\scriptstyle a/\infty = 0$ . 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 $\scriptstyle+\pi/2$ or $\scriptstyle-\pi/2$ 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, $\infty + \infty$ is undefined in the projective line.

Riemann sphere

The set $\mathbb{C}\cup\{\infty\}$ is the Riemann sphere, which is 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. In the Riemann sphere, $1/0=\infty$ , but 0/0 is undefined, as is $0\times\infty$ .

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 $\textstyle\frac{1}{x}$ 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.

Linear algebra

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 0⁺ = 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 (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

2 x = 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.
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 include the axiom 0 ≠ 1 to avoid having to consider the trivial ring or a "field with one element", where the multiplicative identity coincides with the additive identity.

In computer arithmetic

In the SpeedCrunch calculator application, when a number is divided by zero the answer box displays “Error: Divide by zero”.

Most calculators, such as this Texas Instruments TI-86, will halt execution and display an error message when the user or a running program attempts to divide by zero.

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 (not a number). There are two zeroes, +0 (positive zero) and −0 (negative zero) 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.
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 (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. In these cases, if some special behavior is desired for division by zero, the condition must be explicitly tested (for example, using an if statement). 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.
Most calculators will either return an error or state that 1/0 is undefined, however some TI and HP graphing calculators will evaluate (1/0)² to ∞.
More advanced computer algebra systems will return an infinity as a result for division by zero; for instance, Microsoft Math and Mathematica will show an ComplexInfinity result.

Historical accidents

On September 21, 1997, a divide by zero error on board 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]

Footnotes

^ ^a ^b ^c ^d Kaplan, Robert (1999). The nothing that is: A natural history of zero. New York: Oxford University Press. pp. 68–75. ISBN 0195142373.
^ "Sunk by Windows NT". Wired News. 1998-07-24. http://www.wired.com/news/technology/0,1282,13987,00.html.

References

Patrick Suppes 1957 (1999 Dover edition), Introduction to Logic, Dover Publications, Inc., Mineola, New York. ISBN 0-486-40687-3 (pbk.). This book is in print and readily available. Suppes's §8.5 The Problem of Division by Zero begins this way: "That everything is not for the best in this best of all possible worlds, even in mathematics, is well illustrated by the vexing problem of defining the operation of division in the elementary theory of arithmetic" (p. 163). In his §8.7 Five Approaches to Division by Zero he remarks that "...there is no uniformly satisfactory solution" (p. 166)
Charles Seife 2000, Zero: The Biography of a Dangerous Idea, Penguin Books, NY, ISBN 0 14 02.9647 6 (pbk.). This award-winning book is very accessible. Along with the fascinating history of (for some) an abhorent notion and others a cultural asset, describes how zero is misapplied with respect to multiplication and division.
Alfred Tarski 1941 (1995 Dover edition), Introduction to Logic and to the Methodology of Deductive Sciences, Dover Publications, Inc., Mineola, New York. ISBN 0-486-28462-X (pbk.). Tarski's §53 Definitions whose definiendum contains the identity sign discusses how mistakes are made (at least with respect to zero). He ends his chapter "(A discussion of this rather difficult problem [exactly one number satisfying a definiens] will be omitted here.*)" (p. 183). The * points to Exercise #24 (p. 189) wherein he asks for a proof of the following: "In section 53, the definition of the number '0' was stated by way of an example. To be certain this definition does not lead to a contradiction, it should be preceded by the following theorem: There exists exactly one number x such that, for any number y, one has: y + x = y"

Multiplication by Infinity

Thursday, December 16, 2010

Division By Zero

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