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Encyclopedia > Division by zero

In mathematics, a division is called a division by zero if the divisor is zero. Such a division can be formally expressed as textstylefrac{a}{0} where a is the dividend. 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. Hux Flux is a musical group starring the Swede Denis Tapper. ... Division by Zero is the second full-length album from psychedelic trance producers Hux Flux. ... In mathematics, a division is called a division by zero if the divisor is zero. ... 1/0 (also One over Zero) is a webcomic created by Mason Williams (pseudonymously known as Tailsteak). Its name is based on the mathematical concept of division by zero. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... For other senses of this word, see zero or 0. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. ... In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...


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). Programming redirects here. ... The integers are commonly denoted by the above symbol. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... In computing, NaN (Not a Number) is a value or symbol that is usually produced as the result of an operation on invalid input operands, especially in floating-point calculations. ... For the album by Hux Flux, see Division by Zero (album). ...

Contents

Interpretation 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, if you have 10 apples, and you want to distribute them evenly to five people, each person would receive textstylefrac{10}{5} = 2 apples. Similarly, if you have 10 apples to distribute to one person, each person would receive textstylefrac{10}{1} = 10 apples. Elementary arithmetic is the most basic kind of mathematics: it concerns the operations of addition, subtraction, multiplication, and division. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...


We can use this to illustrate the problem of dividing by zero. Say you have 10 apples to distribute to zero people. How many apples does each "person" receive? An attempt to calculate textstylefrac{10}{0} becomes meaningless because the question itself is meaningless -- each "person" doesn't receive zero, or 10, or an infinite number of apples for that matter, because there are simply no people to receive anything in the first place. This is why as far as elementary arithmetic is concerned, division by zero is said to be meaningless, or undefined.


Another way to understand the undefined nature of division by zero is by looking at division as 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 textstylefrac{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. 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ... In mathematics, the result of the division of two integers usually cannot be expressed with an integer quotient, unless a remainder —an amount left over— is also acknowledged. ...


Early attempts

The Brahmasphutasiddhanta of Brahmagupta (598668) 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, The main work of Brahmagupta, Brahmasphutasiddhanta (The Opening of the Universe), written in 628, contains some remarkably advanced ideas, including a good understanding of the mathematical role of zero, rules for manipulating both positive and negative numbers, a method for computing square roots, methods of solving linear and some quadratic... Brahmagupta (ब्रह्मगुप्त) ( ) (589–668) was an Indian mathematician and astronomer. ... Events Aethelfrith of Northumbria possibly defeats the northern British in a major battle at Catraeth. ... Events Childeric II succeeds Clotaire III as Frankish king Constantine IV becomes Byzantine Emperor, succeeding Constans II Theodore of Tarsus made archbishop of Canterbury. ... For other senses of this word, see zero or 0. ...

"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: Events Christian missionary Ansgar visits Birka, trade city of the Swedes. ... Mahavira was a 10th century Indian mathematician from Gulbarga who asserted that the square root of a negative number did not exist. ...

"A number remains unchanged when divided by zero."

Bhaskara II tried to solve the problem by defining textstylefrac{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] Bhāskara (1114-1185), also called Bhāskara II and Bhāskarācārya (Bhaskara the teacher) was an Indian mathematician. ...


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 textstylefrac{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. The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. ... In mathematics, a rational number (or informally fraction) is a ratio of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... Please refer to Real vs. ... The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... In mathematics, multiplication is an elementary arithmetic operation. ...


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 textstylefrac{a}{b} is undefined. Conversely, in a field, the expression textstylefrac{a}{b} is always defined if b is not equal to zero. In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...


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 article is about the branch of mathematics. ... In mathematics, there are a variety of spurious proofs of obvious contradictions. ...


With the following assumptions:

0times 1 = 0
0times 2 = 0

The following must be true:

0times 1 = 0times 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 with 0 / 0 = 1. Look up fallacy in Wiktionary, the free dictionary. ...


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 xx and 2 behind x + x. The above mentioned proof can then be displayed as follows:

(x-x)x = x^2-x^2 = 0,
(x-x)(x+x) = x^2-x^2 = 0,

hence:

(x-x)x = (x-x)(x+x),

Dividing by x-x, gives:

x = x+x,

and dividing by x, gives:

1 = 2,

The "proof" above requires the use of the distributive law. However, this requirement introduces an asymmetry between the two operations in that multiplication distributes over addition, but not the other way around. Thus, the multiplicative identity element, 1, has an additive inverse, -1, but the additive identity element, 0, does not have a multiplicative inverse.


Abstract algebra

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, 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 textstylefrac{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 textstylefrac{2}{2} is undefined. In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ...


Limits and division by zero

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

At first glance it seems possible to define textstylefrac{a}{0} by considering the limit of textstylefrac{a}{b} as b approaches 0. Image File history File links Hyperbola_one_over_x. ... Image File history File links Hyperbola_one_over_x. ... Wikibooks Calculus has a page on the topic of Limits In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements as...


For any positive a, it is known that

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

and for any negative a, A negative number is a number that is less than zero, such as −3. ...

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

Therefore, we might consider defining textstylefrac{a}{0} as +∞ for positive a, and −∞ for negative a. However, this definition can be inconvenient for two reasons.

  1. 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.
  2. Taking the limit from the right is arbitrary. We could just as well have taken limits from the left and defined textstylefrac{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 textstylefrac{0}{0} that can be derived from considering the limit of a ratio. The limit For other uses, see Infinity (disambiguation). ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

 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 textstylefrac{0}{0}. In calculus, lHôpitals rule uses derivatives to help compute limits with indeterminate forms. ...


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 textstylefrac{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: In mathematical logic, a formal calculation is sometimes defined as a calculation which is systematic, but without a rigorous justification. ...

limlimits_{x to 0} {frac{1}{x^2} =frac{limlimits_{x to 0} {1}}{limlimits_{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). A Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ...


Pseudo-division by zero

In algebra for matrices (or linear algebra in general), one can define a pseudo-division, by setting textstylefrac{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. In mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries), which may be numbers or, more generally, any abstract quantities that can be added and multiplied. ... In mathematics, and in particular linear algebra, the pseudoinverse of an matrix is a generalization of the inverse matrix. ... In mathematics, and in particular linear algebra, the pseudoinverse of an matrix is a generalization of the inverse matrix. ...


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 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 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 textstylefrac{a}{0} = infty for nonzero a, and textstylefrac{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. In mathematics, the projective line is a fundamental example of an algebraic curve. ... In mathematics, compactification is applied to topological spaces to make them compact spaces. ...


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. Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with...


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. This set is not a field. The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection — details are given below). ... Plot of the function f(x)=(x2-1)(x-2-i)2/(x2+2+2i). ... The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, . Nota Bene: The real projective line is not equivalent to the extended real number line. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...


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 textstylefrac{a}{0} = infty for positive a.


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. The system of hyperreal numbers represents a rigorous method of treating the ideas about infinite and infinitesimal numbers that had been used casually by mathematicians, scientists, and engineers ever since the invention of calculus by Newton and Leibniz. ... In mathematics, the surreal numbers are a field containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ...


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 ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... Wheels are a kind of algebras where division is always defined. ...


In mathematical analysis

In distribution theory one can extend the function textstylefrac{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. This page deals with mathematical distributions. ... In mathematics, the Cauchy principal value of certain improper integrals is defined as either the finite number where b is a point at which the behavior of the function f is such that for any a < b and for any c > b (one sign is + and the other is −). or... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ...


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. The IEEE Standard for Binary Floating-Point Arithmetic (IEEE 754) is the most widely-used standard for floating-point computation, and is followed by many CPU and FPU implementations. ... A floating point unit (FPU) is a part of a CPU specially designed to carry out operations on floating point numbers. ... A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ... In computing, NaN (Not a Number) is a value or symbol that is usually produced as the result of an operation on invalid input operands, especially in floating-point calculations. ... −0 is the representation of negative zero or minus zero, a number that exists in computing, in some signed number representations for integers, and in most floating point number representations. ... −0 is the representation of negative zero or minus zero, a number that exists in computing, in some signed number representations for integers, and in most floating point number representations. ...


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.) Exception handling is a programming language construct or computer hardware mechanism designed to handle the occurrence of some condition that changes the normal flow of execution. ...


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. A programming language is an artificial language that can be used to control the behavior of a machine, particularly a computer. ... For other uses, see Calculator (disambiguation). ... This article is about a form of limited-precision arithmetic in computing. ... In computer science, undefined behavior is a feature of some programming languages — most famously C. In these languages, to simplify the specification and allow some flexibility in implementation, the specification leaves the results of certain operations specifically undefined. ...


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. The twos complement of a binary number is the value obtained by subtracting the number from a large power of two (specifically, from 2N for an N-bit twos complement). ... In computer science, undefined behavior is a feature of some programming languages — most famously C. In these languages, to simplify the specification and allow some flexibility in implementation, the specification leaves the results of certain operations specifically undefined. ...


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]

is the 264th day of the year (265th in leap years) in the Gregorian calendar. ... For the band, see 1997 (band). ... USS Yorktown (DDG-48/CG-48) was a Ticonderoga-class cruiser in the United States Navy from 1984 to 2004, named for the American Revolutionary War Battle of Yorktown. ...

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, sometimes 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".
  • The Pet Shop Boys on their first album Please, the first song on the recording is "Two divided by zero"

Ttyl redirects here. ... An Internet phenomenon occurs when something becomes extremely popular, often quite suddenly, through the word-of-mouth and self-publishing made feasible by the Internet. ... The end of planet Earth refers to hypotheses of when the Earth either completely ceases to exist as a planet or becomes uninhabitable for life. ... This box:      The ultimate fate of the universe is a topic in physical cosmology. ... Webcomics, also known as online comics and internet comics, are comics that are available to read on the Internet. ... 1/0 (also One over Zero) is a webcomic created by Mason Williams (pseudonymously known as Tailsteak). Its name is based on the mathematical concept of division by zero. ... Metafiction is a kind of fiction which self-consciously addresses the devices of fiction. ... The fourth wall is the imaginary invisible wall at the front of the stage in a proscenium theater, through which the audience sees the action in the world of the play. ... Ted Chiang Ted Chiang (born 1967) is an American science fiction writer. ... Chuck Norris Facts in Rolling Stone. ... Carlos Ray Chuck Norris (born on 10 March 1940) is an American martial artist, action star, Hollywood actor, and recently, an internet phenomenon, who is best known for playing Cordell Walker on Walker, Texas Ranger. ... Pet Shop Boys are a Grammy Award nominated British synthpop/pop music/electronic music duo, consisting of Neil Tennant who provides main vocals, keyboards and very occasionally guitar, and Chris Lowe on keyboards and occasionally on vocals. ... Please is the first album by the UK electronic music group Pet Shop Boys. ...

Footnotes

Wired News, online at Wired. ... Year 1998 (MCMXCVIII) was a common year starting on Thursday (link will display full 1998 Gregorian calendar). ... is the 205th day of the year (206th in leap years) in the Gregorian calendar. ...

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 artihmetic" (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. In order to be certain that 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, we have: y + x = y.

Patrick Colonel Suppes (b. ... // Alfred Tarski (January 14, 1902, Warsaw, Russian-ruled Poland – October 26, 1983, Berkeley, California) was a logician and mathematician who spent four decades as a professor of mathematics at the University of California, Berkeley. ...

Further reading

  • Jakub Czajko (July 2004) "On Cantorian spacetime over number systems with division by zero ", Chaos, Solitons and Fractals, volume 21, number 2, pages 261—271.

Image File history File links WikiNews-Logo. ... Wikinews is a free-content news source and a project of the Wikimedia Foundation. ... Ben Goldacres humourous byline photo Ben Goldacre is an London-based British journalist and doctor. ... Year 2006 (MMVI) was a common year starting on Sunday of the Gregorian calendar. ... is the 341st day of the year (342nd in leap years) in the Gregorian calendar. ...

See also


  Results from FactBites:
 
Division by zero - Wikipedia, the free encyclopedia (1907 words)
Therefore, as far as elementary arithmetic is concerned, division by zero cannot be defined.
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.
Division by zero must be left undefined in any mathematical system that obeys the axioms of a field.
Zero (2872 words)
We can see from this that the early use of zero to denote an empty place is not really the use of zero as a number at all, merely the use of some type of punctuation mark so that the numbers had the correct interpretation.
The sum of zero and a negative number is negative, the sum of a positive number and zero is positive, the sum of zero and zero is zero.
A negative number subtracted from zero is positive, a positive number subtracted from zero is negative, zero subtracted from a negative number is negative, zero subtracted from a positive number is positive, zero subtracted from zero is zero.
  More results at FactBites »

 
 

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