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For the album Division by Zero by Hux Flux, please see Division By Zero (album). In mathematics, a division is called a division by zero if the divisor is zero. Such a division can be formally expressed as 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 often causes a program to terminate (floating point generally does not, see below). 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 of 5 blocks, then you have created 2 equal sets. This would be a demonstration that 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, 13/5 = 2 remainder 3. But in the case of zero, repeated subtraction of zero will never yield a remainder less than zero, so dividing by zero is not defined. 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." Bhaskara II tried to solve the problem by defining . This definition makes a certain degree of sense, as discussed below, but can lead to paradoxes if not treated carefully. It is unlikely that he understood all the intricacies involved, so his solution cannot be considered successful. * 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 is the solution x of the equation 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. 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: This argument is sometimes presented as a riddle; in such cases the 3rd step is usually omitted in an attempt to trick the listener. The fallacy is the assumption in step 4 that -- which is -- simplifies to . This proof is for the special case of dividing by zero when the numerator is zero. The fallacy results from the assumption that -- an assumption that generates the absurdity that . Any other non-zero value assigned to 0/0 leads to similar contradictions. In practice, division by a term in any algebraic argument requires an explicit assumption that the term is not zero or a justification 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 poses problems only when attempting to divide by zero. 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 ? This should be the solution x of the equation . 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 is undefined. Limits and division by zero At first glance it seems possible to define by considering the limit of as b approaches 0. For any positive a, it is known that and for any negative a, Therefore, we might consider defining 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 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) 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 that can be derived from considering the limit of a ratio. The limit does not exist. Limits of the form 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 . 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 as being , provided a is not zero. 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. 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: Real projective line The set is the real projective line. Here means an unsigned infinity, an infinite quantity which is neither positive nor negative. This quantity satisfies which, as we have seen, is necessary in this context. In this structure, we can define for nonzero a, and . 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, 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 π/2 or −π/2 from either direction. Riemann sphere The set is the Riemann sphere, 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; and this set is also not a field. Extended non-negative real number line The negative real numbers can be discarded, and infinity introduced, leading to the set - Where division by zero can be naturally defined as for positive a. Non-standard analysis In hyperreal numbers and surreal numbers, division by zero is still impossible, but division by non-zero infinitesimals is possible. 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. In mathematical analysis 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 ; a sophisticated answer refers to the singular support of the distribution. 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, 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 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 are attended by similar problems, and are handled with the same range of solutions, from explicit error conditions to undefined behavior. 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. * In popular culture See also | |||||||
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